Physics Reference Manual - Geant4 - Cern

Physics Reference ManualDecember 11, 2004

ContentsI Introduction 11 Introduction 21.1 Scope of This Manual . . . . . . . . . . . . . . . . . . . . . . . 21.2 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 32 Monte Carlo Methods 42.1 Status of this document . . . . . . . . . . . . . . . . . . . . . 53 Transportation 6II Particle Decay 74 Decay 84.1 Mean Free Path for Decay in Flight . . . . . . . . . . . . . . . 84.2 Branching Ratios and Decay Channels . . . . . . . . . . . . . 84.2.1 G4PhaseSpaceDecayChannel . . . . . . . . . . . . . . . 94.2.2 G4DalitzDecayChannel . . . . . . . . . . . . . . . . . . 94.2.3 Muon Decay . . . . . . . . . . . . . . . . . . . . . . . . 104.2.4 Leptonic Tau Decay . . . . . . . . . . . . . . . . . . . 114.2.5 Kaon Decay . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 12III Electromagnetic Interactions 135 Particle Transport 145.1 The Interaction Length or Mean Free Path . . . . . . . . . . . 155.2 Determination of the Interaction Point . . . . . . . . . . . . . 165.3 Updating the Particle Lifetime . . . . . . . . . . . . . . . . . . 175.4 Status of this document . . . . . . . . . . . . . . . . . . . . . 17-10

6 Gamma Incident 186.1 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . 196.1.1 Cross Section and Mean Free Path . . . . . . . . . . . 196.1.2 Final State . . . . . . . . . . . . . . . . . . . . . . . . 196.1.3 Status of this document . . . . . . . . . . . . . . . . . 216.2 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . 226.2.1 Cross Section and Mean Free Path . . . . . . . . . . . 226.2.2 Sampling the Final State . . . . . . . . . . . . . . . . . 236.2.3 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2.4 Status of this document . . . . . . . . . . . . . . . . . 246.3 Gamma Conversion into an Electron - Positron Pair . . . . . . 266.3.1 Cross Section and Mean Free Path . . . . . . . . . . . 266.3.2 Final State . . . . . . . . . . . . . . . . . . . . . . . . 276.3.3 Status of this document . . . . . . . . . . . . . . . . . 316.4 Gamma Conversion into a Muon - Anti-muon Pair . . . . . . . 326.4.1 Cross Section and Energy Sharing . . . . . . . . . . . . 326.4.2 Parameterization of the Total Cross Section . . . . . . 356.4.3 Multi-differential Cross Section and Angular Variables 386.4.4 Procedure for the Generation of Muon - Anti-muon Pairs 406.4.5 Status of this document . . . . . . . . . . . . . . . . . 507 Common to All Charged Particles 517.1 Computing the Mean Energy Loss . . . . . . . . . . . . . . . . 527.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 527.1.2 Implementation Details . . . . . . . . . . . . . . . . . . 537.1.3 Energy Loss by Heavy Charged Particles . . . . . . . . 557.1.4 Status of this document . . . . . . . . . . . . . . . . . 567.2 Energy loss fluctuations . . . . . . . . . . . . . . . . . . . . . 577.2.1 Fluctuations in thick absorbers . . . . . . . . . . . . . 577.2.2 Fluctuations in thin absorbers . . . . . . . . . . . . . . 577.2.3 Status of this document . . . . . . . . . . . . . . . . . 607.3 Correcting the cross section for energy variation . . . . . . . . 617.3.1 Status of this document . . . . . . . . . . . . . . . . . 617.4 Conversion from range to kinetic energy . . . . . . . . . . . . 637.4.1 Charged particles . . . . . . . . . . . . . . . . . . . . . 637.4.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . 657.4.3 Status of this document . . . . . . . . . . . . . . . . . 667.5 Multiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . 677.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 677.5.2 Path Length Correction . . . . . . . . . . . . . . . . . 697.5.3 Angular Distribution . . . . . . . . . . . . . . . . . . . 71

7.5.4 Determination of the Model Parameters . . . . . . . . 727.5.5 Nuclear Size Effects . . . . . . . . . . . . . . . . . . . . 747.5.6 Implementation of the Process . . . . . . . . . . . . . . 747.5.7 Status of this document . . . . . . . . . . . . . . . . . 777.6 Transition radiation . . . . . . . . . . . . . . . . . . . . . . . . 787.6.1 The Relationship of Transition Radiation to X-ray CherenkovRadiation . . . . . . . . . . . . . . . . . . . . . . . . . 787.6.2 Calculating the X-ray Transition Radiation Yield . . . 797.6.3 Simulating X-ray Transition Radiation Production . . . 807.6.4 Status of this document . . . . . . . . . . . . . . . . . 827.7 Scintillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.7.1 Status of this document . . . . . . . . . . . . . . . . . 847.8 Čerenkov Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 857.8.1 Status of this document . . . . . . . . . . . . . . . . . 867.9 Photoabsorption ionization model . . . . . . . . . . . . . . . . 877.9.1 Cross Section for Ionizing Collisions . . . . . . . . . . . 877.9.2 Energy Loss Simulation . . . . . . . . . . . . . . . . . 897.9.3 Status of this document . . . . . . . . . . . . . . . . . 907.10 Photoabsorption cross section at low energies . . . . . . . . . . 917.10.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 917.10.2 Status of this document . . . . . . . . . . . . . . . . . 918 Electron Incident 938.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 948.1.2 Continuous Energy Loss . . . . . . . . . . . . . . . . . 948.1.3 Total Cross Section per Atom and Mean Free Path . . 968.1.4 Simulation of Delta-ray Production . . . . . . . . . . . 978.1.5 Status of this document . . . . . . . . . . . . . . . . . 988.2 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . 998.2.1 Cross Section and Energy Loss . . . . . . . . . . . . . 998.2.2 Simulation of Discrete Bremsstrahlung . . . . . . . . . 1058.2.3 Status of this document . . . . . . . . . . . . . . . . . 1098.3 Positron - Electron Annihilation . . . . . . . . . . . . . . . . . 1118.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1118.3.2 Cross Section and Mean Free Path . . . . . . . . . . . 1118.3.3 Sampling the final state . . . . . . . . . . . . . . . . . 1118.3.4 Status of this document . . . . . . . . . . . . . . . . . 1138.4 Positron - Electron Annihilation into Muon - Anti-muon . . . 1158.4.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . 1158.4.2 Sampling of Energies and Angles . . . . . . . . . . . . 115

8.4.3 Status of this document . . . . . . . . . . . . . . . . . 1188.5 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . 1208.5.1 Spectral and Angular Distributions of Synchrotron Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.5.2 Simulating Synchrotron Radiation . . . . . . . . . . . . 1218.5.3 Status of this document . . . . . . . . . . . . . . . . . 1229 Muon Incident 1239.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.1.2 Continuous Energy Loss . . . . . . . . . . . . . . . . . 1249.1.3 Total Cross Section per Atom and Mean Free Path . . 1269.1.4 Simulating Delta-ray Production . . . . . . . . . . . . 1279.1.5 Status of this document . . . . . . . . . . . . . . . . . 1279.2 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.2.1 Differential Cross Section . . . . . . . . . . . . . . . . . 1299.2.2 Continuous Energy Loss . . . . . . . . . . . . . . . . . 1309.2.3 Total Cross Section . . . . . . . . . . . . . . . . . . . . 1309.2.4 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 1319.2.5 Status of this document . . . . . . . . . . . . . . . . . 1329.3 Muon Photonuclear Interaction . . . . . . . . . . . . . . . . . 1349.3.1 Differential Cross Section . . . . . . . . . . . . . . . . . 1349.3.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 1359.3.3 Status of this document . . . . . . . . . . . . . . . . . 1379.4 Positron - Electron Pair Production by Muons . . . . . . . . . 1399.4.1 Differential Cross Section . . . . . . . . . . . . . . . . . 1399.4.2 Total Cross Section and Restricted Energy Loss . . . . 1429.4.3 Sampling of Positron - Electron Pair Production . . . . 1439.4.4 Status of this document . . . . . . . . . . . . . . . . . 14410 Charged Hadron Incident 14610.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14710.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 14710.1.2 Continuous Energy Loss . . . . . . . . . . . . . . . . . 14710.1.3 Total Cross Section per Atom and Mean Free Path . . 14910.1.4 Simulating Delta-ray Production . . . . . . . . . . . . 15010.1.5 Status of this document . . . . . . . . . . . . . . . . . 15111 Low Energy Extensions 15211.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15311.1.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11.1.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . 15311.1.3 Distribution of the Data Sets . . . . . . . . . . . . . . 15411.1.4 Calculation of Total Cross Sections . . . . . . . . . . . 15511.1.5 Sampling of Relevant Physics Quantities . . . . . . . . 15511.1.6 Status of the document . . . . . . . . . . . . . . . . . . 15611.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . 15811.2.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . 15811.2.2 Sampling of the Final State . . . . . . . . . . . . . . . 15811.2.3 Status of the document . . . . . . . . . . . . . . . . . . 15911.3 Compton Scattering by Linearly Polarized Gamma Rays . . . 16011.3.1 The Cross Section . . . . . . . . . . . . . . . . . . . . . 16011.3.2 Angular Distribution . . . . . . . . . . . . . . . . . . . 16011.3.3 Polarization Vector . . . . . . . . . . . . . . . . . . . . 16011.3.4 Unpolarized Photons . . . . . . . . . . . . . . . . . . . 16111.3.5 Status of this document . . . . . . . . . . . . . . . . . 16111.4 Rayleigh Scattering . . . . . . . . . . . . . . . . . . . . . . . . 16211.4.1 Total Cross Section . . . . . . . . . . . . . . . . . . . . 16211.4.2 Sampling of the Final State . . . . . . . . . . . . . . . 16211.4.3 Status of this document . . . . . . . . . . . . . . . . . 16211.5 Gamma Conversion . . . . . . . . . . . . . . . . . . . . . . . . 16411.5.1 Total cross-section . . . . . . . . . . . . . . . . . . . . 16411.5.2 Sampling of the final state . . . . . . . . . . . . . . . . 16411.5.3 Status of the document . . . . . . . . . . . . . . . . . . 16511.6 Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . 16611.6.1 Total cross-section . . . . . . . . . . . . . . . . . . . . 16611.6.2 Sampling of the final state . . . . . . . . . . . . . . . . 16611.6.3 Status of the document . . . . . . . . . . . . . . . . . . 16611.7 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . 16711.7.1 Bremsstrahlung angular distributions . . . . . . . . . . 16811.7.2 Status of the document . . . . . . . . . . . . . . . . . . 17211.8 Electron ionisation . . . . . . . . . . . . . . . . . . . . . . . . 17411.8.1 Status of the document . . . . . . . . . . . . . . . . . . 17511.9 Atomic relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 17611.9.1 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . 17611.9.2 Auger process . . . . . . . . . . . . . . . . . . . . . . . 17711.9.3 Status of the document . . . . . . . . . . . . . . . . . . 17711.10Hadron and Ion Ionisation . . . . . . . . . . . . . . . . . . . . 17911.10.1 Delta-ray production . . . . . . . . . . . . . . . . . . . 17911.10.2 Energy Loss of Fast Hadrons . . . . . . . . . . . . . . . 18011.10.3 Barkas and Bloch effects . . . . . . . . . . . . . . . . . 18211.10.4 Energy losses of slow positive hadrons . . . . . . . . . 183

11.10.5 Energy loss of alpha particles . . . . . . . . . . . . . . 18411.10.6 Effective charge of ions . . . . . . . . . . . . . . . . . . 18511.10.7 Energy losses of slow negative particles . . . . . . . . . 18711.10.8 Energy losses of hadrons in compounds . . . . . . . . . 18711.10.9 Nuclear stopping powers . . . . . . . . . . . . . . . . . 18811.10.10Fluctuations of energy losses of hadrons . . . . . . . . 18911.10.11Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 19111.10.12PIXE . . . . . . . . . . . . . . . . . . . . . . . . . . . 19211.10.13Status of this document . . . . . . . . . . . . . . . . . 19211.11Penelope physics . . . . . . . . . . . . . . . . . . . . . . . . . 19511.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 19511.11.2 Compton scattering . . . . . . . . . . . . . . . . . . . . 19511.11.3 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . 19711.11.4 Gamma conversion . . . . . . . . . . . . . . . . . . . . 19811.11.5 Photoelectric effect . . . . . . . . . . . . . . . . . . . . 20011.11.6 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . 20011.11.7 Ionisation . . . . . . . . . . . . . . . . . . . . . . . . . 20211.11.8 Positron Annihilation . . . . . . . . . . . . . . . . . . . 20811.11.9 Status of the document . . . . . . . . . . . . . . . . . . 20912 Optical Photons 21212.1 Interactions of optical photons . . . . . . . . . . . . . . . . . . 21312.1.1 Physics processes for optical photons . . . . . . . . . . 21312.1.2 Photon polarization . . . . . . . . . . . . . . . . . . . . 21412.1.3 Tracking of the photons . . . . . . . . . . . . . . . . . 21513 Shower Parameterizations 21913.1 Gflash Shower Parameterizations . . . . . . . . . . . . . . . . 22013.1.1 Parameterization Ansatz . . . . . . . . . . . . . . . . . 22013.1.2 Longitudinal Shower Profiles . . . . . . . . . . . . . . 22013.1.3 Radial Shower Profiles . . . . . . . . . . . . . . . . . . 22113.1.4 Gflash Performance . . . . . . . . . . . . . . . . . . . . 22213.1.5 Status of this document . . . . . . . . . . . . . . . . . 223IV Hadronic Interactions 22414 Lepton Hadron Interactions 22514.1 G4MuonNucleusProcess . . . . . . . . . . . . . . . . . . . . . . 22514.1.1 Cross Section Calculation . . . . . . . . . . . . . . . . 22614.2 Status of this document . . . . . . . . . . . . . . . . . . . . . 227

15 Cross-sections in Photonuclear and Electronuclear Reactions22815.1 Approximation of Photonuclear Cross Sections. . . . . . . . . 22815.2 Electronuclear Cross Sections and Reactions . . . . . . . . . . 23115.2.1 Common Notation for Different Approaches to ElectronuclearReactions . . . . . . . . . . . . . . . . . . . 23115.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 23616 Total Reaction Cross Section in Nucleus-nucleus Reactions 23916.1 Sihver Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 23916.2 Kox and Shen Formulae . . . . . . . . . . . . . . . . . . . . . 24016.3 Tripathi formula . . . . . . . . . . . . . . . . . . . . . . . . . 24216.4 Representative Cross Sections . . . . . . . . . . . . . . . . . . 24416.5 Tripathi Formula for ”light” Systems . . . . . . . . . . . . . . 24416.6 Status of this document . . . . . . . . . . . . . . . . . . . . . 24517 Coherent elastic scattering 24917.1 Nucleon-Nucleon elastic Scattering . . . . . . . . . . . . . . . 24918 Hadron-nucleus Elastic Scattering at Medium and High Energy.25018.1 Method of Calculation . . . . . . . . . . . . . . . . . . . . . . 25018.2 Status of this document . . . . . . . . . . . . . . . . . . . . . 25319 Interactions of Stopping Particles 26719.1 Complementary parameterised and theoretical treatment . . . 26719.1.1 Pion absorption at rest . . . . . . . . . . . . . . . . . . 26820 Parametrization Driven Models 27020.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27020.2 Low Energy Model . . . . . . . . . . . . . . . . . . . . . . . . 27120.3 High Energy Model . . . . . . . . . . . . . . . . . . . . . . . . 27220.3.1 Initial Interaction . . . . . . . . . . . . . . . . . . . . . 27220.3.2 Intra-nuclear Cascade . . . . . . . . . . . . . . . . . . . 27220.3.3 High Energy Cascading . . . . . . . . . . . . . . . . . . 27320.3.4 High Energy Cluster Production . . . . . . . . . . . . . 27820.3.5 Medium Energy Cascading . . . . . . . . . . . . . . . . 28020.3.6 Medium Energy Cluster Production . . . . . . . . . . . 28020.3.7 Elastic and Quasi-elastic Scattering . . . . . . . . . . . 28020.4 Status of this document . . . . . . . . . . . . . . . . . . . . . 281

21 Leading Particle Bias 28221.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28221.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28221.2.1 Inclusive hadron production . . . . . . . . . . . . . . . 28321.2.2 Sampling of energy and emission angle of the secondary 28421.2.3 Sampling statistical weight . . . . . . . . . . . . . . . . 28521.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 28522 Parton string model. 28722.1 Reaction initial state simulation. . . . . . . . . . . . . . . . . . 28722.1.1 Allowed projectiles and bombarding energy range forinteraction with nucleon and nuclear targets. . . . . . . 28722.1.2 MC initialization procedure for nucleus. . . . . . . . . 28722.1.3 Random choice of the impact parameter. . . . . . . . . 28922.2 Sample of collision participants in nuclear collisions. . . . . . . 28922.2.1 MC procedure to define collision participants. . . . . . 28922.2.2 Separation of hadron diffraction excitation. . . . . . . . 29022.3 Longitudinal string excitation . . . . . . . . . . . . . . . . . . 29122.3.1 Hadron–nucleon inelastic collision . . . . . . . . . . . . 29122.3.2 The diffractive string excitation . . . . . . . . . . . . . 29122.3.3 The string excitation by parton exchange . . . . . . . . 29122.3.4 Transverse momentum sampling . . . . . . . . . . . . . 29222.3.5 Sampling x-plus and x-minus . . . . . . . . . . . . . . 29222.3.6 The diffractive string excitation . . . . . . . . . . . . . 29222.3.7 The string excitation by parton rearrangement . . . . . 29322.4 Longitudinal string decay. . . . . . . . . . . . . . . . . . . . . 29422.4.1 Hadron production by string fragmentation. . . . . . . 29422.4.2 The hadron formation time and coordinate. . . . . . . 29523 Chiral Invariant Phase Space Decay. 29723.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29723.2 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . 29823.3 Code Development . . . . . . . . . . . . . . . . . . . . . . . . 29923.4 Nucleon-Antinucleon Annihilation at Rest . . . . . . . . . . . 30023.4.1 Meson Production . . . . . . . . . . . . . . . . . . . . 30123.4.2 Baryon Production . . . . . . . . . . . . . . . . . . . . 30823.5 Nuclear Pion Capture at Rest and Photonuclear ReactionsBelow the Delta(3,3) Resonance . . . . . . . . . . . . . . . . . 30823.6 Modeling of real and virtual photon interactions with nucleibelow pion production threshold. . . . . . . . . . . . . . . . . 328

23.7 Chiral invariant phase-space decay in high energy hadron nuclearreactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33723.8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33823.9 Status of this document . . . . . . . . . . . . . . . . . . . . . 33924 Bertini Intranuclear Cascade Model in Geant4 34324.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34324.2 The Geant4 cascade model . . . . . . . . . . . . . . . . . . . . 34424.2.1 Model limits . . . . . . . . . . . . . . . . . . . . . . . . 34424.2.2 Intranuclear cascade model . . . . . . . . . . . . . . . . 34524.2.3 Nuclei model . . . . . . . . . . . . . . . . . . . . . . . 34524.2.4 Pre-equilibrium model . . . . . . . . . . . . . . . . . . 34724.2.5 Break-up models . . . . . . . . . . . . . . . . . . . . . 34824.2.6 Evaporation model . . . . . . . . . . . . . . . . . . . . 34824.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 34824.4 Status of this document . . . . . . . . . . . . . . . . . . . . . 34925 The Geant4 Binary Cascade 35125.1 Modeling overview . . . . . . . . . . . . . . . . . . . . . . . . 35125.1.1 The transport algorithm . . . . . . . . . . . . . . . . . 35125.1.2 The description of the target nucleus and fermi motion 35225.1.3 Optical and phenomenological potentials . . . . . . . . 35325.1.4 Pauli blocking simulation . . . . . . . . . . . . . . . . . 35425.1.5 The scattering term . . . . . . . . . . . . . . . . . . . 35425.1.6 Total inclusive cross-sections . . . . . . . . . . . . . . 35525.1.7 Channel cross-sections . . . . . . . . . . . . . . . . . . 35525.1.8 Mass dependent resonance width and partial width . . 35625.1.9 Resonance production cross-section in the t-channel . . 35625.1.10 Nucleon Nucleon elastic collisions . . . . . . . . . . . . 35725.1.11 Generation of transverse momentum . . . . . . . . . . 35725.1.12 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 35825.1.13 The escaping particle and coherent effects . . . . . . . 35825.1.14 Light ion reactions . . . . . . . . . . . . . . . . . . . . 35925.1.15 Transition to pre-compound modeling . . . . . . . . . . 35925.1.16 Calculation of excitation energies and residuals . . . . 36025.2 Comparison with experiments . . . . . . . . . . . . . . . . . . 36026 Abrasion-ablation Model 37626.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37626.2 Initial nuclear dynamics and impact parameter . . . . . . . . . 37726.3 Abrasion process . . . . . . . . . . . . . . . . . . . . . . . . . 378

26.4 Abraded nucleon spectrum . . . . . . . . . . . . . . . . . . . . 38026.5 De-excitation of the projectile and target nuclear pre-fragmentsby standard Geant4 de-excitation physics . . . . . . . . . . . . 38126.6 De-excitation of the projectile and target nuclear pre-fragmentsby nuclear ablation . . . . . . . . . . . . . . . . . . . . . . . . 38226.7 Definition of the functions P and F used in the abrasion model 38326.8 Status of this document . . . . . . . . . . . . . . . . . . . . . 38427 Electromagnetic Dissociation Model 38727.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38727.2 Status of this document . . . . . . . . . . . . . . . . . . . . . 39028 Precompound model. 39128.1 Reaction initial state. . . . . . . . . . . . . . . . . . . . . . . . 39128.2 Simulation of pre-compound reaction . . . . . . . . . . . . . . 39128.2.1 Statistical equilibrium condition . . . . . . . . . . . . . 39228.2.2 Level density of excited (n-exciton) states . . . . . . . 39228.2.3 Transition probabilities . . . . . . . . . . . . . . . . . . 39228.2.4 Emission probabilities for nucleons . . . . . . . . . . . 39428.2.5 Emission probabilities for complex fragments . . . . . . 39428.2.6 The total probability . . . . . . . . . . . . . . . . . . . 39528.2.7 Calculation of kinetic energies for emitted particle . . . 39528.2.8 Parameters of residual nucleus. . . . . . . . . . . . . . 39528.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 39529 Evaporation Model 39729.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39729.2 Model description. . . . . . . . . . . . . . . . . . . . . . . . . 39729.2.1 Cross sections for inverse reactions. . . . . . . . . . . . 39829.2.2 Coulomb barriers. . . . . . . . . . . . . . . . . . . . . . 39829.2.3 Level densities. . . . . . . . . . . . . . . . . . . . . . . 39929.2.4 Maximum energy available for evaporation. . . . . . . . 39929.2.5 Total decay width. . . . . . . . . . . . . . . . . . . . . 40029.3 GEM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40029.4 Fission probability calculation. . . . . . . . . . . . . . . . . . . 40229.4.1 The fission total probability. . . . . . . . . . . . . . . . 40229.4.2 The fission barrier. . . . . . . . . . . . . . . . . . . . . 40229.5 The Total Probability for Photon Evaporation . . . . . . . . . 40329.5.1 Energy of evaporated photon . . . . . . . . . . . . . . 40329.6 Discrete photon evaporation . . . . . . . . . . . . . . . . . . . 40329.7 Internal conversion electron emission . . . . . . . . . . . . . . 404

29.7.1 Multipolarity . . . . . . . . . . . . . . . . . . . . . . . 40429.7.2 Binding energy . . . . . . . . . . . . . . . . . . . . . . 40529.7.3 Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 40530 Fission model. 40730.1 Reaction initial state. . . . . . . . . . . . . . . . . . . . . . . . 40730.2 Fission process simulation. . . . . . . . . . . . . . . . . . . . . 40730.2.1 Atomic number distribution of fission products. . . . . 40730.2.2 Charge distribution of fission products. . . . . . . . . . 40930.2.3 Kinetic energy distribution of fission products. . . . . . 40930.2.4 Calculation of the excitation energy of fission products. 41030.2.5 Excited fragment momenta. . . . . . . . . . . . . . . . 41031 Fermi break-up model. 41231.1 Fermi break-up simulation for light nuclei. . . . . . . . . . . . 41231.1.1 Allowed channel. . . . . . . . . . . . . . . . . . . . . . 41231.1.2 Break-up probability. . . . . . . . . . . . . . . . . . . . 41331.1.3 Fermi break-up model parameter. . . . . . . . . . . . . 41331.1.4 Fragment characteristics. . . . . . . . . . . . . . . . . 41431.1.5 MC procedure. . . . . . . . . . . . . . . . . . . . . . . 41432 Multifragmentation model. 41532.1 Multifragmentation process simulation. . . . . . . . . . . . . . 41532.1.1 Multifragmentation probability. . . . . . . . . . . . . . 41532.1.2 Direct simulation of the low multiplicity multifragmentdisintegration. . . . . . . . . . . . . . . . . . . . 41732.1.3 Fragment multiplicity distribution. . . . . . . . . . . . 41832.1.4 Atomic number distribution of fragments. . . . . . . . 41832.1.5 Charge distribution of fragments. . . . . . . . . . . . . 41932.1.6 Kinetic energy distribution of fragments. . . . . . . . . 41932.1.7 Calculation of the fragment excitation energies. . . . . 41933 Low Energy Neutron Interactions 42133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42133.2 Physics and Verification . . . . . . . . . . . . . . . . . . . . . 42133.2.1 Inclusive Cross-sections . . . . . . . . . . . . . . . . . . 42133.2.2 Elastic Scattering . . . . . . . . . . . . . . . . . . . . . 42233.2.3 Radiative Capture . . . . . . . . . . . . . . . . . . . . 42433.2.4 Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . 42433.2.5 Inelastic Scattering . . . . . . . . . . . . . . . . . . . . 42733.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

34 Radioactive Decay 43034.1 The Radioactive Decay Module . . . . . . . . . . . . . . . . . 43034.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43034.2.1 Biasing Methods . . . . . . . . . . . . . . . . . . . . . 43134.3 Status of this document . . . . . . . . . . . . . . . . . . . . . 4320

Part IIntroduction1

Chapter 1Introduction1.1 Scope of This ManualThe Physics Reference Manual provides detailed explanations of the physicsimplemented in the Geant4 toolkit. The manual’s purpose is threefold:• to present the theoretical formulation, model, or parameterization ofthe physics interactions included in Geant4,• to describe the probability of the occurrence of an interaction and thesampling mechanisms required to simulate it, and• to serve as a reference for toolkit users and developers who wish toconsult the underlying physics of an interaction.This manual does not discuss code implementation or how to use theimplemented physics interactions in a simulation. These topics are discussedin the User’s Guide for Application Developers. Details of the object-orienteddesign and functionality of the Geant4 toolkit are given in the User’s Guidefor Toolkit Developers. The Installation Guide for Setting up Geant4 inYour Computing Environment describes how to get the Geant4 code, installit, and run it.1.2 Definition of TermsSeveral terms used throughout the Physics Reference Manual have specificmeaning within Geant4, but are not well-defined in general usage. The definitionsof these terms are given here.2

• process - a C++ class which describes how and when a specific kindof physical interaction takes place along a particle track. A given particletype typically has several processes assigned to it. Occaisionally“process” refers to the interaction which the process class describes.• model - a C++ class whose methods implement the details of an interaction,such as its kinematics. One or more models may be assignedto each process. In sections discussing the theory of an interaction,“model” may refer to the formulae or parameterization on which themodel class is based.• Geant3 - a physics simulation tool written in Fortran, and the predecessorof Geant4. Although many references are made to Geant3, noknowledge of it is required to understand this manual.1.3 Status of this document4.12.01 created by D.H. Wright3

Chapter 2Monte Carlo MethodsThe Geant4 toolkit uses a combination of the composition and rejectionMonte Carlo methods. Only the basic formalism of these methods is outlinedhere. For a complete account of the Monte Carlo methods, the interested useris referred to the publications of Butcher and Messel, Messel and Crawford,or Ford and Nelson [1, 2, 3].Suppose we wish to sample x in the interval [x 1 , x 2 ] from the distributionf(x) and the normalised probability density function can be written as :n∑f(x) = N i f i (x)g i (x) (2.1)i=1where N i > 0, f i (x) are normalised density functions on [x 1 , x 2 ] , and 0 ≤g i (x) ≤ 1.According to this method, x can sampled in the following way:1. select a random integer i ∈ {1, 2, · · · n} with probability proportionalto N i2. select a value x 0 from the distribution f i (x)3. calculate g i (x 0 ) and accept x = x 0 with probability g i (x 0 );4. if x 0 is rejected restart from step 1.It can be shown that this scheme is correct and the mean number of tries toaccept a value is ∑ i N i .In practice, a good method of sampling from the distribution f(x) has thefollowing properties:• all the subdistributions f i (x) can be sampled easily;4

• the rejection functions g i (x) can be evaluated easily/quickly;• the mean number of tries is not too large.Thus the different possible decompositions of the distribution f(x) are notequivalent from the practical point of view (e.g. they can be very differentin computational speed) and it can be useful to optimise the decomposition.A remark of practical importance : if our distribution is not normalised∫ x2x 1f(x)dx = C > 0the method can be used in the same manner; the mean number of tries inthis case is ∑ i N i /C.2.1 Status of this document18.01.02 created by M.Maire.Bibliography[1] J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)[2] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[3] R. Ford and W. Nelson. SLAC-265, UC-32 (1985)[4] Particle Data Group. Rev. of Particle Properties. Eur. Phys. J. C15.(2000) 1. http://pdg.lbl.gov5

Chapter 3TransportationThe transportation process is responsible for determining the geometricallimits of a step. It calculates the length of step with which a track will crossinto another volume. When the track actually arrives at a boundary, thetransportation process locates the next volume that it enters.If the particle is charged and there is an electromagnetic (or potentiallyother) field, it is responsible for propagating the particle in this field. It doesthis according to an equation of motion. This equation can be provided byGeant4, for the case a magnetic or EM field, or can be provided by the userfor other fields.The transportation updates the time of flight of a particle, utilising itsinitial velocity.Some additional details on motion in fields:In order to intersect the model Geant4 geometry of a detector or setup,the curved trajectory followed by a charged particle is split into ’chords segments’.A chord is a straight line segment between two trajectory points.Chords are created utilizing a criterion for the maximum estimated distancebetween a curve point and the chord. This distance is also known as thesagitta.The equations of motions are solved utilising Runge Kutta methods.Runge Kutta methods of different can be utilised for fields depending on thenumerical method utilised for approximating the field. Specialised methodsfor near-constant magnetic fields are under development.6

Part IIParticle Decay7

Chapter 4DecayThe decay of particles in flight and at rest is simulated by the G4Decay class.4.1 Mean Free Path for Decay in FlightThe mean free path λ is calculated for each step usingλ = γβcτwhere τ is the lifetime of the particle andγ =1√ 1 − β2 .β and γ are calculated using the momentum at the beginning of the step.The decay time in the rest frame of the particle (proper time) is then sampledand converted to a decay length using β.4.2 Branching Ratios and Decay ChannelsG4Decay selects a decay mode for the particle according to branching ratiosdefined in the G4DecayTable class, which is a member of the G4ParticleDefinitionclass. Each mode is implemented as a class derived from G4VDecayChanneland is responsible for generating the secondaries and the kinematics of thedecay. In a given decay channel the daughter particle momenta are calculatedin the rest frame of the parent and then boosted into the laboratoryframe. Polarization is not currently taken into account for either the parentor its daughters.A large number of specific decay channels may be required to simulatean experiment, ranging from two-body to many-body decays and V − A to8

semi-leptonic decays. Most of these are covered by the five decay channelclasses provided by Geant4:G4PhaseSpaceDecayChannel : phase space decayG4DalitzDecayChannel : dalitz decayG4MuonDecayChannel : muon decayG4TauLeptonicDecayChannel : tau leptonic decayG4KL3DecayChannel : semi-leptonic decays of kaon .4.2.1 G4PhaseSpaceDecayChannelThe majority of decays in Geant4 are implemented using the G4PhaseSpaceDecayChannelclass. It simulates phase space decays with isotropic angular distributions inthe center-of-mass system. Three private methods of G4PhaseSpaceDecayChannelare provided to handle two-, three- and N-body decays:TwoBodyDecayIt()ThreeBodyDecayIt()ManyBodyDecayIt()Some examples of decays handled by this class are:π 0 → γγ,Λ → pπ −andK 0 L → π 0 π + π − .4.2.2 G4DalitzDecayChannelThe Dalitz decayπ 0 → γ + e + + e −and other Dalitz-like decays, such asK 0 L → γ + e + + e −andK 0 L → γ + µ + + µ −9

are simulated by the G4DalitzDecayChannel class. In general, it handles anydecay of the formP 0 → γ + l + + l − ,where P 0 is a spin-0 meson of mass M and l ± are leptons of mass m. Theangular distribution of the γ is isotropic in the center-of-mass system of theparent particle and the leptons are generated isotropically and back-to-backin their center-of-mass frame. The magnitude of the leptons’ momentum issampled from the distribution functionf(t) = (1 −tM 2 ) 3√(1 + 2m2 )t1 − 4m2 ,twhere t is the square of the sum of the leptons’ energy in their center-of-massframe.4.2.3 Muon DecayG4MuonDecayChannel simulates muon decay according to V −A theory. Neglectingthe electron mass, the electron energy is sampled from the followingdistribution:dΓ = G F 2 m µ5192π 3 2ɛ2 (3 − 2ɛ)where: Γ : decay rateɛ : = E e /E maxE e : electron energyE max : maximum electron energy = m µ /2The momenta of the two neutrinos are not sampled from their V − A distributions.Instead they are generated back-to-back and isotropically in theneutrinos’ center-of-mass frame, with the magnitude of the neutrino momentumchosen to conserve energy in the decay. The two neutrinos are thenboosted opposite to the momentum of the decay electron. This approximationis sufficient for most simulations because the neutrino is usually notobserved in any detector.Currently, neither the polarization of the muon or the electron is consideredin this class.10

4.2.4 Leptonic Tau DecayG4TauLeptonicDecayChannel simulates leptonic tau decays according to V −A theory. This class is valid for bothandτ ± → e ± + ν τ + ν eτ ± → µ ± + ν τ + ν µmodes.The energy spectrum is calculated without neglecting lepton mass asfollows:dΓ = G F 2 m τ324π 3 p l E l (3E l m τ 2 − 4E l 2 m τ − 2m τ m l 2 )where: Γ : decay rateE l : daughter lepton energy (total energy)p l : daughter lepton momentum: daughter lepton massm lAs in the case of muon decay, the energies of the two neutrinos are notsampled from their V − A spectra, but are calculated so that energy andmomentum are conserved. Polarization of the τ and final state leptons is nottaken into account in this class.4.2.5 Kaon DecayThe class G4KL3DecayChannel simulates the following four semi-leptonic decaymodes of the kaon:K ± e3K ± µ3K 0 e3K 0 µ3: K ± → π 0 + e ± + ν: K ± → π 0 + µ ± + ν: KL 0 → π± + e ∓ + ν: KL 0 → π ± + µ ∓ + νAssuming that only the vector current contributes to K → lπν decays, thematrix element can be described by using two dimensionless form factors, f +and f − , which depend only on the momentum transfer t = (P K − P π ) 2 .The Dalitz plot density used in this class is as follows [1]:ρ (E π , E µ ) ∝ f 2 + (t)[A + Bξ (t) + Cξ (t)2 ]11

where: A = m K (2E µ E ν − m K E ′ π ) + m µ 2 ( 1 4 E′ π − E ν)B = m µ 2 (E ν − 1 2 E′ π)C = 1 4 m µ 2 E ′ πE ′ π = E π max − E πHere ξ (t) is the ratio of the two form factorsξ (t) = f − (t)/f + (t).f + (t) is assumed to depend linearly on t, i.e.f + (t) = f + (0)[1 + λ + (t/m π 2 )]and f − (t) is assumed to be constant due to time reversal invariance.Two parameters, λ + and ξ (0) are then used for describing the Dalitz plotdensity in this class. The values of these parameters are taken to be theworld average values given by the Particle Data Group [2].4.3 Status of this document10.04.02 re-written by D.H. Wright02.04.02 editing by Hisaya Kurashige14.11.01 editing by Hisaya KurashigeBibliography[1] L.M. Chounet, J.M. Gaillard, and M.K. Gaillard, Phys. Reports 4C, 199(1972).[2] Review of Particle Physics The European Physical Journal C, 15 (2000).12

Part IIIElectromagnetic Interactions13

Chapter 5Particle Transport14

5.1 The Interaction Length or Mean Free Path1) In a simple material the number of atoms per volume is:where:n = N ρANρAAvogadro’s numberdensity of the mediummass of a mole2) In a compound material the number of atoms per volume of the i thelement is:n i = N ρw iA iwhere:Nρw iA iAvogadro’s numberdensity of the mediumproportion by mass of the i th elementmass of a mole of the i th element3) The mean free path of a process, λ, also called the interactionlength, can be given in terms of the total cross section :( ∑λ(E) =i[n i · σ(Z i , E)]) −1where σ(Z, E) is the total cross section per atom of the process and ∑ iruns over all elements composing the material.∑[n i σ(Z i , E)] is also called the macroscopic cross section. The meanifree path is the inverse of the macroscopic cross section.Cross sections per atom and mean free path values are tabulated duringinitialisation.15

5.2 Determination of the Interaction PointThe mean free path, λ, of a particle for a given process depends on themedium and cannot be used directly to sample the probability of an interactionin a heterogeneous detector. The number of mean free paths which aparticle travels is:n λ =∫ x2x 1dxλ(x) , (5.1)which is independent of the material traversed. If n r is a random variabledenoting the number of mean free paths from a given point to the point ofinteraction, it can be shown that n r has the distribution function:P (n r < n λ ) = 1 − e −n λ(5.2)The total number of mean free paths the particle travels before reaching theinteraction point, n λ , is sampled at the beginning of the trajectory as:n λ = − log (η) (5.3)where η is a random number uniformly distributed in the range (0, 1). n λ isupdated after each step ∆x according the formula:n ′ λ = n λ − ∆xλ(x)(5.4)until the step originating from s(x) = n λ·λ(x) is the shortest and this triggersthe specific process.The short description given above is the differential approach to particletransport, which is used in most simulation codes ([2],[1]).In this approach besides the other (discrete) processes the continuous energyloss imposes a limit on the stepsize too, because the cross sections depend ofthe energy of the particle. Then it is assumed that the step is small enoughso that the particle cross sections remain approximately constant during thestep. In principle one must use very small steps in order to insure an accuratesimulation, but computing time increases as the stepsize decreases. A goodcompromise is to limit the stepsize in Geant4 by not allowing the stoppingrange of the particle to decrease by more than 20 % during the step. Thiscondition works well for particles with kinetic energies > 0.5 MeV, but forlower energies it can give very short step sizes. To cure this problem a lowerlimit on the stepsize is also introduced.16

5.3 Updating the Particle LifetimeThe proper and laboratory times of the particle should be updated after eachstep. In the laboratory system:∆t lab =∆x0.5(v 0 + v)(5.5)where∆xv 0vstep travelled by the particleparticle velocity at the beginning of the stepparticle velocity at the end of the stepThis expression is a good approximation if the velocity is not allowed tochange too much during the step.5.4 Status of this document09.10.98 created by L. Urbán.27.07.01 minor revisions by M. Maire01.12.03 integral method subsection added by V. Ivanchenko12.08.04 splitted and partly moved in introduction (mma)Bibliography[1] W.R. Nelson et al. the egs4 Code System. SLAC-Report-265, December1985[2] Geant3 manual, CERN Program Library Long Writeup W5013 (October1994).17

Chapter 6Gamma Incident18

6.1 PhotoElectric effectThe photoelectric effect is the ejection of an electron from a material aftera photon has been absorbed by that material. It is simulated by using aparameterized photon absorption cross section to determine the mean freepath, atomic shell data to determine the energy of the ejected electron, andthe K-shell angular distribution to sample the direction of the electron.6.1.1 Cross Section and Mean Free PathThe parameterization of the photoabsorption cross section proposed by Biggset al. [1] was used :σ(Z, E γ ) = a(Z, E γ)+ b(Z, E γ)+ c(Z, E γ)+ d(Z, E γ)E γ Eγ2 Eγ3 Eγ4(6.1)Using the least-squares method, a separate fit of each of the coefficientsa, b, c, d to the experimental data was performed in several energy intervals[2]. As a rule, the boundaries of these intervals were equal to the correspondingphotoabsorption edges.In a given material the mean free path, λ, for a photon to interact via thephotoelectric effect is given by :( ∑λ(E γ ) =in ati · σ(Z i , E γ )) −1(6.2)where n ati is the number of atoms per volume of the i th element of thematerial. The cross section and mean free path are discontinuous and mustbe computed ’on the fly’ from the formulas 6.1 and 6.2.6.1.2 Final StateChoosing an ElementThe binding energies of the shells depend on the atomic number Z of the material.In compound materials the i th element is chosen randomly accordingto the probability:P rob(Z i , E γ ) = n atiσ(Z i , E γ )∑i[n ati · σ i (E γ )] .19

ShellA quantum can be absorbed if E γ > B shell where the shell energies are takenfrom G4AtomicShells data: the closest available atomic shell is chosen. Thephotoelectron is emitted with kinetic energy :T photoelectron = E γ − B shell (Z i ) (6.3)Theta Distribution of the PhotoelectronThe polar angle of the photoelectron is sampled from the Sauter-Gavriladistribution (for K-shell) [3], which is correct only to zero order in αZ :dσd(cos θ) ∼sin 2 {θ1 + 1 }(1 − β cos θ) 4 2 γ(γ − 1)(γ − 2)(1 − β cos θ)(6.4)where β and γ are the Lorentz factors of the photoelectron.cos θ is sampled from the probability density function :f(cos θ) = 1 − β22β1(1 − β cos θ) 2 =⇒ cos θ =(1 − 2r) + β(1 − 2r)β + 1(6.5)The rejection function is :g(cos θ) =1 − cos2 θ[1 + b(1 − β cos θ)] (6.6)(1 − β cos θ)2with b = γ(γ − 1)(γ − 2)/2It can be shown that g(cos θ) is positive ∀ cos θ ∈ [−1, +1], and can bemajored by :gsup = γ 2 [1 + b(1 − β)] if γ ∈ ]1, 2] (6.7)= γ 2 [1 + b(1 + β)] if γ > 2The efficiency of this method is ∼ 50% if γ < 2, ∼ 25% if γ ∈ [2, 3].RelaxationIn the current implementation the relaxation of the atom is not simulated,but instead is counted as a local energy deposit.20

6.1.3 Status of this document09.10.98 created by M.Maire.08.01.02 updated by mma22.04.02 re-worded by D.H. Wright02.05.02 modifs in total cross section and final state (mma)15.11.02 introduction added by D.H. WrightBibliography[1] Biggs F., and Lighthill R., Preprint Sandia Laboratory, SAND 87-0070(1990)[2] Grichine V.M., Kostin A.P., Kotelnikov S.K. et al., Bulletin of the LebedevInstitute no. 2-3, 34 (1994).[3] Gavrila M. Phys.Rev. 113, 514 (1959).21

6.2 Compton scattering6.2.1 Cross Section and Mean Free PathCross Section per AtomWhen simulating the Compton scattering of a photon from an atomic electron,an empirical cross section formula is used, which reproduces the crosssection data down to 10 keV:Here,σ(Z, E γ ) =[P 1 (Z)log(1 + 2X)X+ P 2(Z) + P 3 (Z)X + P 4 (Z)X 2 ]. (6.8)1 + aX + bX 2 + cX 3Z = atomic number of the mediumE γ = energy of the photonX = E γ /mc 2m = electron massP i (Z) = Z(d i + e i Z + f i Z 2 ).The values of the parameters can be found within the method which computesthe cross section per atom. A fit of the parameters was made to over 511data points [1, 2] chosen from the intervalsand1 ≤ Z ≤ 100E γ ∈ [10 keV, 100 GeV].The accuracy of the fit was estimated to beMean Free Path{∆σ ≈ 10% forσ = Eγ ≃ 10 keV − 20 keV≤ 5 − 6% for E γ > 20 keVIn a given material the mean free path, λ, for a photon to interact via Comptonscattering is given by( ∑λ(E γ ) =in ati · σ i (E γ )) −1(6.9)where n ati is the number of atoms per volume of the i th element of thematerial.22

6.2.2 Sampling the Final StateThe quantum mechanical Klein-Nishina differential cross section per atom is[3] :dσdɛ = m e c 2 [ ] [ 1 πr2 e ZE 0 ɛ + ɛ 1 − ɛ ]sin2 θ(6.10)1 + ɛ 2where r e = classical electron radiusm e c 2 = electron massE 0 = energy of the incident photonE 1 = energy of the scattered photonɛ = E 1 /E 0 .Assuming an elastic collision, the scattering angle θ is defined by the Comptonformula:E 1 = E 0m e c 2m e c 2 + E 0 (1 − cos θ) . (6.11)Sampling the Photon EnergyThe value of ɛ corresponding to the minimum photon energy (backward scattering)is given bym e c 2ɛ 0 =, (6.12)m e c 2 + 2E 0hence ɛ ∈ [ɛ 0 , 1]. Using the combined composition and rejection Monte Carlomethods described in [4, 5, 6] one may setΦ(ɛ) ≃where[ ] [ 1ɛ + ɛ 1 − ɛ ]sin2 θ= f(ɛ)·g(ɛ) = [α1 + ɛ 2 1 f 1 (ɛ) + α 2 f 2 (ɛ)]·g(ɛ), (6.13)α 1 = ln(1/ɛ 0 ) ; f 1 (ɛ) = 1/(α 1 ɛ)α 2 = (1 − ɛ 2 0)/2 ; f 2 (ɛ) = ɛ/α 2 .f 1 and f 2 are probability density functions defined on the interval [ɛ 0 , 1], andg(ɛ) =[1 − ɛ ]1 + ɛ 2 sin2 θis the rejection function ∀ɛ ∈ [ɛ 0 , 1] =⇒ 0 < g(ɛ) ≤ 1.Given a set of 3 random numbers r, r ′ , r ′′ uniformly distributed on the interval[0,1], the sampling procedure for ɛ is the following:1. decide whether to sample from f 1 (ɛ) or f 2 (ɛ):if r < α 1 /(α 1 + α 2 ) select f 1 (ɛ), otherwise select f 2 (ɛ)23

2. sample ɛ from the distributions corresponding to f 1 or f 2 :for f 1 : ɛ = ɛ r′0 (≡ exp(−r ′ α 1 ))for f 2 : ɛ 2 = ɛ 2 0 + (1 − ɛ2 0 )r′3. calculate sin 2 θ = t(2 − t) where t ≡ (1 − cos θ) = m e c 2 (1 − ɛ)/(E 0 ɛ)4. test the rejection function:if g(ɛ) ≥ r ′′ accept ɛ, otherwise go to step 1.Compute the Final State KinematicsAfter the successful sampling of ɛ, the polar angles of the scattered photonwith respect to the direction of the parent photon are generated. The azimuthalangle, φ, is generated isotropically and θ is as defined in the previoussection. The momentum vector of the scattered photon, −→ P γ1 , is then transformedinto the World coordinate system. The kinetic energy and momentumof the recoil electron are then6.2.3 ValidityT el = E 0 − E 1−→P el = −→ P γ0 − −→ P γ1 .The differential cross-section is valid only for those collisions in which theenergy of the recoil electron is large compared to its binding energy (whichis ignored). However, as pointed out by Rossi [7], this has a negligible effectbecause of the small number of recoil electrons produced at very low energies.6.2.4 Status of this document09.10.98 created by M.Maire.14.01.02 minor revision (mma)22.04.02 reworded by D.H. Wright18.03.04 include references for total cross section (mma)Bibliography[1] Hubbell, Gimm and Overbo. J. Phys. Chem. Ref. Data 9 1023 (1980)[2] H. Storm and H.I. Israel Nucl. Data Tables A7 565 (1970)24

[3] O. Klein and Y. Nishina. Z. Physik 52 853 (1929)[4] J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)[5] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[6] R. Ford and W. Nelson. SLAC-265, UC-32 (1985)[7] B. Rossi. High energy particles, Prentice-Hall 77-79 (1952)25

6.3 Gamma Conversion into an Electron - PositronPair6.3.1 Cross Section and Mean Free PathCross Section per AtomThe total cross-section per atom for the conversion of a gamma into an(e + , e − ) pair has been parameterized as[σ(Z, E γ ) = Z(Z + 1) F 1 (X) + F 2 (X) Z + F ]3(X), (6.14)Zwhere E γ is the incident gamma energy and X = ln(E γ /m e c 2 ) . The functionsF n are given byF 1 (X) = a 0 + a 1 X + a 2 X 2 + a 3 X 3 + a 4 X 4 + a 5 X 5 (6.15)F 2 (X) = b 0 + b 1 X + b 2 X 2 + b 3 X 3 + b 4 X 4 + b 5 X 5F 3 (X) = c 0 + c 1 X + c 2 X 2 + c 3 X 3 + c 4 X 4 + c 5 X 5 ,with the parameters a i , b i , c i taken from a least-squares fit to the data [1].Their values can be found in the function which computes formula 6.14.This parameterization describes the data in the rangeand1 ≤ Z ≤ 100E γ ∈ [1.5 MeV, 100 GeV].The accuracy of the fit was estimated to be ∆ σ ≤ 5% with a mean value ofσ≈ 2.2%. Above 100 GeV the cross section is constant. Below E low = 1.5 MeVthe extrapolation( E − 2me c 2 ) 2σ(E) = σ(E low ) ·(6.16)E low − 2m e c 2is used.Mean Free PathIn a given material the mean free path, λ, for a photon to convert into an(e + , e − ) pair is( ∑λ(E γ ) =in ati · σ(Z i , E γ )) −1(6.17)where n ati is the number of atoms per volume of the i th element of thematerial.26

6.3.2 Final StateChoosing an ElementThe differential cross section depends on the atomic number Z of the materialin which the interaction occurs. In a compound material the element i inwhich the interaction occurs is chosen randomly according to the probabilityP rob(Z i , E γ ) = n atiσ(Z i , E γ )∑i[n ati · σ i (E γ )] . (6.18)Corrected Bethe-Heitler Cross SectionAs written in [2], the Bethe-Heitler formula corrected for various effects isdσ(Z, ɛ)dɛ[= αre {[ɛ 2 Z[Z + ξ(Z)] 2 + (1 − ɛ) 2 ] Φ 1 (δ(ɛ)) − F (Z) ]2+ 2 [3 ɛ(1 − ɛ) Φ 2 (δ(ɛ)) − F (Z) ]}(6.19)2where α is the fine-structure constant and r e the classical electron radius.Here ɛ = E/E γ , E γ is the energy of the photon and E is the total energycarried by one particle of the (e + , e − ) pair. The kinematical limits of ɛ arethereforem e c 2E γ= ɛ 0 ≤ ɛ ≤ 1 − ɛ 0 . (6.20)Screening EffectThe screening variable, δ, is a function of ɛδ(ɛ) = 136Z 1/3 ɛ 0ɛ(1 − ɛ) , (6.21)and measures the ’impact parameter’ of the projectile. Two screening functionsare introduced in the Bethe-Heitler formula :for δ ≤ 1 Φ 1 (δ) = 20.867 − 3.242δ + 0.625δ 2 (6.22)Φ 2 (δ) = 20.209 − 1.930δ − 0.086δ 2for δ > 1 Φ 1 (δ) = Φ 2 (δ) = 21.12 − 4.184 ln(δ + 0.952).Because the formula 6.19 is symmetric under the exchange ɛ ↔ (1 − ɛ), therange of ɛ can be restricted toɛ ∈ [ɛ 0 , 1/2]. (6.23)27

Born Approximation The Bethe-Heitler formula is calculated with planewaves, but Coulomb waves should be used instead. To correct for this, aCoulomb correction function is introduced in the Bethe-Heitler formula :withfor E γ < 50 MeV : F (z) = 8/3 ln Z (6.24)for E γ ≥ 50 MeV : F (z) = 8/3 ln Z + 8f c (Z)[f c (Z) = (αZ) 2 1(6.25)1 + (αZ) 2+0.20206 − 0.0369(αZ) 2 + 0.0083(αZ) 4 − 0.0020(αZ) 6 + · · ·].It should be mentioned that, after these additions, the cross section becomesnegative if[ ]42.24 − F (Z)δ > δ max (ɛ 1 ) = exp− 0.952. (6.26)8.368This gives an additional constraint on ɛ :δ ≤ δ max =⇒ ɛ ≥ ɛ 1 = 1 2 − 1 2√1 − δ minδ max(6.27)where(δ min = δ ɛ = 1 )= 1362 Z 4ɛ 1/3 0 (6.28)has been introduced. Finally the range of ɛ becomesɛ ∈ [ɛ min = max(ɛ 0 , ɛ 1 ), 1/2]. (6.29)28

δ(ε)d maxd minε0 ε0 ε11/21Gamma Conversion in the Electron Field The electron cloud gives anadditional contribution to pair creation, proportional to Z (instead of Z 2 ).This is taken into account through the expressionξ(Z) =ln(1440/Z 2/3 )ln(183/Z 1/3 ) − f c (Z) . (6.30)Factorization of the Cross Section ɛ is sampled using the techniques of’composition+rejection’, as treated in [3, 4, 5]. First, two auxiliary screeningfunctions should be introduced:F 1 (δ) = 3Φ 1 (δ) − Φ 2 (δ) − F (Z)F 2 (δ) = 3 2 Φ 1(δ) − 1 2 Φ 2(δ) − F (Z) (6.31)It can be seen that F 1 (δ) and F 2 (δ) are decreasing functions of δ, ∀δ ∈[δ min , δ max ]. They reach their maximum for δ min = δ(ɛ = 1/2) :F 10 = max F 1 (δ) = F 1 (δ min )F 20 = max F 2 (δ) = F 2 (δ min ). (6.32)After some algebraic manipulations the formula 6.19 can be written :dσ(Z, ɛ)= αr 2dɛeZ[Z + ξ(Z)] 2 [ 1min]9 2 − ɛ× [N 1 f 1 (ɛ) g 1 (ɛ) + N 2 f 2 (ɛ) g 2 (ɛ)] , (6.33)29

where[ ] 1 2N 1 =2 − ɛ min F 10 f 1 (ɛ) =[3 1[ 1 2 −ɛ min] − 3 2 ɛ] 2N 2 = 3 2 F 20 f 2 (ɛ) = const =1[ 1 2 −ɛ min]g 1 (ɛ) = F 1(ɛ)F 10g 2 (ɛ) = F 2(ɛ).F 20f 1 (ɛ) and f 2 (ɛ) are probability density functions on the interval ɛ ∈ [ɛ min , 1/2]such that ∫ 1/2ɛ minf i (ɛ) dɛ = 1, and g 1 (ɛ) and g 2 (ɛ) are valid rejection functions: 0 < g i (ɛ) ≤ 1 .Sampling the Energynumbers (r a , r b , r c ) :Given a triplet of uniformly distributed random1. use r a to choose which decomposition term in 6.33 to use:if r a < N 1 /(N 1 + N 2 ) → f 1 (ɛ) g 1 (ɛ) otherwise → f 2 (ɛ) g 2 (ɛ) (6.34)2. sample ɛ from f 1 (ɛ) or f 2 (ɛ) with r b :ɛ = 1 2 − ( 12 − ɛ min)( ) 1r 1/3b or ɛ = ɛ min +2 − ɛ min r b (6.35)3. reject ɛ if g 1 (ɛ)or g 2 (ɛ) < r cnote : below E γ = 2 MeV it is enough to sample ɛ uniformly on [ɛ 0 , 1/2],without rejection.ChargeThe charge of each particle of the pair is fixed randomly.Polar Angle of the Electron or PositronThe polar angle of the electron (or positron) is defined with respect to thedirection of the parent photon. The energy-angle distribution given by Tsai[6] is quite complicated to sample and can be approximated by a densityfunction suggested by Urban [7] :with∀u ∈ [0, ∞[ f(u) =a = 5 89a2 [u exp(−au) + d u exp(−3au)] (6.36)9 + dd = 27and θ ± = mc2E ±u. (6.37)30

A sampling of the distribution 6.36 requires a triplet of random numbers suchthatif r 1 < 99 + d → u = − ln(r 2r 3 )aThe azimuthal angle φ is generated isotropically.otherwise u = − ln(r 2r 3 ). (6.38)3aFinal State The e + and e − momenta are assumed to be coplanar with theparent photon. This information, together with energy conservation, is usedto calculate the momentum vectors of the (e + , e − ) pair and to rotate themto the global reference system.6.3.3 Status of this document12.01.02 created by M.Maire.21.03.02 corrections in angular distribution (mma)22.04.02 re-worded by D.H. WrightBibliography[1] J.H.Hubbell, H.A.Gimm, I.Overbo Jou. Phys. Chem. Ref. Data 9:1023(1980)[2] W. Heitler The Quantum Theory of Radiation, Oxford University Press(1957)[3] R. Ford and W. Nelson. SLAC-210, UC-32 (1978)[4] J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)[5] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[6] Y. S. Tsai, Rev. Mod. Phys. 46 815 (1974), Y. S. Tsai, Rev. Mod. Phys.49 421 (1977)[7] L.Urban in Geant3 writeup, section PHYS-211. Cern Program Library(1993)31

6.4 Gamma Conversion into a Muon - AntimuonPairThe class G4GammaConversionToMuons simulates the process of gamma conversioninto muon pairs. Given the photon energy and Z and A of the materialin which the photon converts, the probability for the conversions to takeplace is calculated according to a parameterized total cross section. Next, thesharing of the photon energy between the µ + and µ − is determined. Finally,the directions of the muons are generated. Details of the implementation aregiven below and can be also found in [1].6.4.1 Cross Section and Energy SharingIn the field of the nucleus, muon pair production on atomic electrons, γ+e →e + µ + + µ − , has a threshold of 2m µ (m µ + m e )/m e ≈ 43.9 GeV . Up toseveral hundred GeV this process has a much lower cross section than thecorresponding process on the nucleus. At higher energies, the cross section onatomic electrons represents a correction of ∼ 1/Z to the total cross section.For the approximately elastic scattering considered here, momentum, butno energy, is transferred to the nucleon. The photon energy is fully sharedby the two muons according toor in terms of energy fractionsE γ = E + µ + E − µ (6.39)x + = E+ µE γ, x − = E− µE γ, x + + x − = 1 .The differential cross section for electromagnetic pair creation of muons interms of the energy fractions of the muons isdσ= 4 α Z 2 rc(1 2 − 4 )dx + 3 x +x − log(W ) , (6.40)where Z is the charge of the nucleus, r c is the classical radius of the particleswhich are pair produced (here muons) andwhereW = W ∞1 + (D n√ e − 2) δ /mµ1 + B Z −1/3 √ e δ /m e(6.41)W ∞ = B Z−1/3D nm µm eδ =m 2 µ2 E γ x + x −√ e = 1.6487 . . . .32

For hydrogen B = 202.4 D n = 1.49and for all other nuclei B = 183 D n = 1.54 A 0.27 . (6.42)These formulae are obtained from the differential cross section for muonbremsstrahlung [2] by means of crossing relations. The formulae take intoaccount the screening of the field of the nucleus by the atomic electrons inthe Thomas-Fermi model, as well as the finite size of the nucleus, which isessential for the problem under consideration. The above parameterizationgives good results for E γ ≫ m µ . The fact that it is approximate closeto threshold is of little practical importance. Close to threshold, the crosssection is small and the few low energy muons produced will not travel veryfar. The cross section calculated from Eq. (6.40) is positive for E γ > 4m µandx min ≤ x ≤ x max with x min = 1 √12 − 4 − m µx max = 1 √1E γ 2 + 4 − m µ,E γ(6.43)except for very asymmetric pair-production, close to threshold, which caneasily be taken care of by explicitly setting σ = 0 whenever σ < 0.Note that the differential cross section is symmetric in x + and x − andthatx + x − = x − x 2where x stands for either x + or x − . By defining a constantσ 0 = 4 α Z 2 r 2 c log(W ∞ ) (6.44)the differential cross section Eq. (6.40) can be rewritten as a normalized andsymmetric as function of x:[1 dσσ 0 dx = 1 − 4 ] log W3 (x − x2 ) . (6.45)log W ∞This is shown in Fig. 6.1 for several elements and a wide range of photonenergies. The asymptotic differential cross section for E γ → ∞is also shown.1σ 0dσ ∞dx = 1 − 4 3 (x − x2 )33

10.90.8E γ → ∞100 TeVH Z=1 A=1.00794Be Z=4 A=9.01218Pb Z=82 A=207.20.70.610 TeV1 TeV0.5dσσ 0 dx0.4100 GeV0.30.210 GeV0.1E γ = 1 GeV00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1xFigure 6.1: Normalized differential cross section for pair production as afunction of x, the energy fraction of the photon energy carried by one ofthe leptons in the pair. The function is shown for three different elements,hydrogen, beryllium and lead, and for a wide range of photon energies.34

6.4.2 Parameterization of the Total Cross SectionThe total cross section is obtained by integration of the differential crosssection Eq. (6.40), that isσ tot (E γ ) =∫ xmaxx mindσdx +dx + = 4 α Z 2 r 2 c∫ xmaxx min(1 − 4 3 x +x −)log(W ) dx + .(6.46)W is a function of (x + , E γ ) and (Z, A) of the element (see Eq. (6.41)). Numericalvalues of W are given in Table 6.1.Table 6.1: Numerical values of W for x + = 0.5 for different elements.E γ W for H W for Be W for Cu W for PbGeV1 2.11 1.594 1.3505 5.21210 19.4 10.85 6.803 43.53100 191.5 102.3 60.10 332.71000 1803 919.3 493.3 1476.110000 11427 4671 1824 1028.1∞ 28087 8549 2607 1339.8Values of the total cross section obtained by numerical integration arelisted in Table 6.2 for four different elements. Units are in µbarn , where1 µbarn = 10 −34 m 2 .Table 6.2: Numerical values for the total cross sectionE γ σ tot , H σ tot , Be σ tot , Cu σ tot , PbGeV µbarn µbarn µbarn µbarn1 0.01559 0.1515 5.047 30.2210 0.09720 1.209 49.56 334.6100 0.1921 2.660 121.7 886.41000 0.2873 4.155 197.6 147610000 0.3715 5.392 253.7 1880∞ 0.4319 6.108 279.0 2042Well above threshold, the total cross section rises about linearly in log(E γ )with the slope1W M = √ (6.47)4 D n e mµ35

10.90.80.7σ / σ ∞0.60.50.40.30.20.1HPb02 3 4 5 6 7 81 10 10 10 10 10 10 10 10E γ in GeVFigure 6.2: Total cross section for the Bethe-Heitler process γ → µ + µ − as afunction of the photon energy E γ in hydrogen and lead, normalized to theasymptotic cross section σ ∞ .36

until it saturates due to screening at σ ∞ . Fig. 6.2 shows the normalized crosssection whereσ ∞ = 7 9 σ 0 and σ 0 = 4 α Z 2 r 2 c log(W ∞) . (6.48)Numerical values of W M are listed in Table 6.3.withandTable 6.3: Numerical values of W M .Element W M1/GeVH 0.963169Be 0.514712Cu 0.303763Pb 0.220771The total cross section can be parameterized asσ par = 28 α Z2 r 2 c9E g =(1 − 4m µE γlog(1 + W M C f E g ) , (6.49)) t (Wssat + E s γ) 1/s. (6.50)W sat = W ∞W M= B Z −1/3 4 √ e m 2 µm e.The threshold behavior in the cross section was found to be well approximatedby t = 1.479 + 0.00799D n and the saturation by s = −0.88. Theagreement at lower energies is improved using an empirical correction factor,applied to the slope W M , of the form[(C f = 1 + 0.04 log 1 + E )]c,E γwhere[E c = −18. + 4347. ]GeV .B Z −1/3A comparison of the parameterized cross section with the numerical integrationof the exact cross section shows that the accuracy of the parametrizationis better than 2%, as seen in Fig. 6.3.37

σ / σ par1.021.0110.990.98PbCuBeH2 3 4 5 6 7 81 10 10 10 10 10 10 10 10E γ in GeVFigure 6.3: Ratio of numerically integrated and parametrized total crosssections as a function of E γ for hydrogen, beryllium, copper and lead.6.4.3 Multi-differential Cross Section and Angular VariablesThe angular distributions are based on the multi-differential cross section forlepton pair production in the field of the Coulomb centerHerewhere[dσdx + du + du − dϕ = 4 Z2 α 3πm 2 µq 4{u 2 + + u2 −(1 + u 2 +) (1 + u 2 −) − 2x +x −u + u −u 2 +(1 + u 2 +) + u 2 ]−− 2u +u − (1 − 2x + x − ) cos ϕ2 (1 + u 2 −) 2 (1 + u 2 +) (1 + u 2 −)}. (6.51)u ± = γ ± θ ± , γ ± = E± µm µ, q 2 = q 2 ‖ + q2 ⊥ , (6.52)q 2 ⊥ = m 2 µq‖ 2 = qmin 2 (1 + x − u 2 + + x + u 2 −) 2 ,[(u+ − u − ) 2 + 2 u + u − (1 − cos ϕ) ] . (6.53)q 2 is the square of the momentum q transferred to the target and q 2 ‖ and q2 ⊥are the squares of the components of the vector q, which are parallel andperpendicular to the initial photon momentum, respectively. The minimummomentum transfer is q min = m 2 µ /(2E γ x + x − ).38

The muon vectors have the componentsp + = p + ( sin θ + cos(ϕ 0 + ϕ/2) , sin θ + sin(ϕ 0 + ϕ/2) , cos θ + ) ,p − = p − (− sin θ − cos(ϕ 0 − ϕ/2) , − sin θ − sin(ϕ 0 − ϕ/2) , cos θ − ) ,(6.54)where p ± = √ E± 2 − m 2 µ . The initial photon direction is taken as the z-axis.The cross section of Eq. (6.51) does not depend on ϕ 0 . Because of azimuthalsymmetry, ϕ 0 can simply be sampled at random in the interval (0, 2 π).Eq. (6.51) is too complicated for efficient Monte Carlo generation. Tosimplify, the cross section is rewritten to be symmetric in u + , u − using anew variable u and small parameters ξ, β, where u ± = u ± ξ/2 and β = u ϕ.When higher powers in small parameters are dropped, the differential crosssection in terms of u, ξ, β becomesdσdx + dξ dβ udu = 4 Z2 α 3 m 2 µ(π q2‖+ m 2 µ(ξ 2 + β 2 ) ) 2(6.55)[ {ξ 2 1(1 + u 2 ) − 2 x (1 − u 2 ) 2 ]}+x 2 − + β2 (1 − 2x + x − ),(1 + u 2 ) 4 (1 + u 2 ) 2where, in this approximation,q 2 ‖ = q2 min (1 + u2 ) 2 .For Monte Carlo generation, it is convenient to replace (ξ, β) by the polarcoordinates (ρ, ψ) with ξ = ρ cos ψ and β = ρ sin ψ. Integrating Eq. 6.55over ψ and using symbolically du 2 where du 2 = 2u du yieldsdσdx + dρ du 2 = 4Z2 α 3m 2 µρ 3 { 1 − x+ x −(q‖ 2 /m2 µ + ρ 2 ) 2 (1 + u 2 ) − x +x − (1 − u 2 ) 2 }.2 (1 + u 2 ) 4Integration with logarithmic accuracy over ρ gives∫(6.56)ρ 3 ∫1( )dρ(q‖ 2 /m2 µ + ρ2 ) ≈ dρ2 ρ = log mµ. (6.57)q ‖q ‖ /m µWithin the logarithmic accuracy, log(m µ /q ‖ ) can be replaced by log(m µ /q min ),so thatdσdx + du = 4 Z2 α 3 { 1 − x+ x −2 m 2 µ (1 + u 2 ) − x +x − (1 − u 2 ) 2 } ( )mµlog . (6.58)2 (1 + u 2 ) 4 q min39

Making the substitution u 2 = 1/t − 1, du 2 = −dt /t 2 givesdσdx + dt = 4 Z2 α 3m 2 µ[1 − 2 x + x − + 4 x + x − t (1 − t)] log( )mµq min. (6.59)Atomic screening and the finite nuclear radius may be taken into account bymultiplying the differential cross section determined by Eq. (6.56) with thefactor(F a (q) − F n (q) ) 2 , (6.60)where F a and F n are atomic and nuclear form factors. Please note that afterintegrating Eq. 6.56 over ρ, the q-dependence is lost.6.4.4 Procedure for the Generation of Muon - AntimuonPairsGiven the photon energy E γ and Z and A of the material in which the γconverts, the probability for the conversions to take place is calculated accordingto the parametrized total cross section Eq. (6.49). The next step,determining how the photon energy is shared between the µ + and µ − , isdone by generating x + according to Eq. (6.40). The directions of the muonsare then generated via the auxilliary variables t, ρ, ψ. In more detail, thefinal state is generated by the following five steps, in which R 1,2,3,4,... are randomnumbers with a flat distribution in the interval [0,1]. The generationproceeds as follows.1) Sampling of the positive muon energy E + µ = x + E γ .This is done using the rejection technique. x + is first sampled from a flatdistribution within kinematic limits usingx + = x min + R 1 (x max − x min )and then brought to the shape of Eq. (6.40) by keeping all x + which satisfy(1 − 4 3 x +x −) log(W )log(W max ) < R 2 .Here W max = W (x + = 1/2) is the maximum value of W , obtained for symmetricpair production at x + = 1/2. About 60% of the events are kept in thisstep. Results of a Monte Carlo generation of x + are illustrated in Fig. 6.4.The shape of the histograms agrees with the differential cross section illustratedin Fig. 6.1.40

30000250002000015000E γ10 GeV100 GeV1000 GeV10000500000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x +Figure 6.4: Histogram of generated x + distributions for beryllium at threedifferent photon energies. The total number of entries at each energy is 10 6 .2) Generate t(= 1γ 2 θ 2 +1 ) .The distribution in t is obtained from Eq.(6.59) asf 1 (t) dt = 1 − 2 x +x − + 4 x + x − t (1 − t)1 + C 1 /t 2 dt , 0 < t ≤ 1 . (6.61)with form factors taken into account byC 1 = (0.35 A0.27 ) 2x + x − E γ /m µ. (6.62)In the interval considered, the function f 1 (t) will always be bounded fromabove bymax[f 1 (t)] = 1 − x +x −1 + C 1.For small x + and large E γ , f 1 (t) approaches unity, as shown in Fig. 6.5.The Monte Carlo generation is done using the rejection technique. About70% of the generated numbers are kept in this step. Generated t-distributionsare shown in Fig. 6.6.3) Generate ψ by the rejection technique using t generated in the previousstep for the frequency distributionf 2 (ψ) = [ 1−2 x + x − +4 x + x − t (1−t) (1+cos(2ψ)) ] , 0 ≤ ψ ≤ 2π . (6.63)41

10.8E γ = 10 GeV0.10.2510.8E γ = 1 TeV0.010.10.25f 1 (t)0.60.40.5f 1 (t)0.60.40.5x +0.20.01x +00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1t0.200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1tFigure 6.5: The function f 1 (t) at E γ = 10 GeV (left) and E γ = 1 TeV (right)in beryllium for different values of x + .30000250002000015000E γ = 10 GeVE γ = 1 TeV10000500000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1tFigure 6.6: Histograms of generated t distributions for E γ = 10 GeV (solidline) and E γ = 100 GeV (dashed line) with 10 6 events each.42

3000025000200001500010000500001 GeV10 GeV100 GeV1000 GeV0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2ψ/πFigure 6.7: Histograms of generated ψ distributions for beryllium at fourdifferent photon energies.43

The maximum of f 2 (ψ) ismax[f 2 (ψ)] = 1 − 2 x + x − [1 − 4 t (1 − t)] . (6.64)Generated distributions in ψ are shown in Fig. 6.7.4) Generate ρ.The distribution in ρ has the formwhereandC 2 =f 3 (ρ) dρ =⎡(4 m√ ⎣ µx+ x − 2E γ x + x − tρ3 dρρ 4 + C 2, 0 ≤ ρ ≤ ρ max , (6.65)ρ 2 max = 1.9A 0.27 ( 1t − 1 ), (6.66)) 2+() ⎤ 2m e ⎦183 Z −1/3 m µ2. (6.67)The ρ distribution is obtained by a direct transformation applied to uniformrandom numbers R i according towhereρ = [C 2 (exp(β R i ) − 1)] 1/4 , (6.68)β = logGenerated distributions of ρ are shown in Fig. 6.85) Calculate θ + , θ − and ϕ from t, ρ, ψ withaccording toγ ± = E± µm µand u =(C2 + ρ 4 )max. (6.69)C 2√1t − 1 . (6.70)θ + = 1 (u + ρ )γ + 2 cos ψ , θ − = 1 (u − ρ )γ − 2 cos ψ and ϕ = ρ u sin ψ .(6.71)The muon vectors can now be constructed from Eq. (6.54), where ϕ 0 is chosenrandomly between 0 and 2π. Fig. 6.9 shows distributions of θ + at differentphoton energies (in beryllium). The spectra peak around 1/γ as expected.The most probable values are θ + ∼ m µ /E µ+ = 1/γ +. In the small angleapproximation used here, the values of θ + and θ − can in principle be any44

x 10020001800160014001200100080060040020001 TeVEγ = 10 GeV0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1ρFigure 6.8: Histograms of generated ρ distributions for beryllium at twodifferent photon energies. The total number of entries at each energy is 10 6 .45

x 10050004500400035003000250020001500100050001TeV100 GeV10 GeV1 GeV10 GeV100 GeV1000 GeV1 GeV0 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.1θ +Figure 6.9: Histograms of generated θ +energies.distributions at different photon46

1.41.2simulatedexact1.00.8Coulomb centre0.60.40.20.00 0.2 0.4 0.6 0.8 12 21 / ( 1 + θ + γ + )Figure 6.10: Angular distribution of positive (or negative) muons. The solidcurve represents the results of the exact calculations. The histogram is thesimulated distribution. The angular distribution for pairs created in the fieldof the Coulomb centre (point-like target) is shown by the dashed curve forcomparison.47

7654321010 -1 1θ + γ +Figure 6.11: Angular distribution in logarithmic scale. The curve correspondsto the exact calculations and the histogram is the simulated distribution.48

4321010 -3 10 -2 10 -1 1| θ + γ + - θ - γ - |Figure 6.12: Distribution of the difference of transverse momenta of positiveand negative muons (with logarithmic x-scale).49

positive value from 0 to ∞. In the simulation, this may lead (with a verysmall probability, of the order of m µ /E γ ) to unphysical events in which θ + orθ − is greater than π. To avoid this, a limiting angle θ cut = π is introduced,and the angular sampling repeated, whenever max(θ + , θ − ) > θ cut .Figs. 6.10,6.11 and 6.12 show distributions of the simulated angular characteristicsof muon pairs in comparison with results of exact calculations.The latter were obtained by means of numerical integration of the squaredmatrix elements with respective nuclear and atomic form factors. All thesecalculations were made for iron, with E γ = 10 GeV and x + = 0.3. As seenfrom Fig. 6.10, wide angle pairs (at low values of the argument in the figure)are suppressed in comparison with the Coulomb center approximation.This is due to the influence of the finite nuclear size which is comparableto the inverse mass of the muon. Typical angles of particle emission are ofthe order of 1/γ ± = m µ /E µ± (Fig. 6.11). Fig. 6.12 illustrates the influence ofthe momentum transferred to the target on the angular characteristics of theproduced pair. In the frame of the often used model which neglects targetrecoil, the pair particles would be symmetric in transverse momenta, andcoplanar with the initial photon.6.4.5 Status of this document28.05.02 created by H.Burkhardt.01.12.02 re-worded by D.H. WrightBibliography[1] H. Burkhardt, S. Kelner, and R. Kokoulin, “Monte Carlo Generator forMuon Pair Production”. CERN-SL-2002-016 (AP) and CLIC Note 511,May 2002.[2] S. R. Kelner, R. P. Kokoulin, and A. A. Petrukhin, “About cross sectionfor high energy muon bremsstrahlung,”. Moscow Phys. Eng. Inst. 024-95, 1995. 31pp.50

Chapter 7Common to All ChargedParticles51

7.1 Computing the Mean Energy LossEnergy loss processes are very similar for e + /e− , µ + /µ− and chargedhadrons, so a common description for them was a natural choice in Geant4.Any energy loss process must calculate the continuous and discrete energyloss in a material. Below a given energy threshold the energy loss is continuousand above it the energy loss is simulated by the explicit production ofsecondary particles - gammas, electrons, and positrons.7.1.1 MethodLetdσ(Z, E, T )dTbe the differential cross-section per atom (atomic number Z) for the ejectionof a secondary particle with kinetic energy T by an incident particle of totalenergy E moving in a material of density ρ. The value of the kinetic energycut-off or production threshold is denoted by T cut . Below this threshold thesoft secondaries ejected are simulated as continuous energy loss by the incidentparticle, and above it they are explicitly generated. The mean rate ofenergy loss is given by:dE soft (E, T cut )dx= n at ·∫ Tcut0dσ(Z, E, T )T dT (7.1)dTwhere n at is the number of atoms per volume in the material. The total crosssection per atom for the ejection of a secondary of energyT > T cut is∫ Tmaxdσ(Z, E, T )σ(Z, E, T cut ) =dT (7.2)T cut dTwhere T max is the maximum energy transferable to the secondary particle.If there are several processes providing energy loss for a given particle, thenthe total continuous part of the energy loss is the sum:dE totsoft (E, T cut)dx= ∑ idE soft,i (E, T cut ). (7.3)dxThese values are pre-calculated during the initialization phase of Geant4and stored in the dE/dx table. Using this table the ranges of the particlein given materials are calculated and stored in the Range table. The Rangetable is then inverted to provide the InverseRange table. At run time, valuesof the particle’s continuous energy loss and range are obtained using these52

tables. Concrete processes contributing to the energy loss are not involved inthe calculation at that moment. In contrast, the production of secondarieswith kinetic energies above the production threshold is sampled by eachconcrete energy loss process.The default energy interval for these tables extends from 100 eV to 100TeV and the default number of bins is 120. For muon energy loss processesmodels are valid for higher energies and this interval can be extended up to1000 PeV. Note that this extention should be done for all three processeswhich contribute to muon energy loss.7.1.2 Implementation DetailsCommon calculations are performed in the class G4V EnergyLossP rocessin which the following methods are implemented:• BuildPhysicsTable;• StorePhysicsTable;• RetrievePhysicsTable;• AlongStepDoIt;• PostStepDoIt;• GetMeanFreePath;• GetContinuousStepLimit;• MicroscopicCrossSection;• GetDEDXDispersion;• SetMinKinEnergy;• SetMaxKinEnergy;• SetDEDXBinning;• SetLambdaBinning;This interface is used by the following processes:• G4eIonisation;• G4eBremsstrahlung;53

• G4hIonisation;• G4ionIonisation;• G4MuIonisation;• G4MuBremsstrahlung;• G4MuPairProduction.These processes mainly provide initialization. The physics models are implementedusing the G4V EmModel interface. Because a model is defined tobe active over a given energy range and for a defined set of G4Regions, anenergy loss process can have one or several models defined for a particle andG4Region. The following models are available:• G4MollerBhabhaModel;• G4eBremmstrahlungModel;• G4BetheBlochModel;• G4BraggModel;• G4PAIModel;• G4MuBetheBlochModel;• G4MuBremmstrahlungModel;• G4MuPairProductionModel;Stepsize Limit Due to Continuous Energy LossContinuous energy loss imposes a limit on the stepsize because of the energydependence of the cross sections. It is generally assumed in MC programsthat the particle cross sections are approximately constant along a step, i.e.the step size should be small enough that the change in cross section, fromthe beginning of the step to the end, is also small. In principle one mustuse very small steps in order to insure an accurate simulation, however thecomputing time increases as the stepsize decreases. A good compromise isto limit the stepsize by not allowing the stopping range of the particle todecrease by more than 20 % during the step. This condition works well forparticles with kinetic energies > 1MeV , but for lower energies it gives veryshort stepsizes.54

To cure this problem a lower limit on the stepsize was introduced. Thereis a natural choice for this limit: the stepsize cannot be smaller than the rangecut parameter of the program. The stepsize limit varies smoothly with decreasingenergy from the value given by the condition ∆range/range = 0.20to the lowest possible value range cut. These are the default step limitationparameters; they can be overwritten using the UI command “/process/eLoss/StepFunction0.2 1 mm”, for example.Energy Loss ComputationThe computation of the mean energy loss after a given step is done by usingthe dE/dx, Range, and InverseRange tables. The dE/dx table is used ifthe energy deposition is less than 5 % of kinetic energy of the particle. Whena larger percentage of energy is lost, the mean loss ∆T can be written as∆T = T 0 − f T (r 0 − step) (7.4)where T 0 is the kinetic energy, r 0 the range at the beginning of the step step,the function f T (r) is the inverse of the Range table (i.e. it gives the kineticenergy of the particle for a range value of r) .After the mean energy loss has been calculated, the process computes theactual energy loss, i.e. the loss with fluctuations. The fluctuation is computedfrom a model described in Section 7.2.7.1.3 Energy Loss by Heavy Charged ParticlesTo save memory in the case of hadron energy loss, dE/dx, Range andInverseRange tables are constructed only for protons. The energy loss forother heavy, charged particles is computed from these tables at the scaledkinetic energy T scaled :T scaled = M protonTM particle, (7.5)where T is the kinetic energy of the particle, and M proton and M particle arethe masses of the proton and particle. Note that in this approach the smalldifferences between T max values calculated for different changed particles areneglected. This is acceptable for hadrons and ions, but is not very accuratefor the simulation of muon energy loss, which is simulated by a separateprocess.55

7.1.4 Status of this document09.10.98 created by L. Urbán.01.12.03 revised by V.Ivanchenko.02.12.03 spelling and grammar check by D.H. Wright56

7.2 Energy loss fluctuations7.2.1 Fluctuations in thick absorbersThe total continuous energy loss of charged particles is a stochastic quantitywith a distribution described in terms of a straggling function. The stragglingis partially taken into account in the simulation of energy loss by theproduction of δ-electrons with energy T > T c . However, continuous energyloss also has fluctuations. Hence in the current GEANT4 implementationtwo different models of fluctuations are applied depending on the value ofthe parameter κ which is the lower limit of the number of interactions of theparticle in the step. The default value chosen is κ = 10. To select a modelfor thick absorbers the following boundary conditions are used:∆E > κT c or T c < κI (7.6)where ∆E is the mean continuous energy loss in a track segment of lengths, T c is the cut kinetic energy of δ-electrons, and I is the average ionizationpotential of the atom. For thick absorbers the straggling function approachesthe Gaussian distribution with Bohr’s variance [4]:Ω 2 = 2πr 2 e m ec 2 N elZ 2 hβ 2 T cs(1 − β22), (7.7)where r e is the classical electron radius, N el is the electron density of themedium, Z h is the charge of the incident particle in units of positron charge,and β is the relativistic velocity.7.2.2 Fluctuations in thin absorbersIf the condition 7.6 is not satisfied the model of energy fluctuations in thinabsorbers is applied. The formulae used to compute the energy loss fluctuation(straggling) are based on a very simple physics model of the atom. It isassumed that the atoms have only two energy levels with binding energies E 1and E 2 . The particle-atom interaction can be an excitation with energy lossE 1 or E 2 , or ionization with energy loss distributed according to a functiong(E) ∼ 1/E 2 :∫ Tup+Ig(E) dE = 1 =⇒ g(E) = I(T up + I) 1IT up E . (7.8)2The macroscopic cross section for excitation (i = 1, 2) isΣ i = C f iE iln[2mc 2 (βγ) 2 /E i ] − β 2ln[2mc 2 (βγ) 2 /I] − β 2 (1 − r) (7.9)57

and the ionization cross section isΣ 3 = CI(T up + I) ln( Tup+I ) r (7.10)Iwhere I denotes the mean ionization energy of the atom, T up is the productionthreshold for delta ray production (or the maximum energy transfer minusmean ionization energy if this value smaller than the production threshold),E i and f i are the energy levels and corresponding oscillator strengths of theatom, and C and r are model parameters.The oscillator strengths f i and energy levels E i should satisfy the constraintsT upf 1 + f 2 = 1 (7.11)f 1· lnE 1 + f 2· lnE 2 = lnI (7.12)The cross section formulae 7.9,7.10 and the sum rule equations 7.11,7.12 canbe found e.g. in Ref. [1].The model parameter C can be defined in the following way. The numbers ofcollisions (n i , i = 1, 2 for excitation and 3 for ionization) follow the Poissondistribution with a mean value < n i >. In a step of length ∆x the meannumber of collisions is given by< n i >= Σ i· ∆x. (7.13)The mean energy loss in a step is the sum of the excitation and ionizationcontributions and can be written as{∫ }dETup+Idx · ∆x = Σ 1 E 1 + Σ 2 E 2 + Eg(E)dE ∆x. (7.14)IFrom this, using eq. 7.9 - 7.12, one can see thatC = dE/dx. (7.15)The other parameters in the fluctuation model have been chosen in the followingway. Z· f 1 and Z· f 2 represent in the model the number of loosely/tightlybound electronsf 2 = 0 for Z = 1 (7.16)f 2 = 2/Z for Z ≥ 2 (7.17)E 2 = 10 eV Z 2 . (7.18)Using these parameter values, E 2 corresponds approximately to the K-shellenergy of the atoms ( and Zf 2 = 2 is the number of K-shell electrons). Theparameters f 1 and E 1 can be obtained from Eqs. 7.11 and 7.12.The parameter r is the only variable in the model which can be tuned. Thisparameter determines the relative contribution of ionization and excitation tothe energy loss. Based on comparisons of simulated energy loss distributionsto experimental data, its value has been fixed at r = 0.4.58

Sampling the energy loss. The energy loss is computed in the modelunder the assumption that the step length (or relative energy loss) is smalland, in consequence, the cross section can be considered constant along thestep. The loss due to the excitation is∆E exc = n 1 E 1 + n 2 E 2 (7.19)where n 1 and n 2 are sampled from a Poisson distribution. The energy lossdue to ionization can be generated from the distribution g(E) by the inversetransformation method :u = F (E) =∫ EIg(x)dx (7.20)E = F −1 I(u) =(7.21)1 − u TupT up+Iwhere u is a uniformly distributed random number ∈ [0, 1]. The contributioncoming from the ionization will then be∆E ion =n3 ∑Tj=1 1 − uupj T up+Iwhere n 3 is the number of ionizations sampled from the Poisson distribution.The total energy loss in a step will be ∆E = ∆E exc + ∆E ion and the energyloss fluctuation comes from the fluctuations of the collision numbers n i .I(7.22)Thin layers : In the case of very small energy loss (small step lengths, ∼ 1mm in gases, ∼ 1 micrometer in solids) this model calculation can give zeroenergy loss for some events. In order to avoid this nonphysical situation, theprobability of zero energy loss is computed asP (∆E = 0) = e −(++) . (7.23)If this probabillity is bigger than a limit (0.01) a special sampling is done,taking into account the fact that in this case the projectile interacts onlywith the outer electrons of the atom. An energy level E 0 = 10 eV has beenchosen to correspond to the outer electrons. The mean number of collisionscan be computed as< n >= 1 dE∆x. (7.24)E 0 dxAll the collisions can be considered as ionizations for this case. The numberof collisions are sampled according to a Poisson distribution and the energyloss is computed from the equation∆E =n∑j=1E 01 − u jT upT up+E 0. (7.25)59

Thick layers : If the mean energy loss and step are in the range of validityof the Gaussian approximation of the fluctuation, the much faster Gaussiansampling is used to compute the actual energy loss.Conclusions. This simple model of the energy loss fluctuations is ratherfast and it can be used for any thickness of material. This has been verified byperforming many simulations and comparing the results with experimentaldata, such as that in Ref.[2].Approaching the limit of the validity of Landau’s theory, the loss distributionapproaches the Landau form smoothly.7.2.3 Status of this document30.01.02 created by L. Urbán.28.08.02 updated by V.Ivanchenko.17.08.04 moved to common to all charged particles (mma)04.12.04 spelling and grammar check by D.H. WrightBibliography[1] H.Bichsel Rev.Mod.Phys. 60 (1988) 663[2] K.Lassila-Perini, L.Urbán Nucl.Inst.Meth. A362(1995) 416[3] geant3 manual Cern Program Library Long Writeup W5013 (1994)[4] ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons andAlpha Particles, ICRU Report 49, 1993.60

7.3 Correcting the cross section for energyvariationAs described in Sections 7.1 and 5.2 the step size limitation is providedby energy loss processes in order to insure the precise calculation of theprobability of particle interaction. It is generally assumed in Monte Carloprograms that the particle cross sections are approximately constant during astep, hence the reaction probability p at the end of the step can be expressedasp = 1 − exp (−nsσ(E i )) , (7.26)where n is the density of atoms in the medium, s is the step length, E i is theenergy of the incident particle at the beginning of the step, and σ(E i ) is thereaction cross section at the beginning of the step.However, it is possible to sample the reaction probability from the exactexpression()p = 1 − exp−∫ EfE inσ(E)ds, (7.27)where E f is the energy of the incident particle at the end of the step, byusing the integral approach to particle transport. This approach is availablefor processes implemented via the G4V EnergyLossP rocess interface.The Monte Carlo method of integration is used for sampling the reactionprobability [1]. It is assumed that the reaction cross section increases withenergy, so that the cross section at the end of the step is always smaller,σ(E f ) < σ(E i ). This assumption is correct for electromagnetic physics.The integral variant of step limitation is the default for the G4eIonisationprocess but is not automatically activated for others. To do so the boolean UIcommand “/process/eLoss/integral true” can be used. The integral variant ofthe energy loss sampling process is less dependent on values of the productioncuts [2], however it should be applied on a case-by-case basis because heavyparticles taking large steps in an absorber can cause inaccurate sampling ofenergy loss fluctuations.7.3.1 Status of this document01.12.03 integral method subsection added by V. Ivanchenko17.08.04 moved to common to all charged particles (mma)61

Bibliography[1] V.N.Ivanchenko et al., Proc. of Int. Conf. MC91: Detector and eventsimulation in high energy physics, Amsterdam 1991, pp. 79-85. (HEPINDEX 30 (1992) No. 3237).[2] V.N.Ivanchenko. Geant4 Workshop (TRIUMF, Canada, 2003)http://www.triumf.ca/geant4-03/talks/04-Thursday-AM-1/02-V.Ivanchenko/eloss03.ppt62

7.4 Conversion from range to kinetic energy7.4.1 Charged particlesThe algorithm which converts the stopping range of a charged particle to thecorresponding kinetic energy is essentially the same for all charged particletypes: given the stopping range of a particle, a vector holding the correspondingkinetic energy for every material is constructed. Only the energyloss formulae are different, depending on whether the particle is an electron,a positron, or a heavy charged particle (muon, pion, proton, etc.). For protonsand anti-protons the above procedure is followed, but for other chargedhadrons, the cut values in kinetic energy are computed using the proton andanti-proton energy loss and range tables.General scheme1. An energy loss table is created and filled for all the elements in theelement table.2. For every material in the material table the following steps are performed:(a) a range vector is constructed using the energy loss table and specificformulae for the low energy part of the calculations,(b) the conversion from stopping range to kinetic energy is performedand the corresponding element of the KineticEnergyCuts vectoris set,(c) the range vector is deleted.3. The energy loss table is deleted at the end of the process.Energy loss formula for heavy charged particlesThe energy loss of the particle is calculated from a simplified Bethe-Blochformula if the kinetic energy of the particle is above the value( )particle massT lim = 2 MeV ×.proton massThe word “simplified” means that the low energy shell correction term andthe high energy Sternheimer density correction term have been omitted. Belowthe energy value T lim a simple parameterized energy loss formula is usedto compute the loss, which reproduces the energy loss values of the stopping63

power tables fairly well. The main reason for using a parameterized formulafor low energy is that the Bethe-Bloch formula breaks down at low energy.The formula has the following form :√ ⎧⎪dE ⎨ a ∗ T + b ∗ T for T ∈ [0, Tdx = M M 0]⎪ √ ⎩c ∗ Tfor T ∈ [TM 0 , T lim ]where :M = particle massT = kinetic energyT 0 = 0.1 MeV × Z 1/3 × M/(proton mass)Z = atomic number.The paramaters a, b and c have been chosen in such a way that dE/dx is acontinuous function of T at T = T lim and T = T 0 , and dE/dx reaches itsmaximum at the correct T value.Energy loss of electrons and positronsThe Berger-Seltzer energy loss formula has been used for T > 10 keV tocompute the energy loss due to ionization. This formula plays the role of theBethe-Bloch equation for electrons (see e.g. the GEANT3 manual). Below10 keV the simple c/(T /mass of electron) parameterization has been used,where c can be determined from the requirement of continuity at T = 10keV. For electrons the radiation loss is important even at relatively low (fewMeV) energies, so a second term has been added to the energy loss formulawhich accounts for radiation losses (losses due to bremsstrahlung). Thissecond term is an empirical, parameterized formula. For positrons a differentformula is used to calculate the ionization loss, while the term accounting forthe radiation losses is the same as that for electrons.Range calculationThe stopping range is defined asR(T ) =∫ T01(dE/dx) dE. The integration has been done analytically for the low energy part andnumerically above an energy limit.64

7.4.2 PhotonsStarting from a particle cut given in absorption lengths, the method constructsa vector holding the cut values in kinetic energy for every material.The main steps of the algorithm are the following :General scheme1. A cross section table is created and filled for all the elements in theelement table.2. For every material in the material table the following steps are performed:(a) an absorption length vector is constructed using the cross sectiontable(b) the conversion from absorption length to kinetic energy is performedand the corresponding element of the KineticEnergyCutsvector is set (It contains the particle cut value in kinetic energyfor the actual material.),(c) the absorption length vector is deleted.3. The cross section table is deleted at the end of the process.Cross section formula for elementsAn approximate empirical formula is used to compute the absorption crosssection of a photon in an element. Here, the absorption cross section meansthe sum of the cross sections of the gamma conversion, Compton scatteringand photoelectric effect. These processes are the “destructive” processes forphotons: they destroy the photon or decrease its energy. (The coherent orRayleigh scattering changes the direction of the gamma only; its cross sectionis not included in the absorption cross section.)Absorption length vectorThe AbsorptionLength vector is calculated for every material as :AbsorptionLength = 5/(macroscopic absorption cross section) .The factor 5 comes from the requirement that the probability of having no’destructive’ interaction should be small, henceexp(−AbsorptionLength * MacroscopicCrossSection) = exp(−5)= 6.7 × 10 −365

Meaningful cuts in absorption lengthThe photon cross section for a material has a minimum at a certain kineticenergy T min . The AbsorptionLength has a maximum at T = T min , thevalue of the maximal AbsorptionLength is the biggest ”meaningful” cut inabsorption length. If the cut given by the user is bigger than this maximum,a warning is printed and the cut in kinetic energy is set to the maximumgamma energy (i.e. all the photons will be killed in the material).7.4.3 Status of this document9.10.98 created by L. Urbán.27.07.01 minor revision M.Maire17.08.04 moved to common to all charged particles (mma)04.12.04 minor re-wording by D.H. Wright66

7.5 Multiple ScatteringGeant4 uses a new multiple scattering (MSC) model to simulate the multiplescattering of charged particles in matter. This model does not use theMoliere formalism [1], but is based on the more complete Lewis theory [2].The model simulates the scattering of the particle after a given step, andalso computes the path length correction and the lateral displacement.7.5.1 IntroductionMultiple Scattering AlgorithmsMSC simulation algorithms can be classified as either ”detailed” or ”condensed”.In the detailed algorithms, all the collisions/interactions experiencedby the particle are simulated. This simulation can be considered asexact; it gives the same results as the solution of the transport equation.However, it can be used only if the number of collisions is not too large,a condition fulfilled only for special geometries (such as thin foils), or lowenough kinetic energies. For larger kinetic energies the average number ofcollisions is very large and the detailed simulation becomes very inefficient.High energy simulation codes use condensed simulation algorithms, in whichthe global effects of the collisions are simulated at the end of a track segment.The global effects generally computed in these codes are the net displacement,energy loss, and change of direction of the charged particle. These quantitiesare computed from the multiple scattering theories used in the codes. Theaccuracy of the condensed simulations is limited by the approximations ofthe multiple scattering theories.Most particle physics simulation codes use the multiple scattering theoriesof Molière [1], Goudsmit and Saunderson [3] and Lewis [2]. The theories ofMolière and Goudsmit-Saunderson give only the angular distribution after astep, while the Lewis theory computes the moments of the spatial distributionas well. None of the these MSC theories gives the probability distributionof the spatial displacement. Therefore each of the MSC simulation codesincorporates its own algorithm to determine the spatial displacement of thecharged particle after a given step. These algorithms are not exact, of course,and are responsible for most of the uncertainties in the MSC codes. Thereforethe simulation results can depend on the value of the step length andgenerally one has to select the value of the step length carefully.A new class of MSC simulation, the ”mixed” simulation algorithms (see67

e.g.[4]), appeared in the literature recently. The mixed algorithm simulatesthe ”hard” collisions one by one and uses a MSC theory to treat the effects ofthe ”soft” collisions at the end of a given step. Such algorithms can preventthe number of steps from becoming too large and also reduce the dependenceon the step length.The MSC model used in Geant4 belongs to the class of condensed simulations.The model is based on Lewis’ MSC theory and uses model functions todetermine the angular and spatial distributions after a step. The functionshave been chosen in such a way as to give the same moments of the (angularand spatial) distributions as the Lewis theory.Definition of TermsIn simulation, a particle is transported by steps through the detector geometry.The shortest distance between the endpoints of a step is called thegeometrical path length, z. In the absence of a magnetic field, this is astraight line. For non-zero fields, z is the shortest distance along a curvedtrajectory. Constraints on z are imposed when particle tracks cross volumeboundaries. The path length of an actual particle, however, is usually longerthan the geometrical path length, due to physical interactions like multiplescattering. This distance is called the true path length, t. Constraints on tare imposed by the physical processes acting on the particle.The properties of the multiple scattering process are completely determinedby the transport mean free paths, λ k , which are functions of the energy in agiven material. The k-th transport mean free path is defined as1λ k= 2πn a∫ 1−1[1 − P k (cosχ)] dσ(χ) d(cosχ) (7.28)dΩwhere dσ(χ)/dΩ is the differential cross section of the scattering, P k (cosχ)is the k-th Legendre polynomial, and n a is the number of atoms per volume.Most of the mean properties of MSC computed in the simulation codes dependonly on the first and second transport mean free paths. The mean valueof the geometrical path length (first moment) corresponding to a given truepath length t is given by< z >= λ 1 ∗( [1 − exp − t ]). (7.29)λ 1Eq. 7.29 is an exact result for the mean value of z if the differential cross68

section has axial symmetry and the energy loss can be neglected. The transformationbetween true and geometrical path lengths is called the path lengthcorrection. This formula and other expressions for the first moments of thespatial distribution were taken from either [4] or [5], but were originally calculatedby Goudsmit and Saunderson [3] and Lewis [2].At the end of the true step length, t, the scattering angle is θ. The meanvalue of cosθ is[< cosθ >= exp − t ]. (7.30)λ 1The variance of cosθ can be written asσ 2 =< cos 2 θ > − < cosθ > 2 = 1 + 2e−2κτ3− e −2τ (7.31)where τ = t/λ 1 and κ = λ 1 /λ 2 . The mean lateral displacement is givenby a more complicated formula [4], but this quantity can also be calculatedrelatively easily and accurately. The square of the mean lateral displacementis[< x 2 + y 2 >= 4λ2 1τ − κ + 1 + κ]13 κ κ − 1 e−τ −κ(κ − 1) e−κτ . (7.32)Here it is assumed that the initial particle direction is parallel to the the zaxis.The transport mean free path values have been calculated by Liljequistet al. [6], [7] for electrons and positrons in the kinetic energy range100 eV - 20 MeV in 15 materials. The MSC model in Geant4 uses these values,and when necessary, linearly interpolates or extrapolates the transportcross section, σ 1 = 1/λ 1 , in atomic number Z and in the square of the particlevelocity, β 2 . The ratio κ is a very slowly varying function of the energy:κ > 2 for T > a few keV, and κ → 3 for very high energies (see [5]). Hence,a constant value of 2.5 is used in the model.7.5.2 Path Length CorrectionAs mentioned above, the path length correction refers to the transformation(true path length ⇒ geometrical path length) and its inverse. The true pathlength −→ geometrical path length transformation is given by eq. 7.29 if thestep is small and the energy loss can be neglected. If the step is not smallthe energy dependence makes the transformation more complicated. For thiscase Eqs. 7.30,7.29 should be modified as69

cosθ >= exp[−∫ t0]duλ 1 (u)(7.33)< z >=∫ t0< cosθ > u du (7.34)where θ is the scattering angle, t and z are the true and geometrical pathlengths, and λ 1 is the transport mean free path.In order to compute Eqs. 7.33,7.34 the t dependence of the transportmean free path must be known. λ 1 depends on the kinetic energy of theparticle which decreases along the step. All computations in the model usea linear approximation for this t dependence:λ 1 (t) = λ 10 (1 − ct) (7.35)Here λ 10 denotes the value of λ 1 at the start of the step, and c is aconstant. It is worth noting that Eq. 7.35 is not a crude approximation.It is rather good at low (< 1 MeV) energy. At higher energies the step isgenerally much smaller than the range of the particle, so the change in energyis small and so is the change in λ 1 . Using Eqs. 7.33 - 7.35 the explicit formulafor the geometrical path length z can be written asz(t) =[]1c(1 + 1 1 − (1 − ct) 1+ 1cλ 10 . (7.36)cλ 10)The value of the constant c can be expressed using λ 10 and λ 11 where λ 11is the value of the transport mean free path at the end of the stepc = λ 10 − λ 11tλ 10. (7.37)At low energies ( T kin < M , M - particle mass) c has a simpler form:c = 1 r 0(7.38)where r 0 denotes the range of the particle at the start of the step.It can easily be seen that for a small step (i.e. for a step with smallrelative energy loss) the formula of z(t) is[z(t) = λ 10 1 − exp (− t ]) . (7.39)λ 10Eq. 7.36 or 7.39 gives the mean value of the geometrical step length fora given true step length.70

The actual geometrical path length is sampled in the model according tothe probability density function defined for z ∈ [0, t] :f(z) = [(k + 1)/t] (z/z 0 ) k for z < z 0f(z) = [(k + 1)/t] [(t − z)/(t − z 0 )] k for z ≥ z 0 . (7.40)As can be seen, f(z) has a maximum at z = z 0 . The value of z 0 depends ont, and this dependence is approximated by the parameterizationz 0 =< z > +d (t− < z >), (7.41)where < z > is the mean value of z, and d is a constant model parameter.The value of the exponent k is computed from the requirement that f(z)should give the same mean value for z as eq. 7.36 or 7.39. Hence,k = 2 < z > −tz 0 − < z > . (7.42)The value of z is sampled according to f(z) if k > 0, otherwise z =< z > isused.The (geometrical path length ⇒ true path length) transformation is performedusing the mean values. The transformation can be written ast =< t >= −λ 1 ∗ log(1 − z ). (7.43)λ 1is the (geometrical) step is small andt(z) = 1 c[ ]11 − (1 − cwz) w(7.44)where w = 1 + 1cλ 10is the step is not small, i.e. the energy loss should betaken into account.This transformation is needed when the particle arrives at a volume boundary,causing the step to be geometry-limited. In this case the true path lengthshould be computed in order to have the correct energy loss of the particleafter the step.7.5.3 Angular DistributionThe quantity u = cosθ is sampled according to a model function g(u). Theshape of this function has been chosen such that Eqs. 7.30 and 7.31 aresatisfied. The functional form of g isg(u) = p[qg 1 (u) + (1 − q)g 3 (u)] + (1 − p)g 2 (u), (7.45)71

where 0 ≤ p, q ≤ 1, and the g i are simple functions of u = cosθ, normalizedover the range u ∈ [−1, 1]. The functions g i have been chosen asg 1 (u) = C 1 e −a(1−u) − 1 ≤ u 0 ≤ u ≤ 1, (7.46)g 2 (u) = C 21(b − u) c − 1 ≤ u ≤ u 0 ≤ 1, (7.47)g 3 (u) = C 3 − 1 ≤ u ≤ 1, (7.48)where a > 0, b > 0,c > 0 and u 0 are model parameters, and the C i arenormalization constants. It is worth noting that for small scattering angles,θ, g 1 (u) is nearly Gaussian (exp(−θ 2 /2θ 2 0 )) if θ2 0 ≈ 1/a, while g 2(u) has aRutherford-like tail for large θ, if b ≈ 1 and c is not far from 2 .7.5.4 Determination of the Model ParametersThe parameters a, b, c, u 0 and p, q are not independent. The requirement thatthe angular distribution function g(u) and its first derivative be continuousat u = u 0 imposes two constraints on the parameters:p g 1 (u 0 ) = (1 − p) g 2 (u 0 ) (7.49)p a g 1 (u 0 ) = (1 − p)cb − u 0g 2 (u 0 ). (7.50)A third constraint comes from Eq. 7.30: g(u) should give the same meanvalue for u as the theory.It follows from Eqs. 7.30 and 7.45 thatq{p < u > 1 +(1 − p) < u > 2 } = e −τ , (7.51)where < u > i denotes the mean value of u computed from the distributiong i (u).The parameter a was chosen according to a modified Highland-Lynch-Dahl formula for the width of the angular distribution [8], [9].where θ 0 isθ 0 = 13.6MeVβcpa =0.51 − cos(θ 0 )(7.52)√z t/X0(1 + 0.038 ln x/X0) (7.53)72

when the original Highland-Lynch-Dahl formula is used. Here θ 0 = θ rmsplaneis the width of the approximate Gaussian projected angle distribution, p,βcand z are the momentum, velocity and charge number of the incident particle,and t/X0 is the thickness of the scattering medium in radiation lengths. Thisvalue of θ 0 is from a fit to the Molière distribution for singly charged particleswith β = 1 for all Z, and is accurate to 11 % or better for 10 −3 ≤ t/X0 ≤ 100(see e.g. Rev. of Particle Properties, section 23.3). The modified formula forθ 0 isθ 0 = 13.6MeVβcpz( tX0 )0.555 (7.54)This formula gives much smaller step dependence in the angular distributionand describes the available electron scattering data better than theHighland form.The value of the parameter u 0 has been chosen asu 0 = 1 − ξ/a (7.55)where ξ is a constant. The numerical value of the parameter ξ has beendetermined from a comparison of the simulated angular distribution withexperimental data. Here, the experiment of Hanson et al.([10]) has beenused for the electron case, where the scattering of 15.7 MeV electrons fromthin gold foils has been measured. The value of the parameter c is set toc = 2 + tλ 0(7.56)The remaining three parameters can be computed from Eqs. 7.49 - 7.51.The numerical value of the parameters can be found in the code.It should be noted that in this model there is no step limitation originatingfrom the multiple scattering process. Another important feature of thismodel is that the sum of the ’true’ step lengths of the particle, that is, thetotal true path length, does not depend on the length of the steps. Mostalgorithms used in simulations do not have these properties.In the case of heavy charged particles (µ, π, p, etc.) the mean transportfree path is calculated from the electron or positron λ 1 values with a ’scaling’applied. This is possible because the transport mean free path λ 1 dependsonly on the variable P β, where P is the momentum, and β is the velocity ofthe particle.73

In its present form the model samples the path length correction andangular distribution from model functions, while for the lateral displacementonly the mean value is used and the correlations are neglected. However,the model is general enough to incorporate other random quantities andcorrelations in the future.7.5.5 Nuclear Size EffectsThe effect of the finite nuclear size is estimated in the Born approximation[6]. In this very simple approximation the scattering cross section can bewritten asdσ(χ)dΩ= dσ B(χ)F (χ) (7.57)dΩwhere dσ B /dΩ is the Born cross section for a screened, point-like nucleus andF (χ) is the squared nuclear form factor. F (χ) ≈ 0 if χ > χ max wheresin( χ max2 ) = 1kR , (7.58)k is the particle wave number, and R is the nuclear radius. This correctionmeans that σ(χ) decreases with energy, so that λ 1 , defined by∫11= 2πn a [1 − P 1 (cosχ)] dσ B(χ)F (χ)d(cosχ) (7.59)λ 1 cosχ max dΩincreases for larger energies.7.5.6 Implementation of the ProcessThe actual step length taken by a particle in the simulation is the smallerof the ’physics step length’, determined by the physics processes, and the’geometrical step length’, determined by the geometry of the detectors. Thephysics step length is the minimum of all the step lengths proposed by the(continuous or discrete) physics processes, and represents the path length ofa particle from the beginning of a step to the interaction point. When multiplescattering is applied, the path length is unaffected but the straight-linedistance (in the absence of a magnetic field) between the beginning of thestep and the interaction point is reduced. It is the responsibility of the MSCmodel to perform the transformation from the ’true’, or physical step length,t, to the ’straight-line’, or geometrical step length, z, so that the step sizelimitations from geometry and physics processes may be fairly compared.This ’t’⇒’z’ transformation can be considered the inverse of the path length74

correction. After the actual step length has been determined and the particlerelocation has been performed, the MSC performs the transformation’z’⇒’t’, so that the true step length t is available for the energy loss andscattering computations.The scattering angle θ of the particle after a step of length t is sampled accordingto the model function given in Eq. 7.45 . The azimuthal angle φ isgenerated uniformly in the range [0, 2π].After the simulation of the scattering, the lateral displacement is computedusing Eq. 7.32. Before doing this, a check is performed to ensure thatthe relocation of the particle with the lateral displacement does not take theparticle beyond the volume boundary.Boundary Crossing AlgorithmIn Geant4 boundary crossing is handled by the transportation code/process.The transportation ensures that the particle does not penetrate a new volumewithout first stopping at the boundary. It must therefore restrict the stepsize when the particle leaves a volume. However, there is no similar steplimitation when a particle enters a volume and this fact does not allow agood backscattering simulation for low energy particles. Low energy particlespenetrate deeply into the volume in the very first step and then, because ofenergy loss, they are not able to return to the boundary in the backwarddirection.A very simple boundary crossing algorithm has been implemented in theMSC code to cure this situation. When entering a new volume the algorithmrestricts the step size to a value f r · max{r, λ}, where r is the range of theparticle and f r is a constant (f r ∈ [0, 1]). This kind of step limitation imposesreal constraints only for low energy particles. Because the parameter f r reducesstep size, performance will be affected. A default value of f r = 0.2 waschosen as a compromise between performance and physics, which providesan approximate simulation of the backscattering. If a better backscatteringsimulation is needed it is easy to set f r to some other small value.Implementation DetailsBecause multiple scattering is very similar for different particles the baseclass G4V MultipleScattering was created to collect and provide significantfeatures of the calculations which are common to different particletypes. The methods implemented in this class are: BuildP hysicsT ables,75

AlongStepDoIt, P ostStepDoIt, AlongStepGetP hysicalInteractionLength,StoreP hysicsT able, and RestoreP hysicsT able. The concrete physics modelis implemented in the class G4MscModel. The G4MultipleScattering processhas only intialization functions, which allows the values of model parameters,such as f r ( SetFacrange), to be defined. It also allows the settingof flags to activate or deactivate the following actions:• Setsamplez(G4bool) - sampling of geometry length;• SetLateralDisplasmentFlag(G4bool) - sampling of lateral displacement;• SetBoundary(G4bool) - boundary algorithm;• SetBuildLambdaTable(G4bool) - build/store/restore a table for the samplingof geometry length.The default initialization is the same for all particles except G4GenericIons,for which all flags are set to false.In the AlongStepGetP hysicalInteractionLength method the minimumstep size due to the physics processes is compared with the step size constraintsimposed by the transportation process and the geometry. In orderto do this, the ’t’ step → ’z’ step transformation must be performed. Therefore,the method should be invoked after the GetP hysicalInteractionLengthmethods of other physics processes, but before the same method of the transportationprocess. The reason for this ordering is that the physics processes’feel’ the true path length t traveled by the particle, while the transportationprocess (geometry) uses the z step length.At this point the program also checks whether the particle has entereda new volume. If it has, the particle steps cannot be bigger than t lim =f r max(r, λ). This step limitation is governed by the physics, because t limdepends on the particle energy and the material.The P ostStepGetP hysicalInteractionLength method of the multiple scatteringprocess simply sets the force flag to ’Forced’ in order to ensure thatP ostStepDoIt is called at every step. It also returns a large value for theinteraction length so that there is no step limitation at this level.The AlongStepDoIt function of the process performs the inverse,’z’ → ’t’ transformation. This function should be invoked after theAlongStepDoIt of the transportation process, that is, after the particle relocationis determined by the geometrical step length, but before applying anyother physics AlongStepDoIts.The P ostStepDoIt method of the process samples the scattering angleand performs the lateral displacement when the particle is not near a boundary.76

7.5.7 Status of this document09.10.98 created by L. Urbán.15.11.01 major revision by L. Urbán.18.04.02 updated by L. Urbán.25.04.02 re-worded by D.H. Wright07.06.02 major revision by L. Urbán.18.11.02 updated by L. Urbán, now it describes the new angle distribution.05.12.02 grammar check and parts re-written by D.H. Wright13.11.03 revision by L. Urbán.01.12.03 revision by V. Ivanchenko.17.05.04 revision by L.Urbán.01.12.04 updated by L.Urbán.Bibliography[1] Z. Naturforsch. 3a (1948) 78.[2] H. W. Lewis. Phys. Rev. 78 (1950) 526.[3] S. Goudsmit and J. L. Saunderson. Phys. Rev. 57 (1940) 24.[4] J. M. Fernandez-Varea et al. NIM B73 (1993) 447.[5] I. Kawrakow and Alex F. Bielajew NIM B 142 (1998) 253.[6] D. Liljequist and M. Ismail. J.Appl.Phys. 62 (1987) 342.[7] D. Liljequist et al. J.Appl.Phys. 68 (1990) 3061.[8] V.L.Highland NIM 129 (1975) 497.[9] G.R. Lynch and O.I. Dahl NIM B58 (1991) 6.[10] A. O. Hanson et al. Phys. Rev. 84 (1951) 634.[11] H.Bichsel Phys. Rev. 111 (1958) 182.[12] B.Gottschalk et al. NIM B74 (1993) 467.77

7.6 Transition radiation7.6.1 The Relationship of Transition Radiation to X-ray Cherenkov RadiationX-ray transition radiation (XTR ) occurs when a relativistic charged particlepasses from one medium to another of a different dielectric permittivity. Inorder to describe this process it is useful to begin with an explanation ofX-ray Cherenkov radiation, which is closely related.The mean number of X-ray Cherenkov radiation (XCR) photons of frequencyω emitted into an angle θ per unit distance along a particle trajectoryis [1]d 3 ¯Nxcr¯hdω dx dθ = α ω2 π¯hc c θ2 Im {Z} . (7.60)Here the quantity Z is introduced as the complex formation zone of XCR inthe medium:Z =L1 − i L , L = c [] −1γ −2 + ω2 pω ω + 2 θ2 , γ −2 = 1 − β 2 . (7.61)lwith l and ω p the photon absorption length and the plasma frequency, respectively,in the medium. For the case of a transparent medium, l → ∞and the complex formation zone reduces to the coherence length L of XCR.The coherence length roughly corresponds to that part of the trajectory inwhich an XCR photon can be created.Introducing a complex quantity Z with its imaginary part proportionalto the absorption cross-section (∼ l −1 ) is required in order to account forabsorption in the medium. Usually, ωp/ω 2 2 ≫ c/ωl. Then it can be seen fromEqs. 7.60 and 7.61 that the number of emitted XCR photons is considerablysuppressed and disappears in the limit of a transparent medium. This iscaused by the destructive interference between the photons emitted fromdifferent parts of the particle trajectory.The destructive interference of X-ray Cherenkov radiation is removed ifthe particle crosses an interface between two media with different dielectricpermittivities, ɛ, whereɛ = 1 − ω2 pω 2 + i c ωl . (7.62)Here the standard high-frequency approximation for the dielectric permittivityhas been used. This is valid for energy transfers larger than the K-shellexcitation potential.78

If layers of media are alternated with spacings of order L, the X-rayradiation yield from a trajectory of unit length can be increased by roughlyl/L times. The radiation produced in this case is called X-ray transitionradiation (XTR).7.6.2 Calculating the X-ray Transition Radiation YieldUsing the methods developed in Ref. [2] one can derive the relation describingthe mean number of XTR photons generated per unit photon frequency andθ 2 inside the radiator for a general XTR radiator consisting of n differentabsorbing media with fluctuating thicknesses:d 2 ¯Nin¯hdω dθ 2 ={αn−1 ∑π¯hc 2 ωθ2 Re (Z i − Z i+1 ) 2 +i=1⎫n−2 ∑ n−k−1 ∑k∏ ⎬+ 2 (Z i − Z i+1 )(Z i+k − Z i+k+1 ) F i+j⎭ . (7.63)k=1 i=1j=1In the case of gamma distributed gap thicknesses the values F j , (j = 1, 2)are:F j =∫ ∞0dt j(νj¯t j) νjt ν j−1 [jΓ(ν j ) exp − ν jt j¯t j− i t ] [j= 1 + i2Z j] −νj¯t j, (7.64)2Z j ν jwhere Z j is the complex formation zone of XTR (similar to relation 7.61for XCR) in the j-th medium [2, 3]. Γ is the Euler gamma function, ¯t j isthe mean thickness of the j-th medium in the radiator and ν j > 0 is theparameter roughly describing the relative fluctuations of t j . In fact, therelative fluctuation is δ j = 1/ √ ν j .In the particular case of n foils of the first medium (Z 1 , F 1 ) interspersedwith gas gaps of the second medium (Z 2 , F 2 ), one obtains:d 2 ¯Nin¯hdω dθ 2 =〈R (n) 〉 = (Z 1 − Z 2 ) 2 {2απ¯hc 2 ωθ2 Re { 〈R (n) 〉 } , F = F 1 F 2 , (7.65)n (1 − F 1)(1 − F 2 )1 − F+ (1 − F 1) 2 F 2 [1 − F n }]. (7.66)(1 − F ) 2This approach allows the implementation of XTR as a GEANT4 parametrization[3], or as a standard electromagnetic process (see below).79

7.6.3 Simulating X-ray Transition Radiation ProductionParameterized models of X-ray transition radiation are implemented by theclasses G4VXTRdEdx, G4VXrayTRadModel, G4VXrayTRmodel and inherited classes.XTR as an electromagnetic process is implemented by G4VXTRenergyLoss,G4RegularXTRadiator and G4GammaXTRadiator.XTR photons generated by a relativistic charged particle intersecting aradiator with 2n interfaces between different media can be simulated by usingthe following algorithm. First the total number of XTR photons is estimatedusing a Poisson distribution about the mean number of photons given by thefollowing expression:¯N (n) =∫ ω2ω 1∫ θmaxdω= α ∫ ω2ωdωπc 2 ω 102θdθ d2 (n) ¯Ndω dθ 2∫ θmax02θ 3 dθRe { 2(Z 1 − Z 2 ) 2 〈R (n) 〉 } . (7.67)Here θ max ∼ 10/γ, ¯hω 1 ∼ 1 keV, ¯hω 2 ∼ 100 keV, and 〈R (n) 〉 correspond to thegeometry of the experiment. For events in which the number of XTR photonsis not equal to zero, the energy and angle of each XTR quantum are simulatedaccording to the integral distributions obtained by the numerical integrationof expression (7.65). For example, the integral energy spectrum of emittedXTR photons, ¯N(n)>ω , is defined from the following integral distribution:¯N (n)>ω = α ∫ ω2∫ θmaxωdω 2θ 3 dθRe { 2(Zπc 2 1 − Z 2 ) 2 〈R (n) 〉 } . (7.68)ω0In Geant4 XTR generation inside radiators is described within the frameworkof the so-called parametrization approach by a family of classes similarto that described in [3]. The base abstract class G4VXTRdEdx is responsiblefor the creation of tables with integral energy and angular distributionsof XTR photons. It also contains the DoIt function providing XTR photongeneration and moving the incident particle through a XTR radiator. Particularmodels like G4IrregularXTRdEdx implement the pure virtual functionGetStackFactor, which calculates the response of the XTR radiator. Theparametrization allows for improved performance and can be used for initialsimulations when one tunes the parametrs of XTR radiator.The same approach is used in the implementation of XTR as a continuouselectromagnetic process. Included below are some comments for thedeclaration of XTR in a user application.80

In DetectorConstruction of an application// Preparation of mixed radiator materialfoilDensity = 1.39*g/cm3; // MylargasDensity = 1.2928*mg/cm3 ; // AirtotDensity = foilDensity*foilGasRatio +gasDensity*(1.0-foilGasRatio) ;fractionFoil = foilDensity*foilGasRatio/totDensity ;fractionGas = gasDensity*(1.0-foilGasRatio)/totDensity ;G4Material* radiatorMat = new G4Material("radiatorMat",totDensity,ncomponents = 2 );radiatorMat->AddMaterial( Mylar, fractionFoil ) ;radiatorMat->AddMaterial( Air, fractionGas ) ;G4cout

However, in the physics list one should pass to the XTR process additionaldetails of the XTR radiator involved:// In PhysicsList of an applicationelse if (particleName == "e-") // Construct processes for electron with XTR{pmanager->AddProcess(new G4MultipleScattering, -1, 1,1 );pmanager->AddProcess(new G4eBremsstrahlung(), -1,-1,1 );pmanager->AddProcess(new Em10StepCut(), -1,-1,1 );pmanager->AddProcess(new G4IonisationByLogicalVolume(particleName,pDet->GetLogicalAbsorber(), // MWPC, straw"IonisationByLogVol"),-1,1,-1);pmanager->AddContinuousProcess(// regular stacknew G4RegularXTRadiator(pDet->GetLogicalRadiator(), // XTR radiatorpDet->GetFoilMaterial(), // real foilpDet->GetGasMaterial(), // real gaspDet->GetFoilThick(),pDet->GetGasThick(),pDet->GetFoilNumber(),// real geometry}"RegularXTRadiator"));7.6.4 Status of this document29.05.02 created by V.Grichine29.11.02 re-written by D.H. WrightBibliography[1] V.M. Grichine, Nucl. Instr. and Meth., A482 (2002) 629.[2] V.M. Grichine, Physics Letters, B525 (2002) 225-23982

[3] J. Apostolakis, S. Giani, V. Grichine et al., Comput. Phys. Commun.132 (2000) 241.83

7.7 ScintillationEvery scintillating material has a characteristic light yield, Y (photons/MeV ),and an intrinsic resolution which generally broadens the statistical distribution,σ i /σ s > 1, due to impurities which are typical for doped crystals likeNaI(Tl) and CsI(Tl). The average yield can have a non-linear dependenceon the local energy deposition. Scintillators also have a time distributionspectrum with one or more exponential decay time constants, τ i , with eachdecay component having its intrinsic photon emission spectrum. These areempirical parameters typical for each material.The generation of scintillation light can be simulated by sampling the numberof photons from a Poisson distribution. This distribution is based on theenergy lost during a step in a material and on the scintillation properties ofthat material. The frequency of each photon is sampled from the empiricalspectra. The photons are generated evenly along the track segment and areemitted uniformly into 4π with a random linear polarization.7.7.1 Status of this document07.12.98 created by P.Gumplinger84

7.8 Čerenkov EffectThe radiation of Čerenkov light occurs when a charged particle moves througha dispersive medium faster than the speed of light in that medium. A dispersivemedium is one whose index of refraction is an increasing function ofphoton energy. Two things happen when such a particle slows down:1. a cone of Čerenkov photons is emitted, with the cone angle (measuredwith respect to the particle momentum) decreasing as the particle losesenergy;2. the momentum of the photons produced increases, while the numberof photons produced decreases.When the particle velocity drops below the local speed of light, photons areno longer emitted. At that point, the Čerenkov cone collapses to zero.In order to simulate Čerenkov radiation the number of photons per tracklength must be calculated. The formulae used for this calculation can befound below and in [1, 2]. Let n be the refractive index of the dielectricmaterial acting as a radiator. Here n = c/c ′ where c ′ is the group velocity oflight in the material, hence 1 ≤ n. In a dispersive material n is an increasingfunction of the photon energy ɛ (dn/dɛ ≥ 0). A particle traveling with speedβ = v/c will emit photons at an angle θ with respect to its direction, whereθ is given bycos θ = 1βn .From this follows the limitation for the momentum of the emitted photons:n(ɛ min ) = 1 β .Photons emitted with an energy beyond a certain value are immediatelyre-absorbed by the material; this is the window of transparency of the radiator.As a consequence, all photons are contained in a cone of opening anglecos θ max = 1/(βn(ɛ max )).The average number of photons produced is given by the relations :dN = αz2¯hc sin2 θdɛdx = αz2¯hc (1 − 1n 2 β 2 )dɛdx≈370z 2 photonseV cm (1 − 1n 2 β 2 )dɛdx85

and the number of photons generated per track length is.∫ (dNdx ≈ ɛmax370z2 dɛ 1 − 1 ) [= 370z 2 ɛɛ min n 2 β 2 max − ɛ min − 1 ∫ ɛmaxβ 2 ɛ min]dɛn 2 (ɛ)The number of photons produced is calculated from a Poisson distributionwith a mean of 〈n〉 = StepLength dN/dx. The energy distribution of thephoton is then sampled from the density function.f(ɛ) =[1 −7.8.1 Status of this document07.12.98 created by P.Gumplinger11.12.01 SI units (mma)08.05.02 re-written by D.H. Wright]1n 2 (ɛ)β 2Bibliography[1] J.D.Jackson, Classical Electrodynamics, John Wiley and Sons (1998)[2] D.E. Groom et al. Particle Data Group . Rev. of Particle Properties.Eur. Phys. J. C15,1 (2000) http://pdg.lbl.gov/86

7.9 Photoabsorption Ionization Model7.9.1 Cross Section for Ionizing CollisionsThe Photoabsorption Ionization (PAI) model describes the ionization energyloss of a relativistic charged particle in matter. For such a particle, thedifferential cross section dσ i /dω for ionizing collisions with energy transfer ωcan be expressed most generally by the following equations [1]:{ [dσ idω = 2πZe4 f(ω)mv 2 ω |ε(ω)| 2 ln− ε 1 − β 2 |ε| 2ε 2arg(1 − β 2 ε ∗ )˜F (ω) =∫ ω0f(ω ′ )|ε(ω ′ )| 2 dω′ ,f(ω) = mωε 2(ω)2π 2 ZN¯h 2 .2mv 2ω |1 − β 2 ε| −]+ ˜F }(ω), (7.69)ω 2Here m and e are the electron mass and charge, ¯h is Planck’s constant,β = v/c is ratio of the particle velocity v to the speed of light c, Z is theeffective atomic number, N is the number of atoms (or molecules) per unitvolume, and ε = ε 1 +iε 2 is the complex dielectric constant of the medium. Inan isotropic non-magnetic medium the dielectric constant can be expressed interms of a complex index of refraction, n(ω) = n 1 +in 2 , ε(ω) = n 2 (ω). In theenergy range above the first ionization potential I 1 for all cases of practicalinterest, and in particular for all gases, n 1 ∼ 1. Therefore the imaginary partof the dielectric constant can be expressed in terms of the photoabsorptioncross section σ γ (ω):ε 2 (ω) = 2n 1 n 2 ∼ 2n 2 = N¯hcω σ γ(ω).The real part of the dielectric constant is calculated in turn from the dispersionrelationε 1 (ω) − 1 = 2N¯hc ∫ ∞πV.p. σ γ (ω ′ )0 ω ′2 − ω 2 dω′ ,where the integral of the pole expression is considered in terms of the principalvalue. In practice it is convenient to calculate the contribution from the87

continuous part of the spectrum only. In this case the normalized photoabsorptioncross section˜σ γ (ω) = 2π2¯he2 [∫Zωmax] −1σ γ (ω) σ γ (ω ′ )dω ′ , ωmax ∼ 100 keVmcI 1is used, which satisfies the quantum mechanical sum rule [2]:∫ ωmaxI 1˜σ γ (ω ′ )dω ′ = 2π2¯he2 Z.mcThe differential cross section for ionizing collisions is expressed by the photoabsorptioncross section in the continuous spectrum region:dσ idω = α { [ ˜σγ (ω) 2mv 2πβ 2 ω |ε(ω)| 2 lnω |1 − β 2 ε| −− ε 1 − β 2 |ε| 2ε 2arg(1 − β 2 ε ∗ )]+ 1 ∫ ω ˜σ γ (ω ′ })ω 2 I 1 |ε(ω ′ )| 2 dω′ , (7.70)ε 2 (ω) = N¯hcω ˜σ γ(ω),ε 1 (ω) − 1 = 2N¯hc ∫πV.p. ωmax˜σ γ (ω ′ )I 1 ω ′2 − ω 2 dω′ .For practical calculations using Eq. 7.69 it is convenient to represent thephotoabsorption cross section as a polynomial in ω −1 as was proposed in [3]:σ γ (ω) =4∑k=1a (i)k ω−k ,where the coefficients, a (i)k result from a separate least-squares fit to experimentaldata in each energy interval i. As a rule the interval borders are equalto the corresponding photoabsorption edges. The dielectric constant can nowbe calculated analytically with elementary functions for all ω, except nearthe photoabsorption edges where there are breaks in the photoabsorptioncross section and the integral for the real part is not defined in the sense ofthe principal value.88

The third term in Eq. (7.69), which can only be integrated numerically,results in a complex calculation of dσ i /dω. However, this term is dominantfor energy transfers ω > 10 keV , where the function |ε(ω)| 2 ∼ 1. This is clearfrom physical reasons, because the third term represents the Rutherford crosssection on atomic electrons which can be considered as quasifree for a givenenergy transfer [4]. In addition, for high energy transfers, ε(ω) = 1−ωp 2/ω2 ∼1, where ω p is the plasma energy of the material. Therefore the factor |ε(ω)| −2can be removed from under the integral and the differential cross section ofionizing collisions can be expressed as:{ [dσ idω = α ˜σγ (ω)πβ 2 |ε(ω)| 2 lnω− ε 1 − β 2 |ε| 2ε 2arg(1 − β 2 ε ∗ )2mv 2ω |1 − β 2 ε| −]+ 1 ∫ ωω 2I 1˜σ γ (ω ′ )dω ′ }This is especially simple in gases when |ε(ω)| −2 ∼ 1 for all ω > I 1 [4].7.9.2 Energy Loss Simulation. (7.71)For a given track length the number of ionizing collisions is simulated by aPoisson distribution whose mean is proportional to the total cross section ofionizing collisions:σ i =∫ ωmaxI 1dσ(ω ′ )dω ′ dω ′ .The energy transfer in each collision is simulated according to a distributionproportional toσ i (> ω) =∫ ωmaxωdσ(ω ′ )dω ′ dω ′ .The sum of the energy transfers is equal to the energy loss. PAI ionisation isimplemented according to the model approach (class G4PAIModel) allowinga user to select specific models in different regions. Here is an example ofphysics list:const G4RegionStore* theRegionStore = G4RegionStore::GetInstance();G4Region* gas = theRegionStore->GetRegion("VertexDetector");...if (particleName == "e-"){89

}G4eIonisation* eion = new G4eIonisation();G4PAIModel* pai = new G4PAIModel(particle,"PAIModel");eion->AddEmModel(0,pai,pai,gas);pmanager->AddProcess(eion,-1, 2, 2);pmanager->AddProcess(new G4MultipleScattering, -1, 1,1);pmanager->AddProcess(new G4eBremsstrahlung,-1,1,3);It shows how to select G4PAIModel to be preferred ionisation model forelectrons in G4Region with name VertexDetector.7.9.3 Status of this document16.11.98 created by V. Grichine08.05.02 re-written by D.H. WrightBibliography[1] Asoskov V.S., Chechin V.A., Grichine V.M. at el, Lebedev Instituteannual report, v. 140, p. 3 (1982)[2] Fano U., and Cooper J.W. Rev.Mod.Phys., v. 40, p. 441 (1968)[3] Biggs F., and Lighthill R., Preprint Sandia Laboratory, SAND 87-0070(1990)[4] Allison W.W.M., and Cobb J. Ann.Rev.Nucl.Part.Sci., v.30,p.253 (1980)90

7.10 Photoabsorption Cross Section at LowEnergies7.10.1 MethodThe photoabsorption cross section, σ γ (ω), where ω is the photon energy, isused in Geant4 for the description of the photo-electric effect, X-ray transportationand ionization effects in very thin absorbers. As mentioned in thediscussion of photoabsorption ionization (see section 7.9), it is convenient torepresent the cross section as a polynomial in ω −1 [1] :σ γ (ω) =4∑k=1a (i)k ω−k . (7.72)Using cross sections from the original Sandia data tables, calculations of primaryionization and energy loss distributions produced by relativistic chargedparticles in gaseous detectors show clear disagreement with experimentaldata, especially for gas mixtures which include xenon.Therefore a special investigation was performed [2] by fitting the coefficientsa (i)k to modern data from synchrotron radiation experiments in the energyrange of 10 − 50 eV . The fits were performed for elements typically usedin detector gas mixtures: hydrogen, fluorine, carbon, nitrogen and oxygen.Parameters for these elements were extracted from data on molecular gasessuch as N 2 , O 2 , CO 2 , CH 4 , and CF 4 [3, 4]. Parameters for the noble gaseswere found using data given in the tables [5, 6].7.10.2 Status of this document18.11.98 created by V. Grichine10.05.02 re-written by D.H. WrightBibliography[1] Biggs F., and Lighthill R., Preprint Sandia Laboratory, SAND 87-0070(1990)[2] Grichine V.M., Kostin A.P., Kotelnikov S.K. et al., Bulletin of the LebedevInstitute no. 2-3, 34 (1994).91

[3] Lee L.C. et al., J.Q.S.R.T., v. 13, p. 1023 (1973).[4] Lee L.C. et al., Journ. of Chem. Phys., v. 67, p. 1237 (1977).[5] G.V. Marr and J.B. West, Atom. Data Nucl. Data Tabl., v. 18, p. 497(1976).[6] J.B. West and J. Morton, Atom. Data Nucl. Data Tabl., v. 30, p. 253(1980).92

Chapter 8Electron Incident93

8.1 Ionization8.1.1 MethodThe G4eIonisation class provides the continuous and discrete energy lossesof electrons and positrons due to ionization in a material according to theapproach described in Section 7.1. The value of the maximum energy transferableto a free electron T max is given by the following relation:T max ={E − mc2for e +(E − mc 2 )/2 for e − (8.1)where mc 2 is the electron mass. Above a given threshold energy the energyloss is simulated by the explicit production of delta rays by Möller scattering(e − e − ), or Bhabha scattering (e + e − ). Below the threshold the soft electronsejected are simulated as continuous energy loss by the incident e ± .8.1.2 Continuous Energy LossThe integration of 7.1 leads to the Berger-Seltzer formula [1]:][]dE= 2πr 2 1 2(γ + 1)edxmc2 n el lnT

N av is Avogadro’s number, ρ is the material density, and A is the mass of amole. In a compound materialn el = ∑ iZ i n ati = ∑ iZ iN av w i ρA i,where w i is the proportion by mass of the i th element, with molar mass A i .The mean excitation energies I for all elements are taken from [2].The functions F ± are given by :F + (τ, τ up ) = ln(ττ up ) (8.3)− τ up2 [τ + 2τ up − 3τ 2 (upy− τ up − τ 3 ) (up τ2y 2 up−τ23 2 − τ τ up33 + τ up4 ) ]y 34F − (τ, τ up ) = −1 − β 2 (8.4)τ+ ln [(τ − τ up )τ up ] + + 1 [ τ2 (upτ − τ up γ 2 2 + (2τ + 1) ln 1 − τ ) ]upτwhere y = 1/(γ + 1).The density effect correction is calculated according to the formalism ofSternheimer [3]:x is a kinetic variable of the particle : x = log 10 (γβ) = ln(γ 2 β 2 )/4.606,and δ(x) is defined byfor x < x 0 : δ(x) = 0for x ∈ [x 0 , x 1 ] : δ(x) = 4.606x − C + a(x 1 − x) mfor x > x 1 : δ(x) = 4.606x − C(8.5)where the matter-dependent constants are calculated as follows:hν p = plasma energy of the medium = √ 4πn el remc 3 2 /α = √ 4πn el r e¯hcC = 1 + 2 ln(I/hν p )x a = C/4.606a = 4.606(x a − x 0 )/(x 1 − x 0 ) mm = 3.(8.6)For condensed media{for C ≤ 3.681 x0 = 0.2 xI < 100 eV1 = 2for C > 3.681 x 0 = 0.326C − 1.0 x 1 = 2{for C ≤ 5.215 x0 = 0.2 xI ≥ 100 eV1 = 3for C > 5.215 x 0 = 0.326C − 1.5 x 1 = 395

and for gaseous mediafor C < 10. x 0 = 1.6 x 1 = 4for C ∈ [10.0, 10.5[ x 0 = 1.7 x 1 = 4for C ∈ [10.5, 11.0[ x 0 = 1.8 x 1 = 4for C ∈ [11.0, 11.5[ x 0 = 1.9 x 1 = 4for C ∈ [11.5, 12.25[ x 0 = 2. x 1 = 4for C ∈ [12.25, 13.804[ x 0 = 2. x 1 = 5for C ≥ 13.804 x 0 = 0.326C − 2.5 x 1 = 5.8.1.3 Total Cross Section per Atom and Mean FreePathThe total cross section per atom for Möller scattering (e − e − ) and Bhabhascattering (e + e − ) is obtained by integrating Eq. 7.2. In Geant4 T cut isalways 1 keV or larger. For delta ray energies much larger than the excitationenergy of the material (T ≫ I), the total cross section becomes [1] for Möllerscattering,σ(Z, E, T cut ) =and for Bhabha scattering (e + e − ),2πreZ2β 2 (γ − 1) × (8.7)[ (γ − 1)2 ( ) 1γ 2 2 − x + 1 x − 11 − x − 2γ − 1 ln 1 − x ],γ 2 xσ(Z, E, T cut ) = 2πr2 eZ(γ − 1) × (8.8)[ ( ) 1 1β 2 x − 1 + B 1 ln x + B 2 (1 − x) − B 32 (1 − x2 ) + B ]43 (1 − x3 ) .Hereγ = E/mc 2 B 1 = 2 − y 2β 2 = 1 − (1/γ 2 ) B 2 = (1 − 2y)(3 + y 2 )x = T cut /(E − mc 2 ) B 3 = (1 − 2y) 2 + (1 − 2y) 3y = 1/(γ + 1) B 4 = (1 − 2y) 3 .The above formulae give the total cross section for scattering above thethreshold energiesT thrMoller = 2T cut and T thrBhabha = T cut . (8.9)In a given material the mean free path is thenλ = (n at · σ) −1 or λ = ( ∑ i n ati · σ i ) −1 . (8.10)96

8.1.4 Simulation of Delta-ray ProductionDifferential Cross SectionFor T ≫ I the differential cross section per atom becomes [1] for Möllerscattering,dσdɛ=2πreZ2β 2 (γ − 1) × (8.11)[ (γ − 1)2+ 1 ( 1γ 2 ɛ ɛ − 2γ − 1 )+ 1 ( 1γ 2 1 − ɛ 1 − ɛ − 2γ − 1 )]γ 2and for Bhabha scattering,dσdɛ = 2πr2 e Z(γ − 1)[ 1β 2 ɛ 2 − B 1ɛ + B 2 − B 3 ɛ + B 4 ɛ 2 ]. (8.12)Here ɛ = T/(E − mc 2 ). The kinematical limits of ɛ areɛ 0 =T cutE − mc 2 ≤ ɛ ≤ 1 2 for e− e − ɛ 0 = T cutE − mc 2 ≤ ɛ ≤ 1 for e+ e − .SamplingThe delta ray energy is sampled according to methods discussed in Chapter2. Apart from normalization, the cross section can be factorized asFor e − e − scatteringdσdɛ= f(ɛ)g(ɛ). (8.13)f(ɛ) = 1 ɛ 0(8.14)ɛ 2 1 − 2ɛ 0[]4g(ɛ) =(γ − 1) 2 ɛ 2 − (2γ 2 ɛ+ 2γ − 1)9γ 2 − 10γ + 51 − ɛ + γ2(8.15)(1 − ɛ) 2and for e + e − scatteringf(ɛ) = 1 ɛ 2 ɛ 01 − ɛ 0(8.16)g(ɛ) = B 0 − B 1 ɛ + B 2 ɛ 2 − B 3 ɛ 3 + B 4 ɛ 4. (8.17)B 0 − B 1 ɛ 0 + B 2 ɛ 2 0 − B 3 ɛ 3 0 + B 4 ɛ 4 0Here B 0 = γ 2 /(γ 2 − 1) and all other quantities have been defined above.To choose ɛ, and hence the delta ray energy,97

1. ɛ is sampled from f(ɛ)2. the rejection function g(ɛ) is calculated using the sampled value of ɛ3. ɛ is accepted with probability g(ɛ).After the successful sampling of ɛ, the direction of the ejected electron isgenerated with respect to the direction of the incident particle. The azimuthalangle φ is generated isotropically and the polar angle θ is calculatedfrom energy-momentum conservation. This information is used to calculatethe energy and momentum of both the scattered incident particle and theejected electron, and to transform them to the global coordinate system.8.1.5 Status of this document9.10.98 created by L. Urbán.29.07.01 revised by M.Maire.13.12.01 minor cosmetic by M.Maire.24.05.02 re-written by D.H. Wright.01.12.03 revised by V. Ivanchenko.Bibliography[1] H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.[2] ICRU Report No. 37 (1984)[3] R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.98

8.2 BremsstrahlungThe class G4eBremsstrahlung provides the energy loss of electrons andpositrons due to the radiation of photons in the field of a nucleus accordingto the approach described in Section 7.1. Above a given threshold energythe energy loss is simulated by the explicit production of photons. Belowthe threshold the emission of soft photons is treated as a continuous energyloss. In GEANT4 the Landau-Pomeranchuk-Migdal effect has also beenimplemented.8.2.1 Cross Section and Energy Lossdσ(Z, T, k)/dk is the differential cross section for the production of a photonof energy k by an electron of kinetic energy T in the field of an atom of chargeZ. If k c is the energy cut-off below which the soft photons are treated ascontinuous energy loss, then the mean value of the energy lost by the electronis∫ELoss brem (Z, T, k kcc) = k0dσ(Z, T, k)dk. (8.18)dkThe total cross section for the emission of a photon of energy larger than k cis∫ T dσ(Z, T, k)σ brem (Z, T, k c ) =dk. (8.19)k c dkParameterization of the Energy Loss and Total Cross SectionThe cross section and energy loss due to bremsstrahlung have been parameterizedusing the EEDL (Evaluated Electrons Data Library) data set [1] asinput.The following parameterization was chosen for the electron bremsstrahlungcross section :[ ] T ασ(Z, T, k c ) = Z(Z + ξ σ )(1 − c sigh Z 1/4 f˙s)(8.20)k c N Avowhere f s is a polynomial in x = lg(T ) with Z-dependent coefficients forx < x l , f s = 1 for x ≥ x l , ξ σ , c sigh , α are constants, N Avo is the Avogadronumber. For the case of low energy electrons (T ≤ T lim = 10MeV ) the aboveexpression should be multiplied by99

( T lim aT )c l l ˙(1 + √ ), (8.21)ZTwith constant c l , a l parameters.The energy loss parameterization is the following :E bremLoss (Z, T, k c) = Z(Z + ξ l)(T + m) 2(T + 2m)[ ] βkca + b T(2 − c lh Z 1 4 )T limT1 + c TT limf lN Avo(8.22)where m is the mass of the electron, ξ l , β, c lh , a, b, c are constants, f l is apolynomial in x = lg(T ) with Z-dependent coefficients for x < x l , f l = 1 forx ≥ x l . For low energies this expression should be divided by( T limT )c l(8.23)and if T < k c the expression should be multiplied by( T k c) a l(8.24)with some constants c l , a l . The numerical values of the parameters andthe coefficients of the polynomyals f s and f l can be found in the class code.The errors of the parameterizations (8.20) and (8.22) were estimated to be∆σσ={6 − 8% for T ≤ 1MeV≤ 4 − 5% for 1MeV < T∆E bremLossE bremLoss={8 − 10% for T ≤ 1MeV5 − 6% for 1MeV < T.When running GEANT4, the energy loss due to soft photon bremsstrahlungis tabulated at initialization time as a function of the medium and of theenergy, as is the mean free path for discrete bremsstrahlung.Corrections for e + e − DifferencesThe preceding section has dealt exclusively with electrons. One might expectthat positrons could be treated the same way. According to reference[9] however,100

“The differences between the radiative loss of positrons and electrons are considerableand cannot be disregarded.[...] The ratio of the radiative energy loss for positrons to that for electronsobeys a simple scaling law, [...] is a function only of the quantity T/Z 2 ”The radiative energy loss for electrons or positrons is given by− 1 ρ( dEdx) ±rad= N Avαr 2 eA (T + m)Z2 Φ ± rad (Z, T )Φ ± rad (Z, T ) = 1αr 2 eZ 2 (T + m)∫ T0k dσ±dk dkand it is the ratioη = Φ+ rad (Z, T )Φ − rad (Z, T ) = η ( TZ 2 )that obeys the scaling law.The authors have calculated this function in the range 10 −7 ≤ T Z 2 ≤ 0.5,where the kinetic energy T is expressed in MeV. Their data can be fairlyaccurately reproduced using a parametrization:η =⎧⎪⎨⎪⎩0 if x ≤ −812 π 1x + a 3 x 3 + a 5 x 5 ) if −8 < x < 91 if x ≥ 9wherex = log(C T )(T in GeV)Z 2C = 7.5221 × 10 6a 1 = 0.415a 3 = 0.0021a 5 = 0.00054.The e + e − energy loss difference is not purely a low-energy phenomenon (atleast for high Z), as shown in Table 8.1.The scaling property will be used to obtain the positron energy loss anddiscrete bremsstrahlung from the corresponding electron values. However,101

TZ 2 (GeV ) T η( )rad. losstotal loss e −10 −9 ∼ 7keV ∼ 0.1 ∼ 0%10 −8 67keV ∼ 0.2 ∼ 1%2 × 10 −7 1.35MeV ∼ 0.5 ∼ 15%2 × 10 −6 13.5MeV ∼ 0.8 ∼ 60%2 × 10 −5 135.MeV ∼ 0.95 > 90%Table 8.1: ratio of the e + e − radiative energy loss in lead (Z=82).while scaling holds for the ratio of the total radiative energy losses, it issignificantly broken for the photon spectrum in the screened case. That is,Φ + ( ) Tdσ +Φ = η dk= does not scale .− Z 2For the case of a point Coulomb charge, scaling would be restored for thephoton spectrum. In order to correct for non-scaling, it is useful to note thatin the photon spectrum from bremsstrahlung reported in [9]:dσ ±dk = S± ( kT)One can further assume thatdσ +dk = f(ɛ)dσ− dk ,and requiredσ −dkS + (k)S − (k) ≤ 1 S+ (1) = 0 S − (1) > 0∫ 10ɛ = k T(8.25)f(ɛ)dɛ = η (8.26)in order to approximately satisfy the scaling law for the ratio of the totalradiative energy loss. From the photon spectra the boundary conditions}f(0) = 1for all Z, T (8.27)f(1) = 0may be inferred. Choosing a simple function for ff(ɛ) = C(1 − ɛ) α C, α > 0, (8.28)the conditions (8.26), (8.27) lead to:C = 1α = 1 − 1 (α > 0 because η < 1)ηf(ɛ) = (1 − ɛ) 1 η −1 .102

Now the weight factors F l and F σ for the positron continuous energy loss andthe discrete bremsstrahlung cross section can be defined:F l = 1 ɛ 0∫ ɛ00f(ɛ)dɛ F σ = 11 − ɛ 0∫ 1ɛ 0f(ɛ)dɛ (8.29)where ɛ 0 = kc and k T c is the photon cut. In this scheme the positron energyloss and discrete bremsstrahlung can be calculated as:(− dE ) + (= F l − dE ) −σ brems + dxdx= F σσbrems−In this approximation the photon spectra are identical, therefore the samesampling is used for generating e − or e + bremsstrahlung. The followingrelations hold:F σ = η(1 − ɛ 0 ) 1 η −1 < ηɛ 0 F l + (1 − ɛ 0 )F σ = η from the def (8.29)⇒ F l = η 1 − (1 − ɛ 0) 1 η )> η 1 − (1 − ɛ {0)Fl > η= η ⇒ɛ 0 ɛ 0 F σ < ηwhich is consistent with the spectra. The effect of the difference in e − ande + bremsstrahlung can also be seen in electromagnetic shower developmentwhen the primary energy is not too high.Landau Pomeranchuk Migdal (LPM) effectThe LPM effect (see for example [3, 4] ) is the suppression of photon productiondue to the multiple scattering of the electron. If an electron undergoesmultiple scattering while traversing the so called “formation zone”, thebremsstrahlung amplitudes from before and after the scattering can interfere,reducing the probability of bremsstrahlung photon emission (a similarsuppression occurs for pair production). The suppression becomes significantfor photon energies below a certain value, given bywherekEE LP MkE

The value of the LPM characteristic energy can be written aswhereαmX 0hcE LP M = αm2 X 0, (8.31)4hcfine structure constantelectron massradiation length in the materialPlanck constantvelocity of light in vacuum.The LPM suppression of the photon spectrum is given by the formulaS LP M =√ELP M · kE 2 , (8.32)while the dielectric suppression (included already in the parameterizations)can be written ask 2S p =k 2 + C p · E , (8.33)2where the quantity C p is given byIn eq. 8.34 the parameters areC p = r 0λ 2 enπ . (8.34)r 0λ enclassical electron radiuselectron Compton wavelengthelectron density in the material.Both suppression effects reduce the effective formation length of the photon,so the suppressions do not simply multiply. For the total suppression S thefollowing equation holds (see [3])1S = 1 + 1 S p+SS 2 LP M(8.35)which can be solved easily for SS =√S4LP M · (1 + 1S p) 2 + 4 · S 2 LP M − S2 LP M · (1 + 12S p). (8.36)104

The LPM effect was implemented by applying to the energy loss a factorSS p, which depends on the energy and material. This is done at initializationtime by computing the correction factorf c =∫ kcut0 n γ (k) · SS pdk∫ kcut0 n γ (k)dk , (8.37)where n γ (k) is the photon spectrum. A similar correction has not beenapplied to the total cross section given by the parameterization 8.20. Insteadthe LPM effect is included in the photon generation algorithm.8.2.2 Simulation of Discrete BremsstrahlungThe energy of the final state photons is sampled according to the spectrum[5] of Seltzer and Berger. They have calculated the bremsstrahlung spectrafor materials with atomic numbers Z = 6, 13, 29, 47, 74 and 92 in the electronkinetic energy range 1 keV - 10 GeV. Their tabulated results have been usedas input in a fit of the parameterized functionS(x) = Ck dσdk ,which will be used to form the rejection function for the sampling process.The parameterization can be written as{(1 − ah ɛ)FS(x) =1 (δ) + b h ɛ 2 F 2 (δ) T ≥ 1MeV1 + a l x + b l x 2 (8.38)T < 1MeVwhereCknormalization constantphoton energyT, E kinetic and total energy of the primary electronx = k Tɛ = k E = x T Eand a h,l and b h,l are the parameters to be fitted. The F i (δ) screening functionsdepend on the screening variableδ = 136meZ 1/3 Eɛ1−ɛF 1 (δ) = F 0 (42.392 − 7.796δ + 1.961δ 2 − F ) δ ≤ 1F 2 (δ) = F 0 (41.734 − 6.484δ + 1.250δ 2 − F ) δ ≤ 1F 1 (δ) = F 2 (δ) = F 0 (42.24 − 8.368 ln(δ + 0.952) − F ) δ > 1142.392−FF 0 =F = 4 ln Z − 0.55(ln Z) 2 .105

The “high energy” (T > 1 MeV) formula is essentially the Coulomb-corrected,screened Bethe-Heitler formula (see e.g. [10, 11, 6]). However, Eq. (8.38) differsfrom Bethe-Heitler in two ways:1. a h , b h depend on T and on the atomic number Z, whereas in the Bethe-Heitler spectrum they are fixed (a h = 1, b h = 0.75);2. the function F is not the same as that in the Bethe-Heitler cross-section;the present function gives a better behavior in the high frequency limit,i.e. when k → T (x → 1).The T and Z dependence of the parameters are described by the equations:a h = 1 + a h1u + a h2u + a h32 u 3b h = 0.75 + b h1u + b h2u + b h32 u 3a l = a l0 + a l1 u + a l2 u 2b l = b l0 + b l1 u + b l2 u 2with( ) Tu = lnm eThe parameters a hi , b hi , a li , b li are polynomials of second order in the variable:v = [Z(Z + 1)] 1/3 .In the limiting case T → ∞, a h → 1, b h → 0.75, Eq. (8.38) gives the Bethe-Heitler cross section.There are altogether 36 linear parameters in the formulae and their valuesare given in the code. This parameterization reproduces the Seltzer-Bergertables to within 2-3 % on average, with the maximum error being less than10-12 %. The original tables, on the other hand, agree well with the experimentaldata and theoretical (low- and high-energy) results (< 10 % below50 MeV and < 5 % above 50 MeV).Apart from the normalization the cross section differential in photon energycan be written asdσdk = 1 1ln 1 x cx g(x) = 1 1 S(x)ln 1 x cxS max106

where x c = k c /T and k c is the photon cut-off energy below which thebremsstrahlung is treated as a continuous energy loss. Using this decompositionof the cross section and two random numbers r 1 , r 2 uniformly distributedin [0, 1], the sampling of x is done as follows:1. sample x from1ln 1 x c1xsettingx = e r 1 ln x c2. calculate the rejection function g(x) and:• if r 2 > g(x) reject x and go back to 1;• if r 2 ≤ g(x) accept x.The application of the dielectric suppression [8] and the LPM effect requiresthat ɛ also be sampled. First, the rejection function must be multiplied by asuppression factorwheren electron density in the mediumC M (ɛ) = 1 + C 0/ɛ 2 c1 + C 0 /ɛ 2C 0 = nr 0λ 2π , ɛ c = k cEr 0classical electron radiusλ reduced Compton wavelength of the electron.Apart from the Migdal correction factor, this is simply expression 8.33 . Thiscorrection decreases the cross-section for low photon energies.While sampling ɛ, the suppression factor f LP M = S S pis also used as a rejectionfunction in order to take into account the LPM effect. Here thesupression factor is compared to a random number r uniformly distributedin the interval [0, 1]. If f LP M ≥ r the simulation continues, otherwise thebremsstrahlung process concludes without photon production. It can be seenthat this procedure performs the LPM suppression correctly.After the successful sampling of ɛ, the polar angles of the radiated photon aregenerated with respect to the parent electron’s momentum. It is difficult to107

find simple formulae for this angle in the literature. For example the doubledifferential cross section reported by Tsai [12, 13] isdσdkdΩ = 2α2 e 2 {[ ]2ɛ − 2πkm 4 (1 + u 2 ) + 12u2 (1 − ɛ)Z(Z + 1)2 (1 + u 2 ) 4[ ] 2 − 2ɛ − ɛ2+− 4u2 (1 − ɛ) [X− 2Z 2 f(1 + u 2 ) 2 (1 + u 2 ) 4 c ((αZ) 2 ) ]}u = EθmX =∫ m 2 (1+u 2 ) 2 [Gelt minZ(t) + G in Z (t) ] t − t mint 2G el,inZ (t) atomic form factors[ km 2 (1 + u 2 ] 2 [) ɛm 2 (1 + u 2 ] 2)t min ==.2E(E − k) 2E(1 − ɛ)The sampling of this distribution is complicated. It is also only an approximationto within a few percent, due at least to the presence of the atomic formfactors. The angular dependence is contained in the variable u = Eθm −1 .For a given value of u the dependence of the shape of the function on Z, Eand ɛ = k/E is very weak. Thus, the distribution can be approximated by afunctionf(u) = C ( ue −au + due −3au) (8.39)whereC =9a2 a = 0.625 d = 279 + dwhere E is in GeV. While this approximation is good at high energies, it becomesless accurate around a few MeV. However in that region the ionizationlosses dominate over the radiative losses.The sampling of the function f(u) can be done with three random numbersr i , uniformly distributed on the interval [0,1]:1. choose between ue −au and due −3au :{a if r1 < 9/(9 + d)b =3a if r 1 ≥ 9/(9 + d)dt2. sample ue −bu :u = − log(r 2r 3 )b108

∫ ∞P = f(u) duu maxE (MeV) P(%)0.511 3.40.6 2.20.8 1.21.0 0.72.0 < 0.1Table 8.2: Angular sampling efficiency3. check that:otherwise go back to 1.u ≤ u max = EπmThe probability of failing the last test is reported in table 8.2.The function f(u) can also be used to describe the angular distribution ofthe photon in µ bremsstrahlung and to describe the angular distribution inphoton pair production.The azimuthal angle φ is generated isotropically. Along with θ, this informationis used to calculate the momentum vectors of the radiated photon andparent electron, and to transform them to the global coordinate system.8.2.3 Status of this document09.10.98 created by L. Urbán.21.03.02 modif in angular distribution (M.Maire)27.05.02 re-written by D.H. Wright01.12.03 minor update by V. Ivanchenko20.05.04 updated by L.UrbanBibliography[1] S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400 Vol.31[2] GEANT3 manual ,CERN Program Library Long Writeup W5013 (October1994).109

[3] V.M.Galitsky and I.I.Gurevich. Nuovo Cimento 32 (1964) 1820.[4] P.L. Anthony et al. SLAC-PUB-7413/LBNL-40054 (February 1997)[5] S.M.Seltzer and M.J.Berger. Nucl.Inst.Meth. 80 (1985) 12.[6] W.R. Nelson et al.:The EGS4 Code System. SLAC-Report-265 , December1985[7] H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.[8] A.B. Migdal. Phys.Rev. 103. (1956) 1811.[9] L. Kim et al. Phys. Rev. A33 (1986) 3002.[10] R.W. Williams, Fundamental Formulas of Physics, vol.2., Dover Pubs.(1960).[11] J. C. Butcher and H. Messel. Nucl.Phys. 20. (1960) 15.[12] Y-S. Tsai, Rev. Mod. Phys. 46. (1974) 815.[13] Y-S. Tsai, Rev. Mod. Phys. 49. (1977) 421.110

8.3 Positron - Electron Annihilation8.3.1 IntroductionThe process G4eplusAnnihilation simulates the in-flight annihilation of apositron with an atomic electron. As is usually done in shower programs [1],it is assumed here that the atomic electron is initially free and at rest. Also,annihilation processes producing one, or three or more, photons are ignoredbecause these processes are negligible compared to the annihilation into twophotons [1, 2].8.3.2 Cross Section and Mean Free PathCross Section per AtomThe annihilation in flight of a positron and electron is described by the crosssection formula of Heitler [3, 1]:σ(Z, E) = Zπr2 eγ + 1whereMean Free Path[ γ 2 + 4γ + 1γ 2 − 1E = total energy of the incident positronγ = E/mc 2r e = classical electron radius( √ )ln γ + γ 2 − 1 − √ γ + 3 ](8.40)γ2 − 1In a given material the mean free path, λ, for a positron to be annihilatedwith an electron is given by( ∑λ(E) =in ati · σ(Z i , E)) −1(8.41)where n ati is the number of atoms per volume of the i th element composingthe material.8.3.3 Sampling the final stateThe final state of the e + e− annihilation processe + e − → γ a γ b111

is simulated by first determining the kinematic limits of the photon energyand then sampling the photon energy within those limits using the differentialcross section. Conservation of energy-momentum is then used to determinethe directions of the final state photons.Kinematic LimitsIf the incident e + has a kinetic energy√T , then the total energy is E e =T + mc 2 and the momentum is P c = T (T + 2mc 2 ). The total availableenergy is E tot = E e + mc 2 = E a + E b and momentum conservation requires⃗P = P ⃗ γa + P ⃗ γb . The fraction of the total energy transferred to one photon(say γ a ) isɛ = E a E a≡E tot T + 2mc . (8.42)2The energy transfered to γ a is largest when γ a is emitted in the direction ofthe incident e + . In that case E amax = (E tot + P c)/2 . The energy transferedto γ a is smallest when γ a is emitted in the opposite direction of the incidente + . Then E amin = (E tot − P c)/2 . Hence,ɛ max = E amax= 1 [ √ ] γ − 11 +E tot 2 γ + 1ɛ min = E amin= 1 [ √ ] γ − 11 −E tot 2 γ + 1(8.43)(8.44)where γ = (T + mc 2 )/mc 2 . Therefore the range of ɛ is [ɛ min ; ɛ max ](≡ [ɛ min ; 1 − ɛ min ]).Sampling the Gamma EnergyA short overview of the sampling method is given in Chapter 2.The differential cross section of the two-photon positron-electron annihilationcan be written as [3, 1]:dσ(Z, ɛ)dɛ= Zπr2 eγ − 11ɛ[1 + 2γ](γ + 1) − ɛ − 1 12 (γ + 1) 2 ɛ(8.45)where Z is the atomic number of the material, r e the classical electron radius,and ɛ ∈ [ɛ min ; ɛ max ] . The differential cross section can be decomposed asdσ(Z, ɛ)dɛ= Zπr2 eαf(ɛ)g(ɛ) (8.46)γ − 1112

whereα = ln(ɛ max /ɛ min )f(ɛ) = 1 [ αɛg(ɛ) = 1 + 2γ](γ + 1) − ɛ − 1 12 (γ + 1) 2 ɛ(8.47)≡ 1 − ɛ + 2γɛ − 1ɛ(γ + 1) 2 (8.48)Given two random numbers r, r ′ ∈ [0, 1], the photon energies are chosen asfollows:1. sample ɛ from f(ɛ) : ɛ = ɛ min(ɛmaxɛ min) r2. test the rejection function: if g(ɛ) ≥ r ′ accept ɛ, otherwise return tostep 1.Then the photon energies are E a = ɛE tot E b = (1 − ɛ)E tot .Computing the Final State KinematicsIf θ is the angle between the incident e + and γ a , then from energy-momentumconservation,cos θ = 1 [T + mc 2 2ɛ − 1 ]=P cɛɛ(γ + 1) − 1ɛ √ γ 2 − 1 . (8.49)The azimuthal angle, φ, is generated isotropically and the photon momentumvectors, Pγa⃗ and P ⃗γb , are computed from energy-momentum conservation andtransformed into the lab coordinate system.Annihilation at RestThe method AtRestDoIt treats the special case when a positron comes torest before annihilating. It generates two photons, each with energy k = mc 2and an isotropic angular distribution.8.3.4 Status of this document09.10.98 created by M.Maire.01.08.01 minor corrections (mma)09.01.02 MeanFreePath (mma)01.12.02 Re-written by D.H. Wright113

Bibliography[1] R. Ford and W. Nelson. SLAC-265, UC-32 (1985)[2] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[3] W. Heitler. The Quantum Theory of Radiation, Clarendon Press, Oxford(1954)114

8.4 Positron - Electron Annihilation into Muon- Anti-muonThe class G4AnnihiToMuPair simulates the electromagnetic production ofmuon pairs by the annihilation of high-energy positrons with atomic electrons.Details of the implementation are given below and can also be foundin Ref. [1].8.4.1 Total Cross SectionThe annihilation of positrons and target electrons producing muon pairs inthe final state (e + e − → µ + µ − ) may give an appreciable contribution to thetotal number of muons produced in high-energy electromagnetic cascades.The threshold positron energy in the laboratory system for this process withthe target electron at rest isE th = 2m 2 µ /m e − m e ≈ 43.69 GeV , (8.50)where m µ and m e are the muon and electron masses, respectively. The totalcross section for the process on the electron isσ = π r2 µ(13 ξ + ξ ) √1 − ξ , (8.51)2where r µ = r e m e /m µ is the classical muon radius, ξ = E th /E, and E is thetotal positron energy in the laboratory frame. In Eq. 8.51, approximationsare made that utilize the inequality m 2 e ≪ m 2 µ.The cross section as a function of the positron energy E is shown in Fig.8.1.It has a maximum at E = 1.396 E th and the value at the maximum is σ max =0.5426 r 2 µ = 1.008 µb.8.4.2 Sampling of Energies and AnglesIt is convenient to simulate the muon kinematic parameters in the center-ofmass(c.m.) system, and then to convert into the laboratory frame.The energies of all particles are the same in the c.m. frame and equal toE cm =√12 m e(E + m e ) . (8.52)The muon momenta in the c.m. frame are P cm = √ E 2 cm − m 2 µ. In whatfollows, let the cosine of the angle between the c.m. momenta of the µ + ande + be denoted as x = cos θ cm .115

10.8σ in µb0.60.40.2050 60 70 80 100 200 300 400 500E in GeVFigure 8.1: Total cross section for the process e + e − → µ + µ − as a function ofthe positron energy E in the laboratory system.116

From the differential cross section it is easy to derive that, apart fromnormalization, the distribution in x is described byf(x) dx = (1 + ξ + x 2 (1 − ξ)) dx , −1 ≤ x ≤ 1 . (8.53)The value of this function is contained in the interval (1 + ξ) ≤ f(x) ≤ 2 andthe generation of x is straightforward using the rejection technique. Fig. 8.2shows both generated and analytic distributions.2Entries per bin1.751.51.2510.750.50.25E = 50 GeV, ξ =.874E= 500 GeV, ξ = .087421 + cos θ cm0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cos θ cmFigure 8.2: Generated histograms with 10 6 entries each and the expectedcos θ cm distributions (dashed lines) at E = 50 and 500 GeV positron energyin the lab frame. The asymptotic 1 + cos θ 2 cm distribution valid for E → ∞is shown as dotted line.The transverse momenta of the µ + and µ − particles are the same, bothin the c.m. and the lab frame, and their absolute values are equal toP ⊥ = P cm sin θ cm = P cm√1 − x2 . (8.54)The energies and longitudinal components of the muon momenta in the labsystem may be obtained by means of a Lorentz transformation. The velocityand Lorentz factor of the center-of-mass in the lab frame may be written asβ =√E − meE + m e,γ ≡1√ 1 − β2 = √E + me2m e= E cmm e. (8.55)117

The laboratory energies and longitudinal components of the momenta of thepositive and negative muons may then be obtained:E + = γ (E cm + x β P cm ) , P +‖ = γ (βE cm + x P cm ) , (8.56)E − = γ (E cm − x β P cm ) , P −‖ = γ (βE cm − x P cm ) . (8.57)Finally, for the vectors of the muon momenta one obtains:P + = (+P ⊥ cos ϕ, +P ⊥ sin ϕ, P +‖ ) , (8.58)P − = (−P ⊥ cos ϕ, −P ⊥ sin ϕ, P −‖ ) , (8.59)where ϕ is a random azimuthal angle chosen between 0 and 2 π. The z-axisis directed along the momentum of the initial positron in the lab frame.The maximum and minimum energies of the muons are given byE min ≈ 1 2 E (1 −E max ≈ 1 2 E (1 +√ )1 − ξ =√ )1 − ξ , (8.60)(2 1 +E th√1 − ξThe fly-out polar angles of the muons are approximately) . (8.61)θ + ≈ P ⊥ /P +‖ , θ − ≈ P ⊥ /P −‖ ; (8.62)the maximal angle θ max ≈ m em µ√1 − ξ is always small compared to 1.ValidityThe process described is assumed to be purely electromagnetic. It is basedon virtual γ exchange, and the Z-boson exchange and γ − Z interferenceprocesses are neglected. The Z-pole corresponds to a positron energy ofE = MZ 2/2m e = 8136 TeV. The validity of the current implementation istherefore restricted to initial positron energies of less than about 1000 TeV.8.4.3 Status of this document05.02.03 created by H.Burkhardt14.04.03 minor re-wording by D.H. Wright118

Bibliography[1] H. Burkhardt, S. Kelner, and R. Kokoulin, “Production of muon pairsin annihilation of high-energy positrons with resting electrons,” CERN-AB-2003-002 (ABP) and CLIC Note 554, January 2003.119

8.5 Synchrotron RadiationSynchrotron radiation photons are produced when ultra-relativistic electronstravel along an approximately circular path. In the following treatment, themagnetic field is assumed to be constant and uniform, and the radius ofcurvature of the electron is assumed to be constant over its trajectory.8.5.1 Spectral and Angular Distributions of SynchrotronRadiationThe spectral distribution of the mean number of synchrotron radiation photons,d ¯N/dω, produced by an ultra-relativistic electron along a circular trajectoryof length L, can be expressed in terms of the mean energy loss spectrumd ¯∆/dω [1]:Here,d ¯Ndω = 1 d ¯∆√ ( ) ∫ 3 Lγ 1 ∞ω dω = 2π α K 5/3 (η)dη. (8.63)R ω c ω/ω cωαRKω c = 1.5β(¯hc/R)γ 3photon energyfine structure constantinstantaneous radius of curvature of the trajectoryMacdonald functioncharacteristic energy of synchrotron radiation.β is the ratio of the electron velocity v to c, γ = 1/ √ 1 − β 2 , and η is an arbitraryintegration variable. In the SI system of units: R(m) = P (GeV/c)/0.3B ⊥ (T ), where B ⊥ is the component of magnetic flux density perpendicular to theelectron velocity, and P is the electron momentum.In order to simulate the energy spectrum of synchrotron radiation usingthe Monte Carlo method, ¯N>ω , the mean number of photons above a givenenergy ω, must be determined. This is done by integrating Eq. 8.63 overenergy, after first transforming d ¯N/dω by using the integral representationof the Macdonald function [2]:¯N >ω ==∫ ∞d ¯Nω dω ′ dω′√ ( ) ∫ 3 Lγ ∞2π α R 0cosh ( 53 t) [cosh 2 (t) exp − ω ]cosh(t) dt. (8.64)ω c120

Here, t is also an arbitrary integration variable. The latter integral is calculatednumerically by the quadrature Laguerre formula [3]. Calculationsindicate that about 50 roots of the Laguerre polynomials are required inorder for the accuracy of the integral estimation to be better than 10 −4 [4].The Monte Carlo method also requires the mean number of synchrotronradiation photons at all energies, ¯N (= ¯N>0 ), in order to determine the nextoccurrence of synchrotron radiation along a trajectory, and to normalize thespectral distribution of the radiation. Setting ω = 0 in Eq. 8.64 yields¯N = ¯N >0 ==√ ( 3 Lγ2π α R(5 Lγ2 √ 3 α R) ∫ ∞0)cosh ( 53 t)cosh 2 (t) dt≈ 10 −2 ( LγR). (8.65)Qualitatively this result can be manipulated using the fact that the meannumber of photons produced along the formation zone length z ≈ R/γ isproportional to α. Then for length L, ¯N ≈ αL/(R/γ). Note that whenγ ≫ 1, and R ∼ γ, ¯N does not depend on the electron energy but is definedby the values of L and B ⊥ only. Instead, it is the mean energy loss dueto synchrotron radiation ¯∆, corresponding to a trajectory of length L, thatdisplays the characteristic relativistic rise:¯∆ =∫ ∞0ω d ¯Ndω dω = 2 ( ) Lγ23 α¯hc βγ 2 =R 28 ¯N15 √ 3 ω c ≈ 0.31 ¯Nω c ∼ γ 2 . (8.66)The angular distribution of synchrotron radiation produced by ultrarelativisticelectrons shows a clear ’searchlight’ effect. Most of the photonsare radiated within an angular range of order 1/γ centered on the electrontrajectory direction. In the interesting region of γ > 10 3 the angular resolutionof X-ray and gamma detectors usually does not allow the details of theangular distribution to be measured. Therefore, the angular distribution isset to be flat in the range 0 − 1/γ.8.5.2 Simulating Synchrotron RadiationThe distance x along the electron/positron trajectory to the next occurrenceof a synchrotron radiation photon is simulated according to the exponentialdistribution, exp(−x ¯N/L). The energy ω of the photon is simulatedaccording to the distribution ¯N >ω / ¯N. The direction of the photon (θ, ϕ) isgenerated relative to the local z-axis which is taken to be along the instantaneousdirection of the electron. θ and ϕ are distributed randomly in theranges [0, 1/γ] and [0, 2π], respectively.121

8.5.3 Status of this document15.10.98 created by V.Grichine02.12.02 re-written by D.H. WrightBibliography[1] R. Maier, Synchrotron Radiation, CERN Report 91-04, 97-115 (1991).[2] Edited by M. Abramovwitz and I.A. Stegan, Handbook of MathemaaticalFunctions, NBS Applied Mathematics Series 55 (1964).[3] G.A. Korn and T.M. Korn, Mathematical Handbook for scientists andengineers, McGRAW-HILL BOOK COMPANY, INC (1961).[4] A.V. Bagulya and V.M. Grichine, Bulletin of Lebedev Institute, no.9-10,7 (1998).122

Chapter 9Muon Incident123

9.1 Ionization9.1.1 MethodThe class G4MuIonisation provides the continuous energy loss due to ionizationand simulates the ’discrete’ part of the ionization, that is delta raysproduced by muons. The approach described in Section 7.1 is used. Thevalue of the maximum energy transferable to a free electron T max is given bythe following relation:T max =2mc 2 (γ 2 − 1)1 + 2γ(m/M) + (m/M) 2 . (9.1)Here m is the electron mass and M the muon mass. The method of calculationof the continuous energy loss and the total cross section are explainedbelow.9.1.2 Continuous Energy LossThe integration of 7.1 leads to the Bethe-Bloch restricted energy loss formula[1] :]dE= 2πr 2 (z p ) 2 [ ( 2mc 2 β 2 γ 2 ) (T upedxn el ln− β 2 1 + T )up− δ − 2C ]eT

w i is the proportion by mass of the i th element, with molar mass A i .The mean excitation energy, I, for all elements is tabulated according tothe ICRU recommended values [2].Density Correctionδ is a correction term which takes into account the reduction in energy lossdue to the so-called density effect. This becomes important at high energybecause media have a tendency to become polarised as the incident particlevelocity increases. As a consequence, the atoms in a medium can no longerbe considered as isolated. To correct for this effect the formulation of Sternheimer[3] is used:x is a kinetic variable of the particle : x = log 10 (γβ) = ln(γ 2 β 2 )/4.606,and δ(x) is defined byfor x < x 0 : δ(x) = 0for x ∈ [x 0 , x 1 ] : δ(x) = 4.606x − C + a(x 1 − x) mfor x > x 1 : δ(x) = 4.606x − C(9.3)where the matter-dependent constants are calculated as follows:hν p = plasma energy of the medium = √ 4πn el remc 3 2 /α = √ 4πn el r e¯hcC = 1 + 2 ln(I/hν p )x a = C/4.606a = 4.606(x a − x 0 )/(x 1 − x 0 ) mm = 3.(9.4)For condensed media{for C ≤ 3.681 x0 = 0.2 xI < 100 eV1 = 2for C > 3.681 x 0 = 0.326C − 1.0 x 1 = 2{for C ≤ 5.215 x0 = 0.2 xI ≥ 100 eV1 = 3for C > 5.215 x 0 = 0.326C − 1.5 x 1 = 3and for gaseous mediafor C < 10. x 0 = 1.6 x 1 = 4for C ∈ [10.0, 10.5[ x 0 = 1.7 x 1 = 4for C ∈ [10.5, 11.0[ x 0 = 1.8 x 1 = 4for C ∈ [11.0, 11.5[ x 0 = 1.9 x 1 = 4for C ∈ [11.5, 12.25[ x 0 = 2. x 1 = 4for C ∈ [12.25, 13.804[ x 0 = 2. x 1 = 5for C ≥ 13.804 x 0 = 0.326C − 2.5 x 1 = 5.125

Shell Correction2C e /Z is the so-called shell correction term which accounts for the fact that,at low energies for light elements and at all energies for heavy ones, theprobability of collision with the electrons of the inner atomic shells (K, L,etc.) is negligible. The semi-empirical formula used in Geant4, applicableto all materials, is due to Barkas [4]:C e (I, βγ) = a(I)(βγ) 2 + b(I)(βγ) 4 + c(I)(βγ) 6 . (9.5)The functions a(I), b(I), c(I) can be found in the source code. This formulabreaks down at low energies, and is valid only when βγ > 0.13 (T > 7.9 MeVfor a proton). For βγ ≤ 0.13 the shell correction term is calculated as:ln(T/TC e (I, βγ)2l )∣ = C e (I, βγ = 0.13)(9.6)βγ≤0.13 ln(7.9 MeV/T 2l )i.e. the correction is switched off logarithmically from T = 7.9 MeV toT = T 2l = 2 MeV.ParameterizationThe mean energy loss can be described by the Bethe-Bloch formula (9.2)only if the projectile velocity is larger than that of the orbital electrons. Inthe low energy region this is not the case, and the parameterization from theICRU’49 report [5] is used in the G4BraggModel class. The Bethe-Blochmodel is applied to muons of higher kinetic energiesT > 2 ∗ M µ /M proton MeV. (9.7)The details of the low energy parameterization are described in Section 11.10.9.1.3 Total Cross Section per Atom and Mean FreePathFor T ≫ I the differential cross section can be written as [1][dσdT = 2πr2 e mc2 Z z2 p 11 − β 2 T+ T 2 ]. (9.8)β 2 T 2 T max 2E 2In Geant4 T cut ≥ 1 keV. Integrating from T cut to T max gives the total crosssectionper atom :σ(Z, E, T cut ) = 2πr2 eZz 2 pβ 2 mc 2 × (9.9)126

In a given material the mean free path is[ ( 1− 1 )−β2ln T max+ T ]max − T cut.T cut T max T max T cut 2E 2λ = (n at · σ) −1 or λ = ( ∑ i n ati · σ i ) −1 . (9.10)The mean free path is tabulated during initialization as a function of thematerial and of the energy of the incident muon.9.1.4 Simulating Delta-ray ProductionA short overview of the sampling method is given in Chapter 2. Apart fromthe normalization, the cross section 9.8 can be factorized :whereThe energy T is chosen bydσdT = f(T )g(T ) with T ∈ [T cut, T max ] (9.11)f(T ) =1. sampling T from f(T )( 1T cut− 1T max) 1T 2 (9.12)g(T ) = 1 − β 2 TT max+ T 22E 2 . (9.13)2. calculating the rejection function g(T ) and accepting the sampled Twith a probability of g(T ).After successful sampling of the energy, the direction of the scattered electronis generated with respect to the direction of the incident muon. Theazimuthal angle φ is generated isotropically. The polar angle θ is calculatedfrom energy-momentum conservation. This information is used to calculatethe energy and momentum of both scattered particles and to transform theminto the global coordinate system.9.1.5 Status of this document09.10.98 created by L. Urbán.14.12.01 revised by M.Maire30.11.02 re-worded by D.H. Wright01.12.03 revised by V. Ivanchenko127

Bibliography[1] Particle Data Group. Rev. of Particle Properties. Eur. Phys. J. C15.(2000) 1. http://pdg.lbl.gov[2] ICRU Report No. 37 (1984)[3] R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.[4] W. H. Barkas. Technical Report 10292,UCRL, August 1962.[5] ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons andAlpha Particles, ICRU Report 49, 1993.128

9.2 BremsstrahlungBremsstrahlung dominates other muon interaction processes in the regionof catastrophic collisions (v ≥ 0.1 ), that is at ”moderate” muon energiesabove the kinematic limit for knock–on electron production. At high energies(E ≥ 1 TeV) this process contributes about 40% of the average muon energyloss.9.2.1 Differential Cross SectionThe differential cross section for muon bremsstrahlung (in units of cm 2 /(g GeV))can be written asdσ(E, ɛ, Z, A)dɛ= 163 αN A( m µ r e) 2 1ɛA Z(ZΦ n + Φ e )(1 − v + 3 4 v2 )= 0 if ɛ ≥ ɛ max = E − µ, (9.14)where µ and m are the muon and electron masses, Z and A are the atomicnumber and atomic weight of the material, and N A is Avogadro’s number.If E and T are the initial total and kinetic energy of the muon, and ɛ isthe emitted photon energy, then ɛ = E − E ′ and the relative energy transferv = ɛ/E.Φ n represents the contribution of the nucleus and can be expressed as√Φ n = ln BZ−1/3 (µ + δ(D n′ e − 2))D n ′ (m + δ√ ;eBZ −1/3 )= 0 if negative.Φ e represents the contribution of the electrons and can be expressed asB ′ Z −2/3 µΦ e = ln (1 + δµ )m 2√ (m + δ √ ;eBe′ Z −2/3 )= 0 if ɛ ≥ ɛ ′ max = E/(1 + µ2 /2mE);= 0 if negative.In Φ n and Φ e , for all nuclei except hydrogen,δ = µ 2 ɛ/2EE ′ = µ 2 v/2(E − ɛ);D n ′ = D n (1−1/Z) , D n = 1.54A 0.27 ;√B = 183, B ′ = 1429, e = 1.648(721271).129

For hydrogen (Z=1) B = 202.4, B ′ = 446, D ′ n = D n.These formulae are taken mostly from Refs. [1] and [2]. They includeimproved nuclear size corrections in comparison with Ref. [3] in the regionv ∼ 1 and low Z. Bremsstrahlung on atomic electrons (taking into accounttarget recoil and atomic binding) is introduced instead of a rough substitutionZ(Z + 1). A correction for processes with nucleus excitation is also included[4].Applicability and Restrictions of the MethodThe above formulae assume that:1. E ≫ µ, hence the ultrarelativistic approximation is used;2. E ≤ 10 20 eV; above this energy, LPM suppression can be expected;3. v ≥ 10 −6 ; below 10 −6 Ter-Mikaelyan suppression takes place. However, inthe latter region the cross section of muon bremsstrahlung is several ordersof magnitude less than that of other processes.The Coulomb correction (for high Z) is not included. However, existingcalculations [5] show that for muon bremsstrahlung this correction is small.9.2.2 Continuous Energy LossThe restricted energy loss for muon bremsstrahlung (dE/dx) rest with relativetransfers v = ɛ/(T + µ) ≤ v cut can be calculated as follows :( ) ∫ dEɛcut∫ vcut= ɛ σ(E, ɛ) dɛ = (T + µ) ɛ σ(E, ɛ) dv .dxrest00If the user cut v cut ≥ v max = T/(T + µ), the total average energy loss iscalculated. Integration is done using Gaussian quadratures, and binningprovides an accuracy better than about 0.03% for T = 1 GeV, Z = 1. Thisrapidly improves with increasing T and Z.9.2.3 Total Cross SectionThe integration of the differential cross section over dɛ gives the total crosssection for muon bremsstrahlung:σ tot (E, ɛ cut ) =∫ ɛmaxɛ cutσ(E, ɛ)dɛ =∫ ln vmaxwhere v max = T/(T + µ). If v cut ≥ v max , σ tot = 0.ln v cutɛσ(E, ɛ)d(ln v), (9.15)130

9.2.4 SamplingThe photon energy ɛ p is found by numerically solving the equation :P =∫ ɛmaxɛ pσ(E, ɛ, Z, A) dɛ/∫ ɛmaxɛ cutσ(E, ɛ, Z, A) dɛ .Here P is the random uniform probability, ɛ max = T , and ɛ cut = (T + µ) ·v cut . v min.cut = 10 −5 is the minimal relative energy transfer adopted in thealgorithm.For fast sampling, the solution of the above equation is tabulated atinitialization time for selected Z, T and P . During simulation, this table isinterpolated in order to find the value of ɛ p corresponding to the probabilityP .The tabulation routine uses accurate functions for the differential crosssection. The table contains values ofx p = ln(v p /v max )/ ln(v max /v cut ), (9.16)where v p = ɛ p /(T + µ) and v max = T/(T + µ). Tabulation is performedin the range 1 ≤ Z ≤ 128, 1 ≤ T ≤ 1000 PeV, 10 −5 ≤ P ≤ 1 with constantlogarithmic steps. Atomic weight (which is a required parameter in thecross section) is estimated here with an iterative solution of the approximaterelation:A = Z (2 + 0.015 A 2/3 ).For Z = 1, A = 1 is used.To find x p (and thus ɛ p ) corresponding to a given probability P , thesampling method performs a linear interpolation in ln Z and ln T , and acubic, 4 point Lagrangian interpolation in ln P . For P ≤ P min , a linearinterpolation in (P, x) coordinates is used, with x = 0 at P = 0. Then theenergy ɛ p is obtained from the inverse transformation of 9.16 :ɛ p = (T + µ)v max (v max /v cut ) xpThe algorithm with the parameters described above has been tested for variousZ and T . It reproduces the differential cross section to within 0.2 –0.7 % for T ≥ 10 GeV. The average total energy loss is accurate to within0.5%. While accuracy improves with increasing T , satisfactory results arealso obtained for 1 ≤ T ≤ 10 GeV.It is important to note that this sampling scheme allows the generationof ɛ p for different user cuts on v which are above v min.cut . To perform such asimulation, it is sufficient to define a new probability variableP ′ = P σ tot (v user.cut )/σ tot (v min.cut )131

and use it in the sampling method. Time consuming re-calculation of the3-dimensional table is therefore not required because only the tabulation ofσ tot (v user.cut ) is needed.The small-angle, ultrarelativistic approximation is used for the simulation(with about 20% accuracy at θ ≤ θ ∗ ≈ 1) of the angular distribution of thefinal state muon and photon. Since the target recoil is small, the muonand photon are directed symmetrically (with equal transverse momenta andcoplanar with the initial muon):p ⊥µ = p ⊥γ , where p ⊥µ = E ′ θ µ , p ⊥γ = ɛθ γ . (9.17)θ µ and θ γ are muon and photon emission angles. The distribution in thevariable r = Eθ γ /µ is given byRandom angles are sampled as follows:f(r)dr ∼ rdr/(1 + r 2 ) 2 . (9.18)θ γ = µ E r θ µ = ɛ E ′ θ γ, (9.19)where√ ar =1 − a , a = ξ rmax2 , r1 + rmax2 max = min(1, E ′ /ɛ) · E θ ∗ /µ ,and ξ is a random number uniformly distributed between 0 and 1.9.2.5 Status of this document09.10.98 created by R.Kokoulin and A.Rybin17.05.00 updated by S.Kelner, R.Kokoulin and A.Rybin30.11.02 re-written by D.H. WrightBibliography[1] S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Preprint MEPhI 024-95,Moscow, 1995; CERN SCAN-9510048.[2] S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Phys. Atomic Nuclei, 60(1997) 576.[3] A.A.Petrukhin, V.V.Shestakov. Canad.J.Phys., 46 (1968) S377.132

[4] Yu.M.Andreyev, L.B.Bezrukov, E.V.Bugaev. Phys. Atomic Nuclei, 57(1994) 2066.[5] Yu.M.Andreev, E.V.Bugaev, Phys. Rev. D, 55 (1997) 1233.133

9.3 Muon Photonuclear InteractionThe inelastic interaction of muons with nuclei is important at high muon energies(E ≥ 10 GeV), and at relatively high energy transfers ν (ν/E ≥ 10 −2 ).It is especially important for light materials and for the study of detector responseto high energy muons, muon propagation and muon-induced hadronicbackground. The average energy loss for this process increases almost linearywith energy, and at TeV muon energies constitutes about 10% of the energyloss rate.The main contribution to the cross section σ(E, ν) and energy loss comesfrom the low Q 2 –region ( Q 2 ≪ 1 GeV 2 ). In this domain, many simplificationscan be made in the theoretical consideration of the process in orderto obtain convenient and simple formulae for the cross section. Most widelyused are the expressions given by Borog and Petrukhin [1], and Bezrukov andBugaev [2]. Results from these authors agree within 10% for the differentialcross section and within about 5% for the average energy loss, provided thesame photonuclear cross section, σ γN , is used in the calculations.9.3.1 Differential Cross SectionThe Borog and Petrukhin formula for the cross section is based on:• Hand’s formalism [3] for inelastic muon scattering,• a semi-phenomenological inelastic form factor, which is a Vector DominanceModel with parameters estimated from experimental data, and• nuclear shadowing effects with a reasonable theoretical parameterization[4].For E ≥ 10 GeV, the Borog and Petrukhin cross section (cm 2 /g GeV), differentialin transferred energy, isσ(E, ν) = Ψ(ν)Φ(E, v), (9.20)Φ(E, v) = v − 1 +[Ψ(ν) = α π1 − v + v22(A eff N AVA1 + 2µ2Λ 2 )]ln134σ γN (ν) 1 ν , (9.21)E 2 ((1 − v)µ 21 + EvΛ1 + µ2 v 2 )Λ 2 (1 − v)(1 + Λ2M + Ev ) ,Λ(9.22)

where ν is the energy lost by the muon, v = ν/E, and µ and M arethe muon and nucleon (proton) masses, respectively. Λ is a Vector DominanceModel parameter in the inelastic form factor which is estimated to beΛ 2 = 0.4 GeV 2 .For A eff , which includes the effect of nuclear shadowing, the parameterization[4]A eff = 0.22A + 0.78A 0.89 (9.23)is chosen.A reasonable choice for the photonuclear cross section, σ γN , is the parameterizationobtained by Caldwell et al. [5] based on the experimental data onphotoproduction by real photons:σ γN = (49.2 + 11.1 ln K + 151.8/ √ K) · 10 −30 cm 2 K in GeV. (9.24)The upper limit of the transferred energy is taken to be ν max = E − M/2.The choice of the lower limit ν min is less certain since the formula 9.20, 9.21,9.22 is not valid in this domain. Fortunately, ν min influences the total crosssection only logarithmically and has no practical effect on the average energyloss for high energy muons. Hence, a reasonable choice for ν min is 0.2 GeV.In Eq. 9.21, A eff and σ γN appear as factors. A more rigorous theoreticalapproach may lead to some dependence of the shadowing effect on ν and E;therefore in the differential cross section and in the sampling procedure, thispossibility is forseen and the atomic weight A of the element is kept as anexplicit parameter.The total cross section is obtained by integration of Eq. 9.20 between ν minand ν max ; to facilitate the computation, a ln(ν)–substitution is used.9.3.2 SamplingSampling the Transferred EnergyThe muon photonuclear interaction is always treated as a discrete processwith its mean free path determined by the total cross section. The totalcross section is obtained by the numerical integration of Eq. 9.20 within thelimits ν min and ν max . The process is considered for muon energies 1GeV ≤T ≤ 1000PeV, though it should be noted that above 100 TeV the extrapolation(Eq. 9.24) of σ γN may be too crude.135

The random transferred energy, ν p , is found from the numerical solution ofthe equation :P =∫ νmaxν pσ(E, ν)dν/∫ νmaxν minσ(E, ν)dν . (9.25)Here P is the random uniform probability, with ν maxν min = 0.2 GeV.= E − M/2 andFor fast sampling, the solution of Eq. 9.25 is tabulated at initialization time.During simulation, the sampling method returns a value of ν p correspondingto the probability P , by interpolating the table. The tabulation routine usesEq. 9.20 for the differential cross section. The table contains values ofx p = ln(ν p /ν max )/ ln(ν max /ν min ), (9.26)calculated at each point on a three-dimensional grid with constant spacings inln(T ), ln(A) and ln(P ) . The sampling uses linear interpolations in ln(T ) andln(A), and a cubic interpolation in ln(P ). Then the transferred energy is calculatedfrom the inverse transformation of Eq. 9.26, ν p = ν max (ν max /ν min ) xp .Tabulated parameters reproduce the theoretical dependence to better than2% for T > 1 GeV and better than 1% for T > 10 GeV.Sampling the Muon Scattering AngleAccording to Refs. [1, 6], in the region where the four-momentum transfer isnot very large (Q 2 ≤ 3GeV 2 ), the t – dependence of the cross section maybe described as:dσdt ∼ (1 − t/t max )t(1 + t/ν 2 )(1 + t/m 2 0) [(1 − y)(1 − t min/t) + y 2 /2], (9.27)where t is the square of the four-momentum transfer, Q 2 = 2(EE ′ −P P ′ cos θ−µ 2 ). Also, t min = (µy) 2 /(1 − y), y = ν/E and t max = 2Mν. ν = E − E ′ isthe energy lost by the muon and E is the total initial muon energy. M isthe nucleon (proton) mass and m 2 0 ≡ Λ2 ≃ 0.4 GeV 2 is a phenomenologicalparameter determing the behavior of the inelastic form factor. Factors whichdepend weakly, or not at all, on t are omitted.To simulate random t and hence the random muon deflection angle, it isconvenient to represent Eq. 9.27 in the form :σ(t) ∼ f(t)g(t), (9.28)136

wheref(t) =g(t) = 1 − t/t max· (1 − y)(1 − t min/t) + y 2 /2,1 + t/t 2 (1 − y) + y 2 /21t(1 + t/t 1 ) , (9.29)andt P is found analytically from Eq. 9.29 :t 1 = min(ν 2 , m 2 0) t 2 = max(ν 2 , m 2 0). (9.30)t P =t max t 1[ ] P,tmax (t min + t 1 )(t max + t 1 )− t maxt min (t max + t 1 )where P is a random uniform number between 0 and 1, which is acceptedwith probability g(t). The conditions of Eq. 9.30 make use of the symmetrybetween ν 2 and m 2 0 in Eq. 9.27 and allow increased selection efficiency, whichis typically ≥ 0.7. The polar muon deflection angle θ can easily be foundfrom 1 sin 2 t P − t min(θ/2) =.4 (EE ′ − µ 2 ) − 2 t minThe hadronic vertex is generated by the hadronic processes taking into accountthe four-momentum transfer.9.3.3 Status of this document12.10.98 created by R.Kokoulin, A.Rybin.18.05.00 edited by S.Kelner, R.Kokoulin, and A.Rybin.07.12.02 re-worded by D.H. Wright30.08.04 correction of eq. 8.24 (to 1/sqrt) from H. AraujoBibliography[1] V.V.Borog and A.A.Petrukhin, Proc. 14th Int.Conf. on Cosmic Rays,Munich,1975, vol.6, p.1949.[2] L.B.Bezrukov and E.V.Bugaev, Sov. J. Nucl. Phys., 33, 1981, p.635.1 This convenient formula has been shown to the authors by D.A. Timashkov.137

[3] L.N.Hand. Phys. Rev., 129, 1834 (1963).[4] S.J.Brodsky, F.E.Close and J.F.Gunion, Phys. Rev. D6, 177 (1972).[5] D.O. Caldwell et al., Phys. Rev. Lett., 42, 553 (1979).[6] V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin, Sov. J. Nucl.Phys., 25, 1977, p.46.138

9.4 Positron - Electron Pair Production byMuonsDirect electron pair production is one of the most important muon interactionprocesses. At TeV muon energies, the pair production cross sectionexceeds those of other muon interaction processes over a range of energytransfers between 100 MeV and 0.1E µ . The average energy loss for pairproduction increases linearly with muon energy, and in the TeV region thisprocess contributes more than half the total energy loss rate.To adequately describe the number of pairs produced, the average energyloss and the stochastic energy loss distribution, the differential cross sectionbehavior over an energy transfer range of 5 MeV ≤ ɛ ≤ 0.1 ·E µ must beaccurately reproduced. This is is because the main contribution to the totalcross section is given by transferred energies 5 MeV ≤ ɛ ≤ 0.01 ·E µ , and becausethe contribution to the average muon energy loss is determined mostlyin the region 0.001 · E µ ≤ ɛ ≤ 0.1 ·E µ .For a theoretical description of the cross section, the formulae of Ref. [1]are used, along with a correction for finite nuclear size [2]. To take intoaccount electron pair production in the field of atomic electrons, the inelasticatomic form factor contribution of Ref. [3] is also applied.9.4.1 Differential Cross SectionDefinitions and ApplicabilityIn the following discussion, these definitions are used:• m and µ are the electron and muon masses, respectively• E ≡ E µ is the total muon energy, E = T + µ• Z and A are the atomic number and weight of the material• ɛ is the total pair energy or, approximately, the muon energy loss (E −E ′ )• v = ɛ/E• e = 2.718 . . .• A ⋆ = 183.The formula for the differential cross section applies when:139

• E µ ≫ µ (E ≥ 2 – 5 GeV) and E µ ≤ 10 15 – 10 17 eV. If muon energiesexceed this limit, the LPM (Landau Pomeranchuk Migdal) effect maybecome important, depending on the material• the muon energy transfer ɛ lies between ɛ min = 4 m and ɛ max = E µ −3 √ e4µ Z 1/3 , although the formal lower limit is ɛ ≫ 2 m, and the formalupper limit requires E ′ µ ≫ µ.• Z ≤ 40 – 50. For higher Z, the Coulomb correction is important buthas not been sufficiently studied theoretically.FormulaeThe differential cross section for electron pair production by muons σ(Z, A, E, ɛ)can be written as :σ(Z, A, E, ɛ) = 43πZ(Z + ζ)AN A (αr 0 ) 2 1 − vɛ∫ ρmax0G(Z, E, v, ρ) dρ, (9.31)whereG(Z, E, v, ρ) = Φ e + (m/µ) 2 Φ µ ,Φ e,µ = B e,µ L ′ e,µandΦ e,µ = 0 whenever Φ e,µ < 0.B e and B µ do not depend on Z, A, and are given byB e = [(2 + ρ 2 )(1 + β) + ξ(3 + ρ 2 )] ln(1 + 1 ξ)+ 1 − ρ2 − β1 + ξ− (3 + ρ 2 );B e ≈ 1 2ξ [(3 − ρ2 ) + 2β(1 + ρ 2 )] for ξ ≥ 10 3 ;[ (B µ = (1 + ρ 2 ) 1 + 3β 2)− 1 ]ξ (1 + 2β)(1 − ρ2 ) ln(1 + ξ)+ ξ(1 − ρ2 − β)1 + ξ+ (1 + 2β)(1 − ρ 2 );B µ ≈ ξ 2 [(5 − ρ2 ) + β(3 + ρ 2 )] for ξ ≤ 10 −3 ;Also,140

ξ = µ2 v 24m 2 (1 − ρ 2 )(1 − v) ; β = v 22(1 − v) ;A ∗ Z −1/3√ (1 + ξ)(1 + YL ′ e )e = ln1 + 2m√ eA ∗ Z −1/3 (1 + ξ)(1 + Y e )Ev(1 − ρ 2 )⎡− 1 ( ) 3mZ1/3 22 ln ⎣1 +(1 + ξ)(1 + Y e )⎤⎦ ;2µL ′ µ = ln (µ/m)A∗ Z −1/3√ (1 + 1/ξ)(1 + Y µ )1 + 2m√ eA ∗ Z −1/3 (1 + ξ)(1 + Y µ )Ev(1 − ρ 2 )[ ]3− ln2 Z1/3√ (1 + 1/ξ)(1 + Y µ ) .For faster computing, the expressions for L ′ e,µ are further algebraically transformed.The functions L ′ e,µ include the nuclear size correction [2] in comparisonwith parameterization [1] :Y e =5 − ρ 2 + 4 β (1 + ρ 2 )2(1 + 3β) ln(3 + 1/ξ) − ρ 2 − 2β(2 − ρ 2 ) ;4 + ρ 2 + 3 β (1 + ρ 2 )Y µ =(1 + ρ 2 )( 3 + 2β) ln(3 + ξ) + 1 − 3 ;2 2 ρ2 √ρ max = [1 − 6µ 2 /E 2 (1 − v)] 1 − 4m/Ev.Comment on the Calculation of the Integral ∫ dρ in Eq. 9.31The integral ρmax ∫After that,0G(Z, E, v, ρ) dρ is computed with the substitutions:∫ ρmax0t = ln(1 − ρ),1 − ρ = exp(t),1 + ρ = 2 − exp(t),1 − ρ 2 = e t (2 − e t ).G(Z, E, v, ρ) dρ =141∫ 0t minG(Z, E, v, ρ) e t dt, (9.32)

wheret min = ln4mɛ(1 ++ 12µ2EE ′ (1 − 4m ɛ)1 − 6µ2EE ′ ) √1 − 4m ɛTo compute the integral of Eq. 9.32 with an accuracy better than 0.5%,Gaussian quadrature with N = 8 points is sufficient.The function ζ(E, Z) in Eq. 9.31 serves to take into account the processon atomic electrons (inelastic atomic form factor contribution). To treatthe energy loss balance correctly, the following approximation, which is analgebraic transformation of the expression in Ref. [3], is used:E/µ0.073 ln1 + γζ(E, Z) =1 Z 2/3 E/µ − 0.26;E/µ0.058 ln1 + γ 2 Z 1/3 E/µ − 0.14ζ(E, Z) = 0if the numerator is negative.For E ≤ 35 µ, ζ(E, Z) = 0. Also γ 1 = 1.95 · 10 −5 and γ 2 = 5.30 · 10 −5 .The above formulae make use of the Thomas-Fermi model which is notgood enough for light elements. For hydrogen (Z = 1) the following parametersmust be changed:A ∗ = 183 ⇒ 202.4;γ 1 = 1.95 · 10 −5 ⇒ 4.4 · 10 −5 ;γ 2 = 5.30 · 10 −5 ⇒ 4.8 · 10 −5 ..9.4.2 Total Cross Section and Restricted Energy LossIf the user’s cut for the energy transfer ɛ cut is greater than ɛ min , the process isrepresented by continuous restricted energy loss for interactions with ɛ ≤ ɛ cut ,and discrete collisions with ɛ > ɛ cut . Respective values of the total crosssection and restricted energy loss rate are defined as:σ tot =∫ ɛmaxɛ cutσ(E, ɛ) dɛ; (dE/dx) restr =∫ ɛcutɛ minɛ σ(E, ɛ) dɛ.For faster computing, ln ɛ substitution and Gaussian quadratures are used.142

9.4.3 Sampling of Positron - Electron Pair ProductionThe e + e − pair energy ɛ P , is found numerically by solving the equationP =∫ ɛmaxɛ Pσ(Z, A, T, ɛ)dɛ /∫ ɛmaxcutσ(Z, A, T, ɛ)dɛ (9.33)or∫ ɛP∫ ɛmax1 − P = σ(Z, A, T, ɛ)dɛ / σ(Z, A, T, ɛ)dɛ (9.34)cutcutTo reach high sampling speed, solutions of Eqs. 9.33, 9.34 are tabulatedat initialization time. Two 3-dimensional tables (referred to here as A andB) of ɛ P (P, T, Z) are created, and then interpolation is used to sample ɛ P .The number and spacing of entries in the table are chosen as follows:• a constant increment in ln T is chosen such that there are four pointsper decade in the range T min − T max . The default range of muon kineticenergies in Geant4 is T = 1 GeV − 1000 PeV.• a constant increment in ln Z is chosen. The shape of the sampling distributiondoes depend on Z, but very weakly, so that eight points in therange 1 ≤ Z ≤ 128 are sufficient. There is practically no dependenceon the atomic weight A.• for probabilities P ≤ 0.5, Eq. 9.33 is used and Table A is computedwith a constant increment in ln P in the range 10 −7 ≤ P ≤ 0.5. Thenumber of points in ln P for Table A is about 100.• for P ≥ 0.5, Eq. 9.34 is used and Table B is computed with a constantincrement in ln(1 − P ) in the range 10 −5 ≤ (1 − P ) ≤ 0.5. In this case50 points are sufficient.The values of ln(ɛ P − cut) are stored in both Table A and Table B.To create the “probability tables” for each (T, Z) pair, the following procedureis used:• a temporary table of ∼ 2000 values of ɛ · σ(Z, A, T, ɛ) is constructedwith a constant increment (∼ 0.02) in ln ɛ in the range (cut, ɛ max ). ɛ istaken in the middle of the corresponding bin in ln ɛ.• the accumulated cross sectionsσ 1 =∫ ln ɛmaxln ɛɛ σ(Z, A, T, ɛ) d(ln ɛ)143

andσ 2 =∫ ln ɛln(cut)ɛ σ(Z, A, T, ɛ) d(ln ɛ)are calculated by summing the temporary table over the values aboveln ɛ (for σ 1 ) and below ln ɛ (for σ 2 ) and then normalizing to obtain theaccumulated probability functions.• finally, values of ln(ɛ P − cut) for corresponding values of ln P andln(1 − P ) are calculated by linear interpolation of the above accumulatedprobabilities to form Tables A and B. The monotonic behavior ofthe accumulated cross sections is very useful in speeding up the interpolationprocedure.The random transferred energy corresponding to a probability P , is thenfound by linear interpolation in ln Z and ln T , and a cubic interpolation inln P for Table A or in ln(1−P ) for Table B. For P ≤ 10 −7 and (1−P ) ≤ 10 −5 ,linear extrapolation using the entries at the edges of the tables may be safelyused. Electron pair energy is related to the auxiliary variable x = ln(ɛ P −cut)found by the trivial interpolation ɛ P = e x + cut.Similar to muon bremsstrahlung (section 9.2), this sampling algorithmdoes not re-initialize the tables for user cuts greater than cut min . Instead,the probability variable is redefined asP ′ = P σ tot (cut user )/σ tot (cut min ),and P ′ is used for sampling.In the simulation of the final state, the muon deflection angle (which isof the order of m/E) is neglected. The procedure for sampling the energypartition between e + and e − and their emission angles is similar to that usedfor the γ → e + e − conversion.9.4.4 Status of this document12.10.98 created by R.Kokoulin and A.Rybin18.05.00 edited by S.Kelner, R.Kokoulin, and A.Rybin27.01.03 re-written by D.H. WrightBibliography[1] R.P.Kokoulin and A.A.Petrukhin, Proc. 11th Intern. Conf. on CosmicRays, Budapest, 1969 [Acta Phys. Acad. Sci. Hung.,29, Suppl.4, p.277,1970].144

[2] R.P.Kokoulin and A.A.Petrukhin, Proc. 12th Int. Conf. on Cosmic Rays,Hobart, 1971, vol.6, p.2436.[3] S.R.Kelner, Phys. Atomic Nuclei, 61 (1998) 448.145

Chapter 10Charged Hadron Incident146

10.1 Ionization10.1.1 MethodThe class G4hIonisation provides the continuous energy loss due to ionizationand simulates the ’discrete’ part of the ionization, that is, delta raysproduced by charged hadrons. The class G4ionIonisation is intended forthe simulation of energy loss by ions and the approach described in Section7.1 is used. The value of the maximum energy transferable to a free electronT max is given by the following relation:T max =2mc 2 (γ 2 − 1)1 + 2γ(m/M) + (m/M) 2 , (10.1)where m is the electron mass and M is the mass of the incident particle. Themethod of calculation of the continuous energy loss and the total cross-sectionare explained below.10.1.2 Continuous Energy LossThe integration of 7.1 leads to the Bethe-Bloch restricted energy loss formula[1] :]dE= 2πr 2 (z p ) 2 [ ( 2mc 2 β 2 γ 2 ) (T upedxn el ln− β 2 1 + T )up− δ − 2C ]eT

w i is the proportion by mass of the i th element, with molar mass A i .The mean excitation energy I for all elements is tabulated according tothe ICRU recommended values [2].Density Correctionδ is a correction term which takes into account the reduction in energy lossdue to the so-called density effect. This becomes important at high energiesbecause media have a tendency to become polarized as the incident particlevelocity increases. As a consequence, the atoms in a medium can no longerbe considered as isolated. To correct for this effect the formulation of Sternheimer[3] is used:x is a kinetic variable of the particle : x = log 10 (γβ) = ln(γ 2 β 2 )/4.606,and δ(x) is defined byfor x < x 0 : δ(x) = 0for x ∈ [x 0 , x 1 ] : δ(x) = 4.606x − C + a(x 1 − x) mfor x > x 1 : δ(x) = 4.606x − C(10.3)where the matter-dependent constants are calculated as follows:hν p = plasma energy of the medium = √ 4πn el remc 3 2 /α = √ 4πn el r e¯hcC = 1 + 2 ln(I/hν p )x a = C/4.606a = 4.606(x a − x 0 )/(x 1 − x 0 ) mm = 3.(10.4)For condensed media{for C ≤ 3.681 x0 = 0.2 xI < 100 eV1 = 2for C > 3.681 x 0 = 0.326C − 1.0 x 1 = 2{for C ≤ 5.215 x0 = 0.2 xI ≥ 100 eV1 = 3for C > 5.215 x 0 = 0.326C − 1.5 x 1 = 3and for gaseous mediafor C < 10. x 0 = 1.6 x 1 = 4for C ∈ [10.0, 10.5[ x 0 = 1.7 x 1 = 4for C ∈ [10.5, 11.0[ x 0 = 1.8 x 1 = 4for C ∈ [11.0, 11.5[ x 0 = 1.9 x 1 = 4for C ∈ [11.5, 12.25[ x 0 = 2. x 1 = 4for C ∈ [12.25, 13.804[ x 0 = 2. x 1 = 5for C ≥ 13.804 x 0 = 0.326C − 2.5 x 1 = 5.148

Shell Correction2C e /Z is the so-called shell correction term which accounts for the fact that,at low energies for light elements and at all energies for heavy ones, theprobability of collision with the electrons of the inner atomic shells (K, L,etc.) is negligible. The semi-empirical formula used in Geant4, applicableto all materials, is due to Barkas [4]:C e (I, βγ) = a(I)(βγ) 2 + b(I)(βγ) 4 + c(I)(βγ) 6 . (10.5)The functions a(I), b(I) and c(I) can be found in the source code. Thisformula breaks down at low energies, and is valid only when βγ > 0.13(T > 7.9 MeV for a proton). For βγ ≤ 0.13 the shell correction term iscalculated as:ln(T/TC e (I, βγ)2l )∣ = C e (I, βγ = 0.13)βγ≤0.13 ln(7.9 MeV/T 2l ) , (10.6)i.e. the correction is switched off logarithmically from T = 7.9 MeV toT = T 2l = 2 MeV.ParameterizationThe mean energy loss can be described by the Bethe-Bloch formula (9.2)only if the projectile velocity is larger than that of the orbital electrons. Inthe low-energy region this is not the case, and the parameterization from theICRU’49 report [5] is used in the G4BraggModel class. The Bethe-Blochmodel is applied for higher kinetic energies of incident particlesT > 2 ∗ M/M proton MeV, (10.7)where M is the particle mass. The details of the low energy parameterizationare described in Section 11.10.10.1.3 Total Cross Section per Atom and Mean FreePathFor T ≫ I the differential cross section can be written as[dσdT = 2πr2 emc 2 Z z2 p 11 − β 2 T+ T 2 ]β 2 T 2 T max 2E 2(10.8)149

[1]. In Geant4 T cut ≥ 1 keV. Integrating from T cut to T max gives the totalcross section per atom :σ(Z, E, T cut ) = 2πr2 eZz 2 pβ 2 mc 2 × (10.9)[ ( 1T cut− 1T max)−β2ln T max+ T ]max − T cutT max T cut 2E 2The last term is for spin 1/2 only. In a given material the mean free path is:λ = (n at · σ) −1 or λ = ( ∑ i n ati · σ i ) −1 (10.10)The mean free path is tabulated during initialization as a function of thematerial and of the energy for all kinds of charged particles.10.1.4 Simulating Delta-ray ProductionA short overview of the sampling method is given in Chapter 2. Apart fromthe normalization, the cross section 10.8 can be factorized :wheredσdT = f(T )g(T ) with T ∈ [T cut, T max ] (10.11)f(T ) =( 1T cut− 1T max) 1T 2 (10.12)g(T ) = 1 − β 2 TT max+ T 22E 2 . (10.13)The last term in g(T ) is for spin 1/2 only. The energy T is chosen by1. sampling T from f(T )2. calculating the rejection function g(T ) and accepting the sampled Twith a probability of g(T ).After the successful sampling of the energy, the direction of the scattered electronis generated with respect to the direction of the incident particle. Theazimuthal angle φ is generated isotropically. The polar angle θ is calculatedfrom energy-momentum conservation. This information is used to calculatethe energy and momentum of both scattered particles and to transform theminto the global coordinate system.150

Ion Effective ChargeAs ions penetrate matter they exchange electrons with the medium. In theimplementation of G4ionIonisation the effective charge approach is used [6].A state of equilibrium between the ion and the medium is assumed, so thatthe ion’s effective charge can be calculated as a function of its kinetic energyin a given material. This is done according to the approximation describedin Section 11.10. Before and after each step the dynamic charge of the ionis recalculated and saved in G4DynamicP article, where it can be used notonly for energy loss calculations but also for the sampling of transportationin an electromagnetic field.10.1.5 Status of this document09.10.98 created by L. Urbán.14.12.01 revised by M.Maire29.11.02 re-worded by D.H. Wright01.12.03 revised by V. IvanchenkoBibliography[1] Particle Data Group. Rev. of Particle Properties. Eur. Phys. J. C15.(2000) 1. http://pdg.lbl.gov[2] ICRU Report No. 37 (1984)[3] R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.[4] W. H. Barkas. Technical Report 10292,UCRL, August 1962.[5] ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons andAlpha Particles, ICRU Report 49, 1993.[6] J.F. Ziegler, J.P. Biersack, U . Littmark, The Stopping and Ranges ofIons in Solids. Vol.1, Pergamon Press, 1985.151

Chapter 11Low Energy Extensions152

11.1 IntroductionAdditional electromagnetic physics processes for photons, electrons, hadronsand ions have been implemented in Geant4 in order to extend the validityrange of particle interactions to lower energies than those available in thestandard Geant4 electromagnetic processes [1, 2, 3]. Because atomic shellstructure is more important in most cases at low energies than it is at higherenergies, the low energy processes make direct use of shell cross section data.The standard processes, which are optimized for high energy physics applications,rely on parameterizations of these data.The low energy processes include the photo-electric effect, Compton scattering,Rayleigh scattering, gamma conversion, bremsstrahlung and ionization.Fluorescence of excited atoms is also considered.Some features common to all low energy processes currently implementedin Geant4 are summarized in this section. Subsequent sections provide moredetailed information for each process.11.1.1 PhysicsThe low energy processes of Geant4 represent electromagnetic interactionsat lower energies than those covered by the equivalent Geant4 standard electromagneticprocesses.The current implementation of low energy processes is valid for energiesdown to 250 eV (and can be used up to approximately 100 GeV), unlessdifferently specified. It covers elements with atomic number between 1 and99.All processes involve two distinct phases:• the calculation and use of total cross sections, and• the generation of the final state.Both phases are based on the theoretical models and on exploitation of evaluateddata.11.1.2 Data SourcesThe data used for the determination of cross-sections and for sampling ofthe final state are extracted from a set of publicly distributed evaluated datalibraries:• EPDL97 (Evaluated Photons Data Library) [4];153

• EEDL (Evaluated Electrons Data Library) [5];• EADL (Evaluated Atomic Data Library) [6];• stopping power data [7, 8, 9, 10];• binding energy values based on data of Scofield [11].Evaluated data sets are produced through the process of critical comparison,selection, renormalization and averaging of the available experimentaldata, normally complemented by model calculations. These libraries providethe following data relevant for the simulation of Geant4 low energy processes:• total cross-sections for photoelectric effect, Compton scattering, Rayleighscattering, pair production and bremsstrahlung,• subshell integrated cross sections for photo-electric effect and ionization,• energy spectra of the secondaries for electron processes,• scattering functions for the Compton effect,• form factors for Rayleigh scattering,• binding energies for electrons for all subshells,• transition probabilities between subshells for fluorescence and the Augereffect, and• stopping power tables.The energy range covered by the data libraries extends from 100 GeVdown to 1 eV for Rayleigh and Compton effects, down to the lowest bindingenergy for each element for photo-electric effect and ionization, and down to10 eV for bremsstrahlung.11.1.3 Distribution of the Data SetsThe author of EPDL97 [4], who is also responsible for the EEDL [5] andEADL [6] data libraries, Dr. Red Cullen, has kindly permitted the librariesand their related documentation to be distributed with the Geant4 toolkit.The data are reformatted for Geant4 input. They can be downloaded fromthe source code section of the Geant4 page: http://cern.ch/geant4/geant4.html.154

The EADL, EEDL and EPDL97 data-sets are also available from severalpublic distribution centres in a format different from the one used byGeant4 [12].Stopping power data are taken from publications [7, 8, 9, 10].11.1.4 Calculation of Total Cross SectionsThe energy dependence of the total cross section is derived for each processfrom the evaluated data libraries. Since the libraries provide cross sectionsfor a set of discrete incident energies, the total cross section at a given energy,E, is obtained by interpolation according to the formula [13]:log(σ(E)) = log(σ 1)log(E 2 /E) + log(σ 2 )log(E/E 1 )log(E 2 /E 1 )(11.1)where E 1 and E 2 are respectively the closest lower and higher energy forwhich data (σ 1 and σ 2 ) are available.For each process a production threshold energy is defined; by default it isset to the low end of the energy validity range of the process (250 eV in thecurrent implementation), but a higher or lower value can be set by the user.For a particle of energy E, the mean free path for interacting via a givenprocess is calculated as:1λ =(11.2)Σ i σ i (E) · n iwhere σ i (E) is the microscopic integrated cross-section of the process consideredat energy E, and n i is the atomic density of the i − th element contributingto the material composition. The sum runs over all the elementsof which the material is composed. The cross sections are determined asdescribed in this section. An exception to this method is the implementationof the chemical effect on hadron/ion stopping powers for a set of materials.11.1.5 Sampling of Relevant Physics QuantitiesThe final state products of the processes are generated by sampling relevantphysical quantities, such as energies and angular distributions of secondaries,from distributions derived from theoretical models and evaluated data. Theenergy dependence of the parameters which characterize the distributions istaken into account either by direct interpolation of the data available in thelibraries, or by interpolation of values obtained from fits to the data.When generating the final state, an atom of the material in which theinteraction occurs is randomly selected and atomic de-excitation is simulated.155

Secondaries which would be produced with energies below their user definedproduction threshold are not created and their energy is depositedlocally.11.1.6 Status of the document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia04.12.2001 reviewed by Vladimir Ivanchenko26.01.2003 minor re-write by D.H. Wright .Bibliography[1] “Geant4 Low Energy Electromagnetic Models for Electrons and Photons”,J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999)[2] V.N. Ivanchenko et al., GEANT4 Simulation of Energy Losses of SlowHadrons, CERN-99-121, INFN/AE-99/20, (September 1999).[3] S. Giani et al., GEANT4 Simulation of Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999).[4] “EPDL97: the Evaluated Photon Data Library, ’97 version”, D.Cullen,J.H.Hubbell, L.Kissel, UCRL–50400, Vol.6, Rev.5[5] “Tables and Graphs of Electron-Interaction Cross-Sections from 10 eVto 100 GeV Derived from the LLNL Evaluated Electron Data Library(EEDL), Z=1-100” S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400Vol.31[6] “Tables and Graphs of Atomic Subshell and Relaxation Data Derivedfrom the LLNL Evaluated Atomic Data Library (EADL), Z=1-100” S.T.Perkins, D.E.Cullen, M.H.Chen, J.H.Hubbell, J.Rathkopf,J.Scofield, UCRL-50400 Vol.30[7] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions inMatter. Vol.3, Pergamon Press, 1977.[8] J.F. Ziegler, The Stopping and Ranges of Ions in Matter. Vol.4, PergamonPress, 1977.156

[9] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Ranges ofIons in Solids. Vol.1, Pergamon Press, 1985.[10] ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons andAlpha Particles, ICRU Report 49, 1993.[11] J.H. Scofield, “Radiative Transitions”, in “Atomic Inner-Shell Processes”,B.Crasemann ed. (Academic Press, New York, 1975),pp.265-292.[12] http://www.nea.fr/html/dbdata/nds evaluated.htm[13] “New Photon, Positron and Electron Interaction Data for Geant in EnergyRange from 1 eV to 10 TeV”, J. Stepanek, Draft to be submittedfor publication157

11.2 Compton Scattering11.2.1 Total Cross SectionThe total cross section for the Compton scattering process is determinedfrom the data as described in section 11.1.4.11.2.2 Sampling of the Final StateFor low energy incident photons, the simulation of the Compton scatteringprocess is performed according to the same procedure used for the “standard”Compton scattering simulation, with the addition that Hubbel’s atomic formfactor [1] or scattering function, SF , is taken into account. The angular andenergy distribution of the incoherently scattered photon is then given bythe product of the Klein-Nishina formula Φ(ɛ) and the scattering function,SF (q) [2]P (ɛ, q) = Φ(ɛ) × SF (q). (11.3)ɛ is the ratio of the scattered photon energy E ′ , and the incident photonenergy E. The momentum transfer is given by q = E × sin 2 (θ/2), where θ isthe polar angle of the scattered photon with respect to the direction of theparent photon. Φ(ɛ) is given byΦ(ɛ) ∼ = [ 1 ɛ + ɛ][1 −ɛ1 + ɛ 2 sin2 θ]. (11.4)The effect of the scattering function becomes significant at low energies,especially in suppressing forward scattering [2].The sampling method of the final state is based on composition and rejectionMonte Carlo methods [3, 4, 5], with the SF function included in therejection functiong(ɛ) =[1 − ɛ ]1 + ɛ 2 sin2 θ × SF (q), (11.5)with 0 < g(ɛ) < Z. Values of the scattering functions at each momentumtransfer, q, are obtained by interpolating the evaluated data for the correspondingatomic number, Z.The polar angle θ is deduced from the sampled ɛ value. In the azimuthaldirection, the angular distributions of both the scattered photon and therecoil electron are considered to be isotropic [6].Since the incoherent scattering occurs mainly on the outermost electronicsubshells, the binding energies can be neglected, as stated in reference [6].158

The momentum vector of the scattered photon, −→ P ′ γ, is transformed into theWorld coordinate system. The kinetic energy and momentum of the recoilelectron are thenT el = E − E ′−→P el = −→ P γ − −→ P ′ γ .11.2.3 Status of the document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia26.01.2003 minor re-write by D.H. WrightBibliography[1] “Summary of Existing Information on the Incoherent Scattering of Photonsparticularly on the Validity of the Use of the Incoherent ScatteringFunction”, Radiat. Phys. Chem. Vol. 50, No 1, pp 113-124 (1997)[2] “A simple model of photon transport”, D.E. Cullen, Nucl. Instr. Meth.in Phys. Res. B 101(1995)499-510[3] J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)[4] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[5] R. Ford and W. Nelson. SLAC-265, UC-32 (1985)[6] “New Photon, Positron and Electron Interaction Data for Geant in EnergyRange from 1 eV to 10 TeV”, J. Stepanek, Draft to be submittedfor publication159

11.3 Compton Scattering by Linearly PolarizedGamma Rays11.3.1 The Cross SectionThe quantum mechanical Klein - Nishina differential cross section for polarizedphotons is [Heitler 1954]:dσdΩ = 1 hν 2 hν 2 [4 r2 o hνo0hν 2 hν + hν]− 2 + 4cos 2 Θhν ohν 2 owhere Θ is the angle between the two polarization vectors. In terms of thepolar and azimuthal angles (θ, φ) this cross section can be written as.dσdΩ = 1 hν 2 hν 2 [2 r2 o hνo0hν 2 hν + hν]− 2cos 2 φsin 2 θhν ohν 2 o11.3.2 Angular DistributionThe integration of this cross section over the azimuthal angle produces thestandard cross section. The angular and energy distribution are then obtainedin the same way as for the standard process. Using these values forthe polar angle and the energy, the azimuthal angle is sampled from thefollowing distribution:P (φ) = 1 − a b cos2 φwhere a = sin 2 θ and b = ɛ + 1/ɛ. ɛ is the ratio between the scattered photonenergy and the incident photon energy.11.3.3 Polarization VectorThe components of the vector polarization of the scattered photon are calculatedfrom⃗ɛ ′ ⊥ = 1 N(ĵcosθ− ˆksinθsinφ)sinβ⃗ɛ ′ ‖[Nî = − 1 N ĵsin2 θsinφcosφ − 1 N ˆksinθcosθcosφ]cosβ160

whereN =√1 − sin 2 θcos 2 φ.cosβ is calculated from cosΘ = Ncosβ, while cosΘ is sampled from the Klein- Nishina distribution.The binding effects and the Compton profile are neglected. The kineticenergy and momentum of the recoil electron are thenT el = E − E ′⃗P el = ⃗ P γ − ⃗ P ′ γ .The momentum vector of the scattered photon P ⃗ γ and its polarizationvector are transformed into the World coordinate system. The polarizationand the direction of the scattered gamma in the final state are calculated inthe reference frame in which the incoming photon is along the z-axis and hasits polarization vector along the x-axis. The transformation to the Worldcoordinate system performs a linear combination of the initial direction, theinitial poalrization and the cross product between them, using the projectionsof the calculated quantities along these axes.11.3.4 Unpolarized PhotonsA special treatment is devoted to unpolarized photons. In this case a randompolarization in the plane perpendicular to the incident photon is selected.11.3.5 Status of this document18.06.2001 created by Gerardo Depaola and Francesco Longo10.06.2002 revision by Francesco Longo26.01.2003 minor re-wording and correction of equations by D.H. WrightBibliography[1] W. Heitler The Quantum Theory of Radiation, Oxford Clarendom Press(1954)161

11.4 Rayleigh Scattering11.4.1 Total Cross SectionThe total cross section for the Rayleigh scattering process is determined fromthe data as described in section 11.1.4.11.4.2 Sampling of the Final StateThe coherent scattered photon angle θ is sampled according to the distributionobtained from the product of the Rayleigh formula (1 + cos 2 θ) sin θ andthe square of Hubbel’s form factor F F 2 (q) [1] [2]Φ(E, θ) = [1 + cos 2 θ] sin θ × F F 2 (q), (11.6)where q = 2E sin(θ/2) is the momentum transfer.Form factors introduce a dependency on the initial energy E of the photonthat is not taken into account in the Rayleigh formula. At low energies, formfactors are isotropic and do not affect angular distribution, while at highenergies they are forward peaked.The sampling procedure is as follows [3]:1. cosθ is chosen from a uniform distribution between -1 and 12. the form factor F F is extracted from the data table for the consideredelement, using logarithmic data interpolation, for q = 2E · sin(θ/2)3. if the value obtained for Φ(E, θ) is larger than a random number uniformlydistributed between 0 and Z 2 , the procedure is repeated fromstep 1, otherwise θ is taken as the photon scattering angle with respectto its incident direction.4. the azimuthal direction of the scattered photon is chosen at random.11.4.3 Status of this document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia10.06.2002 modified by Francesco Longo and Gerardo Depaola26.01.2003 minor re-write and correction of equations by D.H. Wright162

Bibliography[1] ”Relativistic Atom Form Factors and Photon Coherent Scattering CrossSections”, J.H. Hubbell et al., J.Phys.Chem.Ref.Data, 8,69(1979)[2] ”A simple model of photon transport”, D.E. Cullen, Nucl. Instr. Meth.in Phys. Res. B 101(1995)499-510[3] ”New Photon, Positron and Electron Interaction Data for Geant in EnergyRange from 1 eV to 10 TeV”, J. Stepanek, Draft to be submittedfor publication163

11.5 Gamma Conversion11.5.1 Total cross-sectionThe total cross-section of the Gamma Conversion process is determined fromthe data as described in section 11.1.4.11.5.2 Sampling of the final stateFor low energy incident photons, the simulation of the Gamma Conversionfinal state is performed according to [1].The secondary e ± energies are sampled using the Bethe-Heitler crosssectionswith Coulomb correction.The Bethe-Heitler differential cross-section with the Coulomb correctionfor a photon of energy E to produce a pair with one of the particles havingenergy ɛE (ɛ is the fraction of the photon energy carried by one particle ofthe pair) is given by [2]:dσ(Z, E, ɛ)dɛ[(= r2 0αZ(Z + ξ(Z))(ɛ 2 + (1 − ɛ) 2 )E 2+ 2 (3 ɛ(1 − ɛ)Φ 2 (δ) − F (Z) )]2Φ 1 (δ) − F (Z)2)+where Φ i (δ) are the screening functions depending on the screening variableδ [1].The value of ɛ is sampled using composition and rejection Monte Carlomethods [1, 3, 4].After the successful sampling of ɛ, the process generates the polar angles ofthe electron with respect to an axis defined along the direction of the parentphoton. The electron and the positron are assumed to have a symmetricangular distribution. The energy-angle distribution is given by[5]:dσdpdΩ = 2α2 e 2 [( 2x(1 − x)πkm 4 (1 + l)+2−)12lx(1 − x)(Z 2 + Z)+(1 + l) 4( 2x 2 )− 2x + 1 4lx(1 − x)+ (X − 2Z 2 f((αZ) 2 ))(1 + l) 2 (1 + l) 4where k is the photon energy, p the momentum and E the energy of theelectron of the e ± pair x = E/k and l = E 2 θ 2 /m 2 . The sampling of thiscross-section is obtained according to [1].164]

The azimuthal angle φ is generated isotropically.This information together with the momentum conservation is used tocalculate the momentum vectors of both decay products and to transformthem to the GEANT coordinate system. The choice of which particle in thepair is the electron/positron is made randomly.11.5.3 Status of the document18.06.2001 created by Francesco LongoBibliography[1] Urban L., in Brun R. et al. (1993), Geant. Detector Description andSimulation Tool, CERN Program Library, section Phys/211[2] R. Ford and W. Nelson., SLAC-210, UC-32 (1978)[3] J.C. Butcher and H. Messel. Nucl. Phys. 20 15 (1960)[4] H. Messel and D. Crawford. Electron-Photon shower distribution, PergamonPress (1970)[5] Y. S. Tsai, Rev. Mod. Phys. 46 815 (1974), Y. S. Tsai, Rev. Mod. Phys.49 421 (1977)165

11.6 Photoelectric effect11.6.1 Total cross-sectionThe total photoelectric cross-section at a given energy, E, is calculated asdescribed in section 11.1.4. Note that for this process the MeanFreePathTableis not built, since the cross-section is not a smooth function of the energy,therefore in all calculations the cross-section is used directly.11.6.2 Sampling of the final stateThe incident photon is absorbed and an electron is emitted in the samedirection as the incident photon.The electron kinetic energy is the difference between the incident photonenergy and the binding energy of the electron before the interaction. Thesub-shell, from which the electron is emitted, is randomly selected accordingto the relative cross-sections of all subshells, determined at the given energy,T , by interpolating the evaluated cross-section data from the EPDL97 databank [1].The interaction leaves the atom in an excited state. The deexcitation ofthe atom is simulated as described in section 11.9.11.6.3 Status of the document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia13.05.2002 modified by Vladimir IvanchenkoBibliography[1] “EPDL97: the Evaluated Photon Data Library, ’97 version”, D.Cullen,J.H.Hubbell, L.Kissel, UCRL–50400, Vol.6, Rev.5166

11.7 BremsstrahlungThe class G4LowEnergyBremsstrahlung calculates the continuous energy lossdue to low energy gamma emission and simulates the gamma production byelectrons. The gamma production threshold for a given material ω c is usedto separate the continuous and the discrete parts of the process. The energyloss of an electron with the incident energy T are expressed via the integrandover energy of the gammas:∫dEωcdx = σ(T )0.1eV t dσ dω dω∫ T0.1eV dσdωdω, (11.7)where σ(T ) is the total cross-section at a given incident kinetic energy, T ,0.1eV is the low energy limit of the EEDL data. The production cross-sectionis a complimentary function:σ = σ(T )∫ T dσω dωc dω∫ T0.1eV dσdωdω. (11.8)The total cross-section, σ s , is obtained from an interpolation of the evaluatedcross-section data in the EEDL library [1], according to the formula(11.1) in Section 11.1.4.The EEDL data [1] of total cross-sections are parametrised [2] accordingto (11.1). The probability of the emission of a photon with energy, ω, consideringan electron of incident kinetic energy, T , is generated according tothe formula:dσdω = F (x)x , withx = ω T . (11.9)The function, F (x), describing energy spectra of the outcoming photonsis taken from the EEDL library. For each element 15 points in x from 0.01to 1 are used for the linear interpolation of this function. The function Fis normalised by the condition F (0.01) = 1. The energy distributions of theemitted photons available in the EEDL library are for only a few incidentelectron energies (about 10 energy points between 10 eV and 100 GeV). Forother energies a logarithmic interpolation formula (11.1) is used to obtainvalues for the function, F (x). For high energies, the spectral function is veryclose to:F (x) = 1 − x + 0.75x 2 . (11.10)167

11.7.1 Bremsstrahlung angular distributionsThe angular distribution of the emitted photons with respect to the incidentelectron can be sampled according to three alternative generators describedbelow. The direction of the outcoming electron is determined fromthe energy-momentum balance. This generators are currently implementedin G4ModifiedTsai, G4Generator2BS and G4Generator2BN classes.G4ModifiedTsaiThe angular distribution of the emitted photons is obtained from a simplified[3] formula based on the Tsai cross-section [4], which is expected to becomeisotropic in the low energy limit.G4Generator2BSIn G4Generator2BS generator, the angular distribution of the emitted photonsis obtained from the 2BS Koch and Motz bremsstrahlung double differentialcross-section [5]:dσ k , θ = 4Z2 r 2 0137(E 0 + E) 2(y 2 + 1) 2 E 2 0{dk 16y 2k ydy E−(y 2 + 1) 4 E 0[ E2+ 0 + E 2−(y 2 + 1) 2 E024y 2 ] }ElnM(y)(y 2 + 1) 4 E 0where k the photon energy, θ the emission angle, E 0 and E are the initialand final electron energy in units of m e c 2 , r 0 is the classical electron radiusand Z the atomic number of the material. y and M(y) are defined as:1M(y)y = E 0 θ=( k2E 0 E) 2+(Z 1/3111(y 2 + 1)The adopted sampling algorithm is based on the sampling scheme developedby A. F. Bielajew et al. [6], and latter implemented in EGS4. In thissampling algorithm only the angular part of 2BS is used, with the emittedphoton energy, k, determined by GEANT4 ( )dσdk differential cross-section.G4Generator2BNThe angular distribution of the emitted photons is obtained from the 2BNKoch and Motz bremsstrahlung double differential cross-section [5] that canbe written as:168) 2

dσ k , θ = Z2 r02 dk8π137 k{p 8 sin 2 θ(2E0 2 − 1)dΩ k −p 0 p 2 0∆ 4 02(5E0 2 + 2EE 0 + 3)− 2(p2 0 − k2 )p 2 0∆ 2 0Q 2 ∆ 0[ 4E0 sin 2 θ(3k − p 2 0 E) + 4E2 0 (E2 0 + E2 )+p 2 0∆ 4 p 2 0∆ 2 02 − 2(E0 2 − 3EE 0 + E 2 )p 2 0∆ 2 0( ) ( [ 4ɛ ɛQ 4− +p∆0 pQ)∆ 2 0+ 4Ep 2 2∆ 0+ Lpp 0]+ 2k(E2 0 + EE 0 − 1)p 2 0∆ 0− 6k − 2k(p2 0 − k 2 ]})∆ 0 Q 2 ∆ 0in which:L =[ ]EE0 − 1 + pp 0lnEE 0 − 1 − pp 0∆ 0 = E 0 − p 0 cos θQ 2 = p 2 0 + k 2 − 2p 0 k cos θ[ ] E + pɛ = lnɛ Q = lnE − p[ ] Q + pQ − pwhere k is the photon energy, θ the emission angle and (E 0 , p 0 ) and (E, p) arethe total (energy, momentum) of the electron before and after the radiativeemission, all in units of m e c 2 .Since the 2BN cross–section is a 2-dimensional non-factorized distribution anacceptance-rejection technique was the adopted. For the 2BN distribution,two functions g 1 (k) and g 2 (θ) were defined:such that:g 1 (k) = k −b g 2 (θ) =θ1 + cθ 2 (11.11)Ag 1 (k)g 2 (θ) ≥dσ(11.12)dkdθwhere A is a global constant to be completed. Both functions have an analyticalintegral G and an analytical inverse G −1 . The b parameter of g 1 (k) wasempirically tuned and set to 1.2. For positive θ values, g 2 (θ) has a maximum. c parameter controls the function global shape and it was used toat1√(c)tune g 2 (θ) according to the electron kinetic energy.169

To generate photon energy k according to g 1 and θ according to g 2 the inversetransformmethod was used. The integration of these functions givesG 1 = C 1∫ kmaxk mink ′−b dk ′ = C 1k 1−b − k 1−bmin1 − b(11.13)G 2 = C 2∫ θ0θ ′log(1 + cθ 2 )1 + cθ ′2 dθ′ = C 22c(11.14)where C 1 and C 2 are two global constants chosen to normalize the integralin the overall range to the unit. The photon momentum k will range froma minimum cut value k min (required to avoid infrared divergence) to a maximumvalue equal to the electron kinetic energy E k , while the polar angleranges from 0 to π, resulting for C 1 and C 2 :C 1 = 1 − bEk1−bk and θ are then sampled according to:k =[ 1 − bC 1ξ 1 + k 1−bmin]C 2 =2clog(1 + cπ 2 )√√exp ( 2cξ 2θ =2cC 1)(11.15)(11.16)where ξ 1 and ξ 2 are uniformly sampled in the interval (0,1). The event isaccepted if:uAg 1 (k)g 2 (θ) ≤dσ(11.17)dkdθwhere u is a random number with uniform distribution in (0,1). The A andc parameters were computed in a logarithmic grid, ranging from 1 keV to 1.5MeV with 100 points per decade. Since the g 2 (θ) function has a maximumat θ = √ 1 c, the c parameter was computed using the relation c = 1θ max. At thepoint (k min , θ max ) where k min is the k cut value, the double differential crosssectionhas its maximum value, since it is monotonically decreasing in k andthus the global normalization parameter A is estimated from the relation:Ag 1 (k min )g 2 (θ max ) =( d 2 )σdkdθmax(11.18)where g 1 (k min )g 2 (θ max ) = k−b min2 √ . Since A and c can only be retrieved forca fixed number of electron kinetic energies there exists the possibility thatAg 1 (k min )g 2 (θ max ) ≤ ( )d 2 σfor a given E dkdθk. This is a small violation thatmax170

can be corrected introducing an additional multiplicative factor to the A parameter,which was empirically determined to be 1.04 for the entire energyrange.Comparisons between Tsai, 2BS and 2BN generatorsThe currently available generators can be used according to the user requiredprecision and timing requirements. Regarding the energy range, validationresults indicate that for lower energies (≤ 100 keV) there is a significantdeviation on the most probable emission angle between Tsai/2BS generatorsand the 2BN generator - Figure 11.1. The 2BN generator maintains howevera good agreement with Kissel data [7], derived from the work of Tseng andco-workers [8], and it should be used for energies between 1 keV and 100 keV[9]. As the electron kinetic energy increases, the different distributions tendto overlap and all generators present a good agreement with Kissel data.Figure 11.1: Comparison of polar angle distribution of bremsstrahlung photons(k/T = 0.5) for 10 keV (left) and 100 keV (middle) and 500 keV (right)electrons in silver, obtained with Tsai, 2BS and 2BN generatorIn figure 11.2 the sampling efficiency for the different generators are presented.The sampling generation efficiency was defined as the ratio betweenthe number of generated events and the total number of trials. As energiesincreases the sampling efficiency of the 2BN algorithm decreases from 0.65at 1 keV electron kinetic energy down to almost 0.35 at 1 MeV. For energiesup to 10 keV the 2BN sampling efficiency is superior or equivalent tothe one of the 2BS generator. These results are an indication that precisionsimulation of low energy bremsstrahlung can be obtained with little performancedegradation. For energies above 500 keV, Tsai generator can be used,171

etaining a good physics accuracy and a sampling efficiency superior to the2BS generator.Figure 11.2: Sampling efficiency for Tsai generator, 2BS and 2BN Koch andMotz generators.11.7.2 Status of the document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia05.12.2001 modified by Vladimir Ivanchenko13.05.2002 modified by Vladimir Ivanchenko24.11.2003 modified by Andreia Trindade, Pedro Rodrigues and Luis PeraltaBibliography[1] “Geant4 Low Energy Electromagnetic Models for Electrons and Photons”,J.Apostolakis et al., CERN-OPEN-99-034(1999), INFN/AE-99/18(1999)172

[2] “Tables and Graphs of Electron-Interaction Cross-Sections from 10 eVto 100 GeV Derived from the LLNL Evaluated Electron Data Library(EEDL), Z=1-100” S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400Vol.31[3] “GEANT, Detector Description and Simulation Tool”, CERN ApplicationSoftware Group, CERN Program Library Long Writeup W5013[4] “Pair production and bremsstrahlung of charged leptons”, Y. Tsai, Rev.Mod. Phys., Vol.46, 815(1974), Vol.49, 421(1977)[5] “Bremsstrahlung Cross-Section Formulas and Related Data”, H. W.Koch and J. W. Motz, Rev. Mod. Phys., Vol.31, 920(1959)[6] “Improved bremsstrahlung photon angular sampling in the EGS4code system”, A. F. Bielajew, R. Mohan and C.-S. Chui, ReportNRCC/PIRS-0203 (1989)[7] “Bremsstrahlung from electron collisions with neutral atoms”, L. Kissel,C. A. Quarls and R. H. Pratt, At. Data Nucl. Data Tables, Vol. 28,382(1983)[8] “Electron bremsstrahlung angular distributions in the 1-500 keV energyrange”, H. K. Tseng, R. H. Pratt and C. M. Lee , Phys. Rev. A, Vol.19, 187(1979)[9] “GEANT4 Applications and Developments for Medical Physics Experiments”,P. Rodrigues et al. IEEE 2003 NSS/MIC Conference Record173

11.8 Electron ionisationThe class G4LowEnergyIonisation calculates the continuous energy loss dueto electron ionisation and simulates δ-ray production by electrons. The deltaelectronproduction threshold for a given material, T c , is used to separatethe continuous and the discrete parts of the process. The energy loss of anelectron with the incident energy, T , is expressed via the sum over all atomicshells, s, and the integral over the energy, t, of delta-electrons:dEdx = ∑ s(σ s (T )∫ Tc0.1eV t dσ dt dt∫ Tmax0.1eVdσ)dt , (11.19)dtwhere T max = 0.5T is the maximum energy transfered to a δ-electron, σ s (T )is the total cross-section for the shell, s, at a given incident kinetic energy,T , and 0.1eV is the low energy limit of the EEDL data. The δ-electronproduction cross-section is a complimentary function:σ(T ) = ∑ s(σ s (T )∫ Tmax dσT dtc dt∫ Tmax0.1eVdσ)dt . (11.20)dtThe partial sub-shell cross-sections, σ s , are obtained from an interpolationof the evaluated cross-section data in the EEDL library [1], according to theformula (11.1) in Section 11.1.4.The probability of emission of a δ-electron with kinetic energy, t, froma sub-shell, s, of binding energy, B s , as the result of the interaction of anincoming electron with kinetic energy, T , is described by:dσdt = P (x) , withx = t + B s, (11.21)x 2T + B swhere the parameter x is varied from x min = (0.1eV + B s )/(T + B s ) to 0.5.The function, P (x), is parametrised differently in 3 regions of x: from x minto x 1 the linear interpolation with linear scale of 4 points is used; from x 1 tox 2 the linear interpolation with logarithmic scale of 16 points is used; fromx 2 to 0.5 the following interpolation is applied:P (x) = 1 − gx + (1 − g)x 2 +x21 − x ( 1− g) + A ∗ (0.5 − x)/x, (11.22)1 − xwhere A is a fit coefficient, g is expressed via the gamma factor of the incomingelectron:g = (2γ − 1)/γ 2 . (11.23)174

For the high energy case (x >> 1) the formula (11.22) is transformed to theMöller electron-electron scattering formula [2, 3].The value of the coefficient, A, for each element is obtained as a resultof the fit on the spectrum from the EEDL data for those energies whichare available in the database. The values of x 1 and x 2 are chosen for eachatomic shell according to the spectrum of δ-electrons in this shell. Note thatx 1 corresponds to the maximum of the spectrum, if the maximum does notcoincide with x min . The dependence of all 24 parameters on the incidentenergy, T , is evaluated from a logarithmic interpolation (11.1).The sampling of the final state proceeds in three steps. First a shell israndomly selected, then the energy of the delta-electron is sampled, finallythe angle of emission of the scattered electron and of the δ-ray is determinedby energy-momentum conservation taken into account electron motion onthe atomic orbit.The interaction leaves the atom in an excited state. The deexcitation ofthe atom is simulated as described in section 11.9. Sampling of the excitationsis carried out for both the continuous and the discrete parts of the process.11.8.1 Status of the document30.09.1999 created by Alessandra Forti07.02.2000 modified by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia05.12.2001 modified by Vladimir Ivanchenko13.05.2002 modified by Vladimir IvanchenkoBibliography[1] “Tables and Graphs of Electron-Interaction Cross-Sections from 10 eVto 100 GeV Derived from the LLNL Evaluated Electron Data Library(EEDL), Z=1-100” S.T.Perkins, D.E.Cullen, S.M.Seltzer, UCRL-50400Vol.31[2] Geant3 manual ,CERN Program Library Long Writeup W5013 (October1994).[3] H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.175

11.9 Atomic relaxationThe atomic relaxation can be triggered by other electromagnetic interactionssuch as the photoelectric effect or ionisation, which leave the atom in anexcited state.The Livermore Evaluation Atomic Data Library EADL [1] contains datato describe the relaxation of atoms back to neutrality after they are ionised.It is assumed that the binding energy of all subshells are the same forneutral ground state atoms as for ionised atoms [1].The data in EADL includes the radiative and non-radiative transitionprobabilities for each sub-shell of each element, for Z=1 to 100. The atomhas been ionised by a process that has caused an electron to be ejected froman atom, leaving a vacancy or “hole” in a given subshell. The EADL dataare then used to calculate the complete radiative and non-radiative spectrumof X-rays and electrons emitted as the atom relaxes back to neutrality.Non-radiative de-excitation can occur via the Auger effect (the initial andsecondary vacancies are in different shells) or Coster-Kronig effect (transitionswithin the same shell).11.9.1 FluorescenceThe simulation procedure for the fluorescence process is the following:1. If the vacancy subshell is not included in the data, a photon is emittedin a random direction in 4π with an energy equal to the correspondingbinding energy, and the procedure is terminated.2. If the vacancy subshell is included in the data, an outer subshell is randomlyselected taking into account the relative transition probabilitiesfor all possible outer subshells.3. In the case where the energy corresponding to the selected transition islarger than a user defined cut value (equal to zero by default), a photonparticle is created and emitted in a random direction in 4π, with anenergy equal to the transition energy.4. the procedure is repeated from step 1, for the new vacancy subshell.The final local energy deposit is the difference between the binding energyof the initial vacancy subshell and the sum of all transition energies whichwere taken by fluorescence photons. The atom is assumed to be initiallyionised with an electric charge of +1e.176

Sub-shell data are provided in the EADL data bank [1] for Z=1 through100. However, transition probabilities are only explicitly included for Z=6through 100, from the subshells of the K, L, M, N shells and some O subshells.For subshells O,P,Q: transition probabilities are negligible (of theorder of 0.1%) and smaller than the precision with which they are known.Therefore, for the time being, for Z=1 through 5, only a local energy depositcorresponding to the binding energy B of an electron in the ionised subshellis simulated. For subshells of the O, P, and Q shells, a photon is emittedwith that energy B.11.9.2 Auger processThe Auger effect is complimentary to fluorescence, hence the simulation processis the same as for the fluorescence, with the exception that two randomshells are selected, one for the transition electron that fills the original vacancy,and the other for selecting the shell generating the Auger electron.Subshell data are provided in the EADL data bank [1] for Z = 6 through100. Since in EADL no data for elements with Z < 5 are provided, Augereffects are only considered for 5 < Z < 100 and always due to the EADL datatables, only for those transitions which have a probabiliy to occur > 0.1% ofthe total non-radiative transition probability. EADL probability data usedare, however, normalized to one for Fluorescence + Auger.11.9.3 Status of the document08.02.2000 created by Véronique Lefébure08.03.2000 reviewed by Petteri Nieminen and Maria Grazia Pia05.06.2002 added Auger Effect description by Alfonso ManteroBibliography[1] ”Tables and Graphs of Atomic Subshell and Relaxation Data Derivedfrom the LLNL Evaluated Atomic Data Library (EADL), Z=1-100” S.T.Perkins, D.E.Cullen, M.H.Chen, J.H.Hubbell, J.Rathkopf,J.Scofield, UCRL-50400 Vol.30[2] ”A simple model of photon transport”, D.E. Cullen, Nucl. Instr. Meth.in Phys. Res. B 101(1995)499-510177

[3] ”A program to determine the radiation spectra due to a single atomicsubshellionisation by a particle or due to deexcitation or decay of radionuclides”,J. Stepanek, Comp. Phys. Comm. 106(1997)237-257178

11.10 Hadron and Ion IonisationThe class G4hLowEnergyIonisation calculates the continuous energy lossdue to ionisation and simulates the δ-ray production by charged hadrons orions. This represents an extension of the Geant4 physics models down to lowenergy [1, 2].11.10.1 Delta-ray productionIn Geant4, δ-rays are generated generally only above a threshold energy, T c ,the value of which depends on atomic parameters and the cut value, T cut ,calculated from the unique cut in range parameter for all charged particlesin all materials. The total cross-section for the production of a δ-ray electronof kinetic energy T > T c by a particle of kinetic energy E is:σ(E, T c ) =∫ TmaxT cdσ(E, T )dTdT with T c = min(max(I, T cut ), T max ) (11.24)where I is the mean excitation potential of the atom (the formulae of thischarter are precise if T ≫ I), T max is the maximum energy transferable tothe free electronT max =2m e c 2 (γ 2 − 1)1 + 2γ(m e /M) + (m e /M) 2 (11.25)with m e the electron mass, M the mass of the incident particle, and γ is therelativistic factor. For heavy charged particles the differential cross-sectionper atom can be written as [3, 4]:for spin 0for spin 1/2for spin 1dσ = dTdσ = dTdσdT =KZ Z2 hβ 2 T 2 [1 − β 2 TT max][KZZ2 h1 − β 2 T+ T 2 ]β 2 T 2 T max 2E 2KZZ2 h(11.26)[ ( )1 − β 2 T( 1 +T )+ T 2 (1 + T )]β 2 T 2 T max 3Q c 3E 2 2Q cwhere Z is the atomic number, Z h is the effective charge of the incidentparticle in units of positron charge, β is the relativistic velocity, and Q c =(Mc 2 ) 2 /m e c 2 . The factor K is expressed as K = 2πrem 2 e c 2 , where r e is theclassical electron radius. The integration of formula (11.24) gives the totalcross-section, which for particles with spin 0 and 1/2 are the following :(for spin 0 σ(Z, E, T c ) = KZ Z2 h 1 − τ + β 2 )τ ln τ(11.27)β 2 T c179

for spin 1/2 σ(Z, E, T c ) = KZ Z2 hβ 2 ( 1 − τ + β 2 τ ln τT c+ T )max − T c2E 2where τ = T c /T max .The average energy transfer ∆E δ of a particle with spin 0 to δ-electrons withT > T c can be expressed as:∆E δ = N elZ 2 hβ 2 (− ln τ − β 2 (1 − τ) ) (11.28)where N el is the electron density of the medium. Using (11.26) one finds thatthe correction to (11.28) for particles with spin 1/2 is (T 2 max − T 2 c )/4E2 . Thisvalue is very small for low energy and can be neglected. The same conclusioncan be drawn for particles with spin 1.The mean free path of the particle is tabulated during initialisation as afunction of the material and of the energy for all the charged hadrons andstatic ions. Note, that for low energy T c = T max , cross-section is zero andthe mean free path is set to infinity, compatible with the machine precision.11.10.2 Energy Loss of Fast HadronsThe energy lost in soft ionising collisions producing δ-rays below T c are includedin the continuous energy loss. The mean value of the energy loss isgiven by the restricted Bethe-Bloch formula [5, 3] :]dEZh2 = KN eldxT

There exists a variety of phenomenological approximations for parameters inthe Bethe-Bloch formula. In Geant4 the tabulation of the ionisation potentialfrom Ref.[6] is implemented for all the elements. For the density effect theformulation of Sternheimer [7] is used:x is a kinetic variable of the particle : x = log 10 (γβ) = ln(γ 2 β 2 )/4.606,and δ(x) is defined byfor x < x 0 : δ(x) = 0for x ∈ [x 0 , x 1 ] : δ(x) = 4.606x − C + a(x 1 − x) mfor x > x 1 : δ(x) = 4.606x − C(11.30)where the matter-dependent constants are calculated as follows:hν p = plasma energy of the medium = √ 4πn el remc 3 2 /α = √ 4πn el r e¯hcC = 1 + 2 ln(I/hν p )x a = C/4.606a = 4.606(x a − x 0 )/(x 1 − x 0 ) mm = 3.(11.31)For condensed media{for C ≤ 3.681 x0 = 0.2 xI < 100 eV1 = 2for C > 3.681 x 0 = 0.326C − 1.0 x 1 = 2{for C ≤ 5.215 x0 = 0.2 xI ≥ 100 eV1 = 3for C > 5.215 x 0 = 0.326C − 1.5 x 1 = 3and for gaseous mediafor C < 10. x 0 = 1.6 x 1 = 4for C ∈ [10.0, 10.5[ x 0 = 1.7 x 1 = 4for C ∈ [10.5, 11.0[ x 0 = 1.8 x 1 = 4for C ∈ [11.0, 11.5[ x 0 = 1.9 x 1 = 4for C ∈ [11.5, 12.25[ x 0 = 2. x 1 = 4for C ∈ [12.25, 13.804[ x 0 = 2. x 1 = 5for C ≥ 13.804 x 0 = 0.326C − 2.5 x 1 = 5.The semi-empirical formula due to Barkas, which is applicable to all materials,is used for the shell correction term[8]:C e (I, βγ) = a(I)(βγ) 2 + b(I)(βγ) 4 + c(I)(βγ) 6 (11.32)The functions a(I), b(I), c(I) can be found in the source code.This formula breaks down at low energies, and it only applies for βγ > 0.13181

(e.g. T > 7.9 MeV for a proton). For βγ ≤ 0.13 the shell correction term iscalculated as:ln(T/TC e (I, βγ)2l )∣ = C e (I, βγ = 0.13)βγ≤0.13 ln(7.9 MeV/T 2l )hence the correction becomes progressively smaller from T = 7.9 MeV toT = T 2l = 2 MeV.Since M ≫ m e , the ionisation loss does not depend on the hadron mass, buton its velocity. Therefore the energy loss of a charged hadron with kineticenergy, T , is the same as the energy loss of a proton with the same velocity.The corresponding kinetic energy of the proton T p isT proton = M protonMT. (11.33)At initialisation stage of Geant4 the dE/dx tables and range tables for allmaterials are calculated only for protons and antiprotons. During run timethe energy loss and the range of any hadron or ion are recalculated using thescaling relation (11.33).11.10.3 Barkas and Bloch effectsThe accuracy of the Bethe-Bloch stopping power formula (11.33) can beimproved if the higher order terms are taken into account:− dEdx = K Z2 hβ 2 (L 0 + Z h L 1 + Z 2 h L 2), (11.34)where L 1 is the Barkas term [9], describing the difference between ionisationof positively and negatively charged particles, and L 2 is the Bloch term.The Barkas effect for kinetic energy of protons or antiprotons greater than500keV can be described as [10]:L 1 = F (b/√ x)√Zx3, x = β2 c 2(, b = 0.8Z ) 1Zv02 6 1 + 6.02Z−1.19, (11.35)where v 0 is the Bohr velocity (corresponding to proton energy T p = 25keV ),and the function F is tabulated according to [10].The Bloch term [11] can be expressed in the following way:infZh 2 L 2 = −y 2 ∑j=11j(j 2 + y 2 ) , y =182Z h137β . (11.36)

Note, that for y ≪ 1 the simplified expression Z 2 h L 2 = −1.202y 2 can be used.Both the Barkas and Bloch terms break scaling of ionisation losses if theabsolute value of particle charge is different from unity, because the particlecharge Z h is not factorised in the formula (11.34). To take these termsinto account correction is made at each step of the simulation for the valueof dE/dx re-calculated from the proton or antiproton tables. There is thepossibility to switch off the calculation of these terms.11.10.4 Energy losses of slow positive hadronsAt low energies the total energy loss is usually described in terms of electronicstopping power S e = −dE/dx. For charged hadron with velocity β < 0.05(corresponding to 1 MeV for protons), formula (11.30) becomes inaccurate.In this case the velocity of the incident hadron is comparable to the velocityof atomic electrons. At very low energies, when β < 0.01, the model of afree electron gas [12] predicts the stopping power to be proportional to thehadron velocity, but it is not as accurate as the Bethe-Bloch formalism. Theintermediate region 0.01 < β < 0.05 is not covered by precise theories. Inthis energy interval the Bragg peak of ionisation loss occurs.To simulate slow proton energy loss the following parametrisation fromthe review [13] was implemented:S e = A 1 E 1/2 , 1 keV < T p < 10 keV,S e =S low S high,S low + S high10 keV < T p < 1 MeV,S low = A 2 E 0.45 ,S high = A (3E ln 1 + A )4E + A 5E ,S e = A [6ln A 7β 2]4∑β 2 1 − β − 2 β2 − A i+8 (ln E) i , 1 MeV < T p < 100 MeV,i=0(11.37)where S e is the stopping power in [eV/10 15 atoms/cm 2 ], E = T p /M p [keV/amu],A i are twelve fitting parameters found individually for each atom for atomicnumbers from 1 to 92. This parametrisation is used in the interval of protonkinetic energy:T 1 < T p < T 2 , (11.38)where T 1 = 1 keV is the minimal kinetic energy of protons in the tables ofRef.[13], T 2 is an arbitrary value between 2 MeV and 100 MeV, since in thisrange both the parametrisation (11.37) and the Bethe-Bloch formula (11.33)183

Table 11.1: The list of parameterisations available.Name Particle SourceZiegler1977p proton J.F. Ziegler parameterisation [13]Ziegler1977He He 4 J.F. Ziegler parameterisation [15]Ziegler1985p proton TRIM’85 parameterisation [16]ICRU R49p proton ICRU parameterisation [14]ICRU R49He He 4 ICRU parameterisation [14]have practically the same accuracy and are close to each other. Currentlythe value T 2 = 2 MeV is chosen.To avoid problems in computation and to provide a continuous dE/dxfunction, the factor(F = 1 + B T )2(11.39)T pis multiplied by the value of dE/dx for T p > T 2 . The parameter B is determinedfor each element of the material in order to provide continuity atT p = T 2 . The value of B for all atoms is less than 0.01. For the simulationof the stopping power of very slow protons the model of a free electron gas[12] is used:√S e = A T p , T p < T 1 . (11.40)The parameter A is defined for each atom by requiring the stopping powerto be continuous at T p = T 1 . Currently the value used is T 1 = 1 keV .Note that if the cut kinetic energy is small (T c < T max ), then the averageenergy deposit giving rise to δ-electron production (11.28) is subtracted fromthe value of the stopping power S e , which is calculated by formula (11.37).Alternative parametrisations of proton energy loss are also available withinGeant4 (Table 11.1). The parameterisation formulae in Ref.[14] are the sameas in Ref.([13]) for the kinetic energy of protons T p < 1 MeV , but the valuesof the parameters are different. The type of parameterisation is optional andcan be chosen by the user separately for each particle at the initialisationstage of Geant4.11.10.5 Energy loss of alpha particlesThe accuracy of the data for the ionisation losses of α-particles in all elements[14, 15] is comparable to the accuracy of the data for proton energy184

loss [13, 14]. In the GEANT4 energy loss model for α-particles the Bethe-Bloch formula is used for kinetic energy T > T 2 , where T 2 is the arbitraryparameter, currently set to 8 MeV . For lower energies a parameterisation isperformed. In the energy range of the Bragg peak, 1 keV < T < 10 MeV ,the parameterisation is:S e =S low S highS low + S high,S low = A 1 T A 2,S high = A (3T ln 1 + A )4T + A 5T ,(11.41)where S e is the electronic stopping power in [eV/10 15 atoms/cm 2 ], T is thekinetic energy of α-particles in MeV , A i are the five fitting parameters fittedindividually for each atom for atomic numbers from 1 to 92.For higher energies T > 10 MeV , another parametrisation [15] is appliedS e = exp ( A 6 + A 7 E + A 8 E 2 + A 9 E 3) , E = ln(1/T ). (11.42)To ensure a continuous dE/dx function from the energy range of the Bethe-Bloch formula to the energy range of the parameterisation, the factor(F = 1 + B T )2T(11.43)is multiplied by the value of S e as predicted by the Bethe-Bloch formula forT > T 2 . The parameter B is determined for each element of the material inorder to ensure continuity at T p = T 2 . The value of B for different atoms isusually less than 0.01.For kinetic energies of α-particles T < 1 keV the model of free electrongas [12] is usedS e = A √ T , (11.44)The parameter A is defined for each atom by requiring the stopping powerto be continuous at T = 1 keV .11.10.6 Effective charge of ionsFor hadrons or ions the scaling relation can be written asS ei (T ) = Z 2 eff · S ep(T p ), (11.45)185

where S ei is the ion stopping power, S ep is the proton stopping power at theenergy scaled according (11.33), and Z eff is effective charge of the particle,which has to be used in all expressions in place of Z h . For fast particles itis equal to the particle charge Z h , but for slow ions it differs significantlybecause a slow ion picks up electrons from the medium. The ion effectivecharge is expressed via the ion charge Z h and the fractional effective chargeof ion γ i :Z eff = γ i Z h . (11.46)For helium ions fractional effective charge is parameterised for all elementswith good accuracy [16] according to:⎛ ⎡⎤⎞5∑((γ He ) 2 = ⎝1 − exp ⎣− C j Q j ⎦⎠1 + 7 + 0.05Z2exp(−(7.6 − Q) )) 2 ,j=01000Q = max(0, ln T p ), (11.47)where the coefficients C j are the same for all elements, and the helium ionkinetic energy is in keV/amu.The following expression is used for heavy ions [17]:γ i =(q + 1 − q2( v0v F) 2ln(1 + Λ2 )) ( 1 + (0.18 + 0.0015Z) exp(−(7.6 − Q)2 )(11.48)where q is the fractional average charge of the ion, v 0 is the Bohr velocity,v F is the Fermi velocity of the electrons in the target medium, and Λ is theterm taking into account the screening effect. In Ref. [17], Λ is estimated tobe:Λ = 10 v Fv 0(1 − q) 2/3Z 1/3i (6 + q) . (11.49)The Fermi velocity of the medium is of the same order as the Bohr velocity,and its exact value depends on the detailed electronic structure of themedium. Experimental data on the Fermi velocity are taken from Ref.[16].The expression for the fractional average charge of the ion is the following:q = [1 − exp(0.803y 0.3 − 1.3167y 0.6 − 0.38157y − 0.008983y 2 )], (11.50)Z 2 i),where y is a parameter that depends on the ion velocity v iy =v ( )i1 + v2 F. (11.51)v 0 Z 2/3 5vi2186

The parametrisation described in this chapter is only valid if the reducedkinetic energy of the ion is higher than the lower limit of the energy:T p > max(3.25 keV,)25 keV. (11.52)Z 2/3If the ion energy is lower, then the free electron gas model (11.44) is used tocalculate the stopping power.11.10.7 Energy losses of slow negative particlesAt low energies, e.g. below a few MeV for protons/antiprotons, the Bethe-Bloch formula is no longer accurate in describing the energy loss of chargedhadrons and higher Z terms should be taken in account. Odd terms in Zlead to a significant difference between energy loss of positively and negativelycharged particles. The energy loss of negative hadrons is scaled from that ofantiprotons. The antiproton energy loss is calculated in the following way:• if the material is elemental, the quantum harmonic oscillator model isused, as described in [18] and references therein. The lower limit ofapplicability of the model is chosen for all materials at 50 keV . Belowthis value stopping power is set to constant equal to the dE/dx at50 keV .• if the material is not elemental, the energy loss is calculated downto 500 keV using the Barkas correction (11.43) and at lower energiesfitting the proton energy loss curve.11.10.8 Energy losses of hadrons in compoundsTo obtain energy losses in a mixture or compound, the absorber can bethought of as made up of thin layers of pure elements with weights proportionalto the electron density of the element in the absorber (Bragg’s rule):dEdx = ∑ i( ) dE, (11.53)dxiwhere the sum is taken over all elements of the absorber, i is the number ofthe element, ( dE ) dx i is energy loss in the pure i-th element.Bragg’s rule is very accurate for relativistic particles when the interactionof electrons with a nucleus is negligible. But at low energies the accuracy ofBragg’s rule is limited because the energy loss to the electrons in any material187

depends on the detailed orbital and excitation structure of the material. Inthe description of Geant4 materials there is a special attribute: the chemicalformula. It is used in the following way:• if the data on the stopping power for a compound as a function ofthe proton kinetic energy is available (Table 11.2), then the directparametrisation of the data for this material is performed;• if the data on the stopping power for a compound is available for onlyone incident energy (Table 11.3), then the computation is performedbased on Bragg’s rule and the chemical factor for the compound istaken into account;• if there are no data for the compound, the computation is performedbased on Bragg’s rule.In the review [19] the parametrisation stopping power data are presented asS e (T p ) = S Bragg (T p )[1 +( )]f(T p ) Sexp (125 keV )f(125 keV ) S Bragg (125 keV ) − 1 , (11.54)where S exp (125 keV ) is the experimental value of the energy loss for thecompound for 125 keV protons or the reduced experimental value for Heions, S Bragg (T p ) is a value of energy loss calculated according to Bragg’srule, and f(T p ) is a universal function, which describes the disappearance ofdeviations from Bragg’s rule for higher kinetic energies according to:f(T p ) =11 + exp [ β(T1.48( p)− , (11.55)β(25 keV ) 7.0)]where β(T p ) is the relative velocity of the proton with kinetic energy T p .11.10.9 Nuclear stopping powersLow energy ions transfer their energy not only to electrons of a mediumbut also to the nuclei of the medium due to the elastic Coulomb collisions.This contribution to the energy loss is called nuclear stopping power. It isparametrised [15, 16, 14] using a universal parameterisation for reduced ionenergy, ɛ, which depends on ion parameters and on the charge, Z t , and themass, M t , of the target nucleus:ɛ =32.536T M tZ eff Z t (M + M t ) √ Zeff 0.23 + . (11.56)Z0.23 t188

Table 11.2: The list of chemical formulae of compounds for which parametrisationof stopping power as a function of kinetic energy is in Ref.[14].Number Chemical formula1. AlO2. C 2O3. CH 44. (C 2H 4) N-Polyethylene5. (C 2H 4) N-Polypropylene6. (C 8H 8) N7. C 3H 88. SiO 29. H 2O10. H 2O-Gas11. GraphiteThe universal reduced nuclear stopping power, s n , is determined by J. Molierein the framework of Thomas-Fermi potential [20]. The corresponding tabulationfrom Ref.[14] is implemented. To transform the value of nuclear stoppingpower from reduced units to [eV/10 15 atoms/cm 2 ] the following formulais used:8.462Z i Z t M iS n = s n√(M i + M t )Zi 0.23 + Zt0.23. (11.57)The effect of nuclear stopping power is very small at high energies, but it isof the same order of magnitude as electronic stopping power for very slowions (e.g. for protons, T p < 1keV ).11.10.10 Fluctuations of energy losses of hadronsThe total continuous energy loss of charged particles is a stochastic quantitywith a distribution described in terms of a straggling function. The stragglingis partially taken into account by the simulation of energy loss by theproduction of δ-electrons with energy T > T c . However, continuous energyloss also has fluctuations. Hence in the current GEANT4 implementationtwo different models of fluctuations are applied depending on the value ofthe parameter κ which is the lower limit of the number of interactions of theparticle in the step. The default value chosen is κ = 10. To select a modelfor thick absorbers the following boundary conditions are used:∆E > T c κ) or T c < Iκ, (11.58)189

Table 11.3: The list of chemical formulae of compounds for which the chemicalfactor is calculated from the data of Ref.[19].Number Chemical formula Number Chemical formula1. H 2O 28. C 2H 62. C 2H 4O 29. C 2F 63. C 3H 6O 30. C 2H 6O4. C 2H 2 31. C 3H 6O5. C H 3OH 32. C 4H 10O6. C 2H 5OH 33. C 2H 47. C 3H 7OH 34. C 2H 4O8. C 3H 4 35. C 2H 4S9. NH 3 36. SH 210. C 14H 10 37. CH 411. C 6H 6 38. CCLF 312. C 4H 10 39. CCl 2F 213. C 4H 6 40. CHCl 2F14. C 4H 8O 41. (CH 3) 2S15. CCl 4 42. N 2O16. CF 4 43. C 5H 10O17. C 6H 8 44. C 8H 618. C 6H 12 45. (CH 2) N19. C 6H 10O 46. (C 3H 6) N20. C 6H 10 47. (C 8H 8) N21. C 8H 16 48. C 3H 822. C 5H 10 49. C 3H 6-Propylene23. C 5H 8 50. C 3H 6O24. C 3H 6-Cyclopropane 51. C 3H 6S25. C 2H 4F 2 52. C 4H 4S26. C 2H 2F 2 53. C 7H 827. C 4H 8O 2190

where ∆E is the mean continuous energy loss in a track segment of lengths, T c is the cut kinetic energy of δ-electrons, and I is the average ionisationpotential of the atom.For long path lengths the straggling function approaches the Gaussiandistribution with Bohr’s variance [14]:Ω 2 = KN elZ 2 hβ 2 T csf(1 − β22), (11.59)where f is a screening factor, which is equal to unity for fast particles, whereasfor slow positively charged ions with β 2 < 3Z(v 0 /c) 2 f = a + b/Z 2 eff , whereparameters a and b are parametrised for all atoms [22, 23].For short path lengths, when the condition 11.58 is not satisfied, themodel described in the charter 7.2 is applied.11.10.11 SamplingAt each step for a charged hadron or ion in an absorber, the step limit iscalculated using range tables for protons or antiprotons depending on the particlecharge. If the reduced particle energy T p < T 2 the step limit is forced tobe not longer than αR(T 2 ), where R(T 2 ) is the range of the particle with thereduced energy T 2 , α is an arbitrary coefficient, which is currently set to 0.05in order to provide at least 20 steps for particles in the Bragg peak energyrange. In each step continuous energy loss of the particle is calculated usingloss tables for protons or antiprotons for T p > T 2 . For lower energies, continuousenergy loss is calculated using the effective charge approach, chemicalfactors, and nuclear stopping powers. If the step of the particle is limitedby the ionisation process the sampling of δ-electron production is performed.(A short overview of the method is given in Chapter 2.)Apart from the normalisation, the cross-section (11.26) can be written as :with :dσdT ∼ f(T ) g(T ) with T ∈ [T c, T max ] (11.60)f(T ) =( 1T c− 1T max) 1T 2g(T ) = 1 − β 2 TT max+ S(T ),where S(T ) is a spin dependent term (11.26). For a spin-0 particle this termis zero, for a spin- 1 2 particle S(T ) = T 2 /2E 2 , whilst for spin-1 the expressionis more complicated.The energy, T , is sampled by :191

1. Sample T from f(T ).2. Calculate the rejection function g(T ) and accept the sampled T with aprobability of g(T ).After the successful sampling of the energy, the polar angles of the emittedelectron are generated with respect to the direction of the incident particle.The azimuthal angle, φ, is generated isotropically; the polar angle θ is calculatedfrom the energy momentum conservation. This information is usedto calculate the energy and momentum of both particles and to transformthem into the global coordinate system.11.10.12 PIXEPIXE is simulated by calculating cross-sections according to [24] and [25]to identify the primary ionised shell, and generating the subsequent atomicrelaxation as described in 11.9. Sampling of excitations is carried out forboth the continuous and the discrete parts of the process.11.10.13 Status of this document21.11.2000 Created by V.Ivanchenko30.05.2001 Modified by V.Ivanchenko23.11.2001 Modified by M.G. Pia to add PIXE section.19.01.2002 Minor corrections (mma)13.05.2002 Minor corrections (V.Ivanchenko)28.08.2002 Minor corrections (V.Ivanchenko)Bibliography[1] V.N. Ivanchenko et al., GEANT4 Simulation of Energy Losses of SlowHadrons, CERN-99-121, INFN/AE-99/20, (September 1999).[2] S. Giani et al., GEANT4 Simulation of Energy Losses of Ions, CERN-99-300, INFN/AE-99/21, (November 1999).[3] D.E. Groom et al., Eur. Phys. Jour. C15 (2000) 1.[4] B. Rossi, High Energy Particles, Pretice-Hall, Inc., Englewood Cliffs,NJ, 1952.[5] H. Bethe, Ann. Phys. 5 (1930) 325.192

[6] (A. Allisy et al), Stopping Powers for Electrons and Positrons, ICRUReport 37, 1984.[7] R.M. Sternheimer. Phys.Rev. B3 (1971) 3681.[8] W.H. Barkas. Technical Report 10292,UCRL, August 1962.[9] W.H. Barkas, W. Birnbaum, F.M. Smith, Phys. Rev. 101 (1956) 778.[10] J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B5 (1972) 1.[11] F. Bloch, Ann. Phys. 16 (1933) 285.[12] J. Linhard and A. Winther, Mat. Fys. Medd. Dan. Vid. Selsk. 34, No10 (1963).[13] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions inMatter. Vol.3, Pergamon Press, 1977.[14] ICRU (A. Allisy et al), Stopping Powers and Ranges for Protons andAlpha Particles, ICRU Report 49, 1993.[15] J.F. Ziegler, The Stopping and Ranges of Ions in Matter. Vol.4, PergamonPress, 1977.[16] J.F. Ziegler, J.P. Biersack, U . Littmark, The Stopping and Ranges ofIons in Solids. Vol.1, Pergamon Press, 1985.[17] W. Brandt and M. Kitagawa, Phys. Rev. B25 (1982) 5631.[18] P. Sigmund, Nucl. Instr. and Meth. B85 (1994) 541.[19] J.F. Ziegler and J.M. Manoyan, Nucl. Instr. and Meth. B35 (1988) 215.[20] G. Moliere, Theorie der Streuung schneller geladener Teilchen I;Einzelstreuungam abbgeschirmten Coulomb-Feld, Z. f. Naturforsch, A2(1947) 133.[21] GEANT3 manual, CERN Program Library Long Writeup W5013 (October1994).[22] Q. Yang, D.J. O’Connor, Z. Wang, Nucl. Instr. and Meth. B61 (1991)149.[23] W.K. Chu, in: Ion Beam Handbook for Material Analysis, edt.J.W. Mayer and E. Rimini, Academic Press, NY, 1977.193

[24] M. Gryzinski, Phys. Rev. A 135 (1965) 305.[25] M. Gryzinski, Phys. Rev. A 138 (1965) 322.194

11.11 Penelope physics11.11.1 IntroductionA new set of physics processes for photons, electrons and positrons is implementedin Geant4: it includes Compton scattering, photoelectric effect,Rayleigh scattering, gamma conversion, bremsstrahlung, ionization (to bereleased) and positron annihilation (to be released). These processes are theGeant4 implementation of the physics models developed for the PENELOPEcode (PENetration and Energy LOss of Positrons and Electrons), version2001, that are described in detail in Ref. [1]. The Penelope models havebeen specifically developed for Monte Carlo simulation and great care wasgiven to the low energy description (i.e. atomic effects, etc.). Hence, theseimplementations provide reliable results for energies down to a few hundredeV and can be used up to ∼1 GeV [1, 2]. For this reason, they may beused in Geant4 as an alternative to the Low Energy processes. For the samephysics processes, the user now has more alternative descriptions from whichto choose, including the cross section calculation and the final state sampling.11.11.2 Compton scatteringTotal cross sectionThe total cross section of the Compton scattering process is determined froman analytical parameterization. For γ energy E greater than 5 MeV, the usualKlein-Nishina formula is used for σ(E). For E < 5 MeV a more accurateparameterization is used, which takes into account atomic binding effectsand Doppler broadening [3]:σ(E) = 2π∫ 1−1∑shellswhere:r e = classical radius of the electron;m e = mass of the electron;θ = scattering angle;E C = Compton energyre2 EC22 E (E C2 E + E − sin 2 θ) ·E Cf i Θ(E − U i )n i (p maxz ) d(cos θ) (11.61)=E1 + Em ec(1 − cos θ)2195

f i = number of electrons in the i-th atomic shell;U i = ionisation energy of the i-th atomic shell;Θ = Heaviside step function;p maxz = highest possible value of p z (projection of the initial momentum ofthe electron in the direction of the scattering angle)Finally,= E(E − U i)(1 − cos θ) − m e c 2 U i√.c 2E(E − U i )(1 − cos θ) + Ui2n i (x) =12 e[ 1 2 −( 1 2 −√ 2J i0 x) 2 ]if x < 01 − 1 2 e[ 1 2 −( 1 2 +√ 2J i0 x) 2 ]if x > 0(11.62)where J i0 is the value of the p z -distribution profile J i (p z ) for the i-th atomicshell calculated in p z = 0. The values of J i0 for the different shells of thedifferent elements are tabulated from the Hartree-Fock atomic orbitals ofRef. [4].The integration of Eq.(11.61) is performed numerically using the 20-pointGaussian method. For this reason, the initialization of the Penelope Comptonprocess is somewhat slower than the Low Energy process.Sampling of the final stateThe polar deflection cos θ is sampled from the probability density functionP (cos θ) = r2 e2E 2 CE 2 ( E CE + E E C− sin 2 θ ) ∑shellsf i Θ(E − U i )n i (p maxz ) (11.63)(see Ref. [1] for details on the sampling algorithm). Once the direction ofthe emerging photon has been set, the active electron shell i is selected withrelative probability equal to Z i Θ(E − U i )n i [p maxz (E, θ)]. A random value ofp z is generated from the analytical Compton profile [4]. The energy of theemerging photon isE ′ = Eτ1 − τtwhere[ p √ z(1 − τt cos θ) + (1 − τt cos θ)|p z |2 − (1 − tτ 2 )(1 − t) ] ,(11.64)t = ( p ) z 2and τ = E Cm e cE . (11.65)The azimuthal scattering angle φ of the photon is sampled uniformly inthe interval (0,2π). It is assumed that the Compton electron is emitted with196

energy E e = E −E ′ −U i , with polar angle θ e and azimuthal angle φ e = φ+π,relative to the direction of the incident photon. In this case cos θ e is given bycos θ e =E − E ′ cos θ√E2 + E ′ 2− 2EE ′ cos θ . (11.66)Since the active electron shell is known, characteristic x-rays and electronsemitted in the de-excitation of the ionized atom can also be followed. The deexcitationis simulated as described in section 11.9. For further details see [1].11.11.3 Rayleigh scatteringTotal cross sectionThe total cross section of the Rayleigh scattering process is determined froman analytical parameterization. The atomic cross section for coherent scatteringis given approximately by [5]σ(E) = πr 2 e∫ 1−11 + cos 2 θ[F (q, Z)] 2 d cos θ, (11.67)2where F (q, Z) is the atomic form factor, Z is the atomic number and q is themagnitude of the momentum transfer, i.e.q = 2 E csin ( θ ). (11.68)2In the numerical calculation the following analytical approximations are usedfor the form factor:F (q, Z) = f(x, Z) =wherewithormax[f(x, Z), F K (x, Z)] if Z > 10 and f(x, Z) < 2(11.69)Z 1+a 1x 2 +a 2 x 3 +a 3 x 4(1+a 4 x 2 +a 5 x 4 ) 2F K (x, Z) =sin(2b arctan Q)bQ(1 + Q 2 ) b , (11.70)x = 20.6074 qm e c , Q = q2m e ca , b = √ 1 − a 2 , a = α ( Z − 5 ), (11.71)16where α is the fine-structure constant. The function F K (x, Z) is the contributionto the atomic form factor due to the two K-shell electrons (see [6]).197

The parameters of expression f(x, Z) have been determined in Ref. [6] forZ=1 to 92 by numerically fitting the atomic form factors tabulated in Ref.[7]. The integration of Eq.(11.67) is performed numerically using the 20-pointGaussian method. For this reason the initialization of the Penelope Rayleighprocess is somewhat slower than the Low Energy process.Sampling of the final stateThe angular deflection cos θ of the scattered photon is sampled from theprobability distribution functionP (cos θ) = 1 + cos2 θ[F (q, Z)] 2 . (11.72)2For details on the sampling algorithm (which is quite heavy from the computationalpoint of view) see Ref. [1]. The azimuthal scattering angle φ ofthe photon is sampled uniformly in the interval (0,2π).11.11.4 Gamma conversionTotal cross sectionThe total cross section of the γ conversion process is determined from thedata [8], as described in section 11.1.4.Sampling of the final stateThe energies E − and E + of the secondary electron and positron are sampledusing the Bethe-Heitler cross section with the Coulomb correction, using thesemiempirical model of Ref. [6]. Ifɛ = E − + m e c 2Eis the fraction of the γ energy E which is taken away from the electron,(11.73)κ =Em e c 2 and a = αZ, (11.74)the differential cross section, which includes a low-energy correction and ahigh-energy radiative correction, isdσdɛ= re 2 a(Z + η)C 2[ ( 1r 23 2 − ɛ) 2φ1 (ɛ) + φ 2 (ɛ) ] , (11.75)198

where:φ 1 (ɛ) = 7 3 − 2 ln(1 + b2 ) − 6b arctan(b −1 )and−b 2 [4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 )]+4 ln(Rm e c/¯h) − 4f C (Z) + F 0 (κ, Z) (11.76)withφ 2 (ɛ) = 11 6 − 2 ln(1 + b2 ) − 3b arctan(b −1 )+ 1 2 b2 [4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 )]+4 ln(Rm e c/¯h) − 4f C (Z) + F 0 (κ, Z), (11.77)b = Rm ec¯h1 12κ ɛ(1 − ɛ) . (11.78)In this case R is the screening radius for the atom Z (tabulated in [10] forZ=1 to 92) and η is the contribution of pair production in the electron field(rather than in the nuclear field). The parameter η is approximated aswhereη = η ∞ (1 − e −v ), (11.79)v = (0.2840 − 0.1909a) ln(4/κ) + (0.1095 + 0.2206a) ln 2 (4/κ)+(0.02888 − 0.04269a) ln 3 (4/κ)+(0.002527 + 0.002623) ln 4 (4/κ) (11.80)and η ∞ is the contribution for the atom Z in the high-energy limit and istabulated for Z=1 to 92 in Ref. [10]. In the Eq.(11.75), the function f C (Z)is the high-energy Coulomb correction of Ref. [9], given byf C (Z) = a 2 [(1 + a 2 ) −1 + 0.202059 − 0.03693a 2 + 0.00835a 4−0.00201a 6 + 0.00049a 8 − 0.00012a 10 + 0.00003a 12 ]; (11.81)C r = 1.0093 is the high-energy limit of Mork and Olsen’s radiative correction(see Ref. [10]); F 0 (κ, Z) is a Coulomb-like correction function, which has beenanalytically approximated as [1]F 0 (κ, Z) = (−0.1774 − 12.10a + 11.18a 2 )(2/κ) 1/2+(8.523 + 73.26a − 44.41a 2 )(2/κ)−(13.52 + 121.1a − 96.41a 2 )(2/κ) 3/2+(8.946 + 62.05a − 63.41a 2 )(2/κ) 2 . (11.82)199

The kinetic energy E + of the secondary positron is obtained asE + = E − E − − 2m e c 2 . (11.83)The polar angles θ − and θ + of the directions of movement of the electron andthe positron, relative to the direction of the incident photon, are sampledfrom the leading term of the expression obtained from high-energy theory(see Ref. [11])p(cos θ ± ) = a(1 − β ± cos θ ± ) −2 , (11.84)where a is the a normalization constant and β ± is the particle velocity inunits of the speed of light. As the directions of the produced particles andof the incident photon are not necessarily coplanar, the azimuthal angles φ −and φ + of the electron and of the positron are sampled independently anduniformly in the interval (0,2π).11.11.5 Photoelectric effectTotal cross sectionThe total photoelectric cross section at given photon energy E is calculatedfrom the data [12], as described in section 11.1.4.Sampling of the final stateThe incident photon is absorbed and one electron is emitted in the samedirection as the primary photon. The subshell from which the electron isemitted is randomly selected according to the relative cross sections of subshells,determined at the energy E by interpolation of the data of Ref. [11].The electron kinetic energy is the difference between the incident photonenergy and the binding energy of the electron before the interaction in thesampled shell. The interaction leaves the atom in an excited state; the subsequentde-excitation is simulated as described in section 11.9.11.11.6 BremsstrahlungIntroductionThe class G4PenelopeBremsstrahlung calculates the continuous energy lossdue to soft γ emission and simulates the photon production by electrons andpositrons. As usual, the gamma production threshold T c for a given materialis used to separate the continuous and the discrete parts of the process.200

ElectronsThe total cross sections are calculated from the data [14], as described insections 11.1.4 and 11.7.The energy distributiondσ (E), i.e. the probability of the emission of adWphoton with energy W given an incident electron of kinetic energy E, isgenerated according to the formuladσ F (κ)(E) =dW κ , κ = W E . (11.85)The functions F (κ) describing the energy spectra of the outgoing photons aretaken from Ref. [13]. For each element Z from 1 to 92, 32 points in κ, rangingfrom 10 −12 to 1, are used for the linear interpolation of this function. F (κ)is normalized using the condition F (10 −12 ) = 1. The energy distributionof the emitted photons is available in the library [13] for 57 energies of theincident electron between 1 keV and 100 GeV. For other primary energies,logarithmic interpolation is used to obtain the values of the function F (κ).The direction of the emitted bremsstrahlung photon is determined by thepolar angle θ and the azimuthal angle φ. For isotropic media, with randomlyoriented atoms, the bremsstrahlung differential cross section is independentof φ and can be expressed asd 2 σdW d cos θ =dσ p(Z, E, κ; cos θ). (11.86)dWNumerical values of the “shape function” p(Z, E, κ; cos θ), calculated bypartial-wave methods, have been published in Ref. [15] for the followingbenchmark cases: Z= 2, 8, 13, 47, 79 and 92; E= 1, 5, 10, 50, 100 and 500keV; κ= 0, 0.6, 0.8 and 0.95. It was found in Ref. [1] that the benchmarkpartial-wave shape function of Ref. [15] can be closely approximated by theanalytical form (obtained in the Lorentz-dipole approximation)p(cos θ) = A 3 [ ( cos θ − β ′ ) 2 ] 1 − β ′ 21 +8 1 − β ′ cos θ (1 − β ′ cos θ) 2+(1 − A) 3 [ ( cos θ − β ′1 −4 1 − β ′ cos θ m) 2 ] 1 − β ′ 2(1 − β ′ cos θ) , (11.87)2with β ′ = β(1 + B), if one considers A and B as adjustable parameters. Theparameters A and B have been determined, by least squares fitting, for the144 conbinations of atomic numbers, electron energies and reduced photonenergies corresponding to the benchmark shape functions tabulated in [15].The quantities ln(AZβ) and Bβ vary smoothly with Z, β and κ and can201

e obtained by cubic spline interpolation of their values for the benchmarkcases. This permits the fast evaluation of the shape function p(Z, E, κ; cos θ)for any combination of Z, β and κ.The stopping power dE due to soft bremsstrahlung is calculated by interpolatingin E and κ the numerical data of scaled cross sections of Ref. [16]. Thedxenergy and the direction of the outgoing electron are determined by usingenergy-momentum balance.PositronsThe radiative differential cross section dσ+ (E) for positrons reduces to thatdWfor electrons in the high-energy limit, but is smaller for intermediate and lowenergies. Owing to the lack of more accurate calculations, the differentialcross section for positrons is obtained by multiplying the electron differentialcross section dσ− (E) by a κ−indendent factor, i.e.dWdσ +dW= F p(Z, E) dσ−dW . (11.88)The factor F p (Z, E) is set equal to the ratio of the radiative stopping powersfor positrons and electrons, which has been calculated in Ref. [17]. For theactual calculation, the following analytical approximation is used:F p (Z, E) = 1 − exp(−1.2359 · 10 −1 t + 6.1274 · 10 −2 t 2 − 3.1516 · 10 −2 t 3where+7.7446 · 10 −3 t 4 − 1.0595 · 10 −3 t 5 + 7.0568 · 10 −5 t 6t = ln ( 1 + 106Z 2−1.8080 · 10 −6 t(11.89)7 ),Em e c 2 ). (11.90)Because the factor F p (Z, E) is independent on κ, the energy distribution ofthe secondary γ’s has the same shape as electron bremsstrahlung. Similarly,owing to the lack of numerical data for positrons, it is assumed that the shapeof the angular distribution p(Z, E, κ; cos θ) of the bremsstrahlung photons forpositrons is the same as for the electrons.The energy and direction of the outgoing positron are determined fromenergy-momentum balance.11.11.7 IonisationThe G4PenelopeIonisation class calculates the continuous energy loss dueto electron and positron ionisation and simulates the δ-ray production by202

electrons and positrons. The electron production threshold T c for a givenmaterial is used to separate the continuous and the discrete parts of theprocess.The simulation of inelastic collisions of electrons and positrons is performedon the basis of a Generalized Oscillation Strength (GOS) model (see Ref. [1]for a complete description). It is assumed that GOS splits into contributionsfrom the different atomic electron shells.ElectronsThe total cross section σ − (E) for the inelastic collision of electrons of energyE is calculated analytically. It can be split into contributions from distantlongitudinal, distant transverse and close interactions,σ − (E) = σ dis,l + σ dis,t + σ − clo . (11.91)The contributions from distant longitudinal and transverse interactions areσ dis,l = 2πe4m e v 2∑shellsf k1W kln ( W kQ minkQ mink + 2m e c 2W k + 2m e c 2 )Θ(E − Wk ) (11.92)andσ dis,t = 2πe4m e v 2∑shellsf k1W k[ln( 11 − β 2 )− β 2 − δ F]Θ(E − Wk ) (11.93)respectively, where:m e = mass of the electron;v = velocity of the electron;β = velocity of the electron in units of c;f k = number of electrons in the k-th atomic shell;Θ = Heaviside step function;W k = resonance energy of the k-th atomic shell oscillator;Q mink== minimum kinematically allowed recoil energy for energy transfer W k√ [ √√E(E + 2m e c 2 ) − (E − W k )(E − W k + 2m e c 2 ) ] 2+ m2e c 4 − m e c 2 ;δ F = Fermi density effect correction, computed as described in Ref. [18].The value of W k is calculated from the ionisation energy U k of the k-thshell as W k = 1.65 U k . This relation is derived from the hydrogenic model,which is valid for the innermost shells. In this model, the shell ionisationcross sections are only roughly approximated; nevertheless the ionisation of203

inner shells is a low-probability process and the approximation has a weakeffect on the global transport properties 1 .The integrated cross section for close collisions is the Møller cross sectionwhereσ − clo = 2πe4m e v 2∑shellsf k∫ E2W k1W 2 F − (E, W )dW, (11.94)F − (E, W ) = 1 + ( W ) 2 W −E − W E − W + ( E ) 2 ( WE + m e c 2 E − W + W 2 ).E 2(11.95)The integral of Eq.(11.94) can be evaluated analytically. In the final statethere are two indistinguishable free electrons and the fastest one is consideredas the “primary”; accordingly, the maximum allowed energy transfer in closecollisions is E . 2The GOS model also allows evaluation of the spectrum dσ− of the energydWW lost by the primary electron as the sum of distant longitudinal, distanttransverse and close interaction contributions,In particular,dσ −dW= dσ− clodW + dσ dis,ldW+ dσ dis,tdW . (11.96)dσ dis,ldW = 2πe4m e v 2whereQ − =and∑shellsf k1W kln ( W kQ −Q − + 2m e c 2W k + 2m e c 2 )δ(W − Wk )Θ(E − W k ),(11.97)√ [ √√E(E + 2m e c 2 ) − (E − W )(E − W + 2m e c 2 ) ] 2+ m2 e c 4 − m e c 2 ,(11.98)dσ dis,tdW = 2πe4m e v 2dσ − clodW = 2πe4m e v 2∑∑shellsf k1W k[ln( 11 − β 2 )− β 2 − δ F]Θ(E − W k )δ(W − W k ) (11.99)f k1W 2 F − (E, W )Θ(W − W k ). (11.100)shells1 In cases where inner-shell ionisation is directly observed, a more accurate descriptionof the process should be used.204

Eqs. (11.92), (11.93) and (11.94) derive respectively from the integration indW of Eqs. (11.97), (11.99) and (11.100) in the interval [0,W max ], whereW max = E for distant interactions and W max = E for close. The analytical2GOS model provides an accurate average description of inelastic collisions.However, the continuous energy loss spectrum associated with single distantexcitations of a given atomic shell is approximated as a single resonance (aδ distribution). As a consequence, the simulated energy loss spectra showunphysical narrow peaks at energy losses that are multiples of the resonanceenergies. These spurious peaks are automatically smoothed out after multipleinelastic collisions.The explicit expression of dσ− , Eq. (11.96), allows the analytic calculationdWof the partial cross sections for soft and hard ionisation events, i.e.σ − soft =∫ Tc0dσ −dW dW and σ− hard = ∫ WmaxT cdσ −dW. (11.101)dWThe first stage of the simulation is the selection of the active oscillator kand the oscillator branch (distant or close).In distant interactions with the k-th oscillator, the energy loss W of theprimary electron corresponds to the excitation energy W k , i.e. W =W k . If theinteraction is transverse, the angular deflection of the projectile is neglected,i.e. cos θ=1. For longitudinal collisions, the distribution of the recoil energyQ is given byP k (Q) =1Q[1+Q/(2m ecif Q 2 )] − < Q < W max0 otherwise(11.102)Once the energy loss W and the recoil energy Q have been sampled, thepolar scattering angle is determined ascos θ = E(E + 2m ec 2 ) + (E − W )(E − W + 2m e c 2 ) − Q(Q + 2m e c 2 )√.2 E(E + 2m e c 2 )(E − W )(E − W + 2m e c 2 )(11.103)The azimuthal scattering angle φ is sampled uniformly in the interval (0,2π).For close interactions, the distributions for the reduced energy loss κ ≡ W/Efor electrons arePk − (κ) = [ 1κ + 12 (1 − κ) − 12 κ(1 − κ) + ( E ) 2 ( 1 )]1 +E + m e c 2 κ(1 − κ)205Θ(κ − κ c )Θ( 1 2 −(11.104)κ)

with κ c = max(W k , T c )/E. The maximum allowed value of κ is 1/2, consistentwith the indistinguishability of the electrons in the final state. After thesampling of the energy loss W = κE, the polar scattering angle θ is obtainedascos 2 θ = E − WEE + 2m e c 2E − W + 2m e c 2 . (11.105)The azimuthal scattering angle φ is sampled uniformly in the interval (0,2π).According to the GOS model, each oscillator W k corresponds to an atomicshell with f k electrons and ionisation energy U k . In the case of ionisationof an inner shell i (K or L), a secondary electron (δ-ray) is emitted withenergy E s = W − U i and the residual ion is left with a vacancy in the shell(which is then filled with the emission of fluorescence x-rays and/or Augerelectrons). In the case of ionisation of outer shells, the simulated δ-ray isemitted with kinetic energy E s = W and the target atom is assumed toremain in its ground state. The polar angle of emission of the secondaryelectron is calculated ascos 2 θ s =W 2 /β 2 [ Q(Q + 2m e c 2 ) − W 2 ] 21 + (11.106)Q(Q + 2m e c 2 ) 2W (E + m e c 2 )(for close collisions Q = W ), while the azimuthal angle is φ s = φ + π. Inthis model, the Doppler effects on the angular distribution of the δ rays areneglected.The stopping power due to soft interactions of electrons, which is used for thecomputation of the continuous part of the process, is analytically calculatedas∫ TcSin− = N W dσ− dW (11.107)0 dWfrom the expression (11.96), where N is the number of scattering centers(atoms or molecules) per unit volume.PositronsThe total cross section σ + (E) for the inelastic collision of positrons of energyE is calculated analytically. As in the case of electrons, it can be split intocontributions from distant longitudinal, distant transverse and close interactions,σ + (E) = σ dis,l + σ dis,t + σ + clo . (11.108)The contributions from distant longitudinal and transverse interactions arethe same as for electrons, Eq. (11.92) and (11.93), while the integrated cross206

section for close collisions is the Bhabha cross sectionwhereσ + clo = 2πe4m e v 2∑shellsf k∫ EW k1W 2 F + (E, W )dW, (11.109)F + W(E, W ) = 1 − b 1E + b W 22E − b W 32 3E + b W 43 4E ; (11.110)4the Bhabha factors areb 1 = ( γ − 1) 2 2(γ + 1) 2 − 1γ γ 2 − 1b 3 = ( γ − 1γ) 2 2(γ − 1)γ(γ + 1) , b 2 4 = ( γ − 1γb 2 = ( γ − 1) 2 3(γ + 1) 2 + 1,γ (γ + 1) 2) 2 (γ − 1) 2(γ + 1) , (11.111)2(11.112)and γ is the Lorentz factor of the positron. The integral of Eq. (11.109) canbe evaluated analytically. The particles in the final state are not undistinguishableso the maximum energy transfer W max in close collisions is E.As for electrons, the GOS model allows the evaluation of the spectrum dσ+ of dWthe energy W lost by the primary positron as the sum of distant longitudinal,distant transverse and close interaction contributions,dσ +dW= dσ+ clodW + dσ dis,ldW+ dσ dis,tdW , (11.113)where the distant terms dσ dis,land dσ dis,tdWdW(11.99), while the close contribution isare those from Eqs. (11.97) anddσ + clodW = 2πe4m e v 2∑shellsf k1W 2 F + (E, W )Θ(W − W k ). (11.114)Also in this case, the explicit expression of dσ+ , Eq. (11.113), allows andWanalytic calculation of the partial cross sections for soft and hard ionisationevents, i.e.σ + soft =∫ Tc0dσ +∫ EdW dW and σ+ hard = dσ +dW. (11.115)T c dWThe sampling of the final state in the case of distant interactions (transverseor longitudinal) is performed in the same way as for primary electrons, see207

section 11.11.7. For close positron interactions with the k-th oscillator, thedistribution for the reduced energy loss κ ≡ W/E isP + k (κ) = [ 1κ 2 − b 1κ + b 2 − b 3 κ + b 4 κ 2] Θ(κ − κ c )Θ(1 − κ) (11.116)with κ c = max(W k , T c )/E. In this case, the maximum allowed reducedenergy loss κ is 1. After sampling the energy loss W = κE, the polar angleθ and the azimuthal angle φ are obtained using the equations introduced forelectrons in section 11.11.7. Similarly, the generation of δ rays is performedin the same way as for electrons.Finally, the stopping power due to soft interactions of positrons, which isused for the computation of the continuous part of the process, is analyticallycalculated as∫ TcS in+ = N W dσ+ dW (11.117)0 dWfrom the expression (11.113), where N is the number of scattering centersper unit volume.11.11.8 Positron AnnihilationTotal Cross SectionThe total cross section (per target electron) for the annihilation of a positronof energy E into two photons is evaluated from the analytical formula [19, 20]πr 2 eσ(E) =×(γ + 1)(γ 2 − 1){(γ 2 + 4γ + 1) ln [ √γ + γ 2 − 1 ] √− (3 + γ) γ 2 − 1 } . (11.118)where:r e = classical radius of the electron;γ = Lorentz factor of the positron.Sampling of the Final StateThe target electrons are assumed to be free and at rest: binding effects, thatenable one-photon annihilation [19], are neglected. When the annihilationoccurs in flight, the two photons may have different energies, say E − andE + (the photon with lower energy is denoted by the superscript “−”), which208

add to E + 2m e c 2 . Each annihilation event is completely characterized bythe quantityE −ζ =E + 2m e c , (11.119)2which is in the interval ζ min ≤ ζ ≤ 1 2 , withζ min =1γ + 1 + √ γ 2 − 1 . (11.120)The parameter ζ is sampled from the differential distributionP (ζ) =where γ is the Lorentz factor andπre2 [S(ζ) + S(1 − ζ)], (11.121)(γ + 1)(γ 2 − 1)S(ζ) = −(γ + 1) 2 + (γ 2 + 4γ + 1) 1 ζ − 1 ζ 2 . (11.122)From conservation of energy and momentum, it follows that the two photonsare emitted in directions with polar anglescos θ − =1 ( 1)√ γ + 1 −γ2 − 1 ζ(11.123)and1 ( 1 )cos θ + = √ γ + 1 − (11.124)γ2 − 1 1 − ζthat are completely determined by ζ; in particuar, when ζ = ζ min , cos θ − =−1. The azimuthal angles are φ − and φ + = φ − + π; owing to the axialsymmetry of the process, the angle φ − is uniformly distributed in (0, 2π).11.11.9 Status of the document09.06.2003 created by L. Pandola20.06.2003 spelling and grammar check by D.H. Wright07.11.2003 Ionisation and Annihilation section added by L. PandolaBibliography[1] Penelope - A Code System for Monte Carlo Simulation of Electron andPhoton Transport, Workshop Proceedings Issy-les-Moulineaux, France,5−7 November 2001, AEN-NEA;209

[2] J.Sempau et al., Experimental benchmarks of the Monte Carlo codePENELOPE, submitted to NIM B (2002);[3] D.Brusa et al., Fast sampling algorithm for the simulation of photonCompton scattering, NIM A379,167 (1996);[4] F.Biggs et al., Hartree-Fock Compton profiles for the elements, At.DataNucl.Data Tables 16,201 (1975);[5] M.Born, Atomic physics, Ed. Blackie and Sons (1969);[6] J.Baró et al., Analytical cross sections for Monte Carlo simulation ofphoton transport, Radiat.Phys.Chem. 44,531 (1994);[7] J.H.Hubbel et al., Atomic form factors, incoherent scattering functionsand photon scattering cross sections, J. Phys.Chem.Ref.Data 4,471(1975). Erratum: ibid. 6,615 (1977);[8] M.J.Berger and J.H.Hubbel, XCOM: photom cross sections on a personalcomputer, Report NBSIR 87-3597 (National Bureau of Standards)(1987);[9] H.Davies et al., Theory of bremsstrahlung and pair production. II.Integralcross section for pair production, Phys.Rev. 93,788 (1954);[10] J.H.Hubbel et al., Pair, triplet and total atomic cross sections (and massattenuation coefficients) for 1 MeV − 100 GeV photons in element Z=1to 100, J.Phys.Chem.Ref.Data 9,1023 (1980);[11] J.W.Motz et al., Pair production by photons, Rev.Mod.Phys 41,581(1969);[12] D.E.Cullen et al., Tables and graphs of photon-interaction cross sectionsfrom 10 eV to 100 GeV derived from the LLNL evaluated photondata library (EPDL), Report UCRL-50400 (Lawrence Livermore NationalLaboratory) (1989);[13] S.M.Seltzer and M.J.Berger, Bremsstrahlung energy spectra from electronswith kinetic energy 1 keV - 100 GeV incident on screened nucleiand orbital electrons of neutral atoms with Z=1-100, At.Data Nucl.DataTables 35,345 (1986);[14] D.E.Cullen et al., Tables and graphs of electron-interaction cross sectionsfrom 10 eV to 100 GeV derived from the LLNL evaluated photondata library (EEDL), Report UCRL-50400 (Lawrence Livermore NationalLaboratory) (1989);210

[15] L.Kissel et al., Shape functions for atomic-field bremsstrahlung from electronof kinetic energy 1−500 keV on selected neutral atoms 1 ≤ Z ≤ 92,At.Data Nucl.Data.Tab. 28,381 (1983);[16] M.J.Berger and S.M.Seltzer, Stopping power of electrons and positrons,Report NBSIR 82-2550 (National Bureau of Standards) (1982);[17] L.Kim et al., Ratio of positron to electron bremsstrahlung energy loss:an approximate scaling law, Phys.Rev.A 33,3002 (1986);[18] U.Fano, Penetration of protons, alpha particles and mesons,Ann.Rev.Nucl.Sci. 13,1 (1963);[19] W.Heitler, The quantum theory of radiation, Oxford University Press,London (1954);[20] W.R.Nelson et al., The EGS4 code system, Report SLAC-265 (1985).211

Chapter 12Optical Photons212

12.1 Interactions of optical photonsOptical photons are produced when a charged particle traverses:1. a dielectric material with velocity above theČerenkov threshold;2. a scintillating material.12.1.1 Physics processes for optical photonsA photon is called optical when its wavelength is much greater than thetypical atomic spacing, for instance when λ ≥ 10nm which corresponds toan energy E ≤ 100eV . Production of an optical photon in a HEP detectoris primarily due to:1. Čerenkov effect;2. Scintillation.Optical photons undergo three kinds of interactions:1. Elastic (Rayleigh) scattering;2. Absorption;3. Medium boundary interactions.Rayleigh scatteringFor optical photons Rayleigh scattering is usually unimportant. For λ =.2µm we have σ Rayleigh ≈ .2b for N 2 or O 2 which gives a mean free path of≈ 1.7km in air and ≈ 1m in quartz. Two important exceptions are aerogel,which is used as a Čerenkov radiator for some special applications and largewater Čerenkov detectors for neutrino detection.The differential cross section in Rayleigh scattering, dσ/dΩ, is proportionalto cos 2 θ, where θ is the polar angle of the new polarization withrespect to the old polarization.AbsorptionAbsorption is important for optical photons because it determines the lowerλ limit in the window of transparency of the radiator. Absorption competeswith photo-ionization in producing the signal in the detector, so it must betreated properly in the tracking of optical photons.213

Medium boundary effectsWhen a photon arrives at the boundary of a dielectric medium, its behaviourdepends on the nature of the two materials which join at that boundary:• Case dielectric → dielectric.The photon can be transmitted (refracted ray) or reflected (reflectedray). In case where the photon can only be reflected, total internalreflection takes place.• Case dielectric → metal.The photon can be absorbed by the metal or reflected back into thedielectric. If the photon is absorbed it can be detected according tothe photoelectron efficiency of the metal.• Case dielectric → black material.A black material is a tracking medium for which the user has not definedany optical property. In this case the photon is immediately absorbedundetected.12.1.2 Photon polarizationThe photon polarization is defined as a two component vector normal to thedirection of the photon:(a1 e iΦ )1a 2 e iΦ 2= e Φo (a1 e iΦca 2 e −iΦc )where Φ c = (Φ 1 −Φ 2 )/2 is called circularity and Φ o = (Φ 1 +Φ 2 )/2 is calledoverall phase. Circularity gives the left- or right-polarization characteristicof the photon. RICH materials usually do not distinguish between the twopolarizations and photons produced by the Čerenkov effect and scintillationare linearly polarized, that is Φ c = 0.The overall phase is important in determining interference effects betweencoherent waves. These are important only in layers of thickness comparablewith the wavelength, such as interference filters on mirrors. The effects ofsuch coatings can be accounted for by the empirical reflectivity factor forthe surface, and do not require a microscopic simulation. GEANT4 does notkeep track of the overall phase.Vector polarization is described by the polarization angle tan Ψ = a 2 /a 1 .Reflection/transmission probabilities are sensitive to the state of linear polarization,so this has to be taken into account. One parameter is sufficient to214

describe vector polarization, but to avoid too many trigonometrical transformations,a unit vector perpendicular to the direction of the photon is used inGEANT4. The polarization vector is a data member of G4DynamicParticle.12.1.3 Tracking of the photonsOptical photons are subject to in flight absorption, Rayleigh scattering andboundary action. As explained above, the status of the photon is defined bytwo vectors, the photon momentum (⃗p = ¯h ⃗ k) and photon polarization (⃗e).By convention the direction of the polarization vector is that of the electricfield. Let also ⃗u be the normal to the material boundary at the point ofintersection, pointing out of the material which the photon is leaving andtoward the one which the photon is entering. The behaviour of a photon atthe surface boundary is determined by three quantities:1. refraction or reflection angle, this represents the kinematics of the effect;2. amplitude of the reflected and refracted waves, this is the dynamics ofthe effect;3. probability of the photon to be refracted or reflected, this is the quantummechanical effect which we have to take into account if we wantto describe the photon as a particle and not as a wave.As said above, we distinguish three kinds of boundary action, dielectric→ black material, dielectric → metal, dielectric → dielectric. The first caseis trivial, in the sense that the photon is immediately absorbed and it goesundetected.To determine the behaviour of the photon at the boundary, we will atfirst treat it as an homogeneous monochromatic plane wave:Case dielectric → dielectric⃗E = ⃗ E 0 e i⃗ k·⃗x−iωt⃗B = √ µɛ ⃗ k × ⃗ EkIn the classical description the incoming wave splits into a reflected wave(quantities with a double prime) and a refracted wave (quantities with asingle prime). Our problem is solved if we find the following quantities:⃗E ′ = ⃗ E ′ 0 ei⃗ k ′·⃗x−iωt215

⃗E ′′ = ⃗ E ′′0 e i⃗ k ′′·⃗x−iωtFor the wave numbers the following relations hold:| ⃗ k| = | ⃗ k ′′ | = k = ω c√ µɛ| ⃗ k ′ | = k ′ = ω √µc′ ɛ ′Where the speed of the wave in the medium is v = c/ √ µɛ and the quantityn = c/v = √ µɛ is called refractive index of the medium. The condition thatthe three waves, refracted, reflected and incident have the same phase at thesurface of the medium, gives us the well known Fresnel law:( ⃗ k · ⃗x) surf = ( ⃗ k ′ · ⃗x) surf = ( ⃗ k ′′ · ⃗x) surfk sin i = k ′ sin r = k ′′ sin r ′where i, r, r ′ are, respectively, the angle of the incident, refracted andreflected ray with the normal to the surface. From this formula the wellknown condition emerges:i = r ′√sin isin r = µ′ ɛ ′µɛ = n′nThe dynamic properties of the wave at the boundary are derived fromMaxwell’s equations which impose the continuity of the normal componentsof D ⃗ and B ⃗ and of the tangential components of E ⃗ and H ⃗ at the surfaceboundary. The resulting ratios between the amplitudes of the the generatedwaves with respect to the incoming one are expressed in the two followingcases:1. a plane wave with the electric field (polarization vector) perpendicularto the plane defined by the photon direction and the normal to theboundary:E ′ 0E 0=2n cos in cos i = µ µ ′ n ′ cos r =2n cos in cos i + n ′ cos rE 0′′= n cos i − µ n ′ cos rµ ′E 0 n cos i + µ nµ ′ cos r = n cos i − n′ cos rn cos i + n ′ cos r′where we suppose, as it is legitimate for visible or near-visible light,that µ/µ ′ ≈ 1;216

2. a plane wave with the electric field parallel to the above surface:E ′ 0E 0=E ′′0E 0=2n cos iµnµ ′ cos i + n cos r = 2n cos in ′ cos i + n cos r′µn ′ cos i − n cos rµ ′ µnµ ′ cos i + n cos r = n′ cos i − n cos rn ′ cos i + n cos r′with the same approximation as above.We note that in case of photon perpendicular to the surface, the followingrelations hold:E ′ 0E 0=2nn ′ + nE 0′′= n′ − nE 0 n ′ + nwhere the sign convention for the parallel field has been adopted. Thismeans that if n ′ > n there is a phase inversion for the reflected wave.Any incoming wave can be separated into one piece polarized parallel tothe plane and one polarized perpendicular, and the two components treatedaccordingly.To maintain the particle description of the photon, the probability tohave a refracted or reflected photon must be calculated. The constraint isthat the number of photons be conserved, and this can be imposed via theconservation of the energy flux at the boundary, as the number of photons isproportional to the energy. The energy current is given by the expression:⃗S = 1 √c √ µɛE ⃗ × ⃗c ɛ H =2 4π8π µ E2ˆk0and the energy balance on a unit area of the boundary requires that:c 18π µ nE2 0 cos i = c8π⃗S · ⃗u = ⃗ S ′ · ⃗u − ⃗ S ′′ · ⃗uS cos i = S ′ cosr + S ′′ cosi1µ ′ n′ E 0 ′2 cos r + c8π1µ nE′′2 0 cos iIf we set again µ/µ ′ ≈ 1, then the transmission probability for the photonwill be:T = ( E′ 0) 2 n′ cos rE 0 n cos i217

and the corresponding probability to be reflected will be R = 1 − T .In case of reflection, the relation between the incoming photon ( ⃗ k, ⃗e), therefracted one ( ⃗ k ′ , ⃗e ′ ) and the reflected one ( ⃗ k ′′ , ⃗e ′′ ) is given by the followingrelations:⃗q = ⃗ k × ⃗u⃗e · ⃗q⃗e ⊥ = (|⃗q| ) ⃗q|⃗q|⃗e ‖ = ⃗e − ⃗e ⊥e ′ 2n cos i‖ = e ‖n ′ cos i + n cos re ′ 2n cos i⊥| = e ⊥n cos i + n ′ cos re ′′‖ = n′n e′ ‖ − e ‖e ′′ ⊥ = e′ ⊥ − e ⊥After transmission or reflection of the photon, the polarization vectoris re-normalized to 1. In the case where sin r = n sin i/n ′ > 1 then therecannot be a refracted wave, and in this case we have a total internal reflectionaccording to the following formulas:Case dielectric → metal⃗ k ′′ = ⃗ k − 2( ⃗ k · ⃗u)⃗u⃗e ′′ = −⃗e + 2(⃗e · ⃗u)⃗uIn this case the photon cannot be transmitted. So the probability for thephoton to be absorbed by the metal is estimated according to the tableprovided by the user. If the photon is not absorbed, it is reflected.Bibliography[1] J.D. Jackson, Classical Electrodynamics, J. Wiley & Sons Inc., NewYork, 1975.218

Chapter 13Shower Parameterizations219

13.1 Gflash Shower ParameterizationsThe computing time needed for the simulation of high energy electromagneticshowers can become very large, since it increases approximately linearlywith the energy absorbed in the detector. Using parameterizations insteadof individual particle tracking for electromagnetic (sub)showers can speedup the simulations considerably without sacrificing much precision. TheGflash package allows the parameterization of electron and positron showersin homogeneous (for the time being) calorimeters and is based on theparameterization described in Ref. [1] .13.1.1 Parameterization AnsatzThe spatial energy distribution of electromagnetic showers is given by threeprobability density functions (pdf),dE(⃗r) = E f(t)dt f(r)dr f(φ)dφ, (13.1)describing the longitudinal, radial, and azimuthal energy distributions. Heret denotes the longitudinal shower depth in units of radiation length, r measuresthe radial distance from the shower axis in Molière units, and φ is theazimuthal angle. The start of the shower is defined by the space point wherethe electron or positron enters the calorimeter, which is different from theoriginal Gflash. A gamma distribution is used for the parameterization of thelongitudinal shower profile, f(t). The radial distribution f(r), is describedby a two-component ansatz. In φ, it is assumed that the energy is distributeduniformly: f(φ) = 1/2π.13.1.2 Longitudinal Shower ProfilesThe average longitudinal shower profiles can be described by a gamma distribution[2]:〈 1E〉dE(t)dt= f(t) = (βt)α−1 β exp(−βt). (13.2)Γ(α)The center of gravity, 〈t〉, and the depth of the maximum, T , are calculatedfrom the shape parameter α and the scaling parameter β according to:〈t〉 = α β(13.3)T = α − 1β . (13.4)220

In the parameterization all lengths are measured in units of radiationlength (X 0 ), and energies in units of the critical energy (E c = 2.66 ( ) 1.1ZX 0 A ).This allows material independence, since the longitudinal shower momentsare equal in different materials, according to Ref. [3]. The following equationsare used for the energy dependence of T hom and (α hom ), with y = E/E c andt = x/X 0 , x being the longitudinal shower depth:T hom = ln y + t 1 (13.5)α hom = a 1 + (a 2 + a 3 /Z) ln y. (13.6)The y-dependence of the fluctuations can be described by:σ = (s 1 + s 2 ln y) −1 . (13.7)The correlation between ln T hom and ln α hom is given by:ρ(ln T hom , ln α hom ) ≡ ρ = r 1 + r 2 ln y. (13.8)From these formulae, correlated and varying parameters α i and β i are generatedaccording to( ) ( ) ( )ln Ti 〈ln T 〉 z1=+ C(13.9)ln α i 〈ln α〉 z 2withC =(σ(ln T ) 00 σ(ln α)) ⎛ √ √ ⎞1+ρ 1−ρ⎝ √ 2 √ 2 ⎠1+ρ− 1−ρ2 2σ(ln α) and σ(ln T ) are the fluctuations of T hom and (α hom . The values of thecoefficients can be found in Ref. [1].13.1.3 Radial Shower ProfilesFor the description of average radial energy profiles,f(r) = 1dE(t)dE(t, r), (13.10)dra variety of different functions can be found in the literature. In Gflash thefollowing two-component ansatz, an extension of that in Ref.[4], was used:f(r) = pf C (r) + (1 − p)f T (r) (13.11)=2rRC2 p(r 2 + R + (1 − p) 2rRT2C 2 )2 (r 2 + RT 2 )2221

with0 ≤ p ≤ 1.Here R C (R T ) is the median of the core (tail) component and p is a probabilitygiving the relative weight of the core component. The variable τ = t/T ,which measures the shower depth in units of the depth of the shower maximum,is used in order to generalize the radial profiles. This makes theparameterization more convenient and separates the energy and material dependenceof various parameters. The median of the core distribution, R C ,increases linearly with τ. The weight of the core, p, is maximal around theshower maximum, and the width of the tail, R T , is minimal at τ ≈ 1.The following formulae are used to parameterize the radial energy densitydistribution for a given energy and material:R C,hom (τ) = z 1 + z 2 τ (13.12)R T,hom (τ) = k 1 {exp(k 3 (τ − k 2 )) + exp(k 4 (τ − k 2 ))} (13.13){ ( )}p2 − τ p2 − τp hom (τ) = p 1 exp − exp(13.14)p 3p 3The parameters z 1 · · · p 3 are either constant or simple functions of ln E or Z.Radial shape fluctuations are also taken into account. A detailed explanationof this procedure, as well as a list of all the parameters used in Gflash,can be found in Ref. [1].13.1.4 Gflash PerformanceThe parameters used in this Gflash implementation were extracted from fullsimulation studies with Geant 3. They also give good results inside theGeant4 fast shower framework when compared with the full electromagneticshower simulation. However, if more precision or higher particle energies arerequired, retuning may be necessary. For the longitudinal profiles the differencebetween full simulation and Gflash parameterization is at the levelof a few percent. Because the radial profiles are slightly broader in Geant3than in Geant4, the differences may reach > 10%. The gain in speed, on theother hand, is impressive. The simulation of a 1 TeV electron in a P bW O 4cube is 160 times faster with Gflash. Gflash can also be used to parameterizeelectromagnetic showers in sampling calorimeters. So far, however, onlyhomogeneous materials are supported.222

13.1.5 Status of this document02.12.04 created by J.Weng03.12.04 grammar check and minor re-wording by D.H. WrightBibliography[1] G. Grindhammer, S. Peters, The Parameterized Simulation of ElectromagneticShowers in Homogeneous and Sampling Calorimeters, hepex/0001020(1993).[2] E. Longo and I. Sestili,Nucl. Instrum. Meth. 128, 283 (1975).[3] Rossi rentice Hall, New York (1952).[4] G. Grindhammer, M. Rudowicz, and S. Peters, Nucl. Instrum.Meth. A290, 469 (1990).223

Part IVHadronic Interactions224

Chapter 14Lepton Hadron InteractionsThe photonuclear interaction of muons is currently the only process treatedin this category.14.1 G4MuonNucleusProcessThis class simulates the photonuclear interaction of muons in a material.The muon interacts electromagnetically with a nucleus, exchanging a virtualphoton. At energies above a few GeV, the photon interacts hadronically withthe nucleus, producing hadronic secondaries.The outcome of the simulation depends heavily upon the interactionmodel chosen. Hence the model-dependent part of the process is implementedin the G4MuonNucleusInteractionModel class, which can be easilyreplaced by another model.G4MuonNucleusInteractionModel calculates the cross section and finalstates of the muon and hadronic secondaries. The final muon momentum isgiven by a double-differential cross section which depends on the photoabsorptioncross sections for longitudinally and transversely polarized photons.The final hadronic state is determined by replacing the virtual photon with acharged pion of the same Q 2 and then allowing the pion to interact with thenucleus. The charge of the pion is chosen at random. The pion interactionswith the nucleus are modeled by processes derived from the GHEISHA [1]package. These processes are:G4LEPionPlusInelastic, G4LEPionMinusInelasticG4HEPionPlusInelastic, G4HEPionMinusInelasticE ≤ 25 GeVE > 25 GeV225

14.1.1 Cross Section CalculationThe cross section for the above process in a material is given roughly byσ µA = Aσ µNwhere A is the atomic mass number of the material and σ µNsection for the process on a single nucleon:{0.3 (E ≤ 30GeV )σ µN =0.3(E/30) 0.25 (E > 30GeV ) [µb].is the crossThe differential cross section, in terms of muon energy E and emission solidangle Ω, can be expressed as:dσdΩdE = Γ (σ T + ɛσ L )where σ L and σ T are the photoabsorption cross sections for longitudinal andtransverse photons, respectively. Γ is the transverse photon flux and ɛ is thepolarization of the intermediate photon. The photoabsorption cross sectionsare parameterized as:σ L = 0.3(1 − 1)1.868 Q2 νσ T ∼ const = 0.12mbσ Twhile the flux and polarization are given byΓ = Kα2π[ɛ =E ′E11 − ɛ1 + 2 Q2 + ν 2Q 2 tan 2 θ 2] −1.E and E ′ are the initial and final muon energies, Q 2 and ν are the scalingvariablesQ 2 = −q 2 = 2(EE ′ − P P ′ cosθ − m 2 µ )ν = E − E ′ ,and K is given using the Gilman conventionK = ν + Q22ν .226

14.2 Status of this document20.04.02 re-written by D.H. Wright23.10.98 created by M.TakahataBibliography[1] H.Fesefeldt GHEISHA The Simulation of Hadronic Showers 149RWTH/PITHA 85/02 (1985)227

Chapter 15Cross-sections in Photonuclearand Electronuclear Reactions15.1 Approximation of Photonuclear Cross Sections.The photonuclear cross sections parameterized in the G4PhotoNuclearCrossSectionclass cover all incident photon energies from the hadron production thresholdupward. The parameterization is subdivided into five energy regions, eachcorresponding to the physical process that dominates it.• The Giant Dipole Resonance (GDR) region, depending on the nucleus,extends from 10 Mev up to 30 MeV. It usually consists of one largepeak, though for some nuclei several peaks appear.• The “quasi-deuteron” region extends from around 30 MeV up to thepion threshold and is characterized by small cross sections and a broad,low peak.• The ∆ region is characterized by the dominant peak in the cross sectionwhich extends from the pion threshold to 450 MeV.• The Roper resonance region extends from roughly 450 MeV to 1.2 GeV.The cross section in this region is not strictly identified with the realRoper resonance because other processes also occur in this region.• The Reggeon-Pomeron region extends upward from 1.2 GeV.In the GEANT4 photonuclear data base there are about 50 nuclei for whichthe photonuclear absorption cross sections have been measured in the above228

energy ranges. For low energies this number could be enlarged, because forheavy nuclei the neutron photoproduction cross section is close to the totalphoto-absorption cross section. Currently, however, 14 nuclei are used in theparameterization: 1 H, 2 H, 4 He, 6 Li, 7 Li, 9 Be, 12 C, 16 O, 27 Al, 40 Ca, Cu, Sn,Pb, and U. The resulting cross section is a function of A and e = log(E γ ),where E γ is the energy of the incident photon. This function is the sum ofthe components which parameterize each energy region.The cross section in the GDR region can be described as the sum of twopeaks,GDR(e) = th(e, b 1 , s 1 ) · exp(c 1 − p 1 · e) + th(e, b 2 , s 2 ) · exp(c 2 − p 2 · e). (15.1)The exponential parameterizes the falling edge of the resonance which behaveslike a power law in E γ . This behavior is expected from the CHIPSmodel, which includes the nonrelativistic phase space of nucleons to explainevaporation. The functionth(e, b, s) =11 + exp( b−e(15.2)),sdescribes the rising edge of the resonance. It is the nuclear-barrier-reflectionfunction and behaves like a threshold, cutting off the exponential. The exponentialpowers p 1 and p 2 arep 1 = 1, p 2 = 2 for A < 4p 1 = 2, p 2 = 4 for 4 ≤ A < 8p 1 = 3, p 2 = 6 for 8 ≤ A < 12p 1 = 4, p 2 = 8 for A ≥ 12.The A-dependent parameters b i , c i and s i were found for each of the 14 nucleilisted above and interpolated for other nuclei.The ∆ isobar region was parameterized as∆(e, d, f, g, r, q) =d · th(e, f, g)1 + r · (e − q) 2 , (15.3)where d is an overall normalization factor. q can be interpreted as the energyof the ∆ isobar and r can be interpreted as the inverse of the ∆ width. Onceagain th is the threshold function. The A-dependence of these parameters isas follows:229

• d = 0.41 · A (for 1 H it is 0.55, for 2 H it is 0.88), which means that the∆ yield is proportional to A;• f = 5.13 − .00075 · A. exp(f) shows how the pion threshold depends onA. It is clear that the threshold becomes 140 MeV only for uranium;for lighter nuclei it is higher.• g = 0.09 for A ≥ 7 and 0.04 for A < 7;• q = 5.84− .091+.003·A 2 , which means that the “mass” of the ∆ isobar movesto lower energies;• r = 11.9 − 1.24 · log(A). r is 18.0 for 1 H. The inverse width becomessmaller with A, hence the width increases.The A-dependence of the f, q and r parameters is due to the ∆+N → N +Nreaction, which can take place in the nuclear medium below the pion threshold.The quasi-deuteron contribution was parameterized with the same form asthe ∆ contribution but without the threshold function:QD(e, v, w, u) =v1 + w · (e − u) 2 . (15.4)For 1 H and 2 H the quasi-deuteron contribution is almost zero. For thesenuclei the third baryonic resonance was used instead, so the parameters forthese two nuclei are quite different, but trivial. The parameter values aregiven below.• v = exp(−1.7+a·0.84) , where a = log(A). This shows that the A-dependence1+exp(7·(2.38−a))in the quasi-deuteron region is stronger than A 0.84 . It is clear from thedenominator that this contribution is very small for light nuclei (upto 6 Li or 7 Li). For 1 H it is 0.078 and for 2 H it is 0.08, so the deltacontribution does not appear to be growing. Its relative contributiondisappears with A.• u = 3.7 and w = 0.4. The experimental information is not sufficientto determine an A-dependence for these parameters. For both 1 H and2 H u = 6.93 and w = 90, which may indicate contributions from the∆(1600) and ∆(1620).230

The transition Roper contribution was parameterized using the same formas the quasi-deuteron contribution:T r(e, v, w, u) =Using a = log(A), the values of the parameters arev1 + w · (e − u) 2 . (15.5)• v = exp(−2. + a · 0.84). For 1 H it is 0.22 and for 2 H it is 0.34.• u = 6.46 + a · 0.061 (for 1 H and for 2 H it is 6.57), so the “mass” of theRoper moves higher with A.• w = 0.1 + a · 1.65. For 1 H it is 20.0 and for 2 H it is 15.0).The Regge-Pomeron contribution was parametrized as follows:RP (e, h) = h · th(7., 0.2) · (0.0116 · exp(e · 0.16) + 0.4 · exp(−e · 0.2)), (15.6)where h = A · exp(−a · (0.885 + 0.0048 · a)) and, again, a = log(A). The firstexponential in Eq. 15.6 describes the Pomeron contribution while the seconddescribes the Regge contribution.15.2 Electronuclear Cross Sections and ReactionsElectronuclear reactions are so closely connected with photonuclear reactionsthat they are sometimes called “photonuclear” because the one-photonexchange mechanism dominates in electronuclear reactions. In this senseelectrons can be replaced by a flux of equivalent photons. This is not completelytrue, because at high energies the Vector Dominance Model (VDM) ordiffractive mechanisms are possible, but these types of reactions are beyondthe scope of this discussion.15.2.1 Common Notation for Different Approaches toElectronuclear ReactionsThe Equivalent Photon Approximation (EPA) was proposed by E. Fermi [1]and developed by C. Weizsacker and E. Williams [2] and by L. Landau andE. Lifshitz [3]. The covariant form of the EPA method was developed in Refs.[4] and [5]. When using this method it is necessary to take into account thatreal photons are always transversely polarized while virtual photons may231

e longitudinally polarized. In general the differential cross section of theelectronuclear interaction can be written asd 2 σdydQ 2 =απQ 2 (S T L · (σ T + σ L ) − S L · σ L ), (15.7)whereS T L = y 1 − y + y22 + Q24E 2 − m2 eQ 2 (y 2 + Q2E 2 )y 2 + Q2E 2 , (15.8)S L = y 2 (1 − 2m2 e). (15.9)Q2 The differential cross section of the electronuclear scattering can be rewrittenas⎛d 2 σ eAdydQ = αy ⎝ (1 − y 2 )22 πQ 2 y 2 + Q2E 2⎞+ 1 4 − m2 e ⎠ σQ 2 γ ∗ A, (15.10)where σ γ ∗ A = σ γA (ν) for small Q 2 and must be approximated as a function ofɛ, ν, and Q 2 for large Q 2 . Interactions of longitudinal photons are includedin the effective σ γ ∗ A cross section through the ɛ factor, but in the presentGEANT4 method, the cross section of virtual photons is considered to beɛ-independent. The electronuclear problem, with respect to the interactionof virtual photons with nuclei, can thus be split in two. At small Q 2 it ispossible to use the σ γ (ν) cross section. In the Q 2 >> m 2 e region it is necessaryto calculate the effective σ γ ∗(ɛ, ν, Q 2 ) cross section.Following the EPA notation, the differential cross section of electronuclearscattering can be related to the number of equivalent photons dn = dσσ γ ∗ . Fory > Q 2 max ≃ m2 e leads toydn(y)= − α ( 1 + (1 − y)2y 2)ln(dy π 2 1 − y ) + (1 − y) . (15.12)In the y

(where l 1 = lncalculated forQ 2 maxQ 2 min), l 2 = 1 − Q2 maxQ 2 min()y, l 3 = ln2 +Q 2 max/E 2, Q 2 y 2 +Q 2 min /E2 min = m2 ey 2, is 1−yQ 2 max(m e) = 4m2 e1 − y . (15.14)The factor (1 − y) is used arbitrarily to keep Q 2 max(m > e) Q2 min , which canbe considered as a boundary between the low and high Q 2 regions. The fulltransverse photon flux can be calculated as an integral of Eq.(15.13) withthe maximum possible upper limitQ 2 max(max) = 4E2 (1 − y). (15.15)The full transverse photon flux can be approximated byydn(y)= − 2α ( (2 − y) 2 + y 2 )ln(γ) − 1 , (15.16)dy π 2where γ = E m e. It must be pointed out that neither this approximation norEq.(15.13) works at y ≃ 1; at this point Q 2 max(max)becomes smaller thanQ 2 1min . The formal limit of the method is y < 1 − . 2γIn Fig. 15.1(a,b) the energy distribution for the equivalent photons is shown.The low-Q 2 photon flux with the upper limit defined by Eq.(15.14)) is comparedwith the full photon flux. The low-Q 2 photon flux is calculated usingEq.(15.11) (dashed lines) and using Eq.(15.13) (dotted lines). The full photonflux is calculated using Eq.(15.16) (the solid lines) and using Eq.(15.13)with the upper limit defined by Eq.(15.15) (dash-dotted lines, which differfrom the solid lines only at ν ≈ E e ). The conclusion is that in order tocalculate either the number of low-Q 2 equivalent photons or the total numberof equivalent photons one can use the simple approximations given byEq.(15.11) and Eq.(15.16), respectively, instead of using Eq.(15.13), whichcannot be integrated over y analytically. Comparing the low-Q 2 photon fluxand the total photon flux it is possible to show that the low-Q 2 photon flux isabout half of the the total. From the interaction point of view the decrease ofσ γ∗ with increasing Q 2 must be taken into account. The cross section reductionfor the virtual photons with large Q 2 is governed by two factors. First,the cross section drops with Q 2 as the squared dipole nucleonic form-factorG 2 D (Q2 ) ≈(1 +Q 2(843 MeV ) 2 ) −2. (15.17)Second, all the thresholds of the γA reactions are shifted to higher ν by afactor Q2, which is the difference between the K and ν values. Following the2M233

0.050.0450.040.0350.030.0250.020.0150.010.00500.1n(ν)〈nσ γ*〉, mbaE e=1 GeVbE e=10 GeV0.080.06cd0.040.02010 10 210 10 2 10 3ν (MeV)Figure 15.1: Relative contribution of equivalent photons with small Q 2 tothe total “photon flux” for (a) 1 GeV electrons and (b) 10 GeV electrons. Infigures (c) and (d) the equivalent photon distribution dn(ν, Q 2 ) is multipliedby the photonuclear cross section σ γ ∗(K, Q 2 ) and integrated over Q 2 in tworegions: the dashed lines are integrals over the low-Q 2 equivalent photons(under the dashed line in the first two figures), and the solid lines are integralsover the high-Q 2 equivalent photons (above the dashed lines in the first twofigures).234

method proposed in [8] the σ γ ∗ at large Q 2 can be approximated asσ γ∗ = (1 − x)σ γ (K)G 2 D(Q 2 )e b(ɛ,K)·r+c(ɛ,K)·r3 , (15.18)where r = 1ln( Q2 +ν 2). The ɛ-dependence of the a(ɛ, K) and b(ɛ, K) functions2 K 2is weak, so for simplicity the b(K) and c(K) functions are averaged over ɛ.They can be approximated as()K 0.85b(K) ≈, (15.19)185 MeVand()K 3c(K) ≈ −. (15.20)1390 MeVThe result of the integration of the photon flux multiplied by the cross sectionapproximated by Eq.(15.18) is shown in Fig. 15.1(c,d). The integratedcross sections are shown separately for the low-Q 2 region (Q 2 < Q 2 max(m , e)dashed lines) and for the high-Q 2 region (Q 2 > Q 2 max(m e), solid lines). Thesefunctions must be integrated over ln(ν), so it is clear that because of theGiant Dipole Resonance contribution, the low-Q 2 part covers more than halfthe total eA → hadrons cross section. But at ν > 200 MeV , where thehadron multiplicity increases, the large Q 2 part dominates. In this sense, fora better simulation of the production of hadrons by electrons, it is necessaryto simulate the high-Q 2 part as well as the low-Q 2 part.Taking into account the contribution of high-Q 2 photons it is possible to useEq.(15.16) with the over-estimated σ γ ∗ A = σ γA (ν) cross section. The slightlyover-estimated electronuclear cross section isσeA ∗ = (2ln(γ) − 1) · J 1 − ln(γ) (2J 2 − J )3. (15.21)E e E ewhereJ 1 (E e ) = α π∫ EeσγA (ν)dln(ν) (15.22)J 2 (E e ) = α π∫ EeνσγA (ν)dln(ν), (15.23)and∫ Eeν 2 σ γA (ν)dln(ν). (15.24)J 3 (E e ) = α πThe equivalent photon energy ν = yE can be obtained for a particular randomnumber R from the equationR =(2ln(γ) − 1)J 1(ν) − ln(γ)E e(2J 2 (ν) − J 3(ν)E e)(2ln(γ) − 1)J 1 (E e ) − ln(γ)E e(2J 2 (E e ) − J 3(E e)E e) . (15.25)235

Eq.(15.13) is too complicated for the randomization of Q 2 but there is aneasily randomized formula which approximates Eq.(15.13) above the hadronicthreshold (E > 10 MeV ). It readswhereandwithπαD(y)∫ Q 2Q 2 minydn(y, Q 2 )dydQ 2 dQ 2 = −L(y, Q 2 ) − U(y), (15.26)D(y) = 1 − y + y22 , (15.27)()L(y, Q 2 ) = ln F (y) + (e P (y) − 1 +Q2) −1 , (15.28)U(y) = P (y) ·F (y) =(1 − Q2 minQ 2 max(2 − y)(2 − 2y)y 2)Q 2 min· Q2 minQ 2 max, (15.29)(15.30)andP (y) = 1 − yD(y) . (15.31)The Q 2 value can then be calculated asQ 2Q 2 min= 1 − e P (y) + ( e R·L(y,Q2 max )−(1−R)·U(y) − F (y) ) −1, (15.32)where R is a random number. In Fig. 15.2, Eq.(15.13) (solid curve) is comparedto Eq.(15.26) (dashed curve). Because the two curves are almost indistinguishablein the figure, this can be used as an illustration of the Q 2spectrum of virtual photons, which is the derivative of these curves. An alternativeapproach is to use Eq.(15.13) for the randomization with a threedimensional table ydndy (Q2 , y, E e ).After the ν and Q 2 values have been found, the value of σ γ ∗ A(ν, Q 2 ) is calculatedusing Eq.(15.18). If R · σ γA (ν) > σ γ ∗ A(ν, Q 2 ), no interaction occursand the electron keeps going. This “do nothing” process has low probabilityand cannot shadow other processes.15.3 Status of this documentcreated by ?20.05.02 re-written by D.H. Wright01.12.02 expanded section on electronuclear cross sections - H.P. Wellisch236

64E=10, y=0.001E=10, y=0.5E=10, y=0.9520πydn/αdy8642010E=100, y=0.001 E=100, y=0.5 E=100, y=0.95E=1000, y=0.5E=1000, y=0.955E=1000, y=0.001010 -6 10 -4 10 -2 1 10 2 10 4 10 61 10 10 2 10 3 10 4 10 5 10 6 10 10 2 10 3 10 4 10 5Q 2 (MeV 2 )Figure 15.2: Integrals of Q 2 spectra of virtual photons for three energies10 MeV , 100 MeV , and 1 GeV at y = 0.001, y = 0.5, and y = 0.95.The solid line corresponds to Eq.(15.13) and the dashed line (which almosteverywhere coincides with the solid line) corresponds to Eq.(15.13).237

Bibliography[1] E. Fermi, Z. Physik 29, 315 (1924).[2] K. F. von Weizsacker, Z. Physik 88, 612 (1934), E. J. Williams, Phys.Rev. 45, 729 (1934).[3] L. D. Landau and E. M. Lifshitz, Soc. Phys. 6, 244 (1934).[4] I. Ya. Pomeranchuk and I. M. Shmushkevich, Nucl. Phys. 23, 1295(1961).[5] V. N. Gribov et al., ZhETF 41, 1834 (1961).[6] L. D. Landau, E. M. Lifshitz, “Course of Theoretical Physics” v.4, part1, “Relativistic Quantum Theory”, Pergamon Press, p. 351, The methodof equivalent photons.[7] V. M. Budnev et al., Phys. Rep. 15, 181 (1975).[8] F. W. Brasse et al., Nucl. Phys. B 110, 413 (1976).238

Chapter 16Total Reaction Cross Section inNucleus-nucleus ReactionsThe transportation of heavy ions in matter is a subject of much interest inseveral fields of science. An important input for simulations of this processis the total reaction cross section, which is defined as the total (σ T ) minusthe elastic (σ el ) cross section for nucleus-nucleus reactions:σ R = σ T − σ el .The total reaction cross section has been studied both theoretically and experimentallyand several empirical parameterizations of it have been developed.In Geant4 the total reaction cross sections are calculated using foursuch parameterizations: the Sihver[1], Kox[2], Shen[3] and Tripathi[4] formulae.Each of these is discussed in order below.16.1 Sihver FormulaOf the four parameterizations, the Sihver formula has the simplest form:σ R = πr 2 0[A 1/3p + A 1/3t − b 0 [A −1/3p + A −1/3t ]] 2 (16.1)where A p and A t are the mass numbers of the projectile and target nuclei,andb 0 = 1.581 − 0.876(A −1/3p + A −1/3t ),r 0 = 1.36fm.239

It consists of a nuclear geometrical term (Ap 1/3 + A 1/3t ) and an overlap ortransparency parameter (b 0 ) for nucleons in the nucleus. The cross sectionis independent of energy and can be used for incident energies greater than100 MeV/nucleon.16.2 Kox and Shen FormulaeBoth the Kox and Shen formulae are based on the strong absorption model.They express the total reaction cross section in terms of the interaction radiusR, the nucleus-nucleus interaction barrier B, and the center-of-mass energyof the colliding system E CM :σ R = πR 2 [1 −BE CM]. (16.2)Kox formula: Here B is the Coulomb barrier (B c ) of the projectile-targetsystem and is given byZ t Z p e 2B c =r C (A 1/3t + Ap 1/3 ) ,where r C = 1.3 fm, e is the electron charge and Z t , Z p are the atomic numbersof the target and projectile nuclei. R is the interaction radius R int which inthe Kox formula is divided into volume and surface terms:R int = R vol + R surf .R vol and R surf correspond to the energy-independent and energy-dependentcomponents of the reactions, respectively. Collisions which have relativelysmall impact parameters are independent of both energy and mass number.These core collisions are parameterized by R vol . Therefore R vol can dependonly on the volume of the projectile and target nuclei:R vol = r 0 (A 1/3t + A 1/3p ).The second term of the interaction radius is a nuclear surface contributionand is parameterized byR surf = r 0 [a A1/3 t Ap1/3A 1/3t + Ap1/3− c] + D.The first term in brackets is the mass asymmetry which is related tothe volume overlap of the projectile and target. The second term c is240

an energy-dependent parameter which takes into account increasing surfacetransparency as the projectile energy increases. D is the neutron-excesswhich becomes important in collisions of heavy or neutron-rich targets. It isgiven byD = 5(A t − Z t )Z pA p A r.The surface component (R surf ) of the interaction radius is actually not partof the simple framework of the strong absorption model, but a better reproductionof experimental results is possible when it is used.The parameters r 0 , a and c are obtained using a χ 2 minimizing procedurewith the experimental data. In this procedure the parameters r 0 and a werefixed while c was allowed to vary only with the beam energy per nucleon. Thebest χ 2 fit is provided by r 0 = 1.1 fm and a = 1.85 with the correspondingvalues of c listed in Table III and shown in Fig. 12 of Ref. [2] as a functionof beam energy per nucleon. This reference presents the values of c only inchart and figure form, which is not suitable for Monte Carlo calculations.Therefore a simple analytical function is used to calculate c in Geant4. Thefunction is:c = − 10 + 2.0 for x ≥ 1.5x5 c = (− 101.5 5 + 2.0) × ( x1.5 )3 for x < 1.5,x = log(KE),where KE is the projectile kinetic energy in units of MeV/nucleon in thelaboratory system.Shen formula: as mentioned earlier, this formula is also based on the strongabsorption model, therefore it has a structure similar to the Kox formula:σ R = 10πR 2 [1 −BE CM]. (16.3)However, different parameterized forms for R and B are applied. The interactionradius R is given byt Ap1/3A 1/3pR = r 0 [A 1/3t + A 1/3p + 1.85 A1/3+α 5(A t − Z t )Z p+ βE −1/3CMA p A r241− C ′ (KE)]t + A 1/3A 1/3t A 1/3pA 1/3t + Ap1/3

where α, β and r 0 areα = 1fmβ = 0.176MeV 1/3 · fmr 0 = 1.1fmIn Ref. [3] as well, no functional form for C ′ (KE) is given. Hence the samesimple analytical function is used by Geant4 to derive c values.The second term B is called the nuclear-nuclear interaction barrier in theShen formula and is given byB = 1.44Z tZ pr− b R tR pR t + R p(MeV )where r, b, R t and R p are given byr = R t + R p + 3.2fmb = 1MeV · fm −1R i = 1.12A 1/3i − 0.94A −1/3i (i = t, p)The difference between the Kox and Shen formulae appears at energies below30 MeV/nucleon. In this region the Shen formula shows better agreementwith the experimental data in most cases.16.3 Tripathi formulaBecause the Tripathi formula is also based on the strong absorption modelits form is similar to the Kox and Shen formulae:σ R = πr0 2 (A1/3 p + A 1/3t + δ E ) 2 [1 − B ], (16.4)E CMwhere r 0 = 1.1 fm. In the Tripathi formula B and R are given byB = 1.44Z tZ pR242

R = r p + r t + 1.2(A1/3 p + A 1/3t )E 1/3CMwhere r i is the equivalent sphere radius and is related to the r rms,i radius byr i = 1.29r rms,i (i = p, t).δ E represents the energy-dependent term of the reaction cross sectionwhich is due mainly to transparency and Pauli blocking effects. It is givenbyδ E = 1.85S + (0.16S/E 1/3CM) − C KE + [0.91(A t − 2Z t )Z p /(A p A t )],where S is the mass asymmetry term given byS = A1/3 p A 1/3t.Ap 1/3 + A 1/3This is related to the volume overlap of the colliding system. The last termaccounts for the isotope dependence of the reaction cross section and correspondsto the D term in the Kox formula and the second term of R in theShen formula.The term C KE corresponds to c in Kox and C ′ (KE) in Shen and is givenbyC E = D P auli [1 − exp(−KE/40)] − 0.292 exp(−KE/792) × cos(0.229KE 0.453 ).tHere D P auli is related to the density dependence of the colliding system,scaled with respect to the density of the 12 C+ 12 C colliding system:D P auli = 1.75 ρ A p+ ρ Atρ AC + ρ AC.The nuclear density is calculated in the hard sphere model. D P auli simulatesthe modifications of the reaction cross sections caused by Pauli blocking andis being introduced to the Tripathi formula for the first time. The modificationof the reaction cross section due to Pauli blocking plays an importantrole at energies above 100 MeV/nucleon. Different forms of D P auli are usedin the Tripathi formula for alpha-nucleus and lithium-nucleus collisions. Foralpha-nucleus collisions,D P auli = 2.77 − (8.0 × 10 −3 A t ) + (1.8 × 10 −5 A 2 t )−0.8/{1 + exp[(250 − KE)/75]}243

For lithium-nucleus collisions,D P auli = D P auli /3.Note that the Tripathi formula is not fully implemented in Geant4 and canonly be used for projectile energies less than 1 GeV/nucleon.16.4 Representative Cross SectionsRepresentative cross section results from the Sihver, Kox, Shen and Tripathiformulae in Geant4 are displayed in Table I and compared to the experimentalmeasurements of Ref. [2].16.5 Tripathi Formula for ”light” SystemsFor nuclear-nuclear interactions in which the projectile and/or target arelight, Tripathi et al [6] propose an alternative algorithm for determining theinteraction cross section (implemented in the new class G4TripathiLightCrossSection).For such systems, Eq.16.4 becomes:σ R = πr0 2 [A1/3 p + A 1/3t + δ E ] 2 B(1 − R C )X m (16.5)E CMR C is a Coulomb multiplier, which is added since for light systems Eq. 16.4overestimates the interaction distance, causing B (in Eq. 16.4) to be underestimated.Values for R C are given in Table 16.2.where:(X m = 1 − X 1 exp −E )X 1 S L(16.6)X 1 = 2.83 − ( 3.1 × 10 −2) A T + ( 1.7 × 10 −4) A 2 T (16.7)except for neutron interactions with 4 He, for which X 1 is better approximatedto 5.2, and the function S L is given by:[ (S L = 1.2 + 1.6 1 − exp − E )]15(16.8)For light nuclear-nuclear collisions, a slightly more general expression for C Eis used:244

[C E = D 1 − exp(− E )](− 0.292 exp − E )· cos ( 0.229E 0.453) (16.9)T 1 792D and T 1 are dependent on the interaction, and are defined in table 16.3.16.6 Status of this document25.11.03 created by Tatsumi Koi28.11.03 grammar check and re-wording by D.H. Wright18.06.04 light system section added by Peter TruscottBibliography[1] L. Sihver et al., Phys. Rev. C47, 1225 (1993).[2] Kox et al. Phys. Rev. C35, 1678 (1987).[3] Shen et al. Nucl. Phys. A491, 130 (1989).[4] Tripathi et al, NASA Technical Paper 3621 (1997).[5] Jaros et al, Phys. Rev. C 18 2273 (1978).[6] R K Tripathi, F A Cucinotta, and J W Wilson, ”Universal parameterizationof absorption cross-sections - Light systems,” NASA TechnicalPaper TP-1999-209726, 1999.245

Table 16.1: Representative total reaction cross sectionsProj. Target Elab Exp. Results Sihver Kox Shen Tripathi[MeV/n] [mb]12 C 12 C 30 1316±40 — 1295.04 1316.07 1269.2483 965±30 — 957.183 969.107 989.96200 864±45 868.571 885.502 893.854 864.56300 858±60 868.571 871.088 878.293 857.414870 1 939±50 868.571 852.649 857.683 939.412100 1 888±49 868.571 846.337 850.186 936.20527 Al 30 1748±85 — 1801.4 1777.75 1701.0383 1397±40 — 1407.64 1386.82 1405.61200 1270±70 1224.95 1323.46 1301.54 1264.26300 1220±85 1224.95 1306.54 1283.95 1257.6289 Y 30 2724±300 — 2898.61 2725.23 2567.6883 2124±140 — 2478.61 2344.26 2346.54200 1885±120 2156.47 2391.26 2263.77 2206.01300 1885±150 2156.47 2374.17 2247.55 2207.0116 O 27 Al 30 1724±80 — 1965.85 1935.2 1872.2389 Y 30 2707±330 — 3148.27 2957.06 2802.4820 Ne 27 Al 30 2113±100 — 2097.86 2059.4 2016.32100 1446±120 1473.87 1684.01 1658.31 1667.17300 1328±120 1473.87 1611.88 1586.17 1559.16108 Ag 300 2407±200 2 2730.69 3095.18 2939.86 2893.121. Data measured by Jaros et al. [5]2. Natural silver was used in this measurement.246

Table 16.2: Coulomb multiplier for light systems [6].SystemR Cp + d 13.5p + 3 He 21p + 4 He 27p + Li 2.2d + d 13.5d + 4 He 13.5d + C 6.04 He + Ta 0.64 He + Au 0.6247

Table 16.3: Parameters D and T1 for light systems [6].System T1 [MeV] D G [MeV]( 4 He + X only)p + X 23 1.85 +n + X 18 1.85 +0.161+exp( 500−E200 )0.161+exp( 500−E200 )d + X 23 1.65 +0.11+exp( 500−E200 )(Not applicable)(Not applicable)(Not applicable)3 He + X 40 1.55 (Not applicable)4 He + 4 He 40D = 2.77 − 8.0 × 10 −3 A T+1.8 × 10 −5 A 2 T0.8−1+exp( 250−EG )4 He + Be 25 (as for 4 He + 4 He) 3004 He + N 40 (as for 4 He + 4 He) 5004 He + Al 25 (as for 4 He + 4 He) 3004 He + Fe 40 (as for 4 He + 4 He) 3004 He + X (general) 40 (as for 4 He + 4 He) 75300248

Chapter 17Coherent elastic scattering17.1 Nucleon-Nucleon elastic ScatteringThe classes G4LEpp and G4LEnp provide data-driven models for protonproton(or neutron-neutron) and neutron-proton elastic scattering over therange 10-1200 MeV. Final states (primary and recoil particle) are derived bysampling from tables of the cumulative distribution function of the centreof-massscattering angle, tabulated for a discrete set of lab kinetic energiesfrom 10 MeV to 1200 MeV. The CDF’s are tabulated at 1 degree intervalsand sampling is done using bi-linear interpolation in energy and CDF values.The data are derived from differential cross sections obtained from the SAIDdatabase, R. Arndt, 1998.In class G4LEpp there are two data sets: one including Coulomb effects(for p-p scattering) and one with no Coulomb effects (for n-n scatteringor p-p scattering with Coulomb effects suppressed). The methodG4LEpp::SetCoulombEffects can be used to select the desired data set:• SetCoulombEffects(0): No Coulomb effects (the default)• SetCoulombEffects(1): Include Coulomb effectsThe recoil particle will be generated as a new secondary particle. In classG4LEnp, the possiblity of a charge-exchange reaction is included, in whichcase the incident track will be stopped and both the primary and recoilparticles will be generated as secondaries.249

Chapter 18Hadron-nucleus ElasticScattering at Medium and HighEnergy.18.1 Method of CalculationThe Glauber model [1] is used as an alternative method of calculating differentialcross sections for elastic and quasi-elastic hadron-nucleus scatteringat high and intermediate energies.For high energies this includes corrections for inelastic screening and forquasi-elastic scattering the exitation of a discrete level or a state in the continuumis considered.The usual expression for the Glauber model amplitude for multiple scatteringwas usedF (q) = ik ∫2πd 2 be ⃗ q·⃗b M( ⃗ b). (18.1)Here M( ⃗ b) is the hadron-nucleus amplitude in the impact parameter representationM( ⃗ b) = 1 − [1 − e −A ∫ d 3 rΓ( ⃗ b− ⃗ s)ρ(⃗r) ] A , (18.2)k is the incident particle momentum, ⃗q = ⃗ k ′ − ⃗ k is the momentum transfer,and ⃗ k ′ is the scattered particle momentum. Note that |⃗q| 2 = −t - invariantmomentum transfer squared in the center of mass system. Γ( ⃗ b) is thehadron-nucleon amplitude of elastic scattering in the impact-parameter representation250

Γ( ⃗ ∫1b) =2πik hN d⃗qe −⃗q·⃗b f(⃗q). (18.3)The exponential parameterization of the hadron-nucleon amplitude isusually used:f(⃗q) = ikhN σ hNe −0.5q2B . (18.4)2πHere σ hN = σtot hN (1 − iα), σtothN is the total cross section of a hadron-nucleonscattering, B is the slope of the diffraction cone and α is the ratio of the realto imaginary parts of the amplitude at q = 0. The value k hN is the hadronmomentum in the hadron-nucleon coordinate system.The important difference of these calculations from the usual ones is thatthe two-gaussian form of the nuclear density was usedρ(r) = C(e −(r/R 1) 2 − pe −(r/R 2) 2 ), (18.5)where R 1 , R 2 and p are the fitting parameters and C is a normalizationconstant.This density representation allows the expressions for amplitude and differentialcross section to be put into analytical form. It was earlier used forlight [2, 3] and medium [4] nuclei. Described below is an extension of thismethod to heavy nuclei. The form 18.5 is not physical for a heavy nucleus,but nevertheless works rather well (see figures below). The reason is thatthe nucleus absorbs the hadrons very strongly, especially at small impactparameters where the absorption is full. As a result only the peripherial partof the nucleus participates in elastic scattering. Eq. 18.5 therefore describesonly the edge of a heavy nucleus.Substituting Eqs. 18.5 and 18.4 into Eqs. 18.1, 18.2 and 18.3 yields thefollowing formulaF (q) = ikπ2( )A∑ Ak=1(−1) k σ hN( ) [k∑ k[k 2π(R1 3 − pR2) 3 ]km=0(−1) m R13m R1 2 + 2B] k−m×[pR23 ] m (R2 2 + 2BmR 2 2 + 2B +k − m ) −1R1 2 + 2B× exp⎡⎣− −q24(mR2 2 + 2B + k − m ) ⎤ −1⎦ . (18.6)R1 2 + 2B251

An analogous procedure can be used to get the inelastic screening correctionsto the hadron-nucleus amplitude ∆M( ⃗ b) [5]. In this case an intermediateinelastic diffractive state is created which rescatters on the nucleonsof the nucleus and then returns into the initial hadron. Hence it is nessesaryto integrate the production cross section over the mass distribution ofthe exited system dσdiff . The expressions for the corresponding amplitudedtdMx2are quite long and so are not presented here. The corrections for the totalcross-sections can be found in [5].The full amplitude is the sum M( ⃗ b) + ∆M( ⃗ b).The differential cross section is connected with the amplitude in the followingwaydσ= |F (q)| 2 dσ,dΩ CM |dt| = dσ = π |F (q)| 2 . (18.7)dqCM2 kCM2The main energy dependence of the hadron-nucleus elastic scatteringcross section comes from the energy dependence of the parameters of hadronnucleonscattering (σtot hNdσdiffα, B and ). At interesting energies these param-dtdMx 2eters were fixed at their well-known values. The fitting of the nuclear densityparameters was performed over a wide range of atomic numbers (A = 4−208)using experimental data on proton-nuclei elastic scattering at a kinetic energyof T p = 1GeV .The fitting was perfomed both for individual nuclei and for the entire setof nuclei at once.It is necessary to note that for every nucleus an optimal set of densityparameters exists and it differs slightly from the one derived for the full setof nuclei.A comparision of the phenomenological cross sections [6] with experimentis presented in Figs. 18.1 - 18.9In this comparison, the individual nuclei parameters were used. Theexperimental data were obtained in Gatchina (Russia) and in Saclay (France)[6]. The horizontal axis is the scattering angle in the center of mass systemΘ CM and the vertical axis is dσdΩin mbCM Ster .Comparisions were also made for p 4 He elastic scatering at T = 1GeV [7],45GeV and 301GeV [3]. The resulting cross sections dσ are shown in thed|t|Figs. 18.10 - 18.12.In order to generate events the distribution function F of a correspondingprocess must be known. The differential cross section is proportional to thedensity distribution. Therefore to get the distribution function it is sufficientto integrate the differential cross section and normalize it:252

F(q 2 ) =∫ q20q∫max20d(q 2 ) dσd(q 2 )d(q 2 ) dσd(q 2 ) . (18.8)Expressions 18.6 and 18.7 allow analytic integration in Eq. 18.8 but theresult is too long to be given here.For light and medium nuclei the analytic expression is more convenientfor calculations than the numerical integration of Eq. 18.8, but for heavynuclei the latter is preferred due to the large number of terms in the analyticexpression.18.2 Status of this document18.06.04 created by Nikolai Starkov19.06.04 re-written for spelling and grammar by D.H. WrightBibliography[1] R.J. Glauber, in ”High Energy Physics and Nuclear Structure”, editedby S. Devons (Plenum Press, NY 1970).[2] R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968;T. T. Chou, Phys. Rev., 168, 1594, 1968;M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys., A312,pp. 209-216, 1978.[3] Bujak, P. Devensky, A. Kuznetsov et al., Phys. Rev., D23, N 9, pp.1895-1910, 1981.[4] V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., v. 37, N 4,pp. 610-613, 1983;N. T. Ermekov, V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl.Phys., 33, N 6, pp. 775-777, 1981.[5] R.A. Nam, S. I. Nikol’skii, N. I. Starkov et al., Sov. Journ. of Nucl.Phys., v. 26, N 5, pp. 550-555, 1977.253

[6] G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144;[7] J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., Phys. Rev. Lett. 38, N22, pp. 1265-1268;S. J. Wallace. Y. Alexander, Phys. Rev. Lett. 38, N 22, pp. 1269-1272.254

Figure 18.1: Elastic proton scattering on 9 Be at 1 GeV255

Figure 18.2: Elastic proton scattering on 11 B at 1 GeV256

Figure 18.3: Elastic proton scattering on 12 C at 1 GeV257

Figure 18.4: Elastic proton scattering on 16 O at 1 GeV258

Figure 18.5: Elastic proton scattering on 28 Si at 1 GeV259

Figure 18.6: Elastic proton scattering on 40 Ca at 1 GeV260

Figure 18.7: Elastic proton scattering on 58 Ni at 1 GeV261

Figure 18.8: Elastic proton scattering on 90 Zr at 1 GeV262

Figure 18.9: Elastic proton scattering on 208 Pb at 1 GeV263

Figure 18.10: Elastic proton scattering on 4 He at 1 GeV264

Figure 18.11: Elastic proton scattering on 4 He at 45 GeV265

Figure 18.12: Elastic proton scattering on 4 He at 301 GeV266

Chapter 19Interactions of StoppingParticles19.1 Complementary parameterised and theoreticaltreatmentAbsorption of negative pions and kaons at rest from a nucleus is describedin literature [1], [2], [3], [4] as consisting of two main components:• a primary absorption process, involving the interaction of the incidentstopped hadron with one or more nucleons of the target nucleus;• the deexcitation of the remnant nucleus, left in an excitated state as aresult of the occurrence of the primary absorption process.This interpretation is supported by several experiments [5], [6], [7], [8], [9],[10], [11], that have measured various features characterizing these processes.In many cases the experimental measurements are capable to distinguish thefinal products originating from the primary absorption process and thoseresulting from the nuclear deexcitation component.A set of stopped particle absorption processes is implemented in GEANT4,based on this two-component model (PiMinusAbsorptionAtRest and Kaon-MinusAbsorptionAtRest classes, for π − and K − respectively. Both implementationsadopt the same approach: the primary absorption componentof the process is parameterised, based on available experimental data; thenuclear deexcitation component is handled through the theoretical modelsdescribed elsewhere in this Manual.267

19.1.1 Pion absorption at restThe absorption of stopped negative pions in nuclei is interpreted [1], [2],[3], [4] as starting with the absorption of the pion by two or more correlatednucleons; the total energy of the pion is transferred to the absorbing nucleons,which then may leave the nucleus directly, or undergo final-state interactionswith the residual nucleus. The remaining nucleus de-excites by evaporationof low energetic particles.G4PiMinusAbsorptionAtRest generates the primary absorption componentof the process through the parameterisation of existing experimentaldata; the primary absorption component is handled by class G4PiMinusStopAbsorption.In the current implementation only absorption on a nucleon pair is considered,while contributions from absorption on nucleon clusters are neglected;this approximation is supported by experimental results [1], [13] showing thatit is the dominating contribution.Several features of stopped pion absorption are known from experimentalmeasurements on various materials [5], [6], [7], [8], [9], [10], [11], [12]:• the average number of nucleons emitted, as resulting from the primaryabsorption process;• the ratio of nn vs np as nucleon pairs involved in the absorption process;• the energy spectrum of the resulting nucleons emitted and their openingangle distribution.The corresponding final state products and related distributions are generatedaccording to a parameterisation of the available experimental measurementslisted above. The dependence on the material is handled by a strategypattern: the features pertaining to material for which experimental data areavailable are treated in G4PiMinusStopX classes (where X represents an element),inheriting from G4StopMaterial base class. In case of absorption onan element for which experimental data are not available, the experimentaldistributions for the elements closest in Z are used.The excitation energy of the residual nucleus is calculated by differencebetween the initial energy and the energy of the final state products of theprimary absorption process.Another strategy handles the nucleus deexcitation; the current defaultimplementation consists in handling the deexcitatoin component of the processthrough the evaporation model described elsewhere in this Manual.268

Bibliography[1] E. Gadioli and E. Gadioli Erba Phys. Rev. C 36 741 (1987)[2] H.C. Chiang and J. Hufner Nucl. Phys. A352 442 (1981)[3] D. Ashery and J. P. Schiffer Ann. Rev. Nucl. Part. Sci. 36 207 (1986)[4] H. J. Weyer Phys. Rep. 195 295 (1990)[5] R. Hartmann et al., Nucl. Phys. A300 345 (1978)[6] R. Madley et al., Phys. Rev. C 25 3050 (1982)[7] F. W. Schleputz et al., Phys. Rev. C 19 135 (1979)[8] C.J. Orth et al., Phys. Rev. C 21 2524 (1980)[9] H.S. Pruys et al., Nucl. Phys. A316 365 (1979)[10] P. Heusi et al., Nucl. Phys. A407 429 (1983)[11] H.P. Isaak et al., Nucl. Phys. A392 368 (1983)[12] H.P. Isaak et al., Helvetica Physica Acta 55 477 (1982)[13] H. Machner Nucl. Phys. A395 457 (1983)269

Chapter 20Parametrization Driven Models20.1 IntroductionTwo sets of parameterized models are provided for the simulation of highenergy hadron-nucleus interactions. The so-called “low energy model” is intendedfor hadronic projectiles with incident energies between 1 GeV and25 GeV, while the “high energy model” is valid for projectiles between 25GeV and 10 TeV. Both are based on the well-known GHEISHA package ofGEANT3. The physics underlying these models comes from an old-fashionedmulti-chain model in which the incident particle collides with a nucleon insidethe nucleus. The final state of this interaction consists of a recoil nucleon, thescattered incident particle, and possibly many hadronic secondaries. Hadronproduction is approximated by the formation zone concept, in which theinteracting quark-partons require some time and therefore some range tohadronize into real particles. All of these particles are able to re-interactwithin the nucleus, thus developing an intra-nuclear cascade.In these models only the first hadron-nucleon collision is simulated in detail.The remaining interactions within the nucleus are simulated by generatingadditional hadrons and treating them as secondaries from the initial collision.The numbers, types and distributions of the extra hadrons are determinedby functions which were fitted to experimental data or which reproduce generaltrends in hadron-nucleus collisions. Numerous tunable parameters areused throughout these models to obtain reasonable physical behavior. Thisrestricts the use of these models as generators for hadron-nucleus interactionsbecause it is not always clear how the parameters relate to physicalquantities. On the other hand a precise simulation of minimum bias eventsis possible, with significant predictive power for calorimetry.270

20.2 Low Energy ModelIn the low energy parameterized model the mean number of hadrons producedin a hadron-nucleus collision is given byN m = C(s)A 1/3 N ic (20.1)where A is the atomic mass, C(s) is a function only of the center of massenergy s, and N ic is approximately the number of hadrons generated in theinitial collision. Assuming that the collision occurs at the center of the nucleus,each of these hadrons must traverse a distance roughly equal to thenuclear radius. They may therefore potentially interact with a number ofnucleons proportional to A 1/3 . If the energy-dependent cross section for interactionin the nuclear medium is included in C then Eq. 20.1 can beinterpreted as the number of target nucleons excited by the initial collision.Some of these nucleons are added to the intra-nuclear cascade. The rest,especially at higher momenta where nucleon production is suppressed, arereplaced by pions and kaons.Once the mean number of hadrons, N m is calculated, the total number ofhadrons in the intra-nuclear cascade is sampled from a Poisson distributionabout the mean. Sampling from additional distribution functions provides• the combined multiplicity w(⃗a, n i ) for all particles i, i = π + , π 0 , π − , p, n, .....,including the correlations between them,• the additive quantum numbers E (energy), Q (charge), S (strangeness)and B (baryon number) in the entire phase space region, and• the reaction products from nuclear fission and evaporation.A universal function f( ⃗ b, x/p T , m T ) is used for the distribution of the additivequantum numbers, where x is the Feynman variable, p T is the transversemomentum and m T is the transverse mass. ⃗a and ⃗ b are parameter vectors,which depend on the particle type of the incoming beam and the atomicnumber A of the target. Any dependence on the beam energy is completelyrestricted to the multiplicity distribution and the available phase space.The low energy model can be applied to the π + , π − , K + , K − , K 0 and K 0mesons. It can also be applied to the baryons p, n, Λ, Σ + , Σ − , Ξ 0 , Ξ − , Ω − ,and their anti-particles, as well as the light nuclei, d, t and α. The model canin principal be applied down to zero projectile energy, but the assumptionsused to develop it begin to break down in the sub-GeV region.271

20.3 High Energy ModelThe high energy model is valid for incident particle energies from 10-20 GeVup to 10-20 TeV. Individual implementations of the model exist for π + , π − ,K + , K − , KS0 and K0 L mesons, and for p, n, Λ, Σ+ , Σ − , Ξ 0 , Ξ − , and Ω −baryons and their anti-particles.20.3.1 Initial InteractionIn a given implementation, the generation of the final state begins with theselection of a nucleon from the target nucleus. The pion multiplicities resultingfrom the initial interaction of the incident particle and the target nucleonare then calculated. The total pion multiplicity is taken to be a function ofthe log of the available energy in the center of mass of the incident particleand target nucleon, and the π + , π − and π 0 multiplicities are given by theKNO distribution.From this initial set of particles, two are chosen at random to be replacedwith either a kaon-anti-kaon pair, a nucleon-anti-nucleon pair, or a kaon anda hyperon. The relative probabilities of these options are chosen according toa logarithmically interpolated table of strange-pair and nucleon-anti-nucleonpair cross sections. The particle types of the pair are chosen according toaveraged, parameterized cross sections typical at energies of a few GeV. Ifthe increased mass of the new pair causes the total available energy to beexceeded, particles are removed from the initial set as necessary.20.3.2 Intra-nuclear CascadeThe cascade of these particles through the nucleus, and the additional particlesgenerated by the cascade are simulated by several models. These includehigh energy cascading, high energy cluster production, medium energy cascadingand medium energy cluster production. For each event, high energycascading is attempted first. If the available energy is sufficient, this methodwill likely succeed in producing the final state and the interaction will havebeen completely simulated. If it fails due to lack of energy or other reasons,the remaining models are called in succession until the final state is produced.If none of these methods succeeds, quasi-elastic scattering is attempted andfinally, as a last resort, elastic scattering is performed. These models areresponsible for assigning final state momenta to all generated particles, andfor checking that, on average, energy and momentum are conserved.272

20.3.3 High Energy CascadingAs particles from the initial collision cascade through the nucleus more particleswill be generated. The number and type of these particles are parameterizedin terms of the CM energy of the initial particle-nucleon collision.The number of particles produced from the cascade is given roughly byN m = C(s)[A 1/3 − 1]N ic (20.2)where A is the atomic mass, C(s) is a function only of s, the square ofthe center of mass energy, and N ic is approximately the number of hadronsgenerated in the initial collision. This can be understood qualitatively byassuming that the collision occurs, on average, at the center of the nucleus.Then each of the N ic hadrons must traverse a distance roughly equal tothe nuclear radius. They may therefore potentially interact with a numberof nucleons proportional to A 1/3 . If the energy-dependent cross section forinteraction in the nuclear medium is included in C(s) then Eq. 20.2 can beinterpreted as the number of target nucleons excited by the initial collisionand its secondaries.Some of these nucleons are added to the intra-nuclear cascade. The rest,especially at higher momenta where nucleon production is suppressed, arereplaced by pions, kaons and hyperons. The mean of the total number ofhadrons generated in the cascade is partitioned into the mean number ofnucleons, N n , pions, N π and strange particles, N s . Each of these is used asthe mean of a Poisson distribution which produces the randomized numberof each type of particle.The momenta of these particles are generated by first dividing the finalstate phase space into forward and backward hemispheres, where forward isin the direction of the original projectile. Each particle is assigned to onehemisphere or the other according to the particle type and origin:• the original projectile, or its substitute if charge or strangeness exchangeoccurs, is assigned to the forward hemisphere and the targetnucleon is assigned to the backward hemisphere;• the remainder of the particles from the initial collision are assigned atrandom to either hemisphere;• pions and strange particles generated in the intra-nuclear cascade areassigned 80% to the backward hemisphere and 20% to the forwardhemisphere;• nucleons generated in the intra-nuclear cascade are all assigned to thebackward hemisphere.273

It is assumed that energy is separately conserved for each hemisphere. Iftoo many particles have been added to a given hemisphere, randomly chosenparticles are deleted until the energy budget is met. The final state momentaare then generated according to two different algorithms, a cluster model forthe backward nucleons from the intra-nuclear cascade, and a fragmentationmodel for all other particles. Several corrections are then applied to the finalstate particles, including momentum re-scaling, effects due to Fermi motion,and binding energy subtraction. Finally the de-excitation of the residual nucleusis treated by adding lower energy protons, neutrons and light ions tothe final state particle list.Fragmentation Model. This model simulates the fragmentation of thehighly excited hadrons formed in the initial projectile-nucleon collision. Particlemomenta are generated by first sampling the average transverse momentump T from an exponential distribution:whereexp[−ap T b ] (20.3)1.70 ≤ a ≤ 4.00; 1.18 ≤ b ≤ 1.67. (20.4)The values of a and b depend on particle type and result from a parameterizationof experimental data. The value selected for p T is then used to setthe scale for the determination of x, the fraction of the projectile’s momentumcarried by the fragment. The sampling of x assumes that the invariantcross section for the production of fragments can be given byE d3 σdp 3 =K(M 2 x 2 + p T 2) 3/2 (20.5)where E and p are the energy and momentum, respectively, of the producedfragment, and K is a proportionality constant. M is the average transversemass which is parameterized from data and varies from 0.75 GeV to 0.10GeV, depending on particle type. Taking m to be the mass of the fragmentand noting thatp z ≃ xE proj (20.6)in the forward hemisphere andp z ≃ xE targ (20.7)274

in the backward hemisphere, Eq. 20.5 can be re-written to give the samplingfunction for x:d 3 σdp 3 =K1(M 2 x 2 + p 2 T ) 3/2 √, (20.8)m 2 + p 2 T + x 2 Ei2where i = proj or targ.x-sampling is performed for each fragment in the final-state candidatelist. Once a fragment’s momentum is assigned, its total energy is checked tosee if it exceeds the energy budget in its hemisphere. If so, the momentumof the particle is reduced by 10%, as is p T and the integral of the x-samplingfunction, and the momentum selection process is repeated. If the offendingparticle starts out in the forward hemisphere, it is moved to the backwardhemisphere, provided the budget for the backward hemisphere is not exceeded.If, after six iterations, the particle still does not fit, it is removedfrom the candidate list and the kinetic energies of the particles selected up tothis point are reduced by 5%. The entire procedure is repeated up to threetimes for each fragment.The incident and target particles, or their substitutes in the case of chargeorstrangeness-exchange, are guaranteed to be part of the final state. Theyare the last particles to be selected and the remaining energy in their respectivehemispheres is used to set the p z components of their momenta. The p Tcomponents selected by x-sampling are retained.Cluster Model. This model groups the nucleons produced in the intranuclearcascade together with the target nucleon or hyperon, and treatsthem as a cluster moving forward in the center of mass frame. The clusterdisintegrates in such a way that each of its nucleons is given a kineticenergy40 < T nuc < 600MeV (20.9)if the kinetic energy of the original projectile, T inc , is 5 GeV or more. If T incis less than 5 GeV,40(T inc /5GeV) 2 < T nuc < 600(T inc /5GeV) 2 . (20.10)In each range the energy is sampled from a distribution which is skewedstrongly toward the high energy limit. In addition, the angular distributionof the nucleons is skewed forward in order to simulate the forward motion ofthe cluster.Momentum Re-scaling. Up to this point, all final state momenta havebeen generated in the center of mass of the incident projectile and the target275

nucleon. However, the interaction involves more than one nucleon as evidencedby the intra-nuclear cascade. A more correct center of mass shouldthen be defined by the incident projectile and all of the baryons generated bythe cascade, and the final state momenta already calculated must be re-scaledto reflect the new center of mass.This is accomplished by correcting the momentum of each particle in thefinal state candidate list by the factor T 1 /T 2 . T 2 is the total kinetic energy inthe lab frame of all the final state candidates generated assuming a projectilenucleoncenter of mass. T 1 is the total kinetic energy in the lab frame of thesame final state candidates, but whose momenta have been calculated by thephase space decay of an imaginary particle. This particle has the total CMenergy of the original projectile and a cluster consisting of all the baryonsgenerated from the intra-nuclear cascade.Corrections. Part of the Fermi motion of the target nucleons is taken intoaccount by smearing the transverse momentum components of the final stateparticles. The Fermi momentum is first sampled from an average distributionand a random direction for its transverse component is chosen. Thiscomponent, which is proportional to the number of baryons produced in thecascade, is then included in the final state momenta.Each final state particle must escape the nucleus, and in the processreduce its kinetic energy by the nuclear binding energy. The binding energyis parameterized as a function of A:E B = 25MeV( ) A − 1e −(A−1)/120) . (20.11)120Another correction reduces the kinetic energy of final state π 0 s when theincident particle is a π + or π − . This reduction increases as the log of theincident pion energy, and is done to reproduce experimental data. In order toconserve energy on average, the energy removed from the π 0 s is re-distributedamong the final state π + s, π − s and π 0 s.Nuclear De-excitation. After the generation of initial interaction andcascade particles, the target nucleus is left in an excited state. De-excitationis accomplished by evaporating protons, neutrons, deuterons, tritons andalphas from the nucleus according to a parameterized model. The totalkinetic energy given to these particles is a function of the incident particlekinetic energy:T evap = 7.716GeV( ) A − 1F(T)e −F(T)−(A−1)/120 , (20.12)120276

whereandF (T ) = max[0.35 + 0.1304ln(T), 0.15], (20.13)T = 0.1GeV for T inc < 0.1GeV (20.14)T = T inc for 0.1GeV ≤ T inc ≤ 4GeV (20.15)T = 4GeV for T inc > 4GeV. (20.16)The mean energy allocated for proton and neutron emission is T pn and thatfor deuteron, triton and alpha emission is T dta . These are determined bypartitioning T evap :T pn = T evap R , T dta = T evap (1 − R) withR = max[1 − (T/4GeV) 2 , 0.5]. (20.17)The simulated values of T pn and T dta are sampled from normal distributionsabout T pn and T dta and their sum is constrained not to exceed the incidentparticle’s kinetic energy, T inc .The number of proton and neutrons emitted, N pn , is sampled from aPoisson distribution about a mean which depends on R and the number ofbaryons produced in the intranuclear cascade. The average kinetic energyper emitted particle is then T av = T pn /N pn . T av is used to parameterize anexponential which qualitatively describes the nuclear level density as a functionof energy. The simulated kinetic energy of each evaporated proton orneutron is sampled from this exponential. Next, the nuclear binding energyis subtracted and the final momentum is calculated assuming an isotropic angulardistribution. The number of protons and neutrons emitted is (Z/A)N pnand (N/A)N pn , respectively.A similar procedure is followed for the deuterons, tritons and alphas. Thenumber of each species emitted is 0.6N dta , 0.3N dta and 0.1N dta , respectively.Tuning of the High Energy Cascade The final stage of the high energycascade method involves adjusting the momenta of the produced particlesso that they agree better with data. Currently, five such adjustments areperformed, the first three of which apply only to charged particles incidentupon light and medium nuclei at incident energies above ≃ 65 GeV.• If the final state particle is a nucleon or light ion with a momentum ofless than 1.5 GeV/c, its momentum will be set to zero some fractionof the time. This fraction increases with the logarithm of the kineticenergy of the incident particle and decreases with log 10 (A).277

• If the final state particle with the largest momentum happens to be aπ 0 , its momentum is exchanged with either the π + or π − having thelargest momentum, depending on whether the incident particle chargeis positive or negative.• If the number of baryons produced in the cascade is a significant fraction(> 0.3) of A, about 25% of the nucleons and light ions alreadyproduced will be removed from the final particle list, provided theirmomenta are each less than 1.2 GeV/c.• The final state of the interaction is of course heavily influenced by thequantum numbers of the incident particle, particularly in the forwarddirection. This influence is enforced by compiling, for each forwardgoingfinal state particle, the sumS forward = ∆ M + ∆ Q + ∆ S + ∆ B , (20.18)where each ∆ corresponds to the absolute value of the difference of thequantum number between the incident particle and the final state particle.M, Q, S, and B refer to mass, charge, strangeness and baryonnumber, respectively. For final state particles whose character is significantlydifferent from the incident particle (S is large), the momentumcomponent parallel to the incident particle momentum is reduced. Thetransverse component is unchanged. As a result, large-S particles aredriven away from the axis of the hadronic shower. For backward-goingparticles, a similar procedure is followed based on the calculation ofS backward .• Conservation of energy is imposed on the particles in the final state listin one of two ways, depending on whether or not a leading particle hasbeen chosen from the list. If all the particles differ significantly fromthe incident particle in momentum, mass and other quantum numbers,no leading particle is chosen and the kinetic energy of each particle isscaled by the same correction factor. If a leading particle is chosen,its kinetic energy is altered to balance the total energy, while all theremaining particles are unaltered.20.3.4 High Energy Cluster ProductionAs in the high energy cascade model, the high energy cluster model randomlyassigns particles from the initial collision to either a forward- or backwardgoingcluster. Instead of performing the fragmentation process, however,278

the two clusters are treated kinematically as the two-body final state of thehadron-nucleon collision. Each cluster is assigned a kinetic energy T whichis sampled from a distributionexp[−aT 1/b ] (20.19)where both a and b decrease with the number of particles in a cluster. If thecombined total energy of the two clusters is larger than the center of massenergy, the energy of each cluster is reduced accordingly. The center of massmomentum of each cluster can then be found by sampling the 4-momentumtransfer squared, t, from the distributionexp[t(4.0 + 1.6ln(p inc ))] (20.20)where t < 0 and p inc is the incident particle momentum. Then,cosθ = 1 + t − (E c − E i ) 2 + (p c − p i ) 22p c p i, (20.21)where the subscripts c and i refer to the cluster and incident particle, respectively.Once the momentum of each cluster is calculated, the clusteris decomposed into its constituents. The momenta of the constituents aredetermined using a phase space decay algorithm.The particles produced in the intra-nuclear cascade are grouped into athird cluster. They are treated almost exactly as in the high energy cascademodel, where Eq. 20.2 is used to estimate the number of particles produced.The main difference is that the cluster model does not generate strange particlesfrom the cascade. Nucleon suppression is also slightly stronger, leadingto relatively higher pion production at large incident momenta. Kinetic energyand direction are assigned to the cluster as described in the clustermodel paragraph in the previous section.The remaining steps to produce the final state particle list are the sameas those in high energy cascading:• re-scaling of the momenta to reflect a center of mass which involves thecascade baryons,• corrections due to Fermi motion and binding energy,• reduction of final state π 0 energies,• nuclear de-excitation and• high energy tuning.279

20.3.5 Medium Energy CascadingThe medium energy cascade algorithm is very similar to the high energy cascadealgorithm, but it may be invoked for lower incident energies (down to 1GeV). The primary difference between the two codes is the parameterizationof the fragmentation process. The medium energy cascade samples largertransverse momenta for pions and smaller transverse momenta for kaons andbaryons.A second difference is in the treatment of the cluster consisting of particlesgenerated in the cascade. Instead of parameterizing the kinetic energies andangles of the outgoing particles, the phase space decay approach is used.Another difference is that the high energy tuning of the final state distributionis not performed.20.3.6 Medium Energy Cluster ProductionThe medium energy cluster algorithm is nearly identical to the high energycluster algorithm, but it may be invoked for incident energies down to 10MeV. There are three significant differences at medium energy: less nucleonsuppression, fewer particles generated in the intra-nuclear cascade, and nohigh energy tuning of the final state particle distributions.20.3.7 Elastic and Quasi-elastic ScatteringWhen no additional particles are produced in the initial interaction, eitherelastic or quasi-elastic scattering is performed. If there is insufficient energyto induce an intra-nuclear cascade, but enough to excite the target nucleus,quasi-elastic scattering is performed. The final state is calculated using twobodyscattering of the incident particle and the target nucleon, with thescattering angle in the center of mass sampled from an exponential:exp[−2bp in p cm (1 − cosθ)]. (20.22)Here p in is the incident particle momentum, p cm is the momentum in thecenter of mass, and b is a logarithmic function of the incident momentumin the lab frame as parameterized from data. As in the cascade and clusterproduction models, the residual nucleus is then de-excited by evaporatingnucleons and light ions.If the incident energy is too small to excite the nucleus, elastic scatteringis performed. The angular distribution of the scattered particle is sampledfrom the sum of two exponentials whose parameters depend on A.280

20.4 Status of this document7.10.02 re-written by D.H. Wright1.11.04 new section on high energy model by D.H. Wright281

Chapter 21Leading Particle Bias21.1 OverviewG4Mars5GeV is an inclusive event generator for hadron (photon) interactionswith nuclei, and translated from the MARS code system (MARS13 (98)). Toconstruct a cascade tree, only a fixed number of particles are generated ateach vertex. A corresponding statistical weight is assigned to each secondaryparticle according to its type and phase-space. Rarely-produced particles orinteresting phase-space region can be enhanced.N.B. This inclusive simulation is implemented in Geant4 partially for themoment, not completed yet.MARS Code SystemMARS is a set of Monte Carlo programs for inclusive simulation of particleinteractions, and high multiplicity or rare events can be simulated fast withits sophisticated biasing techniques. For the details on the MARS code system,see [1, 2].21.2 MethodIn G4Mars5GeV, three secondary hadrons are generated in the final state ofan hadron(photon)-nucleus inelastic interaction, and a statistical weight isassigned to each particle according to its type, energy and emission angle.In this code, energies, momenta and weights of the secondaries are sampled,and the primary particle is simply terminated at the vertex. The allowedprojectile kinetic energy is E 0 ≤ 5 GeV, and following particles can be sim-282

ulated;p, n, π + , π − , K + , K − , γ, ¯p .Prior to a particle generation, a Coulomb barrier is considered for projectilecharged hadrons (p, π + , K + and ¯p) with kinetic energy of less than 200MeV. The coulomb potential V coulmb is given byV columb = 1.11 × 10 −3 × Z/A 1/3 (GeV), (21.1)where Z and A are atomic and mass number, respectivelly.21.2.1 Inclusive hadron productionThe following three steps are carried out in a sequence to produce secondaryparticles:• nucleon production,• charged pion/kaon production and• neutral pion production.These processes are performed independently, i.e. the energy and momentumconservation law is broken at each event, however, fulfilled on the averageover a number of events simulated.nucleon productionProjectiles K ± and ¯p are replaced with π ± and p, individually to generate thesecondary nucleon. Either of neutron or proton is selected randomly as thesecondary except for the case of gamma projectiling. The gamma is handledas a pion.charged pion/kaon productionIf the incident nucleon does not have enough energy to produce the pion (>280 MeV), charged and neutral pions are not produced. A charged pion isselected with the equal probability, and a bias is eliminated with the appropriateweight which is assigned taking into account the difference betweenπ + and π − both for production probability and for inclusive spectra. It isreplaced with a charged kaon a certain fraction of the time, that depends on283

the projectile energy if E 0 > 2.1 GeV. The ratio of kaon replacement is givenbyR kaon = 1.3 ×{C min + (C − C min ) log(E }0/2)log(100/2)where C min is 0.03 (0.08) for nucleon (others) projectiling, andProduced particle Projectile particleC ={0.071 ( π + )0.083 ( π − )}⎧⎪⎨×⎪⎩1.3 ( π ± )2.0 ( K ± )1.0 ( others )(2.1 ≤ E 0 ≤ 5.2 GeV),⎫⎪⎬⎪⎭(21.2)(21.3)A similar strangeness replacement is not considered for nucleon production.21.2.2 Sampling of energy and emission angle of thesecondaryThe energy and emission angle of the secondary particle depends on projectileenergy. There are formulae depending on whether or not the interactionparticle (IP) is identical to the secondary (JP).For IP ≠ JP, the secondary energy E 2 is simply given byE 2 = E th ×( EmaxE th) ɛ(MeV) , (21.4)where E max = max (E 0 , 0.5 MeV), E th = 1 MeV, and ɛ is a uniform randombetween 0 and 1.For IP = JP,⎧⎪⎨ E th + ɛ (E max − E th )E 0 < 100 E th MeVE 2 = E th × e ɛ (β+99) (MeV) E 0 ≥ 100 E th and ɛ < η⎪⎩E 0 × (β (ɛ − 1) + 1 + 99ɛ)/100 E 0 ≥ 100 E th and ɛ ≥ η(21.5)Here, β = log(E 0 /100 E th ) and η = β/(99 + β). If resulting E 2 is less than0.5 MeV, nothing is generated.Angular distributionThe angular distribution is mainly determined by the energy ratio of thesecondary to the projectile (i.e. the emission angle and probability of the284

occurrence increase as the energy ratio decreases). The emission angle of thesecondary particle with respect to the incident direction is given bywhere τ = E 0 /5(E 0 + 1/2).θ = − log(1 − ɛ(1 − e −π τ ))/τ , (21.6)21.2.3 Sampling statistical weightThe kinematics of the secondary particle are determined randomly using theabove formulae (21.5,21.6). A statistical weight is calculated and assigned toeach generated particle to reproduce a true inclusive spectrum in the event.The weight is given byD2N = V 10(JP) × DW (E) × DA (θ) × V 1 (E, θ, JP), (21.7)where• V 10 is the statistical weight for the production rate based on neutral pionproduction (V 10 = 1).V 10 ={2.0 (2.5) nucleon production (the case of gamma projectile)2.1 charged pion/kaon production(21.8)• DW and DA are dominantly determined by the secondary energy and emissionangle, individually.• V1 is a true double-differential production cross-section (divided by the totalinelastic cross-section) [1], calculated in G4Mars5GeV::D2N2 accordingto the projectile type and energy, target atomic mass, and simulated secondaryenergy, emission angle and particle type.21.3 Status of this document11.06.2002 created by N. Kanaya.20.06.2002 modified by N.V.Mokhov.285

Bibliography[1] N.V. Mokhov, The MARS Code System User’s Guide, Version 13(98),Fermilab-FN-628.[2] http://www-ap.fnal.gov/MARS/286

Chapter 22Parton string model.22.1 Reaction initial state simulation.22.1.1 Allowed projectiles and bombarding energy rangefor interaction with nucleon and nuclear targets.The GEANT4 parton string models are capable to predict final states (producedhadrons which belong to the scalar and vector meson nonets and thebaryon (antibaryon) octet and decuplet) of reactions on nucleon and nucleartargets with nucleon, pion and kaon projectiles. The allowed bombardingenergy √ s > 5 GeV is recommended. Two approaches, based on diffractiveexcitation or soft scattering with diffractive admixture according to crosssection,are considered. Hadron-nucleus collisions in the both approaches(diffractive and parton exchange) are considered as a set of the independenthadron-nucleon collisions. However, the string excitation procedures in theseapproaches are rather different.22.1.2 MC initialization procedure for nucleus.The initialization of each nucleus, consisting from A nucleons and Z protonswith coordinates r i and momenta p i , where i = 1, 2, ..., A is performed.We use the standard initialization Monte Carlo procedure, which is realizedin the most of the high energy nuclear interaction models:• Nucleon radii r i are selected randomly in the rest of nucleus accordingto proton or neutron density ρ(r i ). For heavy nuclei with A > 16 [1]287

nucleon density isρ(r i ) =ρ 01 + exp [(r i − R)/a](22.1)whereρ 0 ≈ 34πR 3 (1 + a2 π 2R 2 )−1 . (22.2)Here R = r 0 A 1/3 fm and r 0 = 1.16(1 − 1.16A −2/3 ) fm and a ≈ 0.545fm. For light nuclei with A < 17 nucleon density is given by a harmonicoscillator shell model [2], e. g.ρ(r i ) = (πR 2 ) −3/2 exp (−r 2 i /R2 ), (22.3)where R 2 = 2/3 < r 2 >= 0.8133A 2/3 fm 2 . To take into accountnucleon repulsive core it is assumed that internucleon distance d > 0.8fm;• The initial momenta of the nucleons are randomly choosen between0 and p maxF , where the maximal momenta of nucleons (in the localThomas-Fermi approximation [3]) depends from the proton or neutrondensity ρ according towith ¯hc = 0.197327 MeVfm;p maxF = ¯hc(3π 2 ρ) 1/3 (22.4)• To obtain coordinate and momentum components, it is assumed thatnucleons are distributed isotropicaly in configuration and momentumspaces;• Then perform shifts of nucleon coordinates r ′ j = r j − 1/A ∑ i r i andmomenta p ′ j = p j − 1/A ∑ i p i of nucleon momenta. The nucleus mustbe centered in configuration space around 0, i. e. ∑ i r i = 0 and thenucleus must be at rest, i. e. ∑ i p i = 0;• We compute energy per nucleon e = E/A = m N + B(A, Z)/A, wherem N is nucleon mass and the nucleus binding energy B(A, Z) is givenby the Bethe-Weizsäcker formula[4]:B(A, Z) == −0.01587A + 0.01834A 2/3 + 0.09286(Z − A 2 )2 + 0.00071Z 2 /A 1/3 ,√and find the effective mass of each nucleon m effi = (E/A) 2 − p 2′i .288(22.5)

22.1.3 Random choice of the impact parameter.The impact parameter 0 ≤ b ≤ R t is randomly selected according to theprobability:P (b)db = bdb, (22.6)where R t is the target radius, respectively. In the case of nuclear projectileor target the nuclear radius is determined from condition:ρ(R)ρ(0)= 0.01. (22.7)22.2 Sample of collision participants in nuclearcollisions.22.2.1 MC procedure to define collision participants.The inelastic hadron–nucleus interactions at ultra–relativistic energies areconsidered as independent hadron–nucleon collisions. It was shown long timeago [5] for the hadron–nucleus collision that such a picture can be obtainedstarting from the Regge–Gribov approach [6], when one assumes that thehadron-nucleus elastic scattering amplitude is a result of reggeon exchangesbetween the initial hadron and nucleons from target–nucleus. This resultleads to simple and efficient MC procedure [7] to define the interaction crosssections and the number of the nucleons participating in the inelastic hadron–nucleus collision:• We should randomly distribute B nucleons from the target-nucleus onthe impact parameter plane according to the weight function T ([ ⃗ b B j ]).This function represents probability density to find sets of the nucleonimpact parameters [ ⃗ b B j ], where j = 1, 2, ..., B.• For each pair of projectile hadron i and target nucleon j with choosenimpact parameters ⃗ b i and ⃗ b B j we should check whether they interactinelastically or not using the probability p ij ( ⃗ b i − ⃗ b B j , s), where s ij =(p i + p j ) 2 is the squared total c.m. energy of the given pair with the4–momenta p i and p j , respectively.In the Regge–Gribov approach[6] the probability for an inelastic collisionof pair of i and j as a function at the squared impact parameter differenceb 2 ij = (⃗ b i − ⃗ b B j )2 and s is given byp ij ( ⃗ b i − ⃗ b B j , s) = c −1 [1 − exp {−2u(b 2 ij, s)}] =289∞∑n=1p (n)ij ( ⃗ b i − ⃗ b B j , s), (22.8)

wherep (n)ij ( ⃗ b i − ⃗ b B j , s) = c −1 exp {−2u(b 2 ij, s)} [2u(b2 ij , s)]n . (22.9)n!is the probability to find the n cut Pomerons or the probability for 2n stringsproduced in an inelastic hadron-nucleon collision. These probabilities are definedin terms of the (eikonal) amplitude of hadron–nucleon elastic scatteringwith Pomeron exchange:u(b 2 z(s)ij , s) =2 exp(−b2 ij /4λ(s)). (22.10)The quantities z(s) and λ(s) are expressed through the parameters of thePomeron trajectory, α ′ P = 0.25 GeV −2 and α P (0) = 1.0808, and the parametersof the Pomeron-hadron vertex R P and γ P :z(s) = 2cγ Pλ(s) (s/s 0) α P (0)−1(22.11)λ(s) = R 2 P + α′ P ln(s/s 0), (22.12)respectively, where s 0 is a dimensional parameter.In Eqs. (22.8,22.9) the so–called shower enhancement coefficient c is introducedto determine the contribution of diffractive dissociation[6]. Thus, theprobability for diffractive dissociation of a pair of nucleons can be computedasp d ij (⃗ b i − ⃗ b B j , s) = c − 1 [p totijc(⃗ b i − ⃗ b B j , s) − p ij( ⃗ b i − ⃗ b B j , s)], (22.13)wherep totij (⃗ b i − ⃗ b B j , s) = (2/c)[1 − exp{−u(b2 ij , s)}]. (22.14)The Pomeron parameters are found from a global fit of the total, elastic,differential elastic and diffractive cross sections of the hadron–nucleoninteraction at different energies.For the nucleon-nucleon, pion-nucleon and kaon-nucleon collisions thePomeron vertex parameters and shower enhancement coefficients are found:RP 2N = 3.56 GeV −2 , γP N = 3.96 GeV −2 , s N 0 = 3.0 GeV 2 , c N = 1.4 andRP2π = 2.36 GeV −2 , γP π = 2.17 GeV −2 , and RP 2K = 1.96 GeV −2 , γP K = 1.92GeV −2 , s K 0 = 2.3 GeV 2 , c π = 1.8.22.2.2 Separation of hadron diffraction excitation.For each pair of target hadron i and projectile nucleon j with choosen impactparameters ⃗ b i and ⃗ b B j we should check whether they interact inelasticallyor not using the probabilityp inij (⃗ b i − ⃗ b B j , s) = p ij( ⃗ b i − ⃗ b B j , s) + pd ij (⃗ b A i − ⃗ b B j , s). (22.15)290

If interaction will be realized, then we have to consider it to be diffractive ornondiffractive with probabilitiesandp d ij (⃗ b i − ⃗ b B j , s)p inij ( ⃗ b A i − ⃗ b B j , s)(22.16)p ij ( ⃗ b i − ⃗ b B j , s)p inij ( ⃗ b A i − ⃗ b B j , s) . (22.17)22.3 Longitudinal string excitation22.3.1 Hadron–nucleon inelastic collisionLet us consider collision of two hadrons with their c. m. momenta P 1 ={E 1 + , m 2 1 /E+ 1 , 0} and P 2 = {E2 − , m 2 2 /E− 2 , 0}, where the light-cone variablesE 1,2 ± = E 1,2 ± P z1,2 are defined through hadron energies E 1,2 = √ m 2 1,2 + Pz1,2,2hadron longitudinal momenta P z1,2 and hadron masses m 1,2 , respectively.Two hadrons collide by two partons with momenta p 1 = {x + E 1 + , 0, 0} andp 2 = {0, x − E2 − , 0}, respectively.22.3.2 The diffractive string excitationIn the diffractive string excitation (the Fritiof approach [9]) only momentumcan be transferred:P 1 ′ = P 1 + qP 2 ′ = P (22.18)2 − q,whereq = {−q 2 t /(x− E − 2 ), q2 t /(x+ E + 1 ), q t} (22.19)is parton momentum transferred and q t is its transverse component. We usethe Fritiof approach to simulate the diffractive excitation of particles.22.3.3 The string excitation by parton exchangeFor this case the parton exchange (rearrangement) and the momentumexchange are allowed [10],[11],[7]:P ′ 1 = P 1 − p 1 + p 2 + qP ′ 2 = P 2 + p 1 − p 2 − q,(22.20)where q = {0, 0, q t } is parton momentum transferred, i. e. only its transversecomponents q t = 0 is taken into account.291

22.3.4 Transverse momentum samplingThe transverse component of the parton momentum transferred is generatedaccording to probabilitywhere parameter a = 0.6 GeV −2 .P (q t )dq t =√ aπ exp (−aq2 t )dq t, (22.21)22.3.5 Sampling x-plus and x-minusLight cone parton quantities x + and x − are generated independently andaccording to distribution:u(x) ∼ x α (1 − x) β , (22.22)where x = x + or x = x − . Parameters α = −1 and β = 0 are chosen forthe FRITIOF approach [9]. In the case of the QGSM approach [7] α = −0.5and β = 1.5 or β = 2.5. Masses of the excited strings should satisfy thekinematical constraints:P 1 ′+ P 1 ′− ≥ m 2 h1 + qt 2 (22.23)andP ′+2 P ′−2 ≥ m 2 h2 + q2 t , (22.24)where hadronic masses m h1 and m h2 (model parameters) are defined by stringquark contents. Thus, the random selection of the values x + and x − is limitedby above constraints.22.3.6 The diffractive string excitationIn the diffractive string excitation (the FRITIOF approach [9]) for eachinelastic hadron–nucleon collision we have to select randomly the transversemomentum transferred q t (in accordance with the probability given by Eq.(22.21)) and select randomly the values of x ± (in accordance with distributiondefined by Eq. (22.22)). Then we have to calculate the parton momentumtransferred q using Eq. (22.19) and update scattered hadron and nucleonor scatterred nucleon and nucleon momenta using Eq. (22.20). For eachcollision we have to check the constraints (22.23) and (22.24), which can bewritten more explicitly:[E + 1 − q2 tx − E − 2][ m2 1E + 1+ q2 tx + E + 1] ≥ m 2 h1 + q 2 t (22.25)292

and[E − 2 + q2 tx − E − 2][ m2 2E − 2q2 t−x + E 1+] ≥ m 2 h1 + q 2 t . (22.26)22.3.7 The string excitation by parton rearrangementIn this approach [7] strings (as result of parton rearrangement) shouldbe spanned not only between valence quarks of colliding hadrons, but alsobetween valence and sea quarks and between sea quarks. The each participanthadron or nucleon should be splitted into set of partons: valencequark and antiquark for meson or valence quark (antiquark) and diquark(antidiquark) for baryon (antibaryon) and additionaly the (n − 1) sea quarkantiquarkpairs (their flavours are selected according to probability ratiosu : d : s = 1 : 1 : 0.35), if hadron or nucleon is participating in the n inelasticcollisions. Thus for each participant hadron or nucleon we have to generatea set of light cone variables x 2n , where x 2n = x + 2n or x 2n = x − 2n according todistribution:2n∏f h (x 1 , x 2 , ..., x 2n ) = f 0 u h q i(x i )δ(1 −i=12n∑i=1x i ), (22.27)where f 0 is the normalization constant. Here, the quark structure functionsu h q i(x i ) for valence quark (antiquark) q v , sea quark and antiquark q s andvalence diquark (antidiquark) qq are:u h q v(x v ) = x αvv, u h q s(x s ) = x αss , u h qq(x qq ) = x βqqqq , (22.28)where α v = −0.5 and α s = −0.5 [10] for the non-strange quarks (antiquarks)and α v = 0 and α s = 0 for strange quarks (antiquarks), β uu = 1.5 andβ ud = 2.5 for proton (antiproton) and β dd = 1.5 and β ud = 2.5 for neutron(antineutron). Usualy x i are selected between x mini ≤ x i ≤ 1, where modelparameter x min is a function of initial energy, to prevent from productionof strings with low masses (less than hadron masses), when whole selectionprocedure should be repeated. Then the transverse momenta of partonsq it are generated according to the Gaussian probability Eq. (22.21) with∑a = 1/4Λ(s) and under the constraint: 2ni=1 q it = 0. The partons areconsidered as the off-shell partons, i. e. m 2 i ≠ 0.293

22.4 Longitudinal string decay.22.4.1 Hadron production by string fragmentation.A string is stretched between flying away constituents: quark and antiquarkor quark and diquark or diquark and antidiquark or antiquark andantidiquark. From knowledge of the constituents longitudinal p 3i = p zi andtransversal p 1i = p xi , p 2i = p yi momenta as well as their energies p 0i = E i ,where i = 1, 2, we can calculate string mass squared:M 2 S = pµ p µ = p 2 0 − p2 1 − p2 2 − p2 3 , (22.29)where p µ = p µ1 + p µ2 is the string four momentum and µ = 0, 1, 2, 3.The fragmentation of a string follows an iterative scheme:string ⇒ hadron + new string, (22.30)i. e. a quark-antiquark (or diquark-antidiquark) pair is created and placedbetween leading quark-antiquark (or diquark-quark or diquark-antidiquarkor antiquark-antidiquark) pair.The values of the strangeness suppression and diquark suppression factorsareu : d : s : qq = 1 : 1 : 0.35 : 0.1. (22.31)A hadron is formed randomly on one of the end-points of the string. Thequark content of the hadrons determines its species and charge. In the chosenfragmentation scheme we can produce not only the groundstates of baryonsand mesons, but also their lowest excited states. If for baryons the quarkcontentdoes not determine whether the state belongs to the lowest octetor to the lowest decuplet, then octet or decuplet are choosen with equalprobabilities. In the case of mesons the multiplet must also be determinedbefore a type of hadron can be assigned. The probability of choosing a certainmultiplet depends on the spin of the multiplet.The zero transverse momentum of created quark-antiquark (or diquarkantidiquark)pair is defined by the sum of an equal and opposite directedtransverse momenta of quark and antiquark.The transverse momentum of created quark is randomly sampled accordingto probability (22.21) with the parameter a = 0.25 GeV −2 . Then ahadron transverse momentum p t is determined by the sum of the transversemomenta of its constituents.The fragmentation function f h (z, p t ) represents the probability distributionfor hadrons with the transverse momenta p t to aquire the light conemomentum fraction z = z ± = (E h ± p h z /(Eq ± p q z ), where Eh and E q294

are the hadron and fragmented quark energies, respectively and p h z and pq zare hadron and fragmented quark longitudinal momenta, respectively, andz min ± ≤ z ± ≤ z max ± , from the fragmenting string. The values of z± min,max aredetermined by hadron m h and constituent transverse masses and the availablestring mass. One of the most common fragmentation function is usedin the LUND model [12]:f h (z, p t ) ∼ 1 z (1 − z)a exp [− b(m2 h + p 2 t )]. (22.32)zOne can use this fragmentation function for the decay of the excited string.One can use also the fragmentation functions are derived in [13]:f h q (z, p t) = [1 + α h q (< p t >)](1 − z) αh q () . (22.33)The advantage of these functions as compared to the LUND fragmentationfunction is that they have correct three–reggeon behaviour at z → 1 [13].22.4.2 The hadron formation time and coordinate.To calculate produced hadron formation times and longitudinal coordinateswe consider the (1 + 1)-string with mass M S and string tension κ,which decays into hadrons at string rest frame. The i-th produced hadronhas energy E i and its longitudinal momentum p zi , respectively. Introducinglight cone variables p ± i = E i ± p iz and numbering string breaking pointsconsecutively from right to left we obtain p + 0 = M S , p + i = κ(z i−1 + − z i + ) andp − i = κx − i .We can identify the hadron formation point coordinate and time as thepoint in space-time, where the quark lines of the quark-antiquark pair formingthe hadron meet for the first time (the so-called ’yo-yo’ formation point [12]):i−1t i = 12κ [M ∑S − 2 p zj + E i − p zi ] (22.34)j=1and coordinatez i = 12κ [M ∑i−1S − 2 E j + p zi − E i ]. (22.35)j=1Bibliography[1] Grypeos M. E., Lalazissis G. A., Massen S. E., Panos C. P., J. Phys.G17 1093 (1991).295

[2] Elton L. R. B., Nuclear Sizes, Oxford University Press, Oxford, 1961.[3] DeShalit A., Feshbach H., Theoretical Nuclear Physics, Vol. 1: NuclearStructure, Wyley, 1974.[4] Bohr A., Mottelson B. R., Nuclear Structure, W. A. Benjamin, NewYork, Vol. 1, 1969.[5] Capella A. and Krzywicki A., Phys. Rev. D18 (1978) 4120.[6] Baker M. and Ter–Martirosyan K. A., Phys. Rep. 28C (1976) 1.[7] Amelin N. S., Gudima K. K., Toneev V. D., Sov. J. Nucl. Phys. 51(1990) 327; Amelin N. S., JINR Report P2-86-56 (1986).[8] Abramovskii V. A., Gribov V. N., Kancheli O. V., Sov. J. Nucl. Phys.18 (1974) 308.[9] Andersson B., Gustafson G., Nielsson-Almquist, Nucl. Phys. 281 289(1987).[10] Kaidalov A. B., Ter-Martirosyan K. A., Phys. Lett. B117 247 (1982).[11] Capella A., Sukhatme U., Tan C. I., Tran Thanh Van. J., Phys. Rep.236 225 (1994).[12] Andersson B., Gustafson G., Ingelman G., Sjöstrand T., Phys. Rep.97 31 (1983).[13] Kaidalov A. B., Sov. J. Nucl. Phys. 45 1452 (1987).296

Chapter 23Chiral Invariant Phase SpaceDecay.23.1 IntroductionThe CHIPS computer code is a quark-level event generator for the fragmentationof hadronic systems into hadrons. In contrast to other partonmodels [1] CHIPS is three-dimensional. It is based on the Chiral InvariantPhase Space model [2, 3, 4] which employs quark-level SU(3); c, b, and tquarks are not implemented. The model can be considered as a generalizationof the chiral bag model of hadrons [5] in which any hadron consists of afew quark-partons. Interactions between hadrons are treated as purely kinematiceffects of quark exchange, and the decay of excited hadronic systems istreated as the fusion of two quark-partons within the system. This approachdoes not pretend to be a dynamical model.An important feature of the model is the homogeneous distribution ofasymptotically free quark-partons over the invariant phase space, as appliedto the fragmentation of various types of excited hadronic systems. The modelmay be considered as a generalization of the well-known hadronic phase spacedistribution [6] approach, because it generates not only angular and momentumdistributions for a given set of hadrons, but also the multiplicitydistributions for different kinds of hadrons.CHIPS may be applied to nucleon excitations, hadronic systems producedin e + e − interactions and high energy nuclear excitations, among others. Despiteits quark nature, the model can also be used successfully at very lowenergies. It is valid for photon and hadron projectiles and for hadron and nucleartargets. Exclusive event generation models multiple hadron productionwhile conserving energy, momentum, and charge. This generally results in297

a good description of particle multiplicities and spectra in multihadron fragmentationprocesses. Thus, it is possible to use the CHIPS event generatorin exclusive modeling of hadron cascades in materials.In this model, the result of a hadronic or nuclear interaction is the creationof a quasmon which is essentially an intermediate state of excited hadronicmatter. When the interaction occurs in vacuum the quasmon can dissipateenergy by radiating particles according to the quark fusion mechanism [2]described in section 23.4. When the interaction occurs in nuclear matter,quasmon fragmentation can occur by quark exchange with surrounding nucleonsor clusters of nucleons [3] (section 23.5), in addition to the abovevacuum mechanism.23.2 Fundamental ConceptsThe CHIPS model is an attempt to use a set of simple rules which govern microscopicquark-level behavior to model macroscopic hadronic systems witha large number of degrees of freedom. The invariant phase space distributionas a paradigm of thermalized chaos is applied to quarks, and simplekinematic mechanisms are used to model the hadronization of quarks intohadrons. Along with relativistic kinematics and the conservation of quantumnumbers, the following concepts are used:• Quasmon: in the CHIPS model, a quasmon is any excited hadronicsystem; it can be viewed as a generalized hadron. At the constituentlevel, a quasmon may be thought of as a bubble of quark-parton plasmain which the quarks are massless and the quark-partons in the quasmonare homogeneously distributed over the invariant phase space. Itmay also be considered as a bubble of the three-dimensional Feynman-Wilson [7] parton gas. The traditional hadron is a particle defined byquantum numbers and a fixed mass or width. The quark content of thehadron is a secondary concept constrained by the quantum numbers.The quasmon, however, is defined by its quark content and mass, andthe concept of a well defined particle with quantum numbers is of secondaryimportance. A given quasmon hadronic state with fixed massand quark content can be considered as a superposition of traditionalhadrons, with the quark content of the superposition being the sameas the quark content of the quasmon.• Quark fusion: the quark fusion hypothesis determines the rules offinal state hadron production, with energy spectra reflecting the momentumdistribution of the quarks in the system. Fusion occurs when298

a quark-parton in a quasmon joins with another quark-parton fromthe same quasmon and forms a hadron. If a neighboring quasmon ispresent, quark-partons may also be exchanged between the two quasmons.The kinematic condition applied to these mechanisms is thatthe resulting hadrons be produced on their mass shells. The modelassumes that the u, d and s quarks are massless, which allows the integralsof the hadronization process to be done easily and the modelingalgorithm to be accelerated. The quark mass is taken into account indirectlyin the masses of outgoing hadrons. The type of the outgoinghadron is selected using combinatoric and kinematic factors consistentwith conservation laws. In the present version of CHIPS all mesons withthree-digit PDG Monte Carlo codes [8] up to spin 4, and all baryonswith four-digit PDG codes up to spin 7 are implemented.2• Critical temperature the only non-kinematic concept of the model isthe hypothesis of the critical temperature of the quasmon. This has a35-year history, starting with Ref. [9] and is based on the experimentalobservation of regularities in the inclusive spectra of hadrons producedin different reactions at high energies. Qualitatively, the hypothesisof a critical temperature assumes that the quark-gluon hadronic system(quasmon) cannot be heated above a certain temperature. Addingmore energy to the hadronic system increases only the number of constituentquark-partons while the temperature remains constant. Thecritical temperature T c = 180 − 200 MeV is the principal parameterof the model and is used to calculate the number of quark-partons ina quasmon. In an infinite thermalized system, for example, the meanenergy of partons is 2T per particle, where T is the temperature of thesystem.23.3 Code DevelopmentBecause the CHIPS event generator was originally developed only for finalstate hadronic fragmentation, the initial interaction of projectiles with targetsrequires further development. Hence, the first applications of CHIPSdescribed interactions at rest, for which the interaction cross section is notimportant [2], [3], and low energy photonuclear reactions, for which the interactioncross section can be calculated easily [4].Formally, the CHIPS event generator can be used for all kinds of hadronicinteraction. This includes photonuclear and electronuclear reactions becausethe photon can be considered as a superposition of vector mesons according to299

the Vector Dominance Model (VDM). As explained below, the photonuclearand electronuclear reactions, as well as the hadron-nuclear cross sections, areparameterized in other GEANT4 classes.The Geant4 String Model interface to the CHIPS generator [10], [11]also makes it possible to use the CHIPS code for nuclear fragmentation atextremely high energies. Applications at intermediate energies (1-10 GeV)require additional tuning of the interaction mechanism.In the first published versions of the CHIPS event generator the classG4Quasmon was the head of the model and all initial interactions were hiddenin its constructor. More complicated applications of the model such as antiprotoncapture at rest and the Geant4 String Model interface to CHIPS ledto the multi-quasmon version of the model. This required a change in thestructure of the CHIPS event generator classes. In the case of at-rest antiprotonannihilation in a nucleus, for example, the first interaction occurs onthe nuclear periphery. After this initial interaction, a fraction (defined by aspecial parameter of the model) of the secondary mesons independently penetratethe nucleus. Each of these mesons can create a separate quasmon inthe interior of the nucleus. In this case the class G4Quasmon can no longer bethe head of the model. A new head class, G4QEnvironment, was developedwhich can adopt a vector of projectile hadrons (G4QHadronVector) and createa vector of quasmons, G4QuasmonVector. All newly created quasmonsthen begin the energy dissipation process in parallel in the same nucleus.The G4QEnvironment instance can be used both for vacuum and for nuclearmatter. If G4QEnvironment is created for vacuum, only one instance ofG4Quasmon is allowed, leaving the model unchanged for hadronic interactions.The convention adopted for the CHIPS model requires all its class namesto use the prefix G4Q in order to distinguish them from GEANT4 classes,most of which use the G4 prefix. The intent is that the G4Q prefix will notbe used by other GEANT4 projects.23.4 Nucleon-Antinucleon Annihilation at RestIn order to generate hadron spectra from the annihilation of a proton withan anti-proton at rest, the number of partons in the system must be found.For a finite system of N partons with a total center-of-mass energy M, theinvariant phase space integral, Φ N , is proportional to M 2N−4 . According to∏the dimensional counting rule, 2N comes from N d 3 p iEi=1 i, and 4 comes from theenergy and momentum conservation function, δ 4 (P¯− ∑ p¯i). At a temperatureT the statistical density of states is proportional to e − M Tso that the300

probability to find a system of N quark-partons in a state with mass M isdW ∝ M 2N−4 e − M T dM. For this kind of probability distribution the meanvalue of M 2 is< M 2 >= 4N(N − 1) · T 2 . (23.1)When N goes to infinity one obtains for massless particles the well-known< M >≡ √ < M 2 > = 2NT result.After a nucleon absorbs an incident quark-parton, such as a real or virtualphoton, for example, the newly formed quasmon has a total of N quarkpartons,where N is determined by Eq. 23.1. Choosing one of these quarkpartonswith energy k in the center of mass system (CMS) of N partons, thespectrum of the remaining N − 1 quark-partons is given bydWkdk ∝ (M N−1) 2N−6 , (23.2)where M N−1 is the effective mass of the N − 1 quark-partons. This resultwas obtained by applying the above phase-space relation (Φ N ∝ M 2N−4 ) tothe residual N − 1 quarks. The effective mass is a function of the total massM,M 2 N−1 = M 2 − 2kM, (23.3)so that the resulting equation for the quark-parton spectrum is:dWkdk23.4.1 Meson Production∝ (1 −2kM )N−3 . (23.4)In this section, only the quark fusion mechanism of hadronization is considered;the quark exchange mechanism can take place only in nuclear matterwhere a quasmon has neighboring nucleons. In order to decompose a quasmoninto an outgoing hadron and a residual quasmon, one needs to calculatethe probability of two quark-partons combining to produce the effective massof the outgoing hadron. This requires that the spectrum of the second quarkpartonbe calculated. This is done by following the same argument used todetermine Eq. 23.4. One quark-parton is chosen from the residual N − 1. Ithas an energy q in the CMS of the N − 1 quark-partons. The spectrum isobtained by substituting N − 1 for N and M N−1 for M in Eq. 23.4 and thenusing Eq. 23.3 to get⎛dWqdq ∝ ⎝1 −⎞2q√ ⎠M 1 − 2kM301N−4. (23.5)

Next, one of the residual quark-partons must be selected from this spectrumsuch that its fusion with the primary quark-parton makes a hadronof mass µ. This selection is performed by the mass shell condition for theoutgoing hadron,µ 2 k= 2√· q · (1 − cos θ). (23.6)1 − 2kMHere θ is the angle between the momenta, k and q of the two quark-partonsin the CMS of N − 1 quarks. Now the kinematic quark fusion probabilitycan be calculated for any primary quark-parton with energy k:P (k, M, µ) =∫ ⎛ ⎝1 −M⎛× δ ⎝µ 2 −2q√1 − 2kM⎞N−4⎠2kq(1 − cos θ)√1 − 2kM⎞⎠ qdqd cos θ. (23.7)Using the δ-function 1 to perform the integration over q one gets:P (k, M, µ) =) N−4∫ ( µ 21 −Mk(1 − cos θ)⎛⎞× ⎝ µ2√ 1 − 2k 2 ( )M ⎠ 1 − cos θd, (23.8)2k(1 − cos θ) µ 2orP (k, M, µ) =) N−4∫M − 2k( µ 21 −4k Mk(1 − cos θ)(µ 2 )× d 1 −. (23.9)Mk(1 − cos θ)After the substitution z = 1 −2qM N−1= 1 −P (k, M, µ) = M − 2k4kµ 2, this becomesMk(1−cos θ)∫z N−4 dz, (23.10)where the limits of integration are 0 when cos θ = 1 − µ2M·k , andz max = 1 − µ22Mk , (23.11)1 If g(x 0 )=0, ∫ f(x)δ [g(x)] dx = ∫ f(x)δ[g(x)]g ′ (x)dg(x) = f(x0)g ′ (x 0)302

when cos θ = −1. The resulting range of θ is therefore −1 < cos θ < 1 − µ2M·k .Integrating from 0 to z yieldsM − 2k4k · (N − 3) · zN−3 , (23.12)and integrating from 0 to z max yields the total kinematic probability forhadronization of a quark-parton with energy k into a hadron with mass µ:M − 2k4k · (N − 3) · zN−3 max . (23.13)The ratio of expressions 23.12 and 23.13 can be treated as a random number,R, uniformly distributed on the interval [0,1]. Solving for z then givesz = N−3√ R · z max . (23.14)In addition to the kinematic selection of the two quark-partons in thefusion process, the quark content of the quasmon and the spin of the candidatefinal hadron are used to determine the probability that a given type ofhadron is produced. Because only the relative hadron formation probabilitiesare necessary, overall normalization factors can be dropped. Hence therelative probability can be written asP h (k, M, µ) = (2s h + 1) · z N−3max · Ch Q . (23.15)Here, only the factor zmax N−3 is used since the other factors in equation 23.13are constant for all candidates for the outgoing hadron. The factor 2s h + 1counts the spin states of a candidate hadron of spin s h , and CQ h is the numberof ways the candidate hadron can be formed from combinations of the quarkswithin the quasmon. In making these combinations, the standard quark wavefunctions for pions and kaons were used. For η and η ′ mesons the quark wavefunctions η =ūu+ ¯dd2− ¯ss √2and η ′ =ūu+ ¯dd2+ ¯ss √2were used. No mixing wasūu+ ¯ddassumed for the ω and φ meson states, hence ω = √2and ϕ = ¯ss.A final model restriction is applied to the hadronization process: after ahadron is emitted, the quark content of the residual quasmon must have aquark content corresponding to either one or two real hadrons. When thequantum numbers of a quasmon, determined by its quark content, cannotbe represented by the quantum numbers of a real hadron, the quasmon isconsidered to be a virtual hadronic molecule such as π + π + or K + π + , in whichcase it is defined in the CHIPS model to be a Chipolino pseudo-particle.To fuse quark-partons and create the decay of a quasmon into a hadronand residual quasmon, one needs to generate randomly the residual quasmon303

mass m, which in fact is the mass of the residual N − 2 quarks. Using anequation similar to 23.3) one finds thatm 2 = z · (M 2 − 2kM). (23.16)Using Eqs. 23.14 and 23.11, the mass of the residual quasmon can be expressedin terms of the random number R:m 2 = (M − 2k) · (M − µ22k ) · N−3√ R. (23.17)At this point, the decay of the original quasmon into a final state hadron anda residual quasmon of mass m has been simulated. The process may now berepeated on the residual quasmon.This iterative hadronization process continues as long as the residualquasmon mass remains greater than m min , whose value depends on the typeof quasmon. For hadron-type residual quasmonsm min = m QCmin + m π 0, (23.18)where m QCmin is the minimum hadron mass for the residual quark content (QC).For Chipolino-type residual quasmons consisting of hadrons h 1 and h 2 ,m min = m h1 + m h2 . (23.19)These conditions insure that the quasmon always has enough energy to decayinto at least two final state hadrons, conserving four-momentum and charge.If the remaining CMS energy of the residual quasmon falls below m min ,then the hadronization process terminates with a final two-particle decay. Ifthe parent quasmon is a Chipolino consisting of hadrons h 1 and h 2 , then abinary decay of the parent quasmon into m h1 and m h2 takes place. If theparent quasmon is not a Chipolino then a decay into m QCmin and m h takesplace. The decay into m QCmin and m 0 π is always possible in this case because ofcondition 23.18.If the residual quasmon is not Chipolino-type, and m > m min , the hadronizationloop can still be finished by the resonance production mechanism, whichis modeled following the concept of parton-hadron duality [12]. If the residualquasmon has a mass in the vicinity of a resonance with the same quarkcontent (ρ or K ∗ for example), there is a probability for the residual quasmonto convert to this resonance. 2 In the present version of the CHIPS eventgenerator the probability of convert to the resonance is given byP res = m2 minm 2 . (23.20)2 When comparing quark contents, the quark content of the quasmon is reduced bycanceling quark-antiquark pairs of the same flavor.304

Hence the resonance with the mass-squared value m 2 r closest to m2 is selected,and the binary decay of the quasmon into m h and m r takes place.With more detailed experimental data, it will be possible to take intoaccount angular momentum conservation, as well as C-, P - and G-parityconservation. In the present version of the generator, η and η ′ are suppressedby a factor of 0.3. This factor was tuned using data from experiments onantiproton annihilation at rest in liquid hydrogen and can be different forother hadronic reactions. It is possible to vary it when describing otherreactions.Another parameter, s/u, controls the suppression of heavy quark production[13]. For proton-antiproton annihilation at rest the strange quarkantiquarksea was found to be suppressed by the factor s/u = 0.1. Inthe JETSET [13] event generator, the default value for this parameter iss/u = 0.3. The lower value may be due to quarks and anti-quarks of collidinghadrons initially forming a non-strange sea, with the strange sea suppressedby the OZI rule [14]. This question is still under discussion [15] and demandsfurther experimental measurements. The s/u parameter may differ for otherreactions. In particular, for e + e − reactions it can be closer to 0.3.Finally, the temperature parameter has been fixed at T = 180 MeV.In earlier versions of the model it was found that this value successfullyreproduced spectra of outgoing hadrons in different types of medium-energyreactions.The above parameters were used to fit not only the spectrum of pionsFig. 23.1,a and the multiplicity distribution for pions Fig. 23.1,b but alsobranching ratios of various measured [16, 17] exclusive channels as shown inFigs. 23.2, 23.3, 23.4. In Fig. 23.4 one can see many decay channels withhigher meson resonances. The relative contribution of events with mesonresonances produced in the final state is 30 - 40 percent, roughly in agreementwith experiment. The agreement between the model and experiment forparticular decay modes is within a factor of 2-3 except for the branchingratios to higher resonances. In these cases it is not completely clear howthe resonance is defined in a concrete experiment. In particular, for thea 2 ω channel the mass sum of final hadrons is 2100 MeV with a full width ofabout 110 MeV while the total initial energy of the p¯p annihilation reaction isonly 1876.5 MeV. This decay channel can be formally simulated by an eventgenerator using the tail of the Breit-Wigner distribution for the a 2 resonance,but it is difficult to imagine how the a 2 resonance can be experimentallyidentified 2Γ away from its mean mass value.305

1/N AdN/dP (Annihilation -1 (GeV/c) -1 )101p – p dataCHIPS MCCharged pionsdN/N A(Annihilation -1 )110 -1p – p dataCHIPS MCπ + +π - +π 0 multiplicity10 -110 -2Kaon Channels0 0.2 0.4 0.6 0.8 1Pion momentum (GeV/c)0 2 4 6 8 10Pion multiplicityFigure 23.1: (a) (left): momentum distribution of charged pions producedin proton-antiproton annihilation at rest. The experimental data are from[16], and the histogram was produced by the CHIPS Monte Carlo. Theexperimental spectrum is normalized to the measured average charged pionmultiplicity, 3.0. (b) (right): pion multiplicity distribution. Data points weretaken from compilations of experimental data [17], and the histogram wasproduced by the CHIPS Monte Carlo. The number of events with kaons inthe final state is shown in pion multiplicity bin 9, where no real 9-pion eventsare generated or observed experimentally. In the model, the percentage ofannihilation events with kaons is close to the experimental value of 6% [17].306

dN/N A(Annihilation -1 )110 -110 -2points: p – p dataCHIPS MCExp. sum of channels = 0.972±0.011MC sum of channels = 1.0000010 -310 -42π 0π + π -3π 0π + π - π 02π + 2π -2π + 2π - π 03π + 3π -3π + 3π - π 0neutrals, excl. 2π 0 , 3π 0π + π - + (neutrals, excl. π 0 )2π + 2π - + (neutrals, excl. π 0 )3π + 3π - + (neutrals, excl. π 0 )4π + 4π - + XK 0 K – 0K + K -K 0 K – 0 π0K + K - π 0K - + K SL π + -K 0 K – 0 + (non-strange, excl. π0 )K + K - + (non-strange, excl. π 0 )K + - K SL + (non-strange, excl. π -+ )2.5 5 7.5 10 12.5 15 17.5 20Annihilation ChannelsFigure 23.2: Branching probabilities for different channels in protonantiprotonannihilation at rest. The experimental data are from [17], andthe histogram was produced by the CHIPS Monte Carlo.307

23.4.2 Baryon ProductionTo model fragmentation into baryons the POPCORN idea [18] was used,which assumes the existence of diquark-partons. The assumption of masslessdiquarks is somewhat inconsistent at low energies, as is the assumption ofmassless s-quarks, but it is simple and it helps to generate baryons in thesame way as mesons.Baryons are heavy, and the baryon production in p¯p annihilation reactionsat medium energies is very sensitive to the value of the temperature. If thetemperature is low, the baryon yield is small, and the mean multiplicity ofpions increases very noticeably with CMS energy as seen in Fig. 23.5. Forhigher temperature values the baryon yield reduces the pion multiplicity athigher energies. The existing experimental data [19], shown in Fig. 23.5, canbe considered as a kind of “thermometer” for the model. This thermometerconfirms that the critical temperature is about 200 MeV.It can be used as a tool for the Monte Carlo simulation of a wide varietyof hadronic reactions. The CHIPS event generator can be used not only for“phase-space background” calculations in place of the standard GENBODroutine [6], but even for taking into account the reflection of resonancesin connected final hadron combinations. Thus it can be useful for physicsanalysis too, even though its main range of application is the simulation ofthe evolution of hadronic and electromagnetic showers in matter at mediumenergies.23.5 Nuclear Pion Capture at Rest and PhotonuclearReactions Below the Delta(3,3)ResonanceWhen compared with the first “in vacuum” version of the model, describedin Section 23.4, modeling hadronic fragmentation in nuclear matter is morecomplicated because of the much greater number of possible secondary fragments.However, the hadronization process itself is simpler in a way. Invacuum, the quark-fusion mechanism requires a quark-parton partner fromthe external (as in JETSET [13]) or internal (the quasmon itself, Section23.4) quark-antiquark sea. In nuclear matter, there is a second possibility:quark exchange with a neighboring hadronic system, which could be a nucleonor multinucleon cluster.In nuclear matter the spectra of secondary hadrons and nuclear fragmentsreflect the quark-parton energy spectrum within a quasmon. In the case of308

1dN/N A(Annihilation -1 )10 -110 -2points: p – p dataCHIPS MCExp. sum of channels = 0.214±0.007MC sum of channels = 0.1994310 -310 -410 -5π 0 π 0 π 0π + π - π 0K 0 K – 0 π0K + K - π 0K - + K SL π + -ηπ 0 π 0ηπ + π -ηηπ 0ωπ 0 π 0ωπ + π -ωηπ 0η′π 0 π 0η′π + π -η′ηπ 0φπ 0 π 0φπ + π -2 4 6 8 10 12 14 16Annihilation Channels with Three-Particle Final StatesFigure 23.3: Branching probabilities for different channels with three-particlefinal states in proton-antiproton annihilation at rest. The points are experimentaldata [17] and the histogram is from the CHIPS Monte Carlo.309

1dN/N A(Annihilation -1 )10 -110 -2points: p – p dataCHIPS MCExp. sum of channels = 0.239±0.009MC sum of channels = 0.1744510 -310 -410 -5π 0 π 0π + π -K 0 K – 0K + K -ηπ 0ηηρ 0 π 0ρ - + π + -ρ 0 ηρ 0 ρ 0ωπ 0ωηωρ 0ωωK* 0 K – 0 + c.c.K* + - K - +K* 0 K – * 0K* + - K* - +η′π 0η′ηη′ρ 0η′ωφπ 0φηφρ 0φωf 2π 0f 2ρ 0f 2ωa 2πa 2ωf 2′π 05 10 15 20 25 30Annihilation Channels with Two-Particle Final StatesFigure 23.4: Branching probabilities for different channels with two-particlefinal states in proton-antiproton annihilation at rest. The points are experimentdata [17] and the histogram is from the CHIPS Monte Carlo.310

14p – p e + e - CHIPS MC100T (MeV)Pion multiplicity121086120140160180 2002204200 1 2 3 4 5 6√s ¬(GeV)Figure 23.5: Average meson multiplicities in proton-antiproton and inelectron-positron annihilation, as a function of the CMS energy of the interactinghadronic system. The points are experimental data [19] and thelines are CHIPS Monte Carlo calculations at several values of the criticaltemperature parameter T .311

inclusive spectra that are decreasing steeply with energy, and correspondinglysteeply decreasing spectra of the quark-partons in a quasmon, onlythose secondary hadrons which get the maximum energy from the primaryquark-parton of energy k contribute to the inclusive spectra. This extremesituation requires the exchanged quark-parton with energy q, coming towardthe quasmon from the cluster, to move in a direction opposite to that of theprimary quark-parton. As a result the hadronization quark exchange processbecomes one-dimensional along the direction of k. If a neighboring nucleonor nucleon cluster with bound mass ˜µ absorbs the primary quark-partonand radiates the exchanged quark-parton in the opposite direction, then theenergy of the outgoing fragment is E = ˜µ + k − q, and the momentum isp = k + q. Both the energy and the momentum of the outgoing nuclear fragmentare known, as is the mass ˜µ of the nuclear fragment in nuclear matter,so the momentum of the primary quark-parton can be reconstructed usingthe approximate relationk = p + E − B · m N2. (23.21)Here B is the baryon number of the outgoing fragment (˜µ ≈ B · m N ) andm N is the nucleon mass. In Ref. [20] it was shown that the invariant inclusivespectra of pions, protons, deuterons, and tritons in proton-nucleusreactions at 400 GeV [21] not only have the same exponential slope but almostcoincide when they are plotted as a function of k = p+E kin. Using data2at 10 GeV [22], it was shown that the parameter k, defined by Eq. 23.21,is also appropriate for the description of secondary antiprotons produced inhigh energy nuclear reactions. This means that the extreme assumption ofone-dimensional hadronization is a good approximation.The same approximation is also valid for the quark fusion mechanism. Inthe one-dimensional case, assuming that q is the momentum of the secondquark fusing with the primary quark-parton of energy k, the total energy ofthe outgoing hadron is E = q + k and the momentum is p = k − q. In theone-dimensional case the secondary quark-parton must move in the oppositedirection with respect to the primary quark-parton, otherwise the mass ofthe outgoing hadron would be zero. So, for mesons k = p+E , in accordance2with Eq. 23.21. In the case of antiproton radiation, the baryon number of thequasmon is increased by one, and the primary antiquark-parton will spendits energy to build up the mass of the antiproton by picking up an antidiquark.Thus, the energy conservation law for antiproton radiation lookslike E + m N = q + k and k = p+E+m N2, which is again in accordance withEq. 23.21.The one-dimensional quark exchange mechanism was proposed in 1984312

Q(M) CRQ(M N-1) RQ(M min)kqPC(µ) ˜ CF(µ c) F(µ)Figure 23.6: Diagram of the quark exchange mechanism.[20]. Even in its approximate form it was useful in the analysis of inclusivespectra of hadrons and nuclear fragments in hadron-nuclear reactions at highenergies. Later the same approach was used in the analysis of nuclear fragmentationin electro-nuclear reactions [23]. Also in 1984 the quark-exchangemechanism developed in the framework of the non-relativistic quark modelwas found to be important for the explanation of the short distance featuresof NN interactions [24]. Later it was successfully applied to K − p interactions[25]. The idea of the quark exchange mechanism between nucleons was usefuleven for the explanation of the EMC effect [26]. For the non-relativisticquark model, the quark exchange technique was developed as an alternativeto the Feynman diagram technique at short distances [27].The CHIPS event generator models quark exchange processes, taking intoaccount kinematic and combinatorial factors for asymptotically free quarkpartons.In the naive picture of the quark-exchange mechanism, one quarkpartontunnels from the asymptotically free region of one hadron to theasymptotically free region of another hadron. To conserve color, anotherquark-parton from the neighboring hadron must replace the first quarkpartonin the quasmon. This makes the tunneling mutual, and the resultingprocess is quark exchange.The experimental data available for multihadron production at high energiesshow regularities in the secondary particle spectra that can be relatedto the simple kinematic, combinatorial, and phase space rules of such quarkexchange and fusion mechanisms. The CHIPS model combines these mechanismsconsistently.Fig. 23.6 shows a quark exchange diagram which helps to keep track ofthe kinematics of the process. It was shown in Section 23.4 that a quasmon,Q is kinematically determined by a few asymptotically free quark-partons313

homogeneously distributed over the invariant phase space. The quasmonmass M is related to the number of quark-partons N through< M 2 >= 4N(N − 1) · T 2 , (23.22)where T is the temperature of the system.The spectrum of quark partons can be calculated as( ) N−3dWk ∗ dk ∝ 1 − 2k∗ , (23.23)∗ Mwhere k ∗ is the energy of the primary quark-parton in the center-of-masssystem (CMS) of the quasmon. After the primary quark-parton is randomizedand the neighboring cluster, or parent cluster, P C, with bound mass ˜µis selected, the quark exchange process begins. To follow the process kinematicallyone should imagine a colored, compound system consisting of astationary, bound parent cluster and the primary quark. The primary quarkhas energy k in the lab system,k = k ∗ · EN + p N · cos(θ k )M N, (23.24)where M N , E N and p N are the mass, energy, and momentum of the √quasmonin the lab frame. The mass of the compound system, CF , is µ c = (˜µ + k) 2 ,where ˜µ and k are the corresponding four-vectors. This colored compoundsystem decays into a free outgoing nuclear fragment, F , with mass µ and arecoiling quark with energy q. q is measured in the CMS of ˜µ, which coincideswith the lab frame in the present version of the model because no clustermotion is considered. At this point one should recall that a colored residualquasmon, CRQ, with mass M N−1 remains after the radiation of k. CRQ isfinally fused with the recoil quark q to form the residual quasmon RQ. Theminimum mass of RQ should be greater than M min , which is determined bythe minimum mass of a hadron (or Chipolino double-hadron as defined inSection 23.4) with the same quark content.All quark-antiquark pairs with the same flavor should be canceled in theminimum mass calculations. This imposes a restriction, which in the CMSof µ c , can be written as2q · (E − p · cos θ qCQ ) + M 2 N−1 > M 2 min, (23.25)where E is the energy and p is the momentum of the colored residual quasmonwith mass M N−1 in the CMS of µ c . The restriction for cos θ qCQ then becomescos θ qCQ < 2qE − M min 2 + MN−12 , (23.26)2qp314

which impliesq > M N−1 2 − Mmin2 . (23.27)2 · (E + p)A second restriction comes from the nuclear Coulomb barrier for chargedparticles. The Coulomb barrier can be calculated in the simple form:E CB = Z F · Z RA 1 3F + A 1 3R(MeV), (23.28)where Z F and A F are the charge and atomic weight of the fragment, andZ R and A R are the charge and atomic weight of the residual nucleus. Theobvious restriction isq < k + ˜µ − µ − E CB . (23.29)In addition to 23.27 and 23.29, the quark exchange mechanism imposesrestrictions which are calculated below. The spectrum of recoiling quarks issimilar to the k ∗ spectrum in the quasmon (23.23):dW(1q dq d cos θ ∝ − 2q˜µ) n−3, (23.30)where n is the number of quark-partons in the nucleon cluster. It is assumedthat n = 3A C , where A C is the atomic weight of the parent cluster. Thetunneling of quarks from one nucleon to another provides a common phasespace for all quark-partons in the cluster.An additional equation follows from the mass shell condition for the outgoingfragment,µ 2 = ˜µ 2 + 2˜µ · k − 2˜µ · q − 2k · q · (1 − cos θ kq ), (23.31)where θ kq is the angle between quark-parton momenta in the lab frame. Fromthis equation q can be calculated asq =˜µ · (k − ∆)˜µ + k · (1 − cos θ kq ) , (23.32)where ∆ is the covariant binding energy of the cluster ∆ = µ2 −˜µ 2. The quark2˜µexchange probability integral can be then written in the form:P (k, ˜µ, µ) =∫δ [ µ 2 − ˜µ 2 − 2˜µ · k + 2˜µ · q + 2k · q · (1 − cos θ kq ) ]×(1 − 2q˜µ ) n−3qdq·d cos θ kq . (23.33)315

Using the δ-function to perform the integration over q one obtainsorP (k, ˜µ, µ) =P (k, ˜µ, µ) =The result of the integration is˜µP (k, ˜µ, µ) =4k(n − 2)⎡(2(k − ∆)× ⎣ 1 −˜µ + 2k) n−3∫ ( 2(k − ∆)1 −˜µ + k(1 − cos θ kq )˜µ(k − ∆)×2[˜µ + k(1 − cos θ kq )] dcos θ 2 kq (23.34)) n−3∫ ( 2(k − ∆)1 −˜µ + k(1 − cos θ kq )() 2˜µ(k − ∆)טµ + k(1 − cos θ kq )( )˜µ + k(1 − cos θkq )× d. (23.35)˜µ(k − ∆)) n−2high−(1 −2(k − ∆)˜µ) ⎤ n−2For randomization it is convenient to make z a random parameterz = 1 −low⎦ . (23.36)2(k − ∆)= 1 − . (23.37)˜µ + k(1 − cos θ kq )2q˜µFrom (23.36) one can find the high and the low limits of the randomization.The first limit is a limit for k: k > ∆. It is similar to the restriction forQuasmon fragmentation in vacuum: k ∗ > µ2µ2. The second limit is k = ,2M 2˜µwhen the low limit of randomization becomes equal to zero. If k < µ2, then2˜µ−1 < cos θ kq < 1 and z low = 1 − 2(k−∆)˜µ. If k > µ22˜µ , then the range of cos θ kqis −1 < cos θ kq < µ2− 1 and z k˜µ low = 0. This value of z low should be correctedusing the Coulomb barrier restriction (23.29), and the value of z high shouldbe corrected using the minimum residual Quasmon restriction (23.27). Inthe case of a Quasmon with momentum much less than k it is possible toimpose tighter restrictions than (23.27) because the direction of motion ofthe CRQ is opposite to k. So cos θ qCQ = − cos θ kq , and from (23.32) one canfind thatcos θ qCQ = 1 −316˜µ · (k − ∆ − q). (23.38)k · q

So in this case the equation (23.27) can be replaced by the more stringentone:q > M N−1 2 − M min 2 + 2 p·˜µ (k − ∆)k2 · (E + p + p·˜µ ) . (23.39)kThe integrated kinematical quark exchange probability (in the range fromz low to z high ) is˜µ4k(n − 2) · zn−2 , (23.40)and the total kinematical probability of hadronization of the quark-partonwith energy k into a nuclear fragment with mass µ is˜µ4k(n − 2) · (z n−2high − ) zn−2 low . (23.41)This can be compared with the vacuum probability of the quark fusion mechanismfrom Section 23.4:M − 2k4k(N − 3) zN−3 max . (23.42)The similarity is very important, as the absolute probabilities define thecompetition between vacuum and nuclear channels.Equations (23.40) and (23.41) can be used for randomization of z:z = z low + n−2√ R · (z high − z low ), (23.43)where R is a random number, uniformly distributed in the interval (0,1).Equation (23.41) can be used to control the competition between differentnuclear fragments and hadrons in the hadronization process, but in contrastto the case of “in vacuum” hadronization it is not enough to take into accountonly quark combinatorics of the Quasmon and the outgoing hadron. Inthe case of hadronization in nuclear matter, different parent bound clustersshould be taken into account as well. For example, tritium can be radiatedas a result of quark exchange with a bound tritium cluster or as a result ofquark exchange with a bound 3 He cluster.To calculate the yield of fragments it is necessary to calculate the probabilityto find a cluster with certain proton and neutron content in a nucleus.One could consider any particular probability as an independent parameter,but in such a case the process of tuning the model would be difficult. Weproposed the following scenario of clusterization. A gas of quasi-free nucleonsis close to the phase transition to a liquid phase bound by strong quarkexchange forces. Precursors of the liquid phase are nuclear clusters, whichmay be considered as “drops” of the liquid phase within the nucleus. Any317

cluster can meet another nucleon and absorb it (making it bigger), or it canrelease one of the nucleons (making it smaller). The first parameter ε 1 is thepercentage of quasi-free nucleons not involved in the clusterization process.The rest of the nucleons (1 − ε 1 ) clusterize. We assume that since on theperiphery of the nucleus the density is lower, one can consider only dibaryonclusters, and neglect triple-baryon clusters. Still we denote the number ofnucleons clusterized in dibaryons on the periphery by the parameter ε 2 . Inthe dense part of the nucleus, strong quark exchange forces make clusters outof quasi-free nucleons with high probability. To characterize the distributionof clusters the parameter ω of clusterization probability was used.If the number of nucleons involved in clusterization is a = (1−ε 1 −ε 2 )·A,then the probability to find a cluster consisting of ν nucleons is defined bythe distributionP ν ∝ C a ν · ων−1 , (23.44)where Cνa is the corresponding binomial coefficient. The coefficient of proportionalitycan be found from the equationa∑a = b · ν · Cν a · ων−1 = b · a · (1 + ω) a−1 . (23.45)ν=1Thus, the number of clusters consisting of ν nucleons isP ν = Ca ν · ων−1. (23.46)(1 + ω)a−1For clusters with an even number of nucleons we used only isotopically symmetricconfigurations (ν = 2n, n protons and n neutrons) and for odd clusters(ν = 2n + 1) we used only two configurations: n neutrons with n + 1 protonsand n + 1 neutrons with n protons. This restriction, which we call “isotopicfocusing”, can be considered as an empirical rule of the CHIPS model whichhelps to describe data. It is applied in the case of nuclear clusterization(isotopically symmetric clusters) and in the case of hadronization in nuclearmatter. In the hadronization process the Quasmon is shifted from isotopicsymmetric state (e.g., by capturing a negative pion) and transfers excessivecharge to the outgoing nuclear cluster. This tendency is symmetric withrespect to the Quasmon and the parent cluster.The temperature parameter used to calculate the number of quark-partonsin a Quasmon (see equation 23.22) was chosen to be T = 180 MeV, which isthe same as in Section 23.4.The CHIPS model is mostly a model of fragmentation conserving energy,momentum, and charge. But to compare it with experimental data oneneeds to model also the first interaction of the projectile with the nucleus.318

For proton-antiproton annihilation this was easy, as we assumed that in theinteraction at rest, a proton and antiproton always create a Quasmon. Inthe case of pion capture the pion can be captured by different clusters. Weassumed that the probability of capture is proportional to the number ofnucleons in a cluster. After the capture the Quasmon is formed, and theCHIPS generator produces fragments consecutively and recursively, choosingat each step the quark-parton four-momentum k, the type of parent andoutgoing fragment, and the four-momentum of the exchange quark-parton q,to produce a final state hadron and the new Quasmon with less energy.In the CHIPS model we consider this process as a chaotic process withlarge number of degrees of freedom and do not take into account any finalstate interactions of outgoing hadrons. Nevertheless, when the excitationenergy dissipates, and in some step the Quasmon mass drops belowmass shell, the quark-parton mechanism of hadronization fails. To modelthe event exclusively, it becomes necessary to continue fragmentation at thehadron level. Such fragmentation process is known as nuclear evaporation.It is modeled using the non-relativistic phase space approach. In the nonrelativisticcase the phase space of nucleons can be integrated as well as inthe ultra-relativistic case of quark-partons.The general formula for the non-relativistic phase space can be foundstarting with the phase space for two particles ˜Φ 2 . It is proportional to theCMS momentum:√˜Φ 2 (W 2 ) ∝ W 2 , (23.47)where W 2 is a total kinetic energy of the two non-relativistic particles. If thephase space integral is known for n−1 hadrons then it is possible to calculatethe phase space integral for n hadrons:∫˜Φ n (W n ) = ˜Φn−1 (W n−1 ) · δ(W n − W n−1 − E kin )×√E kin dE kin dW n−1 . (23.48)Using (23.47) and (23.48) one can find that˜Φ n (W n ) ∝ W 3 2 n− 5 2n (23.49)and the spectrum of hadrons, defined by the phase space of residual n − 1nucleons, can be written as(dN√ ∝ 1 − E ) 3 2kinn−4 . (23.50)Ekin dE kin W nThis spectrum can be randomized. The only problem is from which levelone should measure the thermal kinetic energy when most nucleons in nuclei319

are filling nuclear levels with zero temperature. To model the evaporationprocess we used this unknown level as a parameter U of the evaporationprocess. Comparison with experimental data gives U = 1.7 MeV. Thus, thetotal kinetic energy of A nucleons isW A = U · A + E ex , (23.51)where E ex is the excitation energy of the nucleus.To be radiated, the nucleon should overcome the thresholdU thresh = U + U bind + E CB , (23.52)where U bind is a separation energy of the nucleon, and E CB is the Coulombbarrier energy which is non-zero only for positive particles and can be calculatedusing formula (23.28).Among several experimental investigations of nuclear pion capture at restwe have selected four published results which constitute, in our opinion, a representativedata set covering a wide range of target nuclei, types of producedhadrons and nuclei fragments, and their energy range. In the first publication[38] the spectra of charged fragments (protons, deuterons, tritium, 3 He,4 He) in pion capture were measured on 17 nuclei within one experimentalsetup. To verify the spectra we compared them for a carbon target withdetailed measurements of the spectra of charged fragments given in Ref. [39].In addition, we took 6 Li spectra for a carbon target from the same paper.The neutron spectra were added from Ref. [40] and Ref. [41]. We presentdata and Monte Carlo distributions as the invariant phase space functionf = dσ depending on the variable k = p+E kinas defined in equation (23.21).pdE 2Spectra on 9 Be, 12 C, 28 Si ( 27 Al for secondary neutrons), 59 Co ( 64 Cu forsecondary neutrons), and 181 Ta are shown in Figures 23.7 through 23.11. Thedata are described well, including the total energy spent in the reaction toyield the particular type of fragments.The evaporation process for nucleons is described well, too. It is exponentialin k, and looks especially impressive for Si/Al and Co/Cu data, wherethe Coulomb barrier is low, and one can see proton evaporation as a continuationof the evaporation spectra from secondary neutrons. This way theexponential behavior of the evaporation process can be followed over 3 ordersof magnitude. Clearly seen is the transition region at k ≈ 90 MeV (kineticenergy 15 − 20 MeV) between the quark-level hadronization process andthe hadron-level evaporation process. For light target nuclei the evaporationprocess becomes much less prominent.The 6 Li spectrum on a carbon target exhibits an interesting regularitywhen plotted as a function of k: it practically coincides with the spectrum of320

Table 23.1: Clusterization parameters9 Be 12 C 28 Si 59 Co 181 Taε 1 0.45 0.40 0.35 0.33 0.33ε 2 0.15 0.15 0.05 0.03 0.02ω 5.00 5.00 5.00 5.00 5.004 He fragments, and shows exponential behavior in a wide range of k, correspondingto a few orders of magnitude in the invariant cross section. To keepthe figure readable, we did not plot the 6 Li spectrum generated by CHIPS.It coincides with the 4 He spectrum at k > 200 MeV, and underestimateslithium emission at lower energies, similarly to the 3 He and tritium data.Between the region where hadron-level processes dominate and the kinematiclimit all hadronic spectra slopes become close when plotted as a functionof k. In addition to this general behavior there is an effect of strongproton-neutron splitting. For protons and neutrons it reaches almost anorder of magnitude. To model such splitting in the CHIPS generator, themechanism of “isotopic focusing” was used, which locally transfers the negativecharge from the pion to the first radiated nuclear fragment.Thus, the model qualitatively describes all typical features of the pion captureprocess. The question is what can be extracted from the experimentaldata with this tool. The clusterization parameters are listed in Table 23.1.No formal fitting procedure has been performed. A balanced qualitativeagreement with all data was used to tune the parameters. The differencebetween the ε 2ε 1ratio and the parameter ω (which is the same for all nuclei) isan indication that there is a phase transition between the gas phase and theliquid phase of the nucleus. The large value of the parameter ω, determiningthe average size of a nuclear cluster, is critical in describing the model spectraat large k, where the fragment spectra approach the kinematical limits.Using the same parameters of clusterization we compared the data [42] onγ absorption on Al and Ca nuclei (Fig. 23.12) with the CHIPS results. Onecan see that the spectra of secondary protons and deuterons are qualitativelydescribed by the CHIPS model.The CHIPS model covers a wide spectrum of hadronic reactions with alarge number of degrees of freedom. In the case of nuclear reactions theCHIPS generator helps to understand phenomena such as an order of magnitudesplitting of neutron and proton spectra, high yield of energetic nuclearfragments, and emission of nucleons which kinematically can be producedonly if seven or more nucleons are involved in the reaction.The CHIPS generator allows to extract collective parameters of a nucleus321

dN/pdE (MeV -2 Capture -1 )10 -110 -210 -310 -410 -510 -610 -710 -810 -910 -210 -310 -410 -510 -610 -710 -810 -9protons〈E kin〉 e/m= 6.3 / 5.8 MeVPion capture on 9 Be nucleusneutrons〈E kin〉 e/m= 100.8 / 81.2 MeVdeuterons〈E kin〉 e/m= 5.9 / 5.3 MeVtritium〈E kin〉 e/m= 4.4 / 2.8 MeVHelium-4〈E kin〉 e/m= 1.2 / 5.9 MeVHelium-3〈E kin〉 e/m= 0.4 / 1.3 MeV10 -1 0 200 400 0 200 400k = (p+E kin)/2 (MeV)Figure 23.7: Comparison of the CHIPS model results with experimentaldata on proton, neutron, and nuclear fragment production in the capture ofnegative pions on 9 Be. Proton [38] and neutron [40] experimental spectra areshown in the upper left panel by open circles and open squares, respectively.The model calculations are shown by the two corresponding solid lines. Thesame arrangement is used to present 3 He [38] and tritium [38] spectra in thelower left panel. Deuterium [38] and 4 He [38] spectra are shown in the rightpanels of the figure by open squares and lines (CHIPS model). The averagekinetic energy carried away by each nuclear fragment is shown in the panelsby the two numbers: first is the average calculated using the experimentaldata shown; second is the model result.322

dN/pdE (MeV -2 Capture -1 )10 -110 -210 -310 -410 -510 -610 -710 -810 -910 -210 -310 -410 -510 -610 -710 -810 -9protons〈E kin〉 e/m= 8.7 / 9.8 MeVPion capture on 12 C nucleusneutrons〈E kin〉 e/m= 79.8 / 67.8 MeVdeuterons〈E kin〉 e/m= 5.2 / 5.4 MeVtritium〈E kin〉 e/m= 2.5 / 1.8 MeVHelium-4〈E kin〉 e/m= 5.3 / 5.9 MeVHelium-3Lithium〈E kin〉 e/m= 0.6 / 0.8 MeV10 -1 0 200 400 0 200 400k = (p+E kin)/2 (MeV)Figure 23.8: Same as in Figure 23.7, for pion capture on 12 C. The experimentalneutron spectrum is taken from [41]. In addition, the detailed data oncharged particle production, including the 6 Li spectrum, taken from Ref. [39],are superimposed on the plots as a series of dots.323

dN/pdE (MeV -2 Capture -1 )10 -110 -210 -310 -410 -510 -610 -710 -810 -910 -210 -310 -410 -510 -610 -710 -810 -9Pion capture on 28 Si nucleusneutrons〈E kin〉 e/m= 73.2 / 64.9 MeVprotons〈E kin〉 e/m= 11.9 /13.3 MeVdeuterons〈E kin〉 e/m= 4.1 / 3.9 MeVtritium〈E kin〉 e/m= 1.0 / 1.3 MeVHelium-4〈E kin〉 e/m= 1.0 / 1.5 MeVHelium-3〈E kin〉 e/m= 0.4 / 0.4 MeV10 -1 0 200 400 0 200 400k = (p+E kin)/2 (MeV)Figure 23.9: Same as in Figure 23.7, for pion capture on 28 Si nucleus. Theexperimental neutron spectrum is taken from [41], for the reaction on 27 Al.324

dN/pdE (MeV -2 Capture -1 )10 -110 -210 -310 -410 -510 -610 -710 -810 -910 -210 -310 -410 -510 -610 -710 -810 -9Pion capture on 59 Co nucleusneutrons〈E kin〉 e/m= 73.1 / 68.3 MeVprotons〈E kin〉 e/m= 10.5 /10.0 MeVdeuterons〈E kin〉 e/m= 3.1 / 3.2 MeVtritium〈E kin〉 e/m= 0.8 / 1.1 MeVHelium-4〈E kin〉 e/m= 0.9 / 1.3 MeVHelium-3〈E kin〉 e/m= 0.2 / 0.3 MeV10 -1 0 200 400 0 200 400k = (p+E kin)/2 (MeV)Figure 23.10: Same as in Figure 23.7, for pion capture on 59 Co. The experimentalneutron spectrum is taken from [41], for the reaction on 64 Cu.325

dN/pdE (MeV -2 Capture -1 )10 -110 -210 -310 -410 -510 -610 -710 -810 -910 -210 -310 -410 -510 -610 -710 -810 -9protons〈E kin〉 e/m= 6.6 / 6.9 MeVPion capture on 181 Ta nucleusneutrons〈E kin〉 e/m= 75.4 / 75.6 MeVdeuterons〈E kin〉 e/m= 2.2 / 2.5 MeVtritium〈E kin〉 e/m= 0.8 / 0.9 MeVHelium-4〈E kin〉 e/m= 1.2 / 1.0 MeVHelium-3〈E kin〉 e/m= 0.1 / 0.2 MeV10 -1 0 200 400 0 200 400k = (p+E kin)/2 (MeV)Figure 23.11: Same as in Figure 23.7, for pion capture on 181 Ta. The experimentalneutron spectrum is taken from [41].326

40 Ca(γ,p) spectral cross sectionE γ=60 MeV10 2 80 100 120 140 160 180 200dσ/p pdE pdΩ p(nb MeV -2 sr -1 )10110 -1Θ = 60oΘ = 90oΘ = 150 o10 -2k = (p p+T p)/2 (MeV)Figure 23.12: Comparison of CHIPS model with experimental data [42] onproton and deuteron production at 90 ◦ in photonuclear reactions on 27 Al and40 Ca at 59 – 65 MeV. Open circles and solid squares represent the experimentalproton and deuteron spectra, respectively. Solid and dashed lines showthe results of the corresponding CHIPS model calculation. Statistical errorsin the CHIPS results are not shown and can be judged by the point-to-pointvariations in the lines. The comparison is absolute, using the values of totalphotonuclear cross section 3.6 mb for Al and 5.4 mb for Ca, as given inRef. [43].327

such as clusterization. The qualitative conclusion based on the fit to theexperimental data is that the main fraction of nucleons is clusterized, at leastin heavy nuclei. The nuclear clusters can be considered as drops of a liquidnuclear phase. The quark exchange makes the phase space of quark-partonsof each cluster common, stretching kinematic limits for particle production.The hypothetical quark exchange process is important not only for thenuclear clusterization, but for the nuclear hadronization process, too. Thequark exchange between the excited cluster (Quasmon) and a neighboringnuclear cluster, even at low excitation level, operates with quark-partons atenergies comparable with the nucleon mass. As a result it easily reaches thekinematic limits of the reaction, revealing the multinucleon nature of theprocess.Up to now the most underdeveloped part of the model has been the initialinteraction between projectile and target. That is why we started withproton-antiproton annihilation and pion capture on nuclei at rest, which donot involve any interaction cross section. The further development of themodel will require a better understanding of the mechanism of the first interaction.However, we believe that even the basic model will be useful in theunderstanding the nature of multihadron fragmentation, and because of itsfeatures, is a suitable candidate for the hadron production and hadron cascadeparts of the newly developed event generation and detector simulationMonte Carlo computer codes.23.6 Modeling of real and virtual photon interactionswith nuclei below pion productionthreshold.In the example of the photonuclear reaction discussed in the Appendix D,namely the description of 90 ◦ proton and deuteron spectra in A(γ, X) reactionsat E γ = 59−65 MeV, the assumption on the initial Quasmon excitationmechanism was the same. The description of the 90 ◦ data was satisfactory,but the generated data showed very little angular dependence, as the velocityof Quasmons produced in the initial state was small, and the fragmentationprocess was almost isotropic. Experimentally, the angular dependence of secondaryprotons in photo-nuclear reactions is quite strong even at low energies(see, for example, Ref. [44]). This is a challenging experimental fact whichis difficult to explain in any model. It’s enough to say that if the angulardependence of secondary protons in the γ 40 Ca interaction at 60 MeV is analyzedin terms of relativistic boost, then the velocity of the source should328

each 0.33c; hence the mass of the source should be less than pion mass. Themain subject of the present publication is to show that the quark-exchangemechanism used in the CHIPS model can not only model the clusterizationof nucleons in nuclei and hadronization of intranuclear excitations into nuclearfragments, but can also model complicated mechanisms of interactionof photons and hadrons in nuclear matter.In Ref. Appendix D we defined a quark-exchange diagram which helpsto keep track of the kinematics of the quark-exchange process (see Fig. 1 inApendix D). To apply the same diagram to the first interaction of a photonwith a nucleus, it is necessary to assume that the quark-exchange processtakes place in nuclei continuously, even without any external interaction.Nucleons with high momenta do not leave the nucleus because of the lackof excess energy. The hypothesis of the CHIPS model is that the quarkexchangeforces between nucleons [24] continuously create clusters in normalnuclei. Since a low-energy photon (below the pion production threshold)cannot be absorbed by a free nucleon, other absorption mechanisms involvingmore than one nucleon have to be used.The simplest scenario is photon absorption by a quark-parton in the nucleon.At low energies and in vacuum this does not work because there isno corresponding excited baryonic state. But in nuclear matter there is apossibility to exchange this quark with a neighboring nucleon or a nuclearcluster. The diagram for the process is shown in Fig. 23.13. In this casethe photon is absorbed by a quark-parton from the parent cluster PC 1 , andthen the secondary nucleon or cluster PC 2 absorbs the entire momentum ofthe quark and photon. The exchange quark-parton q restores the balanceof color, producing the final-state hadron F and the residual Quasmon RQ.The process looks like a knockout of a quasi-free nucleon or cluster out ofthe nucleus. It should be emphasized that in this scenario the CHIPS eventgenerator produces not only “quasi-free” nucleons but “quasi-free” fragmentstoo. The yield of these quasi-free nucleons or fragments is concentrated inthe forward direction.The second scenario which provides for an angular dependence is theabsorption of the photon by a colored fragment (CF 2 in Fig. 23.14). In thisscenario, both the primary quark-parton with momentum k and the photonwith momentum q γ are absorbed by a parent cluster (PC 2 in Fig. 23.14),and the recoil quark-parton with momentum q cannot fully compensate themomentum k + q γ . As a result the radiation of the secondary fragment inthe forward direction becomes more probable.In both cases the angular dependence is defined by the first act of hadronization.The further fragmentation of the residual Quasmon is almost isotropic.It was shown in Section 23.4 that the energy spectrum of quark partons329

PC 1(µ ˜ 1) CF 1RQ(M min)(ω,q γ)kqPC 2(µ ˜ 2) CF 2F(µ)Figure 23.13: Diagram of photon absorption in the quark exchange mechanism.PC 1,2 stand for parent clusters with bound masses ˜µ 1,2 , participatingin the quark-exchange. CF 1,2 stand for the colored nuclear fragments in theprocess of quark exchange. F(µ) denotes the outgoing hadron with mass µ inthe final state. RQ is the residual Quasmon which carries the rest of the excitationenergy and momentum. M min characterizes its minimum mass definedby its quark content. Dashed lines indicate colored objects. The photon isabsorbed by a quark-parton k from the parent cluster PC 1 .PC 1(µ ˜ 1) CF 1RQ(M min)kqCF 2PC 2(µ ˜ 2)(ω,q γ)F(µ)Figure 23.14: Diagram of photon absorption in the quark-exchange mechanism.The notation is the same as in Fig. 23.13. The photon is absorbed bythe colored fragment CF 2 .330

in a Quasmon can be calculated as( ) N−3dWk ∗ dk ∝ 1 − 2k∗ , (23.53)∗ Mwhere k ∗ is the energy of the primary quark-parton in the Center of MassSystem (CMS) of the Quasmon, M is the mass of the Quasmon, and N,the number of quark-partons in the Quasmon, can be calculated from theequation< M 2 >= 4 · N · (N − 1) · T 2 . (23.54)Here T is the temperature of the system.In the first scenario of the γA interaction (Fig. 23.13), as both interactingparticles are massless, we assumed that the cross section for the interactionof the photon with a particular quark-parton is proportional to the chargeof the quark-parton squared, and inversely proportional to the mass of thephoton-parton system s, which can be calculated ass = 2ωk(1 − cos(θ k )). (23.55)Here ω is the energy of the photon, and k is the energy of the quark-partonin the Laboratory System (LS):k = k ∗ · EN + p N · cos(θ k )M N. (23.56)In the case of a virtual photon, equation (23.55) can be written ass = 2k(ω − q γ · cos(θ k )), (23.57)where q γ is the momentum of the virtual photon. In both cases equation(23.53) transforms into( ) N−3dWdk ∝ 1 − 2k∗ , (23.58)∗ Mand the angular distribution in cos(θ k ) converges to a δ-function: in the caseof a real photon cos(θ k ) = 1, and in the case of a virtual photon cos(θ k ) = ω q γ.In the second scenario for the photon interaction (Fig. 23.14) we assumedthat both the photon and the primary quark-parton, randomized accordingto equation (23.53), enter the parent cluster PC 2 , and after that the normalprocedure of quark exchange continues, in which the recoiling quark-partonq returns to the first cluster.331

40 Ca(γ,p) spectral cross sectionE γ=60 MeV10 2 80 100 120 140 160 180 200dσ/p pdE pdΩ p(nb MeV -2 sr -1 )10110 -1Θ = 60oΘ = 90oΘ = 150 o10 -2k = (p p+T p)/2 (MeV)Figure 23.15: Comparison of the CHIPS model results (lines in the figure)with the experimental data [42] on proton spectra at 90 ◦ in the photonuclearreactions on 40 Ca at 59–65 MeV (open circles), and proton spectra at 60 ◦(triangles) and 150 ◦ (diamonds). Statistical errors in the CHIPS results arenot shown but can be judged by the point-to-point variations in the lines. Thecomparison is absolute, using the value of total photonuclear cross section of5.4 mb for Ca, as given in Ref. [43].332

12 C(γ,p) reaction at E γ= 123 MeV1.057°0.1dσ/p pdE pdΩ p(nb MeV -2 sr -1 )1.00.11.00.11.00.177°97°117°1.00.1127°180 200 220 240 260 280 300k = (p p+T p)/2 (MeV)Figure 23.16: Comparison of the CHIPS model results (lines in the figure)with the experimental data [46] on proton spectra at 57 ◦ , 77 ◦ , 97 ◦ , 117 ◦ , and127 ◦ in the photonuclear reactions on 12 C at 123 MeV (open circles). Thevalue of the total photonuclear cross section was set at 1.8 mb.333

12 C(γ,p) reaction at E γ= 151 MeV1.057°0.1dσ/p pdE pdΩ p(nb MeV -2 sr -1 )1.00.11.00.11.00.177°97°117°1.00.1127°180 200 220 240 260 280 300k = (p p+T p)/2 (MeV)Figure 23.17: Same as in Fig. 23.16, for the photon energy 151 MeV.334

12 C(γ * ,p) spectral cross sectiondσ/p pdE pdω dΩ p dΩ e (pb MeV-3 sr -2 )110 -110 -2Parallel kinematicsω = 210 MeVq γ= 585 MeV/cΘ qp< 6˚10 -3100 200 300 400 500k = (p p+T p)/2 (MeV)Figure 23.18: Comparison of the CHIPS model results (line in the figure) withthe experimental data [47] (open circles) on the proton spectrum measuredin parallel kinematics in the 12 C(e,e ′ p) reaction at an energy transfer equalto 210 MeV and momentum transfer equal to 585 MeV/c. Statistical errorsin the CHIPS result are not shown but can be judged by the point-to-pointvariations in the line. The relative normalization is arbitrary.335

An additional parameter in the model is the relative contribution of bothmechanisms. As a first approximation we assumed equal probability, butin the future, when more detailed data are obtained, this parameter can beadjusted.We begin the comparison with the data on proton production in the40 Ca(γ, X) reaction at 90 ◦ at 59–65 MeV [42], and at 60 ◦ and 150 ◦ at 60 MeV[45]. We analyzed these data together to compare the angular dependencegenerated by CHIPS with experimental data. The data are presented asa function of the invariant inclusive cross section f = depending ondσp pdE pthe variable k = Tp+pp2, where T p and p p are the kinetic energy and themomentum of the secondary proton. As one can see from Fig. 23.15, theangular dependence of the proton yield in photoproduction on 40 Ca at 60MeV is reproduced quite well by the CHIPS event generator.The second set of measurements that we use for the benchmark comparisondeals with the secondary proton yields in 12 C(γ, X) reactions at 123 and151 MeV [46], which is still below the pion production threshold on a freenucleon. Inclusive spectra of protons have been measured in γ 12 C reactionsat 57 ◦ , 77 ◦ , 97 ◦ , 117 ◦ , and 127 ◦ . Originally, these data were presented as afunction of the missing energy. We present the data in Figs. 23.16 and 23.17together with CHIPS calculations in the form of the invariant inclusive crosssection dependent on k. All parameters of the model such as temperatureT and parameters of clusterization for the particular nucleus were the sameas in Appendix D, where pion capture spectra were fitted. The agreementbetween the experimental data and the CHIPS model results is quite remarkable.Both data and calculations show significant strength in the proton yieldcross section up to the kinematical limits of the reaction. The angular distributionin the model is not as prominent as in the experimental data, butagrees well qualitatively.Using the same parameters, we applied the CHIPS event generator tothe 12 C(e,e ′ p) reaction measured in Ref.[47]. The proton spectra were measuredin parallel kinematics in the interaction of virtual photons with energyω = 210 MeV and momentum q γ = 585 MeV/c. To account for the experimentalconditions in the CHIPS event generator, we have selected protonsgenerated in the forward direction with respect to the direction of the virtualphoton, with the relative angle Θ qp < 6 ◦ . The CHIPS generated distributionand the experimental data are shown in Fig. 23.18 in the form of the invariantinclusive cross section as a function of k. The CHIPS event generatorworks only with ground states of nuclei so we did not expect any narrowpeaks for 1 p 3/2 -shell knockout or for other shells. Nevertheless we found thatthe CHIPS event generator fills in the so-called “ 1 s 1/2 -shell knockout” region,336

which is usually artificially smeared by a Lorentzian [48]. In the regular fragmentationscenario the spectrum of protons below k = 300 MeV is normal;it falls down to the kinematic limit. The additional yield at k > 300 MeV isa reflection of the specific first act of hadronization with the quark exchangekinematics. The slope increase with momentum is approximated well by themodel, but it is obvious that the yield close to the kinematic limit of the2 → 2 reaction can only be described in detail if the excited states of theresidual nucleus are taken into account.The angular dependence of the proton yield in low-energy photo-nuclearreactions is described in the CHIPS model and event generator. The mostimportant assumption in the description is the hypothesis of a direct interactionof the photon with an asymptotically free quark in the nucleus, evenat low energies. This means that asymptotic freedom of QCD and dispersionsum rules [37] can in some way be generalized for low energies. The knockoutof a proton from a nuclear shell or the homogeneous distributions of nuclearevaporation cannot explain significant angular dependences at low energies.The same mechanism appears to be capable of modeling proton yields insuch reactions as the 16 C(e,e ′ p) reaction measured at MIT Bates [47], whereit was shown that the region of missing energy above 50 MeV reflects “twoor-more-particleknockout” (or the “continuum” in terms of the shell model).The CHIPS model may help to understand and model such phenomena.23.7 Chiral invariant phase-space decay in highenergy hadron nuclear reactionsChiral invariant phase-space decay can be used to deeccite an excitedhadronic system. This possibility can be exploited to replace the intranuclearcascading after a high energy primary interaction took place. Thebasic assumption in this is that the energy loss of the high energy hadronin nuclear matter is approximately a constant per unit path length (about0.7 GeV/fm). This energy is extracted from the soft part of the particlespectrum of the primary interaction, and particles with formation times thatplace them within the nuclear boundaries.Several approaches of transfering this energy into quasmons were studied,and comparisons with energy spectra of particles emitted in the backwardhemisphere were made for a range of materials. Best results were achievedwith a model that creates one quasmon per particle absorbed in the nucleus.A first, beta level implementation is released in geant4 4.0. More informationis to be published in a refereed journal.337

23.8 Conclusion.This Manual for the release of CHIPS in GEANT4 is made not only forthe blind im matter simulation of the hadronic processes, but for those users,who would like to improve the interaction part of the CHIPS event generatorfor their own specific reactions. For these users some advice concerning thedata presentation can be useful.This is a good idea to use a normalized invariant function ρ(k)ρ = 2E · d3 σσ tot · d 3 p ∝dσσ tot · pdE ,where σ tot is a total cross section of the reaction. So the simple rule is todivide the distribution over the hadron energy (E) by the momentum andby the reaction cross section. The argument k can be calculated for anyoutgoing hadron or fragment ask = E + p − B · m N2which has a meaning of the energy of the primary quark-parton. As thespectrum of the quark partons is universal for all the secondary hadrons orfragments, the distributions over this parameter have similar shape for all thesecondaries. They should differ only approaching the kinematic limits or inthe evaporation region. This feature is useful for any analysis of experimentaldata independently of the CHIPS model.The released version of the CHIPS event generator is not perfect yet, soin case of an error it is necessary to distinguish between the error of thetest program (CHIPStest.cc) and the error in the body of the generator.Usually the error printing contains the address of the routine, but sometimesthe name is abbreviated so that instead of the G4QEnvironment one canfind G4QE, instead of the G4Quasmon the G4Q is used, or instead of theG4QNucleus only the G4QN appears. The errors in the CHIPStest.cccan be easily analyzed. Even if sometimes energy or charge is not conserved,just exclude this check and keep going. On the other hand, if the error isin the body it is difficult to fix. The normal procedure is to uncomment theflags of the debugging prints in the corresponding part of the source codeand try to find out the reason. Anyway inform authors about the error. Donot forget to attach the CHIPStest.cc and the chipstest.in files.The concluding remarks should be made about the parameters of themodel. The main parameter (the critical temperature T c ) should not bevaried. A big set of data confirms 180 MeV while from the mass spectrumof hadrons it can be found more precisely as 182 MeV. The clusterization338,

parameter is 4. which is just about 4π/3. If the quark exchange starts atthe mean distance between baryons in the dense part of the nucleus, thenthe radius of the clusterization sphere is twice bigger than ”the radius of thespace, occupied by the baryon”. It gives 8 for the parameter, but the space,occupied by the baryon can not be spheric; only cubic subdivision of space ispossible so the factor π/6 appears. But this is a rough estimate, so 4 or even5 can be tried. The surface parameter fD is slightly varying with A growingfrom 0 to 0.04. For the present CHIPS version the recomended parametersfor low energies areA T s/u eta noP fN fD Cp rM sALi 180. 0.1 0.3 223 .4 .00 4. 1.0 0.4Be 180. 0.1 0.3 223 .4 .00 4. 1.0 0.4C 180. 0.1 0.3 223 .4 .00 4. 1.0 0.4O 180. 0.1 0.3 223 .4 .02 4. 1.0 0.4F 180. 0.1 0.3 223 .4 .03 4. 1.0 0.4Al 180. 0.1 0.3 223 .4 .04 4. 1.0 0.4Ca 180. 0.1 0.3 223 .4 .04 4. 1.0 0.4Cu 180. 0.1 0.3 223 .4 .04 4. 1.0 0.4Ta 180. 0.1 0.3 223 .4 .04 4. 1.0 0.4U 180. 0.1 0.3 223 .4 .04 4. 1.0 0.4The parameter of the vacuum hadronization weight can be bigger for thelight nuclei and smaller for the heavy nuclei, but 1. is a good guess. The s/uparameter is not yet tuned, as it demands the strange particles productiondata. A guess is that if a lot of uū and d ¯d pairs come in the reaction asin the p¯p interaction the parameter can be 0.1, in other cases it is closer to0.3 as in other event generators. But the best idea is just do not touch anyparameters for the first experience with the CHIPS event generator. Justchange the incident momentum, the PDG code of the projectile, and theCHIPS stile PDG code of the target.23.9 Status of this document01.01.01 created by M.V. Kossov and H.P. Wellisch26.04.03 first four sections re-written by D.H. WrightBibliography[1] B. Andersson, G. Gustafson, G. Ingelman, T. Sjöstrand, Phys. Rep. 97(1983) 31339

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[36] P.V. Degtyarenko, Applications of the photonuclear fragmentation modelto radiation protection problems, in: Proceedings of Second Specialist’sMeeting on Shielding Aspects of Accelerators, Targets and Irradiation Facilities(SATIF-2), CERN, Geneva, Switzerland, 12-13 October 1995, publishedby Nuclear Energy Agency, Organization for Economic Co-operationand Development, pages 67 - 91 (1996)[37] C. Bernard, A. Duncan, J. LoSecco, and S. Weinberg, Phys. Rev. D 12(1975) 792;E. Poggio, H. Quinn, and S. Weinberg, Phys. Rev. D 13 (1976) 1958[38] A. I. Amelin et al., “Energy spectra of charged particles in the reactionof π − absorption at rest by 6,7 Li, 9 Be, 10,11 B, 12 C, 28 Si, 40 Ca, 59 Co,93 Nb, 114,117,120,124 Sn, 169 Tm, 181 Ta and 209 Bi nuclei”, Moscow Physics andEngineering Institute Preprint No. 034-90, Moscow, 1990.[39] G. Mechtersheimer et al., Nucl. Phys. A324 (1979) 379.[40] C. Cernigoi et al., Nucl. Phys. A456 (1986) 599.[41] R. Madey et al., Phys. Rev. C 25 (1982) 3050.[42] D. Ryckbosch et al., Phys. Rev. C 42 (1990) 444.[43] J. Ahrens et al., Nucl. Phys. A446 (1985) 229c.[44] Jan Ryckebusch et al., Phys. Rev. C 49 (1994) 2704.[45] C. Van den Abeele; private communication cited in the reference:Jan Ryckebusch et al., Phys. Rev. C 49 (1994) 2704.[46] P.D. Harty et al. (unpublished); private communication cited in thereference: Jan Ryckebusch et al., Phys. Rev. C 49 (1994) 2704.[47] L.B. Weinstein et al., Phys. Rev. Lett. 64 (1990) 1646.[48] J.P. Jeukenne and C. Mahaux, Nucl. Phys. A 394 (1983) 445.342

Chapter 24Bertini Intranuclear CascadeModel in Geant424.1 IntroductionWe present here a intranuclear cascade model implemented in Geant4 5.0.The cascade model is based on re-engineering of INUCL code. Models includedare Bertini intra-nuclear cascade model with exitons, pre-equilibriummodel, nucleus explosion model, fission model, and evaporation model. Intermediateenergy nuclear reactions from 100 MeV to 5 GeV energy aretreated for proton, neutron, pions, photon and nuclear isotopes. We representoverview of the models, review results achived from simulations andmake comparisons with experimental data.The intranuclear cascade model (INC) was was first proposed by Serber in1947 [19]. His noticed that, in particle nuclear collisions the deBroglie wavelenghtof the incident particle is comparaple (or shorter) than the averageintra-nucleon distance. Hence, the justification for describing the interactionsin terms of particle-particle collisions.The INC has been succesfully used in the Monte Carlo simulations atintermediate energy region since Goldberger made first calculations by handin 1947 [9]. First computer simulations were done by Metropolis et al. in1958 [16]. Standard methods in INC implementations were formed whenBertini published his results in 1968 [3]. Important addition to INC wasexiton model itroduced by Griffin in 1966 [10].Our presentations describes implementation of Bertini INC model inGeant4 hadronic physics framework [8]. Geant4 is a Monte Carlo particledetector simulation toolkit, having applications also in medical and spacescience. Geant4 provides a flexible framework for the modular implemen-343

tation of various kinds of hadronic interactions. Geant4 exploits advancedSoftware Engineering techniques and Object Oriented technology to achievethe transparency of the physics implementation and to this way provide thepossibility of validating the physics results.The hadronic models framework is based on concepts of physics processesand models. While the process is a general concept, models are allowed tohave restrictions in process type, material, element and energy range. Severalmodels can be utilized by one model class; for instance, a process class forinelastic collisions can use distinct models for different energies.Process classes utilize model classes to determine the secondaries producedin the interaction and to calculate the momenta of the particles. Herewe preset a collection of such models providing medium-energy intranuclearcascade treatment.24.2 The Geant4 cascade modelIn inelastic particle nucleus collision a fast phase (10 −23 − 10 −22 s) of INCresults to highly exited nucleus, and is followed possible by fission and preequilibriumemission. A slower 10 −18 − 10 −16 s compound nucleus phase followswith evaportaion. A Boltzman equation must be solved to treat physicalproces of collision in detail.The intranuclear cascade (INC) model developed by Bertini [3, 4] solvesthe Boltzman equation on the average. The model had been implementedin several codes such as HETC [1]. Our model is based on re-engineering ofINUCL code [20]. Models included are Bertini intranuclear cascade modelwith exitons, pre-equilibrium model, simple nucleus explosion model, fissionmodel, and evaporation model.Nuclear model consist of a three-region approximation to the continuouslychanging density distribution of nuclear matter within nuclei. Reletivistickinematics is applied throughout the cascade. Cascade is stopped when allthe particles, which can escape the nucleus, do it. Then conformity with theenergy - conservation law is checked.24.2.1 Model limitsParticles treated are proton, neutron, pions, photon and nuclear isotopes.Bullet particle can be proton, neutron or pion. Range of targets allowed isarbitrary.The necessary condition of validity of the INC model is λ B /v

elative N-N velocity and ∆t is the time interval between collisions. So thephysical foundation comes approximate at energies less than 200MeV , andneeds to be supported with pre-quilibrium model. also at energies higherthan ≈ 10 GeV) the INC picture breaks down. Model has been tested againstexperimental with bullet kinetic energy between 100 MeV and 5 GeV.24.2.2 Intranuclear cascade modelBasic steps of the INC model are summarised below:1. The spatial point, where the incident particle enters, is selected uniformlyover the projected area of the nucleus.2. Total particle-particle crossections and region-depenent nucleon densitiesare used to select a path lenght for the projectile particle.3. The momentum of the struck nucleon, the type of reaction and fourmomentum of the reaction products are determined.4. Exiton model is updated as the cascade proceeds.5. If Pauli exclusion principle allows and E particle > E cutoff = 2 MeV, step(2) is performed to transport the products.After INC, the residual excitation energy of the resulting nucleus is usedas input for non-equilibrium model.24.2.3 Nuclei modelSome of the basic features of the nuclei are:• The nucleons are assumed to to have a Fermi gas momentum distribution.Fermi energy calculated in a local density approximationi.e. Fermi energy is made radius dependent with Fermi momentump F (r) = ( 3π2 ρ(r)) 1 3 .2• Nucleons binding energies (BE) are calculated using mass formula.A parametrization of the nuclear binding energy uses combination ofKummel mass formula, and experimental data. Also, asymptotic hightemperature mass formula is used if it’s impossible to use experimentaldata.345

InitializationThe initialization phase fixes of nucleus radius and momentum according toFermi gas model.If target is Hydrogen (A = 1) direct particle-particle collision is performed,and no nuclear modelling is used.If 1 < A < 4, a nuclei model consisting one layer with radius of 8.0 fm iscreated.For 4 < A < 11, nuclei model is composed of three consentric spheresi = {1, 2, 3} with radiusr i (α i ) =√C 2 1(1 − 1 A ) + 6.4 √−log(α i )Here α i = {0.01, 0.3, 0.7} and C 1 = 3.3836A 1/3If A > 11, nuclei model with three consentric spheres is also used. Theshere radius is now defined as:r i (α i ) = C 2 log( 1 + e− C 1C 2α i− 1) + C 1where C 2 = 1.7234.The potential energy V for nucleon N isV N =p2 F2m N+ BE N (A, Z)where p f is a Fermi momentum and BE a binding energy.Impulse distribution in each region follows Fermi distribution with zerotemperature.wheref(p) = cp 2 (24.1)∫ pF0f(p)dp = n p orn n (24.2)where n p and n n are the number of protons or neutrons in region. P f isimpulse corresponding the Fermi energyE f =p2 F2m N=¯h2 ( 3π22m N v ) 2 3 (24.3)which depend on the dencity n/v of particles. and which is different foreach particle and each region.346

Pauli exclusion principlePauli exclusion principle forbids interactions where the products would bein occupied states. Following an assumption of completely degenerate Fermigas, the levels are filled from the lowest level. The minimum energy allowedfor the product of collision correspond to the lowest unfilled level of system,which is the Fermi energy in the region. So in practice, Pauli exclusionprinsiple is taken into account by accepting only secondary nucleons whichhave E N > E f .Cross sections and kinematicsPath lengths of nucleons in the nucleus are sampled according to the localdensity and to free N-N cross sections. Angles after collisions are sampledfrom experimental differential cross sections. Tabulated total reactioncross-sections are calculated by Letaw’s formulation [14, 15, 17]. For N-Ncross-sections parametrizations based on the experimental energy and isospindependent data. The parameterization described in [2] is used.For pion the INC crossections are privded to treat elestic collisions andfollowing inelstics channels π − n → π 0 n, π 0 p → π + n and π 0 n → π − p. Multipleparticle production is also implemented.Pion absorption cannels are π + nn → pn, π + pn → pp, π 0 nn → X , π 0 pn→ pn, π 0 pp → pp, π − nn → X , π − pn → nn , and π − pp → pn.24.2.4 Pre-equilibrium modelGeant4 cascade model implements exiton model proposed by Griffin [10,11]. The his model nucleon states are characterized by the number of exitedparticles and holes (the exitons). INC collisions give rise to a sequence ofstates characterized by increasing exciton number, eventually leading to aequilibrated nucleus. For practical implementation of exiton model we useparameters from [18], (level densities) and [13] (matrix elements).In exiton model the posible selection rules for a particle-hole configuarationsin the cource of the cascade are: ∆p = 0, ±1 ∆h = 0, ±1 ∆n = 0, ±2,where p is the number of particle, h is number of holes and n = p + h is thenumber of exitons.Cascade pre-equilibrium model uses target exitation data, exiton configurationfor neutron and proton to produce non-equilibrium evaporation. Theangular distribution is isotropic in the frame of rest of the exiton system.Parametrisations of the level density, is tabulated both with A and Zdependence and with high temperature behaviour (the nuclei binding energy347

using smooth liquid high energy formula).24.2.5 Break-up modelsFermi break-up is allowed only in some extreme cases, i.e. for light nuclei(A < 12 and 3(A − Z) < Z < 6 ) and E exitation > 3E binding Simple explosionmodel decays the nuleus into neutrons and protons and decreases exoticevaporation processes.Fission model is phenomenological model using potential minimization.Binding energy paramerization is used and some features of the fission statisticalmodel are incorporated [7].24.2.6 Evaporation modelStatistical theory for particle emission of the exited nucleus remaining afterthe INC was originally developed by Weisskopf [21]. This model assumescomplete energy equilibration before the particle emission, and reequilibrationof excitation energies between successive evaporations. As aresult the angular distribution of emitted particles is isotropic.Geant4 evaporation model for cascade implementation adapts oftenused computational method developed by Dostrowski [5, 6]. The emissionof particles is computed until the exitation energy fall below some spesificcutoff. If light nucleus is higly exited Fermi break-up model is executed.Also, fission is performed if channel is open. The main chain of evaporationis followed until E exitation falls below E cutoff = 0.1 MeV. The evaporationmodel ends with emission chain which is followed until E exitation < E γ cutoff =10 −15 MeV.24.3 ImplementationCascade model is implemented in Geant4 hadronic physics framework.Source code directory is geant4/source/processes/hadronic/models/cascade/-cascade. All the models are used collectively through interface method Apply-Yourself defined in a class G4CascadeInteface. Geant4 track (G4Track) anda nucleus (G4Nucleus) are given as a parameters.We have tested the cascade models for its first release in Geant4 5.0,for energies 100 MeV – 5 GeV. Detailed comparisons with experimental datahas been made in energy range 160 – 800 MeV.348

24.4 Status of this document01.12.02 created by Aatos Heikkinen, Nikita Stepanov and Hans-Peter WellischBibliography[1] R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann, The high-energytransport code HETC88 and comparisons with experimental data, NuclearInstruments and Methods in Physics Research A 295, (1990), 337–343,[2] V.S. Barashenkov and V.D. Toneev, High Energy interactions of particlesand nuclei with nuclei (In russian), (1972)[3] M. P. Guthrie, R. G. Alsmiller and H. W. Bertini, Nucl. Instr. Meth,66, 1968, 29.[4] H. W. Bertini and P. Guthrie, Results from Medium-EnergyIntranuclear-Cascade Calculation, Nucl. Phys.A169, (1971).[5] I. Dostrovsky, Z. Zraenkel and G. Friedlander, Monte carlo calculationsof high-energy nuclear interactions. III. Application to low-lnergy calculations,Physical Review, 1959, 116, 3, 683-702.[6] I. Dostrovsky and Z. Fraenkel and P. Rabinowitz, Monte Carlo Calculationsof Nuclear Evaporation Processes. V. Emission of Particles HeavierThan 4 He, Physical Review, 1960.[7] P. Fong, Statistical Theory of Fission, 1969, Gordon and Breach, NewYork.[8] Geant4 collaboration, Geant4 general paper (to be published), NuclearInstruments and Methods A, (2003).[9] M. Goldberger, The Interaction of High Energy Neutrons and HevyNuclei, Phys. Rev. 74, (1948), 1269.[10] J. J. Griffin, Statistical Model of Intermediate Structure, Physical ReviewLetters 17, (1966), 478-481.[11] J. J. Griffin, Statistical Model of Intermediate Structure, Physics Letters24B, 1 (1967), 5-7.349

[12] A. S. Iljonov et al., Intermediate-Energy Nuclear Physics, CRC Press1994.[13] C. Kalbach, Exiton Number Dependence of the Griffin Model Two-BodyMatrix Element, Z. Physik A 287, (1978), 319-322.[14] J. R. Letaw et al., The Astrophysical Journal Supplements 51, (1983),271f.[15] J. R. Letaw et al., The Astrophysical Journal 414, 1993, 601.[16] N. Metropolis, R. Bibins, M. Storm, Monte Carlo Calculations onIntranuclear Cascades. I. Low-Energy Studies, Physical Review 110,(1958), 185ff.[17] S. Pearlstein, Medium-energy nuclear data libraries: a case study,neutron- and proton-induced reactions in 5 6Fe, The Astrophysical Journal346, (1989), 1049-1060.[18] I. Ribansky et al., Pre-equilibrium decay and the exiton model, Nucl.Phys. A 205, (1973), 545-560.[19] R. Serber, Nuclear Reactions at High Energies, Phys. Rev. 72, (1947),1114.[20] Experimental and Computer Simulations Study of Radionuclide Productionin Heavy Materials Irradiated by Intermediate Energy Protons,Yu. E. Titarenko et al., nucl-ex/9908012, (1999).[21] V. Weisskopf, Statistics and Nuclear Reactions, Physical Review 52,(1937), 295–302.350

Chapter 25The Geant4 Binary Cascade25.1 Modeling overviewThe Geant4 Binary Cascade is an intranuclear cascade propagating primaryand secondary particles in a nucleus. Interactions are between a primary orsecondary particle and an individual nucleon of the nucleus, leading to thename Binary Cascade. Cross section data are used to select collisions. Whereavailable, experimental cross sections are used by the simulation. Propagatingof particles is the nuclear field is done by numerically solving the equationof motion. The cascade terminates when the average and maximumenergy of secondaries is below threshold. The remaining fragment is treatedby precompound and de-excitation models documented in the correspondingchapters.25.1.1 The transport algorithmFor the primary particle an impact parameter is chosen random in a diskoutside the nucleus perpendicular to a vector passing through the center ofthe nucleus coordinate system an being parallel to the momentum direction.Using a straight line trajectory, the distance of closest approach d mini to eachtarget nucleon i and the corresponding time-of-flight t d i is calculated. Inthis calculation the momentum of the target nucleons is ignored, i.e. thetarget nucleons do not move. The interaction cross section σ i with targetnucleons is calculated using total inclusive cross-sections described below.For calculation of the cross-section the momenta of the nucleons are takeninto account. The primary particle may interact with those target nucleonswhere the distance of closest approach d mini is smaller than d mini < √ σ i. πThese candidate interactions are called collisions, and these collisions are351

stored ordered by time-of-flight t d i . In the case no collision is found, a newimpact parameter is chosen.The primary particle is tracked the time-step given by the time to thefirst collision. As long a particle is outside the nucleus, that is a radius of theoutermost nucleon plus 3fm, particles travel along straight line trajectories.Particles entering the nucleus have their energy corrected for Coulomb effects.Inside the nucleus particles are propagated in the scalar nuclear field. Theequation of motion in the field is solved for a given time-step using a Runge-Kutta integration method.At the end of the step, the primary and the nucleon interact suing thescattering term. The resulting secondaries are checked for the Fermi exclusionprinciple. If any of the two particles has a momentum below Fermi momentum,the interaction is suppressed, and the original primary is tracked tothe next collision. In case interaction is allowed, the secondaries are treatedlike the primary, that is, all possible collisions are calculated like above, withthe addition that these new primary particles may be short-lived and maydecay. A decay is treated like others collisions, the collision time being thetime until the decay of the particle. All secondaries are tracked until theyleave the nucleus, or the until the cascade stops.25.1.2 The description of the target nucleus and fermimotionThe nucleus is constructed from A nucleons and Z protons with nucleoncoordinates r i and momenta p i , with i = 1, 2, ..., A. We use a commoninitialization Monte Carlo procedure, which is realized in the most of thehigh energy nuclear interaction models:• Nucleon radii r i are selected randomly in the nucleus rest frame accordingto nucleon density ρ(r i ). For heavy nuclei with A > 16 [2] nucleondensity isρ 0ρ(r i ) =(25.1)1 + exp [(r i − R)/a]whereρ 0 ≈ 34πR 3 (1 + a2 π 2R 2 )−1 . (25.2)Here R = r 0 A 1/3 fm and r 0 = 1.16(1 − 1.16A −2/3 ) fm and a ≈ 0.545fm. For light nuclei with A < 17 nucleon density is given by a harmonicoscillator shell model [3], e. g.ρ(r i ) = (πR 2 ) −3/2 exp (−r 2 i /R2 ), (25.3)352

where R 2 = 2/3 < r 2 >= 0.8133A 2/3 fm 2 . To take into accountnucleon repulsive core it is assumed that internucleon distance d > 0.8fm;• The nucleus is assumed to be isotropic, i.e. we place each nucleon usinga random direction and the previously determined radius r i .• The initial momenta of the nucleons p i are randomly choosen between0 and p maxF (r), where the maximal momenta of nucleons (in the localThomas-Fermi approximation [4]) depends from the proton or neutrondensity ρ according top maxF (r) = ¯hc(3π 2 ρ(r)) 1/3 (25.4)• To obtain momentum components, it is assumed that nucleons aredistributed isotropic in momentum space; i.e. the momentum directionis chosen at random.• The nucleus must be centered in momentum space around 0, i. e.∑the nucleus must be at rest, i. e. i p i = 0; To achieve this, wechoose one nucleon to compensate the sum the remaining nucleon momentap r est = ∑ i=A−1i=1 . If this sum is larger than maximum momentump maxF (r), we change the direction of the momentum of a few nucleons.If this does not lead to a possible momentum value, than we repeat theprocedure with a different nucleon having a larger maximum momentump maxF (r). In the rare case this fails as well, we choose new momentafor all nucleons.This procedure gives special for hydrogen 1 H, where the proton hasmomentum p = 0, and for deuterium 2 H, where the momenta of protonand neutron are equal, and in opposite direction.• We compute energy per nucleon e = E/A = m N + B(A, Z)/A, wherem N is nucleon mass and the nucleus binding energy B(A, Z) is givenby the tabulation of [5]: and find the effective mass of each nucleonm effi =√(E/A) 2 − p 2′i .25.1.3 Optical and phenomenological potentialsThe effect of collective nuclear elastic interaction upon primary and secondaryparticles is approximated by a nuclear potential.353

For projectile protons and neutrons this scalar potential is given by thelocal Fermi momentum p F (r)V (r) = p2 F (r)2m(25.5)where m is the mass of the neutron m n or the mass of proton m p .For pions the potential is given by the lowest order optical potential [6]V (r) = −2π(¯hc)2 A(1 + m πm π M )b 0ρ(r) (25.6)where A is the nuclear mass number, m π , M are the pion and nucleon mass,m π is the reduced pion mass m π = (m π m N )/(m π +m N ), with m N is the massof the nucleus, and ρ(r) is the nucleon density distribution. The parameterb 0 is the effective s−wave scattering length and is obtained from analysis topion atomic data to be about −0.042fm.25.1.4 Pauli blocking simulationThe cross sections used in this model are cross sections for free particles.In the nucleus these cross sections are reduced to effective cross sections byPauli-blocking due to Fermi statistics.For nucleons created by a collision, ie. an inelastic scattering or fromdecay, we check that all secondary nucleons occupy a state allowed by Fermistatistics. We assume that the nucleus in its ground state and all statesbelow Fermi energy are occupied. All secondary nucleons therefore musthave a momentum p i above local Fermi momentum p F (r), i.e.p i > p maxF (r). (25.7)If any of the nucleons of the collision has a momentum below the localFermi momentum, then the collision is Pauli blocked. The reaction productsare discarded, and the original particles continue the cascade.25.1.5 The scattering termThe basis of the description of the reactive part of the scattering amplitudeare two particle binary collisions (hence binary cascade), resonance production,and decay. Based on the cross-section described later in this paper,collisions will occur when the transverse distance d t of any projectile targetpair becomes smaller than the black disk radium corresponding to the totalcross-section σ tσ tπ > d2 t354

In case of a collision, all particles will be propagated to the estimated timeof the collision, i.e. the time of closest approach, and the collision final stateis produced.25.1.6 Total inclusive cross-sectionsExperimental data are used in the calculation of the total, inelastic andelastic cross-section wherever available.hadron-nucleon scatteringFor the case of proton-proton(pp) and proton-neutron(pn) collisions, as wellas π = and π − nucleon collisions, experimental data are readily available ascollected by the Particle Data Group (PDG) for both elastic and inelasticcollisions. We use a tabulation based on a sub-set of these data for √ Sbelow 3 GeV. For higher energies, parametrizations from the CERN-HERAcollection are included.25.1.7 Channel cross-sectionsA large fraction of the cross-section in individual channels involving mesonnucleon scattering can be modeled as resonance excitation in the s-channel.This kind of interactions show a resonance structure in the energy dependencyof the cross-section, and can be modeled using the Breit-Wigner functionσ res ( √ s) = ∑ F S2J + 1 π Γ I SΓ F S(2S 1 + 1)(2S 2 + 1) k 2 ( √ s − M R ) 2 + Γ/4 ,Where S1 and √ S2 are the spins of the two fusing particles, J is the spin ofthe resonance, (s) the energy in the center of mass system, k the momentumof the fusing particles in the center of mass system, Γ I S and Γ)F S the partialwidth of the resonance for the initial and final state respectively. M R is thenominal mass of the resonance.The initial states included in the model are pion and kaon nucleon scattering.The product resonances taken into account are the Delta resonances withmasses 1232, 1600, 1620, 1700, 1900, 1905, 1910, 1920, 1930, and 1950 MeV,the excited nucleons with masses of 1440, 1520, 1535, 1650, 1675, 1680, 1700,1710, 1720, 1900, 1990, 2090, 2190, 2220, and 2250 MeV, the Lambda, andits excited states at 1520, 1600, 1670, 1690, 1800, 1810, 1820, 1830, 1890,2100, and 2110 MeV, and the Sigma and its excited states at 1660, 1670,1750, 1775, 1915, 1940, and 2030 MeV.355

25.1.8 Mass dependent resonance width and partialwidthDuring the cascading, the resonances produced are assigned reall masses,with values distributed according to the production cross-section describedabove. The concrete (rather than nominal) masses of these resonances maybe small compared to the PDG value, and this implies that some channelsmay not be open for decay. In general it means, that the partial and totalwidth will depend on the concrete mass of the resonance. We are using theUrQMD[13][14] approach for calculating these actual width,Γ R→12 (M) = (1 + r) Γ R→12(M R ) M R p(M) (2l+1). (25.8)p(M R ) (2l+1) M 1 + r(p(M)/p(M R ))2lHere M R is the nominal mass of the resonance, M the actual mass, p isthe momentum in the center of mass system of the particles, L the angularmomentum of the final state, and r=0.2.25.1.9 Resonance production cross-section in the t-channelIn resonance production in the t-channel, single and double resonance excitationin nucleon-nucleon collisions are taken into account. The resonanceproduction cross-sections are as much as possible based on parametrizationsof experimental data[15] for proton proton scattering. The basic formulaused is motivated from the form of the exclusive production cross-section ofthe ∆ 1232 in proton proton collisions:√ √ (√ ) γABs − s0 s0 +σ AB = 2α AB β AB( √ s − √ βAB√s 0 ) 2 + βAB2 sThe parameters of the description for the various channels are given intable25.1. For all other channels, the parametrizations were derived fromthese by adjusting the threshold behavior.The reminder of the cross-section are derived from these, applying detailedbalance. Iso-spin invariance is assumed. The formalism used to applydetailed balance isσ(cd → ab) = ∑ 〈j c m c j d m d ‖ JM〉 2 (2S a + 1)(2S b + 1) 〈p 2 ab 〉J,M 〈j a m a j b m b ‖ JM〉 2 → cd)(2S c + 1)(2S d + 1) 〈p 2 cd 〉σ(ab(25.9)356

Reaction α β γpp → p∆ 1232 25 mbarn 0.4 GeV 3pp → ∆ 1232 ∆ 1232 1.5 mbarn 1 GeV 1pp → pp ∗ 0.55 mbarn 1 GeV 1pp → p∆ ∗ 0.4 mbarn 1 GeV 1pp → ∆ 1232 ∆ ∗ 0.35 mbarn 1 GeV 1pp → ∆ 1232 N ∗ 0.55 mbarn 1 GeV 1Table 25.1: Values of the parameters of the cross-section formula for theindividual channels.25.1.10 Nucleon Nucleon elastic collisionsAngular distributions for elastic scattering of nucleons are taken as closely aspossible from experimental data, i.e. from the result of phase-shift analysis.They are derived from differential cross sections obtained from the SAIDdatabase, R. Arndt, 1998.Final states are derived by sampling from tables of the cumulative distributionfunction of the centre-of-mass scattering angle, tabulated for a discreteset of lab kinetic energies from 10 MeV to 1200 MeV. The CDF’s are tabulatedat 1 degree intervals and sampling is done using bi-linear interpolationin energy and CDF values. Coulomb effects are taken into consideration forpp scattering.25.1.11 Generation of transverse momentumAngular distributions for final states other than nucleon elastic scatteringare calculated analytically, derived from the collision term of the in-mediumrelativistic Boltzmann-Uehling-Uhlenbeck equation, absed on the nucleon nucleonelastic scattering cross-sections:σ NN→NN (s, t) = 1 (D(s, t) + E(s, t) + (invertedt, u))(2π) 2 sHere s, t, u are the Mandelstamm variables, D(s, t) is the direct term,and E(s, t) is the exchange term, withD(s, t) = (gσ NN )4 (t−4m ∗ 2) 22(t−m 2 σ )2 + (gω NN )4 (2s 2 +2st+t 2 −8m ∗2 s+8m ∗4 )(t−m 2 ω )2 +24(g π NN )4 m ∗2 t 2(t−m 2 π) 2− 4(gσ NN gω NN )2 (2s+t−4m ∗2 )m ∗2(t−m 2 σ)(t−m 2 ω),357

andE(s, t) =(gNN σ )4 (t(t+s)+4m ∗2 (s−t))+ (gω 8(t−m 2 σ )(u−m2 σ ) NN )4 (s−2m ∗2 )(s−6m ∗2 ))−2(t−m 2 ω )(u−m2 ω )6(g π NN )4 (4m ∗2 −s−t)m ∗4 t(t−m 2 π )(u=mpi2 )3(g σ NN gπ NN )2 t(t+s)m ∗22(t−m 2 π )(u−m2 σ )+ 3(gσ NN gπ NN )2 m ∗2 (4m ∗2 −s−t)(4m ∗2 −t)(t−m 2 σ )(u−m2 π ) ++ (gσ NN gω NN )2 t 2 −4m ∗2 s−10m ∗2 t+24m ∗44(t−m 2 σ )(u−m2 ω ) +(g σ NN gω NN )2 (t+s) 2 −2m ∗2 s+2m ∗2 t4(t−m 2 ω)(u−m 2 σ)+ 3(gω NN gπ NN )2 (t+s−4m ∗2 )(t+s−2m ∗2 )(t−m 2 ω)(u−m 2 π)+3(g ω NN gπ NN )2 m ∗2 (t 2 −2m ∗2 t)(t−m 2 π )(u−m2 ω ) . (25.10)(25.11)Here, in this first release, the in-medium mass was set to the free mass, andthe nucleon nucleon coupling constants used were 1.434 for the π, 7.54 for theω, and 6.9 for the σ. This formula was used for elementary hadron-nucleondifferential cross-sections by scaling teh center of mass energy squared accordingly.Finite size effects were taken into account at the meson nucleon vertex,using a phenomenological form factor (cut-off) at each vertex.25.1.12 DecayIn the simulation of decay of strong resonances, we use the nominal decaybranching ratios from the particle data book. The stochastic mass of aindividual resonance created is sampled at creation time from the Breit-Wigner form, under the mass constraints posed by center of mass energy ofthe scattering, and the mass in the lightest decay channel. The decay widthfrom the particle data book are then adjusted according to equation 25.8, totake the stochastic mass value into account.All decay channels with nominal branching ratios greater than 1% aresimulated.25.1.13 The escaping particle and coherent effectsWhen a nucleon other than the incident particle leaves the nucleus, theground state of the nucleus changes. The energy of the outgoing particlecannot be such that the total mass of the new nucleus would be below itsground state mass. To avoid this, we reduce the energy of an outgoing nucleonsby the mass-difference of old and new nucleus.Furthermore, the momentum of the final exited nucleus derived fromenergy momentum balance may be such that its mass is below its ground358

state mass. In this case, we arbitrarily scale the momenta of all outgoingparticles by a factor derived from the mass of the nucleus and the mass ofthe system of outgoing particles.25.1.14 Light ion reactionsIn simulating light ion reactions, the initial state of the cascade is preparedin the form of two nuclei, as described in the above section on the nuclearmodel.The lighter of the collision partners is selected to be the projectile. Thenucleons in the projectile are then entered, with position and momenta, intothe initial state of the cascade. Note that before the first scattering of anindividual nucleon, a projectile nucleon’s Fermi-momentum is not taken intoaccount in the tracking inside the target nucleus. The nucleon distributioninside the projectile nucleus is taken to be a representative distribution ofits nucleons in configuration space, rather than an initial state in the senseof QMD. The Fermi momentum and the local field are taken into accountin the calculation of the collision probabilities and final states of the binarycollisions.25.1.15 Transition to pre-compound modelingEventually, the cascade assumptions will break down at low energies, and thestate of affairs has to be treated by means of evaporation and pre-equilibriumdecay. This transition is not at present studied in depth, and an interestingapproach which uses the tracking time, as in the Liege cascade code, remainsto be studied in our context.For this first release, the following algorithm is used to determine whencascading is stopped, and pre-equilibrium decay is called: As long as thereare still particles above the kinetic energy threshold (75 MeV), cascading willcontinue. Otherwise, when the mean kinetic energy of the participants hasdropped below a second threshold (15 MeV), the cascading is stopped.The residual participants, and the nucleus in its current state are thenused to define the initial state, i.e. excitation energy, number of excitons,number of holes, and momentum of the exciton system, for pre-equilibriumdecay.In the case of light ion reactions, the projectile excitation is determinedfrom the binary collision participants (P ) using the statistical approach towardsexcitation energy calculation in an adiabatic abrasion process, as de-359

scribed in [12]:E ex = ∑ P(E P fermi − E P )Given this excitation energy, the projectile fragment is then treated bythe evaporation models described previously.25.1.16 Calculation of excitation energies and residualsAt the end of the cascade, we form a fragment for further treatment inprecompound and nuclear de-excitation models ([16]).These models need information about the nuclear fragment created bythe cascade. The fragment is characterized by the number of nucleons in thefragment, the charge of the fragment, the number of holes, the number of allexcitons, and the number of charged excitons, and the four momentum ofthe fragment.The number of holes is given by the difference of the number of nucleonsin the original nucleus and the number of non-excited nucleons left in thefragment. An exciton is a nucleon captured in the fragment at the end of thecascade.The momentum of the fragment calculated by the difference between themomentum of the primary and the outgoing secondary particles must besplit in two components. The first is the momentum acquired by coherentelastic effects, and the second is the momentum of the excitons in the nucleusrest frame. Only the later part is passed to the de-excitation models.Secondaries arising from de-excitation models, including the final nucleus,are transformed back the frame of the moving fragment.25.2 Comparison with experimentsWe add here a set of preliminary results produced with this code, focusing onneutron and pion production. Given that we are still in the process of writingup the paper, we apologize for the at release time still less then publicationquality plots.Bibliography[1] J. Cugnon, C. Volant, S. Vuillier DAPNIA-SPHN-97-01, Dec 1996. 62pp.Submitted to Nucl.Phys.A,360

cross section (mb/sr.MeV)10110 -1113 MeV p + Al - 7.5 deg- 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.1: Double differential cross-section for neutrons produced in protonscattering off Aluminum. Proton incident energy was 113 MeV.361

cross section (mb/sr.MeV)10110 -1256 MeV p + Al - 7.5 deg- 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.2: Double differential cross-section for neutrons produced in protonscattering off Aluminum. Proton incident energy was 256 MeV. Thepoints are data, the histogram is Binary Cascade prediction.362

cross section (mb/sr.MeV)10110 -1597 MeV p + Al - 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.3: Double differential cross-section for neutrons produced in protonscattering off Aluminum. Proton incident energy was 597 MeV. Thepoints are data, the histogram is Binary Cascade prediction.363

cross section (mb/sr.MeV)10110 -1800 MeV p + Al - 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.4: Double differential cross-section for neutrons produced in protonscattering off Aluminum. Proton incident energy was 800 MeV. Thepoints are data, the histogram is Binary Cascade prediction.364

cross section (mb/sr.MeV)10110 -1113 MeV p + Fe - 7.5 deg- 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.5: Double differential cross-section for neutrons produced in protonscattering off Iron. Proton incident energy was 113 MeV. The points aredata, the histogram is Binary Cascade prediction.365

cross section (mb/sr.MeV)10110 -1256 MeV p + Fe - 7.5 deg- 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.6: Double differential cross-section for neutrons produced in protonscattering off Iron. Proton incident energy was 256 MeV. The points aredata, the histogram is Binary Cascade prediction.366

cross section (mb/sr.MeV)10110 -1597 MeV p + Fe - 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.7: Double differential cross-section for neutrons produced in protonscattering off Iron. Proton incident energy was 597 MeV. The points aredata, the histogram is Binary Cascade prediction.367

cross section (mb/sr.MeV)10110 -1800 MeV p + Fe - 30 deg- 60 deg- 150 deg10 -21 10 10 2Ekin (MeV)Figure 25.8: Double differential cross-section for neutrons produced in protonscattering off Iron. Proton incident energy was 800 MeV. The points aredata, the histogram is Binary Cascade prediction.368

10 3 1 10 10 2cross section (mb/sr.MeV)10 210110 -1113 MeV p + Pb - 7.5 deg- 30 deg- 60 deg- 150 deg10 -2Ekin (MeV)Figure 25.9: Double differential cross-section for neutrons produced in protonscattering off Lead. Proton incident energy was 113 MeV. The pointsare data, the histogram is Binary Cascade prediction.369

10 3 1 10 10 2cross section (mb/sr.MeV)10 210110 -1256 MeV p + Pb - 7.5 deg- 30 deg- 60 deg- 150 deg10 -2Ekin (MeV)Figure 25.10: Double differential cross-section for neutrons produced inproton scattering off Lead. Proton incident energy was 256 MeV. The pointsare data, the histogram is Binary Cascade prediction.370

10 3 1 10 10 2cross section (mb/sr.MeV)10 210110 -1597 MeV p + Pb - 30 deg- 60 deg- 150 deg10 -2Ekin (MeV)Figure 25.11: Double differential cross-section for neutrons produced inproton scattering off Lead. Proton incident energy was 597 MeV. The pointsare data, the histogram is Binary Cascade prediction.371

10 3 1 10 10 2cross section (mb/sr.MeV)10 210110 -1800 MeV p + Pb - 30 deg- 60 deg- 150 deg10 -2Ekin (MeV)Figure 25.12: Double differential cross-section for neutrons produced inproton scattering off Lead. Proton incident energy was 800 MeV. The pointsare data, the histogram is Binary Cascade prediction.372

cross section (mb/sr.MeV)cross section (mb/sr.MeV)110 -110 -210 -310 -4110 -110 -210 -310 -40 100 200 300Ekin (MeV)0 100 200 300Ekin (MeV)cross section (mb/sr.MeV)cross section (mb/sr.MeV)110 -110 -210 -310 -4110 -110 -210 -310 -40 100 200 300Ekin (MeV)p + C 45 degreespi+pi-p + Al 45 degreespi+pi-p + Ni 45 degreespi+pi-p + Pb 45 degreespi+pi-0 100 200 300Ekin (MeV)Figure 25.13: Double differential cross-section for pions produced at 45 ◦ inproton scattering off various materials. Proton incident energy was 597 MeVin each case. The points are data, the histogram is Binary Cascade prediction.373

G. Peter, D. Behrens, C.C. Noack Phys.Rev. C49, 3253, (1994),Hai-Qiao Wang, Xu Cai, Yong Liu High Energy Phys.Nucl.Phys. 16, 101,(1992),A.S. Ilinov, A.B. Botvina, E.S. Golubeva, I.A. Pshenichnov, Sov. J. Nucl.Phys. 55, 734, (1992),and citations therein.[2] Grypeos M. E., Lalazissis G. A., Massen S. E., Panos C. P., J. Phys. G171093 (1991).[3] Elton L. R. B., Nuclear Sizes, Oxford University Press, Oxford, 1961.[4] DeShalit A., Feshbach H., Theoretical Nuclear Physics, Vol. 1: NuclearStructure, Wyley, 1974.[5] reference to be completed[6] K. Stricker, H. McManus, J. A. Carr Nuclear scattering of low energypions, Phys. Rev. C 19, 929, (1979)[7] M. M. Meier et al., Differential neutron production cross sections andneutron yields from stopping-length targets for 113-MeV protons, Nucl.Scien. Engin. 102, 310, (1989)[8] M. M. Meier et al., Differential neutron production cross sections for256-MeV protons, Nucl. Scien. Engin. 110, 289, (1992)[9] W. B. Amian et al., Differential neutron production cross sections for597-MeV protons, Nucl. Scien. Engin. 115, 1, (1993)[10] W. B. Amian et al., Differential neutron production cross sections for800-MeV protons, Nucl. Scien. Engin. 112, 78, (1992)[11] J. F. Crawford et al., Measurement of cross sections and asymmetryparameters for the production of charged pions from various nuclei by585-MeV protons, Phys. Rev. C 22, 1184, (1980)[12] J. J. Gaimard and K. H. Schmidt, “A Reexamination of the abrasion -ablation model for the description of the nuclear fragmentation reaction,”Nucl. Phys. A 531 (1991) 709.[13] reference to be completed.[14] reference to be completed374

[15] reference to be completed[16] reference to be completed375

Chapter 26Abrasion-ablation Model26.1 IntroductionThe abrasion model is a simplified macroscopic model for nuclear-nuclearinteractions based largely on geometric arguments rather than detailed considerationof nucleon-nucleon collisions. As such the speed of the simulationis found to be faster than models such as G4BinaryCascade, but at the costof accuracy. The version of the model implemented is interpreted from theso-called abrasion-ablation model described by Wilson et al [1],[2] togetherwith an algorithm from Cucinotta to approximate the secondary nucleon energyspectrum [3]. By default, instead of performing an ablation process tosimulate the de-excitation of the nuclear pre-fragments, the Geant4 implementationof the abrasion model makes use of existing and more detailed nuclearde-excitation models within Geant4 (G4Evaporation, G4FermiBreakup,G4StatMF) to perform this function (see section 26.5). However, in somecases cross sections for the production of fragments with large ∆A from thepre-abrasion nucleus are more accurately determined using a Geant4 implementationof the ablation model (see section 26.6).The abrasion interaction is the initial fast process in which the overlap regionbetween the projectile and target nuclei is sheered-off (see figure 26.1) Thespectator nucleons in the projectile are assumed to undergo little change inmomentum, and likewise for the spectators in the target nucleus. Some ofthe nucleons in the overlap region do suffer a change in momentum, andare assumed to be part of the original nucleus which then undergoes deexcitation.Less central impacts give rise to an overlap region in which the nucleons cansuffer significant momentum change, and zones in the projectile and targetoutside of the overlap where the nucleons are considered as spectators to the376

initial energetic interaction.The initial description of the interaction must, however, take into considerationchanges in the direction of the projectile and target nuclei due toCoulomb effects, which can then modify the distance of closest approachcompared with the initial impact parameter. Such effects can be importantfor low-energy collisions.26.2 Initial nuclear dynamics and impact parameterFor low-energy collisions, we must consider the deflection of the nuclei as aresult of the Coulomb force (see figure 26.2). Since the dynamics are nonrelativistic,the motion is governed by the conservation of energy equation:E tot = 1 2 µṙ2 +l22µr + Z P Z T e 22 r(26.1)where:E tot = total energy in the centre of mass frame;r,ṙ = distance between nuclei, and rate of change of distance;l = angular momentum;µ = reduced mass of system i.e. m 1 m 2 /(m 1 + m 2 );e = electric charge (units dependent upon the units for E tot and r);Z P , Z T = charge numbers for the projectile and target nuclei.The angular momentum is based on the impact parameter between the nucleiwhen their separation is large, i.e.l 2E tot = 1 ⇒ l 2 = 2E2 µb 2 tot µb 2 (26.2)At the point of closest approach, ṙ=0, therefore:E tot = Etotb2 + Z r 2 P Z T e 2rr 2 = b 2 + Z P Z T e 2E totrRearranging this equation results in the expression:(26.3)where:b 2 = r(r − r m ) (26.4)r m = Z P Z T e 2E tot(26.5)377

In the implementation of the abrasion process in Geant4, the square of thefar-field impact parameter, b, is sampled uniformly subject to the distance ofclosest approach, r, being no greater than r P + r T (the sum of the projectileand target nuclear radii).26.3 Abrasion processIn the abrasion process, as implemented by Wilson et al [1] it is assumedthat the nuclear density for the projectile is constant up to the radius of theprojectile (r P ) and zero outside. This is also assumed to be the case for thetarget nucleus. The amount of nuclear material abraded from the projectileis given by the expression:(∆ abr = F A P[1 − exp − C )]T(26.6)λwhere F is the fraction of the projectile in the interaction zone, λ is thenuclear mean-free-path, assumed to be:λ = 16.6E 0.26 (26.7)E is the energy of the projectile in MeV/nucleon and C T is the chord-lengthat the position in the target nucleus for which the interaction probability ismaximum. For cases where the radius of the target nucleus is greater thanthat of the projectile (i.e. r T > r P ):where:⎧ √⎨ 2 rT 2 C T =− x2 : x > 0√⎩ 2 rT 2 − r 2 : x ≤ 0(26.8)x = r2 P + r 2 − rT22rIn the event that r P > r T then C T is:where:C T ={ √2 rT 2 − x2 : x > 02r T : x ≤ 0x = r2 T + r2 − rP22rThe projectile and target nuclear radii are given by the expression:(26.9)(26.10)(26.11)378

P ≈ 1.29 √ r 2 RMS,P − 0.842r T ≈ 1.29 √ r 2 RMS,T − 0.84 2 (26.12)The excitation energy of the nuclear fragment formed by the spectators inthe projectile is assumed to be determined by the excess surface area, givenby:[∆S = 4πrP2 1 + P − (1 − F ) 2 ]/3(26.13)where the functions P and F are given in section 26.7. Wilson et al equatethis surface area to the excitation to:E S = 0.95∆S (26.14)if the collision is peripheral and there is no significant distortion of the nucleus,orE S = 0.95 {1 + 5F + ΩF 3 } ∆S⎧⎪⎨ 0 : A P > 16Ω =1500 : A P < 12⎪⎩1500 − 320 (A P − 12) : 12 ≤ A P ≤ 16(26.15)if the impact separation is such that r 1.5fm13 · C l : C t ≤ 1.5fm(26.16)379

where:{ √2 rC l =P 2 + 2rr T − r 2 − rT2 r > r T2r P r ≤ r T(26.17)C t = 2√rP 2 − (r2 P + r 2 − rT 2 ) 24r 2 (26.18)For the remaining events, the projectile energy is assumed to be unchanged.Wilson et al assume that the energy required to remove a nucleon is 10MeV,therefore the number of nucleons removed from the projectile by ablation is:∆ abl = E S + E X+ ∆ spc (26.19)10where ∆ spc is the number of loosely-bound spectators in the interaction region,given by:(∆ spc = A P F exp − C )T(26.20)λWilson et al appear to assume that for half of the events the excitationenergy is transferred into one of the nuclei (projectile or target), otherwisethe energy is transferred in to the other (target or projectile respectively).The abrasion process is assumed to occur without preference for the nucleontype, i.e. the probability of a proton being abraded from the projectile isproportional to the fraction of protons in the original projectile, therefore:∆Z abr = ∆ abrZ PA P(26.21)In order to calculate the charge distribution of the final fragment, Wilson et alassume that the products of the interaction lie near to nuclear stability andtherefore can be sampled according to the Rudstam equation (see section26.6). The other obvious condition is that the total charge must remainunchanged.26.4 Abraded nucleon spectrumCucinotta has examined different formulae to represent the secondary protonsspectrum from heavy ion collisions [3]. One of the models (which has beenimplemented to define the final state of the abrasion process) represents themomentum distribution of the secondaries as:380

( )3∑ψ(p) ∝ C i exp − p2 γp+ di=12p 2 0 (26.22)i sinh (γp)where:ψ(p) = number of secondary protons with momentum p per unit ofmomentum phase space [c 3 /MeV 3 ];p = magnitude of the proton momentum in the rest frame of thenucleus from which the particle is projected [MeV/c];C1, C2, C3 = 1.0, 0.03, and 0.0002;p1, p2, p3 = √ 2p 5 F , √ 6p 5 F , 500 [MeV/c]p F = Momentum of nucleons in the nuclei at the Fermi surface [MeV/c]d 0 = 0.11= 90 [MeV/c];γG4WilsonAbrasionModel approximates the momentum distribution for theneutrons to that of the protons, and as mentioned above, the nucleon typesampled is proportional to the fraction of protons or neutrons in the originalnucleus.The angular distribution of the abraded nucleons is assumed to be isotropicin the frame of reference of the nucleus, and therefore those particles from theprojectile are Lorentz-boosted according to the initial projectile momentum.26.5 De-excitation of the projectile and targetnuclear pre-fragments by standardGeant4 de-excitation physicsUnless specified otherwise, G4WilsonAbrasionModel will instantiate the followingde-excitation models to treat subsequent particle emission of the excitednuclear pre-fragments (from both the projectile and the target):1 G4Evaporation, which will perform nuclear evaporation of (α-particles,3 He, 3 H, 2 H, protons and neutrons, in competition with photo-evaporationand nuclear fission (if the nucleus has sufficiently high A).2 G4FermiBreakUp, for nuclei with A ≤ 12 and Z ≤ 6.3 G4StatMF, for multi-fragmentation of the nucleus (minimum energy forthis process set to 5 MeV).As an alternative to using this de-excitation scheme, the user may provideto the G4WilsonAbrasionModel a pointer to her own de-excitation handler,or invoke instantiation of the ablation model (G4WilsonAblationModel).381

26.6 De-excitation of the projectile and targetnuclear pre-fragments by nuclear ablationA nuclear ablation model, based largely on the description provided by Wilsonet al [1], has been developed to provide a better approximation for thefinal nuclear fragment from an abrasion interaction. The algorithm implementedin G4WilsonAblationModel uses the same approach for selecting thefinal-state nucleus as NUCFRG2 and determining the particles evaporatedfrom the pre-fragment in order to achieve that state. However, use is alsomade of classes in Geant4’s evaporation physics to determine the energies ofthe nuclear fragments produced.The number of nucleons ablated from the nuclear pre-fragment (whetheras nucleons or light nuclear fragments) is determined based on the averagebinding energy, assumed by Wilson et al to be 10 MeV, i.e.:A abl ={ ( )IntEx10MeV : AP F > Int ( )E x10MeVA P F : otherwise(26.23)Obviously, the nucleon number of the final fragment, A F , is then determinedby the number of remaining nucleons. The proton number of the final nuclearfragment (Z F ) is sampled stochastically using the Rudstam equation:⎛⎞3∣σ(A F , Z F ) ∝ exp ⎝−R ∣Z F − SA F − T A 2 /2F ∣ ⎠ (26.24)Here R=11.8/AF 0.45 , S=0.486, and T =3.8·10 −4 . Once Z F and A F have beencalculated, the species of the ablated (evaporated) particles are determinedagain using Wilson’s algorithm. The number of α-particles is determinedfirst, on the basis that these have the greatest binding energy:⎧⎨N α =⎩Int ( Z abl2Int ( A abl4) ( ) ( ): IntZabl2 < IntAabl) ( ) ( 4 ): IntZabl≥ IntAabl(26.25)2Calculation of the other ablated nuclear/nucleon species is determined ina similar fashion in order of decreasing binding energy per nucleon of theablated fragment, and subject to conservation of charge and nucleon number.Once the ablated particle species are determined, use is made of the Geant4evaporation classes to sample the order in which the particles are ejected(from G4AlphaEvaporationProbability, G4He3EvaporationProbability, G4TritonEvaporationProbabiG4DeuteronEvaporationProbability, G4ProtonEvaporationProbability and G4NeutronEvaporationPr4382

and the energies and momenta of the evaporated particle and the residualnucleus at each two-body decay (using G4AlphaEvaporationChannel,G4He3EvaporationChannel, G4TritonEvaporationChannel, G4DeuteronEvaporationChannel,G4ProtonEvaporationChannel and G4NeutronEvaporationChannel). If atany stage the probability for evaporation of any of the particles selected bythe ablation process is zero, the evaporation is forced, but no significantmomentum is imparted to the particle/nucleus. Note, however, that anyparticles ejected from the projectile will be Lorentz boosted depending uponthe initial energy per nucleon of the projectile.26.7 Definition of the functions P and F usedin the abrasion modelIn the first instance, the form of the functions P and F used in the abrasionmodel are dependent upon the relative radii of the projectile and target andthe distance of closest approach of the nuclear centres. Four radius condtionsare treated.r T > r P and r T − r P ≤ r ≤ r T + r P :P = 0.125 √ ( ) ( ) 2 [1 1 − βµνµ − 2 −0.125 0.5 √ ( ) ] ( ) 31 1 − βµννµ − 2 + 1ν(26.26)F = 0.75 √ ( ) 21 − βµν − 0.125 [3 √ ( ) 31 − βµν − 1](26.27)ννwhere:r T > r P and r < r T − r P :r Pν =(26.28)r P + r Trβ =(26.29)r P + r Tµ = r Tr P(26.30)P = −1 (26.31)383

P > r T and r P − r T ≤ r ≤ r P + r T :F = 1 (26.32)P = 0.125 √ ( ) ( ) 21 1 − βµνµ − 2 (26.33)ν{ √ ( ) [√ ν 1 1 − µ− 0.125 0.5µ µ − 2 2−ν] √2 − µ− 1} ( 1 − βµ 5 ν) 3F = 0.75 √ µν⎡ [√ 1 − (1 − µ 2 ) 3 ] √ ⎤/2 1 − (1 − µ) 2− 0.125ν⎢ 3 ⎣ µ − µ 3 ⎥⎦( ) 21 − β(26.34)ν( 1 − βν) 3r P > r T and r < r T − r P :[√ ] √ ( ) 1 − µ2 √√1 2βP = − 1 − (26.35)νν⎡F = ⎣1 − ( ⎤( )1 − µ 2)3 / 22⎦√ β1 −(26.36)ν26.8 Status of this document18.06.04 created by Peter TruscottBibliography[1] J W Wilson, R K Tripathi, F A Cucinotta, J K Shinn, F F Badavi,S Y Chun, J W Norbury, C J Zeitlin, L Heilbronn, and J Miller,”NUCFRG2: An evaluation of the semiempirical nuclear fragmentationdatabase,” NASA Technical Paper 3533, 1995.384

Figure 26.1: In the abrasion process, a fraction of the nucleons in the projectileand target nucleons interact to form a fireball region with a velocitybetween that of the projectile and the target. The remaining spectator nucleonsin the projectile and target are not initially affected (although theydo suffer change as a result of longer-term de-excitation).385

Figure 26.2: Illustration clarifying impact parameter in the far-field (b) andactual impact parameter (r).[2] Lawrence W Townsend, John W Wilson, Ram K Tripathi, John WNorbury, Francis F Badavi, and Ferdou Khan, ”HZEFRG1, An energydependentsemiempirical nuclear fragmentation model,” NASA TechnicalPaper 3310, 1993.[3] Francis A Cucinotta, ”Multiple-scattering model for inclusive proton productionin heavy ion collisions,” NASA Technical Paper 3470, 1994.386

Chapter 27Electromagnetic DissociationModel27.1 The ModelThe relative motion of a projectile nucleus travelling at relativistic speedswith respect to another nucleus can give rise to an increasingly hard spectrumof virtual photons. The excitation energy associated with this energyexchange can result in the liberation of nucleons or heavier nuclei (i.e.deuterons, α-particles, etc.). The contribution of this source to the totalinelastic cross section can be important, especially where the proton numberof the nucleus is large. The electromagnetic dissociation (ED) model is implementedin the classes G4EMDissociation, G4EMDissociationCrossSectionand G4EMDissociationSpectrum, with the theory taken from Wilson et al[1], and Bertulani and Baur [2].The number of virtual photons N(E γ , b) per unit area and energy intervalexperienced by the projectile due to the dipole field of the target is given bythe expression [2]:{( ) }N (E γ , b) =αZ2 Txx 2 k 2 2π 2 β 2 b 2 1E (x) + k 2γ γ 2 0 (x)where x is a dimensionless quantity defined as:(27.1)x = bE γ(27.2)γβ¯hcand:α = fine structure constantβ = ratio of the velocity of the projectile in the laboratory frame tothe velocity of light387

γ = Lorentz factor for the projectile in the laboratory frameb = impact parameterc = speed of light¯h = quantum constantE γ = energy of virtual photonk 0 and k 1 = zeroth and first order modified Bessel functions of thesecond kindZ T = atomic number of the target nucleusIntegrating Eq. 27.1 over the impact parameter from b min to ∞ producesthe virtual photon spectrum for the dipole field of:{N E1 (E γ ) = 2αZ2 Tξkπβ 2 0 (ξ)k 1 (ξ) − ξ2 β 2 (k2E γ 2 1 (ξ) − k0 2 (ξ))} (27.3)where, according to the algorithm implemented by Wilson et al in NUCFRG2[1]:ξ = Eγb minγβ¯hcb c = 1.34[b min = (1 + x d )b c + πα 02γα 0 = Z P Z T e 2µβ 2 c 2A 1 /3P + A 1 /3T − 0.75(A −1 /3P + A −1 )]/3T(27.4)and µ is the reduced mass of the projectile/target system, x d = 0.25, and A Pand A T are the projectile and target nucleon numbers. For the last equation,the units of b c are fm. Wilson et al state that there is an equivalent virtualphoton spectrum as a result of the quadrupole field:{N E2 (E γ ) = 2αZ2 T2 ( 1 − β 2) kπβ 4 1 2 E (ξ) + ξ ( 2 − β 2) 2k0 (ξ)k 1 (ξ) − ξ2 β 4 (k2γ 2 1 (ξ) − k0 2 (ξ))}(27.5)The cross section for the interaction of the dipole and quadrupole fields isgiven by:∫σ ED =∫N E1 (E γ ) σ E1 (E γ ) dE γ +N E2 (E γ ) σ E2 (E γ ) dE γ (27.6)388

Wilson et al assume that σ E1 (E γ ) and σ E2 (E γ ) are sharply peaked at thegiant dipole and quadrupole resonance energies:E GDR = ¯hc [ m ∗ c 2 R 2 08J(1 + u −1+ε+3u1+ε+u ε)] − 1 2E GQR = 63A1 /3P(27.7)so that the terms for N E1 and N E2 can be taken out of the integrals in Eq.27.6 and evaluated at the resonances.In Eq. 27.7:u = 3J A −1 /3Q ′ P(27.8)R 0 = r 0 A 1 /3Pɛ = 0.0768, Q ′ = 17MeV, J = 36.8MeV, r 0 = 1.18fm, and m ∗ is 7/10 ofthe nucleon mass (taken as 938.95 MeV/c 2 ). (The dipole and quadrupoleenergies are expressed in units of MeV.)The photonuclear cross sections for the dipole and quadrupole resonances areassumed to be given by:∫σ E1 (E γ ) dE γ = 60 N P Z P(27.9)A Pin units of MeV-mb (N P being the number of neutrons in the projectile) and:∫σ E2 (E γ ) dE γ= 0.22fZEγ2 P A 2 /3P (27.10)in units of µb/MeV. In the latter expression, f is given by:⎧⎪⎨ 0.9 A P > 100f = 0.6 40 < A P ≤ 100(27.11)⎪⎩0.3 40 ≤ A PThe total cross section for electromagnetic dissociation is therefore given byEq. 27.6 with the virtual photon spectra for the dipole and quadrupole fieldscalculated at the resonances:∫σ ED = N E1 (E GDR )∫σ E1 (E γ ) dE γ + N E2 (E GQR ) EGQR2 σE2 (E γ )dE γE 2 γ(27.12)389

where the resonance energies are given by Eq. 27.7 and the integrals for thephotonuclear cross sections given by Eq. 27.9 and Eq. 27.10.The selection of proton or neutron emission is made according to the followingprescription from Wilson et al.⎧⎪⎨σ ED,p = σ ED ×⎪⎩0.5 Z P < 60.6 6 ≤ Z P ≤ 80.7 8 < Z P < 14min [ Z PA P, 1.95 exp(−0.075Z P ) ] Z P ≥ 14σ ED,n = σ ED − σ ED,p(27.13)Note that this implementation of ED interactions only treats the ejectionof single nucleons from the nucleus, and currently does not allow emission ofother light nuclear fragments.⎫⎪⎬⎪⎭27.2 Status of this document19.06.04 created by Peter TruscottBibliography[1] J. W. Wilson, R. K. Tripathi, F. A. Cucinotta, J. K. Shinn, F. F. Badavi,S. Y. Chun, J. W. Norbury, C. J. Zeitlin, L. Heilbronn, and J. Miller,”NUCFRG2: An evaluation of the semiempirical nuclear fragmentationdatabase,” NASA Technical Paper 3533, 1995.[2] C. A. Bertulani, and G. Baur, Electromagnetic processes in relativisticheavy ion collisions, Nucl Phys, A458, 725-744, 1986.390

Chapter 28Precompound model.28.1 Reaction initial state.The GEANT4 precompound model is considered as an extension of thehadron kinetic model. It gives a possibility to extend the low energy rangeof the hadron kinetic model for nucleon-nucleus inelastic collision and it providesa ”smooth” transition from kinetic stage of reaction described by thehadron kinetic model to the equilibrium stage of reaction described by theequilibrium deexcitation models.The initial information for calculation of pre-compound nuclear stageconsists from the atomic mass number A, charge Z of residual nucleus, itsfour momentum P 0 , excitation energy U and number of excitons n equalsthe sum of number of particles p (from them p Z are charged) and number ofholes h.At the preequilibrium stage of reaction, we following the [1] approach,take into account all possible nuclear transition the number of excitons nwith ∆n = +2, −2, 0 [1], which defined by transition probabilities. Onlyemmision of neutrons, protons, deutrons, thritium and helium nuclei aretaken into account.28.2 Simulation of pre-compound reactionThe precompound stage of nuclear reaction is considered until nuclearsystem is not an equilibrium state. Further emission of nuclear fragments orphotons from excited nucleus is simulated using an equilibrium model.391

28.2.1 Statistical equilibrium conditionIn the state of statistical equilibrium, which is characterized by an eqilibriumnumber of excitons n eq , all three type of transitions are equiprobable.Thus n eq is fixed by ω +2 (n eq , U) = ω −2 (n eq , U). From this condition we canget√n eq = 2gU. (28.1)28.2.2 Level density of excited (n-exciton) statesTo obtain Eq. (28.1) it was assumed an equidistant scheme of singleparticlelevels with the density g ≈ 0.595aA, where a is the level densityparameter, when we have the level density of the n-exciton state asρ n (U) =28.2.3 Transition probabilitiesg(gU)n−1p!h!(n − 1)! . (28.2)The partial transition probabilities changing the exciton number by ∆n isdetermined by the squared matrix element averaged over allowed transitions< |M| 2 > and the density of final states ρ ∆n (n, U), which are really accessiblein this transition. It can be defined as following:ω ∆n (n, U) = 2π h < |M|2 > ρ ∆n (n, U). (28.3)The density of final states ρ ∆n (n, U) were derived in paper [2] using the Eq.(28.2) for the level density of the n-exciton state and later corrected for thePauli principle and indistinguishability of identical excitons in paper [3]:ρ ∆n=+2 (n, U) = 1 2[gU − F (p + 1, h + 1)]2g [n + 1gU − F (p + 1, h + 1)] n−1 ,gU − F (p, h)(28.4)ρ ∆n=0 (n, U) = 1 [gU − F (p, h)]g [p(p − 1) + 4ph + h(h − 1)]2 n(28.5)andρ ∆n=−2 (n, U) = 1 gph(n − 2),2(28.6)where F (p, h) = (p 2 + h 2 + p − h)/4 − h/2 and it was taken to be equal zero.To avoid calculation of the averaged squared matrix element < |M| 2 > itwas assumed [1] that transition probability ω ∆n=+2 (n, U) is the same as the392

probability for quasi-free scattering of a nucleon above the Fermi level on anucleon of the target nucleus, i. e.ω ∆n=+2 (n, U) = < σ(v rel)v rel >V int. (28.7)In Eq. (28.7) the interaction volume is estimated as V int = 4π(2r 3 c + λ/2π) 3 ,with the De √Broglie wave length λ/2π corresponding to the relative velocity< v rel >= 2T rel /m, where m is nucleon mass and r c = 0.6 fm.The averaging in < σ(v rel )v rel > is further simplified byFor σ(v rel ) we take approximation:< σ(v rel )v rel >=< σ(v rel ) >< v rel > . (28.8)σ(v rel ) = 0.5[σ pp (v rel ) + σ pn (v rel ]P (T F /T rel ), (28.9)where factor P (T F /T rel ) was introduced to take into account the Pauli principle.It is given byfor T FT rel≤ 0.5 andT FP (T F /T rel ) = 1 − 7 (28.10)5 T relT FT FP (T F /T rel ) = 1 − 7 + 2 (2 − T rel) 5/2 (28.11)5 T rel 5 T rel T Ffor T FT rel> 0.5.The free-particle proton-proton σ pp (v rel ) and proton-neutron σ pn (v rel ) interactioncross sections are estimated using the equations [4]:σ pp (v rel ) = 10.63v 2 rel− 29.93v rel+ 42.9 (28.12)andσ pn (v rel ) = 34.10 − 82.2 + 82.2,vrel2 v rel(28.13)where cross sections are given in mbarn.The mean relative kinetic energy T rel is needed to calculate < v rel >and the factor P (T F /T rel ) was computed as T rel = T p + T n , where meankinetic energies of projectile nucleons T p = T F + U/n and target nucleonsT N = 3T F /5, respecively.393

Combining Eqs. (28.3) - (28.7) and assuming that < |M| 2 > are the samefor transitions with ∆n = 0 and ∆n = ±2 we obtain for another transitionprobabilities:andω ∆n=0 (n, U) == n+1[ gU−F (p,h)V int n gU−F (p+1,h+1) ]n+1 p(p−1)+4ph+h(h−1)gU−F (p,h)ω ∆n=−2 (n, U) == gU−F (p,h)V int[gU−F (p+1,h+1) ]n+1 ph(n+1)(n−2).[gU−F (p,h)] 2(28.14)(28.15)28.2.4 Emission probabilities for nucleonsEmission process probability has been choosen similar as in the classicalequilibrium Weisskopf-Ewing model [5]. Probability to emit nucleon b in theenergy interval (T b , T b + dT b ) is givenW b (n, U, T b ) = σ b (T b ) (2s b + 1)µ bπ 2 h 3 R b (p, h) ρ n−b(E ∗ )ρ n (U) T b, (28.16)where σ b (T b ) is the inverse (absorption of nucleon b) reaction cross section,s b and m b are nucleon spin and reduced mass, the factor R b (p, h) takes intoaccount the condition for the exciton to be a proton or neutron, ρ n−b (E ∗ )and ρ n (U) are level densities of nucleus after and before nucleon emission aredefined in the evaporation model, respectively and E ∗ = U − Q b − T b is theexcitation energy of nucleus after fragment emission.28.2.5 Emission probabilities for complex fragmentsIt was assumed [1] that nucleons inside excited nucleus are able to ”condense”forming complex fragment. The ”condensation” probability to createfragment consisting from N b nucleons inside nucleus with A nucleons is givenbyγ Nb = N 3 b (V b/V ) N b−1 = N 3 b (N b/A) N b−1 , (28.17)where V b and V are fragment b and nucleus volumes, respectively. The lastequation was estimated [1] as the overlap integral of (constant inside a volume)wave function of independent nucleons with that of the fragment.During the prequilibrium stage a ”condense” fragment can be emitted.The probability to emit a fragment can be written as [1]W b (n, U, T b ) = γ Nb R b (p, h) ρ(N b, 0, T b + Q b )σ b (T b ) (2s b + 1)µ b ρ n−b (E ∗ )g b (T b )π 2 h 3 ρ n (U) T b,(28.18)394

whereg b (T b ) = V b(2s b + 1)(2µ b ) 3/24π 2 h 3 (T b + Q b ) 1/2 (28.19)is the single-particle density for complex fragment b, which is obtained byassuming that complex fragment moves inside volume V b in the uniform potentialwell whose depth is equal to be Q b , and the factor R b (p, h) garanteescorrect isotopic composition of a fragment b.28.2.6 The total probabilityThis probability is defined asW tot (n, U) =∑6∑ω ∆n (n, U) + W b (n, U), (28.20)∆n=+2,0,−2b=1where total emission W b (n, U) probabilities to emit fragment b can be obtainedfrom Eqs. (28.16) and (28.18) by integration over T b :W b (n, U) =∫ U−QbV bW b (n, U, T b )dT b . (28.21)28.2.7 Calculation of kinetic energies for emitted particleThe equations (28.16) and (28.18) are used to sample kinetic energies ofemitted fragment.28.2.8 Parameters of residual nucleus.After fragment emission we update parameter of decaying nucleus:A f = A − A b ; Z f = Z − Z b ; P f = P 0 − p b ;E ∗ f = √E 2 f − ⃗ P 2 f − M(A f, Z f ).(28.22)Here p b is the evaporated fragment four momentum.28.3 Status of this document00.00.00 created by Vicente Lara395

Bibliography[1] K.K. Gudima, S.G. Mashnik, V.D. Toneev, Nucl. Phys. A401 329(1983).[2] F. C. Williams, Phys. Lett. B31 180 (1970).[3] I. Ribanský, P. Obloẑinský, E. Bétaˆk, Nucl. Phys. A205 545 (1973).[4] N. Metropolis et al., Phys. Rev. 100 185 (1958).[5] V.E. Weisskopf, D.H. Ewing, Phys. Rev. 57 472 (1940).396

Chapter 29Evaporation Model29.1 Introduction.At the end of the pre-equilibrium stage, or a thermalizing process, theresidual nucleus is supposed to be left in an equilibrium state, in which theexcitation energy E ∗ is shared by a large number of nucleons. Such an equilibratedcompound nucleus is characterized by its mass, charge and excitationenergy with no further memory of the steps which led to its formation. Ifthe excitation energy is higher than the separation energy, it can still ejectnucleons and light fragments (d, t, 3 He, α). These constitute the low energyand most abundant part of the emitted particles in the rest system of theresidual nucleus. The emission of particles by an excited compound nucleushas been successfully described by comparing the nucleus with the evaporationof molecules from a fluid [1]. The first statistical theory of compoundnuclear decay is due to Weisskopf and Ewing[2].29.2 Model description.The Weisskopf treatment is an application of the detailed balance principlethat relates the probabilities to go from a state i to another d and viceversathrough the density of states in the two systems:P i→d ρ(i) = P d→i ρ(d) (29.1)where P d→i is the probability per unit of time of a nucleus d captures a particlej and form a compound nucleus i which is proportional to the compoundnucleus cross section σ inv . Thus, the probability that a parent nucleus i withan excitation energy E ∗ emits a particle j in its ground state with kinetic397

energy ε isP j (ε)dε = g j σ inv (ε) ρ d(E max − ε)εdε (29.2)ρ i (E ∗ )where ρ i (E ∗ ) is the level density of the evaporating nucleus, ρ d (E max −ε) thatof the daugther (residual) nucleus after emission of a fragment j and E max isthe maximum energy that can be carried by the ejectile. With the spin s j andthe mass m j of the emitted particle, g j is expressed as g j = (2s j +1)m j /π 2¯h 2 .This formula must be implemented with a suitable form for the level densityand inverse reaction cross section. We have followed, like many otherimplementations, the original work of Dostrovsky et al. [3] (which representsthe first Monte Carlo code for the evaporation process) with slight modifications.The advantage of the Dostrovsky model is that it leds to a simpleexpression for equation 29.2 that can be analytically integrated and used forMonte Carlo sampling.29.2.1 Cross sections for inverse reactions.The cross section for inverse reaction is expressed by means of empiricalequation [3]( )σ inv (ε) = σ g α1 + β ε(29.3)where σ g = πR 2 is the geometric cross section.In the case of neutrons, α = 0.76 + 2.2A − 1 3 and β = (2.12A − 2 3 − 0.050)/αMeV. This equation gives a good agreement to those calculated from continuumtheory [4] for intermediate nuclei down to ε ∼ 0.05 MeV. For lowerenergies σ inv,n (ε) tends toward infinity, but this causes no difficulty becauseonly the product σ inv,n (ε)ε enters in equation 29.2. It should be noted, thatthe inverse cross section needed in 29.2 is that between a neutron of kineticenergy ε and a nucleus in an excited state.For charged particles (p, d, t, 3 He and α), α = (1 + c j ) and β = −V j ,where c j is a set of parameters calculated by Shapiro [5] in order to providea good fit to the continuum theory [4] cross sections and V j is the Coulombbarrier.29.2.2 Coulomb barriers.Coulomb repulsion, as calculated from elementary electrostatics are notdirectly applicable to the computation of reaction barriers but must be correctedin several ways. The first correction is for the quantum mechanical398

phenomenoon of barrier penetration. The proper quantum mechanical expressionsfor barrier penetration are far too complex to be used if one wishesto retain equation 29.2 in an integrable form. This can be approximatelytaken into account by multiplying the electrostatic Coulomb barrier by acoefficient k j designed to reproduce the barrier penetration approximatelywhose values are tabulated [5].V j = k jZ j Z d e 2R c(29.4)The second correction is for the separation of the centers of the nuclei atcontact, R c . We have computed this separation as R c = R j + R d whereR j,d = r c A 1/3j,d and r c is given [6] by29.2.3 Level densities.r c = 2.173 1 + 0.006103Z jZ d1 + 0.009443Z j Z d(29.5)The simplest and most widely used level density based on the Fermigas model are those of Weisskopf [7] for a completely degenerate Fermi gas.We use this approach with the corrections for nucleon pairing proposed byHurwitz and Bethe [8] which takes into account the displacements of theground state:( √ )ρ(E) = C exp 2 a(E − δ)(29.6)where C is considered as constant and does not need to be specified sinceonly ratios of level densities enter in equation 29.2. δ is the pairing energycorrection of the daughter nucleus evaluated by Cook et al. [9] and Gilbertand Cameron [10] for those values not evaluated by Cook et al.. The leveldensity parameter is calculated according to:{1 + δ }E [1 − exp(−γE)]a(E, A, Z) = ã(A)(29.7)and the parameters calculated by Iljinov et al. [11] and shell corrections ofTruran, Cameron and Hilf [12].29.2.4 Maximum energy available for evaporation.The maximum energy avilable for the evaporation process (i.e. themaximum kinetic energy of the outgoing fragment) is usually computed399

like E ∗ − δ − Q j where is the separation energy of the fragment j: Q j =M i − M d − M j and M i , M d and M j are the nclear masses of the compound,residual and evporated nuclei respectively. However, that expression doesnot consider the recoil energy of the residual nucleus. In order to take intoaccount the recoil energy we use the expressionε maxj= (M i + E ∗ − δ) 2 + M 2 j − M 2 d2(M i + E ∗ − δ)− M j (29.8)29.2.5 Total decay width.The total decay width for evaporation of a fragment j can be obtained byintegrating equation 29.2 over kinetic energy∫ ε maxjΓ j = ¯h P (ε j )dε j (29.9)V jThis integration can be performed analiticaly if we use equation 29.6 for leveldensities and equation 29.3 for inverse reaction cross section. Thus, the totalwidth is given byΓ j = g jm j R 2 d2π¯h 2αa 2 d×⎧{(⎪⎩ βa d − 3 )2{√(2βa d − 3) a d (ε maxj} { √}+ a d (ε maxj − V j ) exp − a i (E ∗ − δ i ) +− V j ) + 2a d (ε maxj − V j )}×{ [ √ √exp 2 a d (ε maxj − V j ) − a i (E ∗ − δ i )]}⎫⎪ ⎭ (29.10)where a d = a(A d , Z d , ε maxj ) and a i = a(A i , Z i , E ∗ ).29.3 GEM ModelAs an alternative model we have implemented the generalized evaporationmodel (GEM) by Furihata [13]. This model considers emission of fragmentsheavier than α particles and uses a more accurate level density function fortotal decay width instead of the approximation used by Dostrovsky. We usethe same set of parameters but for heavy ejectiles the parameters determinedby Matsuse et al. [14] are used.Based on the Fermi gas model, the level density function is expressed as⎧ √⎨ π e 2√ a(E−δ)for E ≥ Eρ(E) = 12 a 1/4 (E−δ) 5/4 x(29.11)⎩ 1T e(E−E 0)/Tfor E < E x400

where E x = U x + δ and U x = 150/M d + 2.5 (M d is the√mass of the daughternucleus). Nuclear temperature T is given as 1/T = a/U x − 1.5U x , and E 0is defined as E 0 = E x − T (log T − log a/4 − (5/4) log U x + 2 √ aU x ).By substituting equation 29.11 into equation 29.2 and integrating overkinetic energy can be obtained the following expressionΓ j =√ πgj πR 2 d α12ρ(E ∗ )⎧⎪⎨×⎪⎩{I 1 (t, t) + (β + V )I 0 (t)} for ε maxj − V j < E x{I 1 (t, t x ) + I 3 (s, s x )e s +(β + V )(I 0 (t x ) + I 2 (s, s x )e s )} for ε maxj − V j ≥ E x .(29.12)I 0 (t), I 1 (t, t x ), I 2 (s, s x ), and I 3 (s, s x ) are expressed as:I 0 (t) = e −E 0/T (e t − 1) (29.13)I 1 (t, t x ) = e −E0/T T {(t − t x + 1)e tx − t − 1} (29.14)I 2 (s, s x ) = 2 √ {2 s −3/2 + 1.5s −5/2 + 3.75s −7/2 −}(sx −3/2 + 1.5sx −5/2 + 3.75sx −7/2 )(29.15)[1I 3 (s, s x ) =2 √ 2s −1/2 + 4s −3/2 + 13.5s −5/2 + 60.0s −7/2 +2{325.125s −9/2 − (s 2 − s 2 x)sx −3/2 + (1.5s 2 + 0.5s 2 x)s −5/2x +(3.75s 2 + 0.25s 2 x )s−7/2 x(59.0625s 2 + 0.9375s 2 x)s −11/2x ++ (12.875s 2 + 0.625s 2 x )s−9/2 x +(324.8s 2 + 3.28s 2 x )s−13/2 x +} ] (29.16)where t = (ε maxj − V j )/T , t x = E x /T , s = 2 √ a(ε maxj − V j − δ j ) and s x =√2 a(E x − δ).Besides light fragments, 60 nuclides up to 28 Mg are considered, not only intheir ground states but also in their exited states, are considered. The excitedstate is assumed to survive if its lifetime T 1/2 is longer than the decay time,i. e., T 1/2 / ln 2 > ¯h/Γ ∗ j , where Γ∗ j is the emission width of the resonancecalculated in the same manner as for ground state particle emission. Thetotal emission width of an ejectile j is summed over its ground state and allits excited states which satisfy the above condition.401

29.4 Fission probability calculation.The fission decay channel (only for nuclei with A > 65) is taken intoaccount as a competitor for fragment and photon evaporation channels.29.4.1 The fission total probability.The fission probability (per unit time) W fis in the Bohr and Wheeler theoryof fission [15] is proportional to the level density ρ fis (T ) ( approximation Eq.(29.6) is used) at the saddle point, i.e.1 ∫ EW fis = ∗ −B fis2π¯hρ fis (E ∗ ) 0 ρ fis (E ∗ − B fis − T )dT == 1+(C f −1) exp (C f )4πa fis exp (2 √ , (29.17)aE ∗ )where B fis is the fission barrier height. The value of C f = 2 √ a fis (E ∗ − B fis )and a, a fis are the level density parameters of the compound and of the fissionsaddle point nuclei, respectively.The value of the level density parameter is large at the saddle point, whenexcitation energy is given by initial excitation energy minus the fission barrierheight, than in the ground state, i. e. a fis > a. a fis = 1.08a for Z < 85,a fis = 1.04a for Z ≥ 89 and a f = a[1.04 + 0.01(89. − Z)] for 85 ≤ Z < 89 isused.29.4.2 The fission barrier.The fission barrier is determined as difference between the saddle-pointand ground state masses.We use simple semiphenomenological approach was suggested by Barashenkovand Gereghi [16]. In their approach fission barrier B fis (A, Z) is approximatedbyB fis = B 0 fis + ∆ g + ∆ p . (29.18)The fission barrier height Bfis 0 (x) varies with the fissility parameter x =Z 2 /A. Bfis(x) 0 is given byfor x ≤ 33.5 andB 0 fis (x) = 12.5 + 4.7(33.5 − x)0.75 (29.19)B 0 fis(x) = 12.5 − 2.7(x − 33.5) 2/3 (29.20)for x > 33.5. The ∆ g = ∆M(N) + ∆M(Z), where ∆M(N) and ∆M(Z) areshell corrections for Cameron’s liquid drop mass formula [17] and the pairingenergy corrections: ∆ p = 1 for odd-odd nuclei, ∆ p = 0 for odd-even nuclei,∆ p = 0.5 for even-odd nuclei and ∆ p = −0.5 for even-even nuclei.402

29.5 The Total Probability for Photon EvaporationAs the first approximation we assume that dipole E1–transitions is themain source of γ–quanta from highly–excited nuclei [11]. The probability toevaporate γ in the energy interval (ɛ γ , ɛ γ + dɛ γ ) per unit of time is givenW γ (ɛ γ ) =1π 2 (¯hc) σ γ(ɛ 3 γ ) ρ(E∗ − ɛ γ )ɛ 2ρ(E ∗ )γ, (29.21)where σ γ (ɛ γ ) is the inverse (absorption of γ) reaction cross section, ρ is anucleus level density is defined by Eq. (29.6).The photoabsorption reaction cross section is given by the expressionσ γ (ɛ γ ) =σ 0 ɛ 2 γ Γ2 R(ɛ 2 γ − , (29.22)E2 GDP ) 2 + Γ 2 Rɛ 2 γwhere σ 0 = 2.5A mb, Γ R = 0.3E GDP and E GDP = 40.3A −1/5 MeV areempirical parameters of the giant dipole resonance [11]. The total radiationprobability isW γ =∫3 E ∗π 2 (¯hc) 3 0The integration is performed numericaly.29.5.1 Energy of evaporated photonσ γ (ɛ γ ) ρ(E∗ − ɛ γ )ɛ 2ρ(E ∗ )γdɛ γ . (29.23)The energy of γ-quantum is sampled according to the Eq. (29.21) distribution.29.6 Discrete photon evaporationThe last step of evaporation cascade consists of evaporation of photonswith discrete energies. The competition between photons and fragments aswell as giant resonance photons is neglected at this step. We consider thediscrete E1, M1 and E2 photon transitions from tabulated isotopes. Thereare large number of isotopes [18] with the experimentally measured exitedlevel energies, spins, parities and relative transitions probabilities. This informationis implemented in the code.403

29.7 Internal conversion electron emissionAn important conpetitive channel to photon emission is internal conversion.To take this into account, the photon evaporation data-base wasentended to include internal conversion coeffficients.The above constitute the first six columns of data in the photon evaporationfiles. The new version of the data base adds eleven new columnscorresponding to:7. ratio of internal conversion to gamma-ray emmission probability8. - 17. internal conversion coefficients for shells K, L1, L2, L3, M1, M2,M3, M4, M5 and N+ respectively. These coefficients are normalised to1.0The calculation of the Internal Conversion Coefficients (ICCs) is done by acubic spline interpolation of tabulalted data for the corresponding transitionenergy. These ICC tables, which we shall label Band [19], Rösel [20] andHager-Seltzer [21], are widely used and were provided in electronic formatby staff at LBNL. The reliability of these tabulated data has been reviewedin Ref. [22]. From tests carried out on these data we find that the ICCscalculated from all three tables are comparable within a 10% uncertainty,which is better than what experimetal measurements are reported to be ableto achive.The range in atomic number covered by these tables is Band: 1

mixed mulltipolarity is considered (M1+E2) and whenever the mixing ratiois provided in the ENSDF file, it is used to calculate the ICCs correspondingto the mixed multipolarity according the the formula:- fraction in M1 = 1/(1 + δ 2 )- fraction in E2 = δ 2 /(1 + δ 2 )where δ is the mixing ratio.29.7.2 Binding energyFor the production of an internal conversion electron, the energy of the transitionmust be at least the binding energy of the shell the electron is beingreleased from. The binding energy corresponding to the various shells in allisotopes used in the ICC calculation has been taken from the Geant4 fileG4AtomicShells.hh.29.7.3 IsotopesThe list of isotopes included in the photon evaporation data base has beenextended from A

[9] J.L. Cook, H. Ferguson and A. R. L. Musgrove. Aust. J. Phys., 20, 477(1967)[10] A. Gilbert and A.G.W. Cameron, Can. J. Phys., 43, 1446 (1965)[11] A. S. Iljinov, M. V. Mevel et al.. Nucl. Phys. A543, 517 (1992).[12] J. W. Truran, A. G. W. Cameron, and E. Hilf. Proc. Int. Conf. on theProperties of Nuclei Far From the Beta-Stability Leysin, Switzerland,August 31 - September 4, 1970, Vol.1, p. 275[13] S. Furihata. Nucl. Instr. and Meth. in Phys. Res. B171, 251 (2000)[14] T. Matsuse, A. Arima, and S. M. Lee. Phys. Rev. C 26, 2338 (1982)[15] Bohr N., Wheeler J. W., Phys. Rev., 56 426 (1939).[16] Barashenkov V. S., Iljinov A. S., Toneev V. D., Gereghi F. G, Nucl.Phys. A206 131 (1973).[17] Cameron A. G. W. Canad. J. Phys., 35 1021 (1957), 36 1040 (1958).[18] Evaluated Nuclear Structure Data File (ENSDF) - a computer fileof evaluated experimental nuclear structure data maintained by theNational Nuclear Data Center, Brookhaven National Laboratory(http://www.nndc.bnl.gov/nndc/nudat/).[19] I.M. Band and M.B. Trzhaskovskaya, Tables of the Gamma-ray InternalConversion Coefficients for the K, L, M Shells, for 1

Chapter 30Fission model.30.1 Reaction initial state.The GEANT4 fission model is capable to predict final excited fragmentsas result of an excited nucleus symmetric or asymmetric fission. The fissionprocess (only for nuclei with atomic number A ≥ 65) is considered as a competitorfor evaporation process, when nucleus transits from an excited stateto the ground state. Here we describe the final state generation. The calculationof the relative probability of fission with respect to the evaporationchannels are described in the chapter concerning evaporation.The initial information for calculation of fission decay consists from theatomic mass number A, charge Z of excited nucleus, its four momentum P 0and excitation energy U.30.2 Fission process simulation.30.2.1 Atomic number distribution of fission products.As follows from experimental data [1] mass distribution of fission productsconsists of the symmetric and the asymmetric components:F (A f ) = F sym (A f ) + ωF asym (A f ), (30.1)where ω(U, A, Z) defines relative contribution of each component and it dependsfrom excitation energy U and A, Z of fissioning nucleus. It was foundin [2] that experimental data can be approximated with a good accuracy, ifone takeF sym (A f ) = exp [− (A f − A sym ) 2] (30.2)4072σ 2 sym

andF asym (A f ) = exp [− (A f −A 2 ) 22σ 2 2+C asym {exp [− (A f −A 1 ) 22σ 2 1] + exp [− A f −(A−A 2 ) 2]+2σ22] + exp [− A f −(A−A 1 ) 2]},2σ22(30.3)where A sym = A/2, A 1 and A 2 are the mean values and σsim 2 , σ2 1 and σ2 2 aredispertions of the Gaussians respectively. From an analysis of experimentaldata [2] the parameter C asym ≈ 0.5 was defined and the next values fordispersions:σsym 2 = exp (0.00553U + 2.1386), (30.4)where U in MeV,for A ≤ 235 and2σ 1 = σ 2 = 5.6 MeV (30.5)2σ 1 = σ 2 = 5.6 + 0.096(A − 235) MeV (30.6)for A > 235 were found.The weight ω(U, A, Z) was approximated as followsω =ω a − F asym (A sym )1 − ω a F sym ((A 1 + A 2 )/2) . (30.7)The values of ω a for nuclei with 96 ≥ Z ≥ 90 were approximated byfor U ≤ 16.25 MeV,for U > 16.25 MeV andω a (U) = exp (0.538U − 9.9564) (30.8)ω a (U) = exp (0.09197U − 2.7003) (30.9)ω a (U) = exp (0.09197U − 1.08808) (30.10)for z = 89. For nuclei with Z ≤ 88 the authors of [2] constracted the followingapproximation:ω a (U) = exp [0.3(227 − a)] exp {0.09197[U − (B fis − 7.5)] − 1.08808},(30.11)where for A > 227 and U < B fis − 7.5 the corresponding factors occuring inexponential functions vanish.408

30.2.2 Charge distribution of fission products.At given mass of fragment A f the experimental data [1] on the charge Z fdistribution of fragments are well approximated by Gaussian with dispertionσ 2 z = 0.36 and the average < Z f > is described by expression:< Z f >= A fZ + ∆Z, (30.12)Awhen parameter ∆Z = −0.45 for A f ≥ 134, ∆Z = −0.45(A f − A/2)/(134 −A/2) for A − 134 < A f < 134 and ∆Z = 0.45 for A ≤ A − 134.After sampling of fragment atomic masses numbers and fragment charges,we have to check that fragment ground state masses do not exceed initialenergy and calculate the maximal fragment kinetic energyT max < U + M(A, Z) − M 1 (A f1 , Z f1 ) − M 2 (A f2 , Z f2 ), (30.13)where U and M(A, Z) are the excitation energy and mass of initial nucleus,M 1 (A f1 , Z f1 ), and M 2 (A f2 , Z f2 ) are masses of the first and second fragment,respectively.30.2.3 Kinetic energy distribution of fission products.We use the empiricaly defined [3] dependence of the average kinetic energy< T kin > (in MeV) of fission fragments on the mass and the charge of afissioning nucleus:< T kin >= 0.1178Z 2 /A 1/3 + 5.8. (30.14)This energy is distributed differently in cases of symmetric and asymmetricmodes of fission. It follows from the analysis of data [2] that in the asymmetricmode, the average kinetic energy of fragments is higher than that inthe symmetric one by approximately 12.5 MeV. To approximate the averagenumbers of kinetic energies < T symasymkin and < Tkin> for the symmetric andasymmetric modes of fission the authors of [2] suggested empirical expressions:where< T symkin >=< T kin > −12.5W asim , (30.15)< T asymkin >=< T kin > +12.5W sim , (30.16)∫W sim = ω∫F sim (A)dA/F (A)dA (30.17)and∫W asim =∫F asim (A)dA/F (A)dA, (30.18)409

espectively. In the symmetric fission the experimental data for the ratio ofthe average kinetic energy of fission fragments < T kin (A f ) > to this maximumenergy < Tkinmax > as a function of the mass of a larger fragment A max can beapproximated by expressions< T kin (A f ) > / < T maxkin >= 1 − k[(A f − A max )/A] 2 (30.19)for A sim ≤ A f ≤ A max + 10 and< T kin (A f ) > / < T maxkin>= 1 − k(10/A) 2 − 2(10/A)k(A f − A max − 10)/A(30.20)for A f > A max + 10, where A max = A sim and k = 5.32 and A max = 134 andk = 23.5 for symmetric and asymmetric fission respectively. For both modesof fission the distribution over the kinetic energy of fragments T kin is choosenGaussian with their own average values < T kin (A f ) >=< T symkin (A f) > or< T kin (A f ) >=< T asymkin (A f) > and dispersions σkin 2 equal 82 MeV or 10 2MeV 2 for symmetrical and asymmetrical modes, respectively.30.2.4 Calculation of the excitation energy of fissionproducts.The total excitation energy of fragments U frag can be defined according toequation:U frag = U + M(A, Z) − M 1 (A f1 , Z f1 ) − M 2 (A f2 , Z f2 ) − T kin , (30.21)where U and M(A, Z) are the excitation energy and mass of initial nucleus,T kin is the fragments kinetic energy, M 1 (A f1 , Z f1 ), and M 2 (A f2 , Z f2 ) aremasses of the first and second fragment, respectively.The value of excitation energy of fragment U f determines the fragmenttemperature (T = √ U f /a f , where a f ∼ A f is the parameter of fragment leveldensity). Assuming that after disintegration fragments have the same temperatureas initial nucleus than the total excitation energy will be distributedbetween fragments in proportion to their mass numbers one obtains30.2.5 Excited fragment momenta.U f = U fragA fA . (30.22)Assuming that fragment kinetic energy T f = Pf 2 /(2(M(A f , Z f + U f ) weare able to calculate the absolute value of fragment c.m. momentumP f = (M 1(A f1 , Z f1 + U f1 )(M 2 (A f2 , Z f2 + U f2 )M 1 (A f1 , Z f1 ) + U f1 + M 2 (A f2 , Z f2 ) + U f2T kin . (30.23)410

and its components, assuming fragment isotropical distribution.Bibliography[1] Vandenbosch R., Huizenga J. R., Nuclear Fission, Academic Press,New York, 1973.[2] Adeev G. D. et al. Preprint INR 816/93, Moscow, 1993.[3] Viola V. E., Kwiatkowski K. and Walker M, Phys. Rev. C31 1550(1985).411

Chapter 31Fermi break-up model.31.1 Fermi break-up simulation for light nuclei.The GEANT4 Fermi break-up model is capable to predict final states asresult of an excited nucleus with atomic number A < 17 statistical break-up.For light nuclei the values of excitation energy per nucleon are oftencomparable with nucleon binding energy. Thus a light excited nucleus breaksinto two or more fragments with branching given by available phase space.To describe a process of nuclear disassembling the so-called Fermi break-upmodel is used [1], [2], [3]. This statistical approach was first used by Fermi[1] to describe the multiple production in high energy nucleon collision.31.1.1 Allowed channel.The channel will be allowed for decay, if the total kinetic energy E kin of allfragments of the given channel at the moment of break-up is positive. Thisenergy can be calculated according to equation:n∑E kin = U + M(A, Z) − E Coulomb − (m b + ɛ b ), (31.1)b=1m b and ɛ b are masses and excitation energies of fragments, respectively,E Coulomb is the Coulomb barrier for a given channel. It is approximatedbyE Coulomb = 3 e 2(1 + V ) −1/3 ( Z25 r 0 V 0 A − ∑ n1/3b=1Z 2A 1/3b), (31.2)where V 0 is the volume of the system corresponding to the normal nuclearmatter density and κ = V V 0is a parameter ( κ = 1 is used).412

31.1.2 Break-up probability.The total probability for nucleus to break-up into n componets (nucleons,deutrons, tritons, alphas etc) in the final state is given byW (E, n) = (V/Ω) n−1 ρ n (E), (31.3)where ρ n (E) is the density of a number of final states, V is the volume ofdecaying system and Ω = (2π¯h) 3 is the normalization volume. The densityρ n (E) can be defined as a product of three factors:ρ n (E) = M n (E)S n G n . (31.4)The first one is the phase space factor defined as∫ +∞ ∫ +∞ n∑n∑M n = ... δ( p b )δ(E −√p n 2 + m 2 b ) ∏d 3 p b , (31.5)−∞−∞b=1where p b is fragment b momentum. The second one is the spin factorn∏S n = (2s b + 1), (31.6)b=1which gives the number of states with different spin orientations. The lastone is the permutation factorG n =k∏j=1b=1b=11n j ! , (31.7)which takes into account identity of components in final state. n j is a numberof components of j- type particles and k is defined by n = ∑ kj=1 n j ).In non-relativistic case (Eq. (31.10) the integration in Eq. (31.5) can beevaluated analiticaly (see e. g. [5]). The probability for a nucleus with energyE disassembling into n fragments with masses m b , where b = 1, 2, 3, ..., nequalsW (E kin , n) = S n G n ( V 1Ω )n−1 ( ∑ nb=1m bwhere Γ(x) is the gamma function.n ∏b=1(2π)3(n−1)/2m b ) 3/2Γ(3(n − 1)/2) E3n/2−5/231.1.3 Fermi break-up model parameter.kin ,(31.8)Thus the Fermi break-up model has only one free parameter V is thevolume of decaying system, which can be calculated as follows:where r 0 = 1.4 fm is used.V = 4πR 3 /3 = 4πr0 3 A/3, (31.9)413

31.1.4 Fragment characteristics.We take into account the formation of fragments in their ground and lowlyingexcited states, which are stable for nucleon emission. However, severalunstable fragments with large lifetimes: 5 He, 5 Li, 8 Be, 9 B etc are also considered.Fragment characteristics A b , Z b , s b and ɛ b are taken from [6].31.1.5 MC procedure.The nucleus break-up is described by the Monte Carlo (MC) procedure.We randomly (according to probability Eq. (31.8) and condition Eq. (31.1))select decay channel. Then for given channel we calculate kinematical quantitiesof each fragment according to n-body phase space distribution:M n =∫ +∞−∞∫ +∞...−∞n∑ n∑ p 2 n∏bδ( p b )δ( − E kin ) d 3 p b . (31.10)b=1 b=12m b b=1The Kopylov’s sampling procedure [7] is applied. The angular distributionsfor emitted fragments are considered to be isotropical.Bibliography[1] Fermi E., Prog. Theor. Phys. 5 1570 (1950).[2] Kretschmar M. Annual Rev. Nucl. Sci. 11 1 (1961).[3] Epherre M., Gradsztajn E., J. Physique 18 48 (1967).[4] Cameron A. G. W. Canad. J. Phys., 35 1021 (1957), 36 1040 (1958).[5] Barashenkov V. S., Barbashov B. M., Bubelev E. G. Nuovo Cimento,7 117 (1958).[6] Ajzenberg-Selone F., Nucl. Phys. 1 360 (1981); A375 (1982); 392(1983); A413 (1984); A433 (1985).[7] Kopylov G. I., Principles of resonance kinematics, Moscow, Nauka,1970 (in Russian).414

Chapter 32Multifragmentation model.32.1 Multifragmentation process simulation.The GEANT4 multifragmentation model is capable to predict final statesas result of an highly excited nucleus statistical break-up.The initial information for calculation of multifragmentation stage consistsfrom the atomic mass number A, charge Z of excited nucleus and itsexcitation energy U. At high excitation energies U/A > 3 MeV the multifragmentationmechanism, when nuclear system can eventually breaks downinto fragments, becomes the dominant. Later on the excited primary fragmentspropagate independently in the mutual Coulomb field and undergode-excitation. Detailed description of multifragmentation mechanism andmodel can be found in review [1].32.1.1 Multifragmentation probability.The probability of a breakup channel b is given by the expression (in theso-called microcanonical approach [1], [2]):W b (U, A, Z) =1∑b exp[S b (U, A, Z)] exp[S b(U, A, Z)], (32.1)where S b (U, A, Z) is the entropy of a multifragment state corresponding to thebreakup channel b. The channels {b} can be parametrized by set of fragmentmultiplicities N Af ,Z ffor fragment with atomic number A f and charge Z f .All partitions {b} should satisfy constraints on the total mass and charge:∑N Af ,Z fA f = A (32.2)f415

and∑N Af ,Z fZ f = Z. (32.3)fIt is assumed [2] that thermodynamic equilibrium is established in everychannel, which can be characterized by the channel temperature T b .The channel temperature T b is determined by the equation constrainingthe average energy E b (T b , V ) associated with partition b:E b (T b , V ) = U + E ground = U + M(A, Z), (32.4)where V is the system volume, E ground is the ground state (at T b = 0) energyof system and M(A, Z) is the mass of nucleus.According to the conventional thermodynamical formulae the average energyof a partitition b is expressed through the system free energy F b asfollowsE b (T b , V ) = F b (T b , V ) + T b S b (T b , V ). (32.5)Thus, if free energy F b of a partition b is known, we can find the channeltemperature T b from Eqs. (32.4) and (32.5), then the entropy S b = −dF b /dT band hence, decay probability W b defined by Eq. (32.1) can be calculated.Calculation of the free energy is based on the use of the liquid-drop descriptionof individual fragments [2]. The free energy of a partition b can besplitted into several terms:F b (T b , V ) = ∑ fF f (T b , V ) + E C (V ), (32.6)where F f (T b , V ) is the average energy of an individual fragment includingthe volumeF V f = [−E 0 − T 2 b /ɛ(A f)]A f , (32.7)surfaceF Surf = β 0 [(T 2 c − T 2 b )/(T 2 c + T 2 b )]5/4 A 2/3f = β(T b )A 2/3f , (32.8)symmetryCoulomband translationalF Simf = γ(A f − 2Z f ) 2 /A f , (32.9)F C f = 3 5Z 2 f e2r 0 A 1/3f[1 − (1 + κ C ) −1/3 ] (32.10)F t f = −T b ln (g f V f /λ 3 T b) + T b ln (N Af ,Z f!)/N Af ,Z f(32.11)416

terms and the last termE C (V ) = 3 Z 2 e 2(32.12)5 Ris the Coulomb energy of the uniformly charged sphere with charge Ze andthe radius R = (3V/4π) 1/3 = r 0 A 1/3 (1 + κ C ) 1/3 , where κ C = 2 [2].Parameters E 0 = 16 MeV, β 0 = 18 MeV, γ = 25 MeV are the coefficientsof the Bethe-Weizsacker mass formula at T b = 0. g f = (2S f + 1)(2I f + 1)is a spin S f and isospin I f degeneracy factor for fragment ( fragments withA f > 1 are treated as the Boltzmann particles), λ Tb = (2πh 2 /m N T b ) 1/2 isthe thermal wavelength, m N is the nucleon mass, r 0 = 1.17 fm, T c = 18MeV is the critical temperature, which corresponds to the liquid-gas phasetransition. ɛ(A f ) = ɛ 0 [1 + 3/(A f − 1)] is the inverse level density of themass A f fragment and ɛ 0 = 16 MeV is considered as a variable modelparameter, whose value depends on the fraction of energy transferred to theinternal degrees of freedom of fragments [2]. The free volume V f = κV =κ 4 3 πr4 0A available to the translational motion of fragment, where κ ≈ 1 andits dependence on the multiplicity of fragments was taken from [2]:κ = [1 + 1.44r 0 A 1/3 (M 1/3 − 1)] 3 − 1. (32.13)For M = 1 κ = 0.The light fragments with A f < 4, which have no excited states, are consideredas elementary particles characterized by the empirical masses M f ,radii R f , binding energies B f , spin degeneracy factors g f of ground states.They contribute to the translation free energy and Coulomb energy.32.1.2 Direct simulation of the low multiplicity multifragmentdisintegration.At comparatively low excitation energy (temperature) system will disintegrateinto a small number of fragments M ≤ 4 and number of channel isnot huge. For such situation a direct (microcanonical) sorting of all decaychannels can be performed. Then, using Eq. (32.1), the average multiplicityvalue < M > can be found. To check that we really have the situation withthe low excitation energy, the obtained value of < M > is examined to obeythe inequality < M >≤ M 0 , where M 0 = 3.3 and M 0 = 2.6 for A ∼ 100and for A ∼ 200, respectively [2]. If the discussed inequality is fulfilled, thenthe set of channels under consideration is belived to be able for a correctdescription of the break up. Then using calculated according Eq. (32.1)probabilities we can randomly select a specific channel with given values ofA f and Z f .417

32.1.3 Fragment multiplicity distribution.The individual fragment multiplicities N Af ,Z fin the so-called macrocanonicalensemble [1] are distributed according to the Poisson distribution:P (N Af ,Z f) = exp (−ω Af ,Z f) ωN A f ,Z fA f ,Z fN Af ,Z f!(32.14)with mean value < N Af ,Z f>= ω Af ,Z fdefined asV f< N Af ,Z f>= g f A 3/2f exp [ 1 (Fλ 3 f (T b , V ) − Ff tT bT (T b, V ) − µA f − νZ f )],b(32.15)where µ and ν are chemical potentials. The chemical potential are found bysubstituting Eq. (32.15) into the system of constraints:andand solving it by iteration.∑< N Af ,Z f> A f = A (32.16)f∑< N Af ,Z f> Z f = Z (32.17)f32.1.4 Atomic number distribution of fragments.Fragment atomic numbers A f > 1 are also distributed according to thePoisson distribution [1] (see Eq. (32.14)) with mean value < N Af > definedasV f< N Af >= A 3/2f exp [ 1 (Fλ 3 f (T b , V ) − Ff tT bT (T f, V ) − µA f − ν < Z f >)],b(32.18)where calculating the internal free energy F f (T b , V ) − Ff t(T b, V ) one has tosubstitute Z f →< Z f >. The average charge < Z f > for fragment havingatomic number A f is given by< Z f (A f ) >=(4γ + ν)A f. (32.19)8γ + 2[1 − (1 + κ) −1/3 ]A 2/3f418

32.1.5 Charge distribution of fragments.At given mass of fragment A f > 1 the charge Z f distribution of fragmentsare described by GaussianP (Z f (A f )) ∼ exp [− (Z f(A f )− < Z f (A f ) >) 22(σ Zf (A f )) 2 ] (32.20)with dispertionAσ Zf (A f ) = √f T b8γ + 2[1 − (1 + κ) −1/3 ]A 2/3f≈√Af T b8γ . (32.21)and the average charge < Z f (A f ) > defined by Eq. (32.17).32.1.6 Kinetic energy distribution of fragments.It is assumed [2] that at the instant of the nucleus break-up the kineticenergy of the fragment T f kin in the rest of nucleus obeys the Boltzmann distributionat given temperature T b :dP (T f kin )dT f kin∼√T f kin exp (−T f kin /T b). (32.22)Under assumption of thermodynamic equilibrium the fragment have isotropicvelocities distribution in the rest frame of nucleus. The total kinetic energyof fragments should be equal 3 MT 2 b, where M is fragment multiplicity, andthe total fragment momentum should be equal zero. These conditions arefullfilled by choosing properly the momenta of two last fragments.The initial conditions for the divergence of the fragment system are determinedby random selection of fragment coordinates distributed with equalprobabilities over the break-up volume V f = κV . It can be a sphere or prolongatedellipsoid. Then Newton’s equations of motion are solved for allfragments in the self-consistent time-dependent Coulomb field [2]. Thus theasymptotic energies of fragments determined as result of this procedure differfrom the initial values by the Coulomb repulsion energy.32.1.7 Calculation of the fragment excitation energies.The temparature T b determines the average excitation energy of each fragment:U f (T b ) = E f (T b ) − E f (0) = T 2 bɛ 0A f + [β(T b ) − T bdβ(T b )dT b− β 0 ]A 2/3f , (32.23)419

where E f (T b ) is the average fragment energy at given temperature T b andβ(T b ) is defined in Eq. (32.8). There is no excitation for fragment withA f < 4, for 4 He excitation energy was taken as U4 He = 4T 2 b /ɛ o.Bibliography[1] Bondorf J. P., Botvina A. S., Iljinov A. S., Mishustin I. N., SneppenK., Phys. Rep. 257 133 (1995).[2] Botvina A. S. et al., Nucl. Phys. A475 663 (1987).420

Chapter 33Low Energy NeutronInteractions33.1 IntroductionThe neutron transport class library described here simulates the interactionsof neutrons with kinetic energies from thermal energies up to O(20 MeV).The upper limit is set by the comprehensive evaluated neutron scatteringdata libraries that the simulation is based on. The result is a set of secondaryparticles that can be passed on to the tracking sub-system for furthergeometric tracking within Geant4.The interactions of neutrons at low energies are split into four parts inanalogy to the other hadronic processes in Geant4. We consider radiativecapture, elastic scattering, fission, and inelastic scattering as separate models.These models comply with the interface for use with the Geant4 hadronicprocesses which enables their transparent use within the Geant4 tool-kittogether with all other Geant4 compliant hadronic shower models.33.2 Physics and Verification33.2.1 Inclusive Cross-sectionsAll cross-section data are taken from the ENDF/B-VI[1] evaluated data library.All inclusive cross-sections are treated as point-wise cross-sections forreasons of performance. For this purpose, the data from the evaluated datalibrary have been processed, to explicitly include all neutron nuclear resonancesin the form of point-like cross-sections rather than in the form of421

parametrisations. The resulting data have been transformed into a linearlyinterpolable format, such that the error due to linear interpolation betweenadjacent data points is smaller than a few percent.The inclusive cross-sections comply with the cross-sections data set interfaceof the Geant4 hadronic design. They are, when registered with thetool-kit at initialisation, used to select the basic process. In the case of fissionand inelastic scattering, point-wise semi-inclusive cross-sections are alsoused in order to decide on the active channel for an individual interaction.As an example, in the case of fission this could be first, second, third, orforth chance fission.33.2.2 Elastic ScatteringThe final state of elastic scattering is described by sampling the differentialscattering cross-sections dσ . Two representations are supported for thedΩnormalised differential cross-section for elastic scattering. The first is a tabulationof the differential cross-section, as a function of the cosine of thescattering angle θ and the kinetic energy E of the incoming neutron.dσdΩ =dσ (cos θ, E)dΩThe tabulations used are normalised by σ/(2π) so the integral of the differentialcross-sections over the scattering angle yields unity.In the second representation, the normalised cross-section are representedas a series of legendre polynomials P l (cos θ), and the legendre coefficients a lare tabulated as a function of the incoming energy of the neutron.n2π dσlσ(E) dΩ (cos θ, E) = ∑l=02l + 1a l (E)P l (cos θ)2Describing the details of the sampling procedures is outside the scope ofthis paper.An example of the result we show in figure 33.1 for the elastic scatteringof 15 MeV neutrons off Uranium a comparison of the simulated angulardistribution of the scattered neutrons with evaluated data. The points arethe evaluated data, the histogram is the Monte Carlo prediction.In order to provide full test-coverage for the algorithms, similar testshave been performed for 72 Ge, 126 Sn, 238 U, 4 He, and 27 Al for a set of neutronkinetic energies. The agreement is very good for all values of scattering angleand neutron energy investigated.422

Figure 33.1: Comparison of data and Monte Carlo for the angular distributionof 15 MeV neutrons scattered elastically off Uranium ( 238 U). The pointsare evaluated data, and the histogram is the Monte Carlo prediction. Thelower plot excludes the forward peak, to better show the Frenel structure ofthe angular distribution of the scattered neutron.423

33.2.3 Radiative CaptureThe final state of radiative capture is described by either photon multiplicities,or photon production cross-sections, and the discrete and continuouscontributions to the photon energy spectra, along with the angular distributionsof the emitted photons.For the description of the photon multiplicity there are two supporteddata representations. It can either be tabulated as a function of the energyof the incoming neutron for each discrete photon as well as the eventualcontinuum contribution, or the full transition probability array is known, andused to determine the photon yields. If photon production cross-sections areused, only a tabulated form is supported.The photon energies E γ are associated to the multiplicities or the crosssectionsfor all discrete photon emissions. For the continuum contribution,the normalised emission probability f is broken down into a weighted sumof normalised distributions g.f (E → E γ ) = ∑ ip i (E)g i (E → E γ )The weights p i are tabulated as a function of the energy E of the incomingneutron. For each neutron energy, the distributions g are tabulated as afunction of the photon energy. As in the ENDF/B-VI data formats[1], severalinterpolation laws are used to minimise the amount of data, and optimise thedescriptive power. All data are derived from evaluated data libraries.The techniques used to describe and sample the angular distributions areidentical to the case of elastic scattering, with the difference that there iseither a tabulation or a set of legendre coefficients for each photon energyand continuum distribution.As an example of the results is shown in figure33.2 the energy distributionof the emitted photons for the radiative capture of 15 MeV neutrons onUranium ( 238 U). Similar comparisons for photon yields, energy and angulardistributions have been performed for capture on 238 U, 235 U, 23 Na, and 14 Nfor a set of incoming neutron energies. In all cases investigated the agreementbetween evaluated data and Monte Carlo is very good.33.2.4 FissionFor neutron induced fission, we take first chance, second chance, third chanceand forth chance fission into account.Neutron yields are tabulated as a function of both the incoming and outgoingneutron energy. The neutron angular distributions are either tabulated,424

or represented in terms of an expansion in legendre polynomials, similar tothe angular distributions for neutron elastic scattering. In case no data areavailable on the angular distribution, isotropic emission in the centre of masssystem of the collision is assumed.There are six different possibilities implemented to represent the neu-Figure 33.2: Comparison of data and Monte Carlo for photon energy distributionsfor radiative capture of 15 MeV neutrons on Uranium ( 238 U). Thepoints are evaluated data, the histogram is the Monte Carlo prediction.425

tron energy distributions. The energy distribution of the fission neutronsf(E → E ′ ) can be tabulated as a normalised function of the incoming andoutgoing neutron energy, again using the ENDF/B-VI interpolation schemesto minimise data volume and maximise precision.The energy distribution can also be represented as a general evaporationspectrum,f(E → E ′ ) = f (E ′ /Θ(E)) .Here E is the energy of the incoming neutron, E ′ is the energy of a fissionneutron, and Θ(E) is effective temperature used to characterise the secondaryneutron energy distribution. Both the effective temperature and thefunctional behaviour of the energy distribution are taken from tabulations.Alternatively energy distribution can be represented as a Maxwell spectrum,f(E → E ′ ) ∝ √ E ′ e E′ /Θ(E) ,or a evaporation spectrumf(E → E ′ ) ∝ E ′ e E′ /Θ(E) .In both these cases, the temperature is tabulated as a function of the incomingneutron energy.The last two options are the energy dependent Watt spectrum, and theMadland Nix spectrum. For the energy dependent Watt spectrum, the energydistribution is represented as√f(E → E ′ ) ∝ e −E′ /a(E) sinh b(E)E ′ .Here both the parameters a, and b are used from tabulation as function ofthe incoming neutron energy. In the case of the Madland Nix spectrum, theenergy distribution is described asf(E → E ′ ) = 1 2 [g(E′ , < K l >) + g(E ′ , < K h >)] .Hereg(E ′ , < K >) =13 √ < K > Θ[u3/22 E 1 (u 2 ) − u 3/21 E 1 (u 1 ) + γ(3/2, u 2 ) − γ(3/2, u 1 ) ] ,u 1 (E ′ , < K >) = (√ E ′ − √ < K >) 2, andΘu 2 (E ′ , < K >) = (√ E ′ + √ < K >) 2.Θ426

Here K l is the kinetic energy of light fragments and K h the kinetic energy ofheavy fragments, E 1 (x) is the exponential integral, and γ(x) is the incompletegamma function. The mean kinetic energies for light and heavy fragmentsare assumed to be energy independent. The temperature Θ is tabulated asa function of the kinetic energy of the incoming neutron.Fission photons are describes in analogy to capture photons, where evaluateddata are available. The measured nuclear excitation levels and transitionprobabilities are used otherwise, if available.As an example of the results is shown in figure33.3 the energy distributionof the fission neutrons in third chance fission of 15 MeV neutrons onUranium ( 238 U). This distribution contains two evaporation spectra and oneWatt spectrum. Similar comparisons for neutron yields, energy and angulardistributions, and well as fission photon yields, energy and angular distributionshave been performed for 238 U, 235 U, 234 U, and 241 Am for a set ofincoming neutron energies. In all cases the agreement between evaluateddata and Monte Carlo is very good.33.2.5 Inelastic ScatteringFor inelastic scattering, the currently supported final states are (nA→) nγs(discrete and continuum), np, nd, nt, n 3 He, nα, nd2α, nt2α, n2p, n2α, npα,n3α, 2n, 2np, 2nd, 2nα, 2n2α, nX, 3n, 3np, 3nα, 4n, p, pd, pα, 2p d, dα,d2α, dt, t, t2α, 3 He, α, 2α, and 3α.The photon distributions are again described as in the case of radiativecapture.The possibility to describe the angular and energy distributions of the finalstate particles as in the case of fission is maintained, except that normallyonly the arbitrary tabulation of secondary energies is applicable.In addition, we support the possibility to describe the energy angularcorrelations explicitly, in analogy with the ENDF/B-VI data formats. Inthis case, the production cross-section for reaction product n can be writtenasσ n (E, E ′ , cos(θ)) = σ(E)Y n (E)p(E, E ′ , cos(θ)).Here Y n (E) is the product multiplicity, σ(E) is the inelastic cross-section,and p(E, E ′ , cos(θ)) is the distribution probability. Azimuthal symmetry isassumed.The representations for the distribution probability supported are isotropicemission, discrete two-body kinematics, N-body phase-space distribution,continuum energy-angle distributions, and continuum angle-energy distributionsin the laboratory system.427

The description of isotropic emission and discrete two-body kinematics ispossible without further information. In the case of N-body phase-space distribution,tabulated values for the number of particles being treated by thelaw, and the total mass of these particles are used. For the continuum energyangledistributions, several options for representing the angular dependenceare available. Apart from the already introduced methods of expansion interms of legendre polynomials, and tabulation (here in both the incomingneutron energy, and the secondary energy), the Kalbach-Mann systematic isavailable. In the case of the continuum angle-energy distributions in the laboratorysystem, only the tabulated form in incoming neutron energy, productenergy, and product angle is implemented.First comparisons for product yields, energy and angular distributionshave been performed for a set of incoming neutron energies, but full test cov-Figure 33.3: Comparison of data and Monte Carlo for fission neutron energydistributions for induced fission by 15 MeV neutrons on Uranium ( 238 U).The curve represents evaluated data and the histogram is the Monte Carloprediction.428

erage is still to be achieved. In all cases currently investigated, the agreementbetween evaluated data and Monte Carlo is very good.33.3 SummaryBy the way of abstraction and code reuse we minimised the amount of codeto be written and maintained. The concept of container-sampling lead toabstraction and encapsulation of data representation and the correspondingrandom number generators. The Object Oriented design allows for easyextension of the cross-section base of the system, and the ENDF-B VI dataevaluations have already been supplemented with evaluated data on nuclearexcitation levels, thus improving the energy spectra of de-excitation photons.Other established data evaluations have been investigated, and extensionsbased on the JENDL[2], CENDL[4], and Brond[5] data libraries are foreseenfor next year.Bibliography[1] ENDF/B-VI: Cross Section Evaluation Working Group, ENDF/B-VI Summary Document, Report BNL-NCS-17541 (ENDF-201)(1991), edited by P.F. Rose, National Nuclear Data Center, BrookhaveNational Laboratory, Upton, NY, USA.[2] JENDL-3: T. Nakagawa, et al., Japanese Evaluated Nuclear Data Library,Version 3, Revision 2, J. Nucl. Sci. Technol. 32, 1259 (1995).[3] Jef-2: C. Nordborg, M. Salvatores, Status of the JEF Evaluated DataLibrary, Nuclear Data for Science and Technology, edited by J.K. Dickens (American Nuclear Society, LaGrange, IL, 1994).[4] CENDL-2: Chinese Nuclear Data Center, CENDL-2, The ChineseEvaluated Nuclear Data Library for Neutron Reaction Data, ReportIAEA-NDS-61, Rev. 3 (1996), International Atomic Energy Agency,Vienna, Austria.[5] Brond-2.2: A.I Blokhin et al., Current status of Russian Nuclear DataLibraries, Nuclear Data for Science and Technology, Volume2,p.695. edited by J. K. Dickens (American Nuclear Society, LaGrange,IL, 1994)429

Chapter 34Radioactive Decay34.1 The Radioactive Decay ModuleG4RadioactiveDecay and associated classes are used to simulate the decay ofradioactive nuclei by α, β + , and β − emission and by electron capture (EC).The simulation model is empirical and data-driven, and uses the EvaluatedNuclear Structure Data File (ENSDF) [1] for information on:nuclear half-lives,nuclear level structure for the parent or daughter nuclide,decay branching ratios, and· the energy of the decay process.If the daughter of a nuclear decay is an excited isomer, its prompt nuclearde-excitation is treated using the G4P hotoEvaporation class [2].34.2 SamplingSampling of the β-spectrum, which includes the coordinated energies andmomenta of the β ± , ν, or ¯ν and residual nucleus, is performed either fromhistogrammed data, or through a three-body decay algorithm. In the lattercase, the effect of the Coulomb barrier on the probability of β ± -emission canalso be taken into account using the Fermi function:√γ ZF (E 0 ) =⎧⎪ 2 (E 0 + 1) 2⎩ + E2 0 + 2E ⎫0⎪⎭1 − e −γ 137 2 41− Z2137 2 −1 . (34.1)430

Here E 0 is the energy of the β-particle given as a fraction of the end-pointenergy, Z is the atomic number of the nucleus, and γ is given by the expression:γ = 2πZ1371 + E 0√E 2 0 + 2E 0. (34.2)Due to the level of imprecision of the rest-mass energy of the nuclei generatedby G4IonT able :: GetNucleusMass, the mass of the parent nucleusis modified to a minor extent just before performing the two- or three-bodydecay so that the Q for the transition process equals that identified in theENSDF data.34.2.1 Biasing MethodsBy default, sampling of the times of radioactive decay and branching ratios isdone according to standard, analogue Monte Carlo modeling. The user mayswitch on one or more of the following variance reduction schemes, which canprovide significant improvement in the modelling efficiency:1. The decays can be biased to occur more frequently at certain times,for example, corresponding to times when measurements are taken in a realexperiment. The statistical weights of the daughter nuclides are reducedaccording to the probability of survival to the time of the event, t, which isdetermined from the decay rate. The decay rate of the n th nuclide in a decaychain is given by the recursive formulae:where:R n (t) =n−1 ∑i=1A n:i f(t, τ i ) + A n:n f(t, τ n ) (34.3)A n:i =τ iτ i − τ nA n:i ∀i < n (34.4)n−1 ∑ τ nA n:n = − A n:i − y n (34.5)i=1τ i − τ nf(t, τ i ) = e− tτ iτ i∫t− infF (t ′ )e t′τ i dt ′ . (34.6)The values τ i are the mean life-times for the nuclei, y i is the yield of thei th nucleus, and F (t) is a function identifying the time profile of the source.The above expression for decay rate is simplified, since it assumes that the431

i th nucleus undergoes 100% of the decays to the (i + 1) th nucleus. Similarexpressions which allow for branching and merging of different decay chainscan be found in Ref. [3].A consequence of the form of equations 34.4 and 34.6 is that the user mayprovide a source time profile so that each decay produced as a result of asimulated source particle incident at time t = 0 is convolved over the sourcetime profile to derive the actual decay rate for that source function.This form of variance reduction is only appropriate if the radionuclei canbe considered to be at rest with respect to the geometry when decay occurs.2. For a given decay mode (α, β + + EC, or β − ) the branching ratios tothe daughter nuclide can be sampled with equal probability, so that somelow probability branches which may have a disproportionately greater effecton the measurement are sampled with increased probability.3. Each parent nuclide can be split into a user-defined number of nuclides(of proportionally lower statistical weight) prior to treating decay in order to increase the sampling of the effects of the daughter products.34.3 Status of this document00.00.00 created by ?21.11.03 bibliography added, minor re-wording by D.H. WrightBibliography[1] J. Tuli, ”Evaluated Nuclear Structure Data File,” BNL-NCS-51655-Rev87, 1987.[2] Chapter 25, Geant4 Physics Reference Manual.[3] P.R. Truscott, PhD Thesis, University of London, 1996.432