PENELOPE 2003 - OECD Nuclear Energy Agency

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72 Chapter 3. Electron and positron interactions are the direct and spin-flip scattering amplitudes, respectively. k ≡ p¯h = 1 [ E(E + 2me c 2 ) ] 1/2 ¯hc (3.7) is the wave number of the projectile, P l (cos θ) are Legendre polynomials, Pl 1 (cos θ) are associated Legendre functions and δ l± are the phase shifts. These are determined from the asymptotic behaviour of the Dirac radial functions for large r (Walker, 1971). Thus, to determine each phase shift we must solve the radial Dirac equations for the potential V (r). The convergence of the partial-wave series (3.6) slows down when the energy of the projectile increases. This makes the calculation difficult for energies larger than a few MeV (in the case of scattering by gold atoms, about 10,000 phase shifts are required at E = 10 MeV). The partial-wave DCS, eq. (3.5), rigourously accounts for spin and other relativistic effects, as well as finite nuclear size effects. A computer code to calculate elastic scattering cross sections by this method has been written by Salvat (2000). Single elastic collisions are determined by the values of the polar and azimuthal scattering angles, θ and φ, respectively. Owing to the assumed spherical symmetry of the scattering centres, single and multiple scattering angular distributions are axially symmetrical about the direction of incidence, i.e. they are independent of the azimuthal scattering angle φ. For simulation purposes, it is convenient to measure polar angular deflections produced by single scattering events in terms of the variable [see eq. (1.61)] µ = (1 − cos θ)/2 (3.8) instead of the scattering angle θ. Notice that µ varies from 0 (forward scattering) to 1 (backward scattering). The DCS per unit angular deflection is dσ el dµ = 4π dσ el dΩ . (3.9) Fig. 3.2 displays DCSs for elastic scattering of electrons and positrons of various energies by aluminium and gold atoms. These numerical results illustrate the variation of the DCS with the atomic number Z, the charge of the projectile and the energy E. Since the interaction V (r) is attractive for electrons and repulsive for positrons, the scattering is more intense for electrons (which can fall deeply into the potential well of the atom) than for positrons (which are repelled from the nucleus and cannot “see” the inner part of the atom). The DCS for low-energy electrons exhibits a diffraction-like structure, while the DCS for positrons decreases monotonously with the deflection µ. The Born approximation (see e.g. Mott and Massey, 1965) predicts a structureless DCS that decreases with µ and is proportional to the squared charge of the projectile (i.e. the same for electrons and positrons). This approximation considers the scattering field as a perturbation (to first order) and, hence, it is valid only for weak fields (low-Z elements). The difference between the (partial wave) DCSs for electrons and positrons gives a clear indication of the applicability of the Born approximation. The total elastic cross section is given by ∫ ∫ dσel σ el = dΩ dΩ = dσel dµ dµ. (3.10)
3.1. Elastic collisions 73 1 E - 1 2 1 E - 1 3 1 E - 1 4 1 E - 1 5 dσ el /dµ (cm 2 ) 1 E - 1 6 1 E - 1 7 1 E - 1 8 1 E - 1 9 1 E - 20 1 E - 21 1 keV 10 10 0 1 E - 22 1 E - 23 A l ( Z = 13 ) el ec t r o n s p o s i t r o n s 10 0 0 1 E - 24 1 E - 6 1 E - 5 1 E - 4 1 E - 3 0.01 0.2 0.4 0.6 0.8 1 .0 µ 1 E - 1 2 1 E - 1 3 1 E - 1 4 1 E - 1 5 dσ el /dµ (cm 2 ) 1 E - 1 6 1 E - 1 7 1 E - 1 8 1 E - 1 9 1 keV 10 10 0 1 E - 20 1 E - 21 1 E - 22 A u ( Z = 7 9 ) el ec t r o n s p o s i t r o n s 10 0 0 1 E - 6 1 E - 5 1 E - 4 1 E - 3 0.01 µ 0.2 0.4 0.6 0.8 1 .0 Figure 3.2: DCS for elastic scattering of electrons and positrons by aluminium and gold atoms as a function of the deflection µ = (1 − cos θ)/2. Notice the change from logarithmic to linear scale at µ = 0.05.
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Data Bank ISBN 92-64-02145-0 PENELO
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FOREWORD The OECD/NEA Data Bank was
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TABLE OF CONTENTS Foreword ........
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4.3 Combined scattering and energy
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PREFACE Radiation transport in matt
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The present version of PENELOPE is
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Chapter 1 Monte Carlo simulation. B
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1.1. Elements of probability theory
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1.1. Elements of probability theory
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1.2. Random sampling methods 7 Tabl
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1.2. Random sampling methods 9 Nume
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1.2. Random sampling methods 11 whe
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1.2. Random sampling methods 13 can
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1.2. Random sampling methods 15 mus
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1.2. Random sampling methods 17 the
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1.3. Monte Carlo integration 19 Con
Page 33 and 34: 1.4. Simulation of radiation transp
Page 37 and 38: ¡ ¢ 1.4. Simulation of radiation
Page 41 and 42: 1.5. Statistical averages and uncer
Page 43 and 44: 1.6. Variance reduction 31 also imp
Page 45 and 46: 1.6. Variance reduction 33 weight a
Page 47 and 48: Chapter 2 Photon interactions In th
Page 49 and 50: 2.1. Coherent (Rayleigh) scattering
Page 51 and 52: 2.1. Coherent (Rayleigh) scattering
Page 53 and 54: 2.2. Photoelectric effect 41 ➤E E
Page 55 and 56: F 2.2. Photoelectric effect 43 1E+7
Page 57 and 58: 2.3. Incoherent (Compton) scatterin
Page 59 and 60: C u 2.3. Incoherent (Compton) scatt
Page 67 and 68: 2.4. Electron-positron pair product
Page 75 and 76: 2.5. Attenuation coefficients 63 di
Page 77 and 78: 2.6. Atomic relaxation 65 2.6 Atomi
Page 79 and 80: 2.6. Atomic relaxation 67 two vacan
Page 81 and 82: Chapter 3 Electron and positron int
Page 83: 3.1. Elastic collisions 71 nucleus
Page 87 and 88: ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ
Page 89 and 90: 3.1. Elastic collisions 77 becomes
Page 91 and 92: B ' B ' 3.1. Elastic collisions 79
Page 93 and 94: 3.2. Inelastic collisions 81 where
Page 95 and 96: 3.2. Inelastic collisions 83 global
Page 97 and 98: 3.2. Inelastic collisions 85 plays
Page 99 and 100: 3.2. Inelastic collisions 87 optica
Page 101 and 102: 3.2. Inelastic collisions 89 The DC
Page 103 and 104: 3.2. Inelastic collisions 91 In the
Page 105 and 106: 3.2. Inelastic collisions 93 where
Page 107 and 108: c b c b 3.2. Inelastic collisions 9
Page 109 and 110: A l A u 3.2. Inelastic collisions 9
Page 111 and 112: 3.2. Inelastic collisions 99 After
Page 113 and 114: 3.2. Inelastic collisions 101 Table
Page 115 and 116: 3.2. Inelastic collisions 103 In re
Page 117 and 118: 3.2. Inelastic collisions 105 1Ε+4
Page 119 and 120: 3.3. Bremsstrahlung emission 107 an
Page 121 and 122: 3.3. Bremsstrahlung emission 109 1
Page 123 and 124: 3.3. Bremsstrahlung emission 111 1
Page 125 and 126: 3.3. Bremsstrahlung emission 113 wh
Page 127 and 128: 3.3. Bremsstrahlung emission 115 Al
Page 129 and 130: 3.3. Bremsstrahlung emission 117 Th
Page 131 and 132: 3.4. Positron annihilation 119 (198
Page 133 and 134: 3.4. Positron annihilation 121 It i
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Chapter 4 Electron/positron transpo
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4.1. Elastic scattering 125 Lewis (
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4.1. Elastic scattering 127 ⎡ ⎛
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4.1. Elastic scattering 129 simulat
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4.1. Elastic scattering 131 The sim
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4.2. Soft energy losses 133 soft in
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4.2. Soft energy losses 135 deviati
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4.2. Soft energy losses 137 scatter
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4.3. Combined scattering and energy
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4.4. Generation of random tracks 14
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Chapter 5 Constructive quadric geom
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5.1. Rotations and translations 155
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5.2. Quadric surfaces 157 Given a f
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5.2. Quadric surfaces 159 Table 5.1
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5.3. Constructive quadric geometry
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5.4. Geometry definition file 163 1
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5.4. Geometry definition file 165
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5.5. The subroutine package pengeom
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5.5. The subroutine package pengeom
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5.6. Debugging and viewing the geom
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5.7. A short tutorial 173 MODULE (
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5.7. A short tutorial 175 Writing a
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Chapter 6 Structure and operation o
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6.1. penelope 179 and/or numbers in
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6.1. penelope 181 1986). The format
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6.1. penelope 183 The connection of
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6.1. penelope 185 that has to be lo
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G y y y y y 6.1. penelope 187 CALL
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6.1. penelope 189 It is the respons
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6.2. Examples of MAIN programs 191
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6.3. Selecting the simulation param
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! 6.3. Selecting the simulation par
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6.5. Installation 205 radiation is
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6.5. Installation 207 To get the ex
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Appendix A Collision kinematics To
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A.1. Two-body reactions 211 Clearly
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A.2. Inelastic collisions of charge
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m A.2. Inelastic collisions of char
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Appendix B Numerical tools B.1 Cubi
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B.1. Cubic spline interpolation 219
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B.2. Numerical quadrature 221 B.2 N
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Appendix C Electron/positron transp
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C.1. Tracking particles in vacuum.
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C.1. Tracking particles in vacuum.
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C.2. Exact tracking in homogeneous
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C.2. Exact tracking in homogeneous
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Bibliography Abramowitz M. and I.A.
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Bibliography 235 Briesmeister J.F.
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Bibliography 237 James F. (1990),
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Bibliography 239 Reimer L. and E.R.
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Bibliography 241 Yates A.C. (1968),
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