PENELOPE 2003 - OECD Nuclear Energy Agency

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104 Chapter 3. Electron and positron interactions cross sections obtained from an optical-data model of the GOS (Mayol and Salvat, 1990). This model assumes the following relationship between the optical oscillator strength (OOS) of the i-th inner shell, df i (W )/dW , and the photoelectric cross section σ ph,i (Z, W ) for absorption of a photon with energy W , df i (W ) dW = m ec 2π 2 e 2¯h σ ph,i(Z, W ). (3.121) This equality holds when the dipole approximation is applicable, i.e. when the wavelength of the photon is much larger than the “size” of the active electron shell. In the calculations we use the photoelectric cross sections from the penelope database, which were extracted from the EPDL (Cullen et al., 1997). The GOS is modelled as a continuous superposition of δ-oscillators weighted by the OOS, df i (Q, W ) dW = ∫ ∞ U i df i (W ′ ) dW ′ F (W ′ ; Q, W ) dW ′ + Z r δ(W − Q) Θ(W − U i ) (3.122) with Z r = Z i − ∫ ∞ This GOS model satisfies the Bethe sum rule ∫ ∞ U i df i (W ′ ) dW ′ dW ′ . (3.123) U i df i (Q, W ) dW dW = Z i if Q > U i . (3.124) For Q < U i the integral of the GOS over W is less than the number Z i of electrons in the active shell (because there is a transfer of oscillator strength to outer shells). As shown by Mayol and Salvat (1990), this model is formally equivalent to the Weizsäcker- Williams method of virtual quanta (see e.g. Jackson, 1975). To compute the inner-shell ionization (“si”) cross section for electrons or positrons of energy E we first consider the energy-loss DCS, which is given by dσ (±) si,i (E) dW + ( Z r + [ ( 1 W Q − + 2m e c 2 ) ( ) ] 1 ln + ln − β 2 W Q − W + 2m e c 2 1 − β 2 df i (W ′ ) } ) 1 dW ′ U i dW ′ W F (±) (E, W ) Θ(W − U 2 i ) Θ(W max − U i ), (3.125) { = 2πe4 dfi (W ) m e v 2 dW ∫ W where Q − is the minimum recoil energy for the energy loss W , eq. (A.31), and F (±) (E, W ) are the Møller (−) and Bhabha (+) factors, eqs. (3.70) and (3.74). It is well known that the Born approximation overestimates the ionization cross sections for incident electrons with kinetic energies near the ionization threshold. This is mainly due to the distortion of the projectile wave function by the electrostatic field of the target atom. This field produces an increase in the effective kinetic energy of the projectile, which is expected to be important in close collisions. To account for this
3.2. Inelastic collisions 105 1Ε+4 1Ε+6 Ar (Z=18) L shells (E) (barn) 1Ε+5 1Ε+4 L shells (E) (barn) 1Ε+3 (−) σ si,i 1Ε+3 K shell (−) σ si,i 1Ε+2 K shell Ag (Z=47) 1Ε+2 1Ε+2 1Ε+4 1Ε+6 1Ε+8 E (eV) 1Ε+3 L shells 1Ε+1 1Ε+3 1Ε+5 1Ε+7 1Ε+9 E (eV) L shells 1Ε+2 (E) (barn) 1Ε+2 K shell (E) (barn) K shell (−) σ si,i 1Ε+1 (−) σ si,i 1Ε+1 Au (Z=79) U (Z=92) 1 1Ε+4 1Ε+6 1Ε+8 E (eV) 1 1Ε+4 1Ε+6 1Ε+8 E (eV) Figure 3.12: Cross sections for ionization of the K shell and the L subshells of argon, silver, gold and uranium atoms by electron impact as functions of the kinetic energy of the projectiles. Solid curves are results from the present optical data model. Circles represent cross-section values calculated by Scofield (1978) using the relativistic plane-wave Born approximation. The dashed curves are cross sections for ionization by impact of positrons calculated from eq. (3.126b).
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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
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1.4. Simulation of radiation transp
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¡ ¢ 1.4. Simulation of radiation
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1.5. Statistical averages and uncer
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1.6. Variance reduction 31 also imp
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1.6. Variance reduction 33 weight a
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Chapter 2 Photon interactions In th
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2.1. Coherent (Rayleigh) scattering
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2.1. Coherent (Rayleigh) scattering
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2.2. Photoelectric effect 41 ➤E E
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F 2.2. Photoelectric effect 43 1E+7
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2.3. Incoherent (Compton) scatterin
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C u 2.3. Incoherent (Compton) scatt
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Page 65 and 66: 2.3. Incoherent (Compton) scatterin
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 and 84: 3.1. Elastic collisions 71 nucleus
Page 85 and 86: 3.1. Elastic collisions 73 1 E - 1
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: 3.2. Inelastic collisions 103 In re
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
Page 135 and 136: Chapter 4 Electron/positron transpo
Page 137 and 138: 4.1. Elastic scattering 125 Lewis (
Page 139 and 140: 4.1. Elastic scattering 127 ⎡ ⎛
Page 141 and 142: 4.1. Elastic scattering 129 simulat
Page 143 and 144: 4.1. Elastic scattering 131 The sim
Page 145 and 146: 4.2. Soft energy losses 133 soft in
Page 147 and 148: 4.2. Soft energy losses 135 deviati
Page 149 and 150: 4.2. Soft energy losses 137 scatter
Page 151 and 152: 4.3. Combined scattering and energy
Page 159 and 160: 4.4. Generation of random tracks 14
Page 165 and 166: 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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