PENELOPE 2003 - OECD Nuclear Energy Agency

More documents

Recommendations

Info

58 Chapter 2. Photon interactions Table 2.2: Reduced screening radius, Rm e c/¯h, and high-energy triplet contribution, η ∞ , for electron-positron pair production obtained from the tables of Hubbell et al. (1980) as described in the text. Notice that ¯h/m e c = 3.8616 × 10 −13 m is the Compton wavelength of the electron. Z Rm e c/¯h η ∞ Z Rm e c/¯h η ∞ Z Rm e c/¯h η ∞ 1 122.81 1.157 32 33.422 1.158 63 26.911 1.194 2 73.167 1.169 33 33.068 1.157 64 26.705 1.196 3 69.228 1.219 34 32.740 1.158 65 26.516 1.197 4 67.301 1.201 35 32.438 1.158 66 26.304 1.196 5 64.696 1.189 36 32.143 1.158 67 26.108 1.197 6 61.228 1.174 37 31.884 1.166 68 25.929 1.197 7 57.524 1.176 38 31.622 1.173 69 25.730 1.198 8 54.033 1.169 39 31.438 1.174 70 25.577 1.198 9 50.787 1.163 40 31.142 1.175 71 25.403 1.200 10 47.851 1.157 41 30.950 1.170 72 25.245 1.201 11 46.373 1.174 42 30.758 1.169 73 25.100 1.202 12 45.401 1.183 43 30.561 1.172 74 24.941 1.204 13 44.503 1.186 44 30.285 1.169 75 24.790 1.205 14 43.815 1.184 45 30.097 1.168 76 24.655 1.206 15 43.074 1.180 46 29.832 1.164 77 24.506 1.208 16 42.321 1.178 47 29.581 1.167 78 24.391 1.207 17 41.586 1.175 48 29.411 1.170 79 24.262 1.208 18 40.953 1.170 49 29.247 1.172 80 24.145 1.212 19 40.524 1.180 50 29.085 1.174 81 24.039 1.215 20 40.256 1.187 51 28.930 1.175 82 23.922 1.218 21 39.756 1.184 52 28.721 1.178 83 23.813 1.221 22 39.144 1.180 53 28.580 1.179 84 23.712 1.224 23 38.462 1.177 54 28.442 1.180 85 23.621 1.227 24 37.778 1.166 55 28.312 1.187 86 23.523 1.230 25 37.174 1.169 56 28.139 1.194 87 23.430 1.237 26 36.663 1.166 57 27.973 1.197 88 23.331 1.243 27 35.986 1.164 58 27.819 1.196 89 23.238 1.247 28 35.317 1.162 59 27.675 1.194 90 23.139 1.250 29 34.688 1.154 60 27.496 1.194 91 23.048 1.251 30 34.197 1.156 61 27.285 1.194 92 22.967 1.252 31 33.786 1.157 62 27.093 1.194
2.4. Electron-positron pair production 59 where 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 + 0.002623a) ln 4 (4/κ). (2.89) Then, the single quantity η ∞ characterizes the triplet production for each element. The approximation given by eq. (2.77) with the fitted value of the screening radius, fails at low energies where it systematically underestimates the total cross section (it can even become negative). To compensate for this fact we introduce an empirical correcting term F 0 (κ, Z), which acts in a way similar to the Coulomb correction. To facilitate the random sampling, the Bethe-Heitler DCS, eq. (2.77), including this lowenergy correction and a high-energy radiative correction, is written in the form [ ( ) dσ pp = r 2 2 1 2 ] dɛ eαZ[Z + η] C r 2 3 2 − ɛ φ 1 (ɛ) + φ 2 (ɛ) , (2.90) where with φ 1 (ɛ) = g 1 (b) + g 0 (κ) φ 2 (ɛ) = g 2 (b) + g 0 (κ) (2.91) g 1 (b) = 1 2 (3Φ 1 − Φ 2 ) − 4 ln(Rm e c/¯h) = 7 3 − 2 ln(1 + b2 ) − 6b arctan(b −1 ) − b 2 [ 4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 ) ] , g 2 (b) = 1 4 (3Φ 1 + Φ 2 ) − 4 ln(Rm e c/¯h) = 11 6 − 2 ln(1 + b2 ) − 3b arctan(b −1 ) + 1 2 b2 [ 4 − 4b arctan(b −1 ) − 3 ln(1 + b −2 ) ] , g 0 (κ) = 4 ln(Rm e c/¯h) − 4f C (Z) + F 0 (κ, Z). (2.92) C r = 1.0093 is the high-energy limit of Mork and Olsen’s radiative correction (Hubbell et al., 1980). The correcting factor F 0 (κ, Z) has been determined by requiring that the total cross section for pair production obtained from the expression given in eq. (2.90) (with η = 0) coincides with the total cross sections for pair production in the field of the nucleus tabulated by Hubbell et al. (1980). By inspection and numerical fitting, we have obtained the following analytical approximation 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 . (2.93)
Page 1 and 2:
Data Bank ISBN 92-64-02145-0 PENELO
Page 3 and 4:
FOREWORD The OECD/NEA Data Bank was
Page 5 and 6:
TABLE OF CONTENTS Foreword ........
Page 7 and 8:
4.3 Combined scattering and energy
Page 9 and 10:
PREFACE Radiation transport in matt
Page 11 and 12:
The present version of PENELOPE is
Page 13 and 14:
Chapter 1 Monte Carlo simulation. B
Page 15 and 16:
1.1. Elements of probability theory
Page 17 and 18:
1.1. Elements of probability theory
Page 19 and 20: 1.2. Random sampling methods 7 Tabl
Page 21 and 22: 1.2. Random sampling methods 9 Nume
Page 23 and 24: 1.2. Random sampling methods 11 whe
Page 25 and 26: 1.2. Random sampling methods 13 can
Page 27 and 28: 1.2. Random sampling methods 15 mus
Page 29 and 30: 1.2. Random sampling methods 17 the
Page 31 and 32: 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 69: 2.4. Electron-positron pair product
Page 73 and 74: 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 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
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 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
4.4. Generation of random tracks 14
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Chapter 5 Constructive quadric geom
Page 167 and 168:
5.1. Rotations and translations 155
Page 169 and 170:
5.2. Quadric surfaces 157 Given a f
Page 171 and 172:
5.2. Quadric surfaces 159 Table 5.1
Page 173 and 174:
5.3. Constructive quadric geometry
Page 175 and 176:
5.4. Geometry definition file 163 1
Page 177 and 178:
5.4. Geometry definition file 165
Page 179 and 180:
5.5. The subroutine package pengeom
Page 181 and 182:
5.5. The subroutine package pengeom
Page 183 and 184:
5.6. Debugging and viewing the geom
Page 185 and 186:
5.7. A short tutorial 173 MODULE (
Page 187 and 188:
5.7. A short tutorial 175 Writing a
Page 189 and 190:
Chapter 6 Structure and operation o
Page 191 and 192:
6.1. penelope 179 and/or numbers in
Page 193 and 194:
6.1. penelope 181 1986). The format
Page 195 and 196:
6.1. penelope 183 The connection of
Page 197 and 198:
6.1. penelope 185 that has to be lo
Page 199 and 200:
G y y y y y 6.1. penelope 187 CALL
Page 201 and 202:
6.1. penelope 189 It is the respons
Page 203 and 204:
6.2. Examples of MAIN programs 191
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
6.3. Selecting the simulation param
Page 215 and 216:
! 6.3. Selecting the simulation par
Page 217 and 218:
6.5. Installation 205 radiation is
Page 219 and 220:
6.5. Installation 207 To get the ex
Page 221 and 222:
Appendix A Collision kinematics To
Page 223 and 224:
A.1. Two-body reactions 211 Clearly
Page 225 and 226:
A.2. Inelastic collisions of charge
Page 227 and 228:
m A.2. Inelastic collisions of char
Page 229 and 230:
Appendix B Numerical tools B.1 Cubi
Page 231 and 232:
B.1. Cubic spline interpolation 219
Page 233 and 234:
B.2. Numerical quadrature 221 B.2 N
Page 235 and 236:
Appendix C Electron/positron transp
Page 237 and 238:
C.1. Tracking particles in vacuum.
Page 239 and 240:
C.1. Tracking particles in vacuum.
Page 241 and 242:
C.2. Exact tracking in homogeneous
Page 243 and 244:
C.2. Exact tracking in homogeneous
Page 245 and 246:
Bibliography Abramowitz M. and I.A.
Page 247 and 248:
Bibliography 235 Briesmeister J.F.
Page 249 and 250:
Bibliography 237 James F. (1990),
Page 251 and 252:
Bibliography 239 Reimer L. and E.R.
Page 253:
Bibliography 241 Yates A.C. (1968),
show all

PENELOPE 2003 - OECD Nuclear Energy Agency

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?