THE EGS5 CODE SYSTEM1

Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANAlex F. Bielajew and Scott J. Wilderman
Department of Nuclear Engineering and Radiological Sciences
The University of Michigan
2355 Bonisteel Boulevard
Ann Arbor, MI 48109, USA
Walter R. Nelson
Department Associate in the Radiation Physics Group (retired)
Radiation Protection Department
Stanford Linear Accelerator Center
Stanford University
2575 Sand Hill Road, Menlo Park, CA 94025, USA

SLAC Report number: SLAC-R-730
KEK Report number: 2005-8

Date of this version: January 13, 2016

1

Work supported by the US Department of Energy under DE-AC02-76SF00515

Stanford University Notices for
SLAC Manual SLAC-R-730
and its included software
known as the EGS5 Code System

Acknowledgement of sponsorship: This manual and its contents, including software, were
produced in part by the Stanford Linear Accelerator Center (SLAC), Stanford University, underContract DE-AC02-76SFO0515 with the U.S. Department of Energy.
Use: The manual and its included software should be used for non-commercial purposes only.
Contact SLAC regarding commercial use.
Government disclaimer of liability: Neither the United States nor the United States Department of Energy, nor any of their employees, makes any warranty, express or implied, or assumes any
legal liability or responsibility for the accuracy, completeness, or usefulness of any data, apparatus,
product, or process disclosed, or represents that its use would not infringe privately owned rights.
Stanford disclaimer of liability: Stanford University makes no representations or warranties,
express or implied, nor assumes any liability for the use of this manual or its contents, including
software.
Maintenance of notices: In the interest of clarity regarding the origin and status of this SLAC
manual and its included software, this and all the preceding Stanford University notices are to:(1) remain affixed to any copy or derivative of this manual or its software made or distributed by
the recipient of this manual or its software; and (2) be affixed to any copy of a document or any
software made or distributed by the recipient that contains a copy or derivative of this manual orits software.
——————————————————
From SLAC Software Notices, Set 3
OTT.002a, 2004 FEB 03——————————————————KEK Reports are available from:
Science Information and Library Services DivisionHigh Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba-shi
Ibaraki-ken, 305-0801
JAPAN

i

Phone:Fax:
E-Mail:Internet:+81-29-864-5137+81-29-864-4604
irdpub@mail.kek.jp
http://www.kek.jp

Contents

1 INTRODUCTION1

1.1

Intent of This Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

History of EGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2.1

Before EGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2.2

EGS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.3

EGS2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.4

EGS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.5

EGS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

Overview of the EGS4 Code System – Vintage 1985 . . . . . . . . . . . . . . . . . . 10

1.4

Improvements to EGS Since 1985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1

Version 5’s Improvements and Enhancements to EGS Electron Physics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2

Improvements and Enhancements to EGS Photon Physics Modeling in Version 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3

Other Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 RADIATION TRANSPORT IN EGS5
2.1

20

Description of Radiation Transport-Shower Processii. . . . . . . . . . . . . . . . . . 20

2.2

Probability Theory and Sampling Methods—A Short Tutorial. . . . . . . . . . . . 21

2.3

Simulating the Physical Processes—An Overview

2.4

Particle Transport Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5

Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6

General Implementation Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7

Bremsstrahlung and Electron-Positron Pair Production. . . . . . . . . . . . . . . . . . . 26

. . . . . . . . . . . . . . . . 37

2.7.1

Bremsstrahlung Photon Angular Distribution . . . . . . . . . . . . . . . . . . 54

2.7.2

Pair Angle Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.8Interactions With Atomic Electrons – General Discussion . . . . . . . . . . . . . . . 61

2.9

Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.10 Møller Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.11 Bhabha Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.12 Two Photon Positron-Electron Annihilation . . . . . . . . . . . . . . . . . . . . . . . 69
2.13 Continuous Electron Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.14 Multiple Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.14.1 The Molière Multiple Scattering Distribution . . . . . . . . . . . . . . . . . . 87
2.14.2 The Goudsmit-Saunderson Multiple Scattering Distribution . . . . . . . . . . 96
2.15 Transport Mechanics in EGS5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.15.1 Random Hinge Transport Mechanics . . . . . . . . . . . . . . . . . . . . . . . 102
2.15.2 Modified Random Hinge Transport Mechanics. . . . . . . . . . . . . . . . . 104

2.15.3 Dual Random Hinge Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.15.4 Boundary Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.15.5 EGS5 Transport Mechanics Algorithm . . . . . . . . . . . . . . . . . . . . . . 109iii

2.15.6 Electron Step-Size Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.15.7 Energy Hinge Step-Size Determination in PEGS . . . . . . . . . . . . . . . . 113
2.15.8 Multiple Scattering Step-size Specification Using Fractional Energy Loss Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.15.9 Multiple Scattering Step-Size Optimization Using Media “Characteristic Dimensions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.15.10 Treatment of Initial Steps of Primary Electrons . . . . . . . . . . . . . . . . . 123
2.16 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.16.1 General Treatment of Photoelectric-Related Phenomena . . . . . . . . . . . . 126
2.16.2 Photoelectron Angular Distribution . . . . . . . . . . . . . . . . . . . . . . . 132
2.17 Coherent (Rayleigh) Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.18 Binding Effects and Doppler Broadening in Compton Scattering . . . . . . . . . . . 134
2.19 Scattering of Linearly Polarized Photons . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.20 Electron Impact Ionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3 A SERIES OF SHORT EGS5 TUTORIALS

145

3.1

Tutorial 1 (Program tutor1.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.2

Tutorial 2 (Program tutor2.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

3.3

Tutorial 3 (Program tutor3.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.4

Tutorial 4 (Program tutor4.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.5

Tutorial 5 (Program tutor5.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.6

Tutorial 6 (Program tutor6.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.7

Tutorial 7 (Program tutor7.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

3.8

Tutorial 8 (Program tutor8.f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

iv

4 ADVANCED EGS5 USER CODES
4.1

204

UCCYL - Cylinder-Slab Geometry and Importance Sampling . . . . . . . . . . . . . 204
4.1.1

Generalized Multi-Cylinder, Multi-Slab Geometry . . . . . . . . . . . . . . . 204

4.1.2

Particle Splitting

4.1.3

Leading Particle Biasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

4.2

UCBEND - Charged Particle Transport in a Magnetic Field. . . . . . . . . . . . . 209

4.3

Using Combinatorial Geometry with EGS5 . . . . . . . . . . . . . . . . . . . . . . . 213

APPENDICES:218

A EGS5 FLOW DIAGRAMS218

B EGS5 USER MANUAL

328

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
B.2 General Description of Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 329
B.3 Variables in EGS5 COMMON Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
B.4 Sequence of Actions Required of User Code MAIN . . . . . . . . . . . . . . . . . . . . 332
B.4.1 Pre-PEGS5 Initializations (Step 1) . . . . . . . . . . . . . . . . . . . . . . . . . 332
B.4.2 PEGS5 Call (Step 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
B.4.3 Pre-HATCH Initializations (Step 3) . . . . . . . . . . . . . . . . . . . . . . . . . 342
B.4.4 Specification of Incident Particle Parameters (Step 4) . . . . . . . . . . . . . 348
B.4.5 HATCH Call (Step 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
B.4.6 Initializations for HOWFAR (Step 6) . . . . . . . . . . . . . . . . . . . . . . . . 350
B.4.7 Initializations for AUSGAB (Step 7) . . . . . . . . . . . . . . . . . . . . . . . . 350v

B.4.8 SHOWER Call (Step 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
B.4.9 Output of Results (Step 9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
B.5 Specifications for HOWFAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

B.5.1 Sample HOWFAR User Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
B.6 Specifications for AUSGAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

B.7 UCSAMPL5 — An Example of a “Complete” EGS5 User Code . . . . . . . . . . . . 361
C PEGS USER MANUAL

370

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
C.2 Structural Organization of PEGS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
C.3 PEGS Options and Input Specifications . . . . . . . . . . . . . . . . . . . . . . . . . 383
C.3.1 Interrelations Between Options . . . . . . . . . . . . . . . . . . . . . . . . . . 383
C.3.2 The ELEM, MIXT, COMP Options . . . . . . . . . . . . . . . . . . . . . . . 388
C.3.3 The ENER Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
C.3.4 The PWLF Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
C.3.5 The DECK Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
C.3.6 The TEST Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
C.3.7 The CALL Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
C.3.8 The PLTI and PLTN Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
C.3.9 The HPLT Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
C.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
D EGS5 INSTALLATION GUIDE

408

D.1 Installation of EGS5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

viD.2 Sample Scripts for Running EGS5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
E CONTENTS OF THE EGS5 DISTRIBUTION

416

E.1 Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
E.2 EGS-Related FORTRAN Source Files . . . . . . . . . . . . . . . . . . . . . . . . . . 417
E.3 PEGS-Related FORTRAN Source Files . . . . . . . . . . . . . . . . . . . . . . . . . 418
E.4 Material Data Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
E.5 Sample User Codes and Run Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
E.6 Auxiliary Subprogram FORTRAN Source Files . . . . . . . . . . . . . . . . . . . . . 424

vii

List of Figures
2.1

Program flow and data control in EGS5. . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2

Feynman diagrams for bremsstrahlung and pair production. . . . . . . . . . . . . . . 37

2.3

Feynman diagrams for two body interactions with electrons. . . . . . . . . . . . . . . 61

2.4

Definition of two-body scattering angles. . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5

Feynman diagram for single photon e+ e− annihilation. . . . . . . . . . . . . . . . . . 73

2.6

Plots of Molière functions f (0) , f (1) , and f (2) . . . . . . . . . . . . . . . . . . . . . . . 92

2.7

Plot of Equation 2.290 (B − `n B = b). . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.8

Schematic of electron transport mechanics model. . . . . . . . . . . . . . . . . . . . . 101

2.9

Random hinge transport mechanics schematic. . . . . . . . . . . . . . . . . . . . . . 103

2.10 Modified random hinge transport mechanics schematic.

. . . . . . . . . . . . . . . . 105

2.11 Dual (energy and angle) hinge transport mechanics schematic.

. . . . . . . . . . . . 106

2.12 Electron transport across region boundaries. . . . . . . . . . . . . . . . . . . . . . . . 108
2.13 Translation steps and transport steps for energy loss hinges. . . . . . . . . . . . . . . 110
2.14 Electron boundary crossing during translation steps. . . . . . . . . . . . . . . . . . . 112
2.15 Schematic illustrating the “broomstick” problem. . . . . . . . . . . . . . . . . . . . . 118
2.16 Schematic illustrating the modified “broomstick” problem as used in EGS5. . . . . . 119

viii

2.17 Convergence of energy deposition as a function of step-size (in terms of fractionalenergy loss) for the broomstick problem with varying diameters D in copper at 5 MeV.121
2.18 Convergence of average lateral displacement as function of step-size (in terms of fractional energy loss) for the broomstick problem with varying diameters D in copperat 5 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.19 Optimal initial scattering strength K1 vs. broomstick diameter (equivalent to thecharacteristic dimension) in titanium at various energies. . . . . . . . . . . . . . . . . 123
2.20 Optimal initial scattering strength K1 vs. broomstick diameter for various elements
at 100 MeV. The upper figure is for values of ρt greater than 0.1, and the lower figure
for smaller characteristic dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.21 Cu GMFP values evaluated by PWLF and LEM . . . . . . . . . . . . . . . . . . . . 127
2.22 Comparison of measured and calculated intensity of K x-rays.

. . . . . . . . . . . . 128

2.23 Photon scattering system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.24 Direction of the polarization vector of the scattered photon . . . . . . . . . . . . . . 139
2.25 Direction of k~0 and e~0 after two rotations by A−1 . . . . . . . . . . . . . . . . . . . . 142
4.1

Diagram used with UCBEND (not to scale). . . . . . . . . . . . . . . . . . . . . . . . 211

4.2

UCBEND simulation at 8.5 MeV (B=2.6 kG).

. . . . . . . . . . . . . . . . . . . . . 212

4.3

UCBEND simulation at 3.5 MeV (B=1.0 kG).

. . . . . . . . . . . . . . . . . . . . . 213

4.4

UCBEND simulation at 8.5 MeV (B=0 kG).

. . . . . . . . . . . . . . . . . . . . . . 214

4.5

Geometry and particle trajectory of UCSAMPCG simulation. . . . . . . . . . . . . . 216

B.1 EGS5 user code control and data flow diagram. . . . . . . . . . . . . . . . . . . . . . 331
B.2 A three-region geometry for a HOWFAR example code. . . . . . . . . . . . . . . . . . . 355
C.1 Flowchart of the PEGS5 subprogram of PEGS, part 1. . . . . . . . . . . . . . . . . . . 373
C.2 Flowchart of the PEGS5 subprogram of PEGS, part 2. . . . . . . . . . . . . . . . . . . 374
C.3 Subprogram relationships in PEGS, part 1. . . . . . . . . . . . . . . . . . . . . . . . 375
ix

C.4 Subprogram relationships in PEGS, part 2. . . . . . . . . . . . . . . . . . . . . . . . 376
C.5 Bremsstrahlung related functions—most accurate form (used to produce the total
cross sections and stopping power). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
C.6 Bremsstrahlung related functions—with run-time approximations (for comparison
with sampled spectra). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
C.7 Pair production related functions—most accurate form (used to produce the total
cross sections and stopping power). . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
C.8 Pair production related functions—with run-time approximations (for comparison
With sampled spectra). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
C.9 Logical relationship between the options of PEGS. . . . . . . . . . . . . . . . . . . . 388

xList of Tables
2.1

Symbols used in EGS5 and PEGS5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2

Default atomic numbers, symbols, weights, densities and I values in PEGS. . . . . . 76

2.3

Default Sternheimer density effect coefficients in PEGS. . . . . . . . . . . . . . . . . 78

2.4

Materials used in reference tables of scattering strength vs. characteristic dimension
at various energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.5

GMFP of Cu at Kβ1 (8.905 keV) and Kβ2 (8.977 keV) energies. . . . . . . . . . . . . . 129

2.6

Data sources for generalized treatment of photoelectric-related phenomena in EGS5. 130

2.7

L x-ray energies and representative intensities (relative to Lα1 ) for lead. . . . . . . . 131

2.8

Total cross section (10−24 cm2 /molecule) for coherent scattering from water.

2.9

Formulas used in various simulation modes employing detailed treatment of Compton
and Rayleigh scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143. . . . 133

B.1 Variable descriptions for COMMON block BOUNDS, include file egs5 bounds.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
B.2 Variable descriptions for COMMON block BREMPR, include file egs5 brempr.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
B.3 Variable descriptions for COMMON block COUNTERS, include file counters.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
B.4 Variable descriptions for COMMON block EDGE2, include file egs5 edge.f of the EGS5
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

xiB.5 Variable descriptions for COMMON block EIICOM, include file egs5 eiicom.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
B.6 Variable descriptions for COMMON block EPCONT, include file egs5 epcont.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
B.7 Variable descriptions for COMMON block MEDIA, include file egs5 media.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
B.8 Variable descriptions for COMMON block MISC, include file egs5 misc.f of the EGS5
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
B.9 Variable descriptions for COMMON block MS, include file egs5 ms.f of the EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
B.10 Variable descriptions for COMMON block RLUXDAT, include file randomm.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
B.11 Variable descriptions for COMMON block STACK, include file egs5 stack.f of the EGS5
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
B.12 Variable descriptions for COMMON block THRESH, include file egs5 thresh.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
B.13 Variable descriptions for COMMON block UPHIOT, include file egs5 uphiot.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
B.14 Variable descriptions for COMMON block USEFUL, include file egs5 useful.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
B.15 Variable descriptions for COMMON block USERSC, include file egs5 usersc.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
B.16 Variable descriptions for COMMON block USERVR, include file egs5 uservr.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
B.17 Variable descriptions for COMMON block USERXT, include file egs5 userxt.f of the
EGS5 distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
B.18 IARG values program status for default AUSGAB calls. . . . . . . . . . . . . . . . . . . 358
B.19 IARG values, IAUSFL indices, and program status for AUSGAB calls. . . . . . . . . . . . 360
C.1 Subroutines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

xiiC.2 Goudsmit-Saunderson-related subroutines in PEGS. . . . . . . . . . . . . . . . . . . 382
C.3 Functions in PEGS, part 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
C.4 Functions in PEGS, part 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
C.5 Functions in PEGS, part 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
C.6 Functions in PEGS, part 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
C.7 ELEM option input data lines in PEGS, part 1. . . . . . . . . . . . . . . . . . . . . . 389
C.8 ELEM option input data lines in PEGS, part 2. . . . . . . . . . . . . . . . . . . . . . 390
C.9 COMP option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 391
C.10 MIXT option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 392
C.11 ENER option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 393
C.12 PWLF option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 394
C.13 DECK option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 395
C.14 TEST option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 395
C.15 CALL option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 395
C.16 PLTI option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . . 395
C.17 PLTN option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 396
C.18 HPLT option input data lines in PEGS. . . . . . . . . . . . . . . . . . . . . . . . . . 396
C.19 Distribution functions available with PLTI and PLTN. . . . . . . . . . . . . . . . . . 405

xiii

PREFACE
In the nineteen years since EGS4 was released, it has been used in a wide variety of applications,particularly in medical physics, radiation measurement studies, and industrial development. Every
new user and every new application bring new challenges for Monte Carlo code designers, andcode refinements and bug fixes eventually result in a code that becomes difficult to maintain.
Several of the code modifications represented significant advances in electron and photon transportphysics, and required a more substantial invocation than code patching. Moreover, the arcane
MORTRAN3[48] computer language of EGS4, was highest on the complaint list of the users of
EGS4. The size of the EGS4 user base is difficult to measure, as there never existed a formal user
registration process. However, some idea of the numbers may be gleaned from the number of EGS4
manuals that were produced and distributed at SLAC: almost three thousand. Consequently, the
EGS5 project was undertaken. It was decided to employ the FORTRAN 77 compiler, yet include
as much as possible, the structural beauty and power of MORTRAN3.
We wish to acknowledge Patrick Lui (Technology Transfer Office at SLAC) for administrating
the business end of this project and securing sufficient funds to enable its completion. We wish toacknowledge Dr. Ray Cowan (MIT/SLAC BaBar) for converting the SLAC-265 document, written
in TEX/Psizzl to LATEX. SLAC 265 served as the foundation upon which this document is built.
Finally, we wish to acknowledge Prof. David W. O. Rogers, co-author of EGS4, for his many
years of EGS4 development and Monte Carlo leadership. We also wish to thank Dr. James Liu(SLAC Radiation Physics Group) for helping to create and run the suite of benchmarking codes
for EGS5. Dr. Francesc Salvat’s assistance in developing the sampling methods for the multiple
elastic scattering distributions is most gratefully acknowledged.
This report consists of four chapters and several appendices. Chapter 1 is an introduction to
EGS5 and to this report in general. We suggest that you read it. Chapter 2 is a major update ofsimilar chapters in the old EGS4 report[126] (SLAC-265) and the old EGS3 report[61] (SLAC-210),
in which all the details of the old physics (i.e., models which were carried over from EGS4) andthe new physics are gathered together. The descriptions of the new physics are extensive, and not
for the faint of heart! Detailed knowledge of the contents of Chapter 2 is not essential in order
to use EGS, but sophisticated users should be aware of its contents. In particular, details of therestrictions on the range of applicability of EGS are dispersed throughout the chapter. First-timeusers of EGS should skip Chapter 2 and come back to it later if necessary.

xivWith the release of the EGS4 version, a deliberate attempt was made to present example
problems in order to help the user “get started,” and we follow that spirit in this report. A seriesof elementary tutorial user codes are presented in Chapter 3, with more sophisticated sample user
codes described in Chapter 4. Novice EGS users will find it helpful to read through the initial
sections of the EGS5 User Manual (provided in Appendix B of this report), proceeding then towork through the tutorials in Chapter 3.
The User Manuals and other materials found in the appendices contain detailed flow charts,
variable lists, and subprogram descriptions of EGS5 and PEGS. Included are step-by-step instructions for developing basic EGS5 user codes and for accessing all of the physics options available
in EGS5 and PEGS. Once acquainted with the basic structure of EGS5, users should find theappendices the most frequently consulted sections of this report.
H. H.
Y. N.
A. F. B.
S. J. W.
W. R. N.
December 2005

xv

Chapter 1

INTRODUCTION
1.1

Intent of This Report

The EGS (Electron-Gamma Shower) code system is a general purpose package for the Monte Carlosimulation of the coupled transport of electrons and photons in an arbitrary geometry for particles
with energies above a few keV up to several hundred GeV (depending on the atomic numbers of
the target materials). This report introduces a new, enhanced version called EGS5. In addition
to explaining and documenting the various enhancements and changes to the previous version(EGS4[126]), this document includes several introductory and advanced tutorials on the use of
EGS5, and also contains the EGS5 User Manual. Our intention has been to make this document
wholly self-contained so that the user need not refer to the original EGS4 manual (SLAC-265[126])
in order to use the code. To this end, we have taken the liberty of incorporating into Chapter 2of this report those portions of Chapter 2 of SLAC-265 which describe physics models of EGS4
retained by EGS5, thereby documenting all the physics contained in EGS5.
On the practical side this report can be used in order to create EGS5 user codes and to run EGS5.
Formerly a stand-alone data preparation code, PEGS (Preprocessor for EGS), is now integrated inthis version of EGS.

1.2

History of EGS

A great deal of the content of the following historical sections of this report has been plagiarized
unashamedly from the original EGS3 document (SLAC-210) authored by Richard Ford and Ralph
Nelson[61]. There are several reasons for this aside from laziness. This history predates one of theauthor’s (AFB) involvement with EGS and he found it very difficult to improve upon the words
penned by Ford and Nelson in that original document. Moreover, the EGS3 manual is now out-1

of-print and this history might have eventually been lost to the ever-burgeoning EGS-community
now estimated to be at least 7000 strong. There had been one previous attempt to give a historicalperspective of EGS[125]. However, this article was very brief and did not convey the large effort
that went into the development of EGS. In this report the historical section on EGS4 as well as
the summary of EGS3 to EGS4 conversion and the overview of EGS4 was taken directly from the
EGS4 manual[126]. The technical descriptions herein are necessarily brief. The new physics of
EGS5 is described in detail in the next Chapter. However, the reader should consult the references
cited in this report for more details regarding motivation and implementation.

1.2.1

Before EGS

The Monte Carlo method was originally suggested by Ulam and von Neumann[174], and was firstused by Goldberger[62] in order to study nuclear disintegrations produced by high-energy particles. The first application of the Monte Carlo technique to study shower production was doneby Wilson[179]. Wilson’s approach was a simple graphical-mechanical one that was described asfollows:“The procedure used was a simple graphical and mechanical one. The distance into thelead was broken into intervals of one-fifth of a radiation length (about 1 mm). The electronsor photons were followed through successive intervals and their fate in passing through agiven interval was decided by spinning a wheel of chance; the fate being read from one of afamily of curves drawn on a cylinder. . .
A word about the wheel of chance; The cylinder, 4 in. outside diameter by 12 in. long isdriven by a high speed motor geared down by a ratio 20 to 1. The motor armature is heavier
than the cylinder and determines where the cylinder stops. The motor was observed to stopat random and, in so far as the cylinder is concerned, its randomness is multiplied by thegear ratio. . . ”

from R. R. Wilson, op. cit.

Although apparently quite tedious, Wilson’s method was still an improvement over the analytic
methods of the time—particularly in studying the average behavior and fluctuations about the
average[141].
The first use of an electronic digital computer in simulating high-energy cascades by Monte
Carlo methods was reported by Butcher and Messel[38, 39], and independently by Varfolomeev andSvetlolobov[176]. These two groups collaborated in a much publicized work[104] that eventually
led to an extensive set of tables describing the shower distribution functions[103]—the so-called2

“shower book”.
For various reasons two completely different codes were written in the early-to-mid-1960’s tosimulate electromagnetic cascades. The first was written by Zerby and Moran[180, 181, 182] at
the Oak Ridge National Laboratory and was motivated by the construction of the Stanford Linear
Accelerator Center and the many physics and engineering problems that were anticipated as aresult of high-energy electron beams showering in various devices and structures at that facility.
This code had been used by Alsmiller and others[5, 6, 7, 2, 8, 3, 4] for a number of studies since
its development1 .
The second code was originally developed by Nagel[115, 112, 113] and several adaptations had
been reported[177, 128, 37]. The original Nagel version, which Ford and Nelson called SHOWER1,
was a FORTRAN code written for high-energy electrons (≤ 1000 MeV) incident upon lead in
cylindrical geometry. Six significant electron and photon interactions (bremsstrahlung, electronelectron scattering, ionization-loss, pair-production, Compton scattering, and the photoelectriceffect) plus multiple Coulomb scattering were accounted for. Except for annihilation, positronsand electrons were treated alike and were followed until they reached a cutoff energy of 1.5 MeV(total energy). Photons were followed down to 0.25 MeV. The cutoff energies were as low as orlower than those used by either Messel and Crawford or by Zerby and Moran.
The availability of Nagel’s dissertation[112] and a copy of his original shower program provided
the incentive for Nicoli[128] to extend the dynamic energy range and flexibility of the code in orderfor it to be made available as a practical tool for the experimental physicist. Nicoli’s modificationsof SHOWER1 fell into three categories:
1. High-energy extensions to the least-squares fits for total interaction probabilities and branching ratios.
2. Provisions for including boundary-condition interrogation in the transport cycle, allowing forparticle marking and/or discarding and the use of generalized energy cutoffs for electrons andphotons.
3. The handling of input/output requirements.
In August, 1966 the Nicoli version (SHOWER2) was brought to SLAC by Nagel, who had beenworking at MIT and had consulted with Nicoli on the above changes and extensions. The SLAC
Computation Group undertook the task of getting the code running on the IBM-360 system andgeneralizing the program to run in elemental media other than just lead. The latter was facilitatedby a set of hand-written notes—brought to SLAC in 1966 by Nagel[114]—on the best way toaccomplish this task and V. Whitis was assigned the job. Whitis left SLAC in the summer of1967 and his work, which consisted mainly of a series of fitting programs written in the ALGOL
language, was passed on to one of us (WRN)2 . Under Nelson’s direction, a programmer (J. Ryder)1

According to Alsmiller[1], the Zerby and Moran source code vanished from ORNL and they were forced to workwith an octal version.
2
Nagel’s computer program was recovered from a trash receptacle at SLAC by WRN. Although no printout of thecode could be found, the punch cards had been sequenced in columns 73-80 and they were easily sorted by machine.

3

constructed SHOWER3 in modular form and wrote a pre-processing code called PREPRO thatcomputed fit-coefficients for the cross-section and branching-ratio data needed by SHOWER3. The
values of these constants depended on the material in which the shower was to be simulated. During
the summer of 1972 the Ryder version of SHOWER3/PREPRO was successfully tested for several
different elements by B. Talwar under the direction of Nelson, thus bringing SHOWER3/PREPRO
into an operational status.
Meanwhile, interest in a computer code capable of simulating electromagnetic cascade showers
had been developing for several years at the High Energy Physics Laboratory (HEPL) at Stanford
University where a group led by R. Hofstadter and E. B. Hughes was continuing their development oflarge NaI(Tl) Total Absorption Shower Counters (TASC’s)[60]. A method of accurately predicting
shower behavior in these counters was needed. A version of Nagel’s code (SHOWER2) was obtained
from Nelson in the fall of 1970; however, efforts to scale from lead to NaI were uncertain and led
to a growing conviction that a generalized code was necessary. Thus it was that Richard Ford
undertook the task of generalizing SHOWER2 to run in any element, mixture, or compound in
September, l971—an effort similar to the one already underway by Ryder, Talwar, and Nelson that
resulted in the final version of SHOWER3/PREPRO.
Ford obtained a copy of a Ryder version of SHOWER3/PREPRO and Nagel’s notes from Nelson. In addition to the references mentioned in Nagel’s notes, the Messel and Crawford “showerbook”[103], as well as the review by Scott[147] on multiple scattering, were found to be very useful sources of information. The essential physics was formulated and the coding was completed
by February, 1972. At that time the HEPL version was called SHOWER (now referred to as
SHOWER4) and the corresponding preprocessor was completely new and was called SHINP (forSHower INPut). Both codes were in FORTRAN and were made operational on the IBM-7700
machine at HEPL—a second generation experimental data-acquisition computer. A number ofinteresting studies were subsequently performed, including calculations of detector resolutions andexpected self-vetoes in gamma detectors due to backscattered photons from shower detectors downstream.
In January, 1974 it appeared likely that HEPL’s computer would be sold. In addition, theHofstadter group was involved in an experiment at SLAC that required shower simulations andthe SHOWER4/SHINP codes were therefore made operational on the considerably faster and more
efficient IBM-360/91 at SLAC. During the calculations that had been performed at HEPL, a coupleof errors were found in the sampling routines that would have been detected earlier if it had beenpossible to test this in a more systematic way. Therefore, it was decided to incorporate into the
new version being brought to SLAC test facilities to insure the correctness of these sampling routines. In order to facilitate comparison between the sampled secondary spectra and the theoretical
distributions, the preprocessing code was split up and modularized into subprograms.

41.2.2

EGS1

About this time Nelson became interested in being able to use Ford’s version of the code and offered
to help support its further development. One of Ford’s objectives was to make the preprocessor
code produce data for the shower code in a form that was directly usable by the shower code witha minimum of input required by the user. In SHOWER3/PREPRO and in SHOWER4/SHINP,
whenever it was desired to create showers in a new medium, it was necessary to look-up the photon
cross sections in the literature and keypunch them for the preprocessing code to use. Subsequent tothis it was necessary to select from several fits produced by the preprocessing code and to include
this new information, consisting of many data cards, with other data used by the shower program.
Ford rewrote the preprocessor to automatically produce all of the data needed by the shower codein a readily acceptable form and, with the assistance of Nelson, obtained photon cross sections forelements 1 to 100 from Storm and Israel[167] on magnetic tape. Ford also separated the showercode’s material-input from its control-input. For flexibility and ease of use, the NAMELIST readfacility of FORTRAN-IV was utilized for reading-in control data in both the preprocessor and theshower codes. The resultant shower code was re-named EGS (Electron-Gamma-Shower) and its
companion code was called PEGS (Preprocessor for EGS). This version, written completely in the
FORTRAN-IV language, is referred to as Version 1 of the EGS Code System (or more simply EGS1
and PEGS1).
The sampling routines were tested using the internal test-procedure facility of EGS1 and, withthe exception of the bremsstrahlung process, were found to be operating very nicely. In the bremsstrahlung case a ripple, amounting to only 5% but still noticeable, was observed when the sampled
data were compared with the theoretical secondary distribution. This effect went away upon selection of another random number generator, and it was concluded that correlations in the original
number generator were the cause. EGS1 was then tested against various experiments in the literature and with other Monte Carlo results that were then available and the authors found reasonable
good agreement in all cases.

1.2.3

EGS2

By the fall of 1974 the Hofstadter group had obtained some hexagonal modular NaI detectors
and the discovery of the J/ψ particle[12, 13] in November, 1974 opened up an exciting area ofhigh-energy gamma-ray spectroscopy for which the modularized NaI detectors were ideally suited.
EGS1, however, could not be readily used to simulate showers in complex geometries such as thosepresented by modular stacks of NaI. A good example of this was the Crystal Ball detector for which
EGS1, under the direction of E. Bloom at SLAC, was modified to handle the particular geometry
in question. Furthermore, Nelson had received a large number of requests from the growing listof EGS users, both at SLAC and elsewhere in the high-energy physics community, to improvefurther EGS1 so that complex geometries could be realized in the near future. Thus it was decided
that EGS1, which was a one-region, one-medium code, should be generalized in order to handle
many-region, many-media, complex, three-dimensional geometries.

5It soon became clear that, in the time available at least, it would not be possible to constructa self-contained code that would have all of the control, scoring, and output options that might
ever be wanted, as well as a geometry package that would automatically handle arbitrary complex
geometries. Therefore, Ford decided to put in only the necessary multi-region structures, to replace
all scoring and output code in EGS1 with a user interface, and to dispense with the EGS1 main
control program completely. Thus EGS1 became a subprogram in itself with two user-callable
subroutines (HATCH and SHOWER) that require two user-written subroutines (HOWFAR and AUSGAB)
in order to define the geometry and do the scoring, respectively.
For added flexibility and portability, EGS1 and PEGS1 were rewritten in an extended FORTRAN language called MORTRAN2, which was translated by a (MORTRAN2) Macro Processor
into standard FORTRAN. The part of EGS1 that was used to test the sampling routines was reconfigured into a separate main program called the TESTSR code, also written in MORTRAN2. These
revisions were completed by the end of 1975 and the new versions of EGS, PEGS, and TESTSR
comprise what is called Version 2 of the EGS Code System, or more simply EGS2, PEGS2, andTESTSR2.

1.2.4

EGS3

One part of EGS2 which seemed aesthetically displeasing was the complex control logic neededin the electron transport routine, ELECTR, in order to transport electrons by variable distances
to interaction points or boundaries using only step lengths taken from a set of 16 discrete values. This procedure had been necessary in order to implement Nagel’s discrete reduced-angle
multiple-scattering scheme[115, 112, 113, 114] in a general multi-region environment. In addition,comparisons of backscattered photon fluence as computed by EGS2 versus unpublished HEPL data,
as well as bremsstrahlung angular distribution calculations comparing EGS2 results with those of
Berger and Seltzer using ETRAN[19], suggested that EGS2 might be predicting values in the backward direction that were low by up to a factor of two. For these reasons, and in order to achieve
greater universality of application (e.g., so that a monoenergetic beam of electrons impinging on avery thin slab would have a continuous angular distribution on exit), Ford decided in the summer of1976 to try to implement a multiple scattering-scheme that would correctly sample the continuous
multiple-scattering distribution for arbitrary step lengths. After some thought, an extension of themethod used by Messel and Crawford[103] was devised. Most of the code for this addition was
written by Ford at Science Applications, Inc., and was brought to SLAC in August 1977 where
it was debugged and tested by Nelson and Ford. The implementation of this system required
some once-only calculations which were made using a stand-alone code called CMS (Continuous
Multiple Scattering)3 . It should be mentioned that the version of PEGS brought to SLAC at thistime had the same physics in it as Version 2, but had been partly rewritten in order to be moremachine independent (e.g., IBM versus CDC), its main remaining machine dependency being its
use of NAMELIST. (NAMELIST is a common extension to FORTRAN employed by many FORTRAN
compilers but is not part of the FORTRAN-IV or FORTRAN 77 standards.) Another option wasadded to the TESTSR code to allow testing of the new EGS multiple-scattering sampling routine,3

Logically, the CMS code should have been included as an option of PEGS, but this has never been done.

6MSCAT.
These versions of EGS, PEGS, and TESTSR comprise what was called Version 3 of the EGS
code system (i.e., EGS3, PEG3, and TESTSR3). Subsequent comparisons of EGS3 calculations
against experiments and other Monte Carlo results were made by the authors (e.g., see SLAC210[61] and/or Jenkins and Nelson[127]) and others and the agreements clearly demonstrated thebasic validity of the code.
The EGS3 Code System released in 1978 contained many features that distinguished it fromNagel’s original code, SHOWER1, the most noteworthy being:
1. Showers could be simulated in any element (Z=1 through 100), compound, or mixture.
2. The energy range for transporting particles was extended so that showers could be initiatedand followed from 100 GeV down to 1 keV for photons, and 1.5 MeV (total energy) for charged
particles.
3. Photons and charged particles were transported in random rather than discrete steps, resulting in a much faster running code.
4. Positrons were allowed to annihilate either in-flight or at rest, and their annihilation quanta
were followed to completion.
5. Electrons and positrons were treated separately using exact, rather than asymptotic, Møller
and Bhabha cross sections, respectively.
6. Sampling schemes were made more efficient.
7. EGS3 became a subroutine package with user interface, allowing much greater flexibility andreducing the necessity for being familiar with the internal details of the code. This also
reduced the likelihood that user edits could introduce bugs into the code.
8. The geometry had to be specified by the user by means of a user-written subprogram called
HOWFAR. However, geometry utilities for determining intersections of trajectories with common
surfaces (e.g., planes, cylinders, cones, spheres and boxes) had also been developed and weremade available.
9. The task of creating media data files was greatly simplified and automated by means of the
PEGS3 preprocessing code, which created output data in a convenient form for direct use by
EGS3.
10. PEGS3 constructed piecewise-linear fits over a large number of energy intervals of the crosssection and branching-ratio data, whereas PREPRO and SHINP both made high-order polynomial fits over a small number of intervals (as did SHOWER1 and SHOWER2).
11. In addition to the options needed to produce data for EGS3, options were made available in
PEGS3 for plotting any of the physical quantities used by EGS3, as well as for comparing
sampled distributions from the TESTSR user code with theoretical spectra. The NAMELIST
read facility of FORTRAN was also introduced at this time.
In particular, for Version 3 versus Version 2
12. The multiple-scattering reduced angle was sampled from a continuous rather than discrete
distribution. This was done for arbitrary step sizes provided that they were not too large toinvalidate the theory. An immediate application of this was the following simplification to7

subroutine ELECTR.
13. The control logic in the charged-particle transport routine, ELECTR, was greatly simplified
and modifications were made to both ELECTR and the photon transport routine, PHOTON, tomake interactions at a boundary impossible.
14. The above changes to the control logic then made it possible for the user to implement
importance-sampling4 techniques into EGS without any furthers “internal” changes to thesystem itself. Examples that come to mind include the production of secondary electron beams
at large angles, photon energy deposition in relatively small (low-Z) absorbers, and deep
penetration (radial and longitudinal) calculations associated with shower counter devices.
15. Provision was made for allowing the density to vary continuously in any given region.
16. A new subroutine (PHOTO) was added in order to treat the photoelectric effect in a mannercomparable to the other interaction processes. The main interest in this was to facilitate
the development of a more general photoelectric routine, such as one that could produce
fluorescent photons and/or Auger electrons for subsequent transport by EGS.
17. Additional calls to AUSGAB, bringing the total from 5 to 23, were made possible in order
to allow for the extraction of additional information without requiring the user to edit the
EGS code itself. For example, the user could determine the number of collision types (e.g.,Compton vs.photoelectric, etc.).
Upon release in 1978, the EGS3 Code System soon became the “industry standard” for designing shower detectors in high-energy physics. Looking back at this period of time several reasons can
explain why EGS became so popular so quickly. Leading the list was the fact that the other codes
mentioned above simply were not available; whereas, anyone could get EGS, together with its documentation, free-of-charge from SLAC. Furthermore, the code had been successfully benchmarked
and support was provided to anyone requesting help. These things provided the fuel for the fire.
What ignited it, however, was the so-called November Revolution[12, 13] of particle physics andthe resulting shift to the use of colliding-beam accelerators. In particular, there was an immediateneed by the high-energy physics community for tools to aid in the design of shower counters for thelarge, vastly-complicated, 4π detector systems associated with the new colliding-beam storage-ringfacilities under construction throughout the world. EGS was there at the right time and right placewhen this happened.
We would be remiss if we did not mention one other code that also was available during this time
period, particularly since published results from it had been used as part of the benchmarking of
EGS3 itself. We refer to the ETRAN Monte Carlo shower code written by Berger and Seltzer[19]5 .
ETRAN treated the low-energy processes (down to 1 keV) in greater detail than EGS3. Insteadof the Molière[107, 108] formulation, ETRAN made use of the Goudsmit-Saunderson[63, 64] approach to multiple scattering, thereby avoiding the small-angle approximations intrinsic to Molière.
ETRAN also treated fluorescence, the effect of atomic binding on atomic electrons, and energy-loss
straggling. Because of the special care taken at low energies, ETRAN, which was initially writ4

For those who may be unfamiliar with the term, importance sampling refers to sampling the most importantregions of a problem and correcting for this bias by means of weight factors (see, for example, the report by Carterand Cashwell[41].)
5
A later version of this program, which contained a fairly general geometry package, was known as SANDYL[46]and was also available at this time.

8ten for energies less than 100 MeV and later extended to 1 GeV[16], ran significantly slower than
EGS3. However, in spite of its accuracy and availability, ETRAN went unnoticed in the world ofhigh-energy physics during this period.

1.2.5

EGS4

EGS3 was designed to simulate electromagnetic cascades in various geometries and at energies up
to a few thousand GeV and down to cutoff kinetic energies of 0.1 MeV (photons) and 1 MeV(electrons and positrons). However, ever since the introduction of the code in 1978 there had been
an increasing need to extend the lower-energy limits—i.e., down to 1 and 10 keV for photons andelectrons, respectively. Essentially, EGS3 had become more and more popular as a general, lowenergy, electron-photon transport code that could be used for a variety of problems in addition to
those generally associated with high-energy electromagnetic cascade showers. It had many featuresthat made it both general as well as versatile, and it was relatively easy to use, so there had been arapid increase in the use of EGS3 both by those outside the high-energy physics community (e.g.,
medical physics) and by those within. Even though other low-energy radiation transport codes wereavailable, most notably ETRAN[15, 18, 19] and its progeny[46, 68], there had been many requests
to extend EGS3 down to lower energies and this was a major, but not the only, reason for creating
EGS4. The various corrections, changes and additions, and new features that were introduced in
the 1985 release of the EGS4 Code System[126] are summarized below.

Summary of EGS3 to EGS4 conversion
As with any widely used code, there had been many extensions made to EGS3 and many small
errors found and corrected as the code was used in new situations. The following lists the mostsignificant differences between EGS3 and EGS4.
• Major Changes and Additions to EGS3.

– Conversion from MORTRAN2 to MORTRAN3.
– Corrections to logic and coding errors in EGS3.
– Extension of electron transport down to 10 keV (kinetic energy).
– Improved Sternheimer treatment of the density effect.
– Improved definition of the radiation length at low atomic numbers.

• New Options and Macros.

– Macro templates for introduction of weighting and biasing techniques.
– Pi-zero option.
– Rayleigh scattering option.
– Compton electron stack position preference (macro).
– Positron discard option (macro) for creation of annihilation gammas.

• Auxiliary Subprograms and Utilities.
9

– Geometry subprograms and corresponding macro packages.
– Miscellaneous energy conservation and event counter utility routines.
– Fixed fractional energy loss subroutines.
• New Applications and Examples.

– Leading-particle biasing macro to increase efficiency.
– Fluorescent-photon transport capability.
– Charged-particle transport in magnetic fields.
– Combinatorial Geometry package.
– Coupling of hadronic and electromagnetic cascade codes.

The most significant changes were made to subroutine ELECTR to correct problems which occurred when lower-energy charged-particle transport was done. The most significant change in thisregard was first brought to attention in the paper “Low energy electron transport with EGS” by
Rogers[136]. Many of the difficulties with the low-energy transport related to the fact that electrontransport sub-steps (multiple scattering and continuous energy loss are modeled at the endpointsof these steps) were too large and various approximations that were valid for high-energy transport(above 10–20 MeV) were invalid for low-energy. Rogers modified the EGS code to allow the user
to control the electron step-size in two ways, one by specifying a maximum allowable energy loss tocontinuous energy-loss processes (ESTEPE) and a geometric step-size control (SMAX) that restricts
the electron step-size to be no larger than some user-specified distance. This allowed low-energy
electron transport to be calculated with some degree of confidence although the user was required
to study the parametric dependence of applications on these two parameters, ESTEPE and SMAX.

1.3

Overview of the EGS4 Code System – Vintage 1985

The following is a summary of the main features of the EGS4 Code System, including statements
about the physics that has been put into it and what can be realistically simulated.
• The radiation transport of electrons (+ or -) or photons can be simulated in any element,
compound, or mixture. That is, the data preparation package, PEGS4, creates data to be
used by EGS4, using cross section tables for elements 1 through 100.
• Both photons and charged particles are transported in random rather than in discrete steps.
• The dynamic range of charged-particle kinetic energies goes from a few tens of keV up to a
few thousand GeV. Conceivably the upper limit can be extended higher, but the validity ofthe physics remains to be checked.
• The dynamic range of photon energies lies between 1 keV and several thousand GeV (seeabove statement).
• The following physics processes are taken into account by the EGS4 Code System:– Bremsstrahlung production (excluding the Elwert correction at low energies).
– Positron annihilation in flight and at rest (the annihilation quanta are followed to completion).
10

– Molière multiple scattering (i.e., Coulomb scattering from nuclei). The reduced angle issampled from a continuous (rather than discrete) distribution. This is done for arbitrary
step sizes, selected randomly, provided that they are not so large or so small as toinvalidate the theory.
– Møller (e− e− ) and Bhabha (e+ e− ) scattering. Exact rather than asymptotic formulas
are used.
– Continuous energy loss applied to charged-particle tracks between discrete interactions.
∗ Total stopping power consists of soft bremsstrahlung and collision loss terms.
∗ Collision loss determined by the (restricted) Bethe-Bloch stopping power with Sternheimer treatment of the density effect.
– Pair production.
– Compton scattering.
– Coherent (Rayleigh) scattering can be included by means of an option.
– Photoelectric effect.
∗ Neither fluorescent photons nor Auger electrons are produced or transported in thedefault version of subroutine PHOTO.
∗ Other user-written versions of PHOTO can be created, however, that allow for the
production and transport of K- and L-edge photons.
• PEGS4 is a stand-alone data preprocessing code consisting of 12 subroutines and 85 functions.
The output is in a form for direct use by EGS4.
– PEGS4 constructs piecewise-linear fits over a large number of energy intervals of thecross section and branching ratio data.
– In general, the user need only use PEGS4 once to obtain the media data files requiredby EGS4.
– PEGS4 control input uses the NAMELIST read facility of the FORTRAN language (inMORTRAN3 form).
– In addition to the options needed to produce data for EGS4, PEGS4 contains options
to plot any of the physical quantities used by EGS4, as well as to compare sampled
distributions produced by the UCTESTSR User Code with theoretical spectra.
– This allows for greater flexibility without requiring one to be overly familiar with theinternal details of the code.
– Together with the macro facility capabilities of the MORTRAN3 language, this reduces
the likelihood that user edits will introduce bugs into the code.
– EGS4 uses material cross section and branching ratio data created and fit by the companion code, PEGS4.
• The geometry for any given problem is specified by a user-written subroutine called HOWFAR
which, in turn, can make use of auxiliary subprograms.
– Auxiliary geometry routines for planes, cylinders, cones, spheres, etc., are provided with
the EGS4 Code System for those who do not wish to write their own.
– Macro versions of these routines are also provided in the set of defining macros (i.e., in
the EGS4MAC file) which, if used, generally result in a faster running simulation.
– The MORSE-CG Combinatorial Geometry package can be incorporated into HOWFAR(e.g., see the UCSAMPCG file on the EGS4 Distribution Tape). However, experience11

indicates that a much slower simulation generally results (of the order of at least a factor
of four).
– Transport can take place in a magnetic field by writing a specially designed HOWFAR
subprogram, or in a more general manner (e.g., including electric field) by making useof MORTRAN3 macro templates that have been appropriately placed for that purposein subroutine ELECTR.
• The user scores and outputs information in the user-written subroutine called AUSGAB.

– Auxiliary subprogram ECNSV1 is provided in order to keep track of energy for conservation (or other) purposes.
– Auxiliary subprogram NTALLY is provided in order to keep track of the number of times
energy has been scored into the ECNSV1 arrays (i.e., an event counter).
– Auxiliary subprogram WATCH is provided in order to allow an event-by-event or step-bystep tracking of the simulation.

• EGS4 allows for the implementation of importance sampling and other variance reduction
techniques (e.g., leading particle biasing, splitting, path length biasing, Russian roulette,etc.).
• Initiation of the radiation transport:– An option exists for initiating a shower with two photons from pi-zero decay (i.e., use
IQI with a value of 2 in the CALL SHOWER statement).
– The user has the choice of initiating the transport by means of a monoenergetic particle,
or by sampling from a known distribution (e.g., a synchrotron radiation spectrum).
– Transport can also be initiated from sources that have spatial and/or angular distributions.

1.4

Improvements to EGS Since 1985

In this section the improvements to EGS since the Version 4 release in December 1985 are described briefly. Only marginal detail is provided and the interested reader is encouraged to consult
the references cited for deeper explanation. Note that many of the updates described here were incorporated in versions of EGS4 that were somewhat generally available, while other improvements
are unique to the current release of EGS5.

1.4.1

Version 5’s Improvements and Enhancements to EGS Electron Physics
Modeling

Electron transport mechanics
A dual random hinge approach, in which energy loss and multiple elastic scattering are fully
decoupled[34], has been adopted for modeling the spatial transport of electrons and positrons in
12

EGS5. The primary advantages of this technique lie in that the random multiple scattering hingepreserves near second order spatial moments of the transport equation over long step lengths[88],and that the hinge mechanics can be formulated so as to permit transport across boundaries between regions of differing media. Additionally, we believe, but have yet to prove analytically, thatthe use of the dual hinges preserves the energy dependent spatial moments.
The algorithm begins with the selection of a distance tE for an energy step and a distance tθ
to the next multiple scattering event. Generalizing the random hinge methodology of PENELOPE[56], both the energy loss and the multiple scattering steps are comprised of two segments. The
lengths of the initial segments are determined by multiplying random numbers ζE and ζθ by tE andtθ , respectively, and the lengths of the second segments are simply the remaining distances. Unlike
EGS4, in which multiple scattering deflection is applied at the end of each step, in EGS5, multiple
scattering is applied at a random “hinge” point (a distance ζθ tθ along the track) during each step.
Likewise, the energy of the electron remains unchanged along its track until the energy “hinge” at
ζE tE is reached. At this point, the electron energy is decremented by the full amount prescribedin the continuous slowing down approximation (CSDA) model for traversing the total pathlengthtE . The two hinge mechanisms are completely independent, as several energy hinges may occur
before a single multiple scattering is applied, and vice versa. In evaluating the multiple scattering
distribution and sampling for the deflection, all parameters which have an energy dependence
are computed using the most recently updated energy for the particle. There is thus an implicit
averaging of the electron energies over the track length, as energy hinges will occur sometimes
before scattering hinges and sometimes after.
Subsequent to reaching a hinge point and undergoing either energy loss or deflection, the electron
must then be transported the distance remaining for the given step, (1 − ζE )tE for energy hinges or(1 − ζθ )tθ for multiple scattering hinges. Once the end of the step is reached, the next step begins
with the determination of a new value of tE or tθ , respectively, a random distance to the next hinge
point is determined and the process is repeated.

Elimination of PRESTA’s boundary crossing logic
The PRESTA (Parameter Reduced Electron Step Transport Algorithm) algortithm was implemented in EGS4[31, 32, 33] to permit accurate treatment of electron transport over long steps.
PRESTA changed the EGS4 transport mechanics model in four principal areas:
1. A refined calculation of the average curvature of the electron sub-step.
2. A lateral correlation algorithm was introduced that added an extra lateral component to thesub-step, correlating it to the multiple-scattering angle selected at the end of the sub-step.
3. A boundary crossing algorithm was introduced. This algorithm causes electron sub-steps to
become shorter in the vicinity of boundaries insuring that no transport artifacts will occur
near interfaces.
4. More careful energy averaging over the electron sub-step was introduced with subsequent
refinements[101, 138].
13

Although the transport mechanics algorithm in PRESTA contains good corrections for longitudinal and lateral displacements (as well as average energy), it has been abandoned in EGS5.
This change was motivated primarily by the desire to eliminate PRESTA’s boundary algorithm,
which, though theoretically the best way for handling elastic-scattering ambiguities in the vicinityof boundaries, is difficult to couple to general purpose combinatorial geometry codes, which may
contain quadric and higher-order surfaces[28]. In the case of quadric surfaces, the determination of
the closest distance to a boundary may involve solving for the roots of a sixth-order polynomial.
While such algorithms may be devised, specifically designed with this application in mind[132],
they are too slow for routine application.
Thus, in addition to providing improved treatment of longitudinal and lateral displacement, the
new transport mechanics of EGS5 also yields advantages in computational speed compared to mostmodels using a boundary sensitive approach. Although applications requiring boundary sensitivity
must exist, we have yet to encounter any in practice. This is certainly an area that would benefit
from more attention, as a boundary sensitive component could be added without much effort.
Moreover, the current EGS5 transport mechanics could be adapted easily to most tracking codes,
which perform ray tracing in combinatorial geometries without boundary sensitivity.

Bremsstrahlung angular distribution
Bielajew et al.[29] modified EGS4 to allow for angular distributions employing the Schiff formulafrom a review article by Koch and Motz[91]. Standard EGS4 makes the approximation that theangle of the bremsstrahlung photon with respect to the initiating charged particle’s direction isΘ = 1/E0 where E0 is the initiating charged particle’s energy in units of the electron rest massenergy. It was acknowledged that this might be a bad approximation for thin-target studies, but
it was expected that there would be no effect in thick-target studies since multiple scattering
would “wash-out” the initial bremsstrahlung angular distribution and that an average value would
be sufficient. However, thick-target studies in the radiotherapy range showed dramatic evidence of
this approximation as a calculation artifact[54]. Angular distributions near the central axis changed
by as much as 40%! Thick-target studies at diagnostic energies also showed the artifact which waseliminated through use of the new sampling technique[124]. This new capability was carried overto EGS5.

ICRU37 collision and radiative stopping powers
Duane et al.[52] modified PEGS4 to give collision stopping powers identical to those of ICRU Report
37[20, 79]. The NBS (now NIST) database EPSTAR[148] which was used to create the ICRU tables
was employed. The modifications also allow the user to input easily an arbitrary density-effectcorrection. This change is relatively small but crucial if one is doing detailed stopping-power-ratio
studies[101, 137].
In a related work, Rogers et al.[139] adapted PEGS4 to make the radiative stopping powers
14

ICRU37-compliant using the NIST database ESPA[148]. Effectively, this modification globallyrenormalizes EGS4’s bremsstrahlung cross section so that the integral of the cross section (theradiative stopping power) agrees with that of ICRU Report 37[20, 79]. This improvement can leadto noticeable changes in the bremsstrahlung cross section for particle energies below 50 MeV[54]and significant differences for energies below a few MeV where bremsstrahlung production is verysmall[124]. These new capabilities were carried over to EGS5.

Low-energy elastic electron cross section modeling
It has long been acknowledged that the Molière multiple scattering distribution used in EGS4
breaks down under certain conditions. In particular: the basic form of the cross section assumed
by Molière is in error in the MeV range, when spin and relativistic effects are important; various
approximations in Molière’s derivation lead to significant errors at pathlengths less than 20 elastic
scattering mean free paths; and the form of Molière’s cross section is incapable of accurately
modeling the structure in the elastic scattering cross section at large angles for low energies andhigh atomic number. It is therefore desirable to have available a more exact treatment, and in
EGS5, we use (in the energy range from 1 keV to 100 MeV) elastic scattering distributions derived
from a state-of-the-art partial-wave analysis (an unpublished work) which includes virtual orbits at
sub-relativistic energies, spin and Pauli effects in the near-relativistic range and nuclear size effects
at higher energies. Additionally, unlike the Molière formalism of EGS4, this model includes explicit
electron-positron differences in multiple scattering, which can be pronounced at low energies.
The multiple scattering distributions are computed using the exact approach of Goudsmit andSaunderson (GS)[63, 64]. Traditionally, sampling from GS distributions has been either prohibitively expensive (requiring computation of several slowly converging series at each sample)
or overly approximate (using pre-computed data tables with limited accuracy). We have developed
here a new fitting and sampling technique which overcomes these drawbacks, based on a scaling
model for multiple scattering distributions which has been known for some time[27]. First, a changeof variables is performed, and a reduced angle χ = (1− cos(θ))/2 is defined. The full range of angles(0 ≤ θ ≤ π, or 0 ≤ χ ≤ 1) is then broken into 256 intervals of equal probability, with the 256th
interval further broken down into 32 sub-intervals of equal spacing. In each of the 287 intervals or
sub-intervals, the distribution is parameterized asf (χ) =

α[1 + β(χ − χ− )(χ+ − χ)](χ + η)2

where α, β and η are parameters of the fit and χ− and χ+ are the endpoints of the interval.
By using a large number of angle bins, this parameterization models the exact form of the
distribution to a very high degree of accuracy, and can be sampled very quickly (see Chapter 2 forthe details of the implementation).

15

Photoelectron angular distribution
In standard EGS4, a photoelectron, the electron produced when a photon is absorbed by an atomby the photoelectric effect, is set in motion in the same direction as the incident photon. In order
to try to refine the comparison with low-energy experiments using thermoluminescent dosimeters,
Bielajew and Rogers[30] employed the theory of Sauter[145] and made it an option for EGS4.
Although Sauter’s theory is relativistic (v ≈ c), it was adopted universally, even though Fischer’s
non-relativistic theory[58] was derived for use in the (v << c) region. Davisson and Evans[50] haveshown that Sauter’s theory is accurate down to 92 keV (v/c = 0.092), and Bielajew and Rogers
found that it had no significant effect relative to the use of Sauter’s formalism in several cases
studied[30]. This option has been carried over into EGS5.

Pair angular distributions
Bielajew[26] also modified6 EGS4 to sample the angular distribution of the electron and positron
emanating from pair production from theoretical distributions taken from the review article by Motzet al.[111]. As a first approximation, pair angles could be sampled from the leading order termin the Sauter-Gluckstern-Hull distribution (Equation 3D-2000 of Motz et al.[111]). Additionally, afast method for sampling the Schiff angular distribution (Motz et al.Equation 3D-2003 et al.[111]),similar to that employed by ETRAN[149, 150], was made available. Use of these angular distributions has noticeable effects at large energies where the pair interaction dominates the photon crosssection and the scoring regions of interest are small enough that multiple scattering does not “washout” the effect of the initial pair distribution. This capability has been carried over into EGS5.

1.4.2

Improvements and Enhancements to EGS Photon Physics Modeling in
Version 5

K- and L-shell fluorescenceStandard EGS4 does not create or transport fluorescent photons. However, a substitute versionof the EGS subroutine PHOTO allowing for the generation of Kα1 and Kβ1 fluorescent photons wasprovided as part of a sample user code with the EGS4 distribution. This version of PHOTO, alongwith an auxiliary subroutine EDGSET (extended by Keith Weaver of the University of California at
San Francisco to 100 elements) defining shell edge energies, was enabled in EGS4 using software
switches, but has become the default in EGS5.
Treating atomic relaxation processes explicitly requires resolving the total photoelectric crosssection into subshell-by-subshell components, which can be cumbersome for compounds containing
6

Development of the pair-angle sampling scheme was motivated by a high-energy physics experiment at SLAC[158].
A secondary result of this effort was the discovery (and correction) of a bug in subroutine UPHI that occurs at energies
above 50 GeV and shows up at small angles.

16

high Z materials. Del Guerra et al.[51] developed a K- and L-edge sampling method for compounds
in EGS4, and Conti et al.[47] used this scheme in investigating the response of HgI2 and CdTe
photon detectors in the diagnostic range.
In 2000, Hirayama and Namito published a general treatment of photoelectric-related phenomena for compounds or mixtures in EGS4[72]. The energy dependent branching ratio of eachsub-shell (L1-, L2- and L3-) was introduced in this improvement by fitting to a quadratic function
in log-log from data provided for limited materials (Ag, Pb and U). For compounds and mixtures,the ratio of photoelectric effect cross section of each element to total photoelectric cross section
was calculated in PEGS4 and output as piece-wise linear fits. In addition to K- and L- x-rays, Kand L- Auger electrons could be treated.
Very recently, Hirayama and Namito published an improvement of their previous generalized
treatment [73]. In this work, K-, L1-, L2-, L3- and other sub-shell photoelectric cross sections takenfrom the PHOTX data base (see below) are fitted to cubic functions in log-log form and the fitted
coefficients and other associated data are initialized in a BLOCK DATA subprogram of EGS. It thus
becomes possible to calculate the branching ratios for each element of compounds and mixturesinside EGS, negating the need to use approximate piece-wise linear-fitted data from PEGS. Thisnew method is included in the default version of EGS5.

Improved photon cross sections
The standard EGS4 photon cross section data was based on the library compiled by Storm andIsrael[167]. Sakamoto[143] updated EGS4 to use the more recent photon cross sections of the
PHOTX library[131]. As discussed by Sakamoto, the principal differences between the Storm andIsrael data and that of PHOTX lie in the low energy photoelectric cross sections. It should be notedthat although the attenuation coefficients in the two photon libraries are different, the practicaleffect in many applications, such as the calculation of exposure buildup factors, is small. PHOTX
is the default library for EGS5.

Binding effects in Compton interactions
In its treatment of Compton scattering, standard EGS4 assumed the electrons in the atomic cloudsof the target atoms to be “free,” ignoring the binding of the atomic electrons to the nucleus.
This is a good approximation for photon energies down to 10’s of keV for most materials. The
lower bound above which this approximation works reasonably well is generally taken as the Kshell energy. Binding effects can be significant at lower energies, particularly for high-Z elements.
Namito and Hirayama included binding effects in the Compton interaction model in EGS4 taking
into account the change in the total Compton cross section as well as the angular distribution of
the emergent particles[119]. The bound Compton modeling was shown to have noticeable effectin low-energy gamma-ray buildup factors in various materials at low energy (40–200 keV) withthe effect being stronger at the lower energies studied[76]. Explicit treatment of binding effects in
17

Compton scattering is included (as an option) in EGS5.

Doppler broadening and linearly-polarized photon scattering
In addition to neglecting the binding of the electrons in the Compton interaction, standard EGS4
also ignored the motion of the electrons in the atomic cloud. Since this motion constitutes a distribution of momenta, bound electrons ejected in Compton interactions emerge not, as assumed in
EGS4, with a single energy dependent on the scattering angle, but with a distribution of possibleenergies. (This effect is usually called Doppler broadening.) Doppler broadening of Compton electron energy spectra has been taken into account in EGS4 by Namito et al.[117], who demonstrated
improved calculations of the measured energy spectra for photon scattering at 40 keV. Explicit
modeling of this effect (on a shell-by-shell basis) is available (as an option) in standard EGS5.
Standard EGS4 also considered all particles to be unpolarized, using cross sections that havebeen summed over incident and out-going particle polarizations. Namito et al.[116] introduced
polarized Compton and Rayleigh scattering in EGS4 and made comparisons with low-energy experiments where gamma ray build-up factors and attenuation coefficients in water, iron and lead
were studied in the energy range 40–250 keV[74]. The effects of binding, Doppler broadening andpolarization are pronounced in the lower energy range studied 7 . Both effects can be modeled as
options in EGS5.

Electron impact ionization
Following the work of Namito and Hirayama[120, 122] K-shell electron-impact ionization (EII) is
included in EGS5. Six different representations of the cross sections for EII are supplied:
1. Casnati[43, 44]
2. Kolbenstvedt (original)[92]
3. Kolbenstvedt (revised)[105]4. Jakoby[80]5. Gryziński[66, 67, 65]
6. Gryziński (relativistic)[66, 67, 65]
7

Recent paper by Flöttmann[59] and others[93, 98] introduced polarization into EGS4, by means of the additionalcalls-to-AUSGAB feature, in order to investigate the development of high-intensity positron sources for future linear
colliders.

18One of these six cross sections is selected by the user by specifying a PEGS5 input parameter. EII
is treated as being a part of Møller scattering in the EGS5 code, and neither the electron mean-free
path nor the stopping power are modified by including EII.
After a K-shell vacancy creation by EII, the emission of a K x-ray is sampled using the Kshell fluorescence yield. This is identical to the calculation after a K-shell vacancy creation by thephotoelectric effect, although no Auger electron or cascade particle is followed in the treatment ofEII. For the case of K x-ray emission, the K x-ray energy is sampled from the ten K x-ray energies
listed in [57] according to the relative yield of the K x-ray, and the difference between EB (K shellbinding energy) and the K x-ray energy is deposited locally. In the case that no K x-ray is generated(i.e., Auger electron emission), EB is deposited locally.

1.4.3

Other Improvements

Merging of EGS and PEGS
Historically, EGS, the simulation code, and PEGS, the data generation code, were completely
separate entities. This left open the possibility that a simulation could work on inappropriate
data, despite the built-in integrity checks. In EGS5, the two codes have been merged, and now
EGS calls PEGS during its initialization procedures. Since the elastic scattering runtime databases
can require large amounts of CPU time in certain cases, the capability to reuse PEGS data files
still exists, but its use is discouraged unless the accurate low energy elastic scattering option is
employed.

Long sequence random number generators
EGS5 employs the RANLUX random number generator[82], with the default “luxury level” set toone. RANLUX has a period of 10155 unique random numbers and at luxury level 1 has a fidelity
approximately equal to that of RANMAR[81], which became the default random number generator for
EGS4 in the interim period between major releases. The user may override the default luxury leveland so employ sequences which pass more stringent tests of randomness, although at a significant
cost in speed.

19Chapter 2

RADIATION TRANSPORT IN
EGS5
2.1

Description of Radiation Transport-Shower Process

Electrons1 , as they traverse matter, lose energy by two basic processes: collision and radiation.
The collision process is one whereby either the atom is left in an excited state or it is ionized. Mostof the time the ejected electron, in the case of ionization, has a small amount of energy that isdeposited locally. On occasion, however, an orbital electron is given a significant amount of kinetic
energy such that it is regarded as a secondary particle called a delta-ray.
Energy loss by radiation (bremsstrahlung) is farily uniformly distributed among secondary
photons of all energies from zero up to the kinetic energy of the primary particle itself. At low
electron energies the collision loss mechanism dominates the electron stopping process, while athigh energies bremsstrahlung events are more important, implying that there must exist an energy
at which the rate of energy loss from the two mechanisms are equivalent. This energy coincides
approximately with the critical energy of the material, a parameter that is used in shower theory for
scaling purposes[141]. Note that the energetic photons produced in bremsstrahlung collisions maythemselves interact in the medium through one of three photon-processes, in relative probabilities
depending on the energy of the photon and the nature of the medium. At high energies, themost likely photon interaction is materialization into an electron-positron pair, while at somewhat
lower energies (in the MeV range), the most prevalent process is Compton scattering from atomic
electrons. Both processes result in a return of energy to the system in the form of electrons which
can generate additional bremsstrahlung photons, resulting in a multiplicative cycle known as anelectromagnetic cascade shower. The third photon interactive process, the photoelectric effect, as
well as multiple Coulomb scattering of the electrons by atoms, perturbs the shower to some degree
1

In this report, we often refer to both positrons and electrons as simply electrons. Distinguishing features will be
noted in context.

20

at low energies. The latter, coupled with the Compton process, gives rise to a lateral spread in
the shower. The net effect in the forward (longitudinal) direction is an increase in the number ofparticles and a decrease in their average energy at each step in the process.
As electrons slow, radiation energy loss through bremsstrahlung collisions become less prevalent,
and the energy of the primary electron is dissipated primarily through excitation and ionization
collisions with atomic electrons. The so-called tail of the shower consists mainly of photons withenergies near the minimum in the mass absorption coefficient for the medium, since Compton
scattered photons predominate at large shower depths.
Analytical descriptions of this shower process generally begin with a set of coupled integrodifferential equations that are prohibitively difficult to solve except under severe approximation.
One such approximation uses asymptotic formulas to describe pair production and bremsstrahlung,and all other processes are ignored. The mathematics in this case is still rather tedious[141], andthe results only apply in the longitudinal direction and for certain energy restrictions. Three
dimensional shower theory is exceedingly more difficult.
The Monte Carlo technique provides a much better way for solving the shower generation
problem, not only because all of the fundamental processes can be included, but because arbitrarygeometries can be modeled. In addition, other minor processes, such as photoneutron production,
can be modeled more readily using Monte Carlo methods when further generalizations of the shower
process are required.
Another fundamental reason for using the Monte Carlo method to simulate showers is theirintrinsic random nature. Since showers develop randomly according to the quantum laws of probability, each shower is different. For applications in which only averages over many showers are ofinterest, analytic solutions of average shower behaviors, if available, would be sufficient. However,for many situations of interest (such as in the use of large NaI crystals to measure the energyof a single high energy electron or gamma ray), the shower-by-shower fluctuations are important.
Applications such as these would require not just computation of mean values, but such quantities
as the probability that a certain amount of energy is contained in a given volume of material. Suchcalculations are much more difficult than computing mean shower behavior, and are beyond ourpresent ability to compute analytically. Thus we again are led to the Monte Carlo method as thebest option for attacking these problems.

2.2

Probability Theory and Sampling Methods—A Short Tutorial

There are many good references on probability theory and Monte Carlo methods (viz. Halmos [69],Hammersley and Handscomb [70], Kingman and Taylor [89], Parzen[130], Loeve[99], Shreider[156],
Spanier and Gelbard[157], Carter and Cashwell[41]) and we shall not try to duplicate their effort
here. Rather, we shall mention only enough to establish our own notation and make the assumption

21

that the reader is already acquainted with the elements of probability theory.
The primary entities of interest will be random variables which take values in certain subsetsof their range with specified probabilities. We shall denote random variables by putting a “hat”( ˆ ) above them (e.g., x̂). If E is a logical expression involving some random variables, then weshall write P r{E} for the probability that E is true. We will call F the distribution function (orcumulative distribution function) of x̂ if
F (x) = P r{ x̂ < x} .

(2.1)

f (x) = F 0 (x)(2.2)

When F (x) is differentiable, thenis the density function (or probability density of x̂ andP r { a < x̂ < b} =Z bf (x)dx .

(2.3)

a

In this case x̂ is called a continuous random variable.
In the other case that is commonly of interest, x̂ takes on discrete values xi with probabilities
pi andX
F (x) =
pi .
(2.4)
xi ≤x

We call P the probability function of x̂ if
P (x) = pi
= 0

if x = xi
if x equals none of the xi .

(2.5)

Such a random variable is called a discrete random variable.
When we have several random variables xi (i = 1, n), we define a joint distribution function F
byF (x1 , . . . , xn ) = P r{ x̂1 ≤ x1 & . . . & x̂n ≤ xn } .

(2.6)

The set of random variables is called independent if
P r{ x̂1 ≤ x1 & . . . & x̂n ≤ xn } =

n
Y

i=1

P r{ x̂1 < xi } .

(2.7)

If F is differentiable in each variable, then we have a joint density function given byf (x1 , . . . , xn ) = ∂ n F (x1 , . . . , xn )/∂x1 , . . . , ∂xn .

(2.8)

Then, if A is some subset of Rn (n-dimensional Euclidean Space),ˆ∈A} =
P r{ x̄
22

ZA

f (x)dn x .

(2.9)

If E1 and E2 are expressions involving random variables and P r{ E2 } 6= 0, we define theconditional probability of E1 given E2 by2
P r{ E1 |E2 } = P r{ E1 & E2 } /P r{ E2 } ,(2.10)

Thus, we can define conditional distribution functions
F3 (x1 , . . . , xj |xj+1 , . . . , xn ) = F1 (x1 , . . . , xn )/F2 (xj+1 , . . . , xn ) .

(2.11)For continuous random variables, we define conditional density functions by
f3 (x1 , . . . , xj |xj+1 , . . . , xn ) = f1 (x1 , . . . , xn )/f2 (xj+1 , . . . , xn ) .

(2.12)

With this preliminary introduction we move now to sampling methods. In practice almost all
sampling is based on the possibility of generating (using computers) sequences of numbers which
behave in many ways like sequences of random variables that are uniformly distributed between 0
and 1. We shall denote uniformly distributed random variables by ζ̂i , and values sampled from theuniform distribution by ζi . Clearly, if F and f are distribution and density functions, respectively,of ζ̂, then they are given by
F (ζ) = 0,

if ζ < 0,= ζ,

if 0 ≤ ζ < 1,

= 1,

(2.13)

if 1 ≤ ζ;

f (ζ) = 1,

if ζ ∈ (0, 1),= 0,

otherwise ,We assume, in what follows, that we have an unlimited number of uniform variables available.
Now, suppose that x̂ and ŷ are related by ŷ = h(x̂) (with h monotonically increasing), and thatx̂ and ŷ have distribution functions F and G; then,F (x) = P r{ x̂ < x } = P r{ h−1 (ŷ) < x}
= P r{ ŷ < h(x)}(2.14)= G(h(x)) .
Thus, we can find F given G and h. In particular, if we define x̂ byx̂ = F −1 (ζ̂) ,then
G(y) = yandh(x) = F (x) ,2

The notation P r{ E1 |E2 } reads “the probability of E1 given E2 .”23(2.15)

so that
P r{ x̂ < x } = G(h(x)) = F (x) .
This is the basis of the so called “direct method” of sampling x̂ in which we set
F (x) = ζand solve for x. The x-values so chosen will have the distribution F .
Another method involves evaluating a function of several variables using uniform random variables for arguments. Thus we letx̂ = h(ζ̂1 , . . . , ζ̂n ) ,(2.16)
then
P r{ x̂ < x}=R~h(ζ)<xdn ζ .

(2.17)

This method has been used to find simple schemes for sampling many distributions. For example,
suppose we pick n + m + 1 values of ζ and number them such that
ζ1 < ζ2 . . . < ζn ≤ ζn+1 < ζn+2 , . . . < ζn+m+1 .

(2.18)

Then, let x = ζn+1 . Using Equations 2.16-2.18 it can be shown thatf (x) = (n + m + 1)! xn (1 − x)m /n!m! ,x ∈ (0, 1) .

A method that we shall frequently use is a combination of the “composition” and “rejection”
techniques (Butler [40], Hammersley and Handscomb[70], Kahn[85]). Suppose f and fi are density
functions, αi are positive real numbers, and gi (x) ∈ [0, 1]. We now sample x̂ as follows:
1. Pick ζ1 and let i be such thati−1X

αj < ζ1

j=1

n
X

j=1

αj ≤

n
X

αj .

(2.19)

j=1

2. Pick x from fi (x), possibly by solving
Z x−∞

fi (x) dx = ζ2 .

(2.20)

3. Pick ζ3 . Terminate the algorithm and accept value of x if
ζ3 < gi (x) .
4. Otherwise, go back to step 1.

24(2.21)

The result of this algorithm is that x̂ will have the density function f given by
f (x) =nX

αi fi (x) gi (x) .

(2.22)

i=1

Thus, if we have a function f which can be put in this form for which the fi can be sampledeasily and the gi evaluated relatively easily, then we have a good method of sampling from f . ItP
can be shown that the mean number of tries to accept a value is αi . If all the gi = 1, we havethe pure “composition” method; and if n = 1, we have the pure “rejection” method. For short we
will call this the “mixed” method. We shall sometimes use the mixed method for some of the fialso. In these cases we will use the notation (for example)
f2 (x) =n2X

α2j f2j (x) g2j (x) .

(2.23)

i=1

and so on.

Finally, we consider the problem of sampling from a joint density function f (x1 , x2 , . . . , xn ).
Define the marginal density functions
gm (x1 , x2 , . . . , xm ) ==ZZf (x1 , x2 , . . . , xn ) dxm+1 dxm+2 . . . dxn
gm+1 (x1 , x2 , . . . , xm , xm+1 ) dxm+1 .
(2.24)We see that gn = f . Now consider hm given byhm (xm |x1 , x2 , . . . , xm−1 ) ≡ gm (x1 , x2 , . . . , xm )/gm−1 (x1 , x2 , . . . , xm−1 ) .
We see thatZhm (xm |x1 , x2 , . . . , xm−1 ) dxm = 1(2.25)(2.26)

from the definition of the g’s. We see that hm is the conditional density function for xm given thespecified values for x1 , x2 , . . . , xm−1 . It can be easily seen that f can be factored into a productof the h’s; namely,f (x1 , x2 , . . . , xn )=h1 (x1 ) h2 (x2 |x1 ) h3 (x3 |x1 , x2 ) . . .

(2.27). . . hn (xn |x1 , x2 , . . . , xn−1 ) .

The procedure then is to get a sample value x1 using density function h1 . Then use this value x1 todetermine a density function h2 (x2 |x1 ) from which to sample x2 . Similarly, the previously sampled
x1 , x2 , . . . , xm determine the density function hm+1 (xm+1 |x1 , x2 , . . . , xm ) for xm+1 . This scheme
is continued until all xi have been sampled. Of course, if the x̂i are independent random variables,
the conditional densities will just be the marginal densities and the variables can be sampled inany order.
There are other methods analogous to the one-dimensional sampling methods which can beused for sampling joint distributions. The reader is referred to the references cited above for moredetails.
25

2.3

Simulating the Physical Processes—An Overview

In some approaches, the Boltzmann transport equation is written for a system and from it a Monte
Carlo simulation of the system is derived. This method gives correct average quantities, such as
fluences, but may not correctly represent fluctuations in the real situation when variance reduction
techniques are employed. The reader is referred to Chapter 3 in the book by Carter and Cashwell[41]for details of this particular method.
In all versions of EGS we have taken a different and more simple-minded approach in that we
attempt to simulate the actual physical processes as closely as possible. We have not introduced
any inherent variance reduction techniques, so that fluctuations in the Monte Carlo results should
truly be representative of real-life fluctuations. For the design of high energy particle detectors,
this is an important consideration. On the other hand, fluctuations are not usually of interestin radiation shielding-type problems, and so EGS5 includes several variance reduction techniques
which may be optionally invoked to make certain classes of calculations run more efficiently. Noneof the variance reduction techniques are invoked by default, however, and so the method of EGS5
can generally be described as analog3 Monte Carlo.
The simulation of an electromagnetic cascade shower can be decomposed into a simulation of thetransport and interactions of single high energy particle, along with some necessary bookkeeping.
A last-in-first-out (LIFO) stack is used to store the properties of particles which have yet to be
simulated. Initially, only the incident particle is on the stack (more correctly, the properties of theincident particle are stored in the first position of corresponding arrays). The basic strategy is totransport the top particle in the stack either until an interaction takes place, until its energy drops
below a predetermined cutoff energy, or until it enters a particular region of space. In the latter
two cases, the particle is taken off the stack and the simulation resumes with the new top particle.
If an interaction takes place, and if there is more than one secondary product particle, the particle
with the lowest available energy is put on the top of the stack. By “available energy” we mean
the maximum energy which can be imparted by a given particle to new secondary particles in acollision: E for photons; E − m for electrons; and E + m for positrons, where E is the particle’s
total energy and m is the electron rest mass energy. By always tracking the lowest energy particle
first, we ensure that the depth of the stack will never exceed log2 (Emax /Ecut ), where Emax is thelargest incident energy to be simulated and Ecut is the lowest cutoff energy. When the final particle
is removed from the stack and none remain, the simulation of the shower event is ended. The
complete simulation of each individual shower event is commonly called a Monte Carlo “history”.
3

The electron transport model in EGS5 is not strictly “analog” in that all scattering collisions are not treated on
an individual basis, but it is “analog” in the sense that the models of the aggregate effects of the large numbers ofcollisions which are grouped together are analytic and can in most circumstances preserve the random nature of thefluctuations in showers.

26

2.4

Particle Transport Simulation

The mean free path, λ, of a particle is given in terms of its total cross section, σt , or alternatively,in terms of its macroscopic total cross section, Σt , according to the expression
λ=

M
1
=
,
Σt
Na ρσt(2.28)

where
Na = Avogadro0 s number,ρ = density,M= molecular weight,

σt = total cross section per molecule.
The probability of an interaction is given by
P r{interaction in distance dx} = dx/λ.
In general, the mean free path may change as the particle moves from one medium to another, or
when it loses energy. The number of mean free paths traversed will be
Nλ =

Z x

dx
.
x0 λ(x)(2.29)

If N̂λ is a random variable denoting the number of mean free paths from a given point until thenext interaction, then it can be shown that N̂λ has the distribution function
P r{N̂λ < Nλ } = 1 − exp(−Nλ )for Nλ > 0.

(2.30)

Using the direct sampling method and the fact that 1 − ζ is also uniform on (0, 1) if ζ is, we cansample Nλ using
Nλ = − ln ζ.
(2.31)
This may be used in Equation 2.29 to obtain the location of the next interaction.

Let us now consider the application of the above to the transport of photons. Pair production,
Compton scattering, and photoelectric processes are treated by default in EGS5. Explicit treatmentof Rayleigh scattering is included as a non-default option. These processes all have cross sections
which are small enough that all interactions may be simulated individually, so that photons in
EGS5 travel in straight lines with constant energies between interactions. Thus, if the overall
volume in which a simulation takes place is composed of a finite number of regions, each of which isa homogeneous material with uniform density, then the integral in Equation 2.29 reduces to a sum.
If x0 , x1 , · · · are the region boundary distances between which λ is constant, then Equation 2.29
becomes
 

i−1 
X
xj − xj−1
x − xi−1Nλ =+,(2.32)λjλi
j=1where x ∈ (xi−1 , xi ). The photon transport procedure is then as follows. First, pick the number ofmean free paths to the next interaction using Equation 2.31. Then perform the following steps:27

1. Compute λ at the current location.
2. Let t1 = λNλ .
3. Compute d, the distance to the nearest boundary along the photon’s direction.
4. Let t2 equal the smaller of t1 and d. Transport by distance t2 .
5. Deduct t2 /λ from Nλ . If the result is zero (this happens when t2 = t1 ), then it is time tointeract—jump out of the loop.
6. This step is reached if t2 = d. Thus, a boundary was reached. Do the necessary bookkeeping.
If the new region is a different material, go to Step 1. Otherwise, go to Step 2.
In regions where there is a vacuum, σ = 0 (λ = ∞) and special coding is used to account for thissituation.
Now let us consider charged particle transport. The relevant interactions considered in EGS5 areelastic Coulomb scattering by nuclei, inelastic scattering by atomic electrons, positron annihilation,and bremsstrahlung. Modeling of charged particle transport is particularly challenging because
analytical expressions for the cross sections for all of the above processes (with the exception ofannihilation) become infinite as the transferred energy approaches zero (the infrared catastrophe,etc.). In actuality, these cross sections, when various corrections are taken into account (i.e.,
screening for nuclear scattering, electron binding for electron scattering, and Landau-PomeranchukMigdal effect corrections for bremsstrahlung), are not infinite, but they are very large and the exact
values for the total cross sections are not well known. Therefore, it is not practical to try to simulate
every interaction. Fortunately, the low momentum transfer events which give rise to the large total
cross section values do not result in large fluctuations in the shower behavior itself. For this reason,large numbers of low momentum transfer collisions can be combined together and modeled ascontinually-occurring processes. Cutoff energies are defined to demark the threshold between whereinteractions are treated as discrete events (sometimes referred to as “hard collisions”) or aggregatedwith other low transfer or “soft” interactions. The electron and photon threshold energiesused by
EGS5 (as set up in PEGS) are defined by the variables AE and AP, respectively, so that any electroninteraction which produces a delta-ray with total energy of at least AE or a photon with energy of atleast AP is considered to be a discrete event. All other interactions are treated as contributing to thecontinuous processes of energy loss and deflection of the electron in between discrete interactions.
Continuous energy losses are due to soft interactions with the atomic electrons (excitation andionization loss) and to the emission of soft bremsstrahlung photons. Deflections are mostly due
to multiple Coulomb elastic scattering from the nucleus, with some contribution coming from soft
electron scattering.
Typically, Monte Carlo methods simulate charged particle transport as a series of “steps” of
distance t over which hundred or thousands (or more) of low momentum transfer elastic and inelastic
events may occur. It is usually assumed that the transport during each step can be modeled asa single, straight-line, mono-energetic translation, at the end of which the aggregate effects of thecontinuous energy loss and deflection incurred over the step t can be accounted for by sampling28

from appropriate distribution functions. Discrete interactions can interrupt these steps at random
distances, just as in the case of photon transport.
Several factors complicate this model. First, because of continuous nature of the energy loss
that occurs over t, the discrete collision cross section varies along the path of the electron. Thus,
mean free paths between between discrete electron interactions vary between the time they are
first computed and the time the interaction position is reached. More importantly, electron paths
over simulated transport distances t are not straight lines, and so in addition to changing particles’
energies and directions, continuous collisions also displace them laterally from their initial paths.
Relatedly, these lateral displacements result in the actual straight-line transport distances being
something less than t. The models used in EGS5 to treat all of the implications of the continuous
energy loss and multiple elastic scattering methodology is examined in section 2.15.

2.5

Particle Interactions

When a point of a discrete interaction has been reached it must be decided which of the competing
processes has occurred. The probability that a given type of interaction occurred is proportional
to its cross section. Suppose the types of interactions possible are numbered 1 to n. Then î, thenumber of the interaction to occur, is a random variable with distribution function
i
X

σj

j=1

,
(2.33)
σtwhere σj is the cross section for the jth type of interaction and σt is the total cross sectionP(= nj=1 σj ). The F (i) are the branching ratios. The number of the interaction to occur, i, isselected by picking a random number and finding the i which satisfies
F (i) =F (i − 1) < ζ < F (i).

(2.34)

Once the type of interaction has been selected, the next step is to determine the parameters forthe product particles. In general, the final state of the interaction can be characterized by, say, nparameters µ1 , µ2 , · · · , µn . The differential cross section is some expression of the form
dn σ = g(~µ) dn µ(2.35)

with the total cross section being given byσ=Zg(~µ) dn µ.

(2.36)

Then f (~µ) = g(~µ)/ g(~µ) dn µ is normalized to 1 and has the properties of a joint density function.
This may be sampled using the method given in a previous section or using some of the moregeneral methods mentioned in the literature. Once the value of ~µ determines the final state, theproperties of the product particles are defined and can be stored on the stack. As mentioned before,
the particle with the least energy is put on top of the stack. The portion of code for transporting
particles of the type corresponding to the top particle is then entered.
R

29

2.6

General Implementation Notes

We have seen in the preceding sections how it is possible, given the total cross sections, branchingratios, final state joint density functions, and an endless supply of random numbers, to simulate
coupled electron and photon transport. Because of the statistical nature of the Monte Carlo method,
the uncertainty in the results will depend on the number of histories run. Generally, statisticaluncertainties are proportional to the inverse square root of the number of histories[156]. Thus, todecrease uncertainties by half it is necessary to run four times as many histories. Also, for given
cutoff energies, the CPU time required to simulate a shower history is in general slightly more thanlinear in the energy of the incident particle. Because of all of these factors, low-uncertainty Monte
Carlo calculations of high-energy shower simulations can be very time consuming.
The most effective way to ease the computational effort of a Monte Carlo simulation is tominimize the number of calculations required at the deepest levels inside the particle transport
loops by precomputing the values of as many of the required variables as possible. It is for thisreason that the computational task of almost all Monte Carlo programs is divided into two parts.
First, a preprocessor code (PEGS, for the EGS5 code system) uses theoretical (and sometimes
empirical) formulas to compute the various physical quantities needed and prepares them in a form
for fast numerical evaluation. A second code (EGS itself, in this work) then uses this data, alongwith user-supplied subroutines which describe the problem geometry and control the scoring ofresults, to perform the actual simulation. The motivation is to perform in advance (in PEGS)
as much computation as possible, so that amount of work done during actual particle transport
simulation is minimized. In earlier versions of EGS, PEGS was a stand-alone program, because withthe slower CPUs in previous generation computers, the data preparation burdens were sometimes
on the same scale as simulation times. As faster computers have become available, users havetypically applied them to longer simulations, and the ratio of data-preparation time to simulation
time has shrunk dramatically. Therefore, beginning with EGS5, PEGS, has been an embedded as
subroutine which can be called by EGS5 user code at the beginning of each simulation.
To aid in debugging and to help those interested in studying the various physics processesmodeled in EGS, starting with EGS4, the EGS Code System was expanded beyond the minimum
coding necessary to simulate radiation transport. With this in mind, PEGS was written in amodular form, and now contains almost 100 subprograms. These include functions to evaluate
physical quantities which are either needed by PEGS or are of interest for other reasons. Otherroutines necessary for use of PEGS within EGS5 include fitting routines and routines to write thedata for a given material onto a data file. Included among the PEGS subprograms not needed forthe operation of EGS5 itself are routines to plot the functions on a lineprinter or a graphic device,
and a routine to compare (on a lineprinter plot) the theoretical final state density functions withsampled final state distributions. Even though PEGS is now a subroutine in the overall EGS5
package, all of the plotting and density function computation routines have been preserved, andmay be accessed by appropriate user code.
The new PEGS driving subroutine, PEGS5, calls some once-only initialization routines and thenenters an option loop. After reading in the option that is desired, a NAMELIST read establishes other30

parameters which may be needed. The action requested is then performed and control returns tothe beginning of the loop. This continues until the control input has been exhausted.
The EGS5 code itself contains, among many other subprograms, four user-callable subroutines,
BLOCK SET, PEGS5, HATCH, and SHOWER. These routines call other subroutines in the EGS5 code,some of which call two user-written subroutines, HOWFAR and AUSGAB, which define the geometry
and output (scoring), respectively. The user communicates with EGS5 by means of the subroutines
described above accessing variables contained in various COMMON blocks.
To use EGS5, the user must write a MAIN program and the subroutines HOWFAR and AUSGAB.
Typically, MAIN performs any initialization needed for the geometry routine HOWFAR and then sets
the values of COMMON block variables which specify such things as the names of the media to be
used, the desired cutoff energies, the unit of distance (e.g., centimeters, radiation lengths, etc.) andso on. MAIN must also call the new initialization routine BLOCK SET, which is used primarily toinitialize data which is not defined in BLOCK DATA subprograms. At this point, MAIN may call the
new PEGS subroutine PEGS5 to prepare material data for the simulation (this may be skipped if
an existing data set is available). A call to the HATCH subroutine then “hatches” EGS5 by doingsome once-only integrity checks and initializations and by reading from a data set the material
data prepared by PEGS5 for the media requested. With the initialization completed, simulationof the shower may begin with MAIN calling subroutine SHOWER when desired. Each call to SHOWER
results in the generation of a single EGS5 particle history, with the arguments to SHOWER specifying
the parameters (energy, position, direction, etc.) of an individual initial particle. This permits
the user the freedom to use any source distribution desired. The flow of control and data when auser-written MAIN program is linked with EGS5 is illustrated in Figure 2.1. Detailed informationneeded to write user programs is provided in the EGS5 User Manual (Appendix B of this report)
As a beginning introduction, however, the reader may find it useful to first study the series of shorttutorials provided in Chapter 3 of this document.
Both EGS5 and PEGS use MeV for the unit of energy, and all EGS5 variables reference the totalenergy (i.e., kinetic plus rest mass) associated with particles. EGS5 primarily employs the CGS
system of units, but PEGS scales its outputs in units of radiation lengths for subsequent (internal)
use by EGS5. By default, the centimeter is the basic working unit of distance used in EGS5, butother transport distance unit may be chosen by setting the variable DUNIT before calling HATCH (seeAppendix B for details).
Table 2.1 defines some of the mathematical and program symbols for entities in EGS and
PEGS, as well as other symbols used in this discussion. The first column gives the item number,the second column shows the mathematical symbol used for the entities, and the third column
shows the FORTRAN name for the same thing in the EGS or PEGS code. A ‘P’, ‘E’, or both incolumn four shows whether the item is used in PEGS, EGS, or both.4 The fifth column contains
the definition, explanation, or name of the item.
A number of physical, mathematical, and derived constants are used by the codes. We have4
Even though PEGS is integrated into EGS, PEGS and EGS have retained the majority of their prior distinctiveness, sharing only a handful of variables.

31

User
Control
Data

USER CODE

MAIN

BLOCK
DATA

Information
Extracted
From Shower

SHOWER

HOWFARAUSGAB

ELECTR

PHOTON(default)

BLOCK
SET
MSCAT

PEGS5

EGS CODE

ANNIH

COMPT

BHABHAPAIR

MOLLER

PHOTO

BREMS

HATCH

UPHI

Figure 2.1: Program flow and data control in EGS5.

32

arrived at their values in a very mnemonic way by means of the PEGS subroutine called PMDCON(Physical, Mathematical, and Derived CONstants). These are items 1 through 16 in Table 2.1.
As illustrated in items 17 through 27, cupped ( ˘ ) energy variables are in MeV, cupped distancevariables are in radiation lengths, and uncupped quantities denote CGS units.
Materials are specified by giving density, ρ, numbers of elements, Ne , atomic symbols of theelements, and the proportions of the atoms which are of each type, pi . PEGS maintains tablesof atomic numbers, Zi , and atomic weights, Ai , for elements 1 through 100. For cases in whichmaterials are composed of single elements, PEGS has a table of default densities which it uses.
For materials which have non-standard isotopes or densities, the values supplied by PEGS may
be overridden. As an alternative to specifying the atomic proportions by number pi , they may be
given in terms of weight, by ρi . The parameters used to define media are summarized as items 28
through 33 in Table 2.1.
The PEGS subroutine, MIX, uses the material parameters described above to compute additional
molecular parameters (items 34 through 53 in Table 2.1) useful in computing cross sections for
mixtures, as will be seen in subsequent sections.

33

Table 2.1: Symbols used in EGS5 and PEGS5.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.

16.

17.
18.
19.
20.
21.
22.
23.
24.
25.

Math
π
c

FORTRAN Program
Definition
PIP
3.1415926536
CP
speed of light= 2.997925 ×1010 cm/secme
RMEPelectron rest mass
= 9.1091 ×10−28 g
h̄
HBAR
P
Planck’s constant/2π= 1.05450 ×10−27 erg-sec
eCGS
ECGS
P
electronic charge (CGS units)= 4.80298 × 10−10 esu
eMKS
EMKS
P
electronic charge (MKS units)= 1.60210 × 10−19 CoulNa
AN
P
Avogadro’s number= 6.02252 × 1023 mole−1(1 Rad)[◦ ]RADDEGP
radian expressed in degrees= 180/πα
FSCPfine structure constant= e2CGS /(h̄c)(1 MeV)[erg] ERGMEVP
one MeV expressed in ergs
= eMKS × 1013
r0
RO
P
classical electron radius= e2CGS /(me c2 )m
RMPEelectron rest energy in MeV= me c2 /ERGMEV
2m
RMT2
PE
2m
m2
RMSQ
PE
m2
‘22.9’
A22P9
P
constant in Nagel’s formula forχc in degrees. Taken from
Molière’s paper in which χc wasexpressed in grads. So to get χcin degrees, should really use 22.9
×360/400.
√ Defined as A22P9 =RADDEG 4πNa× e2CGS /ERGMEV = 22.6960‘6680’
A6680
P
constant in Bethe’s formula for
Molière’s b= 4πNa (h̄/me c)2 (0.885)2/(1.167 × 1.13) = 6702.33
k, k1
Pphoton energy in ergs
k̆, k̆1
P
photon energy in MeV
E, E1
Pelectron total energy in ergs
Ĕ, Ĕ1
P
electron total energy in MeV
TPelectron kinetic energy in ergs
T̆Pelectron kinetic energy in MeVtP
distance in centimeterst̆Pdistance in radiation lengths= t/X0σPcross section in cm2 /(atom ormolecule)34

Table 2.1 (cont.)
Math FORTRAN Program
Definition
26.
ΣP
macroscopic cross section(probability of interaction/cm)= Na ρσ/M27.
Σ̆
P
macroscopic cross section(in X0−1 units) = X0 Σ
28.
ρ
RHO
PE
material density (g/cm3 )29. NeNEPE
number of elements inthe material
30. Zi
Z(I)P
atomic number of the ith elementof the material
31. AiWA(I)P
atomic weight of the ith elementof the material (g/mole)
32.
pi
PZ(I)
P
proportion by number of the ithelement in the material ∝ ρi /Ai33.
ρi
RHOZ(I)P
proportion by weight of the ithelement in the material ∝ pi Ai
34.
M
WM
P
molecular weight (g/mole)P e= N
i=1 pi Ai
35. CM
ZC
P
molecular charge (electrons/
P emolecule) = N
i=1 pi Zi
36.
n
EDEN
P
electron density (electrons/cm3 )= Na ρCM /M
37. X0
RLC
PE
radiation length= (Na ρ4αr02 (ZAB − ZF )/M )−1
P Ne
38. ZTZT
P
i=1 pi Zi (Zi + ξi )
P Ne
−1/3
39. ZBZB
P
pi Zi (Zi + ξi )`n Zi
Pi=1
Ne
40. ZFZF
P
pi Zi (Zi + ξi )fc (Zi )Pi=1
Ne
41. ZSZS
P
i=1 pi Zi (Zi + ξM S )
P Ne
−2/3
42. ZEZE
P
pi Zi (Zi + ξM S )`n Zi
Pi=1
Ne
43. ZX
ZX
P
i=1 pi Zi (Zi + ξM S )×`n (1 + 3.34(αZi )2 )44. ZAZAP
ZT `n 183
45. ZGZGP
ZB /ZT
46. ZPZPPZB /ZA
47. ZVZVP(ZB − ZF )/ZT
48. ZUZUP(ZB − ZF )/ZA
Note: The next four are used in EGS directly and computed in MIX
49.
bc
BLCC
PE
‘6680’ ρZS exp(ZE /ZS ) X0/(M exp(ZX /ZS )) p
50. χcc
XCC
PE(‘22.9’/RADDEG) ZS ρX0 /M
51. tef f0
TEFFOPE(exp(B)/Bmin )/bc
p min
XRO
PE
χcc tef f0 Bmin
52. χr0
53. ξM S FUDGEMS
P
constant used to account for
multiple scattering of atomic
electrons35

Table 2.1 (cont.)
Math
ξi

FORTRAN Program
Definition
54.
XSI(I)
P
constant for bremsstrahlung andpair production of atomic
electrons for element Zi= L0rad (Zi )/[Lrad (Zi ) − fc (Zi )]55. fc (Zi )FCOUL(I)P
bremsstrahlung and pair
production Coulomb correction
constant for element Zi
56.
Bmin
BMIN
P
57.
λ
MFP
P
mean free path
58. A0 (Zi , E) APRIME(I)P
empirical bremsstrahlung
correction factor
59.
U(x)
P
unit step function60.
φ1 (δ)
P
first screening function
61.
φ2 (δ)
P
second screening function
62.
δiP
screening parameter forelement Zi
63.
δ0P
average screening parameter
for a mixture
64.
AP
AP
PE
low energy threshold for soft
bremsstrahlung production
65.
AE
AE
PE
low energy threshold forknock-on electron production66.
ζ̂, ζ̂i ,uniform random variables andsampled value
ζ, ζi ,
67.
αiweight of ith element insampling method68.
λ0Compton wavelength of electron= 2πr0 /α = 2.4262 × 10−10 cm
69.
I¯adj
IEV
P
average adjusted mean
ionization energy
70.
δ
DELTA
P
density effect correction
Note: The following are Sternheimer (density effect) fit parameters
71.
a
AFACT
P
72.
ms
SK
P
73.
x0
X0
P
74.
x1
X1
P
75.
−C
CBAR
P
Note: The following are used in the new definition of radiation length
76.
‘183’
A183
P
constant in Lrad which used
to be 183, now 184.15
77.
‘1440’
A1440
P
constant in Lrad which used
to be 1440, now 1194.
78.
Lrad
ALRAD
P
a) `n184.15Z −1/3 for Z > 4b) tabulated value for Z ≤ 4
0
ALRADPPa) `n1440Z −2/3 for Z > 479.
Lradb) tabulated value for Z ≤ 4
P Ne
80.
ZAB
ZAB
P
i=1 pi Zi (Zi + ξi )Lrad (Zi )36

Figure 2.2: Feynman diagrams for bremsstrahlung and pair production.

2.7

Bremsstrahlung and Electron-Positron Pair Production

The bremsstrahlung and pair production processes are closely related, as can be seen from the
Feynman diagrams in Figure 2.2 . In the case of bremsstrahlung, an electron or positron is scatteredby two photons: a virtual photon from the atomic nucleus and another free photon which is createdby the process. In the case of pair production, an electron traveling backward in time (a positron)is also scattered by two photons, one of which scatters it forward in time making it into an electron.
The net effect is the absorption of the photon and creation of an electron-positron pair.
The discussions and descriptions of these processes which are given here use formulas taken fromthe review articles by Koch and Motz[91] on bremsstrahlung and by Motz, Olsen and Koch[111]on pair production. We also employ some ideas from Butcher and Messel[39] for mixing the crosssections for sampling of the secondary spectra. Below 50 MeV the Born approximation crosssections are used with empirical corrections added to get agreement with experiment. Above 50
MeV the extreme relativistic Coulomb corrected cross sections are used.
The “shower book” by Messel and Crawford[103] takes into account the Landau-PomeranchukMigdal (LPM) “suppression effect”[95, 94, 106] which is important at electron energies greaterthan 100 GeV for bremsstrahlung and greater than a TeV or so for pair production. At theseenergies, the LPM effect, which had been demonstrated experimentally [9], manifests as significant
reductions (“suppression”) in the total bremsstrahlung and pair production cross sections. EGS5
does not currently model the LPM effect. In addition, an effect due to polarization of the medium(which apparently is effective even at ordinary energies) results in the cutoff of the bremsstrahlung
differential cross section at secondary photon energies below a certain fraction of the incident
electron energy. This has been quantified in terms of a factor FP [103], given byFP =nr0 λ20 E02
1+
πk 2

!−1

(2.37)

where n is the electron density, r0 is the classical electron radius, λ0 is the Compton wavelength ofan electron, and E0 and k are the energies of the electron and photon, respectively. If we define acutoff energy byq
kc = E0 nr0 λ20 /π
37(2.38)

then we see that for k  kc , FP goes to one; for k ≈ kc , it is about 1/2; and for k  kc it goes as
k2 /kc2 . In the latter case, the k2 factor multiplied by the usual 1/k dependence results in an overallk dependence as k → 0. Thus we have finite differential and total cross sections and the infraredcatastrophe is averted. It can be seen that the ratio of the cutoff energy to the incident electronenergy is independent of energy, and depends on the medium only through its electron density. For
lead, the ratio is
kc /E0 =q

nr0 λ20 /π(2.39)= 1.195 × 10−4
The natural log of the inverse of this ratio (' 9) is then approximately equal to the total bremsstrahlung cross section in units of inverse radiation lengths (if one takes dσ/dk [X0−1 ] = 1/k for
k > kc , = 0 for k < kc ).
As the corrections presented above have not been implemented in EGS5, it is instructive toinvestigate the magnitude of the error expected this introduces. First, it is clear that at energies
above 100 GeV, ignoring the LPM suppression effect will have a significant impact on the predictedgross behavior of a shower because of the over-estimation of the bremsstrahlung and pair production
cross sections. We therefore set 100 GeV as a safe upper limit to the present EGS5 version. Next,
we see that by ignoring the bremsstrahlung cutoff kc , EGS5 simulations of high energy electrontransport produce too many low energy secondary photons. For example, a 10 GeV electron should
not emit many photons below 1 MeV, whereas EGS5 would continue production down possibly as
low as 1 keV. This should not disturb the general shower behavior much, as there will be many
more low energy electrons than high energy electrons, so that the few extra low energy photons
produced by the high energy electrons should be insignificant compared to the number of low energy
photons produced by the lower energy electrons. It should be clear, however, that if the user wereusing EGS5 to determine thin target bremsstrahlung spectra from high energy electrons, the results
would be in error below the cutoff kc .
Neglecting possible crystal diffraction effects[55, 170], the macroscopic cross section for bremsstrahlung or pair production is given in terms of the microscopic cross sections, σi , for the atomsof type i byP
p i σiNa ρ X.
(2.40)
p i σ i = Na ρ P i
Σ=Mi pi Aii

We see that the macroscopic cross sections do not depend on the absolute normalization of the pi ’s,
only the ratios. With the exception of ionization losses (where polarization effects are important),
Equation 2.40 is also valid for the other interactions that are considered (e.g., Møller, Compton,etc.).
For conciseness in what follows we shall use the notation(E1 if B1 , . . . , En if Bn , En+1 )(2.41)

to denote the conditional expression which takes Ei for its value if Bi is the first true expression,and takes En+1 for its value if no Bi is true. We will also make use of the Kronecker delta function38

defined by
δij = 1 if i = j,0 otherwise.

(2.42)

Using this notation, we start with the following formulas for the bremsstrahlung and pair production
differential cross sections:5 .
dσBrem (Z, Ĕ0 , k̆)A0 (Z, Ĕ0 )r02 αZ(Z + ξ(Z))

k̆

4
2
1 + (Ĕ/Ĕ0 )φ1 (δ) − `n Z − (4fc (Z) if Ĕ0 > 50, 0)
3
! 
)
2
Ĕ
4
−
φ2 (δ) − `n Z − (4 fc (Z) if Ĕ0 > 50, 0)
3 Ĕ0
3
dk̆

×

=





(2.43)

anddσP air (Z, k̆, Ĕ+ )dĔ+=A0p (Z, k̆)r02 αZ(Z + ξ(Z))
k̆3

4
φ1 (δ) − `n Z − (4fc (Z) if k̆ > 50, 0)
×
3


4
2
+ Ĕ+ Ĕ− φ2 (δ) − `n Z − (4 fc (Z) if k̆ > 50, 0)
3
3


2
2
Ĕ+
+ Ĕ−





(2.44)whereδ = 136 Z −1/3 2∆(2.45)

and
∆ =
=

k̆m
2Ĕ0 Ĕk̆m2Ĕ+ Ĕ−(for bremsstrahlung)(2.46)(for pair production) .

(2.47)

To avoid confusion it should be noted that our δ is the same as the δ of Butcher and Messel[39] but
100
we use φi (δ) to denote their fi (δ). Rossi[141] and Koch and Motz[91] use a variable γ = 136
δ. Also,note that our φi (δ) has the same value as the φi (γ) of Koch and Motz[91] (e.g., see their Figure 1)
136
provided “our δ” = 100
times “their γ.” For arbitrary screening, φ1 and φ2 are given by
φ1 (δ) = 4

Z 1

∆
Z 1

(q − ∆)2 [1 − F (q, Z)]24dq
+ 4 + `n Z ,3q3q+ 3∆2 q − 4∆3∆∆
dq 10 4+ `n Z× [1 − F (q, Z)]2 4 +q33

φ2 (δ) = 4



q 3 − 6∆2 q `n

5





(2.48)

(2.49)As described earlier, we have adopted the notation that cupped energy variables (e.g., Ĕ) are in MeV and cupped
distance variables (e.g., t̆) are in radiation lengths. Uncupped variables are in CGS units and constants are defined
explicitly

39where δ = 272 Z −1/3 ∆ as before, and where F (q, Z) is the atomic form factor for an atom withatomic number Z. Following Nagel[112], we have used the Thomas-Fermi form factors, for which
φ1 and φ2 are Z independent and have already been evaluated. Butcher and Messel [39]haveapproximated the screening functions to within 1–2% by the formulas
φ1 (δ) = (20.867 − 3.242δ + 0.625δ2 if δ ≤ 1, 21.12 − 4.184 `n(δ + 0.952))(2.50)φ2 (δ) = (20.029 − 1.930δ − 0.086δ2 if δ ≤ 1, 21.12 − 4.184 `n(δ + 0.952)).

(2.51)

andThe Thomas-Fermi screening is quite accurate for high atomic numbers, but at low atomic numbers
its accuracy decreases. The Hartree form factors are better for low Z. Tsai[172] has given a review
of bremsstrahlung and pair production cross sections including best estimates of form factors andscreening functions, and Seltzer and Berger[153, 154] have reviewed and presented new cross section
data for bremsstrahlung production. EGS has not been modified to reflect these more accurate crosssections except that it redefines the radiation length and ξi to be consistent with the definitions
by Tsai[172] (see discussion of these changes below). We have also checked that, for example, thevalues of φ1 and φ2 , given by Equations 2.50 and 2.51, agree with Tsai’s values within 0.4% for
Z > 4 and within 5% for hydrogen.
A0 (Z, Ĕ0 ) in Equation 2.43 is an empirical correction factor evaluated by the function APRIM
in PEGS. For Ĕ0 > 50 MeV, PEGS takes A0 = 1 since it uses the Coulomb corrected formulas ,which are accurate to about 3% in this energy range. For Ĕ0 < 50 MeV, PEGS5 uses values of A0
generated by Rogers et al. [139]. This effectively renormalizes the bremsstrahlung cross sections
to assure that the total radiative stopping powers (see section 2.13) agree with those publishedin ICRU Report 37 [79]. As an option, the user may request the PEGS4 corrections, which wereinterpolated in Z from the curves of Koch and Motz[91] (see their Figure 23, e.g.). With theavailability of the better cross section data mentioned above, this methodology is somewhat more
approximate than need be, and an improved treatment awaits development by some fresh, energetic
EGS5 user.
The pair production empirical correction factor A0p (Z, k̆) in Equation 2.44 is defined as: A0p (Z, k̆)
is equal to (“The Best Empirical Estimate of the Total Pair Production Cross Section for givenZ, k̆”) divided by (“The Total Pair Production Cross Section obtained by integrating Equation 2.44
over all allowed Ĕ+ values, with A0p = 1”). For k̆ < 50 MeV, we take this best estimate to be thedata compiled by Storm and Israel[167] and in fact we use this data directly without resorting toEquation 2.44 whenever pair production total cross sections are needed for k̆ < 50. For k̆ > 50
MeV, an integration of Equation 2.44 with A0p = 1 is used for the pair production total cross section.
This agrees within a few percent with Storm and Israel[167] up to the limiting energy for which theypresent data but it does lead to a slight discontinuity in the photon cross section at 50 MeV. Thus
A0p (Z, k̆ > 50) is taken to be 1, as it is for bremsstrahlung. Unlike as with bremsstrahlung, however,A0p is never explicitly calculated since it is not needed in determining the total cross section, nor,as will be seen later, is it used in sampling the secondary particle energies.
The fc (Z) in Equations 2.43 and 2.44 is the Coulomb correction term that was derived by40

Davies, Bethe and Maximon[49] (e.g., see their formula 36, p. 791)) and is given by
fc (Z) = a2∞
X1
2 + a2 )ν(νν=1(2.52)

where
a=αZ .
They also suggest a formula accurate to 4 digits up to a = 2/3 (which corresponds to Uranium);
namely,on(2.53)
fc (Z) = a2 (1 + a2 )−1 + 0.20206 − 0.0369a2 + 0.0083a4 − 0.002a6
which function FCOULC of PEGS uses to evaluate fc (Z).

ξ(Z) is a function which is used to take into account bremsstrahlung and pair productionin the field of the atomic electrons. Strictly speaking, these interactions are different from thecorresponding nuclear interaction not only because the mass and charge of an electron are different
from the nuclear mass and charge, but also because of the identity of the electrons. Because of thelightness of the electron, it may be ejected from the atom. In the bremsstrahlung case what wereally have is radiative Møller or Bhabha scattering. In the case of pair production, if the atomic
electron is ejected, we have three rather than two energetic electrons and the reaction is called
triplet production. Because of the electron exchange effects and the γ − e interactions between theexternal photon and the target electron, and also because the target can no longer be treated as
infinitely heavy, the cross section calculations for these interactions are more complicated than forthe corresponding nuclear cases and involve a larger number of approximations (see p. 631 of Motzet al. [111]). As will be seen below, the ratio of cross sections for the interaction in the electron
fields to those in the nuclear field is of the order of 1/Z. Thus, for medium-low to high Z, thecontributions of the atomic electrons are rather minor. On the other hand, for low Z, such as
beryllium and certainly for hydrogen, these interactions are very significant and a more accurate
treatment of these interactions is warranted. Nevertheless, we have not treated the bremsstrahlung
and pair production in the electronic fields in a special way, primarily because most applicationsof interest do not involve only very low Z elements. When low Z elements are involved, they haveusually been mixed with higher Z elements, in which case the pair production and bremsstrahlungin the low Z elements are relatively unimportant. This does limit somewhat the universality of
EGS.
For very high energy incident particles the screening can be considered complete. In thiscase, relatively simple formulas for the interaction in the atomic field can be obtained (Bethe
and Ashkin[24] (formula 59 on p. 263 and formula 119 on p. 332), Koch and Motz[91] (formula
III-8 on p. 949)). The relative values of the radiation integral
φrad ≡

1
E0

Z kmax0kdσBremdk
dk(2.54)

can be used as an estimate of the relative magnitude of the interactions in the electron or nuclear
fields. In the completely screened nuclear field, the radiation integral (formula 4CS of Koch andMotz[91] ) is


1
−1/3
2
2
− fc (Z) .
(2.55))+
φrad,nucleus = 4αr0 Z `n (183 Z
18
41

For φrad in the completely screened electron field Koch and Motz[91] (formula III-8 on p. 949) giveφrad,electron = 4αr02 Z `n (530 Z −2/3 ) .

(2.56)On the other hand, from the formulas of Bethe and Ashkin[24] mentioned above, one would expect
φrad,electron = 4αr02 Z `n (1440 Z −2/3 ) .
Tsai’s work[172] has dealt with these problems more accurately. We have not switched to his
method but continue to define a single parameter, ξ, which is used in a simple Z(Z + ξ) correction
to the cross sections to account for electron effects. However, we have redefined ξ making use of
Tsai’s radiation logarithms:
L0rad (Z)(2.57)
ξ(Z) =
Lrad (Z) − fc (Z)where

L0rad =

`n1194 Z −2/3
 6.144

5.621




5.805

5.924

Lrad =

−1/3
 `n184.15 Z

 5.310

4.790




 4.740
4.710

if Z > 4
if Z = 1
if Z = 2
if Z = 3
if Z = 4
if Z > 4
if Z = 1
if Z = 2
if Z = 3
if Z = 4

These expressions are used by function XSIF of PEGS to compute ξ(Z) for use in Equations 2.43
and 2.44. These definitions replace the simpler `n1440 Z −2/3 and `n183 Z −1/3 used in PEGS3.
Note that for the rest of this chapter, “183” is a variable name representing the value 184.15, andsimilarly, “1440” represents 1194.
We have also changed the definition of radiation length to that of Tsai[172]:
X0−1 =iNa ραr02 h 2
Z [Lrad (Z) − fc (Z)] + ZL0rad (Z) .
A(2.58)

This change has a considerable effect on X0 for very light elements (a 9% increase for hydrogen)but only a small effect for Z ≥ 5.
We have done several comparisons between the pair production cross sections used in PEGS
and the more accurate results of Tsai. In general, they agree to within a few percent except forvery low Z (< about 10) elements at energies below 1 GeV.
Now that we have discussed all items appearing in Equations 2.43 and 2.44, we need to mention
some additional corrections to Equation 2.43 which are neglected in EGS5. First, the differential
42cross section given in Equation 2.43 goes to zero at the maximum photon energy; whereas, in reality
the bremsstrahlung cross section, differential in photon energy, is non-zero at the “high frequency
limit” (Koch and Motz[91] (p. 933)). A more rigorous treatment would modify Equation 2.43 so
as to smoothly approach the proper value at the high frequency limit (bremsstrahlung).
Another correction for Equation 2.43 ignored by EGS5 is the “Elwert factor”, which Koch andMotz[91] recommend be applied below T̆ = 2. For problems in which the bremsstrahlung from low
energy electrons plays an important role, this correction factor is necessary to produce accurate
results. This is therefore a restriction on the use of EGS. However, radiative yield rises rapidly
with electron energy. For example, as an electron slows down from 20 MeV, it loses about 8% of
its energy through radiative loses, but of this, only 0.5% is lost by electrons with energies below 5
MeV. Thus ignoring the Elwert correction factor introduces a negligible error into most problems,
especially those dealing with high energy electrons.
We now discuss the methods we use to sample the secondary energies for bremsstrahlung andpair production interactions. Our methods are based on Equations 2.43 and 2.44 with a coupleof approximations. For the purpose of our discussion let x and x0 be the secondary and incident
particle energies, respectively. Then Equations 2.43 and 2.44 are of the form
dσuncorrected (Z, x0 , x)dσcorrected (Z, x0 , x)
= A0 (Z, x0 ).
dx
dx

(2.59)

According to Equation 2.40 the differential macroscopic cross section, properly weighted with thevarious constituent materials, will beNedσcorrected (Zi , x0 , x)Na ρ XdΣ(x0 , x)
pi
=
dx
M i=1
dx

(2.60)and the total macroscopic cross section will beΣ(x0 ) =Z xmaxxmindΣ(x0 , x)dx
dx

(2.61)

If the incident particle has undergone an interaction of this type, then the probability density
function for x will bedΣ(x0 , x)
f (x) =
Σ(x0 ) .
(2.62)dx
We now observe that any constant factors in the right hand side of Equation 2.60 will cancel out inEquation 2.62, and, in particular, for materials consisting of only one element the correction factor
A0 (Z, x0 ) will be an overall factor and so may be ignored. We now make the approximation thatthe factor A0 (Z, x0 ) can be ignored for secondary sampling purposes even when there is more than
one element. As will be seen, this will allow us to obtain energy independent screening factors.
The way we could avoid making this approximation is to have PEGS generate branching ratios
among all the constituents as a function of particle energy so that whenever an interaction took
place, the first thing done would be to decide with which type of atom the interaction occurred.
For complex mixtures this would take a significantly larger amount of data and running time andwe have chosen to make the above approximation instead.
43

Now suppose that the f (x) in Equation 2.62 is decomposed in a manner that allows for a
combination of “composition” and “rejection” techniques (e.g., see Equation 2.10; that isdΣ(x0 , x)
dx
or



Σ(x0 ) =NeX

α0i fi (x) gi (x)(2.63)i=1

Ne
dΣ(x0 , x) X=
αi fi (x) gi (x) ,dx
i=1(2.64)

where
αi = a0i Σ(x0 ) .
But it can be shown (e.g., see Equation 2.7) that if all αi are multiplied by the same factor, it
will not change the function being sampled. We conclude that to properly sample from the p.d.f.
(Equation 2.62), it is not necessary to obtain Σ(x0 ), but it is sufficient to decompose dΣ(x0 , x)/dx
as shown in Equation 2.64 without bothering to normalize. We seek decompositions of the formnX

αj fj (x)gj (x) =

Ne
X

pi

i=1

j=1

dσuncorrected (Zi , x0 , x).
dx

(2.65)Let us now do this for the bremsstrahlung process (in a manner similar to that of Butcher andMessel[39]) using Equation 2.43 for dσi /dx with A0 (Zi , Ĕ0 ) = 1. The first point to be noted[114] isthat φ1 (δ) and φ2 (δ) → 0 as δ → ∞, so that the expressions in square brackets in Equation 2.43
go to zero, and in fact go negative, at sufficiently large δ. As can be seen from Equation 2.51,
φ2 (δ) = φ1 (δ) for δ > 1. By checking numerical values it is seen that the value of δ for which theexpressions go to zero is greater than 1, so that both [ ] expressions in Equation 2.43 go to zerosimultaneously. Clearly the differential cross section must not be allowed to go negative, so this
imposes an upper kinematic limit δmax (Zi , Ĕ0 ) resulting in the condition (from Equation 2.50)
4
21.12 − 4.184 `n(δmax (Z, Ĕ0 ) + 0.952) − `n Z − (4fc (Z) if Ĕ0 > 50, 0) = 0.
3(2.66)

Solving for δmax (Z, Ĕ0 ) we obtain4δmax (Z, Ĕ0 ) = exp (21.12 − `n Z − (4fc (Z) if Ĕ0 > 50, 0))/4.184 − 0.952 .
3

(2.67)

From PEGS routines BREMDZ, BRMSDZ, and BRMSFZ, which compute dσBrem (Z, Ĕ0 , k̆)/dk̆ as given
by Equation 2.43, it may be seen that the result is set to zero if δ > δmax (Z, Ĕ0 ). Another way oflooking at this is to define

φ0i (Z, Ĕ0 , δ) = φi (δ) if δ ≤ δmax (Z, Ĕ0 ), φi (δmax (Z, Ĕ0 ))We now define

E = k̆/Ĕ044.

(2.68)(2.69)

and use this as the variable to be sampled instead of k̆. The expressions for ∆ and δ then become∆=mE
2Ĕ0 (1 − E)

and

−1/3

δi =

136 Zi(2.70)mE

Ĕ0 (1 − E).

(2.71)Let us also define a variable (corresponding to DEL in EGS)
E

∆E =

,

(2.72)∆E .

(2.73)(1 − E)Ĕ0

so that−1/3

δi = 136 mZi

Since overall factors do not matter let us factorize X0 dΣBrem /dk̆. From Equations 2.60 and 2.43
with A0 = 1, and the definition of X0 in Table 2.1, we obtaindΣ̆BremdE

dΣBrem
dE
!,
Ne
h
i
X
dσuncorrectedpi
4αr02 (ZAB − ZF )
dE
i=1

=

X0

=

Ne
X

=

h

i=1

pi

Zi (Zi + ξ(Zi )) n(1 + (1 − E)2 )E

4φ01 (Zi , Ĕ0 , δi ) − `n Zi −2
4fc (Zi ) if Ĕ0 > 50, 0 − (1 − E)
3
3




4
i
φ02 (Zi , Ĕ0 , δi ) − `n Zi − 4fc (Zi ) if Ĕ0 > 50, 03
1.
4(ZAB − ZF )



×
×
×





(2.74)For some brevity let us use
φ0ji for φ0j (Zi Ĕ0 , δi ), fci0 for (fc (Zi ) if Ĕ0 > 50, 0), and ξi for ξ(Zi ) .
Then after some rearrangement Equation 2.74 becomesdΣ̆BremdE

NeX
1
pi Zi (Zi + ξi )4(ZAB − ZF ) i=1

=
+

−1/3

8(`n Zi

i 1 − E − fci0 )E

2  0
3φ1i − φ02i
3

 

h

−1/3

+ φ01i + 4(`n Zi(2.75)i − fci0 ) E

Now defineφ̂j (∆E ) =

Ne
X

−1/3

pi Zi (Zi + ξi )φ0j (Zi , Ĕ0 , 136 Zi

i=1

45

m∆E ) for j = 1, 2 .

(2.76)

Also, recall from Table 2.1 the definition of ZB and ZF . We then can rewrite Equation 2.76 asdΣ̆Bremdk̆

=n  h
2
3

×





i

3φ̂1 (∆E ) − φ2 (∆E ) + 8 ZB − (ZF if Ĕ0 > 50, 0)

1−E
E





h

i

+ φ̂1 (∆E ) + 4 ZB − (ZF if Ĕ0 > 50, 0)
× 4(ZAB1−ZF ) .

Eo(2.77)

Now letÂ(∆E , Ĕ0 ) = 3φ̂1 (∆E ) − φ̂2 (∆E ) + 8(ZB − (ZF if Ĕ0 > 50, 0))(2.78)

B̂(∆E , Ĕ0 ) = φ̂1 (∆E ) + (ZB − (ZF if Ĕ0 > 50, 0)).

(2.79)

andOne can show that φj (δ) have their maximum values at δ = 0, at which they take values[39]φ1 (0) = 4 `n 183(2.80)

φ2 (0) = φ1 (0) − 2/3 .

(2.81)The φ̂j (∆E ) also are maximum at ∆E = 0 and, in fact (see Table 2.1),
φ̂1 (0) =φ̂2 (0) =

Ne
X

i=1
Ne
X
i=1

pi Zi (Zi + ξi )φ1 (0) = 4ZA ,pi Zi (Zi + ξi )φ2 (0) = 4ZA −(2.82)
2
ZT .
3(2.83)The maximum values of Â and B̂ are now given byÂmax (Ĕ0 ) = Â(0, Ĕ0 ) = 3(4ZA ) − 4ZA −

2
ZT
3







+ 8 ZB − (ZF if Ĕ0 > 50, 0)

2
ZT + 8(ZA + ZB − (ZF if Ĕ0 > 50, 0)) ,3B̂max (Ĕ0 ) = B̂(0, Ĕ0 ) = 4 ZA + ZB − (ZF if Ĕ0 > 50, 0) .
=(2.84)(2.85)

Now define δ0 as the weighted geometric mean of the δi ; that is,
δ0 ≡

Ne
Y

p Z (Zi +ξi )δi i i

i=1

= exp

(

ZT−1

Ne
X

!

PNe

i=1

−1

pi Zi (Zi +ξi )pi Zi (Zi + ξi )`n δi

i=1= 136 m∆E exp

(

ZT−1

Ne
X

)

−1/3
pi Zi (Zi + ξi )`n Zi

i=1(2.86))= 136 m∆E exp (ZB /ZT ) = 136 m eZG ∆E .
Or if we define∆C = 136 m eZG ,46(2.87)

we haveδ 0 = ∆C ∆E .

(2.88)

A(δ0 ) = Â(δ0 /∆C , Ĕ0 )/Âmax (Ĕ0 ) ,(2.89)

Now define00B(δ ) = B̂(δ /∆C , Ĕ0 )/B̂max (Ĕ0 ) .

(2.90)

Then A and B have maximum values of 1 at δ0 = 0, and are thus candidate rejection functions.
We are now ready to explain the reason for introducing the parameter δ0 and to introduce our
final approximation; namely, we assume that the φ̂j can be obtained usingNeX

φ̂j (δ0 /∆c ) =−1/3

pi Zi (Zi + ξi )φ0j (Zi , Ĕ0 , 136 Zi

mδ0 /∆c )i=1≈ φj (δ0 )Ne
X

pi Zi (Zi + ξi ) = φj (δ0 )ZT .

(2.91)

i=1

In order to justify the reasonableness of this approximation, assume for all i that
−1/3

1  136 Zi

mδ0 /∆c < δmax (Zi , Ĕ0 ) .

(2.92)

Then using Equations 2.50 and 2.51 for the φj we obtain
Ne
X

−1/3

pi Zi (Zi + ξi )φj (136 Zi

mδ0 /∆c )

i=1

Ne
X

=

i=1
Ne
X

≈

=
=
=
≈

i=1

h



h



pi Zi (Zi + ξi ) 21.12 − 4.184 `npi Zi (Zi + ξi ) 21.12 − 4.184 `n

21.12 − 4.184 (`n (136 mδ0 /∆c ) + ZG ) ZT



21.12 − 4.184 `n δ0 ZT
21.12 − 4.184 `n (δ0 + 0.952) ZT= φj (δ0 )ZT .

Q.E.D





i

mδ0 /∆c + 0.952−1/3mδ0 /∆c

136 Zi21.12 − 4.184 `n (136 mδ0 /∆c ) ZT − 4.184





−1/3

136 Zi

Ne
X

i

−1/3

pi Zi (Zi + ξi )`n Zi

i=1(2.93)

Butcher and Messel[39] make this approximation, although they don’t mention it explicitly. As
with our previous approximation, this approximation could be avoided (i.e., if we were willing tohave PEGS fit the A and B functions in some convenient way for EGS).
47

We proceed now by using Equation 2.91 to eliminate the φ̂j from Equation 2.78 and Equation 2.79 yieldingÂ(∆E , Ĕ0 )

=3φ1 (δ0 ) − φ2 (δ0 ) ZT

+8 ZB − (ZF if Ĕ0 > 50, 0)

B̂(∆E , Ĕ0 )

,(2.94)

φ1 (δ0 )ZT + 4 ZB − (ZF if Ĕ0 > 50, 0)=

If these are now used in Equations 2.89 and 2.90 we obtain
A(δ0 ) =B(δ0 ) =.

(2.95)

3φ1 (δ0 ) − φ2 (δ0 ) + 8(ZV if Ĕ0 > 50, ZG )
2
3 +8

h

φ (δ0 ) + 4(ZV if Ĕ0 > 50, ZG )h 1(2.96)

i`n 183 + (ZV if Ĕ0 > 50, ZG )(2.97)i

4 `n 183 + (ZV if Ĕ0 > 50, ZG ) .

We now return to Equation 2.77 which we were trying to factor. We havedΣ̆BremdE

2
A(δ0 )Âmax (Ĕ0 )3
14(ZAB − ZF )



=
×

(

=

2
1−E
A(δ0 )
3
E






1−E
E



0

+ B(δ )B̂max (Ĕ0 ) E2
ZT + 8 ZA + ZB − (ZF if Ĕ0 > 50, 0)
30+ B(δ )E [4(ZA + ZB − (ZF if E0 > 50, 0))])14(ZAB − ZF )#

[ZA + ZB − (ZF if Ĕ0 > 50, 0)](ZAB − ZF )

=
×

("

ZT
4
1
+
A(δ0 )
9 [ZA + ZB − (ZF if Ĕ0 > 50, 0)] 3
0+B(δ )E)[ZA + ZB − (ZF if Ĕ0 > 50, 0)](ZAB − ZF )

=
×



("

1
4
+
`n 2
3 9`n 183[1 + (ZU if Ĕ0 > 50, ZP )]



 

1
1−E
1
×
[A(δ0 )] +[2E][B(δ0 )]
`n 2
E
2

1−E
E





!#

.

(2.98)We then see that for Ĕ0 ≤ 50, the case dealt with in Butcher and Messel[39] , we havedΣ̆Brem
dE

=

4
1
+
3 9`n 183(1 + ZP ) 1
(ZA + ZB )0+[2E][B(δ )].
2(ZAB − ZF )



`n 2

 



48

1
`n 2



1−E
E



[A(δ0 )]
(2.99)

This agrees with formula (10) of Butcher and Messel[39] (except that they use ZAB = ZA + ZB anddo not use ZF in their X0 definition), since our ZA , ZB , ZP are the same as their a, b, p. Now
ignoring the factor preceding the {} in Equation 2.98, noting that we require k̆ > AP (the photon
energy cutoff) and also that energy conservation requires k̆ < Ĕ0 − m, we obtain the factorization(see Equation 2.64)




4
1
h
i ,
α1 = `n 2  +3 9 `n 183 1 + (ZU if Ĕ0 > 50, ZP )1
f1 (E) =
`n 2




1−E
E



for E ∈ (0, 1) ,

(2.101)

g1 (E) = A δ0 (E) if E Ĕ0 ∈ AP , Ĕ0 − m , 0





1
,
2
f2 (E) = 2E for E ∈ (0, 1) ,,(2.102)(2.103)α2 =(2.104)

g2 (E) = B δ0 (E) if E Ĕ0 ∈ (AP , Ĕ0 − m), 0

(2.100).

(2.105)

We notice that f2 (E) is properly normalized, but that f1 (E) has infinite integral over (0,1). Instead
we limit the range over which f1 (E) is sampled to 2−NBrem , 1 , where NBrem is chosen such that
2−NBrem ≤APĔ0< 2−(NBrem −1) .

(2.106)

α1j f1j (E) g1j (E)(2.107)

To sample f1 (E) we further factor it to
f1 (E) =

NX
Brem
j=1

where
α1j = 1 ,f1j (E) =

 1

`n 2

2j−1 if E < 2−j , 
1 (1 − E2j−1 )if E ∈ 2−j , 2−j+1 , 0 ,`n 2Eg1j (E) = 1 .

(2.108)(2.109)(2.110)The f1j are properly normalized distributions.
We sample f2 (E) by selecting the larger of two uniform random variables (see Section 2.2);
namely,E = max (ζ1 , ζ2 ) ,(2.111)
where ζ1 and ζ2 are two random numbers drawn uniformly on the interval (0, 1). To sample f1 (E),
we first select the sub-distribution index
j = Integer Part (NBrem ζ1 ) + 1 .
49(2.112)

Then to sample from f1j (E), first let
p = 21−j , E 0 = E/p .

(2.113)We then relate the distributions of E and E 0 , as follows. Suppose x̂, ŷ are random variables withprobability density functions f (x) and g(y) and cumulative distribution functions F (x) and G(y).
Further, suppose that x̂ is related to ŷ byx̂ = h(ŷ)(2.114)

with h monotonic increasing (↑) or decreasing (↓). Then clearly
F (x) = P r{ x̂ < x} = P r{ h(ŷ) < x}
P r{ ŷ < h−1 (x)} if h ↑, P r{ ŷ > h−1 (x)}

=





G(h−1 (x)) if h ↑, 1 − G(h−1 (x)) if h ↓=Letting.

(2.115)x = h(y), y = h−1 (x) ,(2.116)F (h(y)) = (G(y) if h ↑, 1 − G(y) if h ↓) .

(2.117)

we have
Differentiating with respect to y, we obtain,Nowf (h(y))dh(y)= (g(y) if h ↑, −g(y) if h ↓) .
dy

(2.118)

dh(y)
=
dy



dh(y)
dh(y)
if h ↑, −if h ↓dy
dy(2.119)Hencef (h(y)) = g(y)/

dh(y)
dy

,

. dh(h−1 (x))

f (x) = g h−1 (x)dy.

(2.120).

(2.121)As an example, we claim that ifg(E 0 ) =

1
`n 2





1 if E 0 ∈ 0,

1
1 − E0if E 0 ∈
,
2
E0


and
E = h(E 0 ) = E 0 p, E 0 = h−1 (E) = E/p,thenf (E) =

1
`n 2





1
,1
2



dh(E 0 )=p,
dE 0(2.122)(2.123)1
1 − E/p
if E ∈ (0, p/2),
if E ∈ (p/2, p)
p
E



50

,

.

(2.124)

That is, using Equations 2.122 and 2.123 in Equation 2.121, we obtainf (E) =

. dh

g h−1 (E)dy(h−1 (x))(2.125)= g(E/p)/p





E
1
1 − E/p 1
1
1
1
if
∈ 0,
· if E/p ∈,1,=`n 2 p
p
2
E/p
p
2


1
1
1 − E/p=
if E ∈ (0, p/2),
if E ∈ (p/2, p)
Q.E.D.
`n 2 pE
Thus we sample f1j (E) by first sampling E 0 from Equation 2.121 and then letting E = E 0 p.
The g(E 0 ) in Equation 2.122 may be decomposed according tog(E 0 ) =2X

α0i fi0 (E 0 ) ,(2.126)

i=1

with
α01 =
α02 =

1
1
, f10 (E 0 ) = 2 if E 0 ∈ 0,,0
2 `n 22

1
1−, f20 (E 0 ) =2 `n 2













1 − E01
if E 0 ∈1 ·0E(`n 2) − 2

,(2.127)
!

1
,1 ,0
2


.

(2.128)We sample f10 (E 0 ) by letting E 0 = ζ/2, where ζ is a random number uniformly drawn on the interval(0, 1). To sample f20 (E 0 ) we let E 0 = 1− 12 x, x ∈ (0, 1), and sample x from the frequency distributionfunctionh(x) = α00 f 00 (x) g 00 (x)(2.129)
where
α00 =

1
1
0.5
, f 00 (x) = 2x, g 00 (x) == 0(4 `n 2) − 2
2−x
E(2.130)We already know how to sample f 00 (x) (e.g., see Equation 2.111), so this completes the details ofsampling the bremsstrahlung spectrum.
Let us now consider the pair production interaction. The general developments are quite analogous to the bremsstrahlung case and we obtain the following formulas:
E≡

Ĕ
k̆

,

(2.131)

where Ĕ is the energy of one of the secondary electrons, k̆ is the incident photon energy,∆E =1k̆E(1 − E)

δ 0 = ∆C ∆E ,51,

(2.132)(2.133)

dΣ̆P airdE

ZA + ZB − (ZF if k̆ > 50, 0)
ZAB − ZF

=
×

("

!

1
2
−
3 36 `n 183[1 + (ZU if k̆ > 50, ZP )]

"

1
+
12

#

4
1
+
3 9 `n 183[1 + (ZU if k̆ > 50, ZP )]"1
× 12 E −
2


2 #

0

)

A(δ )

[1] C(δ0 )!#,(2.134)A(δ0 ) = same as for bremsstrahlung case,0(2.135)0

C(δ0 )= 3φ21 (δ )+φ2 (δ )+16(ZV if k̆>50, ZG ) ,(2.136)∆C= same as for bremsstrahlung case.

(2.137)− 3 +(16 `n(183+(ZV if k̆>50, ZG ))We sample Equation 2.134 by the following decompositions:
"

1
2
α1 =
−
3 36 `n 183[1 + (ZU if k̆ > 50, ZP )]

f1 (E) = 1, E ∈ (0, 1) ,1
f2 (E) = 12 E −
2

2

,

(2.138),41+
3 9 `n 183[1 + (ZU if k̆ > 50, ZP )]

,(2.139)

g1 (E) = C(δ0 (E)) if k̆E ∈ (m, k̆ − m), 0
1
α2 =
12

#

(2.140)!

(2.141)E ∈ (0, 1) ,

(2.142)

g2 (E) = A(δ0 (E)) if k̆E ∈ (m, k̆ − m), 0.

(2.143)

As with bremsstrahlung, we ignore the factor ahead of the {} in Equation 2.134 when sampling.
This actually has the effect of giving some effective Coulomb correction below 50 MeV, since thefactor neglected would have to be larger than 1 for k̆ < 50
We summarize the “run-time” bremsstrahlung and pair production cross sections:dΣ̆Brem, Run−timedE= `n 2
×

h

1
`n 2



4
1
3 + 9 `n 183[1+(ZU if Ĕ0 >50,ZP )]



1−E
E

52

i

A(δ0 ) +

h i
1
2



[2E] B(δ0 ) ,(2.144)

dΣ̆P air, Run−time
dE

=



1
2
3 − 36 `n 183[1+(ZU if k̆>50,ZP )]

+



1
12





[1] C(δ0 )1
4
3 + 9 `n 183[1+(ZU if k̆>50,ZP )]× 12 E − 21

2 



A(δ0 ) .

(2.145)

When these are divided by Ĕ0 and k̆, respectively, they becomedΣ̆Brem, Run−time
dk̆

and

dΣ̆P air, Run−timedĔ

which are computed by PEGS functions BREMDR and PAIRDR. In general the letter R, as the lastletter of a PEGS cross section function, means “run-time” function. PEGS may be used to plotthese for comparison with the more exact BREMDZ and PAIRDZ which evaluate Equations 2.43 and2.44.
It will be observed that the pair production formulas are symmetric about E = 1/2. One of theelectrons will be given energy k̆E and the other k̆(1 − E). The choice of which one is a positron ismade randomly. Since we need to know which particle has the lower energy so we can put it on
the top stack position, we restrict the range of of E to (0,1/2) and double f1 (E) and f2 (E), thus
guaranteeing that any sampled value of E will correspond to the electron with the lower energy.
We thus sample f1 (E) by letting E = 1/2ζ, and we sample f2 (E) by lettingE=

1(1 − max (ζ1 , ζ2 , ζ3 ))2(2.146)

where the ζ values are drawn uniformly on the interval (0, 1) (see Section 2.2).
A special approximation which has been carried over from previous versions is that if theincident photon has energy less than 2.1 MeV, one of the electrons is made to be at rest and theother given the rest of the available energy. One reason for making this approximation is thatthe pair sampling routine becomes progressively more inefficient as the pair production threshold isapproached. Perhaps a better approximation would be to pick the energy of the low energy electron
uniformly from the interval (m, k̆/2).
We now conclude this section on bremsstrahlung and pair production with a few general remarks.
We first note that for δ0 ≤ 1, that A(δ0 ), B(δ0 ) and C(δ0 ) are all quadratic functions of δ0 . The
three coefficients depend on whether the incident energy is above or below 50 MeV. For δ0 > 1,
A(δ0 ), B(δ0 ) and C(δ0 ) are equal and are given by
A, B, C(δ0 ) =

φ1 (δ0 ) + 4 ZV if Ĕ0 , k̆ > 50, ZG
φ1 (0) + 4 ZV if Ĕ0 , k̆ > 50, ZG



(2.147)

which will have the form c1 + c2 `n(δ0 + c3 ). The A, B, C must not be allowed to go negative.
PEGS computes a maximum allowed ∆E above which the A, B, C are considered to be zero.
53

The various parameters needed to sample the secondaries’ energies for the bremsstrahlung andpair production interactions are computed in the PEGS routine DIFFER. The parameters are stored
for each medium during execution of EGS in COMMON/BREMPR/.
One other use for the bremsstrahlung cross section is for computing the mean energy loss perunit length of an electron due to emission of soft photons (i.e., those with energy below the photon
cutoff energy Ap ). This is given by
dĔ
−
dx

!

=
Sof t Brem

Z AP
0

k̆

dΣ̆Brem
dk̆

!

dk̆

(2.148)which will be discussed in Section 2.13.

2.7.1

Bremsstrahlung Photon Angular Distribution

So far we have only discussed the selection of the energy of the secondaries. Since these are interactions with three body final states, the polar angles of the secondary particles are not uniquelydetermined by the secondary energies, and a complete simulation would sample from some appropriate distributions. However the angles at which products from these reactions are usually emitted
are small compared to angular deviations resulting from multiple scattering. Previous versions of
EGS therefore assumed that the direction of an electron emitting bremsstrahlung is unchanged,that a bremsstrahlung photon is emitted at an angle relative to the incident electron direction,
θ = m/Ĕ0 , and that pair produced particles have production angles relative to the incident photon
direction given by θ = m/k̆. The azimuthal angle for the first product particle is chosen randomly
and the other product particle is given the opposite azimuth.
The above model of the angular distribution of newly created bremsstrahlung photons may beoverly simple for some applications. The angle given by m/k̆ is a good estimate of the expected
average scattering angle, and at high energies, where the distribution is strongly peaked in theforward direction, more accurate modeling of scattering angles does not significantly improve theaccuracy of shower simulations. Additionally, at low energies, particularly in thick targets, the effectof multiple scattering of the initiating electrons greatly overwhelms the impact of photon angular
distributions in defining the development of the shower, and the extra effort and computing timenecessary to implement bremsstrahlung angular distribution sample is not worthwhile.
While it was recognized that the above argument may break down for applications involving
thin target bremsstrahlung spectra, it was discovered that the above assumption about multiple
scattering dominance does not apply even for thick targets at low energies (10 MeV or so) for
narrow beams such as those employed in some medical linacs for producing photon beams forradiotherapy[54]. Thus, options for more accurate modeling of bremsstrahlung scattering angles in
EGS have been developed.

54

Angular distribution formulas
The formula employed for the angular sampling routine is 2BS of Koch and Motz[91], which is the
cross section, differential in photon energy and angle,
4Z 2 r02 dk
dσk,Θ =
ydy
137 k

(

#

"

)

(E0 + E)2
4y 2 E
16y 2 E
E02 + E 2
ln M (y) ,−+−(y 2 + 1)4 E0 (y 2 + 1)2 E02(y 2 + 1)2 E02 (y 2 + 1)4 E0(2.149)where,
1y = E0 Θ;
=
M (y)



k
2E0 E

2

+

Z 1/3
111(y 2 + 1)

!2,

and, the following definitions for the variables apply:

k
Θ
Z
r0
E0 , Eenergy of the photon in units of me c2
angle between the outgoing photon and the incoming electron direction (in radians)
atomic number of the target material≡ e2 /me c2 (classical electron radius)
initial and final electron energy in units of me c2

The following table, copied from the Koch and Motz article, outlines the essential approximations
employed in the derivation of Equation 2.149.

i)
ii)
iii)
iv)
v)

Approximation
Approximate screening potential
Born approximation
Extreme relativistic
Small angles
Approximate e− angular integration

Condition of validity(Ze/r)e−r/a(2πZ/137β0 ), (2πZ/137β)  1
E0 , E, k  1
sin Θ = Θ
Θ < (Z 1/3 /111E0 )It should be noted that only the angular distribution part of Equation 2.149 is employed. The
cross section differential in photon energy employed by the EGS5 code is far less restrictive (Approximation (iii) plus Thomas-Fermi screening factors). For the purposes of modeling electron linacs,
the ultimate test of these approximations is comparison with experiment. In this regard, Koch andMotz present encouraging data (their Figure 17) which exhibits excellent agreement between experiment and Equation 2.149 for 4.54 MeV electrons on Au. Although use of Equation 2.149 violates
constraints ii), iii) and iv) in the cases they showed, the deviation was at worst 10% (at large angles)
and usually much better. In particular, violating the Born approximation constraints seemed notespecially deleterious to the comparison. The conditions of this experiment are similar to those55

used for medical linacs (6–50 MeV, high-Z targets) and therefore the employment of Equation 2.149
seems justified. At lower energies, use of Equation 2.149 still needs to be demonstrated. At higher
energies, the constraints are not so badly violated except for the Born approximation when high-Zmaterials are used. Again, experimental data will judge the suitability of Equation 2.149 in thiscontext.

Sampling procedure
To sample the photon angular distribution, a mixed sampling procedure is employed. Since it is theangular distribution that is required, the overall normalization of Equation 2.149 is unimportant
including any overall energy-dependent factors. The following expression for p(y) is proportional
to Equation 2.149:

p(y)dy = f (y 2 )Nr g(y 2 )dy 2 .
Defining x = y 2 ,f (x) =

(2.150)1 + 1/(πE0 )2,(x + 1)2(2.151)and,g(x) = 3(1 + r 2 ) − 2r − [4 + ln m(x)][(1 + r 2 ) − 4xr/(x + 1)2 ],where,
r = E/E0 ; m(x) =



1−r
2E0 r

2

+

Z 1/3
111(x + 1)

!2

(2.152).

Note that 1/E0 (high frequency limit)≤ r ≤ 1(low frequency limit). Nr is a normalization constantwhich will be discussed later.
The function f (x) will be used for direct sampling. It can be easily verified that this function is
R (πE )2
normalized correctly, (i.e., 0 0 f (x)dx = 1) and the candidate scattering angle is easily foundby inversion to be:
s
1
ζ
Θ̂ =
,
(2.153)E0 1 − ζ + 1/(πE0 )2
where ζ is a random number selected uniformly on the range (0,1) and the “hat” over Θ signifies
that it is a quantity determined by random selection.
The function g is sampled using the rejection technique. In order to employ this technique,the optimum case is to have the location of the maximum of the function, xmax characterized
allowing the most efficient determination Nr = g(xmax )−1 . Failing this, the next best scenario is
to overestimate g(xmax ). The closer this estimate is to the true maximum value, the more efficient
the rejection technique will be. Unfortunately, characterizing g in complete generality proved to bevery difficult. The following observations were made, however. The maximum value of g(x) occurs

56

at either x = 0, x = (πE0 )2 (i.e., at the minimum or maximum values of x), or in the vicinity of
x = 1. Therefore, the rejection function normalization was chosen to be:
Nr = {max[g(0), g(1), g((πE0 )2 )]}−1 .

(2.154)

A more complete discussion of bremsstrahlung angular distributions as adapted to EGS, may
be found documented elsewhere[29].

2.7.2Pair Angle Sampling

In previous versions of EGS, both particles in all newly created e− e+ pairs were set in motion at
fixed angles Θ± with respect to the initiating photon direction. Θ± , the scattering angle of thee+ or e− (in radians), is of the form Θ± = 1/k where k is the energy of the initiating photon inunits of mo c2 , the rest mass of the electron. Defined in this way, Θ± provides an estimate of the
expected average scattering angle6 .
The motivation for employing such a crude approximation is as follows: At high energies the
distribution is so strongly peaked in the forward direction that more accurate modeling will not significantly improve the shower development. At low energies, particularly in thick targets, multiple
scattering of the resultant pair as the particles slow will “wash out” any discernible distribution in
the initial scattering angle. Therefore, the extra effort and computing time necessary to implement
pair angular distributions was not considered worthwhile. It was recognized, however, that theabove argument would break down for applications where the e+ e− pair may be measured before
having a chance to multiple scatter sufficiently and obliterate the initial distribution, and this wasindeed found to be the case.
To address this shortcoming, two new options for sampling the pair angle were introduced, as
described in the two following subsections. Procedures for sampling these formulas are given in
the next sections. The formulas employed in this report were taken from the compilation by Motz,
Olsen and Koch[111].

Leading order approximate distribution
As a first approximation, the leading order multiplicative term of the Sauter-Gluckstern-Hull formula (Equation 3D-2000 of Motz et al.[111]) was used:sin Θ±dP=,dΘ±2p± (E± − p± cos Θ± )2(2.155)

6
The extremely high-energy
form of the leading order approximation discussed later implies that the distribution√
should peak at Θ± = 1/( 3E± ). However, the Bethe-Heitler cross section used in EGS5 peaks at E± = k/2 andthe approximation Θ± = 1/k is a reasonable one on average, given the highly approximate nature of the angularmodeling.

57

where Θ± is the e± scattering angle (in radians), E± and p± are the e± total energy and momentum
in units of the electron rest-mass energy, mo c2 .
The Sauter-Gluckstern-Hull formula, which is used in ETRAN-based codes[149], was derived
under the following approximations:

i)
ii)
iii)

Approximation
No screening
First order Born approximation
Negligible nuclear recoil

Condition of validity
Low-Z elements(2πZ/137β± )  1k  (1/mn ), k  mn (large angles)

where mn is the rest mass energy of the nucleus in units of mo c2 and β± is the magnitude of thevelocity of the e± in units of the speed of light, c.
The full Sauter-Gluckstern-Hull formula differs from the above by a modulating factor which
varies between 0 and approximately 2 and thus it is to be regarded as a crude approximation evenin its region of validity.

The Schiff distribution
This formula employed for the angular sampling routine is Equation 3D-2003 of Motz et al.[111],
which is the cross section, differential in photon energy and angle,2
2αZ 2 r02 E±dσ 2=
dE± dΩ±
π
k3

(

"

#

where,
1
u = E± Θ± ;
=
M (y)



k
2E+ E−

2

+

Z 1/3
111(u2 + 1)

!2,

and, the following definitions for the variables apply:k
E+ , E−
Θ±
dΩ±
Z
r0
α

)

2 + E2E+(E+ − E− )2 16u2 E+ E−
4u2 E+ E−−−−++
ln M (y) ,(u2 + 1)2
(u2 + 1)4
(u2 + 1)2
(u2 + 1)4
(2.156)energy of the photon in units of mo c2
final e± total energy in units of mo c2 (k = E+ + E− )angle between the outgoing e± and the incoming photon direction (in radians)
Differential solid angle of the outgoing e±
atomic number of the target material≡ e2 /mo c2 (classical electron radius)≡ e2 /(h̄c) = 1/137... (fine structure constant)

58

The following table, derived from the Motz et al. article, outlines the essential approximations
involved in the development of Equation 2.156.

i)
ii)
iii)
iv)
v)

Approximation
Approximate screening potential
First order Born approximation
Extreme relativistic
Small angles
Negligible nuclear recoil

Condition of validity(Ze/r)e−r/a(2πZ/137β± )  1
E± , k  1
Θ± = O(E± )k  (1/mn ), k  mn (large angles)

It should be noted that only the angular distribution part of Equation 2.156 is employed. The crosssection differential in electron energy employed by the EGS code is far more widely applicable, as
it was derived under just Approximation (iii) above with Thomas-Fermi screening factors.

Sampling procedure
The distribution expressed by Equation 2.155 is normalized as
Z π
0

dΘ±



dP
dΘ±



= 1,

and may be sampled from the formulasp2 ζ(1 − ζ)
E± (2ζ − 1) + p±
sin Θ± =; cos Θ± =,p± (2ζ − 1) + E±
p± (2ζ − 1) + E±

(2.157)

where ζ is a random number uniform on the range [0, 1]. The sin Θ± and cos Θ± forms are relatedby trigonometric identities but both forms expressed by Equation 2.157 are useful.
To sample the angular distribution expressed by Equation 2.156, a rejection technique is appliedfollowing a change of variables. Consider the change of variables:ξ=1u2 + 1;1
2 π 2 ≤ ξ ≤ 1.
1 + E±

Then, applying the small angle approximation, dΩ± −→ 2πΘ± dΘ± , the angular distribution rejection function, g, has the form:dg= Ng {2 + 3(r + r −1 ) − 4[(r + r −1 + 4ξ − 4ξ 2 ][1 + ln m(ξ)/4]},dξ
wherem(ξ) =(1 + r)(1 + r −1 )
2k

!2

+

59

Z 1/3 ξ111!2

; r = E− /E+ ,(2.158)and Ng is a normalization factor that should be chosen so that"#1g(ξ) ≤ 1 ∀ ξ ∈2 π2 , 1 .
1 + E±The position of the maximum of the function g proved difficult to characterize accurately andso a two-step iterative scheme was developed based upon the slow variation of the logarithmic term
[1+ln m(ξ)/4] in Equation 2.158 and the observation that the position of the maximum is relatively
independent of the value of r. For the purposes of estimating the maximum value of g, r is set
equal to 1, and a satisfactory algorithm for calculating Ng is
"

1
2 π2
1 + E±

Ng = 1.02 max g

!

#

(1)
, g(ξmax),(2.159)(1)

where ξmax is an estimate of the position of the maximum of g after a one-step iteration. The
zeroth-order estimate is:
(0)
ξmax
= max

(

"

222
1
0.01, max
, min 0.5, 2 1/3
2
2
1 + E± π
k Z


#)

,

and the second iteration yields:



where





s

1
1
α0
(1)
0.5, −
ξmax
= max 0.01, max ,min+ sgn(α0 )2 π2
2 3β 0
1 + E±

α0
3β 0

2




1
+ 

4

(0)
(0)
α0 = 1 + ln m(ξmax)/4 − β 0 (ξmax− 1/2),(0)β0 =ξmax (Z 1/3 /111)2andsgn(α0 ) =(0)2m(ξmax )(.

+1 if α0 ≥ 0−1 if α0 < 0The extra 1.02 in Equation 2.159 is a “safety factor”.

The Schiff threshold
The Schiff distribution breaks down mathematically for Eγ < 4.14 MeV. To prevent non-physicalmodeling, if a user has requested the Schiff distribution, the lowest order approximate distributionof Equation 2.155 is used when the photon energy is less than the 4.14 MeV threshold.
A more complete discussion of pair angular distributions as adapted to the EGS code, may befound documented elsewhere[26].
60Figure 2.3: Feynman diagrams for two body interactions with electrons.

Figure 2.4: Definition of two-body scattering angles.

2.8

Interactions With Atomic Electrons – General Discussion

In this section we consider some general aspects of two-body interactions with atomic electrons. The
details of two-body interactions involving specific sets of particles will be examined in subsequent
sections. The reactions we will investigate, Compton scattering, Møller scattering, and Bhabha
scattering, and two photon positron-electron annihilation, are illustrated in the Feynman diagramsin Figure 2.3.
The kinematics of these interactions are described by the four-vector equation
P1 + P2 = P3 + P4 ,(2.160)where we employ the convention that P1 is the incident particle, P2 is the atomic electron (assumed
free and at rest), P3 is the particle whose energy will be calculated, and P4 is the other final stateparticle. Since such collisions are two body reactions, they take place in planes, with scattering
angles illustrated in Figure 2.4 , where we take the incident particle direction to be the z axis andthe interaction occurs in the x − z plane.
Letting mi , Ei , pi denote the mass, energy, and three-momentum of particle i, the four-momenta
61are then given by
P1 = (E1 , 0, 0, p1 )(2.161)P2 = (m, 0, 0, 0)(2.162)P3 = (E3 , p3 sin θ3 , 0, p3 cos θ3 )(2.163)P4 = (E4 , −p4 sin θ4 , 0, p4 cos θ4 ) .

(2.164)

If now we want to find the scattering angle θ3 , assuming we have determined all energy andmomenta, we solve Equation 2.160 for P4 and take its invariant square to get
P42 = P12 + P22 + P32 − 2(P1 + P2 ) · P3 + 2P1 · P2 .

(2.165)Making use of the relation Pi2 = m2i and Equations 2.161 and 2.164, we obtain
m24 = m21 + m2 + m23 + 2[E1 m − (E1 + m)E3 + p1 p3 cos θ3 ] ,(2.166)and for cos θ3 we arrive atcos θ3 =m24 − m21 − m2 − m23 + 2(E1 + m)E3 − 2E1 m.
2p1 p3(2.167)

Clearly, by symmetry, the above equation is still true if we interchange the indices 3 and 4 to obtaina relation for cos θ4 . Thus, we see that the scattering angles are uniquely determined by the final
energies. Azimuthal angles are uniformly distributed, provided, of course, that the two particles
have the opposite azimuth.
In the sections that follow we shall focus on the computation of the differential and total crosssections for these four processes, and we will provide the sampling methods used to determine thesecondary energies. We shall also reduce Equation 2.167 to the specific reactions and derive theexpressions used in EGS to determine the cosines of the scattering angles. In most cases, we get
the sines of the scattering angles using the formulas√sin θ3= 1 − cos2 θ3(2.168)√2(2.169)sin θ4 = − 1 − cos θ4 .
The reason for making sin θ4 negative is that this effectively achieves the opposite azimuth within
the frame work of the EGS routine UPHI, as discussed later. Note that we drop the subscripts on
the scattering angles for simplicity in the discussions below.

2.9

Compton Scattering

The differential and total Compton scattering cross sections are given by formulas originally due
to Klein and Nishina [90]:dΣ̆Compt (k̆0 )dk̆

=

X0 n πr02 m
k̆02



62

C1
+ C2
E



E + C3 + E#(2.170)whereX0 =radiation length (cm),n =electron density (electron/cm 3 ),r0 =classical electron radius (cm2 ),m =electron rest energy (MeV),k̆0 =incident photon energy (MeV),k̆ =scattered photon energy (MeV),Ek̆/k̆0 ,=C1 =(k00 )−2 ,k00 =k̆0 /m,C2 =1 − 2(1 + k00 )/(k00 )2 ,C3 =(1 + 2 k00 )/(k00 )2 .

The Compton cross section integrated over the energy range from k̆1 to k̆2 can be expressed as
Z k̆2
k̆1

dΣ̆Compt (k̆0 )
dk̆
C2 `n

dk̆ =

X0 nπr02
k00



C1



1
1
−
E1 E2



+

E2
+ E2 (C3 + E2 /2) − E1 (C3 + E1 /2)
E1

(2.171)where
E1 = k̆1 /k̆0 ,E2= k̆2 /k̆0 .

The total scattering cross section is obtained from Equation 2.171 with k̆1 and k̆2 set to theminimum and maximum possible scattered photon energies. To see what these are, we use Equation 2.167, noting that m1 = m3 = 0, m4 = m, E1 = p1 = k̆0 , E3 = p3 = k̆, E4 = Ĕ, and p4 = p̆,and arrive at(k̆0 + m)k̆ − k̆0 m.
(2.172)cos θ =k̆0 k̆
Solving for k̆ we get the well-known formula
k̆ =

k̆0
1 + (1 − cos θ)k̆0 /m.

(2.173)The maximum and minimum values of k̆ occur at cos θ = 1, −1, ork̆max = k̆0 ,k̆min =

(2.174)k̆0

1 + 2k̆0 /m
63.

(2.175)

Thus
Σ̆Compt, T otal (k̆0 ) =Equation 2.171 with k̆2 = k̆0 , k̆1 =k̆0
1 + 2k̆0 /m!

.

(2.176)PEGS functions COMPDM, COMPRM, and COMPTM evaluate Equations 2.170, 2.171, and 2.176, respectively. Note that in the discussion above, the atomic electrons are assumed to be unbound and atrest. In section 2.18 we will examine the effects of of this approximation.
We next consider techniques for sampling the energy of the scattered gamma ray. If we define
the variable we wish to sample as
E = k̆/k̆0 ,(2.177)
we can see from Equations 2.174 and 2.175 that E must be in the interval (E0 , 1), where
E0 =

1

.

1 + 2 k̆0 /m(2.178)

We start with a form of the differential cross section similar to that given by Butcher and Messel[39]:dΣ̆Compt
X0 nπr02 m
=
dE
k̆0



1
+E
E

#

"

E sin2 θ∝ f (E)g(E) .
1−
1 + E2h2(2.179)isin θ
as a rejection function.
We will sample f (E) = E1 + E over (E0 , 1) and use g(E) = 1 − E 1+E
2

We factorize

h

1
E +E

i

over (E0 , 1) as follows
f (E) =

2
X
1
+E =αi fi (E)E
i=1(2.180)where1
α1 = `n(1/E0 ), f1 (E) = `n(1/E
0)

α2

 
1
E

, E ∈ (E0 , 1)

2E
= (1 − E02 )/2, f2 (E) = (1−E
2 ) , E ∈ (E0 , 1).

(2.181)(2.182)0

We sample f1 by letting

E = E0 eα1 ζ(2.183)where ζ is a random number drawn uniformly on the interval (0, 1). We could sample f2 by taking
the larger of two random numbers if we were willing to reject sampled values less than E0 . but thiswould get very inefficient for low energy photons. Instead we make a change of variables. Let
E0 =E − E0.
1 − E0(2.184)

Then in order to give E the proper distribution, E 0 must have the distribution
f20 (E 0 ) = f2 (E)dE= α01 f100 (E 0 ) + α02 f200 (E 0 )dE 064(2.185)

whereα01 =k00, f 00 (E 0 ) = 2E 0 , E 0 ∈ (0, 1)0
k0 + 1 1(2.186)

α02 =1, f 00 (E 0 ) = 1, E 0 ∈ (0, 1) .
k00 + 1 2

(2.187)

Both of these sub-distributions are easily sampled.
To compute the rejection function it is necessary to get sin2 θ. Lett=m(1 − E)k̆0 E.

(2.188)Then using Equation 2.172, we havecos θ =

(k̆0 + m)k̆ − k̆0 m
k̆0 k̆

=1+mE − mk̆0 E=1−t .

(2.189)Thus
sin2 θ = 1 − cos2 θ = (1 − cos θ)(1 + cos θ) = t(2 − t) .

(2.190)

When the value of E is accepted, then sin θ and cos θ are obtained viapsin2 θ

(2.191)cos θ = 1 − t .

(2.192)sin θ =The sampling procedure is as follows:
1. Compute the parameters depending on k̆0 , but not E: k00 , E0 , α1 , and α2 .
2. Sample E in the following way: If α1 ≥ (α1 + α2 )ζ1 , use E = E0 eα1 ζ2 . Otherwise, use
E = E0 + (1 − E0 )E 0 , where E 0 is determined from
E 0 = max (ζ3 , ζ4 ) if k00 ≥ (k00 + 1)ζ2
or from
E 0 = ζ3 otherwise.
3. Calculate t and the rejection function g(E). If ζ4 (or ζ5 ) < g(E), reject and return to Step 2.
After determining the secondary energies, Equation 2.167 is used to obtain the scattering angles
and UPHI is called to select random azimuth and to set up the secondary particles in the usual way.
Note again that a more detailed treatment of Compton scattering is provided in section 2.18.

65

2.10

Møller Scattering

The form of the cross sections and the sampling methods that we use for modeling Møller andBhabha scattering follow those given by Messel and Crawford[103] (e.g., see p.13-14 therein) exceptthat various misprints have been corrected.
The differential Møller[109] cross section is given bydΣ̆Møller (Ĕ0 )dĔ

=

X0 n2πr02 m
β 2 T̆02



C1 +

1
E



1
1
− C2 + 0
E
E




1
− C2
E0



(2.193)

where
Ĕ0 =

incident electron energy (MeV),T̆0 =Ĕ0 − m = incident kinetic energy (MeV),Ĕ =energy of scattered electron (MeV),

T̆=

kinetic energy of scattered electron (MeV),

E=

T̆ /T̆0 = fraction of kinetic energy to scattered electron,=

1 − E = fraction of kinetic energy remaining,

E

0

γ =

Ĕ0 /m,C1 =[(γ − 1)/γ]2 ,C2 =beta2 =

(2γ − 1)/γ 2 ,1 − 1/γ 2 = (v/c)2 ,and where the other terms have been defined previously.
Because of the ambiguity in the identity of the initial and final electrons, the cross section issymmetric with respect to the interchange of E with E 0 . Another consequence of this is that E isrestricted to lie in the interval (0, 1/2). It can be seen that Equation 2.193 is singular at E = 0 (alsoat E 0 = 0 but the range of E 0 is now restricted to (1/2, 1)), and the total Møller cross section is
infinite. We get around this by considering as discrete collisions only Møller scattering interactions
for which the scattered electron acquire at least some threshold energy, AE (we also define thethreshold kinetic energy of TE = AE − m). Since the incoming electron energy is Ĕ0 + m and theminimum final energy is 2AE , we see that the threshold initial electron energy at which a discrete
Møller scattering event can take place is given by
ĔTMøller= 2AE − m = 2TE + m .
h(2.194)

Møller scattering interactions in which secondary electrons (δ–rays) are ejected with energies thanless AE are treated as part of the continuous energy loss process (see Section 2.13).
The integral of the Møller cross section over some energy range can be expressed as
Z Ĕ2
Ĕ1

dΣ̆Møller (Ĕ0 )
dĔ

dĔ =

X0 n2πr02 m
β 2 T̆066



C1 (E2 − E1 ) +

1
1
−
E1 E2

+
where

1
E2 E10
1
−
−
C
`n
2
E20
E10
E1 E20Ei = (Ĕi − m)/T̆0 ,Ei0= 1 − Ei ,

i = 1, 2 ,i = 1, 2 ,(2.195)(2.196)(2.197)and other symbols are the same as in Equation 2.193.
The minimum and maximum energies for the scattered electron (by convention, the scatteredelectron is taken to be the one which emerges with the lower energy) are AE and T̆20 +m, respectively.
When these limits are used for Ĕ1 and Ĕ2 in Equation 2.195, we obtain for the total discrete Møller
cross section (which we assume to non-zero only for Ĕ0 > ĔTMøller):hΣ̆Møller, Total (Ĕ0 ) == Equation 2.195 with Ĕ1 = AE and Ĕ2 = (T̆0 /2) + m .

(2.198)PEGS functions AMOLDM, AMOLRM, and AMOLTM evaluate Equations 2.193, 2.195, and 2.198, respectively.
In sampling for the resultant energy, we use the variable E = T̆ /T̆0 and obtainwheredΣ̆Møller (Ĕ0 )X0 n2πr02 m=
f (E)g(E)dE
T̆0 E01
E0,E ∈ (E0 , 1/2),
1 − 2E0 E 2g(E) = g1 [1 + g2 E 2 + r(r − g3 )],f (E) =

E0 = TE /T̆0 ,(2.199)(2.200)(2.201)(2.202)2(2.203)2

g2 = (γ − 1) /γ ,(2.204)2(2.205)r = E/(1 − E) .

(2.206)

g1 = (1 − 2E0 )/β ,2

g3 = (2γ − 1)/γ ,The sampling procedure is as follows:
1. Compute parameters depending on Ĕ0 , but not E: E0 , g1 , g2 , and g3 .
2. Sample E from f (E) by using

E = TE / [T̆0 − (Ĕ0 − ĔTMøller
)ζ1 ]).
h(2.207)

3. Compute r and the rejection function g(E). If ζ2 > g(E), reject and return to Step 2.
After the secondary energies have been determined, Equation 2.167 can be used to obtain the
scattering angles, and EGS routine UPHI can be called to select random azimuthal angles and set
up the secondary particles in the usual way.
67

2.11

Bhabha Scattering

The differential Bhabha[25], cross section, as formulated in PEGS, isdΣ̆Bhabha (Ĕ0 )X0 n2πr02 m
=
dĔ−
T̆02



1
E



1
− B1 + B2 + E (EB4 − B3 )
Eβ 2




(2.208)

where
Ĕ0 =

energy of incident positron (MeV),T̆0 =kinetic energy of incident positron (MeV),β =v/c for incident positron,γ =Ĕ0 /m,

Ĕ− =
E

=

y =

B1 =
B2 =
B3 =
B4 =

energy of secondary electron (MeV),(Ĕ− − m)/T̆0 = T̆− /T̆0 ,1/(γ + 1),
2 − y2 ,(1 − 2y)(3 + y 2 ),B4 + (1 − 2y)2 ,(1 − 2y)3 .

If Equation 2.208 is integrated between Ĕ1 and Ĕ2 , we obtain
Z Ĕ2
Ĕ1

dΣ̆Bhabha (Ĕ0 )dĔ−

X0 n2πr02 m h 1
dĔ− =
β2
T̆02



1
1
−
E1 E2



− B1 `nE2E1i+B2 (E2 − E1 ) + E22 (E2 B4 /3 − B3 /2) − E12 (E1 B4 /3 − B3 /2)(2.209)where
Ei = (Ĕi − m)/T̆0 ,i = 1, 2(2.210)and other symbols are the same as in Equation 2.208.
Unlike in Møller scattering, in Bhabha scattering, the final state particles are distinguishable,
so the upper limit for E is 1. Note that E is the fraction of the kinetic energy that the negative
atomic electron gets. There is still a singularity at E = 0 which is circumvented in the same way as
for Møller by requiring that the energy transfered to the atomic electron be at least TE = AE − m.
It should be noted that there is no singularity at E = 1 as there was for Møller, and in fact, thefinal positron energy may be less than AE (down to m). Thus, the threshold for a discrete Bhabha
interaction is AE , and as long as the positron is above the cutoff energy, it will have some non-zero
Bhabha cross section. Using the minimum and maximum Ĕ− for Ĕ1 and Ĕ2 in Equation 2.209, we
obtain the total cross section asΣ̆Bhabha (Ĕ0 ) = (Equation 2.209 with Ĕ1 = AE & Ĕ2 = Ĕ0 if Ĕ0 > AE , 0).
68(2.211)PEGS functions BHABDM, BHABRM, and BHABTM evaluate Equations 2.208, 2.209, and 2.211, respectively.
In sampling the differential cross section to obtain the electron energy, we take E = T̆− /T̆0 as
the variable to be sampled and obtain
X0 n2πr02 mdΣ̆Bhabha (Ĕ0 )=f (E)g(E)dE
T̆0 E0(2.212)where
1
E0, E ∈ (E0 , 1) ,1 − E0 E 21g(E) = (1 − E0 ) 2 − E (B1 − E (B2 − E(B3 − EB4 ))) ,βf (E) =

E0 = TE /T̆0 ,(2.213)(2.214)(2.215)y = 1/(γ + 1) ,(2.216)3

B4 = (1 − 2y) ,(2.217)2(2.218)2(2.219)B3 = B4 + (1 − 2y) ,B2 = (1 − 2y)(3 + y ),2

B1 = 2 − y .

(2.220)(Note that EGS uses variable YY to avoid conflict with the variable name of the y-coordinate ofparticle.)
The sampling method is as follows:
1. Compute parameters depending on Ĕ0 but not E: E0 , β, γ, B1 , B2 , B3 , and B4 .
2. Sample E from f (E) using

E = E0 /[1 − (1 − E0 )ζ1 ].

(2.221)

3. Compute the rejection function g(E). If ζ2 > g(E), reject and return to Step 2.
The rest of the procedure is similar that used in sampling from the Møller cross section exceptthat now the delta ray may have the most energy, in which case the contents of the two top locationsof the EGS particle stack must be interchanged to ensure that the particle with the lower energy
will be tracked first.

2.12

Two Photon Positron-Electron Annihilation

The two photon positron-electron annihilation cross sections in EGS are taken from Heitler[71] (seep. 268-270 therein). Using Heitler’s formula 6 (on p. 269), translating to the laboratory frame,69integrating over azimuth, and changing from an angle variable to an energy variable, we obtain thefollowing form of the differential cross section, which is used in PEGS:dΣ̆Annih (Ĕ0 )dk̆= S1 (k0 ) + S1 (A − k0 )(2.222)

where
Ĕ0 =

energy of incident positron (MeV),k̆ =energy of secondary photon of lower energy (MeV),γ =Ĕ0 /m,(2.224)A =γ + 1 = (available energy)/m,(2.225)T00

γ − 1 = (kinetic energy)/m,(2.226)

γ2 − 1 =(2.227)=p00=

p̆0 /m =

0

=

k̆/m,

k

S1 (x) =
C1 =
C2 =

q

(2.223)qAT00(2.228)C1 [−1 + (C2 − 1/x)/x],X0 nπr02,
AT00 m
A + 2γ/A .

(2.229)(2.230)(2.231)

We see that Equation 2.222 satisfies, in a manifest way, the symmetry under exchange of theannihilation photons.
Integrating Equation 2.222 between k̆1 and k̆2 we obtain
Z k̆2
k̆1

dΣ̆Annihdk̆dk̆ = S2 (k20 ) − S2 (k10 ) + S2 (A − k10 ) − S2 (A − k20 )(2.232)where
ki0 = k̆i /m,S2 (x) = m

Zi = 1, 2,(2.233)S1 (y)dy = m C1 [−x + C2 `n x + 1/x] .

(2.234)

For the total annihilation cross section we use Heitler’s formula with appropriate changes totake into account units and notation, and we have
X0 nπr02
Σ̆Annih (Ĕ0 ) =
γ+1

"

q
γ 2 + 4γ + 1
γ +3
`n(γ
+
γ 2 − 1) − p
γ2 − 1γ2 − 1#.

(2.235)PEGS functions ANIHDM, ANIHRM, and ANIHTM evaluate Equations 2.222, 2.232, and 2.235, respectively.
In computing the energies of the secondary photons, we sample the parameter E, which isdefined by
k0
k̆
E=
=
.
(2.236)
AĔ0 + m70

To find the limits of E, we first compute the limits for k0 . Using Equation 2.167 with m1 = m, E1 =Ĕ0 , p1 = p̆0 , m3 = m4 = 0, E3 = p3 = k̆, which yields
cos θ =

(Ĕ0 + m)k̆ − Ĕ0 m − m2p̆0 k̆

.

(2.237)

Solving for k̆ and dividing by m we obtain
k0 =

1
A
q
.
=
A − (p̆0 /m) cos θ
1 − T 0 /A cos θ(2.238)0

Setting cos θ3 = ±1 in Equation 2.238 we then see that
1

0
kmin
=

1+

q

1−

q

T00 /A

1

0
kmax
=

We also have
1

0
0
kmin
+ kmax
=

1+
1−

=

q

T00 /Aq

T00 /AA,
A + p00(2.239)=A.
A − p00(2.240)1+
1−

T00 /A + 1 +

= A.

=

1 − T00 /A(2.241)q

T00 /Aq

T00 /A=2A
γ + 1 − (γ − 1)

That is, when one photon is at the minimum, the other is at the maximum and the sum of theirenergies is equal to the available energy (divided by m).
Because of the indistinguishability of the two photons, we have the restriction E < 1/2. Moreover, we have E > E0 , where
E0 =

0
kmin
1
1
q
=
.
=
A
A + p00
A + AT 0(2.242)0Given our change of variables, the cross section to be sampled is given asdΣ̆Annih= mA [S1 (AE) + S1 (A(1 − E))] .
dE(2.243)

Because of the symmetry in E, we can expand the range of E to be sampled from (E0 , 1/2) to(E0 , 1 − E0 ) and we can ignore the second S1 , as we can use 1 − E if we sample a value of E greaterthan 1/2. We can then express the distribution to be sampled asdΣ̆Annih
1= mAC1 −1 + C2 −
dE
AE




71



AE,



, E ∈ (E0 , 1 − E0 ) .

(2.244)

Using the value of C2 and rearranging, we obtain
X0 nπr02 mdΣ̆Annih`n [(1 − E0 )/E0 ] f (E)g(E)
=
dE
T̆0

(2.245)

where
1
1
,
`n[(1 − E0 )/E0 ] E
1
1g(E) = 1 − E + 2 2γ −.
AE

f (E) =

(2.246)(2.247)To see that g(E) is a valid rejection function we find its extrema, as below:
g 0 (E) = −1 +g00 (E) =1
A2 E 2−2<0.
A2 E 3=0⇒E =

1A(2.248)Thus, we see that g(E) has a maximum at E = A1 ≥ E0 with maximum value
1
g
A




=1−

2
<1.
A2

(2.249)The sampling procedure is then as follows:
1. Compute parameters depending on Ĕ0 only:A, γ, T00 , p00 , E0 , `n
2. Sample E usingE = E0 exp ζ1 `n

1 − E0E0

1 − E0
E0





.

.

(2.250)3. Compute g(E). Then reject E and return to Step 2 if ζ2 > g(E).
4. Get larger and smaller E values using
EBigger = max(E, 1 − E) ,ESmaller = 1 − EBigger .

(2.251)(2.252)

5. Use these to set up the photons.
The polar angles are determined by the photon energies using Equation 2.167, and UPHI is called
to select random azimuths.
A special case arises when a positron falls below the cutoff energy set in EGS before annihilating.
When this occurs, the particle is assumed to come to rest before annihilating, and each gamma
72

Figure 2.5: Feynman diagram for single photon e+ e− annihilation.
is assumed to have energy k̆ = m. The gamma angular distribution is taken to be isotropic thephotons moving in opposite directions.
It should be noted that a positron may annihilate and give off only one photon via the process
shown in Figure 2.5. Messel and Crawford[103] make the point that the ratio of one photon
annihilation to two photon annihilation is small until higher energies are reached, at which point
the absolute value of the cross section is small. Thus, EGS ignores one photon positron annihilation.
Note that it is also possible to have annihilation accompanied by three or more photons, but theseinteractions are even less likely than one photon annihilation and are not considered in EGS.

2.13

Continuous Electron Energy Loss

Recall the distinction made earlier between collisions contributing to discrete and to continuous
energy losses in electron and positron scattering. In discrete collisions, secondary particles withenergies above their cutoff energies are created and subsequently transported. Continuous energy
losses result from interactions in which the energy transfered to the secondary particles is below
the discrete transport energy thresholds. The resulting secondary particles, which are either soft
bremsstrahlung photons or low energy atomic electrons. The mean total energy loss per unit length
in these continuous loss collisions is thus given by
−



dE±
dx



Total
Continuous

=−



dE±
dx



Soft
Bremsstrahlung

−



dE±
dx



Sub-Cutoff
Atomic Electrons(2.253)where ± denotes positive or negative electrons. The first term on the right-hand side of Equation 2.253 is the same for electrons and positrons (to the accuracy with which we treat bremsstrahlung) and is given by Equation 2.148.
The second term may be expressed as the integral of the differential cross section for transferringa specified amount of energy, T , to an atomic electron, times the amount of the energy transfer

73

over the range of energy transfers which still give rise to soft final state secondary electrons, as in:
dE±
−
dx




Sub-Cutoff
Atomic Electrons=

Z Tmax

T

0

dΣ±
dT .
dT

(2.254)For values of T on the order of the atomic excitation levels, the frequencies and strengthsof the atomic oscillators must be taken into account and the evaluation of this integral is quite
complicated. For values of T large enough that the atomic electrons may be considered free, theMøller or Bhabha cross sections can be used to describe the scattering. If we let Tmed be a valueof T which is sufficiently above the atomic excitation level, but is still small compared to Tmax , we
have

 Z T
Z TmaxmeddΣMøller
dE−
dΣ−
T
dT +
dT
(2.255)
−
=
T
dx
dT
dE
Tmed
0
and

dE+
−
dx





=

Z Tmed

dΣ+
dT +
T
dT

0

Z TmaxT

TmeddΣBhabhadE
dE(2.256)

When appropriate approximations are made, Equations 2.224 and 2.225 are independent ofTmed . We use the formulas recommended by Berger and Seltzer [17] for restricted stopping power7
which are based on the Bethe-Bloch formula [21, 22, 36]. Note that we have corrected typographic
errors in Berger and Seltzer’s Equations 22-24 for F + (τ, ∆)). Other useful references regarding this
topic are Rohrlich and Carlson[140], Jauch and Rohrlich[83], Turner[173], Sternheimer[159, 160,
161, 162, 163, 165], Evans[53] (for Bhabha and Møller cross sections), Armstrong and Alsmiller[10],and Messel and Crawford[103]. The formula used in PEGS for the restricted stopping power (i.e.,due to sub-cutoff electrons) is
dĔ±
−X0
dx

!

where

Sub-Cutoff
Atomic Electronsγη"2(τ + 2)
+ F ± (τ, ∆) − δ`n ¯(Iadj /m)#

(2.257)=Ĕ0 /m = usual relativistic factor,(2.258)=q

γ 2 − 1 = βγ = p̆0 c/m,

(2.259)1 − γ −2 = v/c for incident particle,(2.260)

q

β

=

TE0

=

TE /m = K.E. cutoff in electron mass units,(2.261)τ=γ − 1,(2.262)−1y=(γ + 1)0
Tmax=

maximum energy transfer=(τ if positron, τ /2 if electron),=

restricted maximum energy transfer=0min(TE0 , Tmax),∆
7

X0 n2πr02 m=
β2(See Bhabha formula),

The concept of “restricted stopping power” is discussed in detail in the book by Kase and Nelson[87].

74(2.263)(2.264)(2.265)

I¯adj=average adjusted mean ionization energy,(2.266)δ=density effect correction,(2.267)−F (τ, ∆)

2=

−1 − β + `n[(τ − ∆)∆] + τ /(τ − ∆)

+

"

∆2
+ (2τ + 1)`n2β2
F (τ, ∆) = `n(τ ∆) −
τ
+

−

(

∆
1−
τ



τ + 2∆ −#/γ 2(2.268)3∆2 y2(∆ − ∆3 /3)y 2 − (∆2 /2 − τ ∆3 /3 + ∆4 /4)y 3o.

(2.269)Since EGS3 was released there has been considerable effort devoted to establishing accurate
values for the parameters in the above equations and considerable changes have occurred, especiallyin the recommended I values. The values recommended in 1983 by Berger and Seltzer[20] havealso been recommend by the International Commission on Radiation Units and Measurements[79].
PEGS contains a table of I values for all elements and these are presented in Table 2.2 along withthe default atomic weights and elemental densities used in PEGS (any of which can be overriddenby the user).
The values of I¯adj for mixtures and compounds are derived from`n I¯adj =NeXpi Zi (`n Iadj (Zi ))

i=1

!,

Ne
X
i=1

pi Zi

!

or taken from Table 2.3 for a select group of materials as discussed below.

75(2.270)Table 2.2: Default atomic numbers, symbols, weights, densities and I values in PEGS.
Z

SymbolAtomicWeightDensity(g/cm3 )

I(eV)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

H
HE
LI
BE
B
C
N
O
F
NE
NA
MG
AL
SI
P
S
CL
AR
K
CA
SC
TI
V
CR
MN
FE
CO
NI
CU
ZN
GA
GE
AS
SE
BR
KR
RB
SR
Y
ZR
NB
MO
TC
RU
RH
PD
AG
CD
IN
SN

1.00797
4.00260
6.93900
9.01220
10.8110012.01115
14.00670
15.99940
18.99840
20.18300
22.98980
24.31200
26.98150
28.08800
30.97380
32.06400
35.45300
39.94800
39.10200
40.08000
44.95600
47.90000
50.94200
51.99800
54.93800
55.84700
58.93320
58.71000
63.54000
65.37000
69.72000
72.5900074.92160
78.96000
79.80800
83.80000
85.47000
87.62000
88.90500
91.22000
92.90600
95.94000
99.00000
101.07000
102.90500
106.40000
107.87000
112.40000
114.82000
118.69000

0.08080.19000.53401.85002.50002.26001.14001.56801.50001.00000.97121.74002.70202.40001.82002.07002.20001.65000.86001.55003.02004.54005.87007.14007.30007.86008.71008.90008.93337.14005.91005.36005.73004.80004.20003.40001.53002.60004.47006.40008.57009.010011.5000
12.2000
12.5000
12.0000
10.5000
8.65007.30007.3100

19.2
41.8
40.0
63.7
76.0
78.0
82.0
95.0
115.0
137.0
149.0
156.0
166.0
173.0
173.0
180.0
174.0
188.0
190.0
191.0
216.0
233.0
245.0
257.0
272.0
286.0
297.0
311.0
322.0
330.0
334.0
350.0
347.0
348.0
357.0
352.0
363.0
366.0
379.0
393.0
417.0
424.0
428.0
441.0
449.0
470.0
470.0
469.0
488.0
488.0

76

Table 2.2 (cont.)
Z

SymbolAtomicWeightDensity(g/cm3 )

I(eV)

51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100

SB
TE
I
XE
CS
BA
LA
CE
PR
ND
PM
SM
EU
GD
TB
DY
HO
ER
TM
YB
LU
HF
TA
W
RE
OS
IR
PT
AU
HG
TL
PB
BI
PO
AT
RN
FR
RA
AC
TH
PA
U
NP
PU
AM
CM
BK
CF
ES
FM

121.75000
127.60000
126.90440
131.30000
132.90500
137.34000
138.91000
140.12000
140.90700
144.24001
147.00000
150.35001
151.98000
157.25000
158.92400
162.50000
164.92999
167.25999
168.93401
173.03999
174.97000
178.49001
180.94800
183.85001
186.20000
190.20000
192.20000
195.08000
196.98700
200.59000
204.37000
207.19000
208.98000
210.00000
210.00000
222.00000
223.00000
226.00000
227.00000
232.03600
231.00000
238.03000
237.00000
242.00000
243.00000
247.00000
247.00000
248.00000
254.00000
253.00000

6.68406.24004.93002.70001.87303.50006.15006.90006.76907.00701.00007.54005.17007.87008.25008.56008.80009.06009.32006.96009.850011.4000
16.6000
19.3000
20.5300
22.4800
22.4200
21.4500
19.3000
14.1900
11.8500
11.3400
9.78009.30001.00004.00001.00005.00001.000011.0000
15.3700
18.9000
20.5000
19.7370
11.7000
7.00001.00001.00001.00001.0000

487.0
485.0
491.0
482.0
488.0
491.0
501.0
523.0
535.0
546.0
560.0
574.0
580.0
591.0
614.0
628.0
650.0
658.0
674.0
684.0
694.0
705.0
718.0
727.0
736.0
746.0
757.0
790.0
790.0
800.0
810.0
823.0
823.0
830.0
825.0
794.0
827.0
826.0
841.0
847.0
878.0
890.0
902.0
921.0
934.0
939.0
952.0
966.0
980.0
994.0

77

Table 2.3: Default Sternheimer density effect coefficients in PEGS.
LABEL
H2-GAS
H2-LIQUIDHE-GAS
LI
BE
B
C-2.265 G/CM**3
C-2.00 G/CM**3
C-1.70 G/CM**3
N2-GAS
O2-GASF
NE-GAS
NA
MG
AL
SI
P
S
CL
AR-GAS
K
CA
SC
TI
V
CR
MN
FE
CO
NI
CU
ZN
GA
GE
AS
SE
BR
KR-GAS
RB
SR
Y
ZR
NI
MO
TC
RU

a
0.14092
0.134830.13443
0.951360.80392
0.56224
0.261420.202400.207620.15349
0.11778
0.110830.080640.07772
0.081630.080240.149210.236100.33992
0.19849
0.197140.19827
0.15643
0.157540.156620.154360.154190.14973
0.146800.14474
0.164960.143390.147140.094400.07188
0.06633
0.06568
0.063350.07446
0.07261
0.07165
0.07138
0.071770.13883
0.10525
0.16572
0.19342ms
5.72735.62495.83472.49932.43392.45122.86973.00362.9532
3.2125
3.29133.2962
3.57713.6452
3.6166
3.63453.25462.91582.64562.97022.96182.92333.07453.05173.03023.01632.98962.97962.96322.95022.84302.90442.8652
3.13143.3306
3.41763.43173.46703.4051
3.41773.44353.45853.4533
3.09303.25492.97382.8707x0
1.86390.47592.20170.13040.05920.0305-0.0178-0.03510.04801.73781.75411.84332.07350.28800.14990.17080.20140.16960.15801.55551.76350.3851
0.32280.16400.09570.06910.03400.0447-0.0012-0.0187-0.0566-0.02540.00490.22670.33760.17670.22581.5262
1.71580.57370.45850.36080.29570.17850.2267
0.09490.059978

x13.27181.92153.6122
1.6397
1.6922
1.96882.34152.48602.53874.13234.32134.40964.6421
3.1962
3.06683.01272.87152.78152.71594.29944.4855
3.1724
3.1191
3.05933.03863.03223.0451
3.1074
3.1531
3.17903.1851
3.2792
3.36683.54343.60963.57023.62644.98995.07483.79953.67783.55423.48903.2201
3.27843.12533.0834

I(eV)
19.2
21.8
41.8
40.0
63.7
76.0
78.0
78.0
78.0
82.0
95.0
115.0
137.0
149.0
156.0
166.0
173.0
173.0
180.0
174.0
188.0
190.0
191.0
216.0
233.0
245.0
257.0
272.0
286.0
297.0
311.0
322.0
330.0
334.0
350.0
347.0
348.0
343.0
352.0
363.0
366.0
379.0
393.0
417.0
424.0
428.0
441.0

−C
9.58353.2632
11.1393
3.12212.78472.84772.86802.9925
3.155010.5400
10.7004
10.9653
11.90415.05264.52974.23954.4351
4.52144.665911.142111.9480
5.64235.03964.69494.44504.26594.17814.27024.29114.2601
4.31154.41904.69064.93535.14115.05105.321011.730712.5115
6.47765.98675.4801
5.17745.01414.87934.77694.7694

δ0
0.00
0.00
0.00
0.14
0.14
0.14
0.12
0.10
0.14
0.00
0.00
0.00
0.00
0.08
0.08
0.12
0.14
0.14
0.14
0.00
0.00
0.10
0.14
0.10
0.12
0.14
0.14
0.14
0.12
0.12
0.10
0.08
0.08
0.14
0.14
0.08
0.10
0.00
0.00
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14

Table 2.3 (cont.)
LABEL
RH
PD
AG
CD
IN
SN
SB
TE
I
XE-GAS
CS
BA
LA
CE
PR
ND
PM
SM
EU
GD
TB
DY
HO
ER
TM
YB
LU
HF
TA
W
RE
OS
IR
PT
AU
HG
TL
PB
BI
PO
RN-GAS
RA
AC
TH
PA
U
NP

a
0.192050.241780.245850.24609
0.238790.18689
0.16652
0.13815
0.23766
0.233140.18233
0.18268
0.18591
0.188850.232650.23530
0.24280
0.24698
0.244480.251090.244530.246650.24638
0.24823
0.248890.252950.24033
0.22918
0.17798
0.15509
0.15184
0.127510.126900.11128
0.09756
0.110140.09455
0.09359
0.094100.092820.20798
0.08804
0.08567
0.08655
0.14770
0.196770.19741ms
2.86332.72392.68992.67722.71442.85762.95193.03542.72762.74142.88662.89062.88282.85922.73312.70502.66742.64032.62452.59772.60562.58492.57262.55732.54692.51412.56432.61552.76232.84472.86272.96082.96583.04173.1101
3.05193.14503.16083.1671
3.18302.74093.24543.26833.26102.98452.81712.8082x0
0.0576
0.05630.06570.12810.24060.28790.31890.3296
0.05491.56300.54730.41900.31610.27130.23330.19840.16270.15200.18880.10580.09470.08220.0761
0.06480.08120.11990.15600.19650.21170.21670.05590.08910.08190.14840.20210.2756
0.34910.3776
0.41520.42671.53680.59910.45590.42020.31440.22600.186979

x1
3.10693.0555
3.1074
3.16673.2032
3.29593.34893.44183.25964.7371
3.59143.45473.32933.3432
3.27733.30633.31993.34603.46333.39323.42243.44743.4782
3.4922
3.50853.6246
3.52183.43373.48053.49603.48453.5414
3.54803.6212
3.69793.72753.80443.80733.82483.82934.98893.94283.79663.7681
3.50793.3721
3.3690

I(eV)
449.0
470.0
470.0
469.0
488.0
488.0
487.0
485.0
491.0
482.0
488.0
491.0
501.0
523.0
535.0
546.0
560.0
574.0
580.0
591.0
614.0
628.0
650.0
658.0
674.0
684.0
694.0
705.0
718.0
727.0
736.0
746.0
757.0
790.0
790.0
800.0
810.0
823.0
823.0
830.0
794.0
826.0
841.0
847.0
878.0
890.0
902.0

−C
4.80084.93585.06305.27275.52115.53405.62415.7131
5.948812.72816.91356.3153
5.78505.78375.80965.82905.82245.8597
6.22785.87385.90455.9183
5.95875.9521
5.9677
6.33255.97855.71395.52625.40595.34455.30835.34185.47325.57475.9605
6.13656.20186.35056.400313.28397.0452
6.3742
6.24736.03275.86945.8149

δ0
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.00
0.00
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.14
0.08
0.10
0.10
0.12
0.14
0.14
0.14
0.14
0.14
0.14
0.00
0.14
0.14
0.14
0.14
0.14
0.14

Table 2.3 (cont.)
LABEL
PU
AM
CM
BK
A 150-PLASTICACETONEACETYLENEADENINE
ADIPOSE TISSUE
AIR-GAS
ALANINE
ALUMINIUM OXIDEAMBERAMMONIAANILINEANTHRACENE
B-100 BONE-EQ. PLASTIC
BAKELITE
BARIUM FLUORIDEBARIUM SULFATEBENZENEBERYLLIUM OXIDEBGOBLOOD (ICRP)
BONE, COMPACT (ICRU)BONE CORTICAL (ICRP)BORON CARBIDEBORN OXIDEBRAIN (ICRP)BUTANE
N-BUTYL ALCHOL
C-552 AIR-EQ. PLASTICCADMIUM TELLURIDECADMIUM TUNGSTATECALCIUM CARBONITECAF2
CALCIUM OXIDECALCIUM SULFATECALCIUM TUNGSTATECARBON DIOXIDECARBON TETRACHLORIDECELLOPHANE
CELLULOSE ACETATE BUTYRACELLULOSE NITRATE
CERIC SURFARE DOSIMETERCESIUM FLUORIDE
CSI

a
0.204190.203080.20257
0.20192
0.10783
0.11100
0.12167
0.20908
0.10278
0.109140.11484
0.08500
0.11934
0.083150.13134
0.14677
0.05268
0.127130.15991
0.11747
0.165190.10755
0.095690.08492
0.058220.06198
0.37087
0.115480.08255
0.108520.100810.10492
0.248400.12861
0.08301
0.069420.121280.07708
0.062100.11768
0.19018
0.11151
0.114440.118130.07666
0.22052
0.25381ms
2.76792.76152.75792.75603.44423.40473.42773.0271
3.48173.39943.35263.54583.40983.64643.34343.28313.73653.34702.88673.04273.21743.49273.0781
3.54063.64193.59192.80763.3832
3.55853.48843.51393.43442.66652.91503.41203.52633.19363.44953.26493.32273.01163.38103.37383.32373.56072.72802.665780x00.15570.22740.24840.23780.13290.21971.60170.12950.18271.74180.13540.04020.13351.68220.16180.11460.12520.1471-0.0098-0.01280.17100.02410.04560.22390.09440.11610.00930.18430.2206
1.37880.19370.15100.04380.01230.04920.0676-0.0172
0.05870.0323
1.62940.17730.15800.17940.1897
0.23630.00840.0395x1
3.39813.50213.51603.51862.62342.60284.00742.42192.65304.27592.63362.86652.56104.11582.58052.52133.04202.60553.3871
3.40692.50912.5846
3.78162.80173.0201
3.09192.10062.73792.80213.75242.64392.70833.28363.5941
3.05493.16833.0171
3.12293.8932
4.18252.91652.67782.68092.72532.87693.3374
3.3353

I(eV)
921.0
934.0
939.0
952.0
65.1
64.2
58.2
71.4
63.2
85.7
71.9
145.2
63.2
53.7
66.2
69.5
85.9
72.4
375.9
285.7
63.4
93.2
534.1
75.2
91.9
106.4
84.7
99.6
73.3
48.3
59.9
86.8
539.3
468.3
136.4
166.0
176.1
152.3
395.0
85.0
166.3
77.6
74.6
87.0
76.7
440.7
553.1

−C
5.87486.28136.3097
6.29123.11003.4341
9.84193.1724
3.2367
10.5961
3.09653.56823.07019.87633.2622
3.15143.45283.25825.41224.89233.32692.98015.74093.4581
3.33903.64882.98593.60273.42798.56333.24253.33385.90965.35943.77384.06534.12093.93885.260310.1537
4.77123.26473.34973.4762
3.52125.9046
6.2807

δ0
0.14
0.14
0.14
0.14
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

Table 2.3 (cont.)
LABEL
CHLOROBENZENECHLOROFORMCONCRETE, PORTLANDCYCLOHEXANE
1,2-DICHLOROBENZENEDICHLORODIETHYL ETHER
1,2-DICHLOROETHANEDIETHYL ETHER
N,N-DIMETHYL FORMAMIDEDIMETHYL SULFOXIDEETHANEETHYL ALCOHOLETHYL CELLULOSEETHYLENE
EYE LENS (ICRP)FERRIC OXIDEFERROBORIDEFERROUS OXIDEFERROUS SULFATE DOSIMETEFREON-12
FREON-12B2
FREON-13
FREON-13B1
FREON-13I1
GADOLINIUM OXYSULFIDEGALLIUM ARSENIDEGEL IN PHOTOGRAPHIC EMULPYREX-GLASSGLASS, LEAD
GLASS, PLATEGLUCOSE
GLUTAMINEGLYCEROLGUANINE
GYPSUM, PLASTER OF PARIS
N-HEPTANE
N-HEXANE
KAPTON, POLYIMIDE FILMLANTHANUM OXYBROMIDELANTHANUM OXYSULFIDELEAD OXIDELITHIUM AMIDELITHIUM CARBONATE
LIF
LITHIUM HYDRIDELIILITHIUM OXIDEa
0.09856
0.16959
0.075150.120350.160100.06799
0.13383
0.10550
0.114700.06619
0.09627
0.09878
0.11077
0.10636
0.096900.104780.129110.12959
0.08759
0.07978
0.051440.07238
0.03925
0.091120.22161
0.071520.101020.08270
0.095440.07678
0.10783
0.119310.10168
0.20530
0.06949
0.11255
0.11085
0.159720.17830
0.21501
0.196450.087400.099360.075930.905670.232740.08035ms
3.37973.06273.54673.42783.08363.52503.16753.45863.37103.57083.60953.48343.40983.53873.45503.13133.02403.01683.49233.46263.55653.5551
3.7194
3.16582.63003.33563.44183.52243.07403.5381
3.3946
3.32543.44813.01863.51343.48853.5027
3.19212.84572.72982.72993.75343.54173.74782.58492.7146
3.787881x00.17140.17860.13010.17280.15870.17730.13750.22310.19770.20211.51070.22180.16831.55280.2070-0.0074-0.0988-0.0279
0.23780.30350.34060.36590.35220.2847-0.1774
0.17640.17090.14790.06140.12370.14110.13470.16530.11630.09950.19280.19840.1509-0.0350-0.0906
0.03560.01980.05510.0171-0.09880.0892-0.0511x1
2.92722.95813.04662.55492.8276
3.15862.95292.67452.6686
3.12633.87432.70522.62573.93272.7446
3.25733.17493.20022.82543.26593.79563.23373.75543.72803.40453.64202.70582.99333.81463.06492.67002.63012.68622.42963.12062.57062.57572.56313.32883.26643.54562.51522.65982.70491.45153.37022.5874

I(eV)
89.1
156.0
135.2
56.4
106.5
103.5
111.9
60.0
66.6
98.6
45.4
62.9
69.3
50.7
73.3
227.3
261.0
248.6
76.4
143.0
284.9
126.6
210.5
293.5
493.3
384.9
74.8
134.0
526.4
145.4
77.2
73.3
72.6
75.0
129.7
54.4
54.0
79.6
439.7
421.2
766.7
55.5
87.9
94.0
36.5
485.1
73.6

−C
3.82014.70553.94643.15444.03484.0135
4.18493.37213.3311
3.98449.10433.36993.24159.43803.37204.22454.20574.31753.51834.8251
5.79764.74835.35555.87745.53475.32993.26873.97085.84764.0602
3.16493.11673.2267
3.1171
3.8382
3.19783.2156
3.34975.46665.44706.21622.79613.20293.16672.35806.26712.9340

δ0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

Table 2.3 (cont.)
LABEL
LITHIUM TETRABORATELUNG (ICRP)
M3 WAXMAGNESIUM CARBONATE
MANESIUM FLUORIDEMAGNESIUM OXIDEMAGNESIUM TETRABORATEMERCURIC IODIDEMETHANEMETHANOLMIX D WAX
MS20 TISSUE SUBSTITUTEMUSCLE, SKELETAL (ICRP)MUSCLE, STRIATED (ICRU)MUSCLE-EQ. LIQ. W SUCROSMUSCLE-EQ. LIQ. W/O SUCR
NAPTHALENENITROBENZENENITROUS OXIDENYLON, DU PONTNYLON, TYPE 6 AND 6/6
NYLON, TYPE 6/10
NYLON, TYPE 11
OCTANE, LIQUID
PARAFFIN WAX
N-PENTANE
PHOTOEMULSION
PLASTIC SCINT.
PLUTONIUM DIOXIDEPOLYCRYLONITRILEPOLYCARBONATEPOLYCHLOROSTYRWNE
POLYETHYLENE
MYLAR
LUCITEPOLYOXYMETHYLENE
POLYPROPYLENE
POLYSTYRENE
TEFLONPOLYTRIFLUOROCHLOROETHY.
POLYVINYL ACETATEPOLYVINYL ALCOHOLPOLYVINYL BUTYRALPOLYVINYL CHLORIDESARANPLOYVINYLIDENE FLUORIDEPOLYVINYL PYRROLIDONEa0.11075
0.08588
0.078640.092190.07934
0.08313
0.09703
0.21513
0.092530.089700.074900.08294
0.086360.085070.094810.09143
0.14766
0.127270.11992
0.115130.118180.118520.14868
0.11387
0.12087
0.10809
0.12399
0.16101
0.20594
0.16275
0.12860
0.07530
0.121080.12679
0.11433
0.108080.150450.16454
0.10606
0.07727
0.114420.11178
0.11544
0.12438
0.15466
0.103160.12504ms
3.43893.53533.64123.50033.64853.59683.48932.72643.62573.54773.68233.6061
3.53303.53833.46993.49823.26543.30913.33183.40443.3826
3.3912
3.25763.47763.42883.52653.0094
3.23932.6522
3.19753.32883.54413.42923.30763.38363.40023.2855
3.22243.40463.50853.3762
3.38933.39833.2104
3.10203.42003.332682

x00.07370.22610.15230.08600.13690.05750.11470.10401.62630.25290.13710.19970.22820.22490.20980.21870.13740.17771.64770.15030.13360.13040.06780.18820.12890.20860.10090.1464-0.23110.15040.16060.12380.13700.15620.18240.15840.15340.16470.16480.17140.17690.14010.15550.15590.13140.17170.1324x1
2.65022.80012.75292.79972.86302.85802.76353.47283.97162.76392.71452.80332.79992.80322.75502.76802.54292.66304.15652.60042.58342.56812.42812.56642.50842.58553.48662.48553.55542.51592.62252.92412.51772.65072.66812.68382.48222.50312.74043.02652.67472.63152.61862.94152.90092.73752.5867

I(eV)
94.6
75.3
67.9
118.0
134.3
143.8
108.3
684.5
41.7
67.6
60.9
75.1
75.3
74.7
74.3
74.2
68.4
75.8
84.9
64.3
63.9
63.2
61.6
54.7
55.9
53.6
331.0
64.7
746.5
69.6
73.1
81.7
57.4
78.7
74.0
77.4
59.2
68.7
99.1
120.7
73.7
69.7
67.2
108.2
134.3
88.8
67.7

−C
3.20933.47083.25403.43193.71053.64043.43286.37879.52433.51603.07803.53413.48093.4636
3.39103.42163.22743.407310.1575
3.12503.06343.03332.7514
3.18342.95513.25045.33193.19975.97193.24593.3201
3.46593.00163.32623.32973.25143.1252
3.29993.41613.85513.33093.11153.18654.05324.25063.37933.1017

δ0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

Table 2.3 (cont.)
LABEL
POTASSIUM IODINE
POTASSIUM OXIDEPROPANE
PROPANE, LIQUID
N-PROPYL ALCOHOLPYRIDINE
RUBBER, BUTYL
RUBBER, NATURAL
RUBBER, NEOPRENE
SIO2
AGBR
AGCL
SILVER HALIDES IN EMUL.
SILVER IODIDESKIN (ICRP)SODIUM CARBONATENAISODIUM MONOXIDESODIUM NITRATESTILBENE
SUCROSETRRPHENYLTESTES (ICRP)TETRACHLOROETHTLENE
THALLIUM CHLORIDETISSUE, SOFT (ICRP)ICRU FOUR-COMP. TISSUETISSUE-EQ. GAS (METHANE)
TISSUE-EQ. GAS (PROPANE)
TITANIUM DIOXIDETOLUENTRICHLOROETHYLENE
TRIETHYL PHOSPHATETUNGSTEN HEXAFLUORIDEURANIUM DICARBIDEURANIUM MONOCARBIDEURANIUM OXIDEUREAVALINE
VITON
H2O
H2O-VAPORXYLENEa0.220530.16789
0.099160.10329
0.09644
0.16399
0.121080.15058
0.097630.084080.245820.22968
0.245930.25059
0.09459
0.087150.125160.07501
0.09391
0.16659
0.11301
0.149640.08533
0.18595
0.185990.08926
0.09629
0.09946
0.098020.08569
0.13284
0.18272
0.069220.03658
0.211200.22972
0.204630.11609
0.11386
0.09965
0.091160.08101
0.13216ms
2.75583.0121
3.59203.56203.5415
3.19773.4296
3.28793.36323.50642.68202.70412.68142.6572
3.46433.56383.03983.69433.5097
3.21683.36303.26853.54283.01562.76903.51103.43713.47083.51593.32673.35583.01373.63023.51342.65772.61692.6711
3.3461
3.37743.45563.47733.59013.3564x0
0.10440.04801.4326
0.28610.20460.16700.13470.15120.15010.13850.0352-0.01390.03530.01480.20190.12870.12030.16520.15340.17340.13410.13220.22740.17130.07050.2211
0.23771.6442
1.5139-0.01190.17220.18030.20540.3020-0.2191-0.2524-0.1938
0.16030.14410.21060.24001.79520.1695x1
3.34423.01103.79982.65682.66812.52452.51542.48152.9461
3.00253.21093.2022
3.21173.29082.75262.8591
3.59202.97932.82212.51422.65582.54292.79882.90833.57162.77992.79084.13993.9916
3.16472.57282.91402.94284.26023.52083.4941
3.52922.65252.62272.78742.80044.34372.5675

I(eV)
431.9
189.9
47.1
52.0
61.1
66.2
56.5
59.8
93.0
139.2
486.6
398.4
487.1
543.5
72.7
125.0
452.0
148.8
114.6
67.7
77.5
71.7
75.0
159.2
690.3
72.3
74.9
61.2
59.5
179.5
62.5
148.1
81.2
354.4
752.0
862.0
720.6
72.8
67.7
98.6
75.0
71.6
61.8

−C
6.10884.64638.78783.55293.29153.31482.9915
3.1272
3.79114.00295.61395.34375.61665.93423.3546
3.71786.0572
4.18923.6502
3.36803.1526
3.26393.46984.66196.30093.43543.50879.95009.35293.9522
3.30264.61483.6242
5.9881
6.0247
6.12105.96053.2032
3.10593.59433.5017
10.59623.2698

δ0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

The density effect has been treated extensively by Sternheimer [159, 160, 161, 162, 163], Sternheimer and Peierls[165], and recently by Berger, Seltzer, and Sternheimer in various collabora83

tions [20, 152, 151, 166, 164]. According to Sternheimer, the density effect is given by the formulas
δ=
where


0


2(`n 10)x + C + a(x12(`n 10)x + C

x =
C =
νP

=

− x)msif x < x0
if x ∈ (x0 , x1 )if x > x1

log10 (p̆c/m) = `n η/`n 10,−2 `n(I¯adj /hνP ) − 1,
plasma frequency =qnr02 c2 /π(2.271)(2.272)(2.273)(2.274)and x0 , x1 , a, ms are parameters obtained by fits made to values of δ as a function of energy
which were explicitly calculated from atomic oscillator strengths for specific materials. Two setsof parameters are available [164, 166], both of which derive from the same program for calculatingδ [20]. As of 1995, PEGS has employed the more extensive data from [164], which includes 98
elements and 180 compounds. A list of the available materials and their parameters is foundin Table 2.3. The composition of the materials is given elsewhere (e.g., Table 7 of Seltzer andBerger[151]).
Since there are such a variety of density effect parameters available, version 4 of PEGS was
modified to allow the user to override the PEGS defaults and to specify directly the six parameters
needed for any material being considered. However, as the six parameters used in PEGS merelyparameterize δ, the stopping powers calculated with these values will be slightly different (≤ ±0.5%)
from those calculated directly from Berger and Seltzer’s original δ values (e.g., see Figure 1 of
Seltzer and Berger[152]). Because some benchmarking application require stopping power values
with extremely fine precision (see, e.g., [101] or [137]), a further override option was developed for
PEGS by Duane et al.[52] in which the values of δ and I¯adj used in [20] can be used in PEGS. When
this option is invoked (through the PEGS input flag EPSTFL), PEGS reproduces exactly values ofcollision stopping power found in ICRU Report [79]. Provided with the implementation of thisoption in PEGS are values of δ and I¯adj for 100 elements and 345 compounds.
For materials not included in Table 2.3 or for which the density effect parameters or values are
not input, the general prescription of Sternheimer and Peierls[165] is used as follows:
1. The value of I¯adj is determined as described above.
2. ms is always taken as 3.
3. C is given by Equations 2.273 and 2.274.
4. For solids and liquids:(a) If I¯adj < 100 eV, then x1 = 2.0 and
x0 = 0.2 when −C < 3.681 or x0 = −0.326C − 1 otherwise.
(b) If I¯adj > 100 eV, then x1 = 3.0 and
x0 = 0.2 if −C < 5.215 or x0 = −0.326C − 1.5 otherwise.
84

5. For gases:(a) If −C < 10.0
and x1 = 4.0.

then x0 = 1.6(b) If 10.0 ≤ −C < 10.5
and x1 = 4.0.

then x0 = 1.7(c) If 10.5 ≤ −C < 11.0
and x1 = 4.0.

then x0 = 1.8(d) If 11.0 ≤ −C < 11.5
and x1 = 4.0.

then x0 = 1.9(e) If 11.5 ≤ −C < 12.25 then x0 = 2.0
and x1 = 4.0.
(f) If 12.25 ≤ −C < 13.804 then x0 = 2.0
and x1 = 5.0.
(g) If −C ≥ 13.804 then x0 = −0.326C − 2.5
and x1 = 5.0.
6. a = [−C − (2`n 10)x0 ]/(x1 − x0 )3 .
The final stage in determining the density effect correction for gases is to scale the results interms of the gas pressure since the tabulated values are all given for NTP (20◦ C, 760mm Hg). If
the pressure correction factor is
GASP = actual pressure/pressure at NTP
then
C(GASP) = C(NTP) + `n(GASP)
andx0 (GASP) = x0 (NTP) − 21 log10 (GASP)
x1 (GASP) = x1 (NTP) − 21 log10 (GASP) .
PEGS routine SPINIT initializes the stopping power routines for a particular medium. Routine
SPIONB, which is a function of Ĕ0 , AE, and POSITR, evaluates Equation 2.257 for a positron if POSITR
is true, and for a electron if POSITR is false.
Note that limitations on the applicability of EGS5 for simulating electron transport at low
energies derive primarily from the breakdown of the Bethe-Bloch formula for the stopping power.
The presence in Equation 2.257 of a term containing the logarithm of the ratio of the electronenergy and the mean ionization energy implies that there must exist some energy (dependent on
85Z) at which the Bethe-Bloch expression will yield physically unrealistic negative values. In addition,
Equation 2.257 was derived under approximations which are strictly valid for high energy particles,
large energy losses and large recoil energies, so the accuracy of the Bethe-Bloch formula begins towane at energies ranging from around 1 keV for low Z materials to several tens of keV for high Z
materials.
As a final comment on continuous energy loss, we note that when an electron is transported agiven distance, it is assumed that its energy loss due to sub-cutoff collisions is equal to the distancetraveled times the mean loss per unit length as evaluated using Equation 2.253. In actuality, theenergy loss over a transported distance is subject to fluctuations and gives rise to a restrictedLandau distribution. Fluctuations due to discrete interactions can be properly accounted for in
EGS in most applications by setting the cutoff energy for charged particle production sufficiently
low. For example, for 20 MeV electrons passing through a thin slab of water, Rogers[136] has shownthat the energy straggling predicted by considering all secondaries down to 1 keV (i.e., AE=512
keV) is in good agreement with the Blunck-Leisegang modification of the energy loss straggling
formalism of Landau (except for large energy loss events where the modified Landau distribution
appears to be wrong). The creation of large numbers of low-energy secondary electrons, even when
they are discarded immediately, adds significant inefficiency to EGS shower simulations, however,and so a correct energy loss fluctuation model is currently under development for EGS5.

2.14

Multiple Scattering

When an electron passes through matter, it undergoes a large number of elastic collisions with theatomic nuclei. These have the effect of changing the electron’s direction, but do not significantly
change its energy. As noted previously, the number of such collisions is so great that direct simulation of individual scattering events is almost never practical8 . Instead, elastic scattering is typicallytreated by bundling large numbers of collisions together into large “steps” and then assuming thatthe electron transport over these larger steps can be characterized by the particle’s longitudinaltranslation and lateral displacement during the step plus its aggregate scattering angle Θ over the
step (the azimuthal angle is assumed to be uniform), which is taken from an appropriate “multiple
scattering” distribution function. The details of the current implementation of this approach areprovided in the next section. The remainder of this section is devoted to discussion of multiple
scattering probability density function (p.d.f.) for Θ. EGS5 currently offers two choices for themultiple scattering p.d.f., one based on the theory of Molière, and a new one based on the approachof Goudsmit and Saunderson (GS) [63, 64].
8
Even in PENELOPE [14], which was originally developed for modeling the transport of electrons at low energies,
only large angle collisions are treated explicitly.

862.14.1

The Molière Multiple Scattering Distribution

All previous versions of EGS have treated electron elastic scattering using Molière’s [107]theory ofmultiple scattering as formulated by Bethe [23]. The details of computing multiple scattering in
mixtures, and a good introduction to the subject is given in the the review article by Scott [147],
to which we make frequent reference in the discussion that follows.
In Versions 1 and 2 of EGS the method of sampling scattering angles was based on a schemeof Nagel’s whereby one of 29 discrete representative reduced angles was selected and then used toobtain the real scattering angle. EGS3 departed from this scheme to use a method similar to thatof Messel and Crawford[103], whereby the scattering angles are chosen in a truly continuous way.
This method also allows us to transport over variable step lengths (usually denoted by t) while stilltaking multiple scattering properly into account.
The cross section for elastic scattering off the nucleus is proportional to Z 2 . Scattering fromatomic electrons is taken into account by replacing Z 2 by Z(Z + ξM S ). Scattering from atomic
electrons which results in discrete delta-rays is already properly taken to account, so the ξM S need
only account for the sub-cutoff scatterings. Scott[147] has outlined the procedures for taking into
account scattering from atomic electrons in a more rigorous way, but we have not implemented it
here. Instead we treat ξM S more as a “fudge factor” to get our multiple scattering as consistent
with experiment as possible. In the developments to follow we shall need the parameters
ZS =

Ne
X

pi Zi (Zi + ξM S ) ,(2.275)

i=1

ZE =

Ne
X

−2/3

pi Zi (Zi + ξM S )`n Zi,(2.276)

i=1

and
ZX =

Ne
X

h

i

pi Zi (Zi + ξM S )`n 1 + 3.34(αZi )2 .

i=1

(2.277)

One of the advantages of Molière’s theory is that the energy-dependent p.d.f. of Θ can be
expressed in terms of an energy-independent p.d.f. of a reduced angle θ, whereθ=Θ,χc B 1/2(2.278)and where χc and B are parameters that depend on energy, material, and pathlength t. The p.d.f.
of Θ is given byf (Θ) = fM (Θ)(sin Θ/Θ)1/2 ,(2.279)which is like Bethe’s formula (58) except that we define our fM (Θ) to be their fM (Θ) times Θ; thatis, we include the phase space factor in ours. The factor (sin Θ/Θ)1/2 is less than one and is used
as a rejection function to correct the Molière distribution at large angles. In addition, we reject all

87

sampled Θ > 180◦ . The p.d.f. fM (Θ) is sampled by first sampling θ, the reduced angle, from its
p.d.f., fr (θ), and then using Equation 2.278 to get Θ. This is equivalent to saying thatfM (Θ)dΘ = fr (θ)dθ .

(2.280)For the reduced angle p.d.f. we use the first three terms of Bethe’s Equation 25; namely,fr (θ) = f

(0)

1
1
(θ) + f (1) (θ) + 2 f (2) (θ) .
BB

(2.281)

The general formula for the f (i) is (Bethe, Equation 26)
f

(n)

−1

(θ) = (n!)

Z ∞
0

in

h

udu J0 (θu) × exp(−u2 /4) 1/4 u2 `n (u2 /4)

For n = 0 this reduces to2

f (0) (θ) = 2e−θ .

.

(2.282)(2.283)Instead of using the somewhat complicated expressions when n = 1 and 2, we have elected to usea) the numerical values presented in Bethe’s paper (for 29 selected values of θ from 0 to 10), b) thefact that f (i) (θ) behaves as θ −2i−2 for large θ, and c) the fact that f (1) (θ) goes over into the single
scattering law at large θ. That is,lim f (1) (θ)θ 4 = 2.
(2.284)
θ→∞This also implies thatlim f (2) (θ)θ 4 = 0.

(2.285)

θ→∞

The f (i) (θ) functions are not needed in EGS directly, but rather PEGS needs the f (i) (θ) tocreate data that EGS does use. Let
η = 1/θ
(2.286)andfη(i) (η) = f (i) (1/η)η −4 = f (i) (θ(η))θ(η)4 .

(2.287)(2)(1)

As a result of Equations 2.285 and 2.286 we see that fη (0) = 2 and fη (0) = 0. We now do a(i)(i)
cubic spline fit to f (i) (θ) for θ(0, 10) and fη (η) for η(0, 5). If we use fˆ(i) (θ) and fˆη (η) to denotethese fits, then we evaluate the f (i) (θ) as
1
f (i) (θ) = fˆ(i) (θ) if θ < 10, 4 fˆη(i) (1/θ)
θ




.

(2.288).

(2.289)(i)

Similarly if we want fη (η) for arbitrary η we usefη(i) (η) =1fˆη(i) (η) if η < 5, 4 fˆ(i) (1/η)
η



88

To complete the mathematical definition of f (Θ) we now give additional formulas for the evaluationof χc and B. We haveB − `n B = b,

(2.290)b = `n Ω0 ,(2.291)2(2.292)Ω0 = bc t/β ,bc =‘66800 ρZS eZE /ZS
M eZX /ZS

(2.293)(Note, PEGS computes b̆c = X0 bc ),‘66800 = 4πNa



h̄
me c

2 "

#

(0.885)2= 6702.33,
1.167 × 1.13

(2.294)

ρ = material mass density (g/cm3 ) ,M = molecular weight =
√
χcc t
,
χc =
ĔM S β 2‘22.90χcc =(180/π)sNeX(2.295)pi Ai ,

(2.296)i=1(2.297)

ρZS(cm−1/2 M eV )M

(2.298)Note, PEGS computes χcc = χcc X0 (r.`.−1/2 M eV )p

‘22.90 = (180/π) 4πNa r0 m = 22.696 (cm M eV ) .
p,(2.299)ĔM S is the energy (in MeV) of the electron that is scattering and may be set equal to the energy atthe beginning or end of the step (or something in between) to try to account for ionization loss overthe step. Equations 2.297 through 2.299 are based on formula 7.4 of Scott[147] which is equivalent
to
# Z
"N
e
t
X
m2 dt0
Na ρ
2
2
4πr0
pi Zi (Zi + ξM S ).
(2.300)χc =M
0 Ĕ(t0 )2 β(t0 )4
i=1√
From this we see that, to be proper, we should replace t/ĔM S β 2 using√t

ĔM S β 2

=

Z t
0

dt0
Ĕ(t0 )2 β(t0 )4

!1/2

,

(2.301)00where Ĕ(tq ) is the particle’s energy (in MeV) after going a distance t along its path. Likewise,β(t0 ) = 1 − m2 /Ĕ(t0 )2 is the particle’s velocity, at the same point, divided by the speed of light.
We assume that our steps are short enough and the energy high enough that Equations 2.297
through 2.299 are sufficiently accurate.

89

For completeness, we give a derivation of Equations 2.292. We start with the definition of Ω0 ,(which differs somewhat from Scott’s definition),Ω0 ≡ eb .

(2.302)

According to Bethe’s formula (22)
eb =

χ2c
χ2c
=
.
2
χα0
‘1.1670 χ2α(2.303)

From the derivation in Bethe it is seen that
‘1.1670 = e2C−1(2.304)

C = 0.577216 is Euler0 s constant.

(2.305)

where
Scott’s formula (7.25) for χα isχ2c `n χα = 4π

Z t
0

Ne
dt0 X
`n χe`
2
i
N
α
`n
χ
+
i
α
i
1
k2 (t0 ) i=1
Zi

"

#

.

(2.306)

where
k

=

p/h̄,

(2.307)

p

=

Ni

=

(2.308)αi=

particle momentum,Na ρ
pi = density of atoms of type i,M
αZi /β = Zi e2 /h̄ν ,χαi=the screening angle for atoms of type i,

=

h

i1/2

χ20 (1.13 + 3.76α2i )(2.309),(2.310)λ̄0,rTF
λ0 /2π = h̄/p = wavelength of electron/2π,

χ0

=

λ̄0

=

rTF

=

Thomas − Fermi radius of atom=

0.885 a0 Zi

−1/3

,
2

(2.311)(2.312)(2.313)2

a0=

Bohr radius = h̄ /me e

χe`
i

=

screening angle for the atomic(2.314)atoms of type i.
The next step is to let β = 1 in the αi that are in the χαi , to delete the term with χe`i , and to let
Zi2 → Zi (Zi + ξM S ). Recalling that p = Eβ/c, Equation 2.306 now becomesNa ρ
χ2c `n χα =
4πe4
M

"N
e
X

pi Zi (Zi + ξM S ) `n χαi

i=1

90

# Z

0

t

dt0
.
E2β4

(2.315)

Since e2 = r0 me c2 , and using Equation 2.301, we obtainNa ρ
χ2c `n χα =
4πr02
M

"N
e
X

pi Zi (Zi + ξM S ) `n χαi

i=1

#

m2 t
.
2 β4
ĔM
S

(2.316)

Dividing by Equation 2.300 and multiplying by 2, we get
"N
e
X

#

ZS−1 .

(2.317)`n χ2αi = `n χ20 + `n 1.13 + 3.76(αZi )2 ,i

(2.318)#(2.319)`n χ2α =pi Zi (Zi + ξM S ) `n χ2αi

i=1

But using Equations 2.310 through 2.314,`n χ20 = `n"h̄2 m2e e4−2/3

p2 (0.885)2 h̄4 Zi

#

h

"

m2 e4
−2/3
− `n Zi,= `n 2 2 e2p h̄ (0.885)`n [1.13 + 3.76(αZi )2 ] = `n 1.13 + `n[1 + 3.34(αZi )2 ] .

(2.320)Hence,`n χ2α ="m2e e4 1.13ZS `np2 h̄2 (0.885)2

so that
χ2α =

!

#

+ ZX − ZE ZS−1 ,m2e e4 1.13 eZX /ZS,
p2 h̄2 (0.885)2 eZE /ZS(2.321)(2.322)

Now, recalling that Equations 2.297 through 2.299 are equivalent to
χ2c =



Na ρ
4πr02 ZS t m2e c2 /E 2 β 4 ,M(2.323)and using Equations 2.302, 2.303, 2.322 and 2.323, we obtain
Ω0 =
=

Na ρ
2
2 4
2 2 2 22 ZE /ZS
M 4πr0 ZS tme c (E β /c )h̄ (0.885) e
m2e (r02 m2e c4 )eZX /ZS E 2 β 4 (1.167)(1.13)"#
ρZS eZE /ZS
t
0
‘6680
β2
M eZX /ZS

= bc t/β 2 .

(2.324)

Q.E.D.

Molière’s B parameter is related to b by the transcendental Equation 2.290. For a given valueof b, the corresponding value of B may be found using Newton’s iteration method. As a roughestimate, B = b+ `n b. It can be seen that b, and hence B, increases logarithmically with increasingtransport distance.
The intuitive meaning of Ω0 is that it may be thought of as the number of scatterings that takeplace in the slab. If this number is too small, then the scattering is not truly multiple scattering and91

Figure 2.6: Plots of Molière functions f (0) , f (1) , and f (2) .
various steps in Molière’s derivation become invalid. In Molière’s original paper[107], he considered
his theory valid forΩ0 (his Ωb ) ≥ 20 ,(2.325)which corresponds toB ≥ 4.5,and b ≥ 3 .

(2.326)

From Equations 2.325 and 2.325 we arrive at the conditiont/β 2 ≥ 20/bc = (tef f )0 .
Another restriction on the validity of Equation 2.279 is mentioned by Bethe[23], namely,χ2c B < 1 .

(2.327)

Resuming our presentation of the method used to sample Θ, we return now to the problem ofsampling θ from fr (θ) given by Equation 2.281. It might at first appear that fr (θ) is alreadydecomposed into sub-distribution functions. However, f (1) (θ) and f (2) (θ) are not always positive,
and thus, are not candidate distribution functions. Graphs of f (1) (θ) and f (2) (θ) are shown in
Figure 2.6. We now adopt a strategy similar to that used by Messel and Crawford[103]; namely,mix enough of f (0) (θ) with f (1) (θ) and f (2) (θ) to make them everywhere positive. Unlike Messel
and Crawford, who dropped the term involving f (2) (θ), we have been able to retain all of the first
three terms in the expansion.

92

The factorization we use isfr (θ) =3X

αi fi (θ)gi (θ) ,(2.328)i=1

where
α1 = 1 − λ/B ,−θ 2

f1 (θ) = 2eθθg2,N ormθ(0, ∞) ,(2.331)α2 = µg2,N orm /B ,(2.332)forθ(0, µ) ,

f3 (θ) = 2µ2 θ −3
θ4g3,N orm(2.333)

λf (0) (θ) + f (1) (θ) + f (2) (θ)/B ,α3 = g3,N orm /2µ2 B ,g3 (θ) =(2.330)g1 (θ) = 1 ,f2 (θ) = 1/µg2 (θ) =for(2.329)for(2.334)(2.335)θ(µ, ∞) ,λf (0) (θ) + f (1) (θ) + f (2) (θ)/B(2.336).

(2.337)

When the third sub-distribution function is selected, we first sample η = 1/θ using fη3 (η) andgη3 (η) given byfη3 (η) = 2µ2 η for η(0, 1/µ) ,(2.338)gη3 (η) =η −4
g3,N ormλf (0) (1/η) + f (1) (1/η) + f (2) (1/η)/B

.

(2.339)Then we let θ = 1/η.
As presented above, this scheme contains four parameters, λ, µ, g2,N orm and g3,N orm ; the latter
two are so chosen that g2 (θ) and gη3 (η) have maximum values (over the specified ranges) which
are not greater than 1. The first sub-distribution is the Gaussian (actually exponential in θ 2 )distribution that dominates for large B (thick slabs). The third sub-distribution represents the“ single scattering tail.” The second sub-distribution can be considered as a correction term forcentral θ values. The parameter µ separates the central region from the tail. The parameter λdetermines the admixture of f (0) in the second and third sub-distribution functions. It must be large
enough to ensure that g2 (θ) and g3 (θ) are always positive. It will also be noted that α1 becomes
negative if B < λ so that this case must be specifically treated. After studying the variation of thetheoretical sampling efficiency with the variation of these parameters, the values
λ = 2,µ = 1, g2,N orm = 1.80, g3,N orm = 4.05

(2.340)were chosen. These values do not give the absolute optimum efficiency, but the optimum µ values
were usually close to one, so we chose µ = 1 for simplicity. λ could not have been chosen much
lower while still maintaining positive rejection functions. Furthermore it was desired to keep λ as
low as possible since this would allow Molière’s distribution to be simulated for as low values of93

B as possible. Although Molière’s theory becomes less reliable for B < 4.5, it was felt that it wasprobably as good an estimate as could easily be obtained even in this range.
Since α1 < 0 for B < λ, some modification of the scheme must√ be devised in this case. What wehave done is to use the computed values of B in computing χc B, but for sampling we set ‘1/B’= ‘1/λ’. This has the effect of causing the Gaussian not to be sampled.
Our next point is best made by means of Figure 2.7 which is a graph of Equation 2.290, thetranscendental equation relating B and b. It will be observed that when viewed as defining afunction of b the resulting function is double valued. We of course reject the part of the curve for
B < 1. We would, however, like to have a value of B for any thickness of transport distance (i.e.,any value of b). In order to obtain a smooth transition to zero thickness we join a straight line fromthe origin, (B = 0, b = 0), to the point on the curve (B = 2, b = 2 − `n 2). B is then determined
by


2


2−`n2 b if b < 2 − `n 2,
B=(2.341)the B > 1 satisfying B − `n B = b,
if b > 2 − `n2 .

For rapid evaluation, B has been fit using a piecewise quadratic fit for b(2, 30); b = 30 corresponding
roughly to a thickness of 107 radiation lengths, which should be sufficient for any application.
Actually, b = 0 does not correspond to t = 0, but rather to t ≈ 2 × 10−6 X0 . We neverthelessset θ = 0 if b ≤ 0. The case where b(0, 3) is not too likely either, since b = 3 roughly corresponds
to t ≈ 10−4 X0 , and is not very important since the scattering angles should be small. However,Rogers[136] has found that in low energy applications, it is possible to take too small a step, thereby
running into this constraint and effectively turning off the multiple scattering.
To complete our discussion on sampling we note that f1 (θ) is sampled directly by means ofθ=p−`n ζ .

(2.342)The p.d.f. of f2 (θ) is sampled by merely choosing a uniformly distributed random number. The
p.d.f. of fˆ3 (η) is sampled by taking the larger of two uniformly distributed random numbers.
Finally, g2 (θ) and gη3 (η) are divided into “B-independent” parts
g2 (θ) = g21 (θ) + g22 (θ)/Bforθ(0, 1) ,(2.343)gη3 (θ) = g31 (η) + g32 (η)/Bforη(0, 1) .

(2.344)

The functions g21 , g22 , g31 , and g32 have been fit by PEGS over the interval (0,1) using a piecewise
quadratic fit. This completes our discussion of the method used to sample θ.
Note that the Bethe condition, χ2c B < 1, places a limit on the length of the electron step size
which can be accurately modeled using Molière’s multiple scattering p.d.f. We can determine themaximum total step size consistent with this constraint, tB , by starting withχ2c (tB )B(tB ) = 1 .

94(2.345)

Figure 2.7: Plot of Equation 2.290 (B − `n B = b).

95

From Equation 2.290 we can write
eb = eB /B ,(2.346)and using Equations 2.291, 2.292, 2.297 with Equations 2.345 and 2.346, we havebc tB
exp[(ĔM S β 2 )2 /χ2cc tB ]χ2cc tB=.
β2(ĔM S β 2 )2Solving for tB we obtaintB =2.14.2(ĔM S β 2 /χcc )2.
`n[bc (ĔM S β/χcc )2 ](2.347)(2.348)The Goudsmit-Saunderson Multiple Scattering Distribution

It has long been acknowledged that Molière’s multiple scattering distribution breaks down undercertain conditions in addition to that given above in 2.348. In particular: the basic form of thecross section assumed by Molière is in error in the MeV range, when spin and relativistic effects areimportant; various approximations in Molière’s derivation lead to significant errors at pathlengths
less than 20 elastic scattering mean free paths (recall Equation 2.325); and the form of Molière’s
cross section is incapable of accurately modeling the structure in the elastic scattering cross section
at large angles for low energies and high atomic number. It is therefore desirable to have available
a more exact treatment, and in EGS5, we use in the energy range from 100 eV to 100 MeV elastic
scattering distributions derived from a state-of-the-art partial-wave analysis (unpublished work)
which includes virtual orbits at sub-relativistic energies, spin and Pauli effects in the near-relativistic
range and nuclear size effects at higher energies. Additionally, unlike the Molière formalism, thismodel includes explicit electron-positron differences in multiple scattering, which can be pronounced
at low energies.
The multiple scattering distributions9 are computed using the exact approach of Goudsmit andSaunderson (GS) [63, 64]. We consider electrons or positrons with kinetic energy T̆0 moving ina hypothetical infinite homogeneous medium, with Na ρ/M scattering centers per unit volume,in which they experience only elastic collisions. We assume that the single-scattering differential
cross section (DCS), dσ(Θ)/dΩ, depends only on the polar scattering angle Θ, i.e. it is axiallysymmetrical about the direction of incidence. This assumption is satisfied as long as the scattering
centers are spherically symmetrical atoms or randomly oriented molecules. Moreover, interference
effects resulting from coherent scattering by several centers are assumed to be negligible. As aconsequence, the theory is applicable only to amorphous materials and to polycrystalline solids.
For the sake of simplicity, we limit our considerations to single-element materials; the generalization
to compounds is straightforward (additivity approximation).
9

The derivation given here is graciously provided by Dr. F. Salvat of Institut de Tècniques Energètiques atUniversitat Politècnica de Catalunya, Barcelona, Spain. Dr. Salvat also provided the source code for computing the
GS distribution which we have adapted for use in PEGS.

96

Denoting σ as the total elastic cross section, the angular distribution f1 (Θ) after a single scattering event is
1 dσ(Θ)f1 (Θ) =,(2.349)σ dΩ
The probability of having a polar scattering angle between Θ and Θ + dΘ in a single collision is
given by 2πf1 (Θ) sin ΘdΘ. It is convenient to write f1 (Θ) in the form of a Legendre series
f1 (Θ) =∞X
2` + 1`=0

4π

F` P` (cos Θ),(2.350)where P` are the Legendre polynomials andF` ≡ 2π

Z 1P` (cos Θ)f1 (Θ) d(cos Θ) ≡ hP` (cos Θ)i.

(2.351)g` ≡ 1 − F` = 2π

Z 1

[1 − P` (cos Θ)] f1 (Θ) d(cos Θ) ≡ h1 − P` (cos Θ)i(2.352)−1

The quantities−1

will be referred to as the transport coefficients. Notice that F0 = 1 and g0 = 0. The value of F`
decreases with ` due to the increasingly faster oscillations of P` (cos Θ) and, hence, g` tends to unity
when ` goes to infinity. The transport mean free paths λ` are defined byλ` ≡ λ/g` .

(2.353)

Assume that an electron starts off from a certain position, which we select as the origin of
our reference frame, moving in the direction of the z-axis. Let f (r, d̂; t) denote the probability
density of finding the electron at the position r = (x, y, z), moving in the direction given by theunit vector d̂ after having traveled a path length t. The diffusion equation for this problem is given
by Lewis [97] asZ hi dσ(χ)∂f+ d̂ · ∇f = N
f (r, d̂0 ; t) − f (r, d̂; t)dΩ,(2.354)∂tdΩwhere χ ≡ cos−1 (d̂ · d̂0 ) is the scattering angle corresponding to the angular deflection d̂0 → d̂.
This equation has to be solved under the boundary condition f (r, d̂; 0) = (1/π)δ(r)δ(1 − cos Θ),
where Θ is the polar angle of the direction d̂. By expanding f (r, d̂; t) in spherical harmonics, Lewis
obtained general expressions for the angular distribution and for the first moments of the spatial
distribution after a given path length t. The angular distribution is given by
FGS (Θ; t) ≡Z

f (r, d̂; t) dr =∞X
2` + 1`=04πexp(−tg` /λ)P` (cos Θ).

(2.355)

It is worth noticing that FGS (Θ; t)dΩ gives the probability of having a final direction in the solid
angle element dΩ around a direction defined by the polar angle Θ. Evidently, the distribution of
Equation 2.355 is symmetrical about the z-axis, i.e., independent of the azimuthal angle of the
final direction.
97

The result given by Equation 2.355 coincides with the distribution obtained by Goudsmit andSaunderson [63] in a more direct way, which we sketch here to make the physical meaning clearer.
Using the Legendre expansion given by Equation 2.350 and a folding property of the Legendre
polynomials, the angular distribution after exactly n collisions is found to be
fn (Θ) =∞X
2` + 14π`=0(F` )n P` (cos Θ).

(2.356)

The probability distribution of the number n of collisions after a path length t is Poissonian withmean t/λ, i.e.
(t/λ)nP (n) = exp(−t/λ).
(2.357)n!
Therefore, the angular distribution after a path length t can be obtained as
FGS (Θ; t) =∞X

P (n)fn (Θ) =

n=0

∞
X
2` + 1`=04π"exp(−t/λ)∞X(t/λ)n

n=0

n!

n

(F` )

#

P` (cos Θ),(2.358)which coincides with expression given in Equation 2.355.
From the orthogonality of the Legendre polynomials, it follows thathP` (cos Θ)iGS ≡ 2π

Z 1

−1

P` (cos Θ)FGS (Θ; t) d(cos Θ) = exp(−tg` /λ).

(2.359)In particular, we havehcos ΘiGS = exp(−t/λ1 )(2.360)and

1[1 + 2 exp(−t/λ2 )] .
(2.361)
3
These expressions can be used to check the accuracy of the calculated Goudsmit-Saunderson distribution.
hcos2 ΘiGS =The Goudsmit and Saunderson expansion of the multiple scattering distribution given in Equation 2.355 is exact for pure elastic scattering without energy loss. To compute it for a given single
scattering DCS, we have to evaluate the transport coefficients g` as defined in Equation 2.352. The
number of terms needed to make the Goudsmit-Saunderson series converge increases as the path
length becomes shorter. In the case of small path lengths, the convergence of the series can be
improved by separating the contributions from electrons that have had no collisions,
FGS (Θ; t) ≡ exp(−t/λ)+δ(cos Θ − 1)
2π∞
X2` + 1`=0

4π[exp(−tg` /λ) − exp(−t/λ)] P` (cos Θ).

(2.362)

The first term on the right-hand side represents unscattered electrons. In the current implementation, angular distributions are expressed in terms of the variableµ ≡ (1 − cos Θ)/2.
98(2.363)

We have
FGS (µ; t) = exp(−t/λ)δ(µ) +

∞
X
`=0

(2` + 1) [exp(−tg` /λ) − exp(−t/λ)] P` (2µ − 1).

(2.364)

Differential cross sections for elastic scattering of electrons and positrons by neutral atoms havebeen calculated for the elements Z =1-95 and projectile kinetic energies from 100 eV to 100 MeV.
The calculations were performed by using the Dirac partial-wave program ELSEPA. The scattering
potentials were obtained from self-consistent atomic electron densities calculated with Desclaux’s
multiconfiguration Dirac-Fock program. A finite nucleus, with Fermi model charge distribution,
was assumed. Exchange effects in electron scattering were described by means of the local exchange
potential of Furness and McCarthy. The DCSs computed in this way account for screening, finite
nuclear size, exchange, spin and relativistic effects in a consistent way.
Electron and positron elastic DCS data is presented in tabulated form using a mesh of kinetic
energies and scattering angles that is dense enough to allow cubic spline interpolation of ln(dσ/dΩ)in ln T and µ. The size of the complete data set, referred to as DCSLIB, is about 125 MB. A seriesof new PEGS subroutines (ELASTINO being the main one) evaluates elastic DCSs for electronsand positrons in elements and compounds (using the additivity approximation) and computes
Goudsmit-Saunderson multiple scattering distributions for specified path lengths. The transportcoefficients g` are obtained by means of Gauss-Legendre quadrature, with the angular range split
into a number of subintervals to improve accuracy.
At small path lengths t, a very large number of coefficients g` may be required to ensure
convergence of the Legendre series given in Equation 2.362. The default value of the number ofcoefficients is set in ELASTINO to 1000, which should yield convergence for reasonably short paths
in most cases. Note that since the time to compute a single distribution function depends on thesquare of the number of coefficients, and, typically, a very large number of distributions must be
computed (the distribution function space is three dimensional, depending on material, energy, andpath length, for both positrons and electrons), the total time to compute a full set of distributionfunctions for some problems can be very long. To speed the computation, ELASTINO works fromlow energy and short paths to higher energies and longer paths, re-setting at each new step thenumber of terms to be computed based on the number required at the previous step. Even so,computation times can be extremely long, even when error tolerances have been set fairly high,and users should be mindful of this whenever invoking the use of this distribution.
Traditionally, sampling from GS distributions has been either prohibitively expensive (requiring
computation of the slowly converging series) or overly approximate (using very large pre-computed
data tables with limited accuracy). We have developed here new fitting and sampling techniquesthat overcome these drawbacks, using a scaling model that have been known for some time [27].
First, the change of variables of Equation 2.363 is performed, and a reduced angle now calledχ = (1 − cos(Θ))/2 is defined. The full range of angles (0 ≤ Θ ≤ π, or 0 ≤ χ ≤ 1) is then broken
into 128 intervals of equal probability, with the 128th interval further broken down into 32 subintervals of equal spacing. Because the distribution is so heavily forward peaked, in most cases thelast 32 sub-intervals cover the entire range from approximately π/2 to π, which is why this region99

is treated specially. In each of the 287 intervals or sub-intervals, the distribution is parameterizedasα[1 + β(χ − χ− )(χ+ − χ)](2.365)f (χ) =
(χ + η)2
where α, β and η are parameters of the fit (which can be determined analytically if it is assumed
that the distribution function conforms the shape given above over each given interval) and χ− andχ+ are the endpoints of the interval.
Using the large number of angle bins which we do, this parameterization models the exact formof the distribution to a very high degree of accuracy, and, in addition, it can be sampled very
quickly. As the 128 bins are equally probable, initial angle intervals can be determined trivially.
When the last (128th ) interval is selected, a linear table search is used to determine which of the 32
equally spaced sub-intervals to use. This table search is usually very fast because the first severalof the 32 sub-intervals typically cover 90% of the probable angles in the final interval. Once thecorrect bin is chosen, the distribution can be sampled quickly. We begin by inverting the first termof Equation 2.365 to produce a value of χ fromχ=ηζ + χ− (χ+ + η)/(χ+ − χ− ),(χ+ + η)/(χ+ − χ− ) − ζ(2.366)

and we then use 1 + β(χ − χ− )(χ+ − χ) as a rejection function. Rejection efficiencies on the orderof 95% are typical. Energy and pathlength interpolation is done by randomly selecting either theupper or lower bounding grid point in the tabulated mesh.

2.15

Transport Mechanics in EGS5

As noted in earlier sections, because of the very large number of scattering collisions per unit path
which electrons undergo as they pass through matter, Monte Carlo simulation of individual electron
collisions (sometimes referred to as “analog Monte Carlo electron transport”) is computationallyfeasible only in limited situations. Computationally realistic simulations of electron transport must
therefore depart from the physical situation of linear, point-to-point, translation between individual
scattering collisions (elastic or inelastic) and instead transport particles through long “multiplescattering steps,” over which thousands of collisions may occur, a technique commonly referred toas the “condensed history” [15] method. If we assume that most electron Monte Carlo simulation
algorithms exhibit Larsen convergence10 [96], then all algorithms exhibit an accuracy/speed tradeoff which is driven solely by the length of the multiple scattering step they can take while still
remaining faithful to the physical processes which might occur during those long steps. Models
describing energy loss, angular deflection, and secondary electron production over long steps arequite well known. Descriptions of the longitudinal and transverse displacement coupled to energy
loss and deflection over a long transport step, even in homogeneous media, however, require complete solutions of the transport equation, and so models which are sometimes quite approximate
10

Larsen noted that electron transport algorithms using multiple-scattering steps ought to converge to the analog
result as the number of collisions in the multiple-scattering steps decreases. It must be noted that in general, this is
not the case.

100

x

Final
DirectionφΘ∆xt

Initial
Direction

z
∆y
s

y

Figure 2.8: Schematic of electron transport mechanics model.
are used. The methods employed in Monte Carlo programs for selecting electron spatial coordinates after transport through multiple-scattering steps have come to be referred to as “transportmechanics.” This is illustrated schematically in Figure 2.8.
As an electron moving in the z direction passes through a semi-infinite region of thickness s,
it will have traveled a total distance of t, will have undergone lateral displacements (relative to its
initial direction) of ∆x and ∆y, and will be traveling in a direction specified by Θ and φ. That is,relative to its initial position and direction, the particle’s final position is (∆x, ∆y, s), and the final
direction vector given by (sin Θ cos φ, sin Θ sin φ, cos Θ). The angle Θ can be determined from anappropriate multiple scattering distribution, the angle φ is assumed to be uniformly distributed, andthe selection of ∆x, ∆y and s are determined from the transport mechanics of a given simulation
algorithm.
The fidelity with which an algorithm’s transport mechanics model reproduces true distributions
for displacement is almost the only factor driving the speed/accuracy trade-off of the program, andhence a substantial effort was made to improve the transport mechanics model in EGS5 relative
to that in EGS4. In this section, we will describe the motivation and development of the EGS5transport mechanics model, discuss the parameters used to control the multiple scattering steps
sizes in EGS5, and then present a novel approach for automatically determining nearly optimalvalues of the parameters based on a single, geometry-related parameter.
Until the introduction of the PRESTA [31] algorithm, previous versions of EGS corrected forthe difference between s and t, but ignored the lateral deflections ∆x and ∆y. Thus, the procedure101used to simulate the transport of the electron was to translate it in a straight line a distance salong its initial direction and then determine its new direction by sampling the scattering angle Θfrom a multiple scattering p.d.f. dependent upon the material, the total distance traveled, t, andthe particle energy.
Ignoring lateral deflections introduces significant errors, however, unless restrictions (often quite
severe) are placed on the maximum sizes of the electron transport tracklengths t. These restrictions were greatly eased by the introduction of the PRESTA algorithm, which treats ∆x and ∆yexplicitly during the transport simulation and which also includes a more accurate prescription for
relating the straight line transport distance s to the actual pathlength t. It should be noted herethat because of the random nature of the particle trajectory, s, ∆x, and ∆y are actually randomvariables, dependent upon the scattering angle Θ and the tracklength t. In PRESTA, ∆x and ∆yrange between 0 and t/2 and s is given by some fraction of t. There are two major drawbacks
to the PRESTA formalism, however. First, in situations where an electron is traveling close to aregion boundary, translating it lateral distances ∆x and ∆y perpendicular to its initial direction
can sometimes result in moving it across the boundary and into a region with different material
properties. Thus PRESTA required computationally expensive interrogation of the problem geometry and sometimes resulted in very small steps when particles were traveling roughly parallel
to nearby region boundaries. Second, PRESTA is not adept at modeling backscattering. Electronbackscattering in general results from a single, very large angle collision and not as the aggregate
effect of a large number of small-angle collisions. It is clear from Figure 2.8 that physically, if
an electron were to experience a 180 degree collision immediately at t = 0, it could potentiallytravel a distance s = −t in the backward direction. Thus the set of all possible final positions
for an electron traveling a pathlength t is a sphere of radius t (this is sometimes referred to as
the “transport sphere”). Because it always sets s to be a positive value, the PRESTA formalism
tends to overestimate the penetration prior to the large-angle backscatter event, which can lead tosignificant errors in computations of energy distributions of backscattered electrons, among otherquantities.

2.15.1

Random Hinge Transport Mechanics

In EGS5 a new transport mechanics model, the random hinge, is introduced to address theseshortcomings of PRESTA and thus permit very long step-sizes. The random hinge model, firstused in PENELOPE [14], derives a large part of its success from being formulated so that the set of
all possible termination points is in fact the full transport sphere of radius t. In this methodology,
instead of transporting the particle a distance s and then displacing the particle by ∆x and ∆yand updating its direction according to the sampled scattering angles Θ and φ, the particle track t
is first split randomly into two parts of lengths ζt and (1 − ζ)t, where ζ is a random number. The
electron is transported along its initial direction a distance ζt, at which point (the hinge point) its
direction is updated (using Θ and φ as the scattering angles) and the particle is then translated inthis new direction the remainder of it original tracklength, given by (1 − ζ)t.
It is clearly seen from the schematic of Figure 2.9 that the case of ζ = 0 (i.e., the hinge point

102

FinalDirectionxφΘ∆x(1−ζ) t
t
Θ

z

ζt

y

∆y
s

Figure 2.9: Random hinge transport mechanics schematic.
falling at the initial position) provides a mechanism to simulate track termination points over thefull transport sphere. Formally, the advantages of this method can be demonstrated by examining
the moments of the random hinge model. Ignoring energy losses, Kawrakow and Bielajew [88] haveshown that this version of the random hinge model yields the correct values of the average straightline path hsi/t and lateral displacement h∆x2 + ∆y 2 i/t, and also comes very near to preserving
many higher angle spatial moments.
Energy losses, because they result in changes in the single scattering cross section along t,
reduce the accuracy of the random hinge. It has been shown by Fernández-Varea et al. [56] thatwhen the single elastic scattering cross section is dependent on t, the first moments for axial
and lateral displacements in the random hinge model differ from the exact moments in that theyeffectively evaluate first and second transport cross sections at slightly different distances along t.
The transport cross sections (or inverse transport mean free paths), G` , are given by the Legendre
moments of the single scattering cross sections, as inG` (t) = 2π

Z 1

−1

dµΣ(µ; t)[1 − P` (µ)],(2.367)

where Σ(µ; t) is the spatially dependent macroscopic single elastic scattering cross section and µ iscos(Θ). Whereas the correct expressions for hsi/t and h∆x2 + ∆y 2 i/t in the energy dependent case
are
t
t
hsi
= 1 − G1 ( )(2.368)
t
2
3
and
2t
t
h∆x2 + ∆y 2 i= G2 ( ),(2.369)t
94103

the average axial and lateral displacements in the random hinge model arehsitt= 1 − G1 ( )t2
2(2.370)and2tth∆x2 + ∆y 2 i= G2 ( )(2.371)t92respectively [56]. Since the G` (t)’s increase as t increases (and energy decreases), the random
hinge model slightly underestimates the average straight-line path s and overestimates the lateraldeflections.

2.15.2Modified Random Hinge Transport Mechanics

The analysis above suggests a modification to correct the random hinge method which was first
employed in the Monte Carlo program DPM [155]. In the original random hinge methodology,energy loss is accounted for by simply evaluating the multiple scattering p.d.f. at the hinge point,
which, on average, will occur at t/2. In the new methodology, the location of the random hinge
is not a randomly selected fraction of the total distance to be traveled, t, but instead it is basedon a randomly selected fraction of the integral over t of G1 , the first transport mean free path,commonly referred to as the “scattering power.” This integral quantity is commonly called the“scattering strength,” and denoted by K1 (t), as inK1 (t) =Z t
0

dt0 G1 (t0 ).

(2.372)Thus, instead of the hinge consisting of steps ζt and (1 − ζ)t, it consists of steps of the distances
corresponding to the accumulation of scattering strengths ζK1 (t) and (1 − ζ)K1 (t). In this way,
the total scattering strength K1 (t) over the step is preserved, and the average location of the hinge
will correspond to that location which preserves the first moments of spatial displacements. The
modified random hinge transport mechanics of EGS5, illustrated in Figure 2.10, is therefore capableof simulating accurately the average final positions of electrons moving long pathlengths throughmaterials, even when energy loss occurs continuously along the track. The length of an electron
step is therefore limited only by the accuracy of the multiple scattering p.d.f., any need to higher
order spatial moments (as would be required for some differential tallies), and the accuracy of themethod used to compute the integral in Equation 2.372 along the paths between hinge points.
Values of the scattering powers G1 for electrons and positrons as a function of energy arecomputed in PEGS in a new function G1E. For kinetic energies less than 100 MeV, values are takenfrom cubic spline fits of data provided in the DCSLIB package described in the previous section.
Above 100 MeV, the Wentzel-shaped scattering cross section used by Molière is assumed, in whichcase Equation 2.367 for ` = 1 can be integrated analytically to yield"Na ρ 2πZ 2 e4ln
G1 (E) =M p2 v 2104

π2
π 2 + χ2a−χ2aπ 2 + χ2a!#,(2.373)FinalDirection
φ

x

Θ
∆x
t( (1 − ζ) K1)
t(K1)
Θ

z

t(ζ K1)y∆ysFigure 2.10: Modified random hinge transport mechanics schematic.
where χ2a is taken from Equation 2.322. Discontinuities at 100 MeV between values of G1 computedfrom the partial-wave cross section data of DCSLIB and from the Wentzel cross section above areusually small (on the order of fractions of 1 percent), and hence are smoothed only crudely.
Since the Goudsmit-Saunderson multiple scattering distributions are pre-computed in PEGS,
and as we noted earlier, they are dependent on step size, the modified random hinge methodology
requires that they be tabulated in terms of K1 instead of t. When this option is invoked, a new
PEGS routine MAKEK1, which uses a Gauss-Legendre quadrature to solve Equation 2.372 with G1
expressed in terms of energy, is called to prepare a table of values of K1 ranging between the
maximum and minimum seen in the current problem. Note that because in EGS5 we marchacross boundaries rather than stop and apply multiple scattering first, it is never the case that anarbitrarily small multiple scattering step will be taken, and the minimum step is known in advance.

2.15.3

Dual Random Hinge Approach

The accuracy with which the integral over the scattering power G1 is computed over the hinge canbe stated more plainly as the accuracy of the computation of the hinge distances in cm given the
scattering strength, ζK1 and (1 − ζ)K1 . Inverting integrals over pathlength such as Equation 2.372
is a common process in Monte Carlo electron transport programs, analogous to the computation in
EGS4, for example, of the pathlength to be taken given a specified fractional energy loss to occur
over the step. In typical Monte Carlo programs this type of calculation is done by some form of
the trapezoid rule, which requires that the integrand (stopping power, scattering power, etc.) vary105

FinalDirectionφxEnergy HingesΘt(ζ∆ E)∆xt( (1 − ζ) K1)
t(K1)
Θ

z

t(ζ K1)∆yMono−energetic transport∆t =∆K1/G1 in each segmentt((1−ζ)∆E)ys

Figure 2.11: Dual (energy and angle) hinge transport mechanics schematic.
no more than linearly through the step.
EGS5 employs a different approach to this problem in an effort to retain as much as possible thevery large steps permitted with the multiple scattering hinge as long as average values of K1 arepreserved over the steps. Thus, rather than require small steps to assure accurate integration of G1
over energy, EGS5 completely decouples energy loss and multiple scattering by employing a second
random hinge to describe energy loss, relying on the random position of the energy loss hinges
to yield the correct energy-averaged computation of K1 . A schematic of the dual random hinge
approach is presented in Figure 2.11. Independent energy loss steps tE (∆E) (the distance being
determined from some initially specified fractional energy loss) and multiple scattering steps tΘ (K1 )(here the distance is determined from an initially specified scattering strength) are simultaneously
processed, using random hinges. The lengths of the initial segments are determined by multiplying
random numbers ζE and ζΘ by ∆E and K1 respectively, as before. A particle is then transported
linearly until it reaches a hinge point at either tE (ζE ∆E) or tΘ (ζΘ K1 ). When the multiple scattering
hinge point is reached, the deflection is modeled assuming that the entire multiple scattering pathtΘ (K1 ) has been traversed, and transport continues. Similarly, the energy of the electron remains
unchanged along its track until the energy hinge point tE (ζE ∆E) is reached, at which time theelectron energy is decremented by the full amount ∆E prescribed. Subsequent to a reaching ahinge point and undergoing either energy loss or deflection, the electron must then be transported
the distance remaining for the given step, tE [(1 − ζE )∆E) for energy hinges or tΘ [(1 − ζΘ )K1
for multiple scattering hinges. The two hinge mechanisms are completely independent, as several
energy hinges may occur before a single multiple scattering is applied, and vice versa.

106

In evaluating the multiple scattering distribution and sampling for the deflection, all parameterswhich have an energy dependence are computed using the most recently updated energy for theparticle. Since energy hinges will occur sometimes before scattering hinges and sometimes after,
there is thus an implicit averaging of the electron energies over the full hinge distance. It is in thisway that the random energy hinge provides accurate integration of K1 by, on average, using G1 at
the correct energy. This same methodology is employed when computing the energy loss from theintegral of the stopping power: rather than attempt to approximate the average stopping powerover the step, EGS5 relies on the random energy hinge to provide the correct average result for the
total energy loss.
Note that the quantities that remain fixed over the hinges are ∆E and K1 , and not the distancestE and tΘ . Whenever an event occurs in which values of the scattering power or stopping powerchange, such as an energy hinge, the energy loss and scattering strength remain fixed, and any
unused hinge distances tE and tΘ must be updated using the new values of the scattering andstopping power. Energy deposition along the track is done using the continuous slowing down
approximation (CSDA) model, with updates made whenever a hinge point is reached or any otherevent occurs.

2.15.4

Boundary Crossing

One extremely important implication of the use of the modified random hinge for multiple scatteringis that it lends itself to a method for seamlessly crossing region boundaries. In all previous versionsof EGS, including EGS4 with PRESTA, when a particle boundary was reached multiple elastic
scattering was applied and transport started anew in the new region. With the modified random
hinge however, multiple scattering occurs only at hinge points. If a boundary is crossed during
either the pre-hinge (ζK1 (t)) or post-hinge ((1 − ζ)K1 (t)) portion of the step, the value of G1 usedin updating the accumulated scattering strength is simply changed to reflect the new value of thescattering power in the new media. Thus it is not necessary to apply multiple scattering at region
boundaries, and the expensive re-interrogation of the problem geometry required by PRESTA iscompletely avoided. Inherent in this is the implication that multiple scattering distributions areequivalent for different materials at a given energy and for pathlengths which correspond to thesame scattering strength K1 . This, of course, is not strictly true. It can be shown formally, however,that for a multiple scattering distribution expressed as a sum of Legendre polynomials in Θ,
hcos(Θ)i = exp(−K1 )(2.374)

K1 ' 1 − hcos(Θ)i.

(2.375)so that for small K1
Thus, in preserving K1 for cross boundary transport, the EGS5 method also roughly preserves theaverage cosine of the scattering angle over the boundary.
Inspection of the implementation details reveals that the boundary crossing in EGS5 is analogous
to an energy hinge without energy loss. All step-size variables (rates and distances) need to be
updated, but otherwise transport to the next event is uninterrupted, as shown in Figure 2.12.
107

FinalDirectionφxEnergy HingesΘt(ζ∆ E)∆xt( (1 − ζ) K1)
t(K1)
Θ

z

t(ζ K1)∆yMono−energetic transport∆t =∆K1/G1 in each segmentt((1−ζ)∆E)ys

Material A

Material B

Figure 2.12: Electron transport across region boundaries.

1082.15.5

EGS5 Transport Mechanics Algorithm

The transport between hard collisions (bremsstrahlung or delta-ray collisions) is superimposed on
the decoupled hinge mechanics as an independent, third possible transport process. To retain thedecoupling of geometry from all physics processes, for hard collisions EGS5 holds fixed over all
boundary crossing an initially sampled number of mean free paths before the next hard collision,
updating the corresponding distance (computed from the new total cross section), when enteringa new region. Again, while the random energy hinge preserves the average distance between hardcollisions, it does not preserve the exact distribution of collision distances if the hard collision crosssection exhibits an energy dependence between the energy hinges. In practice, however, this leads
to only small errors in cases where the energy hinge steps are very large and the hard collision
mean free path is sharply varying with energy.
Thus the dual random hinge transport mechanics can be described as follows: at the beginningof the particle simulation, four possible events are considered: an energy loss hinge (determinedby a hinge on specified energy loss ∆E); a multiple scattering hinge (determined by a hinge ona specified scattering strength K1 ); a hard collision (specified by a randomly sampled number ofmean free paths); and boundary crossing (specified by the problem geometry). The distances toeach of these possible events is computed using the appropriate stopping power, scattering power,
total cross section or region geometry, and the particle is transported linearly through the shortestof those 4 distances. The appropriate process is applied, values of the stopping power, scattering
power, cross section and boundary condition are updated if need be, and new values of the distances
to the varies events are computed to reflect any changes. Transport along all hinges then continues
through to the next event. Effectively, there are four transport processes occurring simultaneously
at each single translation of the electron.
The details of the implementation of the dual random hinge, because it is such a radical departure from other transport mechanics models, can sometimes lead to some confusion (and, in any
case, the energy hinge definitely leads to important implications for many common tallies), and so
we present an expanded explication here. Ignoring hard collisions and boundary crossings for the
moment and generalizing here for the sake of brevity, we note that a step t involves transport overthe initial step prior to the hinge a distance tinit given by (ζt) followed by transport through theresidual step distance tresid by ((1 − ζ)t). In practice, once a particle reaches the hinge point at
tinit , we do not simply transport the particle through tresid to the end of the current step, because
nothing actually occurs there, as the physics was applied at the hinge point. So instead, immediately after each hinge the distance to the next hinge point is determined and total step that theparticle must be transported before it reaches its next hinge is given by tstep = t1resid + t2init , where
the superscripts refer to the 1st and 2nd hinges steps. We thus have the somewhat counter-intuitive
situation in that when a particle is translated between two hinge points, it is actually being moved
a distance which corresponds to the residual part of one transport step plus the initial part of thesecond transport step. Thus we distinguish between translation hinge steps, over which particles
are moved from one hinge point to the next, linearly and with constant energy, and transport steps,
which refer to the conventional condensed history Monte Carlo distances over which energy losses
and multiple scattering angles are computed and applied. Translation hinge steps and transport109Transport Steps,∆ E = E x ESTEPE

transport step 1

transport step 2∆ E1= DEINITIAL1+ DERESID1 ∆ E2 = DEINITIAL2 + DERESID2
DEINITIAL1

DERESID1

energy hinge 1

DEINITIAL2

DERESID2

energy hinge 2ΚΕ = ΚΕ − ∆ Ε1

initial translation step,
DEINITIAL1

KE = KE − ∆ Ε2translation step 2,
DERESID1 + DEINITIAL2

translation step 3,
DERESID2 + DEINITIAL3

Translation Steps, between random hinge points

Figure 2.13: Translation steps and transport steps for energy loss hinges. The top half of the
figure illustrates the step size (in terms of energy loss) for consecutive conventional Monte Carlotransport steps, with the energy loss set at a constant fraction of the current kinetic energy (Etimes ESTEPE as in EGS4, for example). The lower schematic shows how these steps are broken
into a series of translation steps between randomly determined hinge distances. Transport throughthe translation steps is mono-energetic, with full energy loss being applied at the hinge points.
Note that the second translation step, for which the electron kinetic energy is constant, actually
involves moving the electron through pieces of 2 different transport steps. Multiple scattering could
interrupt this energy step translation at any point or at several points, but does not impact theenergy transport mechanics.
steps thus overlap rather than correspond, as illustrated in Figure 2.13.
Variables which contain information about the full distance to the next hinge (the translationstep), the part of that distance which is the initial part of the current transport step, and theresidual (post-hinge) distance remaining to complete the current transport step, for both energy
loss and deflection, must now be stored while the particle is being transported. Again, it is not the
distances themselves but rather the energy losses and scattering strengths which matter, and in
EGS5, these variables are called DENSTEP, DEINITIAL and DERESID, for the energy loss hinge and
K1STEP, K1INIT and K1RSD, for the the multiple scattering hinge. For reasons discussed below,
only the scattering strength variables become part of the particle stack; the energy loss hinge
variables are all local to the current particle only.
Several interesting consequences arise from the use of energy hinge mechanics. Even though
the electron energy is changed only at the energy hinge point, energy deposition is modeled as
occurring along the entire electron transport step, and the EGS4 energy loss variable EDEP, which
has been retained in EGS5, is computed along every translation step, and passed to the user for
110

tally, as in EGS4. Depositing energy continually in this manner, rather than only in discrete,large, chunks at the hinge points, obviates requiring very small values of DEINITIAL to attainsmall statistical uncertainty in most energy deposition problems. It does introduce some artifacts,
however, in that while translating a particle to an energy hinge point, energy is being deposited
into the medium without being decremented from the particle. Similarly, along the track segment
between the hinge and the end of the energy transport step (the part described by DERESID), thefull transport step energy loss will have been decremented from the electron, but not all of that will
have been deposited via continuous stopping loss until the end of the transport step. Thus energy
is not strictly conserved, in that a particle’s total energy plus the energy of its daughter particles
plus the energy that it has deposited do not sum to the particle’s initial energy except at the exacttransitions between hinge steps (which are not stopping points on the translation steps). This has
some interesting consequences. For instance, if an electron escapes the problem geometry before
reaching its next energy hinge, a hinge must be imposed at the boundary to and the amount ofenergy deposited prior to the escape of the particle much be decremented from the particle before
it is tallied as an escaped particle. Similarly, if an electron escapes during the residual part of aprevious hinge (i.e., before the full DERESID has been deposited), its kinetic energy must adjusted
upward to account for the fact that the full hinge energy has already been decremented from theparticle, but a portion of it has yet to be deposited, because the end of the hinge step had yet tobe reached. See Figure 2.14 for a schematic illustrating this problem.
Likewise, when a hard collision occurs, a particle will be somewhere in the midst of sometransport step, on either the residual side of the previous energy hinge or the initial side of theupcoming hinge, and so the kinetic energy available for the collision must be adjusted to accountfor the energy deposited during the translation step. If the electron has yet to reach the end of
the previous hinge still (i.e., it’s still on the residual side), we must adjust the particle kinetic
energy by increasing it to account for energy decremented but not deposited before proceeding thecollision analysis. And, as with the case of escaping particles, if the electron has passed the end of
the previous hinge step and is on the initial part of the next hinge step, we must impose a hinge
and decrement the already deposited energy from the electron. These adjustments are necessary
not only to preserve energy deposition, but also to determine the kinetic energy available for the
hard collision. In terminating hinges before hard collisions, we adjust the electron energy exactlyas was done for escaping electrons, by setting E = E − DEINITIAL + DENSTEP.
A final interesting problem occurs at final electron energy hinge. Because the electron energy
is set to be equivalent to ECUT at the final hinge point, which is reached before the electron has
been translated through the DERESID portion of its transport step, some provision must be madefor tracking electrons during the residual parts of their final hinge steps. This requirement iscomplicated somewhat in that such electrons will have total energy which is exactly equal to the
problem cutoff energy. In the present EGS5 algorithm, when the last energy hinge point is reached,
the electron energy is set to ECUT, DENSTEP is set to DERESID, and DEINITIAL is set to zero (DERESID
is then set to zero as well). The multiple scattering hinge step at this point is set to infinity, as
is the distance to the next hard collision. The particle is thus transported linearly through the
distance corresponding to DERESID, depositing the appropriate CSDA energy along the way.

111

tranport step 1

transport step 2

DEINITIAL1 DERESID1

DEINITIAL2

DERESID2translation step,
DERESID1 + DEINITIAL2

A.) Energy decremented at last hinge,
but not yet deposited in translation
BoundaryA.) Escape prior to endof transport stepA.) DENSTEP

next energy hingeprevious energy hingeB.) Energy deposited intranslation, but not yet
decremented

BoundaryB.) DENSTEPEnergy deposition

B.) Escape after end oftransport step

Figure 2.14: Electron boundary crossing during translation steps. In the top schematic (case A),
the electron crosses escapes the problem volume before traversing the full distance corresponding
to the initial transport step (i.e., through both legs of the hinge), while in the bottom schematic(case B), the particle escapes the volume after the completion of initial transport step. In case A,
the full transport step CSDA energy loss is decremented from the particle’s energy at the time it
passes through the hinge point, but as the particle has not traveled the distance corresponding tothat amount of energy loss, some kinetic energy must be added back to the particle. In case B,
the particle crosses the boundary during the first leg of the second energy hinge step, thus it has
deposited energy into the system even though its energy has not been updated, so some kinetic
energy must be subtracted at that point. In either instance, the final energy of the electron can beshown to be E = E − DEINITIAL + DENSTEP.
112

2.15.6

Electron Step-Size Selection

User control of multiple scattering step-sizes in EGS4 was accomplished through the specificationof the variable ESTEPE, the fractional kinetic energy loss desired over the steps. Because of thecomplete decoupling of the energy loss and elastic scattering models in EGS5, however, there are
two separate step-size mechanisms, one based on energy loss and controlling energy hinge steps, andone based on scattering strength and controlling multiple-scattering hinge steps. Of the two, themultiple-scattering step is almost always the more limiting because it is what drives the accuracy of
the transport mechanics model. As stated earlier, energy hinges are required primarily to provide
accurate numerical integration of energy-dependent quantities. Because hard collisions impose defacto energy hinges whenever they are encountered, for situations in which their cross sections arelarge because the PEGS thresholds AE and AP are small, the numerous hard collisions often provide
the full accuracy required in energy integration, thus rendering energy hinges superfluous. Only incases where hard collision probabilities are small and material cross sections, stopping powers, etc..,vary rapidly with energy do energy hinge steps actually need to be imposed, and for such instances,a mechanism has been developed by which these hinge steps are automatically determined in PEGS,
as described below.

2.15.7

Energy Hinge Step-Size Determination in PEGSAs noted above, in the dual hinge formalism of EGS5, the energy steps provide no function otherthan assuring accurate numerical integration over energy-dependent variables (such as energy loss,
scattering strength, etc.). For any such quantity f which varies with energy through a step of total
length t, if the energy hinge occurs at a distance h, the EGS5 random energy hinge methodology
gives for the integration of f over distance variable s through t
Z t

F (t : h) =ds f (s)(2.376)0= hf (E0 ) + (t − h)f (E1 )where E0 is the initial energy and E1 the energy at the end of the energy hinge step t. As
implemented in EGS5, the energy hinges distances are uniformly distributed in energy, so if ζ is a
−1 and (t − h) = (1 − ζ)∆E| dE |−1 , giving us in practicerandom number, we have h = ζ∆E| dEdx |dx
F (t : h) = ζ∆E f (E0 )

dE −1
dE −1
+ (1 − ζ)∆E f (E1 )
dx E0
dx E1

(2.377)Since the energy hinges are uniformly distributed over t, the average values of the integrated
quantities are given by
F (t) =Z t

dh F (t : h) p(h)(2.378)

0

=

dE −1
dE −1
+ (1 − ζ)∆E f (E1 )dζ ζ∆E f (E0 )
dx E0
dx E1
0

Z 1

113

!

∆E
2

=

dE −1
dE −1
+ f (E1 )f (E0 )
dx E0
dx E1

!

Thus, the random energy hinge step distances are limited by the accuracy which can be achieved
in numerically integrating energy dependent quantities of interest using the trapezoid rule. This
limit suggests a prescription for determining the energy hinge step-sizes in EGS5: we take t as
the longest step-size which assures that Equation 2.379 is accurate to within a given tolerance f E
when applied to the integration of the following: the stopping power to compute the energy loss;
the scattering power to compute the scattering strength; and the hard collision cross section tocompute the hard collision total scattering probability and mean free path. Thus, in general, if∆F |anal is the analytic integral of one of our functions f over t, we wish to satisfy∆F |anal − F (t) ≤ f E
or
or

Z t

(2.379)t−
ds f (t)(f (E0 ) + f (E1 )) ≤ f E
2
0
anal




(2.380)dE −1
RC (E0 ) − RC (E1 )−(f (E0 ) + f (E1 )) ≤ f E
dEf (E)
dx
2
E0
anal

Z E1





(2.381)

where RC (E) is the CSDA range for an electron with energy E.
For the scattering strength, the function f is the scattering power G1 , and for energy loss
f is the stopping power, in which case the analytical expression reduces simply to ∆E. Notethat in the case of the electron mean free path and total scattering probability, the expressions
for both the analytical function and the random hinge results are somewhat different from theresults described above, as the integrands for those quantities contain the spatial distribution of
the collision distances. For the random energy hinge methodology, the probability per unit path of
an interaction taking place over a step of length t is given byp(s : h) =

Σ0 e−sΣ0s ≤ h,

Σ1 e−hΣ0 e−(s−h)Σ1(2.382)s>hwhere h is the hinge distance, Σ0 the cross section at the initial energy, and Σ1 the cross section
after the energy hinge. The random hinge mean free path over t is then given as
λEh (t) =
=
=
=

=

1

Z t

ds s p(s)PEh (t) 0
Z t
Z s
1
ds s
dh p(s : h) p(h)PEh (t) 0
0


Z t
Z s
1
Σ0 (t − s) −sΣ0 Σ1 −hΣ0 −(s−h)Σ1
e
+
e
e
ds s
dh
PEh (t) 0
st
t
0


−sΣ1

Σ0 (t − s) −sΣ0 Σ1 ee+
ds s 
PEh (t) 0
t
1

Z t


−tΣ1

e
1

−
PEh (t) Σ0
1







1 − e−s(Σ0 −Σ1 )t(Σ0 − Σ1 )(2.383)








 (Σ − Σ ) 1 − e−tΣ01 − e−t(Σ0 −Σ1 ) 
0
1
1

+
1+
(Σ0 − Σ1 )tΣ1tΣ20 Σ1114

where PEh (t) is the probability that there is a hard collision of any kind over t, p(h) is uniform andgiven by 1/s, with the probability for a given s that we have yet to encounter the hinge given by(t − s)/t and the probability that we are past the hinge given by s/t. PEh (t) is given byPEh (t) =

Z t

ds

0

=

Z t

=

Z t

Z s

dh p(s : h) p(h)(2.384)0Z s

Σ0 (t − s) −sΣ0 Σ1 −hΣ0 −(s−h)Σ1
dh
ds
e
+
e
e
st
t
0
0
0



ds 

= 1−



Σ0 (t − s) −sΣ0e+t
e−tΣ1 1 − e−t(Σ0 −Σ1 )t(Σ0 − Σ1 )

Σ1 e−sΣ1 1 − e−s(Σ0 −Σ1 )t(Σ0 − Σ1 )







Note that the distribution of collision distances, p(s), for the random energy hinge can be seen fromthe integrand in the above expressions to be−sΣ1

Σ0 (t − s) −sΣ0 Σ1 ep(s) =e+t

1 − e−s(Σ0 −Σ1 )t(Σ0 − Σ1 )



(2.385)In the exact case for electrons passing through media with varying cross sections, we have, ofcourse,
 Z s

Z t
1
λ(t) =
ds s Σ(s) exp −ds0 Σ(s0 )(2.386)
P (t) 00with the expression for P (t), the probability of any scatter,
P (t) =Z t0

ds Σ(s) exp −

Z s

0

0



ds Σ(s )0(2.387)

Expressed in terms of energy loss steps rather than distance these becomeEdE 0 −1dE −1
Σ(E) exp −Σ(E 0 )dE 0
dE (RC (E0 ) − RC (E))λ(t) =
dx
dx
E0
E0

(

Z E1

and

)

Z

E
dE 0 −1
dE −1
dE
dE 0
P (t) =
Σ(E) exp −Σ(E 0 )
dx
dx
E0
E0

(

Z E1

Z

)

(2.388)(2.389)Note that in the above, we have described energy hinge steps in both terms of the change inenergy loss (from E0 to E1 ) and also in terms of distance traveled t, as convenient. In EGS5 we
use a simple prescription for relating the two and for switching back and forth. For a given initialenergy E0 and a pathlength t, E1 is given as E0 − ∆E(t) with ∆E(t) computed as follows. A tableof electron CSDA ranges RC (E) is constructed as a function of energy as
RC (E) =

Z E

dE 0

0

115

dE −1
.
dx

(2.390)Since the CSDA range is uniquely defined monotonic function of energy, its inverse, the energy ofan electron with a given CSDA range, EC (R), can be trivially determined. Thus we have∆E(t) = E0 − EC (RC (E0 ) − t)

(2.391)

By interpolating tabulated values of RC (E) and EC (R), relating energy loss to distance traveled isstraightforward.

Implementation We begin with the energy loss integration, for which case we are looking forthe largest ∆E for which
1 ∆E
1−
t 2

−1
−1
dE
dE
(E0 )
(E1 )
+
dx
dx

!

< E

(2.392)where t = RC (E0 ) − RC (E1 ), the pathlength as determined from the range tables, and represents
the analytical value we wish to preserve within a relative error tolerance given by E . We use an
iterative process, beginning with a value of ∆E that is 50% of E0 and step down in 5% increments
until the inequality is satisfied. We next look at scattering power, starting with the value of ∆E
required by the stopping power integration. In this case, we numerically compute the integral ofstored values of G1 (E0 ) times the stopping power for K1 (∆E) and compare that value to that fromthe energy hinge trapezoidal integration,
∆E
2

!

dE −1
dE −1
(E0 )G1 (E0 ) +(E1 )G1 (E1 )dx
dx

If the difference is greater than E , we reduce ∆E by 5% and continue until the difference is lessthan E .
A treatment for the maximum hinge steps which preserve the mean free path (using Equations 2.386 and 2.384) and total scattering probability (using Equations 2.385 and 2.387) for hardcollisions to within E is still being developed.

2.15.8Multiple Scattering Step-size Specification Using Fractional Energy Loss
Parameters

To control multiple scattering step-sizes, it would seem logical for EGS5 to require specificationof cos Θ, because K1 is very close to 1 − hcos Θi (see Equation 2.375). However, because electron
scattering power changes (increases) much more rapidly than stopping power as an electron slows
below an MeV or so, fixing K1 for the entire electron trajectory results in taking much, muchsmaller steps for lower energy electrons than the fixed fractional energy loss method using ESTEPE
of EGS4, given the same step size at the initial (higher) energy. For example, the value of K1
corresponding to a 2% energy loss for a 10 MeV electron in water corresponds to a 0.3% energy loss
at 500 keV. Thus a step-size control mechanism based on constant scattering strength would force
so many small steps at lower energies that EGS5 could be slower than EGS4 for certain problems,
116despite being able to take much longer steps at high energies because of the random hinge transportmechanics. To circumvent this problem, EGS5 reverts to specifying the multiple scattering step-size
based on a fractional energy loss over the step, which is now called EFRACH, to distinguish it fromESTEPE in EGS4. EFRACH is material dependent and specified in the PEGS input NAMELIST.
One consequence of the work done to assure that EGS5 expends most of its computational
effort where it was most important (i.e., at higher energies), was the recognition that additional
efficiency might be achieved by allowing the fractional energy loss of the multiple scattering steps
to vary over the energy range of the problem. Thus, in addition to EFRACH, which is now defined tobe the fractional energy loss over a multiple scattering step at the highest problem energy, a second
parameter EFRACL, corresponding to the fractional energy loss over a multiple scattering step at
the lowest problem energy, is used. Like EFRACH, EFRACL is also material dependent and is specified
in the PEGS input NAMELIST, and the fractional energy loss permitting for multiple-scattering
steps swings logarithmically over the energy range of the problem.
Since the computational effort required to solve electron transport problems by Monte Carlo isdirectly related to the number (and therefore the size) of the multiple scattering steps, an “optimal”
step-size in a simulation would therefore be the longest one for which the transport mechanics error
is less than the desired accuracy. Users thus need to judiciously select EFRACH and EFRACL to glean
maximum peformance.
In practice, however, optimal selection of step-sizes is difficult because the intricate interplay
between electron step-size and accuracy almost always depends on not only the particular quantityof interest being computed, but also on the fineness or granularity of desired output tallies. For
example, a simulation of the spatial energy deposition distribution in a voxelized geometry is amuch more fine grain simulation than a computation of bulk energy deposited in a large detector
module. Even given that optimization of step-size selection necessarily requires some input fromthe user because of it is inherently problem-dependent, it is clear that even expert users would be
unduly burdened by the need to optimally select both EFRACH and EFRACL in order to maximize theperformance of EGS5. To address this problem, a method has been devised which will automaticallyselect step-sizes based on a single user input parameter based on the problem geometry, as described
below.

2.15.9Multiple Scattering Step-Size Optimization Using Media “Characteristic
Dimensions”

As optimization always involves a speed/accuracy trade-off, development of an automated method
for optimization of step-size selection first requires the adoption of a practical standard defining“accuracy.” Perhaps the most severe test of a Monte Carlo program’s electron transport algorithm
is the “broomstick” problem, in which the tracks of electrons normally incident on the planar facesof semi-infinite right-circular cylinders of progressive smaller radii (to isolate the effects of the transport mechanics, hard-collisions are usually ignored in this problem) are simulated. For cylinders ofradii approaching infinitesimal thinness, the average total tracklength of incident electrons before117

short steps accurate, but slow

step too long, inaccurate

Full CSDA Range"optimal" step

Figure 2.15: Schematic illustrating the “broomstick” problem.
they scatter out of the cylinder is given by the single elastic scattering mean-free path, and any
Monte Carlo algorithm using a larger step-size would over-estimate the penetration (and hence
average total tracklength) in the cylinder. It should be clear then that for cylinders with arbitrary
radii, if we begin with results generated by using very small step-sizes and then gradually increase
the steps, we will eventually encounter divergence in the computed average electron track-length
inside the cylinder, as our model will at some point over-estimate the penetration prior to deflectionout toward the sides of the “broomstick.” This is illustrated in Figure 2.15.
Since spatial energy deposition profiles are essentially maps of region-dependent electron tracklengths, and because most problem tallies will be correct if the spatial distribution of electron
tracklengths is correct, the largest value of scattering strength K1 which produces converged results for energy deposition for a given material in a reference geometric volume was initially chosen
to be the standard for the EGS5 step-size selection algorithm. Subsequent studies showed, however, that many problems require a more stringent characterization of electron tracks than average
length, and so a standard based on the average position of track end points for electrons traversing
geometric volumes of given reference sizes was adopted as the standard. In this new adaptation,
instead of computing energy deposition in semi-infinite cylinders, we look at the average lateraldeflection hri of electrons emerging from the far face of cylinders with lengths L equal to their diameters D. Values of hri over finite cylinders are clearly more sensitive to multiple scattering step-sizes
than values of hti, leading to more conservative estimates of the maximum permitted scatteringstrength and thus better assuring accurate results for problems other than those involving energy
deposition.
Using this criteria for defining accuracy, tables of the material and energy dependent valuesof the largest scattering strengths K1 yielding, for volumes of a given size, values of hri which
are within 1% of the converged results for small K1 were compiled. Given this data, EGS5 isable to provide a step-size control mechanism based on a single user input parameter in units oflength which characterizes the geometric granularity of the problem tallies. This value is called the“characteristic dimension” for the problem, and is set in the user’s MAIN program by specifying a
non-zero value of the variable CHARD, which is material dependent. Given this parameter, EGS5
automatically selects the optimal energy dependent values of K1 by interpolating the compiled datatables of maximum scattering strengths in dimension, material and energy. The tables range in118

Different scattering angles
L (= D)D (= L)

Correct hinge step

Hinge steps too long, <r> too small

Figure 2.16: Schematic illustrating the modified “broomstick” problem as used in EGS5.
energy from 2 keV11 up to 1 TeV at values of 2, 3, 5, 7 and 10 in each of the 9 energy decades
spanning the energy range. The characteristic geometric dimensions in the data sets range from10−6 times the electron CSDA range at the low end, and up to half of the CSDA range at the highend12 . (Note that ignoring hard collisions and using unrestricted stopping powers at the upper
end of the energy range in question is physically unrealistic. Computations in this energy range
were made nonetheless to fill out the tables with overly-conservative estimates of the appropriate
step-size.)
Because K1 is the integral over distance of scattering power, which is proportional to ρZ 2 /A
times the integral of the shape of the differential elastic scattering cross section, K1 should beroughly proportional to tρZ 2 /A, if t is the distance, and that was generally found to be the case.
Interpolation in the geometric dimension variable is therefore done in terms of tρ so that interpolation between materials can be performed in terms of Z 2 /A. (To account for the effect ofsoft collision electron scattering, interpolations are actually done in terms of Z(Z + 1) instead of
Z 2 .) Positron K1 values are determined by scaling electron scattering strength by the ratio of thepositron and electron scattering power. The list of reference materials is given in Table 2.4.
11

For high Z materials for which the Bethe stopping power formula is inaccurate at 2 keV, the tables stop at 10
keV.
12
An additional constraint on the minimum characteristic dimension in EGS5 is the smallest pathlength for which
the Molière multiple scattering distribution produces viable results. Bethe [23] has suggested that paths which
encompass at least 20 elastic scattering collisions are necessary, though EGS5 will compute the distribution using as
few as e collisions, which is a numerical limit that simply assures positivity.

119Table 2.4: Materials used in reference tables of scattering strength vs. characteristic dimension at
various energies.
Material
Li
C
H2 O
Al
S
Ti
Cu
Ge
Zr
Ag
La
Gd
Hf
W
Au
U

Z
3
6
10
13
16
22
29
32
40
47
57
64
72
74
79
92

Z(Z + 1)
12
42
76
182
272
506
870
1056
1640
2256
3306
4160
5256
5550
6320
8556

A
6.93900
12.01115
18.01534
26.98150
32.06435
47.90000
63.54000
72.59000
91.22000
107.87000
138.91000
157.25000
178.49000
183.85000
196.98700
232.03600

ρ
0.53402.26001.00002.70202.07004.54008.93335.36006.400010.5000
6.15007.870011.4000
19.3000
19.3000
18.9000

Z(Z + 1)/A
1.72943.49684.21866.74548.482910.5637
13.6922
14.5475
17.9785
20.9141
23.799626.4547
29.4470
30.187732.0833
36.8736

To generate the data sets, then, for each of the 16 reference materials, 45 reference energies,
and 29 broomstick lengths and diameters (i.e., characteristic dimension) a series of Monte Carlosimulations were performed, using up to 25 different values of fractional energy loss (called EFRACH
in EGS5), covering the range from 30% to 0.001% (except when such steps were less than thetheoretical lower limits of the Molière distribution). Energy loss hinges were set to the lesser of
EFRACH and 4% fractional energy loss, and 100,000 histories were simulated, resulting in relative
statistical uncertainties in the computed values of hri at 2σ of around 0.3%. Tallies were made of
the the average track length inside the volume, the average lateral displacement of the particles
escaping the end of the volume, the average longitudinal displacement of particles escaping thesides of the broomstick, and the fractional energy deposited, backscattered and escaping from theside of the broomstick. Computations of the number of hinges expected for the scattering strength
being tested given the broomstick dimension, were also made for each run, and the anticipatednumber of collisions per hinge were also determined and stored.
Illustrative plots showing the divergence in the results as step-sizes are increased in Copperat 5 MeV for several different broomstick thicknesses are shown in Figures 2.17 (results of energy
deposition) and 2.18 (results of lateral spread).
Approximately 20,000 such plots were generated from over 500,000 simulations to encompass thedesired ranges of materials, energies, and characteristic dimensions. The data was then analyzed
to determine the maximum fractional energy loss which showed convergence within the statisticaluncertainty of the data, using a least-squares fit to a line with slope zero and intercept given by the1205 MeV electrons on Cu, Range = 3.796037e-01 cm
D = 1.8980e-01 cm
D = 7.5921e-02 cm
D = 3.7960e-02 cm
D = 1.8980e-02 cm
D = 7.5921e-03 cm
D = 3.7960e-03 cm
D = 1.8980e-03 cm
D = 7.5921e-04 cm
D = 3.7960e-04 cm

Fractional Energy Deposited

1

0.8

0.6

0.4

0.2

0
0.001

0.01
EfracH

0.1

Figure 2.17: Convergence of energy deposition as a function of step-size (in terms of fractionalenergy loss) for the broomstick problem with varying diameters D in copper at 5 MeV.
converged value at short paths. The initial scattering strengths K1 corresponding to the determinedmaximum fractional energy losses were computed using Equation 2.372 cast in terms of an integral
over energy instead of over pathlength,
K1 (E0 ) =

Z E0
E1

dE 0 G1 (E 0 )

dE −1
,
dx

(2.393)where E0 is the initial energy, E1 the energy after the determined maximum fractional energy loss,dE
the stopping power of the medium. Plots of these maximum
G1 the scattering strength anddx
values of K1 to assure convergence as a function of broomstick diameter for several energies intitanium are shown in Figure 2.19.
The expected nearly linearly relationship between K1 and t and the appropriate scaling of K1
with E is clearly evident in Figure 2.19. We also see in that figure, however, several artifacts ofour method. First, we see that our estimation process did not always produce monotonic results,
primarily because of noise in the data due to the Monte Carlo statistics (given the large number ofruns, some outlier points were to be expected). Additionally, the plots exhibit some discrete jumps
because of the finite number of possible K1 values tested in the parameter study. An additional
artifact can be seen in the top plot of Figure 2.20, which shows converged scattering strength data
vs. characteristic dimension for a variety of elements at 100 MeV. Because we limited the test runs
to a maximum of 30% fractional energy loss, we see a plateauing of the plots for high energies.
Other data sets show a corresponding artifact caused by numerical limits on the minimum step-size
121

5 MeV electrons on Cu, Range = 3.80E-01 cm
0.1
D = 5e-02 of range
D = 3e-02 of range
D = 2e-02 of range
D = 1e-02 of range
D = 7e-03 of range
D = 5e-03 of range
D = 3e-03 of range
D = 2e-03 of range
D = 1e-03 of range

Average Lateral Displacement (cm)

0.01

0.001

0.0001

1e-05

1e-06
0.001

0.01

0.1

EfracH

Figure 2.18: Convergence of average lateral displacement as function of step-size (in terms offractional energy loss) for the broomstick problem with varying diameters D in copper at 5 MeV.
at low energies. Note also from Figure 2.20 that the scaling of K1 in Z(Z + 1)/A rather than Z isevident in the comparison of the plots for water and carbon.
Despite the approximations involved in the definition and determination of convergence, theplots of our computed values of the maximum K1 which still assures accurate hri as a function of
the broomstick dimension exhibits for the most part the behavior we expected. Especially for agiven element at a given energy, the log-log K1 plots can generally be described as being roughly
linear in t, though possibly plateauing at either end. Thus each curve (representing one energy
for a given element) can be wholly defined by the characteristic dimensions corresponding to theonset (if any) of plateaus at either end, the values of K1 at those points, and the slope of line on
a log-log between those plateau points. All of the curves were inspected numerically and corrected
to assure monotonicity, to eliminate the more significant artifacts caused by having a limited setof discrete data points and also to eliminate any physically unrealistic trends in K1 as a functionof t or E for the same material. Using these corrected plots, the five parameters defining eachline were determined, with a least squares fit applied to calculate the slopes. The full set of theseparameters has been compiled into a single data file K1.dat provided with the EGS5 distribution.
Thus, for any material, characteristic dimension, and energy, an easy three-way linear interpolation
can be performed to determine the appropriate value of K1 . This is done in a new EGS5 routine
RK1 which is called by HATCH, and which maps a piece-wise linear fit of K1 (E) onto the same global
energy ladder used by the other electron data variables in EGS.

122

Electrons on Ti
1
20MeV
10MeV
7MeV
5MeV
3MeV
2MeV
1MeV
700keV
500keV
300keV

0.1

K1

0.01

0.001

0.0001

1e-05
1e-05

0.0001

0.001
0.01
0.1
Characteristic Dimension (cm)1

10Figure 2.19: Optimal initial scattering strength K1 vs. broomstick diameter (equivalent to thecharacteristic dimension) in titanium at various energies.
The sample problem tutor4 described in the next chapter investigates and illustrates the effectiveness of the automated step-size selection method based on characteristic dimension.

2.15.10Treatment of Initial Steps of Primary Electrons

It must be noted that not all classes of problems are guaranteed to be modeled accurately whenever
the average tracklength or average lateral deflection is modeled correctly in given region volumes. In
particular, problems using tallies which have a spatial or directional dependence on secondary particle production occurring prior to the first multiple scattering hinge point (such as deep penetration
shower simulations) can exhibit step-size artifacts not present in EGS4, since high energy bremsstrahlung directions can be correlated with electron directions. In such cases, while EGS4 always
imposes multiple scattering prior to secondary particle production, the random hinge methodology
of EGS5 does not always assure at least some deflection prior to secondary particle generating,sometimes leading to over-estimation of particles in the forward direction.
To counter this problem and still permit EGS5 to take very very long steps as often as possible,
for all problems in which secondary particles are being produced and the primary source particles areelectrons, a mechanism has been introduced in EGS5 to force very small initial multiple scattering
hinge steps. If and only if an electron is determined by EGS subroutine SHOWER to be a primary123

100 MeV Electrons
Li
C
LW
Al
S
Ti
Ge
Zr
Ag
La
Hf
Au

0.01

K1

0.001

0.0001

1e-05
0.1

1
Characteristic Dimension (gm/cm^2)

10

100 MeV Electrons
Li
C
LW
Al
S
Ti
Ge
Zr
Ag
La
Hf
Au

0.0001

K1

1e-05

1e-06

1e-07
0.001

0.01
Characteristic Dimension (gm/cm^2)

0.1

Figure 2.20: Optimal initial scattering strength K1 vs. broomstick diameter for various elements
at 100 MeV. The upper figure is for values of ρt greater than 0.1, and the lower figure for smaller
characteristic dimensions.

124particle on its very first track, its initial multiple scattering step is automatically set to be thatused for the smallest characteristic dimension treated in the data set. (This in effect is usually the
distance corresponding to the smallest value of K1 for which the Molière distribution is defined.)
Subsequent multiple scattering steps for such particles are then taken to be the minimum of twice
the previous step and the default step given the characteristic dimension of the problem. Thisapproach is still approximate, however, and may be replaced by a single scattering model.

2.16

Photoelectric Effect

The total photoelectric cross sections used in the standard version PEGS4 were taken from Stormand Israel[167]. In PEGS5 we use the more recent compilation in the PHOTX library[131], as
originally implemented by Sakamoto[143] as a modification to PEGS4. PHOTX provides data forelements 1 through 100 in units of barns/atom, and PEGS subroutine PHOTTZ computes
Na ρ
X0Σ̆photo,partial (Z, k̆) =M

1 × 10−24 cm2
barn!

σphoto (Z, k̆) (barns) ,(2.394)

where σphoto (Z, k̆) is obtained by using PEGS function AINTP to do a log-log interpolation in energyof the cross sections in the data base. The total cross section, as computed by PEGS routine PHOTTE,
is given byΣ̆photo (k̆) =NeXpi Σ̆photo,partial (Zi , k̆) .

(2.395)i=1This total photoelectric cross section is then used in the computation of the photon mean free path.
The run-time model of the photoelectric effect was rather simple in early versions of EGS,
which treated photoelectric event directly within subroutine PHOTON. Starting with EGS3, however,subroutine PHOTO was created to provide flexibility in modeling the energies and angles of the ejected
secondary electrons. Since some of the photon energy imparted in a photoelectric absorption isconsumed in ejecting the electron from its orbit, the kinetic energy of a photoelectron is given by
the difference between the incident photon energy and the edge energy of the electron’s sub-shell.
For applications involving high energy gammas, the edge energy is negligible, but for applications
involving photons with energies on the order of several hundred keV or less, treating sub-shell edgeenergies can be important. Thus the default version of EGS4 provided the weighted average K-edgeenergy given byPNepi Σ̆photo (AP )ĔK−edge (Zi )˘Ē K−edge = i=1 PNe.
(2.396)i=1 pi Σ̆photo (AP )In this implementation, photoelectrons are created with total energy˘Ĕ = k̆ − Ē
K−edge + m,

(2.397)˘
provided, of course, that the initial photon energy is greater than Ē
K−edge . To preserve the energy
˘balance, a photon of energy Ē K−edge is created and then forcibly discarded. Photoelectric inter125

˘
actions involving photons with energies below Ē
K−edge are treated as being completely absorbed,
discarded by a call to user routine AUSGAB with IARG=4.
For applications involving energies on the order of the K-edge energy of the materials being
modeled, this treatment is not suitable. Thus, provided with the EGS4 distribution as part of asample user code, was a substitute version of PHOTO[45] which allowed for more explicit modelingof K-shell interactions, including the generation of Kα1 and Kβ1 fluorescent photons. This versionof PHOTO is the basis for the much more generalized version of PHOTO which has become the default
in EGS5.
The microscopic photoelectric absorption cross section σphoto (Z, k̆) of Equation 2.394 is actually
the sum over all the constituent atomic sub-shells of the cross section for each sub-shell s which
has edge energy less than k̆, as in
σphoto (Z, k̆) =Xsσphoto(Z, k̆)(2.398)ss
where σphoto(Z, k̆) is the photoelectric cross section for sub-shell s of element Z at energy k̆.

Evaluation of cross sections near absorption edges The energy dependence of gamma crosssection is modeled in PEGS and EGS using a piece-wise linear fit, which can result in large errors in
the vicinity of photon absorption edges. For example, material data created by PEGS for element
copper with UP=1.0 MeV,AP=0.001 MeV and 200 energy bins produces errors of 60% and 79% in
the gamma mean-free path (GMFP) at the energies of the Cu Kβ1 and Kβ2 x-rays as shown in
Table 2.5. From Figure 2.21, we can see that the linearly fitted gamma mean free path (GMFP)
differs significantly from its exact value in the 64th energy interval in the fitted data, which contains
the absorption edge. This GMFP error leads to an underestimation of Kβ x-ray production by afactor of 2, as shown in Figure 2.22.
To circumvent this problem, a method called the “local extrapolation method” (LEM), hasbeen devised to specially treat energy intervals containing absorption edges [121]. For such energy
intervals, an extrapolation is performed using either the next higher or next lower energy bin,depending on whether the gamma energy is higher or lower than the edge energy. also employed.
The gamma mean free path of Cu at Kβ1 and Kβ2 energies evaluated using LEM agree withexact values to within 1% as shown in Table 2.5. Figure 2.21 shows that accurate prediction of
the intensity of the characteristic x-rays can be achieved by using LEM. EGS5 employs the LEM
method for K-, L1-, L2- and L3- edges by default.

2.16.1

General Treatment of Photoelectric-Related Phenomena

Accurate modeling of the energy of ejected photoelectrons in the general case thus requires resolving
the total photoelectric cross section σphoto (Z, k̆) for a given species into the appropriate sub-shell˘cross sections, so that the correct sub-shell atomic binding energy can be substituted for Ē
K−edge in126

50
PWLF-LEM

GMFP (µ
µm)

Exact

Kβ2
β2

Kβ1
β1

40

PWLF

30

20
K-Edge

10
65th
63rd

0
8.4

8.6

64th
8.8

66th
9

9.2

Energy (keV)

Figure 2.21: Cu GMFP values evaluated by PWLF and LEM

127

9.4

-1

10

Counts (/keV/sr/source)

Cu 40 keVExp
EGS PWLF
EGS PWLF-LEM

-2

10

Kα
α
Kβ
β

-3

10

-4

10

(Fe Kα
α)

-5

10

5

6

7
8
Energy Deposition (keV)

9

Figure 2.22: Comparison of measured and calculated intensity of K x-rays.

128

10

Equation 2.397. Additionally, the vacancies created in atomic sub-shells subsequent to the ejectionof photoelectrons can give rise to either characteristic x-rays or Auger electrons (and additional
vacancies in lower energy sub-shells) when the atom de-excites. Modeling the photoelectric effectwith this level of detail is crucial in many low-energy applications, such as the simulation of detector
response at low energies.
A general treatment of photoelectric-related phenomena in elements, compounds and mixtures
was introduced into EGS4, and an improved method has been implemented in EGS5 by Hirayama
and Namito[72, 73]. K-, L1-, L2-, L3- and other sub-shell photoelectric cross sections taken fromthe PHOTX data base are fitted to cubic functions in log-log form,sln(σphoto(Z, k̆)) = M0s (Z) + M1s (Z) ln(k̆) + M2s (Z) ln(k̆)2 + M3s (Z) ln(k̆)3 .

(2.399)It thus becomes possible to calculate the ratios of sub-shell photoelectric cross sections for eachsub-shell of each constituent element of any compound or mixture quickly and accurately inside
EGS, rather than approximately via the piece-wise linear fits supplied by PEGS. Once the correctelement and sub-shell have been determined (by sampling the discrete distributions of the branchingratios), the photoelectron energy is given byĔ = k̆ − Ĕs−edge (Z) + m ,(2.400)given that Ĕs−edge (Z) is the binding energy of the s-shell of element Z. Since the sub-shell vacancy
is thus known, atomic relaxation can be modeled and additional secondary particles generated
based on fluorescence and Auger transition probabilities and energies.
The fitted coefficients M0s (Z), M1s (Z), M2s (Z) and M3s (Z) and the other associated data required
to model sub-shell level photoelectric effect and secondary particles from atomic relaxation are
initialized in a new BLOCK DATA subprogram of EGS5. The full data set required and the sources
for the data in EGS5 are given in Table 2.6.
Of the more than 50 possible transitions which may occur during the relaxation of L-shellvacancies, 20 of the most important can be modeled in EGS5. All have relative intensities larger
than 1% of the Lα1 transition for Fermium (Z = 100). Table 2.7 lists the atomic transitions which
produce these x-rays, along with their energies and their intensities relative to the Lα1 line for lead.

Table 2.5: GMFP of Cu at Kβ1 (8.905 keV) and Kβ2 (8.977 keV) energies.

Exact∗PWLF
PWLF-LEM

Kβ1
29.84
11.80
29.69

GMFP (µm)(Error) Kβ2
30.53(-60%) 6.295(-0.5%) 30.30(Error)(-79%)(-0.8%)∗ Obtained using CALL option of PEGS.

129

Table 2.6: Data sources for generalized treatment of photoelectric-related phenomena in EGS5.
Data
K-edge energies
Probabilities of x-ray emission at Kand L-shell absorption
K x-ray energies
K x-ray emission probabilities
L1, L2, and L3 edge energies
L x-ray energies
Probability of Coster-Kronig L1- andL2-shell absorption
L x-ray emission probabilities
Average M edge energies
Auger electron energies

K-Auger intensities
L-Auger intensities
M0s (Z), M1s (Z), M2s (Z) and M3s (Z)SourceTable 2 of Table of Isotopes, Eighth Edition [57]Table 3 of Table of Isotopes, Eighth Edition
Table 7 Table of Isotopes, Eighth Edition
Table 7 Table of Isotopes, Eighth Edition, Adjusted
to experimental data by Salem et al.[144]
Table 2 of Table of Isotopes, Eighth Edition
Table 7b of Table of Isotopes, Eighth Edition (Stormand Israel [167] is used if data is not available in [57])Table 3 of Table of Isotopes, Eighth Edition
Theoretical data by Scofield[146], adjusted to experimental data by Salem et al.
Calculated from sub-shell binding energy in Table 2of Table of Isotopes, Eighth Edition
Calculated neglecting correction term by using atomic
electron binding energy in Table 2 of Table of Isotopes, Eighth Edition
Z = 12 − 17, Table 1 of Assad[11]Z > 17 Table 8 of Table of Isotopes, Eighth Edition
Table 2 from McGuire[102]PHOTX[131]130Table 2.7: L x-ray energies and representative intensities (relative to Lα1 ) for lead. The relative
intensities in this table are taken from Storm and Israel [167], and were derived using a representative energy. EGS5 explicitly models the energy dependence of the relative frequency at which Lsub-shell vacancies are created.
L X-ray
L1 -M2 =β4
L1 -M3 =β3L1 -M9 =β10
L1 -M5 =β9/1
L1 -N2 =γ2
L1 -N3 =γ3
L1 -O2 =γ4/1
L1 -O3 =γ4/2
L2 -M1 =η
L2 -M4 =β1
L2 -N1 =γ5
L2 -N4 =γ1
L2 -O4 =γ6
L3 -M1 =l
L3 -M4 =α2
L3 -M5 =α1
L3 -N1 =β6
L3 -N4 =β15
L3 -N5 =β2
L3 -O4,5 =β5Energy (keV)
12.30712.79413.27513.37715.09715.21615.75715.775
11.34912.61414.30914.76515.1789.184
10.45010.551
12.14212.60012.62213.015131

Intensity (%)
31.6
34.6
1.15
1.71
8.13
9.67
1.59
1.86
3.56
130.
.917
26.7
3.26
5.91
11.4
100.
1.45
2.14
19.3
2.57

2.16.2

Photoelectron Angular Distribution

In previous versions of EGS newly created photoelectrons were set in motion in the same direction
as the initiating photon. This proved to be too approximate for some applications, and so to address
this shortcoming, EGS4 was modified by Bielajew and Rogers to use the theory of Sauter[145] todetermine photoelectron angles. Empirical justification for the use of this distribution has been
given by Davisson and Evans[50], who showed that it applies even in the non-relativistic realm,
despite being derived for relativistic electrons. The implementation of a non-relativistic formuladue to Fischer[58] did not significantly impact simulation results, and so Sauter’s formula has been
applied universally in the EGS code.

Sampling the Sauter angular distribution
The Sauter distribution[145] as given by Davisson and Evans[50] may be integrated over the azimuthal angle and cast in the form:
f (µ)dµ =1 − µ2[1 + K(1 − βµ)]dµ(1 − βµ)4(2.401)whereµ = cos Θ
β = v/cqγ = 1/ 1 − β 2

K = (γ/2)(γ − 1)(γ − 2).
Here µ is the cosine of the angle that the electron is ejected (with respect to the initiating photon
direction), β is the speed of the electron relative to the speed of light and γ is the familiar relativistic factor. Although Equation 2.401 may be integrated easily, its integral can not be inverted
analytically and so a direct sampling approach is not feasible. In addition, equation 2.401 may also
be very sharply peaked in the forward direction, making rejection sampling inefficient. Therefore,
we employ a mixed technique to sample for photoelectron angles. We make the separation:

whereg(µ) =
is the directly sampled part, andf (µ) = g(µ)h(µ)(2.402)1[1 + K(1 − βµ)](1 − βµ)3(2.403)

1 − µ2
1 − βµ(2.404)h(µ) =

1
is sampled via the rejection method. g(µ) and h(µ) may be easily normalized so that −1
g(µ)dµ = 1
and h(µ) ≤ 1 for all µ, resulting in efficient sampling. A more complete discussion of photoelectron
angular distributions as adapted for EGS may be found elsewhere[30].

132RTable 2.8: Total cross section (10−24 cm2 /molecule) for coherent scattering from water.
Photon Energy Free O + Free H2 O Liquid(keV)
2 Free H(b) Molecule Water
20
2.65
2.92
2.46
60
0.417
0.444
0.392
100
0.161
0.170
0.151
(a ) From Johns and Yaffe [84]. Note that effects on scattering angle are more dramatic than for total crosssections.
(b) Default values used in EGS/PEGS.

2.17

Coherent (Rayleigh) Scattering

The total coherent (Rayleigh) scattering cross sections used in EGS are from Storm and Israel[167]and are available for elements 1 through 100. As with the photoelectric effect cross sections, thedata file is in units of barns/atom. The PEGS routine COHETZ computes
Na ρX0Σ̆coher,partial (Z, k̆) =M

1 × 10−24 cm2barn!

σcoher (Z, k̆) (barns) ,(2.405)where σcoher (Z, k̆) is obtained by using the PEGS function AINTP to do a log-log interpolation inenergy of the Storm and Israel cross sections. To obtain the total cross section, we treat all theatoms as if they act independently. That is, in PEGS routine COHETM we computeΣ̆coher (k̆) =NeXpi Σ̆coher,partial (Zi , k̆) .

(2.406)i=1

To permit PEGS to retain complete generality when treating compounds and mixtures, theindependent atom approximation is also used when calculating form factors for coherent scattering.
However, as shown in Table 2.8, this assumption is known to be poor, as both the molecular
structure (row 3 of Table 2.8) and the structure of the medium (row 4 of Table 2.8) can affectcoherent scattering [110, 84]. For simulations involving problems in which the correlation effectbetween molecules on coherent scattering is important, PEGS5 provides for the direct input ofinterference coherent cross sections and form factors (if such data is available). Interference coherent
cross sections and form factors for selected materials, including water, are provided in the EGS5
distribution, and the procedure for invoking this option is described in Appendix C of this report,
The PEGS5 User Manual.
Rather than passing Σ̆coher (k̆) to the EGS code, PEGS passes the ratio (Σ̆tot − Σ̆coher )/Σ̆tot ,which EGS uses as a correction factor to include coherent scattering only if the user requests it.
Let us now develop a method for sampling the coherent scattering angle. The differential
coherent scattering cross section is given byr2
dσR (θ)= 0 (1 + cos2 θ)[FT (q)]2 ,
dΩ
2
133

(2.407)

where r0 is the classical electron radius. FT (q) is the total molecular form factor calculated under
the independent atoms assumption discussed above. That is,[FT (q)]2 =NeXpi [F (q, Zi )]2(2.408)i=1

where F (q, Zi ) is the atomic form factor for element Zi and the momentum transfer, q, is given by
q = 2k sinθ √= 2k(1 − cos θ)1/2 .
2(2.409)Using dΩ = 2πd(cos θ), defining µ = cos θ, and with q 2 = 2k2 (1 − µ), we can writedσR (q 2 )
dq 2

=
=

πr02  1 + µ2 
[FT (q)]2k2
2
 1 + µ2  [F (q)]2
πr02
T
2
A(q
)
,
max
2
k2
2
A(qmax)where
2
A(qmax
)=

Z q2

max

0

[FT (q)]2 d(q 2 ) .

(2.410)(2.411)

2
Using this decomposition, we take [FT (q)]2 /A(qmax) as a probability density function and (1 +2µ )/2 as a rejection function (e.g., see Section 2.2). The variable q 2 is sampled fromζ1 =A(q 2 ),
2)
A(qmax(2.412)where ζ1 is a random number drawn uniformly on the interval (0, 1) and the µ value corresponding
to the q 2 value is obtained fromq2µ=1− 2 .
(2.413)2k
If a second random number, ζ2 , is chosen such
1 + µ2≥ ζ2 ,
2(2.414)then the value of µ is accepted for the scattering angle. Otherwise µ is rejected and the sampling
process is repeated. Tabulated values of F (q, Z) given by Hubbell and Overbø[77] have been usedin PEGS.

2.18

Binding Effects and Doppler Broadening in Compton Scattering

The treatment of Compton scattering presented earlier in this chapter assumed that atomic electrons in two-body collisions are unbound and at rest. For high energy photons these assumptions
134

are reasonable, but at lower initial energies, atomic electron binding has the effect of decreasing
the Compton scattering cross section given by Equation 2.170, particularly in the forward direction. Additionally, because bound atomic electrons are in motion, they emerge from Compton
interactions not at energies wholly defined by the scattering angle as given in Equation 2.173, butwith a distribution of possible energies (this effect is usually referred to as “Doppler broadening.”)
Treatments of both atomic binding effects and Doppler broadening were introduced into EGS4 by
Namito and co-workers [119, 117], and all of those methods have been incorporated into the default
version of EGS5, as options initiated through flags specified by the user.
To examine the effects of atomic binding and electron motion, we start with a more generalized treatment of photon scattering than that of Klein and Nishina. Ribberfors derived a doubly
differential Compton scattering cross section for unpolarized photons impingent on bound atomic
electrons using the relativistic impulse approximation [134]. His result can be expressed as
d2 σ

!

r2
= 0
2
dΩdk̆ bC,ik̆c k̆
k̆02

!

dpz

k̆c

dk̆

k̆0

+

k̆0
k̆c

2

!

− sin θ Ji (pz )(2.415)wherepz = −137
dpz
dk̆

=

k̆c =

k̆0 − k̆ − k̆0 k̆(1 − cos θ)/m,~ ~h̄c|k̆0 − k̆|

137k̆0pz (k̆ − k̆0 cos θ)
−
,
~ ~
~ ~
h̄c|k̆0 − k̆|k̆c(h̄c)2 |k̆0 − k̆|2
k̆0

1 + k̆m0 (1 − cos θ),(2.416)(2.417)(2.418)and~ ~h̄c|k̆0 − k̆| =qk̆02 + k̆2 − 2k̆0 k̆ cos θ.

(2.419)Here, the subscript “bC ” denotes Compton scattering by a bound electron; subscript “i ” denotes
the sub-shell number corresponding to the (n, l, m)-th sub-shells; r0 is the classical electron radius
as before; k̆0 and k̆ are the incident and scattered photon energies, respectively, and k̆c is theCompton scattered photon energy for an electron at rest (Equation 2.173); pz is the projectionof the electron pre-collision momentum on the photon scattering vector in atomic units; Ji (pz ) is
the Compton profile of the i-th sub-shell[35]; θ is the scattering polar angle; and m is the electron
rest mass. Note that as we are dealing with bound electrons, it is implicit in the above that thecross section given by Equation 2.415 is 0 when k̆ > k̆0 − Ii , where Ii is the binding energy of anelectron in the i-th shell. Note also that by substituting Equation 2.417 into Equation 2.415 after
eliminating the second term on the right-hand side of Equation 2.417, one obtains an equivalent
formula to Ribberfors’ Equation 3 [134].
The singly-differential Compton cross section (in solid angle) for the scattering from a boundelectron is obtained by integrating Equation 2.415 over k̆ with the assumption that k̆ = k̆c in the135second term on the right-hand side to yield


dσ
dΩ

r2
= 0
2
bC,i



k̆c
k̆0

!2

k̆c
k̆0

where
SiIA (k̆0 , θ, Z) =

+

k̆0
k̆c

!

2

− sin θ SiIA (k̆0 , θ, Z),Z pi,max−∞Ji (pz )dpz .

(2.420)(2.421)

Here, Z is the atomic number and SiIA (k̆0 , θ, Z) is the called the incoherent scattering function of
the i -th shell electrons in the impulse approximation calculated by Ribberfors and Berggren[135],
and pi,max is obtained by putting k̆ = k̆0 − Ii in Equation 2.416. Note that SiIA (k̆0 , θ, Z) converges
to the number of electrons in each sub-shell when pi,max → ∞. The singly-differential Compton
cross section of a whole atom is obtained by summing Equation 2.420 for all of the sub-shells,


dσ
dΩ

IA

r2
= 0
2
bC

k̆c
k̆0

!2

k̆c
k̆0

k̆0

+

k̆c

!

2

− sin θ S IA (k̆0 , θ, Z),(2.422)

where
S IA (k̆0 , θ, Z) =X

SiIA (k̆0 , θ, Z).

(2.423)iHere, S IA (k̆0 , θ, Z) is the incoherent scattering function of the atom in the impulse approximation.
Note that an alternative computation of the incoherent scattering function based on Waller-Hartree
theory[178] and denoted as S W H (x, Z) has been widely used in modeling electron binding effects
on the angular distribution of Compton scattered photons. In this representation of the incoherent
scattering function, x is the momentum transfer in Å, given byθk̆0 (keV)sin,x=12.3992
 (2.424)and equivalent to q of Equation 2.409. Using S W H (x, Z) as the incoherent scattering function, thedifferential Compton scattering cross section is given by


dσ
dΩ

W H
bC

r2
= 0
2

k̆c
k̆0

!2

k̆c
k̆0

+

k̆0
k̆c

2

!

− sin θ S W H (x, Z),(2.425)which is the simply the Klein-Nishina cross section from before multiplied by the incoherent scattering function. Close agreement between S IA (k̆0 , θ, Z) and S W H (x, Z) for several atoms has beenshown by Ribberfors [135], though Namito et al.[118] have pointed out differences between S IA andS W H at low energies. As noted earlier, as S W H (x, Z) increases from a value of 0 at x=0 to Z asx → ∞, the net effect of atomic binding as defined through the incoherent scattering function isto decrease the Klein-Nishina cross section in the forward direction for low energies, especially forhigh Z materials.
The total bound Compton scattering cross section of an atom can be obtained by integrating
Equation 2.425 over the solid angle (Ω),
WH
σbC
=

Z 4π 

dσ
dΩ

136

W H
bC

dΩ.

(2.426)

Implementation in EGS5 If the user requests that binding effects be taken into account, The
total Compton scattering cross section σbC from Equation 2.426 is used in computing the total photon scattering cross section, with values taken from the DLC-99/HUGO[142, 78, 77] library. If theuser further requests that incoherent scattering functions be used in determining Compton scattering angles, θ is sampled according to Equation 2.425, using S W H (x, Z) (taken from HUGO [142] asa rejection function. (Note that the methodology presented earlier for sampling the Klein-Nishina
distribution sampled the scattered photon energy, rather than the scattering angle.)
If the user further requests that shell-wise Compton profiles be used to simulate Doppler broadening, θ is determined as above and then the electron sub-shell number i is randomly selected, taken
in proportion to the number of electrons in each sub-shell. Next, pi,max is calculated by settingk̆ = k̆0 − Ii in Equation 2.416 and using the sampled θ. Then, pz is sampled in the interval (0, 100)from the normalized cumulative density function of Ji (x) for the i-th sub-shell, according toR pzoζ = Fi (pz ) = R 1000

Ji (p0z )dp0z.
Ji (p0z )dp0z(2.427)

where ζ is a random number between (0, 1). PEGS computes and prints piece-wise linear fits of
Fi−1 (ζ), the inverse of Fi (pz ), so that pz can be determined trivially as pz = Fi−1 (ζ). The valueof “100” in Equation 2.427 comes from the upper limit of the pz values taken from the Compton
profile data in Biggs [35]. The speed penalty involved in using shell-wise Compton scattering in
EGS5 is negligible, as the sampling of pz is rapid. Note that when pz > pi,max , it is rejected, and thesub-shell number i and pz are sampled again. Lastly, another rejection by k̆ , which correspondsk̆0
to the second term on the right-hand side of Equation 2.415, is performed.

Limitations of EGS5 in modeling bound Compton scattering and Doppler broadening
The following theoretical restrictions apply to the detailed model of Compton scattering in EGS5:• The Compton profile of a free atom is used. To treat compounds, an amorphous mixture offree atoms is assumed, and any molecular effects on the Compton profiles are ignored.
• The Compton scattered electron energy and direction are calculated using the energy andmomentum conservation laws assuming that no energy absorption by atoms in the Compton
scattering occurs. The electron binding energy is then subtracted from the recoil electronenergy. Electron binding energy is deposited locally with IARG=4.

2.19

Scattering of Linearly Polarized Photons

All of the treatments of Compton and Rayleigh photon scattering presented earlier in this chapter
have assumed that the incoming photons are unpolarized. Since the scattering of polarized photons
is not isotropic in the azimuthal angle, simulations involving polarized photons will not be accurate
using the methods described above. To overcome this limitation, Namito et al. [116] developed a
137

Z

Y
k

θ

O

φ
e0
X
k0

Figure 2.23: Photon scattering system. An incident photon toward the Z-direction is scattered atpoint O. The propagation vector k~0 and polarization vector e~0 of an incident photon are parallel to
e~z and e~x , respectively. Here, e~z and e~x are unit vectors along the z- and x-axis. The scattering polar
angle is θ and the scattering azimuth angle from the plane of e~0 is φ. The scattered propagation
vector is ~k.

method for modeling the scattering of linearly polarized photons in EGS4, and that treatment has
been included in EGS5.
Consider a photon scattering system of Figure 2.23, in which a completely linearly polarized
photon, whose propagation vector and polarization vector are k~0 and e~0 , is scattered at point
O, The propagation vector of the scattered photon is ~k, and the polar and azimuth scattering
angles are θ and φ. Using the methodology of Heitler [71], we consider two components of thedirection vector ~e, one in the same plane as e~0 (which we denote as e~k ) and the other component
perpendicular to the plane of e~0 , (called e~⊥ ). Figure 2.24 shows e~k in the plane S defined by ~k ande~0 , and e~⊥ perpendicular to the plane S. Under the condition that k~0 k e~z and e~0 k e~x (as shownin Figures 2.23 and 2.24), these two polarization vectors, e~k and e~⊥ , can be determined to be thefollowing functions of θ and φ :e~k = N e~x −

1sin2 θ cos φ sin φ e~y −
N

and
e~⊥ =





1
cos θ e~y −
N






1
cos θ sin θ cos φ e~zN

1
sin θ sin φ e~z .
N(2.428)(2.429)q

Here, N = cos2 θ cos2 φ + sin2 φ , e~x , e~y , e~z are unit vectors along the x-, y- and z-axis, respectively,
and e~k and e~⊥ are treated as normalized vectors.

The Compton scattering cross section for linearly polarized photons Ribberfors derived
a doubly differential Compton scattering cross section for an unpolarized photon using the relativis138

X

S
Z
=

e
k
e0

O

k0
Y

Figure 2.24: Direction of the polarization vector of the scattered photon. Plane S contains e~0
and ~k. e~k is in plane S, and is perpendicular to ~k. e~⊥ is perpendicular to plane S.

tic impulse approximation [134]. By modifying Ribberfors’ formula, a doubly differential Compton
scattering cross section for a linearly polarized photon can be derived as:
d2 σ
dΩdk

!

=
bC,i

r02
4



kc k
k02



k0
dpz
kc
+
− 2 + 4 cos2 Θ ×Ji (pz ),
k0
kc
dk


(2.430)where Θ is the angle between the incident polarization vector e~0 and the scattered polarization vector ~e. Note that Equation 2.415 can be obtained by integrating Equation 2.430 over the azimuthal
angle, and that the Compton-scattering cross section of a free electron (i.e., ignoring binding effects) for linearly polarized photons is obtained by integrating Equation 2.430 over k̆ and putting
Ii = 0,!2!
dσ
1 2 k̆ck̆ck̆02+− 2 + 4 cos Θ .
(2.431)= r0
dΩ f C
4
k̆0
k̆0
k̆c
The Rayleigh scattering cross section for linearly polarized photons The Rayleigh scattering cross section for linearly polarized photons is the product of the square of the atomic form
factors (as defined earlier in Equation 2.408) and the single electron elastic scattering (Thomson)cross section for linear polarized photons. The polarized Thomson scattering cross section perelectron is given by [86]dσ
= r02 cos2 Θ,(2.432)dΩ T
and so the polarized Rayleigh scattering cross section is thus


dσR
dΩ



R

= r02 cos2 Θ[FT (q)]2 .
139(2.433)

Note the similarities and differences between this expression and Equation 2.407 for the unpolarized
Rayleigh cross section.
The parallel and perpendicular components of the single electron Compton and Thomson crosssections are given by


dσ
dΩ



1 2
r
4 0

=

f C,k

k̆c
k̆0



!2 (

k̆c
k̆0

+

k̆0
k̆c

−2

o

+4 1 − sin2 θ cos2 φ


dσ
dΩ



1 2
r
4 0

=

f C,⊥

k̆c
k̆0

!2

k̆c
k̆0

+

k̆0
k̆c

,

(2.434)!

−2 ,(2.435)anddσ= r02 (1 − sin2 θ cos2 φ),dΩ T,kdσ= 0.
dΩ T,⊥


(2.436)(2.437)

By adding Equations 2.434 and 2.435, Equations 2.436 and 2.437, respectively, the following Compton scattering cross section and the Thomson scattering cross section for θ and φ are obtained:


dσ
dΩ

1
= r02
2
fC

k̆c



and



dσ
dΩ

k̆0



T

!2

k̆c
k̆0

+

k̆0
k̆c

2

!

2

− 2 sin θ cos φ= r02 (1 − sin2 θ cos2 φ).

(2.438)(2.439)Sampling of the scattering azimuth angle and polarization vector The azimuthal angle
φ is sampled for the determined θ according to Equations 2.438 and 2.439. The direction of thescattered polarization vector is then calculated. It is shown in Equations 2.434 and 2.435 thata completely linearly polarized photon is de-polarized in Compton scattering according to somede-polarization probability. This de-polarization probability, 1 − P , is
1−P =

k̆c
k̆0

+

k̆0
k̆c

!,

−2

k̆c
k̆0

+

k̆0
k̆c

2

2

!

− 2 sin θ cos φ .

(2.440)In Compton scattering, either a polarized or de-polarized photon is sampled according to this depolarization probability. When the scattered photon is polarized, the direction of the polarization
vector is calculated according to Equations 2.428. When the scattered photon is de-polarized,
the direction of the polarization vector is sampled from the direction between e~k and e~⊥ , shownas Equations 2.428 and 2.429, respectively, at random. In Rayleigh scattering, since there is noprobability for de-polarization, the direction of ~e is always calculated according to Equation 2.428.
140

Transformation to the laboratory system In the scattering system used here, k~0 and e~0 are
in the direction of e~z and e~x , respectively, as shown in Figure 2.23, whereas k~0 and e~0 may be in anarbitrary direction in the laboratory system. The scattering and laboratory systems are connected
via three rotations, which are calculated from the direction of k~0 and e~0 in the laboratory system.
Using these three rotations, ~k and ~e are transformed from the scattering system to the laboratory
system.
Here, we describe the relation of the laboratory system, which is used in the default EGS4
simulation, to the scattering system used in Compton and Rayleigh scattering routines for linearlypolarized photons.
The laboratory system, in which k~0 and e~0 are in arbitrary directions, and the scattering system,
in which k~0 is parallel to e~z and e~0 is parallel to e~x , is transformed to each other by three rotations.
e~z and e~x are unit vectors parallel to the z- and x-axes. Two rotations are necessary to makek~0 k e~z . These rotations were described by Cashwell and Everett [42]. The default version of EGS4
already treats these rotations. Using Cashwell’s notation, this A−1 matrix is
vw
ρ
u
ρ

 uw

 ρ
A−1 =  −v
ρ

u

v



−ρ

0 .
w

(2.441)√
Here, ρ = 1 − w2 . This matrix is written with an inverse sign, since A is mainly used for atransformation from the scattering system to the laboratory system. It is clear thatu0
A−1  v  =  0  .
w
1




 

(2.442)In the laboratory system, e~0 ⊥ k~0 ; rotation by A−1 does not change this relation. As A−1 makes
k~0 k e~z , A−1 moves e~0 onto the x-y plane. In Figure 2.25, k~0 and e~0 after two rotations by the A−1
matrix is shown. Another rotation by an angle (−ω) along the z-axis is necessary to make e~0 k e~x. The cos ω and sin ω are calculated usingcos ω
A−1 e~0 =  sin ω  .
0

(2.443)

By these three rotations, k~0 and e~0 in the laboratory system are transferred to those in the scatteringsystem. The scattered photon propagation vector (~k) and the polarization vector (~e) are transferredfrom the scattering system to the laboratory system by an inverse of these three rotations afterCompton or Rayleigh scattering.
The relationship of k~0 , ~k, e~0 and ~e in laboratory system and those in scatter system are:k~0 (lab) = A · B · k~0 (scatter),e~0 (lab) = A · B · e~0 (scatter),~k (lab) = A · B · ~k (scatter),~e (lab) = A · B · ~e (scatter),
141(2.444)

X

Z

ω
k0

O

e0

Y

Figure 2.25: Direction of k~0 and e~0 after two rotations by A−1 . While k~0 is already parallel to
e~z , another rotation along the z-axis is necessary to make e~0 k e~x .
where

cos ω
B =  sin ω0− sin ω
cos ω
0



0
0.
1


(2.445)

Limitations of the present code for modeling linearly polarized photon scattering• Circularly polarized photon scattering is ignored and elliptically polarized photon scattering
is treated as partially linearly polarized photon scattering.
• Characteristic x-rays and bremsstrahlung photons are assumed to be unpolarized.
The formulas used in each simulation mode are summarized in Table 2.9.

2.20

Electron Impact Ionization

Because of the interest in modeling the generation in the production of characteristic radiation,
it is desirable to treat explicit δ-ray collisions involving inner shells atomic electrons. Inelastic
electron scattering collisions which result in the ejection of a bound atomic electron is typicallycalled electron impact ionization (EII), and a modification to EGS4 by Namito et al. [120] allowing
the treatment of K-shell electron impact ionization has been retained in EGS5. Six different crosssections describing EII are available, as given below. A detail discussion of the cross sections anda guide for selecting the most appropriate one for given applications can be found in [120].
142

Table 2.9: Formulas used in various simulation modes employing detailed treatment of Compton
and Rayleigh scattering.
Equation
2.430
2.431
2.432
2.415
2.425
2.426

Simulation mode
Compton scattering with LP, σbC , S(x, Z) and DB.
Compton scattering with LP.
Rayleigh scattering with LP.
Compton scattering with σbC , S(x, Z) and DB.
Compton scattering with σbC and S(x, Z).
Compton scattering with σbC .

1. Casnati [43, 44]
2. Kolbenstvedt-revised [105]3. Kolbenstvedt-original [92]4. Jakoby [80]5. Gryziński [65] Equation 21
6. Gryziński-relativistic [65] Equation 23
EII is treated as a subset of Møller scattering in EGS, so neither the electron mean-free path nor
the stopping power are modified when EII is treated. Molecular binding effects are ignored, andelectron impact ionization in L and higher shells is treated as free electron Møller scattering.
The ratio of the K-shell EII cross section of J-th element in a material to the Møller scatteringcross section is calculated by the following equation:
R(E, J) =PJ

j=1 ΣEII,j (E)ΣMoller(E)ΣEII,j (E) = pj σEII,j (E) ρ,(2.446)N0,W(2.447)

where
R(E, J) = the cumulative distribution function of the ratio of the K-shell EII cross section of theJ-th element in a material to the Møller scattering cross section at electron energy E,
ΣMoller (E) = macroscopic Møller scattering cross section at electron energy E,
ΣEII,j (E) = macroscopic EII cross section of the j-th element at electron energy E,
σEII,j (E) = microscopic EII cross section of the j-th element at electron energy E,
pj = proportion by number of the j-th element in the material,ρ = density of a material,
N0 = Avogadro’s number,

143

W = Atomic, molecular and mixture weight for an element, for a compound, and for a mixture.

In EGS, K-shell vacancy creation by EII is sampled using R(E, J) values calculated in PEGS
and included in the material data file.

K-x ray emission and energy deposition After K-shell vacancy creation by EII, emission of Kx rays is sampled using the K-shell fluorescence yield [57], as in the treatment of the photoelectriceffect [73]. However, in the current EGS implementation, neither Auger electron emission nor
atomic relaxation cascades are modeled for K-shell vacancies generated by EII. K-shell x-ray energies
are sampled from the ten possible lines given in [57], and the difference between the K-shell binding
energy EB and the emitted x-ray energy is deposited locally. In the case when no x-ray is generated(i.e., an Auger electron emission occurred), the full binding energy EB is deposited locally.

Secondary electron energies and angles The energy and direction of ejected electrons following EII are treated in an approximate manner. The binding energy EB is subtracted at random
from the energy of either one of the two electrons after energies have been determined from thestandard Møller scattering analysis. In the case that neither of the two electron has kinetic energy
greater than EB , EB is subtracted from the energy of both electrons while keeping the ratio of the
kinetic energies unchanged. The directions of electrons after EII are those given from the Møller
scattering collision analysis.

144

Chapter 3

A SERIES OF SHORT EGS5
TUTORIALS
EGS is a powerful system which can be used to produce very complex Monte Carlo simulations.
In spite of some complexity, the user’s interface with the system is, in principle, very simple. In
the following series of tutorial programs, we use various aspects of the user interface in what werefer to as “EGS5user codes.” In these user codes we will introduce some basic scoring techniques.
Formal documentation in the form of EGS5 and PEGS user manuals can be found in Appendices B
and C, respectively.
These tutorials are written under the assumption that the reader is generally familiar with thecontents of the EGS5 and PEGS user manuals, although a complete understanding of the manuals is
not required. In fact, the purpose of these tutorials is to make these manuals more understandable.
Although the programs presented here are very simple in construction, it should become clear thatwith various extensions (generally of a bookkeeping nature), a wide range of powerful programs canbe constructed from these tutorial examples. For brevity, we sometimes present only partial source
listings of these user codes in the following sections. The complete source code for each tutorial canbe found in the EGS5 distribution. Note also that the results from these tutorial programs may
be slightly different on machines with different word lengths, different floating-point hardware, ordifferent compiler optimizations.

3.1

Tutorial 1 (Program tutor1.f)

The geometry of the eight tutorials is the same. Namely, a semi-infinite slab of material is placedin a vacuum and a pencil beam of photons or electrons is incident normal to the surface. The slab
is in the X-Y plane and the particles are incident at the origin traveling along the Z-axis. In thefirst problem, a beam of 20 MeV electrons is incident on a 1 mm thick plate of tantalum. In order145to use EGS5 to answer the question “What comes out the far side of the plate?”, we have createdthe user code (tutor1.f) shown below. Also provided is the PEGS5 input file required for this run(see Appendix C for a description of how to construct PEGS5 input files).
!***********************************************************************!
!
**************!
**!
* tutor1.f *!
**!
**************!
! An EGS5 user code. It lists the particles escaping from the back! of a 1 mm Ta plate when a pencil beam of 20 MeV electrons! is incident on it normally.
!
! For SLAC-R-730/KEK Report 2005-8: A simple example which ’scores’! by listing particles!
! The following units are used: unit 6 for output!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!----------------------------------------------------------------------! Step 1: Initialization!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/randomm.f’
!
!
!
!
!

bounds contains ecut and pcut
media contains the array media
misc contains med
thresh contains ae and ap
useful contains RM

146

!

usersc contains emaxe!

common/geom/zboundreal*8 zboundgeom passes info to our howfar routine*

real*8 ein,xin,yin,zin,
uin,vin,win,wtin
integer iqin,irin! Arguments

integer i,jcharacter*24 medarr(1)! Local variables!
!
!

---------Open files---------open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)!

====================
call counters_out(0)====================!

!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=1
medarr(1)=’TA’
do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend do! nmed and dunit default to 1, i.e. one medium and we work in cm
chard(1) = 0.1d0

!
!
!
!

!
!

optional, but recommended to invoke
automatic step-size control--------------------------------------------Run KEK version of PEGS5 before calling HATCH(method was developed by Y. Namito - 010306)---------------------------------------------

147

100
!
!

write(6,100)
FORMAT(’ PEGS5-call comes next’/)==========
call pegs5==========!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------nreg=3!
nreg : number of region
med(1)=0
med(3)=0
med(2)=1! Vacuum in regions 1 and 3, ta in region 2
ecut(2)=1.5! Terminate electron histories at 1.5 MeV in the platepcut(2)=0.1! Terminate
photon histories at 0.1 MeV in the plate!
Only needed for region 2 since no transport elsewhere!
ecut is total energy = 0.989
MeV kinetic energy
!
!
!
!

120

!
!

-------------------------------------------------------Random number seeds. Must be defined before call hatch
or defaults will be used. inseed (1- 2^31)-------------------------------------------------------luxlev = 1
inseed=1
write(6,120) inseed
FORMAT(/,’ inseed=’,I12,5X,*’ (seed for generating unique sequences of Ranlux)’)=============
call rluxinit ! Initialize the Ranlux random-number generator=============!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!----------------------------------------------------------------------! Define initial variables for 20 MeV beam of electrons incident! perpendicular to the slab
iqin=-1!
Incident charge - electrons!
20 MeV kinetic energy
ein=20.0d0 + RM
xin=0.0
yin=0.0
zin=0.0!
Incident at origin

148

!
!
!
!

uin=0.0
vin=0.0
win=1.0
Moving along z axis
irin=2Starts in region 2, could be 1
weight = 1 since no variance reduction used
wtin=1.0
Weight = 1 since no variance reduction used!----------------------------------------------------------------------! Step 5:
hatch-call!----------------------------------------------------------------------! Maximum total energy of an electron for this problem must be! defined before hatch callemaxe = ein

130
!
!
!

140
!
!
!
!
!

write(6,130)
format(/’ Start tutor1’/’ Call hatch to get cross-section data’)-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,140)
FORMAT(/,’ HATCH-call comes next’,/)==========
call hatch==========-----------------------------Close files (after HATCH call)-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)!

Pick up cross section data for ta
write(6,150) ae(1)-RM, ap(1)150format(/’ Knock-on electrons can be created and any electron ’,*’followed down to’ /T40,F8.3,’ MeV kinetic energy’/*’ Brem photons can be created and any photon followed down to’,*/T40,F8.3,’ MeV’)! Compton events can create electrons and photons below these cutoffs!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------zbound=0.1

149

!

plate is 1 mm thick!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------! Print header for output - which is all ausgab does in this casewrite(6,160)
160
format(/T19,’Kinetic energy(MeV)’,T40,’charge’,T48,*’angle w.r.t. z axis-degrees’)!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------! Initiate the shower 10 times
do i=1,10
write(6,170) i
170
format(’ Start history’,I4)
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)!----------------------------------------------------------------------! Step 9: Output-of-results!----------------------------------------------------------------------! Note output is at the end of each history in subroutine ausgab
end dostop
end!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In general, ausgab is a routine which is called under a series! of well defined conditions specified by the value of iarg (see the! egs5 manual for the list). This is a particularly simple ausgab.
! Whenever this routine is called with iarg=3 , a particle has! been discarded by the user in howfar! we get ausgab to print the required information at that point!
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_stack.f’
include ’include/egs5_useful.f’! Main EGS "header" file
! COMMONs required by EGS5 code150

integer iarg

! Arguments

real*8 angle,ekine! Local variables

if (iarg.eq.3.and.ir(np).eq.3) thenAngle w.r.t. z axis in degrees
angle=acos(w(np))*180./3.14159
if (iq(np).eq.0) thenekine=e(np)elseekine=e(np)-RM! Get kinetic energy
end ifwrite(6,100) ekine,iq(np),angle
100
format(T21,F10.3,T33,I10,T49,F10.1)
end ifreturnend!

!--------------------------last line of ausgab.f-----------------------!-------------------------------howfar.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required (geometry) subroutine for use with the EGS5 Code System!***********************************************************************!
! The following is a general specification of howfar!
given a particle at (x,y,z) in region ir and going in direction!
(u,v,w), this routine answers the question, can the particle go!
a distance ustep without crossing a boundary!
If yes, it merely returns!
If no, it sets ustep=distance to boundary in the current!
direction and sets irnew to the region number
on the!
far side of the boundary (this can be messy in general!)!
!
The user can terminate a history by setting idisc>0. here we!
terminate all histories which enter region 3 or are going!
backwards in region 1
!
!
|
|
!
Region 1|
Region 2|
Region 3!
||!
e- =========>|| e- or photon ====>!
||!
vacuum|Ta|
vacuum!
!***********************************************************************
subroutine howfar

151

implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’!

! Main EGS "header" file
! COMMONs required by EGS5 codecommon/geom/zboundreal*8 zboundgeom passes info to our howfar routine
real*8 tval

!
!

!
!

!

!
!
!

!

!
!

!

! Local variable

if (ir(np).eq.3) thenidisc=1
returnTerminate this history: it is past the plate
We are in the Ta plate - check the geometryelse if (ir(np).eq.2) thenif (w(np).gt.0.0) thenGoing forward - consider first since most frequent
tval is dist to boundary in this direction
tval=(zbound-z(np))/w(np)
if (tval.gt.ustep) thenreturnCan take currently requested step
else
ustep=tval
irnew=3
returnend ifend of w(np)>0 case
Going back towards origin
else if (w(np).lt.0.0) thenDistance to plane at origin
tval=-z(np)/w(np)
if (tval.gt.ustep) thenreturnCan take currently requested step
else
ustep=tval
irnew=1
returnend if
End w(np)<0 caseCannot hit boundary
else if (w(np).eq.0.0) thenreturnend if
End of region 2 case

152

!
!

In regon with sourceThis must be a source particle on z=0 boundary
else if (ir(np).eq.1) thenif (w(np).gt.0.0) thenustep=0.0
irnew=2
returnelse! It must be a reflected particle-discard it
idisc=1
returnend if! End region 1 caseend ifend!--------------------------last line of howfar.f------------------------ELEM&INP IRAYL=0,IBOUND=0,INCOH=0,ICPROF=0,IMPACT=0 &END
TA
TA
TA
ENER
&INP AE=1.50,AP=0.10,UE=20.611,UP=20.0 &END
TEST
&INP &END
PWLF&INP &END
DECK&INP &END

This user code produces the following output file called egs5job.out6 (a copy of this file is included
with the EGS5 distribution, named tutor1.out).
PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor1
Call hatch to get cross-section data
HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to153

0.989 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.100 MeV

Start history

Start history

Start history

Start history

Start history

chargeangle w.r.t. z axis-degrees

1.867
0.680
15.455

0
0
-1

2.3
10.8
27.0

0.400
1.451
16.141

0
0
-1

28.4
28.2
21.6

0.301
1.402
16.128

0
0
-1

12.4
33.3
73.4

7.623
10.344

0
-1

25.7
31.4

0.678
1.131
16.286

0
0
-1

3.2
5.8
28.1

17.973

-1

35.0

2.874
1.654
2.588
10.489

0
0
-1
-1

2.9
16.5
53.4
12.0

0.180
17.685

0
-1

83.7
21.7

16.818

-1

12.2

1.697
0.440
1.905
2.210
11.533

0
0
0
0
-1

4.0
7.5
9.7
41.9
48.2

2

3

4

5

Start history

6

Start history

7

Start history

Kinetic energy(MeV)
1

8

Start history

9

Start history

10

By keeping track of many of these histories, we could answer a lot of questions about what comesout the far side of the plate, but it should be recognized that these are all bookkeeping extensions to
the problem – the physics itself already accomplished with EGS5 and the relatively small amountof user code listed above. The scoring routine for this problem is the simplest possible; namely, it
outputs on the file some of the parameters of the various particles leaving the plate.

154

In addition, this user code includes examples of the following items that are discussed in detailin the EGS5 User Manual (Appendix B).

• The use of include statements to use values defined by parameter statements and to allow
easy insertion of COMMONS.
• The technique required in order to define the array MEDIA.
• The definition of calling PEGS5 to produce material data used by user code.
• The definition of seeds for the RANLUX random number generator.
• The definition of calling parameters for the SHOWER routine.
• A very simple AUSGAB routine.
• A simple HOWFAR routine.

3.2

Tutorial 2 (Program tutor2.f)

In this example we use the same geometry as above, but we want the fraction of the incident energythat is reflected from, transmitted through, and deposited in the plate. The coding is essentiallythe same as tutor1 except that COMMON/SCORE/ and a new array ENCORE are defined at Step 1 inthe sequence of steps required in the construction of a user code MAIN program, as described in
the EGS5 User Manual of Appendix B. The latter is initialized to zero (Step 7) and subsequently
output on the file (Step 9). The AUSGAB routine is considerably different as shown below.

!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR2, we score the energy deposited! in the various regions. This amounts to the total energy! reflected, deposited and transmitted by the slab.
!
! For IARG=0, an electron or photon step is about to occur and we! score the energy deposited, if any. Note that only electrons! deposit energy during a step, and due to our geometry, electrons! only take steps in region 2 - however there is no need to check.
! For IARG=1,2 and 4, particles have been discarded for falling! below various energy cutoffs and all their energy is deposited155

! locally (in fact EDEP = particles kinetic energy).
! For IARG=3, we are discarding the particle since it is in! region 1 or 3, so score its energy.
!
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’! Main EGS "header" file! epcont contains edep! stack contains x, y, z, u, v,! w, ir and np

common/score/escore(3)real*8 escore
integer iarg

! Arguments

integer irl! Local variables

if (iarg.le.4) then
irl=ir(np)!
Pick up current region number
escore(irl)=escore(irl)+edep
end ifreturnend!--------------------------last line of ausgab.f------------------------AUSGAB is still very simple since all we need to do is to keep track of the energy depositedin the three regions. The variable EDEP (available through COMMON/EPCONT/) contains the energy
deposited during a particular step for a variety of different IARG-situations, as described in thecomments above and further elaborated upon in Appendix B. In this example, but not always,
we can sum EDEP for any value of IARG up to 4. The following is the output provided by tutor2(named tutor2.out in distribution file).

PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor2
Call hatch to get cross-section data

156

HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to0.989 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.100 MeV
Fraction of energy reflected from plate=
Fraction of energy deposited in plate=
Fraction of energy transmitted through plate=
Total fraction of energy accounted for=3.3

0.655%
12.958%86.388%----------100.000%

Tutorial 3 (Program tutor3.f)

The geometry in this example is similar to the previous two but the problem is very different. Here
we investigate the energy response function for a 2.54 cm thick slab of NaI when a 5 MeV beamof photons is incident on it. In this case the final scoring and binning is done at the end of eachhistory (i.e., after all the descendants from each initial photon have been tracked completely). The
following shows the change required (at Step 8 and 9) and the new AUSGAB routine.

!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------! Initiate the shower ncase times
ncase=10000
do i=1,ncase
ehist = 0.0! Zero energy deposited in this history
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)! Increment bin corresponding to energy deposited in this history
ibin= min0 (int(ehist/bwidth + 0.999), 25)
if (ibin.ne.0) thenebin(ibin)=ebin(ibin)+1
end ifend do!----------------------------------------------------------------------! Step 9: Output-of-results!----------------------------------------------------------------------! Pick up maximum bin for normalization
binmax=0.0
do j=1,25

157

binmax=max(binmax,ebin(j))
end dowrite(6,160) ein,zbound160
format(/’ Response function’/’ for a’,F8.2,’ MeV pencil beam of’,*’photons on a’,F7.2,’ cm thick slab of NaI’/ T6,*’Energy counts/incident photon’)do j=1,48
line(j)=’ ’
end do! Blank entire output arraydo j=1,25
icol=int(ebin(j)/binmax*48.0+0.999)
if (icol.eq.0) icol=1line(icol)=’*’! Load output array at desired location
write(6,170) bwidth*j,ebin(j)/float(ncase),line
170
format(F10.2,F10.4,48A1)line(icol)=’ ’! Reblankend dostop
end!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR3, we score the energy deposited! in the detector region, region 2!
! For IARG=0, an electron or photon step is about to occur and we! score the energy deposited, if any. Note that only electrons! deposit energy during a step, and due to our geometry, electrons! only take steps in region 2 - however there is no need to check!
this here! For IARG=1,2 and 4,particles have been discarded for falling below! various energy cutoffs and all their energy is deposited locally! (in fact EDEP = particles kinetic energy). This only happens in! region 2. For IARG=3, we are discarding the particle since it is!
in region 1 or 3, so we do not score its energy!
! EHIST keeps track of the total energy deposited during each! history. In the main routine it is zeroed at the start of each! history and binned at the end of each history.
!***********************************************************************

158

subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’! Main EGS "header" file! epcont contains edep! stack contains x, y, z, u, v,! w, ir and np

common/score/ehist
real*8 ehist
integer iarg

! Arguments

if (iarg.le.2 .or. iarg.eq.4) thenehist=ehist + edep
end ifreturnend!--------------------------last line of ausgab.f------------------------

The following is the output provided by tutor3 (named tutor3.out in distribution file).
PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor3
Call hatch to get cross-section data
HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.189 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.010 MeV
Response function
for a5.00 MeV pencil beam ofphotons on a
Energy counts/incident photon
0.20
0.0060*159

2.54 cm thick slab of NaI

0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60
3.80
4.00
4.20
4.40
4.60
4.80
5.00

3.4

0.00580.00540.00500.00580.00650.00470.00430.00600.00470.00610.00530.00590.00590.00610.00530.00690.00790.01030.02730.02260.01550.03520.02090.0371

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

Tutorial 4 (Program tutor4.f)

This program examines the dependence of EGS5 results on electron step-size. Recall that for electrons with low initial energies, the limitations inherent in the EGS4 transport mechanics mandated
that the user specify quite small electron step-sizes (defined in terms of the fractional energy loss
over a step, ESTEPE) in order to assure converged results. As noted in Chapter 2, the transportmechanics algorithm of EGS5 naturally mitigates these dependencies. In addition, EGS5 provides
several prescriptions for user selection of step-sizes based on material and region geometries.
As in EGS4, the program tutor4.f is based on tutor2.f, but with a 2 mm slab of silicon as themedium and 2.0 MeV for the incident electron energy. In addition to scoring transmitted, depositedand reflected energy, the number of transmitted and reflected electrons are tallied in tutor4.f. The
example problem actually consists of three distinct runs, two using step-size specification based on
fractional energy loss (EFRACH and EFRACL) and one using a user specified “characteristic dimension”
for the given medium. The main purpose of this tutorial is to demonstrate effectiveness of setting
electron multiple scattering step sizes by specifying the characteristic dimension of the problem.
This approach is much easier, even for expert users, than controling step sizes through EFRACH and
EFRACL.
Recall that multiple-scattering step-sizes in EGS5 are defined in terms of the scattering strength160

accumulated over the given distance, and can be specified by the parameters EFRACH and EFRACL,
which correspond to the fractional energy loss over the step at the upper energy range of the
problem UE and the lower energy limit AE, respectively. Thus EFRACH and EFRACL, which are set in
the PEGS input file, pgs5job.pegs5inp, correspond to ESTEPE from EGS4. In the first tutor4.f
example, the material in the slab is named “SI with long steps” and EFRACH and EFRACL are set
to 0.30. It should be noted that these values were chosen to be artificially high (the defaults in
EGS5 are 0.05 and 0.20, respectively) to help illustrate step-size dependence, since on this problem,
EGS5 shows little dependence on step-size with the default values). In the second example, thematerial is switched to one named “SI with short steps,” and EFRACH and EFRACL are set to be 0.01
and 0.02, respectively. (Thus this example problem also illustrates a method for creating different“media” which are actually the same material with different physics options invoked.)
In the third pass through tutor4.f, the first material is again used, but the step-size is selectedby the specification of the characteristic dimension, CHARD. When CHARD is positive, EGS5 ignoresstep-sizes which correspond to EFRACH and EFRACL and instead automatically chooses values of theinitial scattering strength which provide converged values for electron tracklength (to within 1%
accuracy) for electrons impinging on a cylinders of diameter and thickness CHARD. For the current
problem, CHARD can be taken to be equivalent to the slab thickness, as seen below.
Following is the source code (except for subroutine HOWFAR, which is not changed from tutor2.f)
used by tutor4.f.

!***********************************************************************!
!
**************!
**!
* tutor4.f *!
**!
**************!
! An EGS5 user code. It lists the particles escaping from the back! of a 2 mm Si plate when a pencil beam of 2 MeV electrons! is incident on it normally.
!
! For SLAC-R-730/KEK Report 2005-8: A simple example which scores! reflected, deposited, and transmitted particles and energy and! demonstrates step-size selection!
! The following units are used: unit 6 for output!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!----------------------------------------------------------------------! Step 1: Initialization161

!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/randomm.f’
!
!
!
!
!
!

bounds contains ecut and pcut
media contains the array media
misc contains med
thresh contains ae and ap
useful contains RM
usersc contains emaxe!

common/geom/zboundreal*8 zboundgeom passes info to our howfar routine
common/score/escore(3),iscore(3)real*8 escore
integer iscore*

real*8 ein,xin,yin,zin,
uin,vin,win,wtin
integer iqin,irin! Arguments

real*8 anorm,totalreal* tarray(2),tt,tt0,tt1,cputime
integer loop,i,j,ncase
character*24 medarr(2)! Local variables

real etime!
!
!

---------Open files---------open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)
do loop = 1,3

162

!
!

====================
call counters_out(0)====================!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============! nmed and dunit default to 1, i.e. one medium and we work in cm
if(loop.eq.3) thenchard(1) = 0.20d0
chard(2) = 0.20d0
else
chard(1) = 0.00d0
chard(2) = 0.00d0
endif

100

!
!

optional, but recommended to invoke
automatic step-size control!
!

optional, but recommended to invoke
automatic step-size control

write(6,100) loop, chard(1)FORMAT(72(’*’),/,*’Initializing EGS5, loop = ’,I1,’: charD = ’,f5.2,/,*72(’*’),/)
if(loop.eq.1) then!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=2
medarr(1)=’SI with long steps’
medarr(2)=’SI with short steps
’

!
!
!
!

do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend do--------------------------------------------Run KEK version of PEGS5 before calling HATCH(method was developed by Y. Namito - 010306)---------------------------------------------

110

write(6,110)
FORMAT(’ PEGS5-call comes next’/)!
!

==========
call pegs5==========

163

endif120

if(loop.lt.3) thenwrite(6,120) loop,medarr(loop)
FORMAT(’ Using media number ’,i1,’, ’,a24,’ for this run’,/)endif!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------nreg=3!
nreg : number of region
med(1)=0
med(3)=0if(loop.eq.2) thenmed(2)=2
else
med(2)=1
endif! Vacuum in regions 1 and 3, Si in region 2
ecut(2)=0.700! Terminate electron histories at .700 MeV in the platepcut(2)=0.010! Terminate
photon histories at 0.01 MeV in the plate!
Only needed for region 2 since no transport elsewhere!
ecut is total energy = 0.189
MeV kinetic energy
!
!
!
!

150

!
!

-------------------------------------------------------Random number seeds. Must be defined before call hatch
or defaults will be used. inseed (1- 2^31)-------------------------------------------------------luxlev=1
inseed=1
kount=0
mkount=0
do i = 1, 25
isdext(i) = 0
end dowrite(6,150) inseed
FORMAT(/,’ inseed=’,I12,5X,*’ (seed for generating unique sequences of Ranlux)’)=============
call rluxinit ! Initialize the Ranlux random-number generator=============!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!-----------------------------------------------------------------------

164

! Define initial variables for 2 MeV beam of electrons incident! perpendicular to the slab
iqin=-1!
Incident charge - electrons!
2 MeV kinetic energy
ein=2.d0 + RM
xin=0.0
yin=0.0
zin=0.0!
Incident at origin
uin=0.0
vin=0.0
win=1.0!
Moving along z axis
irin=2!
Starts in region 2, could be 1!
weight = 1 since no variance reduction used
wtin=1.0!
Weight = 1 since no variance reduction used!----------------------------------------------------------------------! Step 5:
hatch-call!----------------------------------------------------------------------! Maximum total energy of an electron for this problem must be! defined before hatch callemaxe = ein

160
!
!
!

170
!
!

write(6,160)
FORMAT(/’ Start tutor4’/’ Call hatch to get cross-section data’)-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,170)
FORMAT(/,’ HATCH-call comes next’,/)==========
call hatch==========!
!
!

-----------------------------Close files (after HATCH call)-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)!

Pick up cross section data for ta
write(6,180) ae(1)-RM, ap(1)

165

180

FORMAT(/’ Knock-on electrons can be created and any electron ’,*’followed down to’ /T40,F8.3,’ MeV kinetic energy’/*’ Brem photons can be created and any photon followed down to’,*/T40,F8.3,’ MeV’)! Compton events can create electrons and photons below these cutoffs!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------zbound=0.2!
plate is 2 mm thick!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------do i=1,3
iscore(i)=0
escore(i)=0.d0! Zero scoring array before startingend do!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------tt=etime(tarray)tt0=tarray(1)! Initiate the shower ncase times
ncase=50000
do i=1,ncase
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)
end dott=etime(tarray)tt1=tarray(1)cputime=tt1-tt0!----------------------------------------------------------------------! Step 9: Output-of-results!----------------------------------------------------------------------write(6,190) cputime,ncase
190
FORMAT(’CPU time = ’,1X,G15.5,’ sec for ’,I8,’ cases’)200

anorm = 100./float(ncase)write(6,200) iscore(1)*anorm,iscore(3)*anorm
FORMAT(/,*’ Fraction of electrons reflected from plate=’,T50,F10.1,’%’,/,*’ Fraction of electrons transmitted through plate=’,T50,F10.1,’%’)! Normalize to % of total input energy
anorm = 100./((ein-RM)*float(ncase))

166

210

total=0.0
do i=1,3
total=total+escore(i)end dowrite(6,210) (escore(i)*anorm,i=1,3),total*anorm
FORMAT(/,/,* ’ Fraction of energy reflected from plate=’,T50,F10.1,’%’*/ ’ Fraction of energy deposited in plate=’,T50,F10.1,’%’/*’ Fraction of energy transmitted through plate=’,T50,F10.1,’%’/*T50,11(’-’)/’ Total fraction of energy accounted for=’, T50,
*F10.1,’%’/)
end do! do four times through

stop
end
!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR4, we score the energy deposited! in the various regions and count transmitted and reflected! electrons.
!
! For IARG=0, an electron or photon step is about to occur and we! score the energy deposited, if any. Note that only electrons! deposit energy during a step, and due to our geometry, electrons! only take steps in region 2 - however there is no need to check.
! For IARG=1,2 and 4, particles have been discarded for falling! below various energy cutoffs and all their energy is deposited! locally (in fact EDEP = particles kinetic energy).
! For IARG=3, we are discarding the particle since it is in! region 1 or 3, so score its energy, and if it is an electron,! score it’s region.
!
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 code167

common/score/escore(3), iscore(3)real*8 escore
integer iscore!
!

!

integer iarg

! Arguments

integer irl! Local variables

if (iarg.le.4) then
irl=ir(np)Pick up current region number
escore(irl)=escore(irl)+edep
Pick up energy deposition/transmission/reflection
if (iarg.eq.3 .and. iq(np).eq.-1) theniscore(irl)=iscore(irl)+1
Pick up electron transmission/reflection
end ifend ifreturnend!--------------------------last line of ausgab.f------------------------

Following is the PEGS input file required by tutor4.f, which specifies two versions of the samematerial (silicon), one with long step-sizes and one with short step-sizes.

ELEM&INP EFRACH=0.30,EFRACL=0.30,
IRAYL=0,IBOUND=0,INCOH=0,ICPROF=0,IMPACT=0 &END
SI with long steps
SI
SI
ENER
&INP AE=0.700,AP=0.010,UE=2.521,UP=2.1 &END
PWLF
&INP &END
DECK&INP &END
ELEM&INP EFRACH=0.01,EFRACL=0.01,
IRAYL=0,IBOUND=0,INCOH=0,ICPROF=0,IMPACT=0 &END
SI with short steps
SI
SI
ENER
&INP AE=0.700,AP=0.010,UE=2.521,UP=2.1 &END
PWLF
&INP &END
DECK&INP &END

168

Following is the output produced by tutor4.f, showing that the severe step-size dependence exhibited by EGS4 on this problem is greatly diminished in EGS5. Recall that in the tutor4 exampleof EGS4, runs using the default step-size algorithm predicted 1.3% reflection and 49.21.0%, EGS4
returned values of 6.4% reflection and 61.3% transmission. In contrast, an EGS5 run using very
long multiple scattering steps (corresponding to 30% energy loss over the steps, and labeled “loop1” in the output file) yields values for reflection (8.1%) and transmission (66.5%) which are fairly
close to EGS5 results generated with 1% energy loss steps (7.3% and 64.4% for reflection andtransmission, respectively, “loop 2” in the output). This level of agreement (roughly 3% error in
the transmission fraction, even when using 30illustrates the power of the modified random hinge
approach. In addition, the significant discrepancies between the 1% energy loss step results of
EGS4 and EGS5 forcefully demonstrate the shortcomings of the EGS4 transport mechanics model,
even for very small electron step sizes.
The results from the third part of this example (“loop 3”) show that the expected 1% accuracy
is obtained when the multiple scattering steps sizes are chosen automatically by EGS5 based on
the given characteristic dimension of 2 mm for this problem. Note that for this dimension and at
this energy, the step-sizes selected by EGS5 for silicon are roughly 10-15 times as large as thoseused in the 1% energy loss run (loop 2), thus providing significant speedup (a factor of three, as
seen from the output) for calculations at this level of accuracy.
************************************************************************
Initializing EGS5, loop = 1: charD = 0.00************************************************************************PEGS5-call comes next
Using media number 1, SI with long steps

for this run

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor4
Call hatch to get cross-section data
HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR
2 MEDIA.
WARNING in RMSFIT: no characteristic dimension input for media
Using old data from gsdist.dat with:
efrach, efrachl =3.0000E-01
3.0000E-01
WARNING in RMSFIT: no characteristic dimension input for media
Using old data from gsdist.dat with:
efrach, efrachl =1.0000E-02
1.0000E-02

12Knock-on electrons can be created and any electron followed down to
0.189 MeV kinetic energy

169Brem photons can be created and any photon followed down to
0.010 MeV
CPU time =13.368sec for
50000 cases
Fraction of electrons reflected from plate=
Fraction of electrons transmitted through plate=

Fraction of energy reflected from plate=
Fraction of energy deposited in plate=
Fraction of energy transmitted through plate=
Total fraction of energy accounted for=

8.1%
66.5%

3.9%
59.2%
36.9%
----------100.0%************************************************************************
Initializing EGS5, loop = 2: charD = 0.00************************************************************************
Using media number 2, SI with short steps

for this run

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor4
Call hatch to get cross-section data
HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR
2 MEDIA.
WARNING in RMSFIT: no characteristic dimension input for media
Using old data from gsdist.dat with:
efrach, efrachl =3.0000E-01
3.0000E-01
WARNING in RMSFIT: no characteristic dimension input for media
Using old data from gsdist.dat with:
efrach, efrachl =1.0000E-02
1.0000E-02

12Knock-on electrons can be created and any electron followed down to
0.189 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.010 MeV
CPU time =111.67
sec for
50000 cases
Fraction of electrons reflected from plate=
Fraction of electrons transmitted through plate=

7.3%
64.4%

Fraction of energy reflected from plate=
Fraction of energy deposited in plate=
Fraction of energy transmitted through plate=

3.2%
62.2%
34.6%

170

----------100.0%

Total fraction of energy accounted for=************************************************************************
Initializing EGS5, loop = 3: charD = 0.20************************************************************************

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor4
Call hatch to get cross-section data
HATCH-call comes next
EGS SUCCESSFULLY ’HATCHED’ FOR

2 MEDIA.

Knock-on electrons can be created and any electron followed down to
0.189 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.010 MeV
CPU time =36.373sec for
50000 cases
Fraction of electrons reflected from plate=
Fraction of electrons transmitted through plate=

Fraction of energy reflected from plate=
Fraction of energy deposited in plate=
Fraction of energy transmitted through plate=
Total fraction of energy accounted for=

3.5

7.4%
64.8%

3.3%
62.0%
34.7%
----------100.0%

Tutorial 5 (Program tutor5.f)

In this program we give an example that includes Rayleigh scattering and which makes use of thevariable called LATCH (contained in COMMON/STACK/). LATCH can be set for any particle on the“stack” of particles being transported, and is passed on to all its progeny. This provides a simple
procedure for keeping track of the histories of particles. In this example we make use of LATCH to
keep track of how often photons from an incident 50 keV beam are Compton or Rayleigh scattered
while passing through a 0.5 cm slab of water.
This user program also demonstrates the use of the IAUSFL array of flags (in COMMON/EPCONT/).
By setting the appropriate flags, the user can cause the EGS5 system to call the AUSGAB subroutine
171

in any combination of 31 well specified situations (see Appendix B). By default, EGS calls AUSGAB
only 5 out of the possible 31 situations. Here, by setting IAUSFL(18) and IAUSFL(24) from 0(default) to 1 in the main program, we cause EGS to call AUSGAB with IARG=17 and IARG=23 (i.e.,just before a Compton or a Rayleigh scattering event, respectively). We make use of these calls to
set some flags associated with each photon rather than for scoring any variables. A complete listingof tutor5.f, except for HOWFAR routine which is similar to the other examples, is given below.
!***********************************************************************!
!
**************!
**!
* tutor5.f *!
**!
**************!
! An EGS5 user code which scores the number and average energy of the! primary, Rayleigh scattered and Compton scattered photons passing! through a 5 cm thick slab of water when a 50 keV pencil beam of! photons is incident normally!
!
! For SLAC-R-730/KEK Report 2005-8: Example of including Rayleigh!
scattering, and use of the LATCH feature!
! The following units are used: unit 6 for output!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!----------------------------------------------------------------------! Step 1: Initialization!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_epcont.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_stack.f’
include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’172

include ’include/randomm.f’
!
!
!
!
!
!
!
!

bounds contains ecut and pcut
epcont contains iausfl
media contains the array media
misc contains med
stack contains latchi
thresh contains ae and ap
useful contains RM
usersc contains emaxe!

common/geom/zboundreal*8 zboundgeom passes info to our howfar routine
common/score/count(3),entot(3)real*8 count,entot*

real*8 ein,xin,yin,zin,
uin,vin,win,wtin
integer iqin,irin! Arguments

real*8 anorm
integer i,j,ncase
character*24 medarr(1)! Local variables!
!
!

---------Open files---------open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)!

====================
call counters_out(0)====================!

!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=1
medarr(1)=’H2O’
do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)

173

end doend do! nmed and dunit default to 1, i.e. one medium and we work in cm
chard(1) = 0.5d0

!
!
!
!
100
!
!

!
!

optional, but recommended to invoke
automatic step-size control--------------------------------------------Run KEK version of PEGS5 before calling HATCH(method was developed by Y. Namito - 010306)--------------------------------------------write(6,100)
FORMAT(’ PEGS5-call comes next’/)==========
call pegs5==========!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------nreg=3!
nreg : number of region
med(1)=0
med(3)=0
med(2)=1! Regions 1 and 3 are vacuum, region 2, H2O
ecut(2)=1.5!
Terminate electron histories at 1.5 MeV in the slab
pcut(2)=0.010!
Terminatephoton histories at 0.01 MeV in the slab
iraylr(2)=1!
Turn on rayleigh scattering in the slab
! Note, above three parameters need to be set for all regions in which! there is particle transport - just region 2 in this case
!
!
!
!

120

!
!

-------------------------------------------------------Random number seeds. Must be defined before call hatch
or defaults will be used. inseed (1- 2^31)-------------------------------------------------------luxlev=1
inseed=1
write(6,120) inseed
FORMAT(/,’ inseed=’,I12,5X,*’ (seed for generating unique sequences of Ranlux)’)=============
call rluxinit ! Initialize the Ranlux random-number generator=============

174

!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!----------------------------------------------------------------------! Define initial variables for 50 keV beam of photons normally incident! on the slab
iqin=0!
Incident photons!
50 keV
ein=0.050
xin=0.0
yin=0.0
zin=0.0!
Incident at origin
uin=0.0
vin=0.0
win=1.0!
Moving along z axis
irin=2!
Starts in region 2, could be 1
wtin=1.0!
weight = 1 since no variance reduction usedlatchi=0!
latch set to zero at start of each history!----------------------------------------------------------------------! Step 5:
hatch-call!----------------------------------------------------------------------! Maximum total energy of an electron for this problem must be! defined before hatch callemaxe = ein + RM

130
!
!
!

140
!

write(6,130)
format(/’ Start tutor5’/’ Call hatch to get cross-section data’)-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,140)
format(/,’ HATCH-call comes next’,/)!

==========
call hatch==========!
!

-----------------------------Close files (after HATCH call)

175

!

-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)!

Pick up cross section data for water
write(6,150) ae(1)-RM, ap(1)150format(/’ Knock-on electrons can be created and any electron ’,*’followed down to’ /T40,F8.3,’ MeV kinetic energy’/*’ Brem photons can be created and any photon followed down to’,*/T40,F8.3,’ MeV’)! Compton events can create electrons and photons below these cutoffs!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------zbound=0.5!
Plate is 0.5 cm thick!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------do i=1,3
count(i)=0.0
entot(i)=0.0! Zero scoring array at startend do!
!
!

We want to set flags in ausgab every time a rayleigh scatteringor Compton scattering occurs. Set the flags in iausfl(comin
epcont) to signal the egs system to make the appropriate calls
iausfl(18)=1
iausfl(24)=1!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------! Initiate the shower ncase times
ncase=10000
do i=1,NCASE
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)
end do!----------------------------------------------------------------------! Step 9: Output-of-results!----------------------------------------------------------------------! Normalize to % of photon number
anorm = 100./float(ncase)
do i=1,3
if (count(i).ne.0) thenentot(i)=entot(i)/count(i)!
Get average energies

176

160

end ifend dowrite(6,160) ein*1000.,zbound, pcut(2), (anorm*count(i),entot(i),*i=1,3)
format(/’ For’,F6.1,’ keV photons incident on’,F4.1,’cm of H2O’,*’ with PCUT=’,F5.3,’ MeV’ //’ Transmitted primaries=’,T40,F8.2,*’% ave energy=’,F10.3,’ MeV’// ’ Fraction Rayleigh scattering=’,*T40,F8.2,’% ave energy=’,F10.3,’ MeV’ //*’ Fraction Compton scattering only=’,T40,F8.2,’% ave energy=’,*F10.3, ’ MeV’//)

stop
end
!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR5 we both set flags whenever there is! a scattering event and then count histories when they have come! through the slab , according to what kind of scattering they have
! undergone.
! The logic is as follows!
set FLAG1 if a Compton event occurs!
set FLAG2 if a Rayleigh event occurs! The FLAGS are the units and thousands digits in the parameter LATCH!
! When a history is terminated, increment various counters according! to whether no flags are set - i.e. its a primary, FLAG2 is set,
! i.e. it has Rayleigh scattered or FLAG1 is set and FLAG2 is not set!
i.e. only Compton scattering has occurred.
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/score/count(3),entot(3)real*8 count,entot
integer iarg

! Arguments

integer jj! Local variable177

if (iarg.eq.17) then
A Compton scatter is about to occur
latch(np)=latch(np)+1
else if (iarg.eq.23) then
! A Rayleigh scatter is about to occur
latch(np)=latch(np)+1000!
If a history has terminated because leaving the slab, score it! Particle has left slab
else if (iarg .eq. 3) thenif (ir(np).eq.3 .or. ir(np) .eq. 1) then!
It is transmitted or reflected
jj=0
if (latch(np) .eq. 0) then!
No scattering - a primary
jj=1else if (mod(latch(np),10000)-mod(latch(np),100) .ne. 0) then!
at least one Rayleigh scatter
jj=2
else if (mod(latch(np),100) .ne. 0) then!
at least one Compton scatter without Rayleigh
jj=3
!
debug
else
write(6,1080) jj,latch(np)1080format(’ jj,latch(np)=’,2I10)
end ifif (jj .ne. 0) thencount(jj)=count(jj) + 1.
entot(jj) = entot(jj) + e(np)end if!
End region 3 blockend if! End iarg 3 blockend ifreturnend!--------------------------last line of ausgab.f-----------------------!

Following is the output provided by tutor5.f.

PEGS5-call comes next
inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor5
Call hatch to get cross-section data

178

HATCH-call comes next
RAYLEIGH OPTION REQUESTED FOR MEDIUM NUMBER

1

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.010 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.010 MeV
For

50.0 keV photons incident on 0.5cm of H2O with PCUT=0.010 MeV

Transmitted primaries=

88.89%

ave energy=

0.050 MeV

Fraction Rayleigh scattering=

0.95%

ave energy=

0.049 MeV

Fraction Compton scattering only=

8.60%

ave energy=

0.046 MeV

3.6

Tutorial 6 (Program tutor6.f)

One of the important features of the EGS5 Code System is that the user has direct control overthe geometry in which the radiation transport takes place, and rather complex geometries can be
described in a very simple manner with the aid of geometry subprograms. Subroutine HOWFAR, in
the previous examples, was made more detailed than necessary for demonstration purposes. One
can greatly simplify things, particularly for the simple slab geometry case, as shown in the following
excerpts from tutor6.

!***********************************************************************!
!
**************!
**!
* tutor6.f *!
**!
**************!
! An EGS5 user code. It lists the particles escaping from the back! of a 1 mm Ta plate when a pencil beam of 20 MeV electrons! is incident on it normally.
!
! NOTE: This program is the same as TUTOR1.f except that the!
geometry subroutine (HOWFAR) is simplified by the use of

179

!

the general purpose geometry subroutines PLAN2P.
.
.
.
!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------! Define the coordinates and the normal vectors for the two planes.
! Information required by howfar (and auxiliary geometry subprograms)! and passed through common/pladta/!
! First plane (the x-y plane through the origin)pcoord(1,1)=0.0
pcoord(2,1)=0.0
pcoord(3,1)=0.0! Coordinates
pnorm(1,1) =0.0
pnorm(2,1) =0.0
pnorm(3,1)= 1.0! Normal vectors! Second plane (note: slab is 1 mm thick)pcoord(1,2)=0.0
pcoord(2,2)=0.0
pcoord(3,2)=0.1! Coordinates
pnorm(1,2) =0.0
pnorm(2,2) =0.0
pnorm(3,2)= 1.0! Normal vectors.
.
.
!-------------------------------howfar.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required (geometry) subroutine for use with the EGS5 Code System!***********************************************************************!
! The following is a general specification of howfar. Essentially! it is the same as that given in tutor1.f with the following! exception: 1) Particles must be initially begin in region 2 and are!
discarded when that enter region 1 or 3 (no check!
is made on w(np)).
!
2) The coding is much simplified (i.e., modular)!
As a result of using auxiliary geometry subprogram!
plan2p (which calls plane1 and chgtr which require!
commons epcont, pladta, and stack).
!
!
The user can terminate a history by setting idisc>0. Here we!
terminate all histories which enter region 3 or are going180

!
backwards in region 1
!
!
|
|
!
Region 1|
Region 2|
Region 3!
||!
e- =========>|| e- or photon ====>!
||!
vacuum|Ta|
vacuum!
! DESCRIPTION - PLAN2P is generally called from subroutine HOWFAR!
whenever a particle is in a region bounded by two planes that!
ARE parallel. Both subroutines PLANE1 and CHGTR are called!
by PLAN2P (the second PLANE1 call is not made if the first!
plane is hit, or if the trajectory is parallel).
!-----------------------------------------------------------------!
NPL1= ID number assigned to plane called first (input)!
NRG1= ID number assigned to region particle trajectory!
will lead into!
ISD1
= 1 normal points towards current region (input)!
= -1 normal points away from current region (input)!
NPL2= Same (but for plane called second)!
NRG2= Same (but for plane called second)!
ISD2= Same (but for plane called second)!***********************************************************************
subroutine howfar
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’!
!
!

---------------------Auxiliary-code COMMONs---------------------include ’auxcommons/aux_h.f’! Main EGS "header" file! epcont contains irnew, ustep! and idisc! stack contains x, y, z, u, v,! w, ir and np! Auxiliary-code "header" fileinclude ’auxcommons/pladta.f’
integer irl! Local variable

irl=ir(np)! Set local variable
if (irl.ne.2) thenidisc=1! Terminate this history if not in plateelse! We are in the Ta plate - check the geometry
call plan2p(irl,irl+1,1,irl-1,irl-1,-1)
end if181

returnend!--------------------------last line of howfar.f------------------------

The actual HOWFAR code is now less than 10 lines long, the rest consisting of COMMON’s and comments.
For a complete understanding of how PLAN2P and its related subroutines PLANE1 and CHGTR are
called and used, the reader should refer to comments in the appropriate subroutine (distributedwith the EGS5 Code System). For a description of the concepts involved in modeling geometryfor Monte Carlo programs in general (and EGS4 in particular), the reader may wish to refer tothe document “How to Code Geometry: Writing Subroutine HOWFAR,” which is provided with the
EGS5 distribution.

3.7

Tutorial 7 (Program tutor7.f)

In this program we give an example that includes K- and L-fluorescence photons, which can begenerated in any material, including compounds or mixtures. Three problems are considered, allof which two compute the reflected photon spectrum generated when a 100 keV beam of photons
is incident upon a 1.0 cm lead slab. If the IEDGFL flag is set to 1, fluorescence photons may be
produced after K- or L-photoelectric effect interactions in that region. A complete listing of tutor7(except the HOWFAR routine, which is the same as tutor6.f) is given below.
!***********************************************************************!
!
**************!
**!
* tutor7.f *!
**!
**************!
! An EGS5 user code which scores the spectrum of reflection from! 1.0 cm thick slab of lead when a 100 keV beam of photons is incident! on it with or without fluorescence photons and Doppler broadening.
!
! For SLAC-R-730/KEK Report 2005-8: Example of including fluorescence!
! The following units are used: unit 6 for output!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!-----------------------------------------------------------------------

182

! Step 1: Initialization!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_edge.f’include ’include/egs5_epcont.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/randomm.f’
!
!
!
!
!
!
!
!

bounds contains ecut and pcutedge contains iedgfl
epcont contains iausfl
media contains the array media
misc contains med
thresh contains ae and ap
useful contains RM
usersc contains emaxe!
!
!

---------------------Auxiliary-code COMMONs---------------------include ’auxcommons/aux_h.f’! Auxiliary-code "header" fileinclude ’auxcommons/pladta.f’common/score/bwidth,ebin(50)real*8 bwidth,ebin*

real*8 ein,xin,yin,zin,
uin,vin,win,wtin
integer iqin,irin! Arguments

real*8 binmax
integer i,icol,j,ncases
character*24 medarr(1)character*4 line(48)!
!
!

! Local variables---------Open files----------

183

open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)!
!

====================
call counters_out(0)====================!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=1
medarr(1)=’PB’
do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend do! nmed and dunit default to 1, i.e. one medium and we work in cm
chard(1) = 1.0d0

!
!
!
!
100
!
!

!
!

optional, but recommended to invoke
automatic step-size control--------------------------------------------Run KEK version of PEGS5 before calling HATCH(method was developed by Y. Namito - 010306)--------------------------------------------write(6,100)
FORMAT(’ PEGS5-call comes next’/)==========
call pegs5==========!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------nreg=3!
nreg : number of region
med(1)=0
med(3)=0
med(2)=1! Regions 1 and 3 are vacuum, region 2, lead184

iraylr(2)=1
Turn on rayleigh scattering in the slab
iedgfl(2)=1!
1: Turn on fluorescence production in the slab!
0: Turn off fluorescence production in the slab
! Note, above three parameters need to be set for all regions in which! there is particle transport - just region 2 in this caseincohr(2)=1!
1: Turn on incoherent scattering function in the slab!
0: Turn off incoherent scattering function in the slab
iprofr(2)=0!
1: Turn on Doppler broadening in the slab!
0: Turn off Doppler broadening in the slab
!

!
!
!
!

120

!
!

-------------------------------------------------------Random number seeds. Must be defined before call hatch
or defaults will be used. inseed (1- 2^31)-------------------------------------------------------luxlev=1
inseed=1
write(6,120) inseed
FORMAT(/,’ inseed=’,I12,5X,*’ (seed for generating unique sequences of Ranlux)’)=============
call rluxinit ! Initialize the Ranlux random-number generator=============!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!----------------------------------------------------------------------! Define initial variables for 100 keV beam of photons normally incident! on the slab
iqin=0!
Incident photons!
100 keV
ein=0.100
xin=0.0
yin=0.0
zin=0.0!
Incident at origin
uin=0.0
vin=0.0
win=1.0!
Moving along z axis
irin=2!
Starts in region 2, could be 1
wtin=1.0!
weight = 1 since no variance reduction used!-----------------------------------------------------------------------

185

! Step 5:
hatch-call!----------------------------------------------------------------------! Maximum total energy of an electron for this problem must be! defined before hatch callemaxe = ein + RM

130
!
!
!

140
!
!
!
!
!

write(6,130)
format(/’ Start tutor7’/’ Call hatch to get cross-section data’)-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,140)
FORMAT(/,’ HATCH-call comes next’,/)==========
call hatch==========-----------------------------Close files (after HATCH call)-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)!

Pick up cross section data for lead
write(6,150) ae(1)-RM, ap(1)150format(/’ Knock-on electrons can be created and any electron ’,*’followed down to’ /T40,F8.3,’ MeV kinetic energy’/*’ Brem photons can be created and any photon followed down to’,*/T40,F8.3,’ MeV’)! Compton events can create electrons and photons below these cutoffs!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------! Define the coordinates and the normal vectors for the two planes.
! Information required by howfar (and auxiliary geometry subprograms)! and passed through common/pladta/!
! First plane (the x-y plane through the origin)pcoord(1,1)=0.0
pcoord(2,1)=0.0
pcoord(3,1)=0.0! Coordinates
pnorm(1,1) =0.0
pnorm(2,1) =0.0
pnorm(3,1)= 1.0

186

! Normal vectors! Second plane (note: slab is 1 cm thick)pcoord(1,2)=0.0
pcoord(2,2)=0.0
pcoord(3,2)=1.0! Coordinates
pnorm(1,2) =0.0
pnorm(2,2) =0.0
pnorm(3,2)= 1.0! Normal vectors!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------do i=1,50
ebin(i) = 0.0! Zero scoring array before startingend dobwidth = 0.002!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------! Initiate the shower ncase times
ncases=300000
do i=1,NCASES
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)
end do!----------------------------------------------------------------------! Step 8: Output-of-results!----------------------------------------------------------------------! Use log10(ncases) as maximum value
binmax=alog10(float(ncases))

160

170

if (iedgfl(2).eq.1) thenwrite(6,160) ein,pcoord(3,2),ncases
format(/’ Reflected photon spectrum’/’ for a’,F8.2,* ’ MeV pencil beam of photons on a’,F7.2,* ’ cm thick slab of lead’/’ with fluorescence photon’//T6,* ’Energy counts/incident photon’/* 25X,’ log(counts for ’,I8,’ incident photons)’)
else
write(6,170) ein,pcoord(3,2),ncases
format(’ Reflected photon spectrum’/’ for a’,F8.2,* ’ MeV pencil beam of photons on a’,F7.2,* ’ cm thick slab of lead’/’ without fluorescence photon’//T6,* ’Energy counts/incident photon’/* 25X,’ log(counts for ’,I8,’ incident photons)’)
end if187

do j=1,48
line(j)=’ ’
end do! Blank entire output arraydo j=1,50
if(ebin(j).gt.0) thenicol=*int(dlog10(ebin(j))/binmax*48.0+0.999)
if (icol.eq.0) icol=1
else
icol = 1
endifline(icol)=’*’! Load output array at desired location
write(6,180) bwidth*j,ebin(j)/float(ncases),line
180
format(F10.4,F12.6,48A1)line(icol)=’ ’! Reblankend dowrite(6,190)incohr(2),iprofr(2)format(’ incohr(2)=’,i2,’ iprofr(2)=’,i2)
stop
end!-------------------------last line of main code-----------------------190!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR7 we score photons reflected! from the slab (ir(np)=1 and iq(np)=0).
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/score/bwidth,ebin(50)real*8 bwidth,ebin
integer iarg

! Arguments

integer ibin,irl! Local variable

188

!

irl=ir(np)! Local variable
if(irl.eq.1.and.iq(np).eq.0) then! Photon is reflected
Increment bin corresponding to photon energy
ibin= min0 (int(e(np)/bwidth + 0.999), 50)
if (ibin.ne.0) thenebin(ibin)=ebin(ibin)+1
end ifend ifreturnend!--------------------------last line of ausgab.f------------------------

The following are the outputs generated by tutor7 with and without the fluorescence option(named tutor7 w.out and tutor7 wo.out in the distribution, respectively).

PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor7
Call hatch to get cross-section data
HATCH-call comes next
RAYLEIGH OPTION REQUESTED FOR MEDIUM NUMBER

1

INCOHERENT OPTION REQUESTED FOR MEDIUM NUMBER

1

SHELL COMPTON PROFILE DATA AVAILABLE FOR MEDIUM

1 BUT OPTION NOT REQUESTED.

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.010 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.001 MeV
Reflected photon spectrum
for a0.10 MeV pencil beam of photons on awith fluorescence photon
Energy0.0020counts/incident photonlog(counts for
0.000000*

1.00 cm thick slab of lead

300000 incident photons)

1890.00400.000000*
0.00600.000013*
0.00800.000003*
0.01000.000093
0.01200.001990
0.01400.003020
0.01600.000350
0.01800.000000*
0.02000.000007 *0.02200.000013*
0.02400.000010*
0.02600.000010*
0.02800.000013*
0.03000.000013*
0.03200.000013*
0.03400.000023*
0.03600.000007 *0.03800.000000*
0.04000.000017*
0.04200.000017*
0.04400.000010*
0.04600.000007 *0.04800.000000*
0.05000.000010*
0.05200.000007 *0.05400.000013*
0.05600.000007 *0.05800.000070
0.06000.000287
0.06200.000287
0.06400.000257
0.06600.000273
0.06800.000370
0.07000.000370
0.07200.000493
0.07400.050430
0.07600.086820
0.07800.000730
0.08000.000657
0.08200.000537
0.08400.000233
0.08600.034873
0.08800.010270
0.09000.000033*
0.09200.000003*
0.09400.000000*
0.09600.000000*
0.09800.000000*
0.10000.000747
incohr(2)= 1 iprofr(2)= 0

*
*
*
*

*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*

*

190

PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor7
Call hatch to get cross-section data
HATCH-call comes next
RAYLEIGH OPTION REQUESTED FOR MEDIUM NUMBER

1

INCOHERENT OPTION REQUESTED FOR MEDIUM NUMBER

1

SHELL COMPTON PROFILE DATA AVAILABLE FOR MEDIUM

1 BUT OPTION NOT REQUESTED.

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.010 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.001 MeV
Reflected photon spectrum
for a0.10 MeV pencil beam of photons on a
1.00 cm thick slab of lead
without fluorescence photon
Energy0.00200.00400.00600.00800.01000.01200.01400.01600.01800.02000.02200.02400.02600.02800.03000.03200.03400.03600.03800.04000.0420counts/incident photonlog(counts for
0.000000*
0.000003*
0.000003*
0.000007 *0.000003*
0.000007 *0.000010*
0.000007 *0.000003*
0.000000*
0.000007 *0.000000*
0.000010*
0.000000*
0.000017*
0.000007 *0.000010*
0.000010*
0.000017*
0.000003*
0.000007 *300000 incident photons)

191

0.04400.000007 *0.04600.000007 *0.04800.000007 *0.05000.000007 *0.05200.000020*
0.05400.000003*
0.05600.000007 *0.05800.000007 *0.06000.000003*
0.06200.000007 *0.06400.000000*
0.06600.000013*
0.06800.000010*
0.07000.000020*
0.07200.000103
0.07400.001370
0.07600.001033
0.07800.000863
0.08000.000583
0.08200.000333
0.08400.000127
0.08600.000020*
0.08800.000010*
0.09000.000003*
0.09200.000000*
0.09400.000000*
0.09600.000000*
0.09800.000000*
0.10000.000830
incohr(2)= 1 iprofr(2)= 0

*
*
*
*
*
*
*

*

Also examined in tutor7 is the effect of Doppler broadening of Compton scattered photon
energies on the energy spectrum of reflected photons. To turn on Doppler broadening, the IPROFR
flag is set to 1 in tutor7.f. The output generated for this tutorial, without fluorescence photons
but with Doppler broadening, is given below.
PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48
ranlux initialized by rluxgo from seed1
Start tutor7
Call hatch to get cross-section data
HATCH-call comes next
RAYLEIGH OPTION REQUESTED FOR MEDIUM NUMBER

1192

INCOHERENT OPTION REQUESTED FOR MEDIUM NUMBER

1

COMPTON PROFILE OPTION REQUESTED FOR MEDIUM NUMBER

1

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.010 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.001 MeV
Reflected photon spectrum
for a0.10 MeV pencil beam of photons on a
1.00 cm thick slab of lead
without fluorescence photon
Energy0.00200.00400.00600.00800.01000.01200.01400.01600.01800.02000.02200.02400.02600.02800.03000.03200.03400.03600.03800.04000.04200.04400.04600.04800.05000.05200.05400.05600.05800.06000.06200.06400.06600.0680counts/incident photonlog(counts for
0.000000*
0.000000*
0.000000*
0.000007 *0.000007 *0.000003*
0.000010*
0.000007 *0.000007 *0.000000*
0.000007 *0.000000*
0.000007 *0.000000*
0.000010*
0.000007 *0.000010*
0.000003*
0.000003*
0.000003*
0.000003*
0.000003*
0.000007 *0.000003*
0.000003*
0.000017*
0.000017*
0.000017*
0.000030*
0.000040*
0.000040*
0.000080*
0.000063*
0.000107*

300000 incident photons)

193

0.0700
0.000223
0.07200.000373
0.07400.000620
0.07600.000633
0.07800.000593
0.08000.000557
0.08200.000393
0.08400.000250
0.08600.000207
0.08800.000087
0.09000.000030*
0.09200.000017*
0.09400.000010*
0.09600.000023*
0.09800.000007 *0.10000.000860
incohr(2)= 1 iprofr(2)= 1

3.8

*
*
*
*
*
*
*
*
*
*

*

Tutorial 8 (Program tutor8.f)

The final tutorial is a variation of tutor7 in which particle splitting can be invoked to speed
the simulation of infrequently occuring events. As in the last tutorial, we wish to compute theenergy spectra of photons reflected from a lead slab, but in this case, the incident paricles are 100
keV electrons. Emitted photons will therefore be bremsstrahlung photons and photons emitted
subsequent to electron impact ionization events. Since neither process is high probability, particle
splitting will be invoked. This is done by setting the flags IEISPL and IBRSPL to 1 (note thatthe bremsstrahlung splitting flag is a region-dependent array), and choosing suitable values of thenumber of split particles, NEISPL and NBRSPL. The source code and output results for this problem
are shown below.
!***********************************************************************!
!
**************!
**!
* tutor8.f *!
**!
**************!
! An EGS5 user code which scores the spectrum of photons reflected from! a 1.0 cm thick slab of lead when a 100 keV beam of electrons is! incident on it, with or without fluorescence photons from electron! impact ionization and/or photo-ionization of bremsstrahlung photons.
! Particle splitting is demonstrated.
!
! For SLAC-R-730/KEK Report 2005-8: Example of splitting!

194

! The following units are used: unit 6 for output!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!----------------------------------------------------------------------! Step 1: Initialization!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_brempr.f’include ’include/egs5_edge.f’include ’include/egs5_eiicom.f’include ’include/egs5_epcont.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/randomm.f’
!
!
!
!
!
!
!
!

bounds contains ecut and pcutedge contains iedgfl
epcont contains iausfl
media contains the array media
misc contains med
thresh contains ae and ap
useful contains RM
usersc contains emaxe!
!
!

---------------------Auxiliary-code COMMONs---------------------include ’auxcommons/aux_h.f’! Auxiliary-code "header" fileinclude ’auxcommons/pladta.f’common/score/bwidth,ebin(50)real*8 bwidth,ebin*

real*8 ein,xin,yin,zin,
uin,vin,win,wtin! Arguments

195integer iqin,irin
real*8 binmax
integer i,icol,j,ncases
character*24 medarr(1)character*4 line(48)! Local variables!
!
!

---------Open files---------open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)!

====================
call counters_out(0)====================!

!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=1
medarr(1)=’PB’
do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend do! nmed and dunit default to 1, i.e. one medium and we work in cm
chard(1) = 1.0d0

!
!
!
!
100
!
!

!
!

optional, but recommended to invoke
automatic step-size control--------------------------------------------Run KEK version of PEGS5 before calling HATCH(method was developed by Y. Namito - 010306)--------------------------------------------write(6,100)
FORMAT(’ PEGS5-call comes next’/)==========
call pegs5==========

196

!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------nreg=3!
nreg : number of region
med(1)=0
med(3)=0
med(2)=1! Regions 1 and 3 are vacuum, region 2, leadiraylr(2)=0!
Turn on rayleigh scattering in the slab
iedgfl(2)=1!
1: Turn on fluorescence production in the slab!
0: Turn off fluorescence production in the slab
! Note, above three parameters need to be set for all regions in which! there is particle transport - just region 2 in this caseincohr(2)=0!
1: Turn on incoherent scattering function in the slab!
0: Turn off incoherent scattering function in the slab
iprofr(2)=0!
1: Turn on Doppler broadening in the slab!
0: Turn off Doppler broadening in the slab
impacr(2)=1!
1: Turn on electron impact ionization in the slab!
0: Turn off electron impact ionization in the slab
ieispl=1!
1: Turn on eii photon splitting!
0: Turn off eii photon splitting
neispl=20!
neispl: number of splitting for eii photon
ibrspl(1)=0ibrspl(2)=1
ibrspl(3)=0!
1: Turn on brems photon splitting!
0: Turn off brems photon splitting
nbrspl=30!
nbrspl: number of splitting for brems photon
ibrdst=1!
1: Turn on brems angular distribution!
0: Turn off brems angular distribution
!
!
!
!

120

-------------------------------------------------------Random number seeds. Must be defined before call hatch
or defaults will be used. inseed (1- 2^31)-------------------------------------------------------luxlev=1
inseed=1
write(6,120) inseed
FORMAT(/,’ inseed=’,I12,5X,*’ (seed for generating unique sequences of Ranlux)’)

197

!
!

=============
call rluxinit ! Initialize the Ranlux random-number generator=============!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!----------------------------------------------------------------------! Define initial variables for 100 keV beam of electrons normally incident! on the slab
iqin=-1!
Incident electrons!
100 keV
ein=0.611
xin=0.0
yin=0.0
zin=0.0!
Incident at origin
uin=0.0
vin=0.0
win=1.0!
Moving along z axis
irin=2!
Starts in region 2, could be 1
wtin=1.0!
weight = 1 since no variance reduction used!----------------------------------------------------------------------! Step 5:
hatch-call!----------------------------------------------------------------------! Maximum total energy of an electron for this problem must be! defined before hatch callemaxe = ein

130
!
!
!

140
!

write(6,130)
format(/’ Start tutor8’/’ Call hatch to get cross-section data’)-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,140)
FORMAT(/,’ HATCH-call comes next’,/)!

==========
call hatch==========!

------------------------------

198

!
!

Close files (after HATCH call)-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)!

Pick up cross section data for lead
write(6,150) ae(1)-RM, ap(1)150format(/’ Knock-on electrons can be created and any electron ’,*’followed down to’ /T40,F8.3,’ MeV kinetic energy’/*’ Brem photons can be created and any photon followed down to’,*/T40,F8.3,’ MeV’)! Compton events can create electrons and photons below these cutoffs!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------! Define the coordinates and the normal vectors for the two planes.
! Information required by howfar (and auxiliary geometry subprograms)! and passed through common/pladta/!
! First plane (the x-y plane through the origin)pcoord(1,1)=0.0
pcoord(2,1)=0.0
pcoord(3,1)=0.0! Coordinates
pnorm(1,1) =0.0
pnorm(2,1) =0.0
pnorm(3,1)= 1.0! Normal vectors! Second plane (note: slab is 1 cm thick)pcoord(1,2)=0.0
pcoord(2,2)=0.0
pcoord(3,2)=1.0! Coordinates
pnorm(1,2) =0.0
pnorm(2,2) =0.0
pnorm(3,2)= 1.0! Normal vectors!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------do i=1,50
ebin(i) = 0.0! Zero scoring array before startingend dobwidth = 0.002!----------------------------------------------------------------------! Step 8: Shower-call!-----------------------------------------------------------------------

199

! Initiate the shower ncase times
ncases=100000
do i=1,NCASES
call shower(iqin,ein,xin,yin,zin,uin,vin,win,irin,wtin)
end do!----------------------------------------------------------------------! Step 8: Output-of-results!----------------------------------------------------------------------! Use log10(ncases) as maximum value
binmax=alog10(float(ncases))

160

170

if (iedgfl(2).eq.1) thenwrite(6,160) ein,pcoord(3,2)
format(/’ Reflected photon spectrum’/’ for a’,F8.2,* ’ MeV pencil beam of electrons on a’,F7.2,* ’ cm thick slab of lead’/’ with fluorescence photon’//T6,* ’Energy counts/incident electron’/* 25X,’ log(counts for 10^6 incident electrons)’)
else
write(6,170) ein,pcoord(3,2)
format(’ Reflected photon spectrum’/’ for a’,F8.2,* ’ MeV pencil beam of electrons on a’,F7.2,* ’ cm thick slab of lead’/’ without fluorescence photon’//T6,* ’Energy counts/incident electron’/* 25X,’ log(counts for 10^6 incident electrons)’)
end ifdo j=1,48
line(j)=’ ’
end do! Blank entire output arraydo j=1,50
if(ebin(j).gt.0) thenicol=*int(dlog10(ebin(j))/binmax*48.0+0.999)
if (icol.le.0) icol=1
else
icol = 1
endifline(icol)=’*’! Load output array at desired location
write(6,180) bwidth*j,ebin(j)/float(ncases),line
180
format(F10.4,F12.6,48A1)line(icol)=’ ’! Reblankend do190

write(6,190)impacr(2),ieispl,neispl
format(’ impacr(2)=’,i2,’ ieispl=’,i2,’ neispl=’,i3)
write(6,200)ibrspl(2),nbrspl,ibrdst

200

200

format(’ ibrspl=’,i2,’ nbrspl=’,i3,’ ibrdst=’,i2)
stop
end!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------!***********************************************************************!
! In this AUSGAB routine for TUTOR8 we score photons reflected! from the slab (ir(np)=1 and iq(np)=0).
!***********************************************************************
subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/score/bwidth,ebin(50)real*8 bwidth,ebin
integer iarg

! Arguments

integer ibin,irl! Local variable

irl=ir(np)! Local variable
if(irl.eq.1.and.iq(np).eq.0) then! Photon is reflected (emitted backward relative to source)!
Increment bin corresponding to photon energy
ibin= min0 (int(e(np)/bwidth + 0.999), 50)
if (ibin.ne.0) thenebin(ibin)=ebin(ibin)+wt(np)end ifend ifreturnend!--------------------------last line of ausgab.f------------------------PEGS5-call comes next

inseed=1(seed for generating unique sequences of Ranlux)
ranlux luxury level set by rluxgo : 1
p= 48

201

ranlux initialized by rluxgo from seed1

Start tutor8
Call hatch to get cross-section data
HATCH-call comes next
E- IMPACT IONIZATION OPTION REQUESTED FOR MEDIUM NUMBER
RAYLEIGH DATA AVAILABLE FOR MEDIUM

1

1 BUT OPTION NOT REQUESTED.

INCOHERENT DATA AVAILABLE FOR MEDIUM

1 BUT OPTION NOT REQUESTED.

SHELL COMPTON PROFILE DATA AVAILABLE FOR MEDIUM

1 BUT OPTION NOT REQUESTED.

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
Knock-on electrons can be created and any electron followed down to
0.010 MeV kinetic energy
Brem photons can be created and any photon followed down to
0.001 MeV
Reflected photon spectrum
for a0.61 MeV pencil beam of electrons on awith fluorescence photon
Energy0.00200.00400.00600.00800.01000.01200.01400.01600.01800.02000.02200.02400.02600.02800.03000.03200.03400.03600.03800.04000.04200.04400.04601.00 cm thick slab of leadcounts/incident electronlog(counts for 10^6 incident electrons)
0.000118*
0.000230*
0.000279*
0.000451*
0.000545*
0.000797*
0.000710*
0.000371*
0.000284*
0.000307*0.000298*
0.000294*
0.000269*
0.000256*0.000235*
0.000240*
0.000212*
0.000202*
0.000192*
0.000189*
0.000167*
0.000153*
0.000146*202

0.04800.000135*
0.05000.000123*
0.05200.000107*
0.05400.000113*
0.05600.000102*
0.05800.000090*
0.06000.000083*
0.06200.000073*
0.06400.000073*
0.06600.000067*
0.06800.000062*
0.07000.000050*
0.07200.000050*
0.07400.000063*
0.07600.000055*
0.07800.000033*
0.08000.000033*
0.08200.000027*
0.08400.000024*
0.08600.000025*
0.08800.000020 *0.09000.000011*
0.09200.000008*
0.09400.000009*
0.09600.000004*
0.09800.000003*
0.10000.000002*
impacr(2)= 1 ieispl= 1 neispl= 20
ibrspl= 1 nbrspl= 30 ibrdst= 1

203

Chapter 4

ADVANCED EGS5 USER CODESIn this chapter we present user codes which demonstrate some of the more advanced featuresand capabilities of the EGS5 Code System. We will show how to transport charged particles ina magnetic field and how to produce and transport fluorescent photons following a photoelectricinteraction. A generalized multi-cylinder, multi-slab geometryis presented in which both splitting
and leading particle biasing are used in order to reduce the variance and speed up the calculation ofenergy deposition in accelerator targets and surrounding devices. We also show how to incorporate
into EGS5 combinatorial geometry(CG) subprograms derived from MORSE-CG.
All of the user codes presented here are provided with the EGS5 distribution and the reader
should study the appropriate program listings themselves in order to gain a more complete understanding of what they have been designed to do.

4.1

UCCYL - Cylinder-Slab Geometry and Importance Sampling

4.1.1

Generalized Multi-Cylinder, Multi-Slab Geometry

The user code UCCYL provides an example of how to set up a geometry consisting of multiple
cylindrical shells that are orthogonal to multiple slabs. The groundwork for this generalized scheme
was provided in Tutorials 6 and 7 of Chapter 3, where the geometry subprograms PLANE1 and
PLAN2P were presented. Several other geometry routines are required in the present case, including
CYLNDR, which is the cylinder analog to PLANE1. For a description of the implementation anduse of these and other auxiliary geometry routines provided with EGS5, the user is referred tothe document “How to Code Geometry: Writing Subroutine HOWFAR,” which is part of the EGS5
distribution.

204

The main purpose of this section, however, is to demonstrate how to incorporate importancesampling into an EGS5 problem, so we will not discuss the geometry techniques any further. Instead, those interested in the cylinder-slab generalization are referred to the commented FORTRAN
listing of UCCYL.

4.1.2

Particle Splitting

A version of UCCYL was used to answer a number of questions about energy deposition in apositron production target, as well as regions surrounding such a target. Two of these questions
are: what is the temperature rise in the target and what is the radiation damage to nearby vacuum
O-rings, when 1013 electrons/sec with energies of 33 GeV strike the target?
One of the first things observed during initial EGS runs was that the statistics in the (backward)
region designated as an “O-ring” were quite low because
1. the region of interest was too small,
2. the majority of the shower was forward-directed and only low energy radiation headed backwards towards the region, and3. there was too much shielding material separating the the region of interest from the source.
However, by studying the energy deposition block diagrams (from auxiliary routine ECNSV1) andthe associated event counters (from NTALLY), obtained during the initial computer runs, we wereable to make the following observations and recommendations:• Most of the charged particles depositing energy come from photons interacting in the regionof interest–not from charged particles entering from surrounding regions (ı.e. the “cavity” isfairly big).
• The region (“cavity”) is small enough, however, that only a fraction of the photons traversing
the region interact (verified by simple calculation).
• Suggestion: Every time a photon first enters the region, split it into 10 identical photons,
1
to the progeny; include the weight in the EDEP (and any other)each carrying a weight of 10
scoring.
This was accomplished by placing the following statement in HOWFAR:
if (irsplt(irl).eq.1.and.irl.ne.irold.and.iq(np).eq.0) then
! Apply particle splitting
if (lsplt.ne.0) then205

wt(np)=wt(np)/lsplt
do isplt=1,nsplt
x(np+isplt)=x(np)y(np+isplt)=y(np)z(np+isplt)=z(np)u(np+isplt)=u(np)v(np+isplt)=v(np)w(np+isplt)=w(np)ir(np+isplt)=ir(np)wt(np+isplt)=wt(np)dnear(np+isplt)=dnear(np)latch(np+isplt)=latch(np)e(np+isplt)=e(np)iq(np+isplt)=iq(np)k1step(np+isplt)=0.d0
k1rsd(np+isplt)=0.d0
k1init(np+isplt)=0.d0
end donp=np+nsplt
end ifend ifwhere IRL is a local variable (IR(NP)= region of interest), IRSPLT(IRL) is a flag telling us thatsplitting is “turned on”, and IRL.NE.IRLOLD provides us with a way of doing the splitting only
when the photon (IQ(NP)=0) first enters the region.
Most of the variables above are associated with the current (ı.e. NP) particle on the “stack” (ı.e.
COMMON/STACK/), and with the number of splits (LSPLT=10 in UCCYL).
For convenience, the common
common/passit/irsplt(66),lsplt,nsplt,nreg
integer irsplt,lsplt,nsplt,nreg
was also included in the user code (NSPLT=LSPLT-1).
The result of this effort was an increase in the number of energy deposition events in theimportant region of interest and, as a result, a decrease in the variance (as judged by batch-run
statistics.

2064.1.3

Leading Particle Biasing

The second application of importance sampling that we will discuss primarily concerns the deposition of energy in an electromagnetic shower initiated by a high energy electron (or photon).
The analog approach that is used throughout EGS5, in which each and every particle is generally
followed to completion (ı.e. energy cutoff guarantees that high energy shower calculations will takelots of time. For the most energetic particle energies under consideration at present day accelerators (> 50 GeV), one can barely manage to simulate showers in this fashion because of computer
time limitations (Recall that execution time per incident particle grows linearly with energy). Fortunately, a certain class of problems involving the calculation of energy deposition is well-suited
for a non-analog treatment known as leading particle biasing[175]. Examples of such problems areradiobiological dose, heating effects, and radiation damage, although some care must be taken in
not being too general with this statement (more on this later).
As a rule, variance reduction techniques of this type should only be used when there is someprior knowledge of the physical processes that are the most (or least) important to the answer one
is looking for. Leading particle biasing is a classic example of this. The most important processes inthe development of an EM shower, at least in terms of total energy deposition, are bremsstrahlung
and pair production. Furthermore, after every one of these interactions the particle with the higherof the two energies is expected to contribute most to the total energy deposition.
Leading particle biasing is very easily implemented within the framework of EGS5 by means of
the following statements:
if (iarg.eq.7) then
! Apply Leading Particle Biasing for brems.
eks=e(np)+e(np-1)-RM! Kinetic energy before brems.
ekenp=e(np)if(iq(np).ne.0) ekenp=e(np)-RM
call randomset(rnnolp)
if (rnnolp.lt.ekenp/eks) then ! Follow npe(np-1)=e(np)iq(np-1)=iq(np)u(np-1)=u(np)v(np-1)=v(np)w(np-1)=w(np)end ifekenp=e(np-1)
if (iq(np-1).ne.0) ekenp=e(np-1)-RM
wt(np-1)=wt(np-1)*eks/ekenp
np=np-1
end ifif (iarg.eq.16) then ! Apply Leading Particle Biasing for pair.
eks=e(np)+e(np-1)-2.0*RM207

ekenp=e(np)-RM
call randomset(rnnolp)
if (rnnolp.lt.ekenp/eks) then ! Follow npe(np-1)=e(np)iq(np-1)=iq(np)u(np-1)=u(np)v(np-1)=v(np)w(np-1)=w(np)end ifekenp=e(np-1)-RM
wt(np-1)=wt(np-1)*eks/ekenp
np=np-1
end if

These statements to apply leading particle biasing for bremsstrahlung and pair are included in a
version of UCCYL (called UC LP) at the proper locations within subroutine AUSGAB.
If IAUSFL(8) or IAUSFL(17) is set as 1, AUSGAB is called for IARG of 7 and 16 respectively,
then Leading Particle Biasing is applied after bremsstrahlung and pair production, respectively.
The COMMON/EPCONT/ is now needed in MAIN in order to pass the IAUSFL(8) and IAUSFL(17) flags(1=on, 0=off). The subroutine RANDOMSET is explained in the EGS5 User Manual (see Appendix B).
The other quantities are best understood with the help of the EGS5 Flow Diagrams for ELECTR
and PHOTON (see Appendix A).
Either statement can be explained as follows. An interaction of the proper type occurs and theflag has been turned “on”. A random number, uniformly distributed between 0 and 1, is drawn andcompared with the fraction of the kinetic energy that was assigned to the current particle—i.e.,
the lower energy particle of the two produced in the interaction. If the random number is less than
this fraction, the lower energy (NP) particle is kept and the one below it on the stack (NP-1) isthrown away. Otherwise the higher energy particle (NP-1) is selected and the current particle (NP)is tossed out. Obviously the particles in the shower with the highest energy will preferentially be
selected by such a scheme and, to assure that the Monte Carlo game is “played fairly”, we have
imposed two necessary requirements:• The lowest energy particles must be selected some of the time (which we have done bysampling).
• The proper weight must be assigned to whichever particle chosen.
To complete the process, one should make use of the particle weight when scoring information in
subroutine AUSGAB. Namely,• Sum particle weights (WT(NP)) when “counting” particles.
208• Sum the weighted energy deposition (WT(NP)*EDEP).
The easiest way to see that the weights have been properly assigned in the statements above is
by example. Assume that the incident particle has a kinetic energy of 1000 MeV and that one ofthe progeny has energy 100 MeV and the other has 900 MeV. Then clearly the 100 MeV particle
will be chosen 10% of the time and will have a weight of 1000
100 , whereas the 900 MeV particle will
be chosen 90% of the time and will have a weight of 1000.
But
the total particle count will average
900
1000
1000
out to two (ı.e. 0.1 × 100 + 0.9 × 900 = 2).
This scheme has been found to increase the speed of shower calculations by a factor of 300 at
33 GeV. However, because the biasing can be somewhat severe at times during the calculation,
the weights that are assigned tend to become rather large, and the net result is that the overall
1) is not usually 300 times better. Nevertheless, factors of 20 or more are
efficiency ( variance×time
generally obtained for many problems[75]and this technique can be invaluable.
As an example, at one time it took about one minute of CPU time on the IBM-3081 to completely
generate one 50 GeV shower in a large absorber. When simulating “real” beams of particles having
spatial and/or angular distributions, analog EGS calculations gave only 60 incident events in one
hour runs, which was inadequate. The use of leading particle biasing with a factor of 300 increasein speed, on the other hand, produced 18,000 events/hour, which was sufficient.
As a final note, it should be understood that leading particle biasing, or any importance sampling
scheme for that matter, should not be attempted in an arbitrary manner. For example, one should
not use leading particle biasing for the “O-ring” problem above since, even though many more
incident shower events will certainly be generated, the weights that are assigned to the low energy
particles heading in the direction of the “O-ring” region will tend to be quite high, and no significant
reduction in the variance will result. Quite the contrary, biasing of this sort can lead to very
erroneous results, and one should really have a full grasp (ı.e. pre-knowledge) of the important
aspects of the radiation transport before attempting to apply any of these variance reduction
methods with EGS5 (or any Monte Carlo program for that matter).

4.2

UCBEND - Charged Particle Transport in a Magnetic Field

Charged particle motion in a magnetic field is governed by the Lorentz force equationp~ = q~v × B~ = d~
Fdt(4.1)~ is the magnetic field strength vector, ~v is the velocity vector, and q is the electric charge
where Bof the particle. This equation can be expanded into its Cartesian components to givemẍ = q(ẏBz − żBy )mÿ = q(żBx − ẋBz )mz̈ = q(ẋBy − ẏBx )
209

(4.2)

where ẋ and ẍ (for example) are the x-components of velocity and acceleration, respectively. No
attempt is made to solve these coupled differential equations. Instead, in the example user code
called UCBEND, we use a simple vector expression for the change in the direction of motion aftera very small translation, ∆l, in a constant magnetic field strength, B, in which the particle energy
is assumed to remain constant. Namely,v̂ = B̂(v̂0 · B̂) + [v̂0 − B̂(v̂0 · B̂)] cos α − v̂0 × B̂ sin α

(4.3)

where
α = ∆l/R,
R = radius of curvature = qp/B,
p = particle momentum,B̂ = magnetic field unit vector,v̂0 = particle direction unit vector (before)= U0 î + V0 ĵ + W0 k̂,v̂ = particle direction unit vector (after)= U î + V ĵ + W k̂.
The problem that we attempt to solve with UCBEND relates to a series of spectral measurements
performed at SLAC [100]. An RF cavity was tested and found to emit high levels of radiation.
Because the cavity was subjected to very high power levels, surface electrons were found to be
produced from two locations within the structure and were accelerated to kinetic energies of 8.5
and 3.5 MeV, respectively, depending on their point of origin. Upon exiting the cavity they passed
through a 0.015 inch copper window, where they lost energy primarily by ionization and were
multiply scattered. The experimentalists attempted to measure the spectrum of the electrons
using a magnet, a lead slit, and a Faraday cup. A strong peak corresponding to the 8.5 MeV group
was observed, but the 3.5 MeV peak was not observed. The object of the EGS5 simulation was
to try to understand these observations, particularly with the help of graphics showing particle
trajectories.
Subroutine HOWFAR of UCBEND was designed using the diagram shown in Figure 4.1. Electrons
start from the origin (at plane 1), transport through the copper window (Region 2), emanate out
into the air (Region 3), and pass through the magnetic field, which is constant and along the positive
y-direction (in Region 4 only). They are then bent in the direction of the lead wall (Regions 6 and
8) where some of them pass through the slit (Region 7) and get scored (at Plane 10). By varying
the field strength of the magnet one should be able to re-create the observations for both 3.5 MeV
and 8.5 MeV.
In order to gain a more complete understanding of the geometry, it is suggested that the reader
study the listing for UCBEND, which is included with the EGS5 distribution. What is of interest
here is the method that we employ in order to transport the electrons through the magnetic fieldregion. Using Equation 4.3 with ∆l/R << 1, we obtainU= (U0 + αW0 )F ,
210

Vac

z

11

6
Pb
5

4

6
5
Air

7

4

9

10

Vac

Vac

Air

12
Air

Vac

8
7
3
8

11

9

Air

3

Pb

10

2
2
1

Cu

1

e-

x

Vac

Figure 4.1: Diagram used with UCBEND (not to scale).
V= V0 ,

W= (W0 − αU0 )F ,

(4.4)

where
F

= unit vector renormalization factor= (1 + α2 )−1/2 .

(4.5)

The following code listing, which pertains only to that HOWFAR section relating to Region 4, should
be fairly self-explanatory (Note that USTEP has been limited to 0.1 cm in this example, but thiscan be changed by the user).
else if (irl.eq.4) thenif (ifield.ne.0.and.iq(np).ne.0) then
psq=e(np)*e(np)-RMSQ
if (psq.lt.0.0) psq=1.0E-9
p=sqrt(psq)211

Figure 4.2: UCBEND simulation at 8.5 MeV (B=2.6 kG).
poverb=p/(bfield*0.3)
u0=u(np)v0=v(np)w0=w(np)
rcurv=-iq(np)*poverb
if (ustep.gt.0.1) thenustep=dmin1(ustep,0.1*abs(poverb))
end ifustep0=ustep
irnew=irlalpha=ustep/rcurv
renorf=1./sqrt(1.+alpha*alpha)w(np)=(w0-u0*alpha)*renorf
u(np)=(u0+w0*alpha)*renorf
v(np)=v0
call plan2p(6,11,1,3,3,-1)
call plan2p(7,5,1,11,12,-1)
else
call plan2p(6,11,1,3,3,-1)
call plan2p(7,5,1,11,12,-1)
end if
The result of running UCBEND at 8.5 and 3.5 MeV is shown in Figure 4.2 and Figure 4.3,
respectively, for 100 incident electrons (the red lines are electrons and the yellow lines are photons).1
1

These figures were created by using a freely distributed program called “CGVIEW,” which is briefly describedin section 4.3.

212

Figure 4.3: UCBEND simulation at 3.5 MeV (B=1.0 kG).
It becomes clear from pictures such as these that it is easier to focus the 8.5 MeV electrons throughthe slit than the 3.5 MeV ones. A number of EGS5 “experiments” can be performed rather easily
now that the basic geometry and magnetic field transport is established. For example, one canreplace the air regions and/or the copper window with vacuum in order to determine what effectthey have. What was discovered by doing this is that the copper window has the greatest effecton the transmission through the slit. Note that Figure 4.4, illustrates what happens when themagnetic field is turned off. Users interested in charged particle transport in an electric field should
see Rawlinson, Bielajew, Munro and Galbraith [133].

4.3

Using Combinatorial Geometry with EGS5

All of the user codes presented thus far have treated problems with simple geometries involving
few regions and only basic shapes (parallel slabs and right circular cylinders). Construction of subroutine HOWFAR for such geometries is rather straight-forward, as distances to boundaries are easily
determined for planar and cylindrical surfaces, and identifying new region numbers subsequent toboundary crossings is almost trivial when there are few regions with limited intersections. However,for simulations involving complex devices such as radiotherapy accelerator heads or high energy
physics detectors, subroutine HOWFAR can be quite cumbersome to construct. In such applications,regions can have complex shapes for which distance to boundary computations are more involved(e.g., in concave regions there can be multiple intersection points on the same surface), and/or havesurfaces which are bordered on one side by a large number of distinct volumes (e.g., at a collimatorinterface), making region identification more difficult.
As the need to simulate complex geometry is a common problem in Monte Carlo modeling,213

Figure 4.4: UCBEND simulation at 8.5 MeV (B=0 kG).
programs have been developed to assist in the construction of problem geometries by defining
regions to be the unions and/or intersections of sets of basic volume elements of varying sizes. In
Monte Carlo applications, this method for creating geometries through the combining of smaller
volumes has come to be referred to as “combinatorial geometry,”, or CG. Note that this usage is notexact, as the term combinatorial geometry actually denotes a much more extensive mathematicalfield (see, for example, the book by Pach and Agarwal[129]).
In this section we describe the advanced EGS5 user code UCSAMPCG, which exploits the features of combinatorial geometry. The CG package invoked by UCSAMPCG is included in the EGS5
distribution in the auxiliary source code file cg related.f, and is based on the well-known MORSECG[168] combinatorial geometry Monte Carlo program. The MORSE-CG package was originallyadapted to EGS4-PRESTA by Torii and Sugita [171], and Sugita and Torii have further modified it
for EGS5 [169]. The subprograms of cg related.f have retained the MORSE-CG methodology forconstructing and evaluating geometry, and the EGS5 implementation is compatible with MORSECG input files. Five elemental volume shapes can be modeled: rectangular parallelepipeds (RPP),
spheres (SPH), right circular cylinders (RCC), truncated right angle cones (TRC), and tori (TOR).
As mentioned above, more complicated volumes can be specified by appropriately intersecting these
five basic bodies. The reader is referred to the MORSE-CG manual [168] and the Torii and Sugita
citations [171, 169] for detailed descriptions of the CG adaptation and implementation, as well as
for instructions on creating input files.
User code UCSAMPCG uses combinatorial geometry to simulate the same problem which thesample user code UCSAMPL5, presented in great detail in the EGS5 User Manual (see Appendix Bof this report), treats with a simple geometry model. The coupling of the CG package to EGS5 isaccomplished entirely within subroutine HOWFAR, which performs the actual boundary and region
checks. A once-only initialization is done by calling the CG routine GEOMGT, which reads in the214necessary geometry input. The following data, which is in the familiar MORSE-CG format [168]and which is included as file ucsampcg.data of the EGS5 distribution, is used by UCSAMPCG
to model with CG the same geometry treated by UCSAMPL5 in a non-CG implementation.
RPP

1

RPP

2

END
Z1
Z2
END
1

-1.5
3.0
-2.5
4.0
+1
+2

1.5

-1.5

1.5

0.0

2.5

-2.5

2.5

-1.0

-1

0

The HOWFAR routine supplied with UCSAMPCG is general and can be used for any CG geometry
which can be specified by the five types of bodies listed above. All that is required of the user isthe creation of an appropriately formatted input data file. It must be noted, though, that becauseof the computational overhead introduced in generalizing HOWFAR, for problems involving simple,
regular geometries (such as cylindrical slabs or rectangular pixels), EGS5 user codes using CG will
execute 1-2 times more slowly than user codes tailored to the particular problem geometries. (Thisis a substantial improvement relative to the CG implementation in EGS4, which was “significantly
less efficient than the macro-geometry methods” [171].) For many problems, the greatly facilitatedease in setting up the simulation geometry with CG will more than offset the slight penalty incomputational efficiency.
To facilitate the construction of CG geometry input files and to provide for the display of particle
trajectories from EGS5 simulations, Namito and co-workers [123] developed a stand-alone visualinterface program, CGVIEW, which is freely distributed on the Internet at:
http://rcwww.kek.jp/research/egs/kek/cgview/.
The geometry used in UCSAMPCG and some sample output particle trajectories drawn with
CGVIEW are shown in Fig. 4.5.
CGVIEW can also be used to assist in the construction of CG input data files. For geometries
with simple shapes, users can select and position volume elements visually using tools provided with
CGVIEW, and later have the program create output data files in the correct MORSE-CG format.
Additionally, the integrity of existing CG input data files can be checked through the use of the
CGVIEW pseudo particle feature. This function allows the user to specify locations and directions
for pseudo source particles, which are then tracked by CGVIEW through the input geometry,searching for regions of overlapping and undefined zones. Thus, errors in geometry specification
may be corrected prior to being used in Monte Carlo simulations.

215

Figure 4.5: Geometry and particle trajectory of UCSAMPCG simulation. The stand alone program
CGVIEW was used to draw the picture.

216

CG modeling of thin (∼ 1µm) regions in EGS5 In the CG implementation of EGS5, whena particle has approached within a given small distance of a surface, it is assumed to have reached
the surface, and it is immediately transported a small distance into the region on the opposite sideof the boundary. The variables describing the minimum distance of approach (which actually isallowed to vary by surface type) and the length of the cross-boundary step are contained in common
block epstbl (found in auxiliary common file geom common.f) and their values are by default
set to 1 µm (in block data cgtype of auxiliary code cg related.f).
To prevent errors in region identification when modeling small volumes, the values of the epstbl
variables cgmnst, cgeps1 and cgeps2, along with those of the variables pertaining to any of the
body types in the problem (variable rcceps for right circular cylinders, for example) must be
modified to be no more than 1% of the minimum dimension of any body in the problem geometry.
This can be done in either of two ways:
1. directly altering the values assigned to the variables in block data cgtype of cg related.f,
as in changing:data cgmnst/1.0d-4/...
data geleps/1.0d-4/
to, for example
data cgmnst/1.0d-6/...
data geleps/1.0d-6/
2. including auxcommons/geom common.f in the main program of the user code and explicitly assigning values to the necessary variables of epstbl, as in:
cgmnst=1.0d-6...
geleps=1.0d-6
Note that as CGVIEW uses a default minimum distance to boundary of 1.0 × 10−6 , region
identification errors of this type may not appear in CGVIEW geometry checks of simulations
involving small regions.

217

Appendix A

EGS5 FLOW DIAGRAMS
Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANThis EGS5 Flow Chart Compilation is Appendix A of a document called
SLAC-R-730/KEK-2005-8, which can be obtained from the SLAC and KEK web sites.

218

KEY for EGS5 Flow Diagrams

subroutine
annih

Beginning of subprogram.

Statement.

iannih=iannih+1

rnnow
.gt.
rejf

yes

Conditional statement.

no

i=1
i=i+1

i>10

yes

DO loop.

nocall
randomset(rnnow)

return(to photon)

write(6,102) medium

stop

Call to external subprogram.

Return statement.

Output (or Input) related statement.

Stop statement.

Jump over flow chart line.

219

subroutine
annihVersion051219-1435

iannih=iannih+1

avip = e(np) + RM
a = avip/RM
g = a - 1.0
t = g - 1.0
p = sqrt(a*t)
pot = p/t
ep0 = 1.0/(a + p)
"Sample 1/ep from ep = ep0 to 1 - ep0"

"Rejection sampling loop"

call
randomset(rnnow)
ep = ep0*exp(rnnow*log((1.0 - ep0)/ep0))
call
randomset(rnnow)
rejf = 1.0 - ep + (2.0*g - 1.0/ep)/a**2

yes

rnnow
.gt.
rejf
no

ep
= max(ep,1.D0 - ep)
esg1= avip*ep
e(np) = esg1
esg2
= avip - esg1
e(np + 1) = esg2
iq(np) = 0
costhe = (esg1 - RM)*pot/esg1
costhe = min(1.D0,costhe)sinthe = sqrt((1.0 - costhe)*(1.0 + costhe))calluphi(2,1)

np = np + 1
iq(np) = 0
costhe = (esg2 - RM)*pot/esg2
costhe = min(1.D0,costhe)sinthe = - sqrt((1.0 - costhe)*(1.0 + costhe))calluphi(3,2)

return(to photon)220

Version
051219-1435

subroutineaphi(br)

iaphi = iaphi +1
iarg=21

iausfl(iarg+1)
ne
0
no

yes

call
ausgab(iarg)

pnorm0 = sqrt(u(np)*u(np) + v(np)*v(np) + w(np)*w(np))u(np) = u(np)/pnorm0v(np) = v(np)/pnorm0w(np) = w(np)/pnorm0
valmax = br + 1./br"Rejection sampling loop"

call
randomset(rnnow)

ph0 = rnnow*twopi
sinph0 = sin(ph0)cph0 = pi5d2 - ph0
cosph0 = sin(cph0)
valloc = sqrt(sinph0*sinph0 + cosph0*cosph0)
sinph0 = sinph0/valloc
cosph0 = cosph0/valloc
val = (valmax - 2.*sinthe*sinthe*cosph0*cosph0)/valmax

callrandomset(rnnow)

no

rnnow
le
val
yes

anorm2 =costhe*costhe*cosph0*cosph0 + sinph0*sinph0
call
randomset(rnnow)(valmax - 2.)/(valmax - 2. + 2.*anorm2) .gt. rnnow.or.
ldpola = 1
anorm2 .lt. 1.E-10
yesnoldpola = 0
1

221

1

anormr = 1./sqrt(anorm2)
sineta = -anormr*sinph0
coseta = anormr*costhe*cosph0

ldpola eq 
1

no

yes

call
randomset(rnnow)

eta = rnnow*twopi
sineta = sin(eta)ceta = pi5d2 - eta
coseta = sin(ceta)

ufa = costhe*cosph0*coseta - sinph0*sineta
vfa = costhe*sinph0*coseta + cosph0*sineta
wfa = -sinthe*coseta
asav = u(np)bsav = v(np)csav = w(np)sinps2 = asav*asav + bsav*bsav

sinps2 lt.
1.E-20noyes

sinpsi = sqrt(sinps2)
sindel = bsav/sinpsi
cosdel = asav/sinpsi
cosomg = cosdel*csav*uf(np) +sindel*csav*vf(np) - sinpsi*wf(np)sinomg = -sindel*uf(np) + cosdel*vf(np)cosomg = uf(np)sinomg = vf(np)enorm = sqrt(uf(np)*uf(np) + vf(np)*vf(np) + wf(np)*wf(np))

yes

enorm
 lt.
1.E-4

call
randomset(rnnow)

nocosphi = cosomg*cosph0 - sinomg*sinph0
sinphi = sinomg*cosph0 + cosomg*sinph0
ufb = cosomg*ufa - sinomg*vfa
vfb = sinomg*ufa + cosomg*vfa
wfb = wfa

yessinps2 lt.
1.E-20uf(np) = ufb
vf(np) = vfb
wf(np) = wfb

nouf(np) = cosdel*csav*ufb - sindel*vfb + asav*wfb
vf(np) = sindel*csav*ufb + cosdel*vfb + bsav*wfb
wf(np) = -sinpsi*ufb + csav*wfb

enorm = sqrt(uf(np)*uf(np) + vf(np)*vf(np) + wf(np)*wf(np))
uf(np) = uf(np)/enorm
vf(np) = vf(np)/enorm
wf(np) = wf(np)/enorm

return(to photon)222

omg = rnnow*twopi
sinomg = sin(omg)
comg = pi5d2 - omg
cosomg = sin(comg)

Version
051219-1435

subroutine
bhabhaibhabha = ibhabha +1"Rejection sampling loop"

eip = e(np);ekin = eip - RM
t0 = ekin/RM;e0 = t0 + 1.0
yy = 1.0/(t0 + 2.0)
;e02 = e0*e0
betai2= e02/(e02-1.0) ;ep0 = te(medium)/ekin
ep0c = 1.0 - ep0;y2 = yy*yy
yp = 1.0 - 2.0*yy;yp2 = yp*yp
b4 = yp2*yp;b3 = b4 + yp2
b2 = yp*(3.0 + y2);b1 = 2.0 - y2
br = ep0
re1 = ep0c*(betai2-br*(b1-br*(b2-br*(b3-br*b4))))
br = 1.0
re2 = ep0c*(betai2-br*(b1-br*(b2-br*(b3-br*b4))))remax = max(re1,re2)

callrandomset(rnnow)
br = ep0/(1.0 - ep0c*rnnow)
callrandomset(rnnow)
rejf = ep0c*(betai2-br*(b1-br*(b2-br*(b3-br*b4))))
rejf = rejf/remax

yes

rnnow
 gt.
rejf
"Put e- on topof stack"

no
yes

br
lt.
0.5


"Put e+ on topof stack"noiq(np) = -1
iq(np+1) = 1
br = 1.0 - br
k1step(np+1) = k1step(np)k1init(np+1) = k1init(np)k1rsd(np+1) = k1rsd(np)k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.

1

223

iq(np+1) = -1
k1step(np+1) = 0.
k1init(np+1) = 0.
k1rsd(np+1) = 0.

1
br = max(br,0.D0)
ekse2 = br*ekinese1 = eip - ekse2
ese2 = ekse2 + RM
e(np) = ese1
e(np+1) = ese2
h1 = (eip + RM)/ekindcosth = h1*(ese1 - RM)/(ese1 + RM)
ttt = 1.0 - dcosth

ttt
le.
0.0

yes



sinthe = 0.0

nosinthe = sqrt(1.0 - dcosth)
costhe = sqrt(dcosth)calluphi(2,1)
np = np + 1
dcosth = h1*(ese2 - RM)/(ese2 + RM)
ttt = 1.0 - dcosth

ttt
le.
0.0

yes



no
sinthe = -sqrt(1.0 - dcosth)
costhe = sqrt(dcosth)calluphi(3,2)

return(to electr)

224

sinthe = 0.0

Version
051219-1435

subroutine
brems

ibrems = ibrems +1
eie = e(np)np = np + 1"Coulomb-corrected
Bethe Heitler distribution"

yes

eie
.lt.
50.0

"Bethe-Heitler distribution"
lvx = 1
lvl0 = 0nolvx = 2lvl0 = 3

abrems = float(int(AILN2*log(eie/ap(medium))))call
randomset(rnnow)

0.5
no
.lt.
(abrems*alphi(lvx,medium) + 0.5)*rnnow"Rejection sampling loop""(1-br)/br
subdistribution
sampling""2br
subdistribution
sampling"

yes
call
randomset(rnnow)

callrandomset(rnnow1)

idistr = abrems*rnnow
p = pwr2i(idistr+1)
lvl = lvl0 + 1

call
randomset(rnnow2)

callrandomset(rnnow)"br <0.5"

rnnow
.ge.
AI2LN2yes

call
randomset(rnnow)

callrandomset(rnnow)

br = rnnow*0.5

call
randomset(rnnow1

call
randomset(rnnow2)
h = max(rnnow1,rnnow2)
br = 1.0 - 0.5*h
rnnow
.gt.
0.5/br"Rejection sampling loop"nobr = max(rnnow1,rnnow2)
lvl = lvl0 + 2

yes

no
1

2

225

3

1

2

3

br = br*pesg = eie*bresg.lt.
ap(medium)

yes

no

"Rejection sampling loop"

ese = eie - esg
ese
.lt.
RM

yes

no
del = br/ese
del.ge.
delpos(lvx,medium)yesnodelta = delcm(medium)*del
rejf = dl1(lvl,medium) + delta*(dl2(lvl,medium) + delta*
dl3(lvl,medium))

delta
.lt.
1.0

yes

no

rejf = dl4(lvl,medium) +dl5(lvl,medium)*log(delta + dl6(lvl,medium))call
randomset(rnnow)

rnnow
.gt.
rejf

yes

no

"Photon energy accepted"

ibrdst
.ne.
1

yes

no

theta = RM/eie

ztarg = zbrang(medium)tteie = eie/RM
ttese = ese/RM
esedei = ttese/tteie
y2max = (PI*tteie)**2
rjarg1 = 1.0 + esedei**2
rjarg2 = 3.0*rjarg1 - 2.0*esedei
rjarg3 = ((1.0 - esedei)/(2.0*tteie*esedei))**2
y2tst1 = (1.0 + 0.0e0)**2
rejmin = (4.0 + log(rjarg3 + ztarg/y2tst1))*(4.0*esedei*0.0e0/y2tst1 - rjarg1) + rjarg2
y2tst1 = (1.0 + 1.0e0)**2
rejmid = (4.0 + log(rjarg3 + ztarg/y2tst1))*(4.0*esedei*1.0e0/y2tst1 - rjarg1) + rjarg2
y2tst1 = (1.0 + y2max)**2
rejmax = (4.0 + log(rjarg3 + ztarg/y2tst1))*(4.0*esedei*y2max/y2tst1 - rjarg1) + rjarg2
rejtop = max(rejmin,rejmid,rejmax)

4

226

5

5

"Rejection sampling loop"4

call
randomset(rnnow)

y2tst =rnnow/(1.0 - rnnow + 1.0/y2max)
y2tst1 = (1.0 + y2tst)**2
rejtst = (4.0 + log(rjarg3 + ztarg/y2tst1))*(4.0*esedei*y2tst/y2tst1 - rjarg1) + rjarg2
callrandomset(rnnow)

yes

rnnow
.gt.
(rejtst/rejtop)
notheta = sqrt(y2tst)/tteiecalluphi(1,3)

esg
.le.
ese

yes

no
iq(np) = iq(np-1)
iq(np-1) = 0
e(np) = ese
e(np-1) = esg
t = u(np)u(np) = u(np-1)
u(np-1) = t
t = v(np)v(np) = v(np-1)
v(np-1) = t
t = w(np)w(np) = w(np-1)
w(np-1) = t
k1step(np) = k1step(np-1)
k1step(np-1) = 0.
k1init(np) = k1init(np-1)
k1init(np-1) = 0.
k1rsd(np) = k1rsd(np-1)
k1rsd(np-1) = 0.

return(to electr)

227

iq(np) = 0
e(np) = esg
e(np-1) = ese
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.

subroutine
collis(lelec,irl,sig0,go1)

Version070808-1230

icollis = icollis +1
go1 = .false.
eke = e(np) - RM
elke = log(eke)lelke = eke1(medium)*elke + eke0(medium)"electron"
yes

lelec
.lt.
0
"positron"

ebr1 = ebr11(lelke+iextp,medium)*elke +ebr10(lelke+iextp,medium)

no

eke
.le.
ap(med(irl))

pbr1 = pbr11(lelke,medium)*elke +pbr10(lelke,medium)yes
ebr1 = 0.

no
eke
.le.
ap(med(irl))

yes
pbr1 = 0.

call
randomset(rnnow)

no

rnnow
.lt.
ebr1

call
randomset(rnnow)

yes"Brems"

no
rnnow
.lt.
pbr1

yes

no

"Not Moller"
ebr1
.le.
0.

no

"Not Brems either"yes
go1 = .true.

no

yes

"Bhabha"

e(np)
.le.
thmoll(medium)yespbr2 = pbr21(lelke,medium)*elke +pbr20(lelke,medium)

rnnow
.lt.
pbr2

no

iarg = 8"Moller"iarg = 10

yes

call
ausgab(iarg)

iausfl(iarg+1)
 ne 
0noiausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

callmollercallbhabhaiarg =9iarg =11

1

2

3

228

4

5

1

yes

call
ausgab(iarg)

23

iausfl(iarg+1)
 ne 
0noiausfl(iarg+1)
 ne 
0nogo1 = .true.

iq(np).eq.
0"Annihilation-in-flight"
iarg = 12

4

5

7

8

yes

call
ausgab(iarg)
yesnogo1 = .true.
yes

call
ausgab(iarg)

iausfl(iarg+1)
 ne 
0no"Brems"iarg = 6

call
annih
iausfl(iarg+1)
 ne 
0nouf(np) = 0. ;uf(np+1) = 0.
vf(np) = 0. ;vf(np+1) = 0.
wf(np) = 0.;wf(np+1) = 0

iarg = 13

yes

call
ausgab(iarg)

yesfbrspl
.eq.
0.d0
no

no

nbrspl
.gt.
0

yes

yes

call
ausgab(iarg)

call
brems

iausfl(iarg+1)
 ne 
0nouf(np) = 0. ;uf(np+1) = 0.
vf(np) = 0. ;vf(np+1) = 0.
wf(np) = 0. ;wf(np+1) = 0"Splitting has been requested"

ibrspl.eq.
1

nonbrspl .gt. 1
no.and.
(np + nbrspl) .ge. MXSTACK
yes
nbrspl = (2*nbrspl + 1)/3
fbrspl = 1./float(nbrspl)

yesfbrspl = 1./float(nbrspl)

nbrspl.eq.
1write(6,105)

yesibrspl=0

no
stop

no

(np+nbrspl).lt.
MXSTACK
yes
6

229

6
iq(np)
.eq.
0

yes

no
npstrt = np

wt(np-1) = wt(np-1)*fbrspl
frstbr = e(np-1)e(np) = e(np) + e(np-1)
iarg = 7yes

callausgab(iarg)

iausfl(iarg+1)
 ne 
0
no

7

8

12

13

npstrt = np - 1;fdummy = u(np-1)
u(np-1) = u(np) ;u(np) = fdummy
fdummy = v(np-1) ;v(np-1) = v(np)v(np) = fdummy ;fdummy = w(np-1)
w(np-1) = w(np) ;w(np) = fdummy
fdummy = e(np-1) ;e(np-1) = e(np)e(np) = fdummy ;fdummy = wt(np-1)
wt(np-1) = wt(np) ;wt(np) = fdummy
idummy = iq(np-1);iq(np-1) = iq(np)iq(np) = idummy
idummy = latch(np-1)
latch(np-1) = latch(np)latch(np) = idummy
fdummy = uf(np-1);uf(np-1) = uf(np)uf(np) = fdummy ;fdummy = vf(np-1)
vf(np-1) = vf(np) ;vf(np) = fdummy
fdummy = wf(np-1);wf(np-1) = wf(np)wf(np) = fdummy
k1step(np) = k1step(np-1)
k1step(np-1) = 0.
k1init(np) = k1init(np-1)
k1init(np-1) = 0.
k1rsd(np) = k1rsd(np-1)
k1rsd(np-1) = 0.

icsplt = 1

yes

icsplt.ge.
nbrsplnoicsplt = icsplt+1
iarg = 6yescallausgab(iarg)

iausfl(iarg+1)
 ne 
0
no

call
brems

iq(np)
.eq.
0

yes

no
wt(np-1) = wt(np-1)*fbrspl
e(np) = e(np) + e(np-1)

iarg = 79

fdummy = u(np-1) ;u(np-1) = u(np)u(np) = fdummy ;fdummy = v(np-1)
v(np-1) = v(np) ;v(np) = fdummy
fdummy = w(np-1);w(np-1) = w(np)w(np) = fdummy ;fdummy = e(np-1)
e(np-1) = e(np) ;e(np) = fdummy
fdummy = wt(np-1);wt(np-1) = wt(np)wt(np) = fdummy ;idummy = iq(np-1)
iq(np-1) = iq(np) ;iq(np) = idummy
idummy = latch(np-1)
latch(np-1) = latch(np)latch(np) = idummy
fdummy = uf(np-1);uf(np-1) = uf(np)uf(np) = fdummy ;fdummy = vf(np-1)
vf(np-1) = vf(np) ;vf(np) = fdummy
fdummy = wf(np-1);wf(np-1) = wf(np)wf(np) = fdummy
k1step(np) = k1step(np-1)
k1step(np-1) = 0.
k1init(np) = k1init(np-1)
k1init(np-1) = 0.
k1rsd(np) = k1rsd(np-1)
k1rsd(np-1) = 0.

10
11

230

9

11

10

yes

call
ausgab(iarg)

12e(np) = e(np) - frstbr;fdummy = u(np)u(np) = u(npstrt);u(npstrt) = fdummy
fdummy = v(np);v(np) = v(npstrt)v(npstrt) = fdummy ;fdummy = w(np)w(np) = w(npstrt);w(npstrt) = fdummy
fdummy = e(np);e(np) = e(npstrt)e(npstrt) = fdummy ;fdummy = wt(np)wt(np) = wt(npstrt) ;wt(npstrt) =fdummyidummy = iq(np);iq(np) = iq(npstrt)
iq(npstrt) = idummy ;idummy = latch(np)latch(np) = latch(npstrt)
latch(npstrt) = idummy
fdummy = uf(np);uf(np) = uf(npstrt)
uf(npstrt) = fdummy ;fdummy = vf(np)vf(np) = vf(npstrt) ;vf(npstrt) = fdummy
fdummy = wf(np);wf(np) = wf(npstrt)
wf(npstrt) = fdummy
k1step(np) = k1step(npstrt)
k1step(npstrt) = 0.0
k1init(np) = k1init(npstrt)
k1init(npstrt) = 0.0
k1rsd(np) = k1rsd(npstrt)
k1rsd(npstrt) = 0.0

iausfl(iarg+1)
 ne 
0noiarg = 7yes

callausgab(iarg)

iausfl(iarg+1)
 ne 
0noiq(np)
.eq.
0

yes

no
go1 = .true.

return(to electr)

231

13

Version
051219-1435

subroutine
compt

icompt = icompt +1
irloc = ir(np)incohr(irloc) .ne. 1.and.
iprofr(irloc) .eq. 1

yes

stop

no
eig = e(np)egp = eig/RM
br0i =1. + 2.*egp
alph1 = log(br0i)
alph2 = egp*(br0i + 1.)/(br0i*br0i)
sumalp = alph1 + alph2
callrandomset(rnnow)

alph1
.ge.
sumalp*rnnow

yesno"br distribution"callrandomset(rnnow1)"1/br distribution"callrandomset(rnnow)

br = exp(alph1*rnnow)/br0i"Rejection sampling loop"

call
randomset(rnnow)

no

egp
.ge.
(egp + 1.)*rnnow
yescall
randomset(rnnow)

rnnow1 = max(rnnow1,rnnow)

br = ((br0i - 1.)*rnnow1 + 1.)/br0i
esg = br*eig
a1mibr = 1. - bresedef = eig*a1mibr + RM
temp = RM*a1mibr/esg
sinthe = max(0.D0,temp*(2. - temp))

call
randomset(rnnow)

1

2

232

"Determine seconday photon energy
and polar angle from kinematics"

2

1

rejf3 = 1. - br*sinthe/(1. + br*br)
irloc = ir(np)"Rejection sampling loop"no"Calculate rejection function"

incohr(irloc)
.eq.
1
yes

medium = med(irloc)
alamb = 0.012398520
xval = sqrt(temp/2.)*eig/alambxval .ge. 5.E-3.and.
xval .le. 80.

xval
.gt.
80.

no

no

yes

yes
xlv = log(xval)
lxlv = sco1(medium)*xlv + sco0(medium)sxz = sxz1(lxlv,medium)*xlv + sxz0(lxlv,medium)sxz = 1.

sxz = 0.

rejf3 = rejf3*sxz

yes

rnnow
.gt.
rejf3
no

"Photon energy accepted"

sinthe = sqrt(sinthe)
costhe = 1. - temp
irloc = ir(np)noiprofr(irloc)
.eq.
1
yes

"Consider Compton profile"esgmax = eig - cpimev
valloc = sqrt(eig*eig + esgmax*esgmax - 2.*eig*esgmax*costhe)qvalmx = (eig - esgmax - eig*esgmax*(1. - costhe)/RM)*
137./valloc

yes

qvalmx.ge.
100.

esg = eig/(1. + eig/RM*(1. - costhe))nomedium = med(irloc)
call
randomset(rnnow)

icprof(medium)
.eq.
1
no
3

4

5

yes

lvallc = cco1(medium)*rnnow + cco0(medium)cpr = cpr1(lvallc,medium)*rnnow + cpr0(lvallc,medium)

233

6

3

4

5

6

icprof(medium)
.eq.
3

no

yes
call
randomset(rnnow1)

yesishell=1
ishell=ishell+1

ishell>mxshel(medium)nornnow1.le.
elecno(ishell,medium)

no

yes

lvallc = ccos1(medium)*rnnow + ccos0(medium)cpr = cprs1(lvallc,ishell,medium)*rnnow + cprs0(lvallc,ishell,medium)

f1 = (cpr/137.)*(cpr/137.)
f2 = (1. - costhe)/RM
f3 = (1. + f2*eig)*(1. + f2*eig) - f1
f4 = (f1*costhe - 1. - f2*eig)
f5 = f4*f4 - f3 + f1*f3
eps1 = 0.0

yes

f5
.lt.
eps1
no

esg1 = (-f4 - sqrt(f5))/f3*eig
esg2 = (-f4 + sqrt(f5))/f3*eig
call
randomset(rnnow2)

rnnow2
.lt.
0.5

yes
esg = esg1

noesg = esg2

icprof(medium)
.eq.
3

yes
esgmax = eig - capio(ishell,medium)noyes

esg .gt. esgmax.or.
esg .lt. 0.

no callrandomset(rnnow3)

esg
no
.lt.
esgmax*rnnow3
yes

7

234

7
ese = eig - esg + RM

yesiprofr(irloc)
.eq.
1
no

ese = ese - capio(ishell,medium)edep = capio(ishell,medium)etmp = e(np)e(np) = edep
iqtmp = iq(np)iq(np) = 0
iarg = 4

yes

lpolar(irloc).eq.
0no "Include linear
polarization"calluphi(2,1)

iausfl(iarg+1)
 ne 
0nocall
aphi(br)e(np) = etmp
iq(np) = iqtmp

calluphi(3,1)

np = np + 1
psq = esedef**2 - RMSQ

yes

costhe = 0.0
sinthe = -1.0

psq
.le.
0.0
no

costhe = (ese + esg)*a1mibr/sqrt(psq)sinthe = -sqrt(max(0.D0,1.D0 - costhe*costhe))calluphi(3,2)
uf(np) = 0.
vf(np) = 0.
wf(np) = 0.
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.
k1step(np-1) = 0.
k1init(np-1) = 0.
k1rsd(np-1) = 0.

"Photon lowest energy"
iq(np) = 0
iq(np-1) = -1e(np) = esg
e(np-1) = ese
t = u(np)u(np) = u(np-1)
u(np-1) = t
t = v(np)v(np) = v(np-1)
v(np-1) = t
t = w(np)w(np) = w(np-1)
w(np-1) = t
t=uf(np)
uf(np) = uf(np-1)
uf(np-1) = t
t = vf(np)vf(np) = vf(np-1)
vf(np-1) = t
t = wf(np)wf(np) = wf(np-1)
wf(np-1) = t

ese
.le.
esg

no

yes

"Electron lowest energy"

iq(np) = -1e(np) = ese
e(np-1) = esg

return(to photon)

235

yes

call
ausgab(iarg)

Version
051227-1600

subroutine
counters_out

OPEN(UNIT=99,FILE='egs5job.out99',
STATUS='UNKNOWN')

iannih = 0iaphi=0ibhabha = 0
ibrems = 0
icollis = 0
icompt = 0
iedgbin = 0ieii=0ielectr = 0
ihatch = 0
ihardx = 0
ikauger = 0ikshell = 0
ikxray = 0
ilauger = 0
ilshell = 0
ilxray = 0
imoller = 0
imscat = 0
ipair=0iphoto = 0
iphoton = 0
iraylei = 0
ishower= 0
iuphi = 0
itmxs = 0
noscat = 0
iblock = 0

yesioflag.eq.
0noioflag
.eq.
1

no

yes
write(99,*) 'Values for subroutine-entry counters:'
write(99,*)
write(99,*) 'iannih =',iannih
write(99,*) 'iaphi =',iaphi
write(99,*) 'ibhabha=',ibhabha
write(99,*) 'ibrems=',ibrems
write(99,*) 'icollis =',icollis
write(99,*) 'icompt =',icompt
write(99,*) 'iedgbin=',iedgbin
write(99,*) 'ieii =',ieii
write(99,*) 'ielectr=',ielectr
write(99,*) 'ihatch =',ihatch
write(99,*) 'ihardx =',ihardx
write(99,*) 'ikauger=',ikauger
write(99,*) 'ikshell=',ikshell
write(99,*) 'ikxray =',ikxray
write(99,*) 'ilauger=',ilauger
write(99,*) 'ilshell=',ilshell
write(99,*) 'ilxray =',ilxray
write(99,*) 'imoller=',imoller
write(99,*) 'imscat =',imscat
write(99,*) 'ipair =',ipair
write(99,*) 'iphoto =',iphoto
write(99,*) 'iphoton=',iphoton
write(99,*) 'iraylei=',iraylei
write(99,*) 'ishower=',ishower
write(99,*) 'iuphi =',iuphi
write(99,*) 'itmxs =',itmxs
write(99,*) 'noscat =',noscat
write(99,*) 'iblock =',iblock
CLOSE(UNIT=99)

return236

write(6,*) ' *** Error using
subroutine counters_out'
write(6,*) 'ioflag=',ioflag
CLOSE(UNIT=99)
write(6,*) ' Program stopped'stopVersion
051219-1435

subroutine
edgbiniedgbin = iedgbin +1
yes

medium=1

medium>nmed

medium=medium+1
noner = nepm(medium)

ner
.gt.
20

yes
stop

no
nedgb(medium) = 0
izn=1
izn=izn+1

yes

izn>nernoiz1 = zelem(medium,izn)
yes

ikl=1

ikl>4

ikl=ikl+1

noeee = eedge(ikl,iz1)/1000.0eee.gt.
ap(medium)noyes
nedgb(medium) = nedgb(medium) + 1
eig = log(eee)ledgb(nedgb(medium),medium) =ge1(medium)*eig +ge0(medium)edgb(nedgb(medium),medium) = eee

nedgb(medium)
.gt.
0

no

yes
ii=1
ii=ii+1

yesii>nedgb(medium)nojj=1
jj=jj+1

yesjj>nedgb(medium)
no

1

2

3

4

5

237

6

7

1

2

3

4

no

5

ii
.ne.
jj
yes

no

ledgb(ii,medium).eq.
ledgb(jj,medium)
yeswrite(6,102) medium

return(to hatch)

238

6

7

Version
080425-1100

subroutineeii

ieii = ieii +1
nxray = 0
nauger = 0
capbind = eedge(1,iz)*0.001
ese1 = e(np-1)
ese2 = e(np)esesum = e(np) + e(np-1) - 2.*RM

ese1 - RM .gt. capbind.or.
ese2 - RM .gt. capbind

noratio = 1. - capbind/esesume(np-1) = (ese1 - rm)*ratio + RM
e(np) = (ese2 - rm)*ratio + RM

yes
call
randomset(rnnow)

rnnow
.gt.
0.5

yes

e(np-1) = ese1 - capbinde(np) = ese2noe(np-1) = ese1
e(np) = ese2 - capbindnoe(np-1) .le. RM.or.
e(np) .le. RM
yes
rnnow
.le.
0.5

yes

e(np-1) = ese1 - capbinde(np) = ese2noe(np-1) = ese1
e(np) = ese2 - capbind

yes

yes

feispl.eq.
0.d0

ieispl.eq.
1

noneispl = 1
feispl = 1.

nononeispl.gt.
0neispl .gt.1
no.and.
(np + neispl) .ge. MXSTACK

yes
feispl = 1./float(neispl)

write(6,105)

yes

stop
1

239

2

3

2

1

3

neispl = (2*neispl + 1)/3
feispl = 1./float(neispl)

neispl.eq.
1

yes

ieispl = 0

nono

np + neispl.lt.
MXSTACK
yes
callkshellnxray .ge.1
yes
.and.
exray(1) .gt. eedge(2,iz)*0.001

enew = exray(1)ieie = 1

noenew = 0.0
ieie = 0edep = capbind - enew
etmp = e(np)e(np)= edep
iqtmp = iq(np)iq(np) = -1

iarg=4

iausfl(iarg+1)
ne
0
no

yes

call
ausgab(iarg)

e(np) = etmp
iq(np) = iqtmp

no

ieie
.eq.
1
yes

np = np + 1
e(np) = enew
iq(np) = 0
wt(np-1) = wt(np-1)*feispl

5

4

240

4

5

jnp=1
jnp=jnp+1

jnp>neispl
noiq(np-1+jnp) = iq(np)e(np-1+jnp) = e(np)callrandomset(rnnow)

costhe = 2.*rnnow - 1.
sinthe = sqrt(1. - costhe*costhe)u(np) = 0.
v(np) = 0.
w(np) = 1.

calluphi(2,1)u(np-1+jnp) = u(np)v(np-1+jnp) = v(np)w(np-1+jnp) = w(np)x(np-1+jnp) = x(np-1)
y(np-1+jnp) = y(np-1)z(np-1+jnp) = z(np-1)
ir(np-1+jnp) = ir(np-1)
wt(np-1+jnp) = wt(np-1)
time(np-1+jnp) = time(np-1)
dnear(np-1+jnp) = dnear(np-1)
latch(np-1+jnp) = latch(np-1)
k1step(np-1+jnp) = 0.
k1init(np-1+jnp) = 0.
k1rsd(np-1+jnp) = 0.

wt(np-1) = wt(np-1)/feispl
np = np + neispl - 1

return(to moller)

241

yes

Version
091105-0835

subroutine
electr(ircode)
deresid = 0.d0
deinitial = 0.d0
denstep = 0.d0
hardstep = 0.d0
dok1s0 = 0ielectr = ielectr +1

ircode
.eq.
-1

yes

no

no

ircode = 1
irold = ir(np)irl = ir(np)"New electron loop"

ek1s1(1,1).ne.0.d0.or.
ek1s0(1,1).ne.0.d0
yes1

dok1s0 = 1
nk1i = 0

lelec = iq(np)eie = e(np)medium = med(irl)medium.eq.
0"Tstep loop"

yeststep = vacdst"go to 6"

no2e(np).le.ecut(irl).and.
deresid.eq.0.d0

yes"go to 14"

nohardstep
.eq.
0.d0

no

yes
call
randomset(rnnow)

hardstep = max(-log(rnnow),EPSEMFP)"Ustep loop"3rhof=rhor(irl)/rhom(medium)eke = eie - RM
elke = log(eke)lelke = eke1(medium)*elke + eke0(medium)

1 2 3

4

242

5 6

4

1 2 3
dedx0 = pdedx1(lelke,medium)*elke+ pdedx0(lelke,medium)
scpow0 = pscpw1(lelke,medium)*elke+ pscpw0(lelke,medium)
range0 = prang1(lelke,medium)*elke+ prang0(lelke,medium)
range0 = range0 - pctrng(irl)

5 6

lelec
.eq.
-1

no

yes
dedx0 = ededx1(lelke,medium)*elke+ ededx0(lelke,medium)
scpow0 = escpw1(lelke,medium)*elke+ escpw0(lelke,medium)
range0 = erang1(lelke,medium)*elke+ erang0(lelke,medium)
range0 = range0 - ectrng(irl)dedx = rhof*dedx0
scpow = rhof*scpow0
range = rhof*range0 + ENEPS/dedx

useGSD(medium).eq.
0"get Moliereparameters"

no

yes

ems = e(np)bms = (ems - RM)* (ems + RM)/ems**2
ams = blcc(medium)/bms
gms = rhof*(xcc(medium)/(ems*bms))**2
tmxs = 1.d0 / (log(ams/gms) * gms)

call hardx(lelec,eke,lelke,elke,sig0)

sig = rhof*sig0

yes

sig
.le.
0.

thard = vacdst

nothard = hardstep / sig"set energy hinge distance"
denstep
.eq.
0.d0

no

yes
denstep = deresid
estepe = estep1(lelke,medium)*elke + estep0(lelke,medium)
detot = eke * estepenoestepr(irl)
.ne.
0
yes
detot = detot * estepr(irl)

7 8 9

10

243

11

12 13

7 8 9

10

yes

11

12 13

(e(np)-detot).lt.
ecut(irl)

detot = e(np)-ecut(irl)nocallrandomset(rnnow)

deinitial = rnnow * detot
deresid = detot - deinitialdenstep = denstep + deinitial

yes

dedx
.le.
0.

tinel = vacdst
range = vacdst

notinel = denstep / dedx
4
nok1step(np)
.eq.
0.d0
yes
k1step(np) = k1rsd(np)

lelec
.eq.
-1

no

kinit0 = pkini1(lelke,medium)*elke+ pkini0(lelke,medium)yeskinit0 = ekini1(lelke,medium)*elke+ ekini0(lelke,medium)nok1Lscl(irl)
.ne.
0.d0
yes

kinit0 = kinit0 * (k1Lscl(irl) + k1Hscl(irl) * elke)

dok1s0
.eq.
1
no

yes

no

lelec
.eq.
-1
yes

k1s0 = ek1s1(lelke,medium)*elke+ ek1s0(lelke,medium)14 15 16 17

18

19

20

244

k1s0 = pk1s1(lelke,medium)*elke+ pk1s0(lelke,medium)21 22

14 15 16 17

18

20

19

21 22

k1s0 = k1s0*(2**nk1i)
nk1i = nk1i + 1
nodok1s0=0

kinit0
.gt.
k1s0

yes
kinit0 = k1s0

ktotal = rhof*kinit0

useGSD(medium) no.eq.
0

itmxs = itmxs + 1

yes
false
tmxsetyes

truektotal/scpow
.gt.
tmxs
no

ktotal = tmxs*scpow

omega0 = ams * ktotal/scpownoomega0
.lt.
2.80
yes

ktotal = scpow * 2.80 / ams

call
randomset(rnnow)

k1init(np) = rnnow * ktotal
k1rsd(np) = ktotal - k1init(np)k1step(np) = k1step(np) + k1init(np)yestmscat = vacdst

scpow
.le.
0.
no

tmscat = k1step(np) / scpow
5
tstep = MIN(tmscat,tinel,thard)
6

irnew = ir(np)idisc = 0
ustep0 = tstep
ustep=ustep0
edep=0.d0
medold = medium
23 24 25 26 27 28

29

245

30

23 24 25 26 27 28

29

30

ustep
.gt.
dnear(np)yescallhowfarnorange .lt. dnear(np).and.
e(np) .le. esave(irl).and.
medium .ne. 0

no

yes

"user-range-discard"

lelec
.eq.
-1

no
idisc = 99

yesidisc = 1

idisc
.gt.
0

"imedeate discard: go to 16"

yes

no

"chck negative USTEP"

ustep
.lt.
0.

yes

ustep
.lt.
-1.D-9nono

no
ustep=o.do

ustep
.ne.
0.
yes

no

medium
.ne.
0
yes

no

yes

ierust=ierust+1
write(6,102)
ierust,ustep,ir(np),irnew, irold,x(np),y(np),z(np),sqrt(x(np)**2 +y(np)**2)ierust.gt.
1000

edep = ustep * dedx
write(6,103)
yes
iarg=0e(np) = e(np) - deinitial + denstep

iausfl(iarg+1)
ne
0
no

yes

stop

call
ausgab(iarg)

e(np) = e(np) + deinitial - denstep
x(np) = x(np) + u(np)*ustep
y(np) = y(np) + v(np)*ustep
z(np) = z(np) + w(np)*ustep
time(np) = time(np) +ustep*eie/(2.99792458d10*sqrt((eie-RM)*(eie+RM)))
31 32 33 34 35 36

37

38

246

39

40

31 32 33 34 35 36

37

38

39

40

48

49

dnear(np) = dnear(np) - ustep
irold = ir(np)nomedium.ne.
0
yesthard = thard - ustep
tmscat = tmscat - ustep
tinel = tinel - ustep
hardstep = thard * sig
k1step(np) = tmscat * scpow
denstep = tinel * dedx

ustep.ne.ustep0.or.
ustep.eq.0.D0

noyes
ir(np) = irnew
irl = irnew
medium = med(irl)

ustep
.ne.
0.d0

no

yes
iarg=5e(np) = e(np) - deinitial + denstep

iausfl(iarg+1)
ne
0
no

yes

call
ausgab(iarg)

e(np) = e(np) + deinitial - denstep

yes
tstep = vacdst

medium
.eq.
0
no
irnew
.ne.
irold

no

yes
41 42 43 44 45

46

247

47

41 42 43 44 45

46

47

ecut(irnew).gt.ecut(irold).or.
med(irold).eq.0

48

49

56

57

no

yes
ecsda = eie - deinitial + denstep

no

ecsda
.le.
ecut(irnew)
yese(np) = ecsda
deinitial = 0.d0
denstep = 0.d0
deresid = 0.d0

ecsda - denstep.le.
ecut(irnew)"go to 14"

noyes
eie = ecsda
e(np) = eie
detot = e(np) - ecut(irnew)

callrandomset(rnnow)"go to 3"deinitial = rnnow * detot
deresid = detot - deinitialdenstep = deinitialeie - (deinitial + deresid).le.
noecut(irnew)
yes"go to 3"

deresid = eie - deinitial - ecut(irnew)"go to 5"nomedium.ne.medold.or.
rhor(irnew).ne.rhom(medium)yes"go to 2"

yesdenstep.eq.
0.d0"go to 3"

50 51 52 53 54

55

248

55

50 51 52 53 54

tinel
.eq.
0.
yes

56

57

63

64

no

"energy loss hinge"

de = deinitial + deresid

ekef = eke - deeold = eie
enew = eold - deedep = deiarg=27

iausfl(iarg+1)
ne
0
no

yes

call
ausgab(iarg)

yes

call
ausgab(iarg)

yes"go to 14"

de = edep
eie = eie - dee(np) = eie
iarg=28

iausfl(iarg+1)
ne0
no"go to 3"abs(eie - ecut(irl)).lt.
1.d-14noyes
eie = ecut(irl)e(np) = eie

denstep = deresid
deinitial = 0.d0
deresid = 0.d0

58 59

60 61

no

deresid
.eq.
0.d0

62

249

5859

60 61

62

tmscat
.eq.
0.0
yes

63

64

71

72

no

"multiple scattering hinge"

tmstep = (k1init(np) + k1rsd(np)) / scpow

useGSD(medium)
.eq.
0
yes

no

omega0 = ams*tmstep*rhof

omega0.le.
2.718282

yes
iskpms = 1

noiskpms = 0
blc = log(omega0)
blcold = blc

b = -10.27666 +blc* (17.82596 6.468813*blc)

yes

blc
.lt.
1.306853
noib = b0bgb + blc*b1bgb

yes
write(6,101) ib

ib
.gt.
nbgb

ib = nbgb

nob = bgb0(ib) + blc*(bgb1(ib) + blc*bgb2(ib))

iarg=29

iausfl(iarg+1)
ne
0
no

yes

call
ausgab(iarg)

call
mscat

iarg=30
65 66

67 68

69

250

70

65 66

67 68

69

70

iausfl(iarg+1)
 ne 
0no"go to 4"

yes

71

call
ausgab(iarg)

calluphi(2,1)"go to 5"

thard
.eq.
0.

no

yes

"hard collision"e(np) = e(np) - deinitial + denstep
deresid = 0.d0
deinitial = 0.d0
denstep = 0.d0

call collis(lelec,irl,sig0,go1)

no"go to 1"
true

go1

iq(np).eq.
0

yes

return(to shower)

false"go to 2"

eie = e(np)14noidr = 2eie.gt.
ae(medium)yes
idr = 1edep = e(np) - RM

yes

edep
.ne.
0

iarg = idr

noiausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

15

no

lelec
.eq.
1
yes

73

74

75

251

76

72

73

74

76

75

no

edep
.lt.
e(np)
"positron anihiration
at rest"yes

call
randomset(rnnow)

costhe = rnnow
call
randomset(rnnow)

yes

rnnow
.le.
0.5

costhe = -costhenosinthe = sqrt(1. - costhe**2)
e(np) = RM
iq(np) = 0
u(np) = 0.
v(np) = 0.
w(np) = 1.
calluphi(2,1)"first photon"uf(np) = 0.
;uf(np+1) = 0.
vf(np) = 0.
;vf(np+1) = 0.
wf(np) = 0.
;wf(np+1) = 0.
hardstep = 0. ;denstep = 0.
deinitial = 0. ;deresid = 0.
k1step(np) = 0.:k1init(np) = 0.
k1rsd(np) = 0.
"second photon"
np = np + 1;e(np) = RM
iq(np) = 0;x(np) = x(np-1)
y(np) = y(np-1) ;z(np) = z(np-1)
ir(np) = ir(np-1) ;wt(np) = wt(np-1)
dnear(np) = dnear(np-1)
latch(np) = latch(np-1)
u(np) = -u(np-1);v(np) = -v(np-1)
w(np) = -w(np-1)
hardstep = 0. ;denstep = 0.
deinitial = 0.
;k1step(np) = 0.
k1init(np) = 0. ;k1rsd(np) = 0.
time(np) = time(np-1)

iarg = 14
np = np - 1
ircode = 2
iausfl(iarg+1)
 ne 
0
no
77

78

252

yes

call
ausgab(iarg)

79

77

79

78

return(to shower)

16"Ecut discard"e(np) = e(np) - deinitial + denstep
deinitial = 0.d0
denstep = 0.d0
deresid = 0.d0

idisc = abs(idisc)edep = eie + RM

nolelec .eq. -1.or.
idisc .eq. 99
yesedep = eie - RM

iarg = 3

iausfl(iarg+1)
 ne 
0no"go to 15"

yes

idisc
.eq.
99
no
np = np - 1
ircode = 2

return(to shower)

253

yes

call
ausgab(iarg)

subroutine
hardx(charge,kEnergy,keIndex,keFraction,sig0)

Version090303-1415

ihardx = ihardx +1

charge.lt.
0

no

sig0 = psig1(keIndex,medium)*keFraction +psig0(keIndex,medium)yes
iextp = 0

kEnergyno.lt.
(thmoll(medium)-RM)*1.5
yesmollerThresh = thmoll(medium) - RM
logMollerThresh = log(mollerThresh)mollerIndex = eke1(medium)*logMollerThresh + eke0(medium)

mollerIndex.eq.
keIndexnoyesthmoll(medium)-RM.le.
nokEnergy

iextp = -1

yes
iextp = 1

sig0 = esig1(keIndex+iextp,medium)*keFraction +esig0(keIndex+iextp,medium)

sig0
.le.
0.0

no

yes
sig0=1.e-10

return(to electr)

254

Version
060318-1555

subroutine
hatchihatch = ihatch +1

iblock.ne.
1

yeswrite(6,5000)

stop

no
i1st
.ne.
0

no

yes
i1st=0true.not.
rluxsetwrite(6,*)'RNG ranlux not
initialized: doing
so in HATCH'

callrluxgofalse
latchi = 0.0
nisub = mxsinc_loc - 2
fnsss = nsinss
wid = PI5D2/float(nisub)
wss = wid/(fnsss - 1.0)
zeros(1) = 0.
zeros(2) = PIzeros(3) = TWOPI

isub=1
isub=isub+1

isub >mxsinc_loc

yesnosx = 0.
sy = 0.
sxx = 0.
sxy = 0.
xs0 = wid*float(isub - 2)
xs1 = xs0 + wid
iz = 0izz=1izz >3yes

izz=izz+1
nono

xs0 .le. zeros(izz).and.
zeros(izz) .le. xs1
yes

1

2

3

255

4

5

1

2

5

4

3
iz = izz

yes

iz
.eq.
0

xsi = xs0
no

xsi = zeros(iz)iss=1

iss >nsinss

yes

iss=iss+1
noxs = wid*float(isub - 2) + wss*float(iss - 1) -xsi
ys = sin(xs + xsi)
sx = sx + xs
sy = sy + ys
sxx = sxx + xs*xs
sxy = sxy + xs*ys

yes

iz
.ne.
0

sin1(isub) = sxy/sxx
sin0(isub) = -sin1(isub)*xsi
nodel = fnsss*sxx-sx*sx
sin1(isub) = (fnsss*sxy - sy*sx)/delsin0(isub) = (sy*sxx - sx*sxy)/del - sin1(isub)*xsi

sinc0 = 2.
sinc1 = 1./wid

noistest.ne.
0
yesadev = 0.
rdev = 0.
s2c2mn = 10.
s2c2mx = 0.

isub=1isub >nisub

yesisub=isub+1
noiss=1
iss=iss+1

iss >nsinss

yes

no

6

7

8

9

10

256

11

12

6

7

8

10

9

11

theta = wid*float(isub - 1) + wss*float(iss - 1)
cthet = PI5D2 - theta
sinthe = sin(theta)
costhe = sin(cthet)sint = sin(theta)
cost = cos(theta)
asd = abs(sinthe - sint)
acd = abs(costhe - cost)adev = max(adev,asd,acd)

yes

sint
.ne.
0.

rdev = max(rdev,asd/abs(sint))

no

yes

cost
.ne.
0.

rdev = max(rdev,acd/abs(cost))

nos2c2 = sinthe**2 + costhe**2
s2c2mn = min(s2c2mn,s2c2)
s2c2mx = max(s2c2mx,s2c2)

yes

isub
.lt.
11

write(6,1340) theta,sinthe,sint,
costhe,costnowrite(6,1350) mxsinc_loc,nsinss
write(6,1360) adev,rdev,s2c2mn,s2c2mx
adev = 0.
rdev = 0.
s2c2mn = 10.
s2c2mx = 0.

irn=1irn >nrna

yes

irn=irn+1
nocall
randomset(rnnow)

theta = rnnow*PI5D2
cthet = PI5D2 - theta
sinthe = sin(theta)
costhe = sin(cthet)sint = sin(theta)
cost = cos(theta)
asd = abs(sinthe - sint)
acd = abs(costhe - cost)adev = max(adev,asd,acd)
13

14

15

16

257

17

12

13

14

15

16

17

yes

sint
.ne.
0.

rdev = max(rdev,asd/abs(sint))

no
yes

cost
.ne.
0.

rdev = max(rdev,acd/abs(cost))

nos2c2 = sinthe**2 + costhe**2
s2c2mn = min(s2c2mn,s2c2)
s2c2mx = max(s2c2mx,s2c2)

write(6,1380) nrna
write(6,1390) adev,rdev,s2c2mn,s2c2mx

p = 1.
yes

i=1

i >50

i=i+1

no
pwr2i(i) = p
p = p/2.

j=1

yes

j >nmed

j=j+1

no
i=1

yes

i >nreg

i=i+1

no
no

iraylr(i) .eq. 1.and.
med(i) .eq. j

j=1

yes
iraylm(j) = 1

yesj >nmed

j=j+1

no
i=1
i=i+1

i >nreg

yes

no
no

incohr(i) .eq. 1.and.
med(i).eq.jyesincohm(j) = 1

18

258

18

j=1

yes

j >nmed

j=j+1

no
i=1

yes

i >nreg

i=i+1

no
no

iprofr(i) .eq. 1.and.
med(i) .eq. j

j=1

yes
iprofm(j) = 1

yesj >nmed

j=j+1
no
i=1

yes

i >nreg

i=i+1

no
no

impacr(i).eq.1.and.
med(i) .eq. jyesimpacm(j) = 1

rewind kmpi
nm = 0

im=1

yes

im >nmed

im=im+1
nolok(im) = 0

no

yes

iraylm(im)
.eq.
1

im=1

write(6,1440) im

yes

im >nmed

im=im+1nonoincohm(im)
.eq.
1

yes
write(6,4090) im

19

259

19

im=1

yes

im >nmed

im=im+1
noiprofm(im)
.eq.
1

no

im=1

yes

write(6,4110) im

yes

im >nmed

im=im+1
nonoyes

impacm(im).eq.
1write(6,4130) im

5read(kmpi,1270,end=1470) mbuf

ib=1

yes

ib >lmdl

ib=ib+1nonombuf(ib).ne.
mdlabl(ib)

im=1

yes

yes

im >nmed

im=im+1
noib=1
ib=ib+1

yes

i b>lmdnnoil = lmdl + ibyesmbuf(il).ne.
media(ib,im)

no

ib
.eq.
lmdn

yes

no

yes

lok(im)
.ne.
0
no

20

21

260

21

20

no
lok(im) = 1
nm = nm + 1
write(kmpo,1510) im,mbuf
read(kmpi,2000,err=1000) (mbuf(i),i=1,5),rhom(im),nne(im),iunrst(im),epstfl(im),iaprim(im)"1000"
backspace(kmpi)

read(kmpi,2001) (mbuf(i),i=1,5),rhom(im),nne(im),iunrst(im),epstfl(im),iaprim(im)write(kmpo,1540) (mbuf(i),i=1,5),rhom(im),nne(im)ie=1ie >nne(im)yes

ie=ie+1noread(kmpi,1560) (mbuf(i),i=1,6),(asym(im,ie,i),i=1,2),
zelem(im,ie),wa(im,ie),pz(im,ie),rhoz(im,ie)
write(kmpo,1570) (mbuf(i),i=1,6),(asym(im,ie,i),i=1,2),
zelem(im,ie),wa(im,ie),pz(im,ie),rhoz(im,ie)

write(kmpo,1580)
read(kmpi,1260) rlcm(im),ae(im),ap(im),ue(im),up(im)write(kmpo,1260)rlcm(im),ae(im),ap(im),ue(im),up(im)te(im) = ae(im) - RM
thmoll(im) = te(im)*2. + RM
write(kmpo,4280)
read(kmpi,1250) msge(im),mge(im),mseke(im),meke(im),mleke(im),mcmfp(im),mrange(im),irayl,ibound,incoh,icprof(im),impact
write(kmpo,1250) msge(im),mge(im),mseke(im),meke(im),mleke(im),mcmfp(im),mrange(im),irayl,ibound,incoh,icprof(im),impact

nsge = msge(im)nge = mge(im)nseke = mseke(im)neke = meke(im)nleke = mleke(im)ncmfp = mcmfp(im)nrange = mrange(im)write(kmpo,1600)
read(kmpi,1260) (dl1(i,im),dl2(i,im),dl3(i,im),dl4(i,im),dl5(i,im),dl6(i,im),i=1,6)
write(kmpo,1260) (dl1(i,im),dl2(i,im),dl3(i,im),dl4(i,im),dl5(i,im),dl6(i,im),i=1,6)
22

23

261

23

22

write(kmpo,1610)
read(kmpi,1260) delcm(im),(alphi(i,im),bpar(i,im),delpos(i,im),i=1,2)
write(kmpo,1260) delcm(im),(alphi(i,im),bpar(i,im),delpos(i,im),i=1,2)
write(kmpo,1620)
read(kmpi,1260) xr0(im),teff0(im),blcc(im),xcc(im)write(kmpo,1260) xr0(im),teff0(im),blcc(im),xcc(im)write(kmpo,1630)
read(kmpi,1260) eke0(im),eke1(im)write(kmpo,1260) eke0(im),eke1(im)write(kmpo,1640)
read(kmpi,1260) (esig0(i,im),esig1(i,im),psig0(i,im),psig1(i,im),
ededx0(i,im),ededx1(i,im),pdedx0(i,im),pdedx1(i,im),ebr10(i,im),
ebr11(i,im),pbr10(i,im),pbr11(i,im),pbr20(i,im),pbr21(i,im),
tmxs0(i,im),tmxs1(i,im),escpw0(i,im),escpw1(i,im),pscpw0(i,im),
pscpw1(i,im),ekini0(i,im),ekini1(i,im),pkini0(i,im),pkini1(i,im),erang0(i,im),erang1(i,im),prang0(i,im),prang1(i,im),estep0(i,im),estep1(i,im),i=1,neke)
write(kmpo,1260) (esig0(i,im),esig1(i,im),psig0(i,im),psig1(i,im),
ededx0(i,im),ededx1(i,im),pdedx0(i,im),pdedx1(i,im),ebr10(i,im),
ebr11(i,im),pbr10(i,im),pbr11(i,im),pbr20(i,im),pbr21(i,im),
tmxs0(i,im),tmxs1(i,im),escpw0(i,im),escpw1(i,im),pscpw0(i,im),
pscpw1(i,im),ekini0(i,im),ekini1(i,im),pkini0(i,im),pkini1(i,im),erang0(i,im),erang1(i,im),prang0(i,im),prang1(i,im),estep0(i,im),estep1(i,im), i=1,neke)

write(kmpo,4340)
read(kmpi,1260) tebinda,ge0(im),ge1(im)write(kmpo,1260) tebinda,ge0(im),ge1(im)write(kmpo,1660)
read(kmpi,1260) (gmfp0(i,im),gmfp1(i,im),gbr10(i,im),gbr11(i,im),gbr20(i,im),gbr21(i,im),i=1,nge)
write(kmpo,1260) (gmfp0(i,im),gmfp1(i,im),gbr10(i,im),gbr11(i,im),gbr20(i,im),gbr21(i,im),i=1,nge)iraylm(im).eq.1
yes
.and.
(irayl.lt.1.or.irayl .gt.2)

stop

write(6,4360) im

stop

write(6,4370) im

stop

write(6,4375) imnoyes
incohm(im).eq.1.and.
ibound.ne.1

noincohm(im).eq.1.and.
incoh.ne.1

yeswrite(6,4380) im

no

yes
yes

stop

iprofm(im).eq.
1

icprof(im)
.ne.
3
no

no
write(6,4400) im

incohm(im)
.eq.
0

yes
impacm(im).eq.1.and.
impact.eq.0
no24

write(6,4390) im

stop

write(6,4400) im

stop

yes

no
no

ibound
.eq.
0

25

262

yes

24

25

irayl .eq.1.or.irayl .eq.2.or.
irayl.eq.3

noyes
write(kmpo,1680)
read(kmpi,1250) ngr(im)write(kmpo,1250) ngr(im)ngrim = ngr(im)write(kmpo,1690)
read(kmpi,1260) rco0(im),rco1(im)write(kmpo,1260) rco0(im),rco1(im)write(kmpo,1700)
read(kmpi,1260) (rsct0(i,im),rsct1(i,im),i=1,ngrim)
write(kmpo,1260) (rsct0(i,im),rsct1(i,im),i=1,ngrim)
write(kmpo,1710)
read(kmpi,1260) (cohe0(i,im),cohe1(i,im),i=1,nge)
write(kmpo,1260) (cohe0(i,im),cohe1(i,im),i=1,nge)iraylm(im)
.ne.
1
no

yes

incoh
.eq.
1

no

write(6,1720) im

yes
write(kmpo,4450)
read(kmpi,1250) ngs(im)write(kmpo,1250) ngs(im)ngsim = ngs(im)write(kmpo,4460)
read(kmpi,1260) sco0(im),sco1(im)write(kmpo,1260) sco0(im),sco1(im)write(kmpo,4470)
read(kmpi,1260) (sxz0(i,im),sxz1(i,im),i=1,ngsim)
write(kmpo,1260) (sxz0(i,im),sxz1(i,im),i=1,ngsim)

incohm(im)
.ne.
1
no

yes

icprof(im).eq.1.or.
icprof(im).eq.2

nowrite(6,4480) im

yes
write(kmpo,4490)
read(kmpi,1250) ngc(im)write(kmpo,1250) ngc(im)ngcim = ngc(im)write(kmpo,4500)
read(kmpi,1260) cco0(im),cco1(im),cpimev
write(kmpo,1260) cco0(im),cco1(im),cpimev
write(kmpo,4510)
read(kmpi,1260) (cpr0(i,im),cpr1(i,im),i=1,ngcim)
write(kmpo,1260) (cpr0(i,im),cpr1(i,im),i=1,ngcim)
26

27

263

28

26

27

28

iprofm(im)
.ne.
1
no
icprof(im).eq.3.or.
icprof(im) .eq. 4

yeswrite(6,4520) imnoyes
write(kmpo,4530)
read(kmpi,1250) mxshel(im),ngc(im)write(kmpo,1250) mxshel(im),ngc(im)ngcim = ngc(im)mxsim = mxshel(im)write(kmpo,4540)
read(kmpi,1260) (elecno(i,im),i=1,mxsim)
write(kmpo,1260) (elecno(i,im),i=1,mxsim)
write(kmpo,4550)
read(kmpi,1260) (capio(i,im),i=1,mxsim)
write(kmpo,1260) (capio(i,im),i=1,mxsim)
write(kmpo,4560)
read(kmpi,1260) ccos0(im),ccos1(im)write(kmpo,1260) ccos0(im),ccos1(im)write(kmpo,4570)
read(kmpi,1260) ((cprs0(i,is,im),cprs1(i,is,im),is=1,mxsim),
i=1,ngcim)
write(kmpo,1260)((cprs0(i,is,im),cprs1(i,is,im),is=1,mxsim),
i=1,ngcim)

iprofm(im)
.ne.
1
no

yes

impact.ge.
1

nowrite(6,4580) im

yes
write(kmpo,4590)
read(kmpi,1250) nepm(im)write(kmpo,1250) nepm(im)ner = nepm(im)write(kmpo,4610)
read(kmpi,1250) neii(im)write(kmpo,1250) neii(im)write(kmpo,4620)
read(kmpi,1260) eico0(im),eico1(im)write(kmpo,1260) eico0(im),eico1(im)write(kmpo,4630)
read(kmpi,1260) ((eii0(i,ifun,im),eii1(i,ifun,im),
ifun=1,ner),i=1,neii(im))
write(kmpo,1260) ((eii0(i,ifun,im),eii1(i,ifun,im),
ifun=1,ner),i=1,neii(im))

noimpacm(im)
.ne.
1
no

yes

nm
.ge.
nmed

yes

write(6,4640) im

29

264

29

yes

emaxe
.eq.
0.d0

emaxe=1.d50

j=1j >nmed

yes

j=j+1
no

no

emaxe=min(ue(j),up(j)+RM,emaxe)
yes
stop

emaxe
.lt.
1.d-4

write(6,5007) emaxe

write(6,5006) emaxe

no

j=1

yes

j >nmed

j=j+1

yes

emaxe
.gt.
ue(j)

stop

write(6,5008) emaxe, ue(j), j

no
stop
yes
write(6,5009) emaxe-RM, up(j), j

emaxe-RM
.gt.
up(j)
no

dunitr = dunit

no

dunit
.lt.
0.
yes

id = max0(1,min0(MXMED,int(-dunit)))dunit = rlc(id)

yes

dunit
.ne.
1.

write(6,1730) dunitr,dunit

no
im=1

im >nmedyes

im=im+1
nodfact = rlc(im)/dunit
dfacti = 1.0/dfact
i=1
i=i+1

i - meke(im)
.gt.
0

yes

no
30

31

33

32

265

34

30

31

33

32

esig0(i,im) = esig0(i,im)*dfacti
esig1(i,im) = esig1(i,im)*dfacti
psig0(i,im) = psig0(i,im)*dfacti
psig1(i,im) = psig1(i,im)*dfacti
ededx0(i,im) = ededx0(i,im)*dfacti
ededx1(i,im) = ededx1(i,im)*dfacti
pdedx0(i,im) = pdedx0(i,im)*dfacti
pdedx1(i,im) = pdedx1(i,im)*dfacti
tmxs0(i,im) = tmxs0(i,im)*dfact
tmxs1(i,im) = tmxs1(i,im)*dfact

i=1
i=i+1

i - meke(im)
.gt.
0

yes

no
dunitr
.eq.
1
no

yes
dfactr = 1.d0

dfactr = 1.d0 / dunit
escpw0(i,im) = escpw0(i,im)*dfactr
escpw1(i,im) = escpw1(i,im)*dfactr
pscpw0(i,im) = pscpw0(i,im)*dfactr
pscpw1(i,im) = pscpw1(i,im)*dfactr
ekini0(i,im) = ekini0(i,im)*dfactr
ekini1(i,im) = ekini1(i,im)*dfactr
pkini0(i,im) = pkini0(i,im)*dfactr
pkini1(i,im) = pkini1(i,im)*dfactr
erang0(i,im) = erang0(i,im)*dfact
erang1(i,im) = erang1(i,im)*dfact
prang0(i,im) = prang0(i,im)*dfact
prang1(i,im) = prang1(i,im)*dfact

teff0(im) = teff0(im)*dfact
blcc(im) = blcc(im)*dfacti
xcc(im) = xcc(im)*sqrt(dfacti)
rldu(im) = rlc(im)/dunit
i=1
i=i+1

i - mge(im)
.gt.
0

yes

no
gmfp0(i,im) = gmfp0(i,im)*dfact
gmfp1(i,im) = gmfp1(i,im)*dfact

35

266

34

35
ecutmn = 1.d20
vacdst = vacdst*dunito/dunit
dunito = dunit

jr=1

yes

jr >nreg

jr=jr+1
nomd = med(jr)md.ge.1.and.
md.le.nmednoyesecut(jr) = max(ecut(jr),ae(md))
pcut(jr) = max(pcut(jr),ap(md))
ecutmn = min(ecutmn,ecut(jr))
eke = ecut(jr) - RM
elke = log(eke)lelke = eke1(md)*elke + eke0(md)ectrng(jr) = erang1(lelke,md)*elke + erang0(lelke,md)
pctrng(jr) = prang1(lelke,md)*elke + prang0(lelke,md)rhor(jr)
.eq.
0.0

yes
rhor(jr) = rho(md)noibrdst.eq.1.or.
iprdst .gt. 0

no

yes
im=1

im >nmedyes

im=im+1
nozbrang(im) = 0.
pznorm = 0.

ie=1ie >nne(im)yes

ie=ie+1
nozbrang(im) = zbrang(im) + pz(im,ie)*zelem(im,ie)*(zelem(im,ie) + 1.0)
pznorm = pznorm + pz(im,ie)zbrang(im) = (8.116224E-05)*(zbrang(im)/pznorm)**(1./3.)

ii=1

ii >nregyes

ii=ii+1
no
36

37

38

267

37

36

no

38

yes

iedgfl(ii).ne.0.or.
iauger(ii).ne.0

call
edgbin

yes

nmed
.eq.
1

write(6,1830)nowrite(6,1840) nmed

call
rk1

call
rmsfiti=1i >nmed

yes

i=i+1
no
no emaxe/ue(i).lt.0.1d0.and.
charD(i).eq.0.d0

write(6,5005) (media(j,i),j=1,24)

return(to shower)

1470
write(6,1850) kmpi
im=1

im >nmed

im=im+1nono

lok(im).ne.
1
yes
write(6,1870)(media(i,im),i=1,lmdn)

stop

268

yes

subroutine
kaugerVersion051219-1435

ikauger = ikauger + 1

yes

dfkaug(13,iz)
.eq.
0.
no
nauger = nauger + 1
call
randomset (rnnow)yeskaug=1kaug=kaug+1

kaug>13
no

no

rnnow
.le.
dfkaug(kaug,iz)

kaug
.eq.
1

eauger(nauger) = ekaug(14,iz)*1.E-3
yeseauger(nauger) = ekaug(kaug,iz)*1.E-3"K-L1L1"
yes

calllshell(1)calllshell(1)calllshell(1)calllshell(2)calllshell(1)calllshell(3)calllshell(2)calllshell(2)calllshell(2)calllshaell(3)

no
kaug
.eq.
2

"K-L1L2"
yes

no
kaug
.eq.
3

"K-L1L3"
yes

no
kaug
.eq.
4

"K-L2L2"
yes

no
kaug
.eq.
5

"K-L2L3"

yes

no

2

1

269

1

2

kaug
.eq.
6

"K-L3L3"

yes

calllshell(3)

no
kaug.
eq.
7

"K-L1M"

yes

no
kaug
.eq.
10

call
lshell(1)"KL1N"

yes

no
kaug
.eq.
8

"K-L2M"yescalllshell(2)

no
kaug
.eq.
11

"KL2N"
yes

no
kaug
.eq.
9

"K-L3M"yescalllshell(3)

no
kaug
.eq.
12

"KL3N"
yesnoreturn(to kshell)

270

call
lshell(3)Version
051219-1435

subroutine
kshellikshell = ikshell + 1
call
randomset (rnnow)

rnnow
.gt.
omegak(iz)no"K-auger"yes"K-xray"

call
kxray

return(to photo)

271

call
kaugerVersion
051219-1435

subroutine
kxray

ikxray = ikxray + 1

yesdfkx(9,iz)
.eq.
0.
no
nxray = nxray + 1
call
randomset (rnnow)

yes

ik=1

exray(nxray) = ekx(10,iz)*1.E-3

ik>9

ik=ik+1
no

no

rnnow
.le.
dfkx(ik,iz)

ik
.eq.
1

yes

exray(nxray) = ekx(ik,iz)*1.E-3"K-L3"
yescalllshell(3)

no
ik
.eq.
2

"K-L2"
yescalllshell(2)

no
ik
.eq.
3
no

"K-L1"
yes

return(to kshell)

272

call
lshell(1)Version
051219-1435

subroutine
laugerilauger = ilauger + 1
call
randomset (rnnow)

no

no

ii
.eq.
3

yes

ii
.eq.
2

ii
.eq.
1

no

yes

yes

dfl1aug(5,iz)
.eq.
0.

yes

no
nauger = nauger + 1

laug=1laug>5

laug=laug+1

yeseauger(nauger) = el1aug(6,iz)*1.E-3

no

no

rnnow
.le.
dfl1aug(laug,iz)

yeseauger(nauger) = el1aug(laug,iz)*1.E-3

df21aug(5,iz)
.eq.
0.

yes

no
nauger = nauger + 1

laug=1laug>5

laug=laug+1

no

no

yes

eauger(nauger) = el2aug(6,iz)*1.E-3

rnnow
.le.
dfl2aug(laug,iz)

yeseauger(nauger) = el2aug(laug,iz)*1.E-3

1

2

3

273

1

3

2

dfl3aug(5,iz)
.eq.
0.

yes

no
nauger = nauger + 1

laug=1
laug=laug+1

laug>5

no

no

yes

eauger(nauger) = el3aug(6,iz)*1.E-3

rnnow
.le.
dfl3aug(laug,iz)

yeseauger(nauger) = el3aug(laug,iz)*1.E-3

return(to kshell)

274

Version
051219-1435

subroutine
lshell(ll)ilshell = ilshell + 1
ickflg=0"L2-hole is created"

ll
.eq.
2

yes

no
"L3-hole is created"

ll
.eq.
3

yes

no

"L1-hole is created"callrandomset (rnnow)

ebind = eedge(2,iz)*1.E-3
rnnow
.gt.
omegal1(iz))"L1-X-ray"nocall
lxray(1)yes"Coster-Kronig
rnnow
f12"
.le.
yes
omegal1(iz) + f12(iz)ickflg=1

no"Coster-Kronig
rnnow.
f13"
le.
omegal1(iz) + f12(iz) + f13(iz) yes
noickflg=1"L1-Auger"

calllauger(1)callrandomset (rnnow)

ickflg
.eq.
0
no
rnnow
.gt.
omegal2(iz)yes"L2-X-ray"noebind = eedge(3,iz)*1.E-3

calllxray(2)

yes
1

2

3

275

4

2

1

3

rnnow
"Coster-Kronig
.le.
f23"
omegal2(iz) + f23(iz)yesickflg=1

nocall
lauger(2)ickflg
.eq.
0
no

yes

ebind = eedge(4,iz)*1.E-3

call
randomset (rnnow)

rnnow
.gt.
omegal3(iz)yes"L3-X-ray"no"L3-Auger"

calllauger(3)return(to photo)

276

call
lxray(3)4Version
051219-1435

subroutine
lxray(ll)ilxray = ilxray + 1
call
randomset (rnnow)

ll
.eq.
2

"L2-Xray"yesno"L3-Xray"

ll
.eq.
3

yes

no

"L1-Xray"dflx1(7,iz)
.eq.
0.

yes

no
nxray=nxray+1

lx=1
lx>7

lx=lx+1

yes

exray(nxray) = elx1(8,iz)*1.E-3

no
rnnow
.le.
dflx1(lx,iz)

nodflx2(4,iz).eq.
0.

exray(nxray) = elx1(lx,iz)*1.E-3
yes

yes

no
nxray=nxray+1

lx=1

lx>4

lx=lx+1

exray(nxray) = elx2(5,iz)*1.E-3
yes

no

no

rnnow
.le.
dflx2(lx,iz)

yes

exray(nxray) = elx2(lx,iz)*1.E-3

2

1

277

1

2

dflx3(6,iz)
.eq.
0.

yes

no
nxray=nxray+1

yes

lx=1
lx=lx+1

exray(nxray) = elx3(7,iz)*1.E-3

lx>6
no

no

rnnow
.le.
dflx3(lx,iz)

yes
exray(nxray) = elx3(lx,iz)*1.E-3

return(to lshell)

278subroutinemollerVersion051219-1435

imoller = imoller +1
eie = e(np)ekin = eie - RM
t0 = ekin/RM
e0 = t0 + 1.0
extrae= eie - thmoll(medium)e02 = e0*e0
ep0 = te(medium)/eking2 = t0*t0/e02
g3 = (2.0*t0 + 1.0)/e02
gmax = (1.0 + 1.25*g2)"Rejection sampling loop"

call
randomset (rnnow)

br = te(medium)/(ekin - extrae*rnnow)r = br/(1.0 - br)call
randomset (rnnow)

rejf = 1.0 + g2*br*br + r*(r - g3)
rnnow = gmax*rnnow

yes

rnnow
.gt.
rejf
no
ekse2 = br*ekinese1 = eie - ekse2
ese2 = ekse2 + RM
e(np) = ese1
e(np+1) = ese2

h1 = (eie + RM)/ekindcosth = h1*(ese1 - RM)/(ese1 + RM)
sinthe = sqrt(1.0 - dcosth)
costhe = sqrt(dcosth)calluphi(2,1)
np = np + 1
iq(np)=-1
dcosth = h1*(ese2 - RM)/(ese2 + RM)
sinthe = - sqrt(1.0 - dcosth)
costhe = sqrt(dcosth)calluphi(3,2)
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.
1

279

1

impacr(ir(np)).eq.1.and.
iedgfl(ir(np)).ne.0

no

yes
call
randomset (rnnow)

eke = eie - RM
elke2 = log(eke)lelke2 = eico1(medium)*elke2 + eico0(medium)yes

ifun=1
ifun=ifun+1

ifun>nepm(medium)noeiir = eii1(lelke2,ifun,medium)*elke2 +eii0(lelke2,ifun,medium)

no

rnnow
.lt.
eiir
yes
iarg=25

iausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

iz = zelem(medium,ifun)
call
eii

iarg=26

iausfl(iarg+1)
 ne 
0noreturn(to electrl)

280

yes

call
ausgab(iarg)

Version
091105-0835

subroutine
mscat

imscat = imscat +1
im = medium
nouseGSD(im).ne.0.and.
eke.lt.msgrid(nmsgrd(im),im)
yes
iq(np).eq.
-1

noiprt = 2

yes"find the correct energy interval

iport = 1

delog = DLOG10(eke*1.d6)
demod = MOD(delog,1.d0)
decade = delog - demod
iegrid = nmsdec(im) * (decade - initde(im) + demod) - jskip(im)iegrid.ge.
nmsgrd(im)noyesiegrid = nmsgrd(im) - 1
fject = (eke - msgrid(iegrid,im)) /(msgrid(iegrid+1,im) - msgrid(iegrid,im))callrandomset (xi)

xi
.lt.
fject

yes
iegrid = iegrid +1

noktot = k1rsd(np) + k1init(np)ik1 = DLOG(ktot/k1grd(iprt,1)) / dk1log(iprt) + 1

ik1
.gt.
NK1

yes
ik1 = NK1-1

no
ik1
.le.
0
no
1

2

281

yes
ik1 = 1

1

2
fject = (ktot - k1grd(iprt,ik1)) /(k1grd(iprt,ik1+1) - k1grd(iprt,ik1))
call
randomset (xi)

xi
.lt.
fject

yes
ik1 = ik1 +1

nocall
randomset (xi)iamu = xi * neqp(im) + 1
iamu
.eq.
neqp(im)

no

yes
call
randomset (xi)

xi
yes
.lt.
ecdf(2,iprt,iegrid,ik1,im)

iamu = 1

nocall findi(ecdf(1,iprt,iegrid,ik1,im),xi,neqa(im)+1,iamu)

iamu = iamu + neqp(im) - 1

b1 = ebms(iamu,iprt,iegrid,ik1,im)
eta = eetams(iamu,iprt,iegrid,ik1,im)
x1 = eamu(iamu,iprt,iegrid,ik1,im)
x2 = eamu(iamu+1,iprt,iegrid,ik1,im)
c1 = (x2 + eta) / (x2 - x1)
fmax = 1.d0 + b1 * (x2 - x1)**2"Rejection sampling loop""Sample Wentzel shape part of fit"callrandomset (xi)xmu = ((eta * xi) + (x1 * c1)) / (c1 - xi)
fject = 1.d0 + b1 * (xmu - x1) * (x2 - xmu)
callrandomset (xi)yesxi * fmax
.gt.
fject
no

3

4

282

3

4

costhe = 1.d0 - 2.d0 * xmu
sinthe = DSQRT(1.d0 - costhe * costhe)iskpms = 0

nomsct(ir(np)).eq.1.or.
iskpms.ne.0

yes

no

sinthe = 0.
costhe = 1.
theta = 0.
noscat = noscat + 1
iskpms = 0

xr = sqrt(gms*tmstep*b)"Set bi (B-inverse) that will be
used in sampling (bi must not
be larger than 1/lambda=1/2)"

yes

b
.gt.
2.
no

bi = 1./bbi = 0.5

bmd = 1. + 1.75*bi
bm1 = (1. - 2./b)/bmd
bm2 = (1. + 0.025*bi)/bmd" Loop for Bethe correction
factor (or other) rejection"callrandomset (rms1)"Gaussian, F1"
rms1
.le.
bm1
no
no

yes

call
randomset (rms2)

yes

yes

"Tail, F3"

no

rms2
.eq.
0.

rms1
.le.
bm2

rms2 = 1.E-30
callrandomset (rms3)
thr = sqrt(max(0.D0,-log(rms2)))"Rejection sampling loop"

call
randomset (rms4)

eta = max(rms3,rms4)
i31 = b0g31 + eta*b1g31
g31 = g310(i31) + eta*(g311(i31) + eta*g312(i31))
i32 = b0g32 + eta*b1g32
g32 = g320(i32) + eta*(g321(i32) + eta*g322(i32))
g3 = g31 + g32*bi
call
randomset (rms5)

yes

rms5
.gt.
g3
no

5

6

8

7

283

9

5

6

7

8

thr = 1./eta"Rejection sampling loop""Central correction, F2" callrandomset (rms6)

thr = rms6
i21 = b0g21 + thr*b1g21
g21 = g210(i21) + thr*(g211(i21) + thr*g212(i21))
i22 = b0g22 + thr*b1g22
g22 = g220(i22) + thr*(g221(i22) + thr*g222(i22))
g2 = g21 + g22*bi
call
randomset (rms7)

yes

rms7
.gt.
g2
no
theta = thr*xr

yes

theta
.ge.
PI
no
sinthe = sin(theta)
callrandomset (rms8)

norms8**2*theta.le.
sintheyescthet = PI5D2 - theta
costhe = sin(cthet)return(to electr)

284"Real angle (thr is the
reduced angle)"

9subroutine
pairVersion051219-1435

ipair = ipair +1
eig = e(np)"Approximate:smoothing
connection at boundary"
eig
.le.
2.1

yes

no

call
randomset(rnnow1)
ese2 = RM + rnnow*(eig/2. - RM)

yes
eig
.lt.
50.

lvx = 1
lvl0 = 0nolvx = 2lvl0 = 3

callrandomset(rnnow1)

call
randomset(rnnow)"Uniform distribution""Rejection sampling loop"

rnnow
.ge.
bpar(lvx,medium)nolvl = lvl0 + 3
br = rnnow1*0.5

yes"12(br-0.5)**2
subdistribution"

lvl = lvl0 + 1
call
randomset(rnnow2)

callrandomset(rnnow3)

br = 0.5*(1.0 - max(rnnow1,rnnow2,rnnow3))

yes

eig*br.lt.
RM
nodel = 1.0/(eig*br*(1.0 - br))

yes

del
.ge.
delpos(lvx,medium)
no

1

2

3

285

2

1

3

delta = delcm(medium)*del"Rejection sampling loop"

delta
.lt.
1.0

no

rejf = dl4(lvl,medium) +dl5(lvl,medium)*log(delta +dl6(lvl,medium))yesrejf = dl1(lvl,medium) +delta*(dl2(lvl,medium) +delta*dl3(lvl,medium))call
randomset(rnnow)

yes

rnnow
.gt.
rejf
no
ese2 = br*eig"Set up secondary electron #1(electron #2 has lower energy)"

ese1 = eig - ese2
e(np) = ese1
e(np+1) = ese2
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.
k1step(np+1) = 0.
k1init(np+1) = 0.
k1rsd(np+1) = 0.

"Sample to get polar angles
of secondary electrons"(iprdst.eq.1).or.
((iprdst.eq.2).and.
(eig.lt.4.14))"Sample lowest-orderangular distribution"ichrg=1
ichrg>2yesichrg=ichrg+1
no

ichrg
.eq.
1

no
ese = ese2

yes
ese = ese1

pse = sqrt(max(0.D0,(ese - RM)*(ese + RM)))
callrandomset(rnnow)
costhe = 1.0 - 2.0*rnnow
sinthe = RM*sqrt((1.0 - costhe)*(1.0 + costhe))/(pse*costhe + ese)costhe = (ese*costhe + pse)/(pse*costhe + ese)
4

5

6

286

7

4

6

5

7

no

ichrg
.eq.
1

np = np + 1
sinthe = -sinthe

yes

call
uphi(3,2)

calluphi(2,1)

callrandomset(rnnow)

rnnow
.le.
0.5
yes

no

iq(np) = -1
iq(np-1) = 1

iq(np) = 1
iq(np-1) = -1

no"Sample fromMotz-Olsen-Koch(1969) distribution"(iprdst.eq.2).and.
(eig.ge.4.14)
yesztarg = zbrang(medium)tteig = eig/RM

ichrg=1
ichrg>2yesichrg=ichrg+1
no

no

ichrg
.eq.
1

ese = ese2

yes
ese = ese1
ttese = ese/RM
ttpse = sqrt((ttese - 1.0)*(ttese + 1.0))
esedei = ttese/(tteig - ttese)
eseder = 1.0/esedei
ximin = 1.0/(1.0 + (PI*ttese)**2)
rejmin = 2.0 + 3.0*(esedei + eseder) - 4.00*(esedei +eseder + 1.0 - 4.0*(ximin - 0.5)**2)*(1.0 +0.25*log(((1.0 + eseder)*(1.0 + esedei)/(2.0*tteig))**2 + ztarg*ximin**2))ya = (2.0/tteig)**2
xitry = max(0.01D0,max(ximin,min(0.5D0,sqrt(ya/ztarg))))galpha = 1.0 + 0.25*log(ya + ztarg*xitry**2)
gbeta = 0.5*ztarg*xitry/(ya + ztarg*xitry**2)
galpha = galpha - gbeta*(xitry - 0.5)
ximid = galpha/(3.0*gbeta)
8

9

11

10

287

12

8

9

10

11

12

no

galpha
.ge.
0.0
yes

ximid = 0.5 - ximid - sqrt(ximid**2+0.25)

ximid = 0.5 - ximid + sqrt(ximid**2 + 0.25)

ximid = max(0.01D0,max(ximin,min(0.5D0,ximid)))rejmid = 2.0 + 3.0*(esedei + eseder) - 4.0*(esedei +eseder + 1.0 - 4.0*(ximid - 0.5)**2)*(1.0 +0.25*log(((1.0 + eseder)*(1.0 + esedei)/(2.0*tteig))**2 + ztarg*ximid**2))
rejtop = 1.02*max(rejmin,rejmid)"Rejection sampling loop"

call
randomset(xitest)
rejtst = 2.0 + 3.0*(esedei + eseder) - 4.0*(esedei +eseder + 1.0 - 4.0*(xitst - 0.5)**2)*(1.0 +0.25*log(((1.0 + eseder)*(1.0 + esedei)/(2.0*tteig))**2 + ztarg*xitst**2))
callrandomset(rtest)

theta = sqrt(1.0/xitst - 1.0)/ttese

yes rtest.gt.(rejtst/rejtop).or.
(theta .ge. PI)nosinthe=sin(theta)
costhe=cos(theta)

no

ichrg
.eq.
1
yes

np = np + 1
sinthe = -sinthe

calluphi(3,2)

calluphi(2,1)

callrandomset(rnnow)

rnnow
.le.
0.5
yes

no

iq(np) = -1
iq(np-1) = 1

iq(np) = 1
iq(np-1) = -1

13

14

288

13

14

theta=RM/eig"Polar angle ism/E (default)"calluphi(1,1)

np = np + 1
sinthe = -sinthe

calluphi(3,2)

callrandomset(rnnow)

rnnow
.le.
0.5
yes
iq(np) = 1
iq(np-1) = -1

return(to photon)

289

no

iq(np) = -1
iq(np-1) = 1

Version
080425-1100

subroutine
photo

iphoto = iphoto +1"Calculate energy
dependent sub-shell ratio"

nxray = 0
nauger = 0
irl = ir(np)eig = e(np)phol = log(eig)medium = med(irl)eigk = eig*1000.D0
pholk = log(eigk)
pholk2 = pholk*pholk
pholk3 = pholk2*pholk
total = 0.
"Shell-wise photoelectric calculation"
i=1
i=i+1

i>nne(medium)yesnoiz = zelem(medium,i)

eigk
.le.
eedge(1,iz)

yescrosk(i)=0.0

nocrosk(i) = exp(pm0(1,iz) + pm1(1,iz)*pholk +pm2(1,iz)*pholk2 + pm3(1,iz)*pholk3)

pm0(2,iz).eq.0.
yes.or.
eigk .le. eedge(2,iz)

crosl1(i) = 0.

nocrosl1(i) = exp(pm0(2,iz) + pm1(2,iz)*pholk +pm2(2,iz)*pholk2 + pm3(2,iz)*pholk3)

pm0(3,iz).eq.0.
yes.or.
eigk .le. eedge(3,iz)

crosl2(i) = 0.

nocrosl2(i) = exp(pm0(3,iz) + pm1(3,iz)*pholk +pm2(3,iz)*pholk2 + pm3(3,iz)*pholk3)

pm0(4,iz).eq.0.
yes.or.
eigk .le. eedge(4,iz)

crosl3(i) = 0.

nocrosl3(i) = exp(pm0(4,iz) + pm1(4,iz)*pholk +pm2(4,iz)*pholk2 + pm3(4,iz)*pholk3)

1

2

290

3

3

2

1

eigk
.le.
embind(iz)yestcros(i) = 0.

noyes

pm0(5,iz).eq.
0.

crosm(i) = 0.

nocrosm(i) = exp(pm0(5,iz) + pm1(5,iz)*pholk +pm2(5,iz)*pholk2 + pm3(5,iz)*pholk3)

tcros(i) = crosk(i) + crosl1(i) + crosl2(i) +crosl3(i) + crosm(i)bshk(i) = crosk(i)/tcros(i)bshl1(i) = (crosk(i) + crosl1(i))/tcros(i)bshl2(i) = (crosk(i) + crosl1(i) + crosl2(i))/tcros(i)bshl3(i) = (tcros(i) - crosm(i))/tcros(i)tcros(i) = tcros(i)*pz(medium,i)total = total + tcros(i)

total
.eq.
0.0

"below M-edge"yesiarg=4noiz = zelem(medium,1)
noel = 1nne(medium).eq.
1"element"
yes

iausfl(iarg+1)
 ne 
0nono "compoundor
mixture"i=1
i=i+1

edep = eigi>nne(medium)-1

yes

call
ausgab(iarg)

e(np) = 0.

yes

no

i
.eq.
1

yes

pbran(i) = tcros(i)/totalnopbran(i) = pbran(i-1) + tcros(i)/total

call
randomset (rnnow)

i=1
i=i+1

i>nne(medium)-1

yes

no
4

5

6

7

291

8

4

5

6

8

7

rnnow
.le.
pbran(i)yesiz = zelem(medium,i)noel = inoiz = zelem(medium,nne(medium))
noel = nne(medium)

eigk
.le.
eedge(4,iz)"M, N ..-absoption"yesebind = embind(iz)*1.D-3
edep = ebind"Sample to decide shell"

no

call
randomset (rnnow)

rnnow
.gt.
bshl3(noel)"M, N ..-absoption"yesebind = embind(iz)*1.D-3
edep = ebindno"K or L photoelectric""K
rnnow
photoelectric".le.
bshk(noel)yesebind = eedge(1,iz)*1.D-3

no
rnnow
.le.
bshl1(noel)callksahell"L1
photoelectric"
yes

call
lshell(1)

no
rnnow
.le.
bshl2(noel)"L2
photoelectric"
yes

call
lshell(2)"L3photoelectric"

no

call
lshell(3)iedgfl(irl)
.le.
0
no

iauger(irl)
.le.
0
no

yes

yes

nxray=0

nauger=0

edep = ebind
10

9

292

11

10

9

nxray
.ge.
1

11

no

yes
iphot=1
iphot=iphot+1 iphot>nxraynoyesedep = edep - exray(iphot)nonauger
.ge.
1
yes

yes

ielec=1
ielec=ielec+1

ielec>naugernoedep = edep - eauger(ielec)

edep
.lt.
0.

yes
edep=0.0noe(np)=edep
iarg=4

iausfl(iarg+1)
 ne 
0noyes

iq(np) = -1e(np) = eig - ebind + RM

no

call
ausgab(iarg)"Photoelectron (always set up)

iphter(ir(np)).eq.
1"Select photoelectron direction"yeseelec = e(np)

no

eelec
.gt.
ecut(ir(np))
yes

12

13

14

293

12

14

13

beta = sqrt((eelec - RM)*(eelec + RM))/eelec
gamma = eelec/RM
alpha = 0.5D0*gamma - 0.5D0 + 1.D0/gamma
ratio = beta/alphacallrandomset (rnnow)"Rejection sampling loop"

rnpht = 2.D0*rnnow - 1.D0

ratio
.le.
0.2D0

no

yes

xi = gamma*gamma*(1.D0 + alpha*(sqrt(1.D0 +ratio*(2.D0*rnpht + ratio))- 1.D0))
costhe = (1.D0 - 1.D0/xi)/betafkappa = rnpht + 0.5D0*ratio*(1.D0 - rnpht)*(1.D0 + rnpht)
costhe = (beta + fkappa)/(1.D0 + beta*fkappa)
xi = 1.D0/(1.D0 - beta*costhe)sinth2 = max(0.D0,(1.D0 - costhe)*(1.D0 + costhe))callrandomset (rnnow)

rnnow
no
.le.
0.5D0*(1.D0 + gamma)*sinth2*xi/gamma
yessinthe = sqrt(sinth2)
calluphi(2,1)

nonauger.ne.
0
yes"Set up Auger electrons"
ielec=1
ielec>nauger
ielec=ielec+1

yes

nonp = np + 1
e(np) = eauger(ielec) + RM
iq(np) = -1
call
randomset (rnnow)

15

16

17

294

18

19

15

17

16
costhe = 2.D0*rnnow - 1.D0
sinthe = sqrt(1.D0 -costhe*costhe)u(np) = 0.
v(np) = 0.
w(np) = 1.D0
calluphi(2,1)x(np) = x(np-1)
y(np) = y(np-1)z(np) = z(np-1)
ir(np) = ir(np-1)
wt(np) = wt(np-1)
time(np) = time(np-1)
dnear(np) = dnear(np-1)
latch(np) = latch(np-1)
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.

no

nxray
.ne.
0
"Set up fluorescent photons"
iphot=1
iphoto=iphot+1

yes
iphot>nxray

yesnonp = np + 1
e(np) = exray(iphot)
iq(np) = 0
call
randomset (rnnow)

costhe = 2.D0*rnnow - 1.D0
sinthe = sqrt(1.D0 - costhe*costhe)u(np) = 0.
v(np) = 0.
w(np) = 1.D0
calluphi(2,1)x(np) = x(np-1)
y(np) = y(np-1)z(np) = z(np-1)
ir(np) = ir(np-1)
wt(np) = wt(np-1)
time(np) = time(np-1)
dnear(np) = dnear(np-1)
latch(np) = latch(np-1)
k1step(np) = 0.
k1init(np) = 0.
k1rsd(np) = 0.

295

return(to photon)

18

19subroutine
photonVersion091105-0835

iphoton = iphoton +1
ircode = 1
eig = e(np)irl = ir(np)medium = med(irl)

eig
.le.
pcut(irl)

yesnogle = log(eig)callrandomset (rnnow)

rnnow
.eq.
0.

yes
rnnow = 1.E-30

no"NEW-ENERGY loop"

dpmfp = -log(rnnow)

cexptr.ne.
0.
"exponential
transform"

no

yes

w(np)
.gt.
0.

no

yes
temp = cexptr*w(np)bexptr = 1./(1. - temp)
dpmfp = dpmfp*bexptr
wt(np) = wt(np)*bexptr*exp(-dpmfp*temp)

irold = ir(np)"NEW-MEDIUM loop"

medium
.ne.
0

1

no

yes
lgle = ge1(medium)*gle + ge0(medium)iextp=0

2

3

296

4

5

2

1

4

3

eig
.lt.
0.15

5

no

yes
iij=1
iij=iij+1

iij>nedgb(medium)yesnoledgb(iij,medium)
.eq.
lgle

no

yes
edgb(iij,medium) no
.le.
eig

iextp = -1

yes
iextp = 1

"NEW-MEDIUM loop""NEW-ENERGY loop"

gmfpr0 = gmfp1(lgle+iextp,medium)*gle +gmfp0(lgle+iextp,medium)medium.eq.
0

yes

tstep = vacdst

no"PHOTON-TRANSPORT loop"

rhof = rhor(irl)/rho(medium)
gmfp = gmfpr0/rhof

iraylr(irl)
.eq.
1

no

yes

"Apply Rayleigh correction"cohfac = cohe1(lgle+iextp,medium)*gle +cohe0(lgle+iextp,medium)
gmfp = gmfp*cohfac

tstep = gmfp*dpmfp

irnew = ir(np)idisc = 0
ustep = tstep

ustep
.gt.
dnear(np)

yes

call
howfar

no
6

7

8

9

297

10

6

7

8

9

10

idisc
.gt.
0

yes

no
edep = 0.

iarg = 0
iausfl(iarg+1)
.ne.
0

yes

call
ausgab(iarg)"PHOTON-TRANSPORT loop""NEW-MEDIUM loop""NEW-ENERGY loop"nox(np) = x(np) + u(np)*ustep
y(np) = y(np) + v(np)*ustep
z(np) = z(np) + w(np)*ustep
time(np) = time(np) +ustep/2.99792458d10
dnear(np) = dnear(np) - ustep

medium
.ne.
0

no

yes
dpmfp = max(0.D0,dpmfp - ustep/gmfp)

irold = ir(np)medold = medium

irnew
.ne.
irold

no

yes
ir(np) = irnew
irl = irnew
medium = med(irl)iarg=5

iausfl(iarg+1)
 ne 
0
no

yes

eig
.le.
pcut(irl)

yes

call
ausgab(iarg)

no
idisc
.lt.
0

yes
edep = eig

noiarg=3
11

12

13

14

298

15

16

11

12

14

13

15

medium.ne.
medoldyes

iausfl(iarg+1)
 ne 
0
no

no
yes

16
yes

call
ausgab(iarg)

ircode = 2
np = np - 1

medium .eq. 0.or.
dpmfp .gt. EPSGMFP
noiraylr(irl)
.eq.
1

no

yes
call
randomset (rnnow)

rnnow
.le.
(1.0 - cohfac)

no

yes

"Rayleigh scattering"

iausfl(iarg+1)
 ne 
0no"NEW-MEDIUM loop""NEW-ENERGY loop"

iarg=23yes

call
ausgab(iarg)

yes

call
ausgab(iarg)

calllayleiiarg=24

iausfl(iarg+1)
 ne 
0nocall
randomset (rnnow)

gbr1 = gbr11(lgle+iextp,medium)*gle +gbr10(lgle+iextp,medium)nornnow.le.gbr1.and.
e(np).gt.RMT2
yes"Pair production"

iarg=15

17

18

19

299

20

21

17

18

19

iausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

yes

call
ausgab(iarg)

20

21

25

26

call
pair

iarg=16

iausfl(iarg+1)
 ne 
0no"NEW-ENERGY loop"

gbr2 = gbr21(lgle+iextp,medium)*gle +gbr20(lgle+iextp,medium)

no

rnnow
.lt.
gbr2
"Compton scattering"yesiarg=17

iausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

call
compt

iq(np).eq.0
no.and.
e(np-1).lt.ecut(ir(np-1))
yes
iq(np) = iq(np-1)
iq(np-1) = 0
t = e(np); e(np) = e(np-1)
e(np-1) = t;t = u(np);u(np) = u(np-1)
u(np-1) = t
t = v(np);v(np) = v(np-1)
v(np-1) = t
t = w(np);w(np) = w(np-1)
w(np-1) = t
t = uf(np);uf(np) = uf(np-1)
uf(np-1) = t
t = vf(np);vf(np) = vf(np-1)
vf(np-1) = t
t = wf(np);wf(np) = wf(np-1)
wf(np-1) = t

iarg=18
22

23

24

300

22

23

25

24

iausfl(iarg+1)
 ne 
0nono

iq(np)
.ne.
0

yes

26

call
ausgab(iarg)

yes"Photo-electric"
iarg=19"NEW-ENERGY loop"

iausfl(iarg+1)
 ne 
0
no

yes

call
ausgab(iarg)

call
photo

np
.eq.
0

yes
ircode = 2

noiarg=20

iausfl(iarg+1)
 ne 
0noyes

iq(np)
.eq.
-1

call
ausgab(iarg)

yesnoeig = e(np)

no

eig
.lt.
pcut(irl)
yes
eig
.gt.
ap(medium)noidr = 2yes
idr = 1edep = eig
iarg=idr
27

301

28

27

iausfl(iarg+1)
 ne 
0
no

28

yes

call
ausgab(iarg)

ircode = 2
np = np - 1

return(to shower)302

Version
051219-1435

subroutine
raylei

iraylei = iraylei +1

call
randomset (rnnow)

lxxx = rco1(medium)*rnnow + rco0(medium)x2 = rsct1(lxxx,medium)*rnnow + rsct0(lxxx,medium)q2 = x2*RMSQ/(20.60744*20.60744)
costhe = 1.-q2/(2.*e(np)*e(np))

yesabs(costhe)abs(costhe)
.gt.
.gt.
1.1.
no

csqthe = costhe*costhe
rejf = (1. + csqthe)/2.
call
randomset (rnnow)

no

rnnow
.le.
rejf
yes

sinthe = sqrt(1. - csqthe)

lpolar(ir(np))
.eq.
0
no
br = 1.
callaphi(br)call
uphi(3,1)

return(to photon)

303

yes

call
uphi(2,1)

Version
060313-0945

subroutine
rk1

open(17,file='data/k1.dat',',STATUS='old')read(17,*) nmatk1

yes

i=1
i=i+1

i>nmatk1
noread(17,*) z, z2k1w(i), watot, rhok1(i)z2k1w(i) = z2k1w(i) / watot
read(17,*) nek1(i)yesj=nek1(i)
j=j-1

j<1
no

read(17,*) ek0k1(j,i), slopek1(j,i), bk1(j,i),dlowk1(j,i),k1low(j,i),dhighk1(j,i),k1high(j,i)

bk1(j,i) = bk1(j,i) - slopek1(j,i) * DLOG(rhok1(i))
dhighk1(j,i) = dhighk1(j,i) * rhok1(i)dlowk1(j,i) = dlowk1(j,i) * rhok1(i)close(17)im=1im>nmed
im=im+1

yes

no

yes

charD(im)
.eq.
0.d0
no

lcharD = dlog(charD(im)*rhom(im))
watot = 0.d0
z2w = 0.d0

1

2

304

3

1

2

3

yes

ie=1
ie>nne(im)ie=ie+1
noz2w = z2w + pz(im,ie) * zelem(im,ie) * (zelem(im,ie) + 1.d0)
watot = watot + pz(im,ie) * wa(im,ie)

z2w = z2w / watot
z2w.gt.
z2k1w(nmatk1)

yes

nz2 = 1iz2 = nmatk1
zfrac = 0.d0

yes

nz2 = 1iz2 = 1
zfrac = 0.d0

no
z2w
.lt.
z2k1w(1)nonz2 = 2call findi(z2k1w,z2w,nmatk1,iz2)

zfrac = (z2w - z2k1w(iz2)) / (z2k1w(iz2+1) - z2k1w(iz2))

yes

n=1
n>nz2
n=n+1
no

idx = iz2 + n - 1
eke = ek0k1(nek1(idx),idx)ue(im)-RM
.gt.
eke

no
scpeMax(n) = 0.0

yeselke = log(eke)j = eke0(im) + elke * eke1(im)scpeMax(n) = escpw1(j,im)*elke + escpw0(j,im)neke = meke(im)

4

5

305

6

4

5

j=1
j=j+1

6

yes
j>neke-1
no
j
.eq.
1

no

j
.eq.
neke-1

yes

no

yes
eke = ue(im) - RM
elke = log(eke)eke = ae(im) - RM
elke = log(eke)elke = (j + 1 - eke0(im)) / eke1(im)eke = dexp(elke)scpe = escpw1(j,im)*elke + escpw0(j,im)
scpp = pscpw1(j,im)*elke + pscpw0(j,im)
scprat = scpp/scpe

n=1
n=n+1

yes
n>nz2
no

extrape = 1.d0
idx = iz2 + n - 1

no
eke
.gt.
ek0k1(nek1(idx),idx)

eke
.lt.
ek0k1(1,idx)

yes

no

yes
niek = 1
frace = 0.d0
iek1 = 1

niek = 1
iek1 = nek1(idx)frace = 0.d0
extrape = scpe/scpeMax(n)niek = 2

call findi(ek0k1(1,idx),eke,nek1(idx),iek1)

frace = (eke - ek0k1(iek1,idx)) /(ek0k1(iek1+1,idx) - ek0k1(iek1,idx))

m=1
m=m+1

yes
m>nieknoiedx = iek1 + m - 1

charD(im)*rhom(im).gt.
dhighk1(iedx,idx) no

charD(im)*rhom(im).lt.
dlowk1(iedx,idx) no

yes
7

8

9

10

11

306

yes
12

13 14

15 16 17

7

8

9

10

13 14

12

11
k1ez(m) = k1high(iedx,idx)

15 16 17k1ez(m) = k1low(iedx,idx)

k1ez(m) = DEXP(slopek1(iedx,idx)*
lcharD + bk1(iedx,idx))

k1sez(m) = k1low(iedx,idx)

k1z(n) = k1ez(1) + frace * (k1ez(2) - k1ez(1))
k1z(n) = extrape * k1z(n)k1sz(n) = k1sez(1) + frace * (k1sez(2) - k1sez(1))
k1sz(n) = extrape * k1sz(n)k1new = k1z(1) + zfrac * (k1z(2) - k1z(1))
k1s = k1sz(1) + zfrac * (k1sz(2) - k1sz(1))
k1s = k1s

k1new
.gt.
k1maxe

no

k1new
.lt.
k1mineyes

yes

k1maxe = k1new

k1mine = k1new

k1new*scprat.gt.
k1maxpk1new*scprat.lt.
k1minp

no

yes

no

yes

k1maxp = k1new*scprat

k1s
.lt.
k1minenok1minp = k1new*scpratnoyes
k1mine = k1s

k1s*scprat.lt.
k1minpnoyes
k1minp = k1s*scprat

j
.gt.
1

no

yes
delke = elke - elkeoldekini1(j,im) = (k1new - k1old) / delke
ekini0(j,im) = (k1old * elke - k1new * elkeold) / delke
pkini1(j,im) = ekini1(j,im) * scprat
pkini0(j,im) = ekini0(j,im) * scprat
ek1s1(j,im) = (k1s - k1sold) / delke
ek1s0(j,im) = (k1sold * elke - k1s * elkeold) / delke
pk1s1(j,im) = ek1s1(j,im) * scprat
pk1s0(j,im) = ek1s0(j,im) * scprat
18

19

20

307

21

22 23

18

19

20

21

elkeold = elke
k1old = k1new
k1sold = k1s

ekini1(1,im) = ekini1(2,im)
ekini0(1,im) = ekini0(2,im)
pkini1(1,im) = pkini1(2,im)
pkini0(1,im) = pkini0(2,im)
ekini1(neke,im) = ekini1(neke-1,im)
ekini0(neke,im) = ekini0(neke-1,im)
pkini1(neke,im) = pkini1(neke-1,im)
pkini0(neke,im) = pkini0(neke-1,im)
ek1s1(1,im) = ek1s1(2,im)
ek1s0(1,im) = ek1s0(2,im)
pk1s1(1,im) = pk1s1(2,im)
pk1s0(1,im) = pk1s0(2,im)
ek1s1(neke,im) = ek1s1(neke-1,im)
ek1s0(neke,im) = ek1s0(neke-1,im)
pk1s1(neke,im) = pk1s1(neke-1,im)
pk1s0(neke,im) = pk1s0(neke-1,im)
10

j=1
j=j+1

yes
j>nregnok1Lscl(j).gt.0.d0.and.
k1Hscl(j).gt.0.d0

nok1Lscl(j)
.ne.
0.d0

yes
im = med(j)c2 = (k1Lscl(j) - k1Hscl(j)) /dlog( (ae(im) - RM)/(ue(im) - RM) )c1 = k1Hscl(j) - dlog(ue(im) - RM) * c2
k1Lscl(j) = c1
k1Hscl(j) = c2

return(to hatch)308

k1Lscl(j) = 0.d0

22 23

subroutine
rmsfitVersion060314-0855

openMS = .false.
useGS = .false.
yes

k=1
k=k+1

k>nmednocharD(k)
.eq.
0.d0

yes
openMS = .true.

no
yes
useGSD(k)
.eq.
0
no

nmsgrd(k) = 1
msgrid(1,k) = 0.d0

useGS = .true.

.not.
(useGS.or.
openMS)

true

false
false

useGS
true

k=1
k=k+1

yes
k>nmednonouseGSD(k)
.eq.
0.d0
yes
write(6,1001) k
useGSD(k) = 1

GSFile = .false.
doGS=.false.
open(17,file='gsdist.dat',STATUS='old,ERR=14')
GSFile = .true.
13

close(17)open(17,file='pgs5job.msfit',STATUS='old')
1

2

309

return12"end of file; go to 13"
read(17,*,end=13) nfmeds

j=1
j=j+1

yes
j>nfmedsnoread(17,'(72a1)',end=1000) bufferread(17,*) hasGS, charD0, efrch0, efrcl0, ue0, ae0

k=1
k=k+1

yes
k>nmednoib=1ib>lmdnib=ib+1
noil = lmdl + ib
yesbuffer(il).ne.
media(ib,k)

no

no
ib
.eq.
lmdn

no

yes
useGSD(k)
useGS
.and.
.eq.
hasGS.eq.0
0.d0

yes
write(6,1002) k
doGS = .true.

nouseGSD(k)charD(k)
.eq.
0.d0

no

yes
useGSD(k)charD0
.ne.
.eq.
0.d0

no

write(6,1005) k, efrch0, efrcl0

yes
false
useGS

write(6,1004) k

true

stop

write(6,1003) k, charD0

useGS
true
.and.
(ue0.lt.ue(j) .or. ae0.gt.ae(j))

write(6,1006) k, ae0, ue0, ae(j), ue(j)doGS = .true.

false

3

310

3

true

returnclose(17).not.useGS
false
yes

j=1
j=j+1

j>nregnok = med(j)
k
.ne.
0

no

yes
no

rhom(k).ne.
rhor(j)yes
stopwrite(6,1007) j,k

j=1
j=j+1

yes
j>nregno(k1Hscl(j) + k1Lscl(j))
.gt.
no
0.d0
yes
k1Hscl(j) = 0.d0
k1Lscl(j) = 0.d0
write(6,1008) j

false

GSFile.and.
.not.doGS
trueread(17,'(72a1)',end=1000) bufferread(17,'(72a1)',end=1000) buffer

yes

i=1
i=i+1

i>NK1
no

read(17,*) idummy, k1grd(1,i), k1grd(2,1)

5

4

311

5

4

"go to 30"
noabs(k1mine-k1grd(1,1))/k1mine.gt.1.d-6.or.
abs(k1minp-k1grd(2,1))/k1minp.gt.1.d-6.or.
abs(k1maxe-k1grd(1,NK1))/k1maxe.gt.1.d-6.or.
abs(k1maxp-k1grd(2,NK1))/k1maxp.gt.1.d-6
yeswrite(6,1009)
doGS = .true.

false
doGS
true
close(17)open(UNIT=17,FILE='gsdist.dat',STATUS='unknown')callelastino
close(17)open(UNIT=17,FILE='gsdist.dat',STATUS='old')

j=1
j=j+1

yes
j>nfmeds*2+1
noread(17,'(72a1)',end=1000) bufferread(17,'(72a1)') bufferread(17,'(72a1)') buffer

yes

i=1

i>NK1

i=i+1
no
read(17,*) idummy, k1grd(1,i), k1grd(2,i)
30
dk1log(1) = dlog(k1grd(1,2)/k1grd(1,1))
dk1log(2) = dlog(k1grd(2,2)/k1grd(2,1))
nm = 0
6

312

6

yes

k=1
k=k+1

k>nmednolok(k) = 0
5read(17,'(72a1)',end=1000) buffer

yes

ib=1
ib=ib+1

ib>lmdlnonobuffer(ib).ne.
mdlabl(ib)
yes

k=1
k=k+1

k>nmed

yes

no
yes

ib=1
ib=ib+1

ib>lmdnnoil = lmdl + ib

buffer(il).ne.
media(ib,k)
no
no

ib
.eq.
lmdn
yes

yes

lok(k).ne.
0
nolok(k) = 1
nm = nm + 1

8

7

313

yes

8

7

read(17,'(72a1)') bufferread(17,'(72a1)') bufferread(17,*) nmsgrd
read(17,'(72a1)') bufferread(17,*) initde
read(17,'(72a1)') bufferread(17,*) nmsdec
read(17,'(72a1)') bufferread(17,*) jskip
read(17,'(72a1)') bufferread(17,*) neqp
read(17,'(72a1)') bufferread(17,*) neqa

jskip = jskip - 1

iprt=1
iprt=iprt+1

iprt>2yesnoread(17,'(72a1)') buffer

i=1
i=i+1

i>nmsgrd(k)yesnoread(17,*) msgrid(i,k)

yes

ik1=1

ik1>NK1

ik1=ik1+1
noread(17,'(72a1)') bufferread(17,*) pnoscat(iprt,i,ik1,k)read(17,'(72a1)') buffereamu(1,iprt,i,ik1,k) = 0.0

j=1

yes

j>neqp(k)+neqa(k)-1

j=j+1
no
read(17,*) dummy, eamu(j+1,iprt,i,ik1,k),ebms(j,iprt,i,ik1,k),eetams(j,iprt,i,ik1,k)ebms(iprt,i,ik1,j,k) = ebms(iprt,i,ik1,j,k) * .25d0

read(17,'(72a1)') buffer

j=1

j>neqa(k) + 1

yes

j=j+1
no
9

10

11

12

13

14

314

15

16

17

19

9

10

11

12

13

15

14

16

read(17,*) ecdf(j,iprt,i,ik1,k)ecdf(neqa(k)+1,iprt,i,ik1,k) = 1.d0

nm
.ge.
nmed

no

yes
no

nmed
.eq.
1

yes
write(6,5001)

write(6,5002) nmed

close(17)return1000
write(6,5003)
write(6,5004)

yes

k=1
k=k+1

k>nmednonolok(k).ne.
1
yes

write(6,'(24a1)') (media(i,k),i=1,lmdn)

close(17)

stop

315

17

19

subroutine shower(iqi,ei,xi,yi,zi,ui,vi,wi,iri,wti)

Version080425-1100

ishower = ishowe +1
np = 1;dneari = 0.0
iq(1) = iqi;e(1) = eiu(1) = ui;v(1) = vi
w(1) = wi;x(1) = xiy(1) = yi;z(1) = zi
ir(1) = iri;wt(1) = wti
dnear(1) = dnearilatch(1) = latchi
deresid = 0.d0;deinitial = 0.d0
denstep = 0.d0;k1step(1) = 0.d0
k1init(1) = 0.d0;k1rsd(1) = 0.d0
time(1) = 0.d0

iqi
.eq.
2

no

yes

"π0-decay"

yes

ei**2
.le.
PI0MSQ

write(6,5002) nmed

nocall
randomset (rnnow)

dcsth = rnnow
dei = eidpi = dsqrt(dei*dei - PI0MSQ)
deg = dei + dpi*dcsth
dpgl = dpi + dei*dcsth
dcosth = dpgl/deg
costhe = dcosth
sinthe = dsqrt(1.d0 - dcosth*dcosth)
iq(1) = 0e(1) = deg/2.

calluphi(2,1)

np = 2
deg = dei - dpi*dcsth
dpgl = dpi - dei*dcsth
dcosth = dpgl/deg
costhe = dcosth
sinthe = -dsqrt(1.d0 - dcosth*dcosth)
iq(2) = 0e(2) = deg/2.

1

2

316

stop

1

2
call
uphi(3,2)

ircode = -1"e+,e- or photon"

iq(np)
.eq.
0

yes

no
call
electr(ircode)

ircode
.eq.
2
no
call
photon(ircode)noircode
.eq.
2
yes

yes

np
.gt.
0

return317

yessubroutineuphi(ientry,lvl)

Version080425-1100

iuphi = iuphi + 1
iarg=21

iausfl(iarg+1)
 ne 
0noyes
sinthe = sin(theta)
cthet = PI5D2 - theta
costhe = sin(cthet)

yes

call
ausgab(iarg)

ientry
.eq.
1
no

call
randomset (rnnow)

yesientry.eq.
2nophi = rnnow*TWOPI
sinphi = sin(phi)cphi = PI5D2 - phi
cosphi = sin(cphi)

ientry
.eq.
3

no

yes
write(6,100) ientry,lvl
usav = u(np)vsav = v(np)wsav = w(np)

yes

lvl
.eq.
1

stop

no
usav = u(np-1)
vsav = v(np-1)
wsav = w(np-1)

yesx(np) = x(np-1)
y(np) = y(np-1)
yesz(np) = z(np-1)
ir(np) = ir(np-1)
wt(np) = wt(np-1)
dnear(np) =dnear(np-1)
latch(np) = latch(np-1)
time(np) = time(np-1)
k1step(np) = 0.d0
k1init(np) = 0.d0
k1rsd(np) = 0.d0

lvl
.eq.
2
no
lvl
.eq.
3

no

sinps2 = usav*usav + vsav*vsav

sinpsi = sqrt(sinps2)
us = sinthe*cosphi
vs = sinthe*sinphi
sindel = vsav/sinpsi
cosdel = usav/sinpsi
u(np) = wsav*cosdel*us - sindel*vs + usav*costhev(np) = wsav*sindel*us + cosdel*vs + vsav*costhew(np) = -sinpsi*us + wsav*costhenosinps2.lt.
1.e-20

1

318

yes

u(np) = sinthe*cosphi
v(np) = sinthe*sinphi
w(np) = wsav*costhe

1
iarg=22

iausfl(iarg+1)
ne
0
no


yes



return319

call
ausgab(iarg)

subroutine
randomset(rndum)

Version051219-1435

uni = seeds(j24) - seeds(i24) - carryuni = uni + 1.0
carry = twom24

uni
.lt.
0.

yes

no
carry = 0.

seeds(i24) = uni
i24 = next(i24)j24 = next(j24)rndum = uni

no

uni
.lt.
twom12yesrndum = rndum + twom24*seeds(j24)

rndum
.eq.
0.

yes
rndum = twom48

noin24 = in24 + 1

no

in24
.eq.
24
yes

in24 = 0
kount = kount + nskip

isk=1
isk=isk+1

yesisk>nskipnouni = seeds(j24) - seeds(i24) - carry

uni
.lt.
0.

yes
uni = uni + 1.0
carry = twom24
nocarry = 0.

seeds(i24) = uni
i24 = next(i24)j24 = next(j24)

1

320

1
kount = kount + 1

no

kount
.ge.
igiga
yes
mkount = mkount + 1
kount = kount - igiga

return321

Version
051219-1435

subroutine
rluxinit

tisdext = 0

yes

i=1
i=i+1

i>25
no

tisdext = tisdext + isdext(i)tisdext
.ne.
0

yes

no
call
rluxgoreturn322

call
rluxinsubroutine
rluxgoVersion051219-1435

luxlev.le 0.or.
luxlev.gt.maxlev

yeswrite (6,'(a,i7)')' illegal ranlux level in rluxgo: ',luxlev

no
stop

nskip = ndskip(luxlev)

write(6,'(a,i2,a,i4)') ' ranlux luxury level set by rluxgo :',luxlev,' p=', nskip+24

in24 = 0
yes

inseed.lt.
0write (6,'(a)') ' Illegal initialization in
rluxgo, negative input seed'nojseed = inseed

stop

yes

inseed.gt.
0write(6,'(a,i12)') ' ranlux initialized byrluxgo from seed', jseed

nojseed = jsdflt
write(6,'(a)')' ranlux initialized byrluxgo from default seed'

inseed = jseed
twom24 = 1.

yes

i=1
i=i+1

i>24
no

twom24 = twom24 * 0.5
k = jseed/53668
jseed = 40014*(jseed-k*53668) -k*12211

yes
jseed = jseed+icons

jseed
.lt.
0
no
iseeds(i) = mod(jseed,itwo24)

itwom12 = twom24 * 4096.

1
1

323

1
yes

i=1
i=i+1

i>24
no

seeds(i) = real(iseeds(i))*twom24
next(i) = i-1

next(1) = 24i24 = 24
j24 = 10carry = 0.
yes
seeds(24).eq.
0.

jscarry = twom24

nono

kount+mkount
.ne.
0
yes

write(6,'(a,i,a,i)') ' Restarting ranlux with kount = ',kount,' and mkount = ',mkountiouter=1

iouter>mkount+1

iouter=iouter+1
nojinner = igiga
yes
inner = kountiouter.eq.
mkount+1
no

isk=1
isk=isk+1

yesisk>innernouni = seeds(j24) - seeds(i24) - carryuni = uni + 1.0
carry = twom24

yes

uni
.lt.
0.
no
carry = 0.

seeds(i24) = uni
i24 = next(i24)j24 = next(j24)

2

324

yes

2
in24 = mod(kount, nskip+24)
in24 = mod(kount, nskip+24)

nomkount.gt.
0
yes

izip = mod(igiga, nskip+24)
izip2 = mkount*izip + in24
in24 = mod(izip2, nskip+24)

yes

in24
.gt.
23

write (6,'(a/a,3i11,a,i5)') ' Error in RESTARTING
with RLUXGO:',' The values', inseed, kount,mkount, ' cannot occur at luxury level', luxlev

norluxset = .true.
stop
returnsubroutine
rluxout

Version051219-1435

yes

i=1
i=i+1

i>24
no

isdext(i) = int(seeds(i)*twop12*twop12)

isdext(25) = i24 + 100*j24 + 10000*in24 + 1000000*luxlev

carry
.gt.
0.

yes

cisdext(25) = -isdext(25)arry = twom24

noreturn325

subroutine
rluxinVersion051219-1435

write(6,'(a)') ' full initialization of ranlux with 25 integers:'
write(6,'(5x,5i12)') isdext
twom24 = 1.

yes

i=1
i=i+1

i>24
no

next(i) = i-1
twom24 = twom24 * 0.5

next(1) = 24
twom12 = twom24 * 4096.
yes

i=1
i=i+1

i>24
no

seeds(i) = real(isdext(i))*twom24

carry = 0.

isdext(25)
.lt.
0

yes

carry = twom24

noisd = iabs(isdext(25))
i24 = mod(isd,100)
isd = isd/100
j24 = mod(isd,100)
isd = isd/100
in24 = mod(isd,100)
isd = isd/100
luxlev = isdluxlev.le.
maxlevnowrite (6,'(a,i5)') ' ranlux illegal luxury
rluxin: ',luxlev

yes
stop

nskip = ndskip(luxlev)

write (6,'(a,i2)') ' ranlux luxury level set byrluxin to: ', luxlev

inseed = -1
rluxset = .true.

return326

subroutine
rluxout

Version051219-1435

yes

i=1
i=i+1

i>24
no

isdext(i) = int(seeds(i)*twop12*twop12)

isdext(25) = i24 + 100*j24 + 10000*in24 + 1000000*luxlev

no

carry
.gt.
0.
yes
isdext(25) = -isdext(25)return327

Appendix B

EGS5 USER MANUAL
Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANAlex F. Bielajew and Scott J. Wilderman
Department of Nuclear Engineering and Radiological Sciences
The University of Michigan
2355 Bonisteel Boulevard
Ann Arbor, MI 48109, USA
Walter R. Nelson
Department Associate in the Radiation Physics Group (retired)
Radiation Protection Department
Stanford Linear Accelerator Center
2575 Sand Hill Road Menlo Park, CA 94025, USA
This EGS5 User Manual is Appendix B of a document called
SLAC-R-730/KEK-2005-8, which can be obtained from the SLAC and KEK web sites.

328

B.1

Introduction

Version 5 of the EGS code system is written exclusively in FORTRAN, marking a departure fromthe use of the MORTRAN programming language, first introduced with Version 2. To retainsome of the functionality and flexibility that MORTRAN provided, EGS5 employs some common
extensions of FORTRAN-77, most notably “include” statements, which are used to import identicalversions of all the EGS5 COMMON blocks into all appropriate subroutines. Users can thus alter thevalues of parameters set in the COMMON block files which are “included,” in the various source codes,
thus emulating the use of MORTRAN macros in specifying array dimensions. Each of the COMMON
block files contains the declarations for just one COMMON block, with all files containing EGS-related
COMMON blocks located in a directory named include and all PEGS-related files in a directory called
pegscommons.
Additionally, many of the features and options in EGS4 which were invoked through MORTRAN
macro substitutions have been retained in the base shower code in EGS5 and can be“turned on”
by user specification of the appropriate flags and parameters.

B.2

General Description of Implementation

As described in Chapter 2 of SLAC-R-730/KEK-2005-8 (“The EGS5 Code System”), to use EGS
the user must write a “user code” consisting of a MAIN program and subroutines HOWFAR and
AUSGAB. The user defines and controls an EGS5 shower simulation by initializing, tallying, and insome cases altering variables found in COMMON blocks shared by user code MAIN and a set of four
EGS5 subroutines which MAIN must call (BLOCK SET, PEGS5, HATCH, and SHOWER). The user canaccess and manipulate variables located in many additional COMMON blocks which are shared by
EGS5 subroutines which call the user subroutines HOWFAR and AUSGAB at points in the simulation
specified by the user.
The user’s MAIN program first calls the EGS5 BLOCK SET subroutine to set default values forvariables in EGS5 COMMON blocks which are too large to be defined in BLOCK DATA. MAIN also
initializes variables needed by HOWFAR, and defines the values of EGS5 COMMON block variables
corresponding to such things as names of the media to be used, the desired cutoff energies, and the
distance unit to be used (e.g., inches, centimeters, radiation lengths, etc.). MAIN next calls EGS5
subroutine PEGS5 (to create basic material data) and then calls EGS5 subroutine HATCH, which“hatches” EGS by reading the material data created by PEGS for the media in the given problem.
Once the initialization is complete, MAIN then calls the EGS5 subroutine SHOWER, with each call to
SHOWER resulting in the simulation of one history (often referred to as a “case”). The arguments to
SHOWER specify the parameters of the incident particle initiating the cascade. The user subroutine
HOWFAR is required for modeling the problem geometry (which it does primarily by keeping trackof and reporting to EGS) the regions in which the particles lie), while user subroutine AUSGAB istypically used to score the results of the simulation.

329

Table B.1: Variable descriptions for COMMON block BOUNDS, include file egs5 bounds.f of the EGS5
distribution.
ECUT
Array of region-dependent charged particle cutoff energies in MeV.
PCUT
Array of region-dependent photon cutoff energies in MeV.
VACDST Distance to transport in vacuum (default=1.E8).
In addition to MAIN, HOWFAR, and AUSGAB, additional subprograms may be included in the user
code to facilitate the geometry computations of HOWFAR, among other things. (Sample “auxiliary”
subroutines useful in performing distance-to-boundary computations and in moving particles acrossregions in a variety of common geometries are provided with the EGS5 distribution.)
The interaction between the user code and the EGS5 modules is best illustrated in Figure B.1.
In summary, the user controls an EGS5 simulation by means of:

PEGS5
HATCH
SHOWER
HOWFARAUSGABparameters
variables

Calls to subroutines:
to create media data
to establish media data
to initiate the cascade
Calls from EGS to user subroutines:
to specify the geometry
to score and output the results
Altering elements of COMMON blocks:
inside EGS5 source code, to set array dimensions
inside user code, to specify problem data

The following sections discuss the above mechanisms in greater detail.

B.3

Variables in EGS5 COMMON Blocks

Listed in Tables B.1 through B.17 are the variables in EGS5 COMMON blocks which may be relevantto the user, along with a brief description of their functions. Methods for manipulating thesevariables to either control EGS5 shower simulations or to retrieve results will be discussed insubsequent sections. Note that an asterisk (*) after a variable name in any of the tables indicatesa change from the original EGS4 default.

330

+---------++-------------+| User|| Information || Control || Extracted || Data|| From Shower |+---------++-------------+
U
|
/|\
S
|
|
E
\|/
|
R
+----------------++--------++--------+|MAIN|| HOWFAR || AUSGAB |<----- + C+----------------++--------++--------+| O||/|\ /|\
/|\
/|\
| D
|
|
|
|
|
|
| E
|
|
|
+-----------+ ||==== | ======== | ============ | ========== | | | ======= |==
|
|
|
| | |
|
|
|
|
+--------+ | |
|
\|/
\|/
|
|
| |
|
+----------+ +--------++--------++--------+||BLOCK DATA| | SHOWER |--->| ELECTR |+->| PHOTON |--> + ||BLOCK_SET | +--------++--------+| +--------+| ||(Defaults)|
|
|
|
| | E
+----------++-------> | --------> +
| | G
|
+--+
| | S
|
|
+-------++-------+ | |\|/+-->| MSCAT |+--| COMPT |<-+ | C+----------+|+-------+| +-------+ | | O|
PEGS5 ||+--| PAIR |<-+ | D| (Create ||+-------+| +-------+ | | E|
Media |+-->| ANNIH |--+ || ||Data) ||+-------+ | | +-------+ | |+----------++-->| BHABHA|--+ | | PHOTO |<-+ |||+-------+ | | +-------+||+-->| MOLLER|--+ |
|
|
\|/
|
+-------+ | |||+----------++-->| BREMS |--+ |+------> +|HATCH ||+-------+ | ||| (Access ||\|/\|/||PEGS5 ||+------+||Data) |+----------> | UPHI |-----------> ++----------++------+
Figure B.1: EGS5 user code control and data flow diagram.

331

Table B.2: Variable descriptions for COMMON block BREMPR, include file egs5 brempr.f of the EGS5
distribution.
IBRDST* Flag for turning on (=1) sampling of bremsstrahlung polar angle from (default=0
implies angle given by m/E).
IPRDST* Flag for specifying order of sampling of polar angles of pair electrons (default=0,
implies angles given by m/k).
IBRSPL* Array of flags for turning on (=1) splitting of bremsstrahlung photons (default=0
implies no splitting).
NBRSPL* Number of bremsstrahlung photons for splitting when IBRSPL=1

B.4

Sequence of Actions Required of User Code MAIN

The exact sequence of procedures required of user code MAIN for the specification and control ofan EGS5 simulation is listed below. Details for implementing the necessary steps are provided insubsequent subsections.
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Step 8
Step 9Pre-PEGS5 initializations
PEGS5 call
Pre-HATCH initializations
Specification of incident particle parameters
HATCH call
Initializations for HOWFAR
Initializations for AUSGABSHOWER call
Output of results

Steps 4, 6, and 7 may actually fall anywhere after step 1 and before step 8, and step 8 must be
executed at least once prior to step 9. Step 2 may be skipped if an existing PEGS5 data file has
been prepared and properly linked.

B.4.1

Pre-PEGS5 Initializations (Step 1)

Prior to calling PEGS5 , users must define certain variables and may, at their discretion, override
some of the EGS5 parameter defaults. As noted earlier, all EGS5 variables are readily accessed
through COMMON blocks which are imported into user code through “include” statements, as in:include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’ ! bounds contains ecut and pcut
include ’include/egs5_edge.f’! edge contains iedgfl
332

Table B.3: Variable descriptions for COMMON block COUNTERS, include file counters.f of the EGS5
distribution. All variables in COMMON block COUNTERS are initialized to 0 by a call to subroutine
COUNTERS OUT(0) with argument of 0.
IANNIH*
IAPHI*IBHABHA*IBREMS*ICOLLIS*ICOMPT*
IEDGBIN*IEII*IELECTR*IHARDX*IHATCH*IKAUGER*IKSHELL*IKXRAY*ILAUGER*ILSHELL*ILXRAY*IMOLLER*IMSCAT*
IPAIR*IPHOTO*IPHOTON*IRAYLEI*ISHOWER*IUPHI*ITMXS*NOSCAT

IBLOCK*

Number of times calling subprogram ANNIH.
Number of times calling subprogram APHI.
Number of times calling subprogram BHABHA.
Number of times calling subprogram BREMS.
Number of times calling subprogram COLLIS.
Number of times calling subprogram COMPT.
Number of times calling subprogram EDGBIN.
Number of times calling subprogram EII.
Number of times calling subprogram ELECTR.
Number of times calling subprogram HARDX.
Number of times calling subprogram HATCH.
Number of times calling subprogram KAUGER.
Number of times calling subprogram KSHELL.
Number of times calling subprogram KXRAY.
Number of times calling subprogram LAUGER.
Number of times calling subprogram LSHELL.
Number of times calling subprogram LXRAY.
Number of times calling subprogram MOLLER.
Number of times calling subprogram MSCAT.
Number of times calling subprogram PAIR.
Number of times calling subprogram PHOTO.
Number of times calling subprogram PHOTON.
Number of times calling subprogram RAYLEI.
Number of times calling subprogram SHOWER.
Number of times calling subprogram UPHI.
Number of times requested multiple scattering step was truncated in ELECTR because
pathlength was too long.
Number of times multiple scattering was aborted in MSCAT because the pathlength
was too small (Note change in that NOSCAT has been moved here from EGS4 COMMON
block MISC).
Number of times calling subprogram BLOCK SET.

Table B.4: Variable descriptions for COMMON block EDGE2, include file egs5 edge.f of the EGS5
distribution.
IEDGFL* Array of flags for turning on (=1) explicit treatment of K and L-edge fluorescentphotons (default=0).
IAUGER* Array of flags for turning on (=1) explicit treatment of K and L Auger electrons(default=0).

333

Table B.5: Variable descriptions for COMMON block EIICOM, include file egs5 eiicom.f of the EGS5
distribution.
IEISPL* Flag for turning on (=1) splitting of x-rays generated by electron-impact ionization(default=0 implies no splitting).
NEISPL* Number of electron impact ionization x-rays for splitting when IEISPL=1.

Table B.6: Variable descriptions for COMMON block EPCONT, include file egs5 epcont.f of the EGS5
distribution.
EDEP
Energy deposited in MeV.
TSTEPDistance to next event: interaction, energy hinge, or multiple scattering hinge (cm).
USTEP
Initially, user step length requested (from HOWFAR), and then actual transport steptaken. Thus, USTEP should be scored when estimating track length.
TMSTEP
Total step over both legs of a multiple scattering hinge (replaces the EGS4 variable
TVSTEP). Note that under the random hinge transport mechanics scheme of EGS5,TMSTEP is not reflective of any transport distance, and that the EGS4 variables
TVSTEP, TUSTEP and VSTEP are redundant and so have been removed from thiscommon (as has the variable TSCAT). For legacy purposes, the variable TVSTEP isretained as a local variable declared in egs5 epcont.f and equivalenced to USTEP.
RHOF
Value of density scaling correction (default=1).
EOLD
Charged particle (total) energy at beginning of step in MeV.
ENEW
Charged particle (total) energy at end of step in MeV.
EKE
Kinetic energy of charged particle in MeV.
ELKE
Natural logarithm of EKE.
BETA2
β 2 for present particle. (Note that EGS4 variable BETA is no longer included.)
GLE
Natural logarithm of photon energy.
IDISC
User discard request flag (to be set in HOWFAR). IDISC > 0 means user requests immediate discard, IDISC < 0 means user requests discard after completion of transport,
and IDISC = 0 (default) means no user discard requested.
IROLD
Index of previous region.
IRNEW
Index of new region.
IAUSFL Array of flags for turning on various calls to AUSGAB.

334

Table B.7: Variable descriptions for COMMON block MEDIA, include file egs5 media.f of the EGS5
distribution.
RLCM
Array containing radiation lengths of the media in cm. (Note the name change
necessitated by combining EGS and PEGS.)RLDU
Array containing radiation lengths of the media in distance units established by
DUNIT.
RHOM
Array containing density of the media in g/cm3 . (Note the name change necessitatedby combining EGS and PEGS.)
NMED
Number of media being used (default=1).
MEDIA
Array containing names of media (default is NaI).
IRAYLM
Array of flags for turning on (=1) coherent (Rayleigh) scattering in various media.
Set in HATCH based on values of IRAYLR.
INCOHM* Array of flags for turning on (=1) use of incoherent scattering function for Compton
scattering angles in various media. Set in HATCH based on values of INCOHR.
IPROFM* Array of flags for turning on (=1) Doppler broadening of Compton scattering energies in various media. Set in HATCH based on values of IPROFR.
IMPACM* Array of flags for turning on (=1) electron impact ionization in various media. Set
in HATCH based on values of IMPACR.
CHARD*
Array of “characteristic dimensions,” or representative size (in cm) of scoring regionsin various media. Set by user code MAIN prior to PEGS5 call to invoke automatedelectron step-size selection.
USEGSD* Array of flags indicating (on =1) whether given media uses Goudsmit-Saunderson
multiple scattering distribution. Set by user code MAIN prior to HATCH call (default=0). Note that in the current implementation, it is a requirement that if one
elects to use this option in one media, one must use it in all media.

335

Table B.8: Variable descriptions for COMMON block MISC, include file egs5 misc.f of the EGS5
distribution.
NREG*
Number of regions for the problem, set by user code MAIN prior to HATCH call.
MED
Array containing medium index for each region, set by user code MAIN prior to HATCH
call.
DUNIT
The distance unit to be used. DUNIT=1 (default) establishes all distances in cm,
whereas DUNIT=2.54 establishes all distances in inches.
KMPI
FORTRAN unit number (default=12) from which to read material data.
KMPO
FORTRAN unit number (default=8) on which to “echo” material data (e.g., printed
output, “dummy” output, etc.).
RHOR
Array containing the density for each region (g/cm3 ). If this is different than thedefault density of the material in that region, the cross sections and stopping powers(with the exception of the density effect) are scaled appropriately.
NOMSCT* Array of flags forcing multiple scattering to be bypassed (on =1) in subroutine MSCAT
for various regions (default=0, off).
IRAYLR
Array of flags for turning on (=1) coherent (Rayleigh) scattering in various regions(default=0).
LPOLAR* Array of flags for turning on (=1) linearly polarized photon scattering in various
regions (default=0). When LPOLAR is turned on in a region, the electric field vectors
variables UF(1), VF(1) and WF(1) in the particle stack (see below) must be
specified.
INCOHR* Array of flags for turning on (=1) use of incoherent scattering function for Compton
scattering angles in various regions (default=0).
IPROFR* Array of flags for turning on (=1) Doppler broadening of Compton scattering energies in various regions (default=0).
IMPACR* Array of flags for turning on (=1) electron impact ionization in various regions(default=0).
K1HSCL* Array of parameters for scaling region scattering strength at highest problem energy,
set in user code MAIN prior to HATCH call.
K1LSCL* Array of parameters for scaling region scattering strength at lowest problem energy,
set in user code MAIN prior to HATCH call.

Table B.9: Variable descriptions for COMMON block MS, include file egs5 ms.f of the EGS5 distribution.
TMXSET* Flag to force truncation of requested multiple scattering steps which violate Bethe
criteria (default=.true., enforce limit).

336

Table B.10: Variable descriptions for COMMON block RLUXDAT, include file randomm.f of the EGS5
distribution.
LUXLEV* Luxury level of random number generator RANLUX (called RANDOMSET in EGS5) (default=1).
INSEED* Initial seed used with RANLUX random number generator (default = 314159265).
KOUNT*
Number of random numbers delivered plus number skipped at any point in simulation (up to 109 ).
MKOUNT* Number of sets of 109 random numbers delivered at any point in simulation.
ISDEXT* Array of integer representations of the current RANLUX seeds at any point in simulation.

Table B.11: Variable descriptions for COMMON block STACK, include file egs5 stack.f of the EGS5
distribution. This COMMON contains information about particles currently in the shower. All variables are arrays except for NP, LATCHI, DEINITIAL, DERESID and DENSTEP.
E
X,Y,Z
U,V,W
UF,VF,WF*
DNEAR
WTK1STEP*
K1RSD*K1INIT*
TIME
DENSTEP*
DERESID*
DEINITIAL*
IQ
IR
LATCH*LATCHI*NP

Total energy in MeV.
Position of particle in units established by DUNIT.
Direction cosines of particle.
Electric field vectors of polarized photon.
A lower bound on the distance from the coordinates (X,Y,Z) to nearest surfaceof current region.
Statistical weight of current particle (default=1.0). Used in conjunction withvariance reduction techniques as determined by user.
Scattering strength remaining before the next multiple scattering hinge.
Scattering strength remaining after the current multiple scattering hinge to theend of the full, current multiple scattering step.
Scattering strength from the end of the previous multiple scattering step to thecurrent multiple scattering hinge.
Time (in seconds) since the start of the current history.
Energy loss remaining before the next energy loss hinge.
Energy loss remaining after the current energy loss hinge to the end of the full,current energy loss step.
Energy loss from the end of the previous energy loss step to the current energy
loss hinge.
Integer charge of particle, +1,0,-1, for positrons, photons, and electrons, respectively.
Index of particle’s current region.
Latching variable
Initialization for latch
The stack pointer (i.e., the particle currently being pointed to). Also, the numberof particles on the stack.

337

Table B.12: Variable descriptions for COMMON block THRESH, include file egs5 thresh.f of the EGS5
distribution.
RMT2
Twice the electron rest mass energy in MeV.
RMSQElectron rest mass energy squared in MeV2 .
AP
Array containing PEGS lower photon cutoff energy for each medium in MeV.
UP
Array containing PEGS upper photon cutoff energy for each medium in MeV.
AE
Array containing PEGS lower charged particle cutoff energy for each medium in
MeV.
UE
Array containing PEGS upper charged particle cutoff energy for each medium in
MeV.
TESame as AE except kinetic energy rather than total energy.
THMOLL Array containing the Møller threshold energy (THMOLL=AE+TE) for each medium in
MeV.

Table B.13: Variable descriptions for COMMON block UPHIOT, include file egs5 uphiot.f of the
EGS5 distribution.
THETA
Collision scattering angle (polar).
SINTHE Sine of THETA.
COSTHE Cosine of THETA.
SINPHI Sine of PHI (the azimuthal scattering angle of the collision).
COSPHI Cosine of PHI.
PI
π
TWOPI
2π
PI5D2* 5π/2

Table B.14: Variable descriptions for COMMON block USEFUL, include file egs5 useful.f of the EGS5
distribution.
MEDIUM Index of current medium. If vacuum, then MEDIUM=0.
MEDOLD Index of previous medium.
RM
Electron rest mass energy in MeV.
IBLOBE Flag indicating if photon is below binding energy (EBINDA) after a photoelectricinteraction (yes=1).

338

Table B.15: Variable descriptions for COMMON block USERSC, include file egs5 usersc.f of the EGS5
distribution.
ESTEPR*
Array of factors by which to scale the energy hinge steps in various regions (default=0, implying no scaling).
ESAVE*
Array of energies below which to discard electrons which have ranges less than theperpendicular distances to their current region boundaries (default=0., implying norange-based discard).
EMAXE*
Maximum total energy (in MeV) of any electron in the simulation.
Table B.16: Variable descriptions for COMMON block USERVR, include file egs5 uservr.f of the EGS5
distribution.
CEXPTR* Constant used in exponential transform of photon collision distance (default=0, notransformation).
include ’include/egs5_epcont.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_thresh.f’include ’include/egs5_uphiot.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/randomm.f’! epcont contains iausfl! media contains the array media! misc contains med! thresh contains ae and ap! uphiot contains PI! useful contains RM! usersc contains emaxe

Note that most of the variables accessed in a typical user code MAIN program can be found in the
COMMON files referenced by the include statements in the above example. Other variables which
a user code might wish to access and the EGS5 include files which contain them were given in
Tables B.1 through B.17 of the previous section.
Note that all EGS5 variables are explicitly declared (all EGS5 subroutines and functions begin
with the statement IMPLICT NONE), and that all floating-point variables (except some of those usedin the random number generator and in sample user codes which call intrinsic functions to compute
CPU time) are declared as REAL*8.

Table B.17: Variable descriptions for COMMON block USERXT, include file egs5 userxt.f of the EGS5
distribution.
IPHTER*
Array of flags for turning on (=1) sampling of angular distributions of photoelectrons
in various regions (default=0, implying emission in direction of incident photon).

339

Optional parameter modifications
The EGS5 file include/egs5 h.f is different from the other files in the include directory in that it
contains not COMMON blocks, but rather declarations and specifications of the FORTRAN parameters
used by the other EGS5 include files to define array dimensions. This is done so that users may
trivially update the dimensions of all arrays throughout the EGS5 code system simply by changing
the values of the appropriate variables in the PARAMETER statements of include/egs5 h.f. The
principal parameters defined in include/egs5 h.f which users may wish to adjust are MXMED (themaximum number of media for the problem), MXREG (the maximum number of regions), and MXSTACK(the maximum stack size). Most of the other parameters defined in include/egs5 h.f should bealtered only under exceptional circumstances. Some examples of parameter modifications are givenin the comments in include/egs5 h.f, as seen below:! Maximum number of different media (excluding vacuum)
integer MXMED
parameter (MXMED = 4)!
parameter (MXMED = 10)! Maximum number of regions allocatedinteger MXREG
parameter (MXREG = 2000)!
parameter (MXREG = 2097153)

Required initializations
Two sets of initializations must be performed in the user’s MAIN program. First, MAIN must call the
EGS5 subroutine BLOCK SET to initialize common block variables not defined in BLOCK DATA. Thisis done simply by including the statement:!
!

==============
call block_set==============! Initialize some general variables

Also, if the user is interested in tracking the number of calls to the various subroutines of EGS5,
the counters in common block COUNTERS may also be initialized at this point by calling subroutine
COUNTERS OUT with argument 0, as in:!
!

====================
call counters_out(0)====================
340

Second, because of the way PEGS and EGS are linked in EGS5, the specification of the namesof the problem media prior to calling PEGS5 is now a requirement of EGS5 user codes. The COMMONMEDIA variables NMED (the number of media for the current problem) and MEDIA (a character arrayof the names of the media) must be set prior to a call to PEGS5. Note that the media names must
be exactly 24 characters long. An example of a typical method for filling the MEDIA array (using
lead, steel, and air at NTP as the media), is shown below. First, a local array is declared andinitialized in MAIN , and then copied into MEDIA as in:character*24 medarr(3)medarr(1)=’PB
medarr(2)=’STEEL
medarr(3)=’AIR AT NTP’’’

nmed=3!Number of media used
do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend doOne final variable, which is optional but recommended, must be set prior to PEGS5 being called
if it is to be used. As described in chapter 2 of SLAC-R-730/KEK-2005-8, EGS5 provides amethod for selecting nearly optimal electron multiple-scattering step-sizes in most applications.
The method requires the input specification of a material-dependent parameter CHARD, dimensioned
CHARD(MXMED) and related to the size (in cm) of the smallest scoring region for a given material.
Values (in cm) of CHARD, which is part of COMMON MEDIA, can be passed to PEGS5 simply by assigningvalues, as in:chard(1) = .60d0
chard(2) = .10d0
chard(3) = .85!
!

optional, but recommended to invoke
automatic step-size control

If CHARD is not specified or is set to 0 (the default) for a given material, PEGS5 will use a methodfor determining scattering strengths (and hence step-sizes) for electron multiple scattering basedon fractional energy losses, also described in chapter 2 of the EGS5 Code System report.

B.4.2

PEGS5 Call (Step 2)

MAIN may now call PEGS5 to create material data files for the problem. Specifications for the PEGS
input is found in the “PEGS User Manual,” Appendix C of SLAC-R-730/KEK-2005-8. Note that341

the call to PEGS5 may be skipped if the working user code directory contains an existing PEGS
data file generated with parameters compatible with the current EGS5 simulation specifications.
Checks for compatibility are performed in HATCH.

B.4.3

Pre-HATCH Initializations (Step 3)

Users are strictly required to define the following variables prior to HATCH being called: NREG, thenumber of regions in the geometry; MED, an array containing the material numbers (as set prior
to the call to PEGS5) of each region, and EMAXE, the maximum total energy of any electron in the
problem. No other variables used by HATCH (and then by the EGS5 system in simulating showers),
need be explicitly specified. However, if the user wishes to use any of the non-default options orfeatures of EGS5, the appropriate flags for invoking such requests must be specified prior to thecall to HATCH , even if the data needed to execute such options has been generated by PEGS. Allof the variables processed by HATCH in setting up an EGS5 simulation are described below.

Variables required by HATCHEMAXE This variable, the maximum energy of an electron in the problem, is located in COMMON
USERSC found in include/egs5 usersc.f, and is used by HATCH to perform checks on the compatibility of the EGS5 problem specification and the PEGS data file being used.

NREG HATCH uses the variable NREG when verifying and loading region-dependent options for the
problem materials.

MED The array MED, dimensioned MED(MXREG), contains the medium indices for each region (default
values are 1 for all MXREG). A medium index of zero means a region is vacuum. Indices are definedby the order specified by the user, and are independent of the order in which the materials aredefined in the PEGS data file being used. Consider the three media example above (from thepre-PEGS5 initialization section with vacuum defined as a fourth regions. The EGS5 user code toaccomplish this might look like:med(1)=3
med(2)=1
med(3)=0
med(4)=2!First region is AIR AT NTP!Second region is LEAD
!Third region is VACUUM!Fourth region is STEEL

Optional variables and flags processed by HATCH

342

ECUT and PCUT The ECUT and PCUT arrays contain the cutoff energies (in MeV) for the termination of the tracking of charged particles and photons, respectively, for each region. They are
dimensioned ECUT(MXREG) and PCUT(MXREG) and are initialized to 0.0 in BLOCK SET. Note that
HATCH will override any user defined values of ECUT and PCUT if these values are lower than thethreshold energies set in PEGS for the generation of secondary electrons and photons (the parameters AE and AP). Thus, by assigning values of ECUT and PCUT prior to the HATCH call, the user canraise (but not lower) the cutoff energies. This can be illustrated by considering the four region
example from above. The statements
do i=1,3
ecut(i)=10.0
pcut(i)=100.0
end dowhen put in Step 3 of the user code result in charged particle histories being terminated at 10.0
MeV (total energy) and photon histories being terminated at 100.0 MeV in the first three regions
only. In the fourth region the respective cutoffs will be determined by the values of AE and AP asestablished by PEGS. ECUT and PCUT are elements of COMMON BOUNDS .

IRAYLR The elements of this array (dimensioned IRAYLR(MXREG) and contained in COMMON/MISC/),are set to 1 prior to calling HATCH when coherent (Rayleigh) scattering is to be modeled in particularregions. Execution of EGS is terminated if Rayleigh scattering data is not included in the PEGS
data file, however.

INCOHR The elements of this array (dimensioned INCOHR(MXREG) and found in COMMON/MISC/),are set to 1 prior to calling HATCH when incoherent scattering functions are to be used in sampling
Compton scattering angles in particular regions. Execution of EGS5 is terminated if the appropriate
incoherent scattering function data is not found in the PEGS data file being used, however. Notethat when INCOHR(I)=1, it is necessary to have used IBOUND=1 for the corresponding materials
when PEGS was run.

IPROFR The elements of this array (dimensioned IPROFR(MXREG) and accessed via COMMON/MISC/),are set to 1 prior to calling HATCH if Doppler broadening of the energies of Compton scatteredphotons is to be modeling in particular regions. EGS5 execution is terminated if the Doppler
broadening data is not found by HATCH in the PEGS data file being used, however. Note that when
IPROFR(I)=1, it is necessary to have set IBOUND=1 and INCOH=1 in the corresponding materials
when PEGS was run, and that MAIN must set INCOHR(I)=1 for the corresponding regions as well.

IMPACR The elements of this array (dimensioned IMPACR(MXREG) and found in COMMON/MISC/),are set to 1 prior to calling HATCH when electron impact ionization is to be simulated in particular343

regions. Execution of EGS5 is terminated if the electron impact ionization data is not found in the
PEGS data, however.

IPHTER The elements of this array (dimensioned IPHTER(MXREG) and located in COMMON/USERXT/),
are set to 1 if photoelectron angles are to be sampled in particular regions. The default (IPHTER=0)
assumes emission in the direction of the incident photon.

IEDGFL The elements of this array (dimensioned IEDGFL(MXREG) and passed in COMMON/EDGE2/),
are set to 1 if K and L-edge fluorescence is be explicitly modeled in specific regions.

IAUGER The elements of this array (dimensioned IAUGER(MXREG) and found in COMMON/EDGE2/),
are set to 1 if K and L-edge Auger electrons are to be generated in given regions.

LPOLAR The elements of this array (dimensioned LPOLAR(MXREG) and contained in COMMON/MISC/),are set to 1 if linearly polarized photon scattering is to be modeled in specified regions. WhenLPOLAR turned on and the source particle is a photon, the initial electric field vector must be
specified by setting stack variables UF(1), VF(1) and WF(1) inside the loop over histories inwhich SHOWER is called. Otherwise, linearly polarized photon scattering will occur using arbitraryvalues of the electric vector, and meaningless results may be produced. See KEK Internal 95-4 for
guidance on setting initial values of the electric vector.

DUNIT The parameter DUNIT defines the unit of distance to be used in the shower simulation (thedefault is cm if DUNIT=1). On input to HATCH, DUNIT is interpreted as follows:
1. DUNIT > 0 means that DUNIT is the length of the distance unit expressed in centimeters. For
example, setting DUNIT=2.54 would mean that the distance unit would be one inch.
2. DUNIT < 0 means that the absolute value of DUNIT will be interpreted as a medium index. The
distance unit used will then be the radiation length for the medium, and on exit from HATCH,
DUNIT will be equal to the radiation length of that medium in centimeters. The obvious useof this feature is for the case of only one medium with DUNIT=-1, which results in the shower
being expressed entirely in radiation lengths of the first medium.
Note that the unit of distance used in PEGS is the radiation length. After HATCH interprets DUNIT, it
scales all PEGS data by units of distance as specified by the user, so that all subsequent operations
in EGS will be performed with distances in units of DUNIT (default value: 1.0 cm).

344

K1HSCL and K1LSCL
The parameters K1HSCL and K1LSCL permit the user to apply energydependent scaling of the material-dependent scattering strength (which is roughly proportional tothe multiple-scattering step-size distance) on a region-by-region basis. When K1HSCL and K1LSCL
are non-zero for a region, the scattering strength at EMAXE is scaled by the factor K1HSCL andthe scattering strength at ECUT for the region is scaled by K1LSCL. Scaling at other electron energies is determined by logarithm interpolation. K1HSCL(MXREG) and K1LSCL(MXREG) are found in
COMMON/MISC/ and are initialized to 0.0 in BLOCK SET, which implies no scaling.

USEGSD If the user wishes to use the Goudsmit-Saunderson multiple-scattering distribution function instead of the Molière distribution function for a material, USEGSD(MXMED) must be set to
be non-zero prior to the call to HATCH. In the current version of EGS5, all regions must use theGoudsmit-Saunderson distribution if any of them do. USEGSD is a part of COMMON block MEDIA.

RHOR Media of similar materials but with varying density in different regions can be defined by
setting non-zero values of the region density in the variable RHOR(MXREG) of COMMON/MISC/ prior
to calls to HATCH . This feature eliminates the need for the user to create a distinct new media
for each region which has a a given material but with a different density. Values of RHOR shouldbe specified in terms of the actual density in each region, not the density relative to the reference
density. RHOR is initialized to 0 in BLOCK SET and assigned the default density of the medium by
HATCH unless specified by the user prior to HATCH being called.

Flags and variables which may be set either before or after HATCH is called
The following variables can be set either before or after the call to HATCH.

TMXSET When TMXSET is .true., any multiple-scattering step, whether selected by the user ordetermined by EGS5 using CHARD, which violate the Bethe criteria for the maximum allowed step
length (see chapter 2 of SLAC-R-730/KEK-2005-8) will be truncated in ELECTR to the maximum.
If the user wishes to over-ride this limit, TMXSET (which is material dependent and part of COMMON
MS and defaults to .true.) can be set to .false. at any point in an EGS5 user code.

ESTEPR Electron energy hinge steps are scaled on a region-dependent basis when users set nonzero values of ESTEPR(MXREG) prior to a call to SHOWER. Since energy hinge step sizes are determinedin PEGS, ESTEPR provides the user the capability to take smaller or larger steps in certain materials
or regions for increased accuracy or efficiency, respectively. ESTEPR, which is part of EGS5 COMMON
USERSC, is initialized to 0.0 in BLOCK SET and ignored in ELECTR unless set by the user.

345

ESAVE The variable ESAVE, dimensioned ESAVE(MXREG) and part of COMMON USERSC, can be employed by users to speed computations for applications which involve the transport of electrons
across boundaries between scoring and non-scoring regions. For example, if a user is interested inenergy deposition in a gas detector, only those electrons which are energetic enough to escape thesolid walls surrounding the gas of the detector have a chance to be scored. Thus the simulationof the transport in the walls of electrons with ranges less than the closest normal distances to theouter walls adds nothing but CPU time to the simulation. If, however, the user specifies a non-zerovalue of ESAVE for a given region, ELECTR will discard the electron if its energy is less than ESAVE
and its range is less than DNEAR (see below), thus speeding the computation. This technique iscommonly called “range rejection,” and is most effective when ESAVE is much larger than ECUT.
Note that the “range” of the electron is defined very crudely here, as simply E(NP) divided by the
stopping power of the medium. This assures that the decision to discard a particle based on range
rejection will be conservative as long as the stopping power of the medium increases at energies
below ESAVE.

IBRDST The parameter IBRDST, which has a default value of 0 and is part of COMMON BREMPR,
determines the procedure for determining the angle of bremsstrahlung photons (relative to theincident electrons), as described below:
IBRDST
0
1

Method for determining θ
fixed at m/Ĕ0sampled from Koch and Motz formula 2BS

Values of IBRDST set by the user apply to all media and regions in a simulation.

IPRDST The value of the parameter IPRDST determines the method used for determining the angles
of electron and positron pairs resulting from photon pair-production in the same way that IBRDST
is used to select the sampling method for bremsstrahlung photon angles. IPRDST, which is part ofCOMMON BREMPR, has a default value of 0 and controls pair electron angles (relative to the incident
photon direction) as follows:
IPRDST
0
1
2

Method for determining θ
fixed at m/ksampled from Motz, Olsen and Koch formula 3D-2000
sampled from Motz, Olsen and Koch formula 3D-2003

IEISPL and NEISPL In order to speed up EGS5 simulations for applications involving x-rays generated from electron-impact ionizations, a method for creating additional x-rays using the familiar
Monte Carlo technique of splitting is provided (see chapter 4 of SLAC-R-730/KEK-2005-8 for thedescription of an EGS5 application involving splitting of particles). If the flag IEISPL, which is346

part of COMMON EIICOM and defaulted to 0, is set to be 1, each electron-impact ionization event
which leads to the production of a characteristic x-ray will result in NEISPL appropriately weighted
x-rays being produced.

IBRSPL and NBRSPL The parameters IBRSPL (a region-dependent array dimensioned IBRSPL(MXREG))
and NBRSPL allow the user to improve the efficiencies of simulations in which low-probability bremsstrahlung photon production is important by splitting the secondary particles. The method is similar to that described above for splitting in x-ray production following electron-impact ionization.
When the parameter IBRSPL, which has a default value of 0 and is found in COMMON BREMPR, is set
to 1 for a given region, each electron bremsstrahlung event taking place in that region will resultin the generation of NBRSPL appropriately weighted photons.

CEXPTR The parameter CEXPTR, found in COMMON USERVR, is a scaling factor which can be used
to either force or inhibit photon collisions in regions with cross section that are very small or verylarge. If λ is the photon mean free path and we use C to represent the scaling factor CEXPTR, wehave for the interaction probability distribution:p̃(λ)dλ = (1 − Cµ)e−λ(1−Cµ) dλ,where theoverall multiplier 1 − Cµ is introduced to ensure that the probability is correctly normalR
ized, i.e. 0∞ p̃(λ)dλ = 1. For C = 0, we have the unbiased probability distribution e−λ dλ. One seesthat for 0 < C < 1, the average distance to an interaction is stretched and for C < 0, the average
distance to the next interaction is shortened.
Note that the average number of mean free paths toR∞1.
an interaction, hλi, is given by hλi = 0 λp̃(λ)dλ = 1−CµNOMSCT The user may override all treatment of electron multiple-scattering in a given region by
setting the switch NOMSCT(MXREG) to be 1 for that region. NOMSCT which is a part of COMMON MISC
and is initialized 0, is used primarily as a debugging and code development tool, and is included inthis description for completeness only.

Random number generator initialization
Whenever EGS (including any part of the user code) requires a floating point random number taken
uniformly from the interval (0,1) to be returned to a variable (all EGS5 routines use the variablename RNNOW), the following statement is required:call randomset(rnnow)
EGS5 employs the random number generator RANLUX, implemented by James. Depending on
the input specification, called the “luxury level,” RANLUX provides random sequences which pass347

different levels of tests for randomness and execute at different speeds. Independent random sequences for the same luxury level can be generated with RANLUX by simply specifying a different
input “seed,” any integer in the range from 1 to 231 . The default luxury level, as defined in the variable LUXLEV of COMMON RLUXDAT in file include/randomm.f, is 1, and the default seed, INSEED,
is 314159265. RANLUX is initialized by HATCH using the defaults for LUXLEV and INSEED unless theuser specifies different values prior to the HATCH call. In addition, users may initialize the generator
themselves at any time by invoking
call rluxinit
after specifying LUXLEV and INSEED.
The user may also restart RANLUX at any desired point in a previously used sequence of random
numbers using either of two ways. RANLUX keeps a tally of the number of random numbers delivered
through the variables MKOUNT and KOUNT as MKOUNT*100000000 + KOUNT, and MKOUNT and KOUNT
are accessible at all times through COMMON RLUXDAT . If the user calls RLUXINIT and supplies valuesof MKOUNT and KOUNT in addition to LUXLEV and INSEED, the RANLUX will be restarted at exactlythat point in the sequence defined by MKOUNT and KOUNT.
Alternatively, the user may execute the following statement
call rluxout
at any time, at which point integer representations of the current values of the seeds in RANLUX will
be returned via the array ISDEXT of COMMON RLUXDAT . A call to RLUXINIT at any time when thevalues of ISDEXT are non-zero will result in a restart of RANLUX based on the seeds in ISDEXT.
Thus the restart options can be summarized as follows:
1. A brute-force method involves calling RLUXINIT with the values of the luxury level, initialseed, and number of delivered randoms up to the time of the desired restart. The values of
MKOUNT and KOUNT can be obtained at any time directly from COMMON RLUXDAT.
2. A more elegant restart using the actual seeds can be done by passing the integer seeds at thetime of the restart to RLUXINIT via ISDEXT in COMMON RLUXDAT. The seeds in ISDEXT can beobtained for later restart at any convenient time (such as the end of a shower, or the end ofa batch) by a call to RLUXOUT.

B.4.4

Specification of Incident Particle Parameters (Step 4)This step required in constructing a MAIN user code is self-explanatory. An example of suitable
coding is given as follows:348

iqi=-1xi=0.0yi=0.0zi=0.0ui=0.0vi=0.0wi=1.0!Incident particle is an electron!Particle coordinates

iri=2
wti=1.0
ncases=10
idinc=-1!Region number 2 is the incident region!Weight factor in importance sampling!Number of histories to run!Direction cosines

ei=1000.d0!Total energy (MeV)
ekin=ei+iqi*RM !Incident kinetic energy
Note that the variables initialized above are the ones passed to EGS5 subroutine SHOWER, as described below in step 8.

B.4.5

HATCH Call (Step 5)

When the user code MAIN calls the EGS HATCH subroutine, EGS is “hatched” by executing somenecessary once-only initializations and reading material data for the media from a data set thatcreated by PEGS. The required call is, trivially:!
!

==========
call hatch==========

Some examples of reports from HATCH are shown below. The following is a typical output
message when DUNIT has not been changed (and Rayleigh data is included in the file):
RAYLEIGH DATA AVAILABLE FOR MEDIUM

1 BUT OPTION NOT REQUESTED.

EGS SUCCESSFULLY ’HATCHED’ FOR ONE MEDIUM.
For a non-default specification of DUNIT (DUNIT=2.54, for example), the output report from
HATCH would look like the following (for two media and no Rayleigh data):
DUNIT REQUESTED&USED ARE:
2.54000E+00
2.54000E+00(CM.)
EGS SUCCESSFULLY ’HATCHED’ FOR
2 MEDIA.
349

Failure to successfully “hatch” a medium because it could not be found in the PEGS data fileresults in message below, and execution is terminated by HATCH .
END OF FILE ON UNIT 12
PROGRAM STOPPED IN HATCH BECAUSE THE
FOLLOWING NAMES WERE NOT RECOGNIZED:(list of names)
Note that one cannot ask for the same medium twice, though one can define two media which arephysically identical to be distinct for the purposes of EGS by using different names for them in
PEGS input files.

B.4.6

Initializations for HOWFAR (Step 6)

As stated previously, HOWFAR is the routine that describes the translation of particles throughthe geometry of the various regions in the problem. Note that initialization of data required by
HOWFAR may be done at any step prior to calling SHOWER in Step 8, and that in fact, for sometrivial versions of HOWFAR, no initializations are required at all. For versions of HOWFAR which model
realistic geometries, however, it is likely that some initialization will be required in MAIN or auxiliary
user subprograms called by MAIN. In such cases it will also be necessary that local auxiliary COMMON
blocks be defined to pass geometry data to HOWFAR.

B.4.7

Initializations for AUSGAB (Step 7)

This step is similar to initialization for HOWFAR above in that it could actually be done anywhere in
MAIN prior to SHOWER being called. An example initialization based on a three region geometry isgiven here. Suppose that we wish to know the total energy deposited in each of the three regions.
We could declare a scoring array, ESUM in a COMMON block TOTALS in both MAIN and in AUSGAB as:common/totals/esum(3)This array would be initialized in MAIN by the statements:
do i=1,3
esum(i)=0.0
end doThen the statement350

esum(ir(np))=esum(ir(np)) + edepin AUSGAB would keep a running total of the energy deposited in each region under consideration.
Note that global auxiliary subroutines ECNSV1 and NTALLY are provided with the EGS5 distribution
to facilitate scoring of energy deposition and the numbers of various types of events, respectively.

B.4.8

SHOWER Call (Step 8)

The calling sequence for SHOWER is:call shower(iqi,ei,xi,yi,zi,ui,vi,wi,iri,wti)
All of the arguments in this call are declared real*8 in SHOWER , except for iqi and iri which areinteger. These variables, which can have any names the user wishes in MAIN , specify the charge,
total energy, position, direction, region index, and statistical weight of the incident particle, and areused to fill the corresponding stack variables (see the listing in Table B.11). In a typical problem
SHOWER is called repeatedly in a loop over a number of “histories” or “cases” as in
do i=1,NCASES
call shower(iqi,ei,xi,....,etc.)
end doThe statistical weight WTI of the incident particle is generally taken as unity unless variance reduction techniques are employed by the user. Note that if IQI is assigned the value of 2, subroutine
SHOWER recognizes this as a pi-zero meson decay event, and two photons are added to the stackwith energies and direction cosines appropriately obtained by sampling.

Specification of electric vector of photon for SHOWER
This is necessary only if the incident particle is a photon and the scattering of linearly polarized
photons is being modeled. The following 3 examples illustrate the specification of the photon
electric field vector and the passing of that data to SHOWER.

Example 1.

Completely linearly polarized photon source with electric vector along +y-direction:

ufi=0.0
vfi=1.0351

wfi=0.0
do i=1,ncases
uf(1)=ufi
vf(1)=vfi
wf(1)=wfi
call shower(iqi,e,xi,yi,zi,ui,vi,wi,iri,wti)
end doExample 2. Partially linearly polarized photon source with source propagation vector along
the z-direction and polarization vector along the y-axis with P=0.85 (P is the degree of linear
polarization):
ui=0.0vi=0.0wi=1.0pval=0.85
pratio=0.5+pval*0.5! Degree of linear polarization! Ratio of y-polarization

do i=1,ncases
call randomset(value)if(value.lt.pratio) thenufi=0.0
vfi=1.0
wfi=0.0
else
ufi=1.0
vfi=0.0
wfi=0.0
end ifuf(1)=ufi
vf(1)=vfi
wf(1)=wfi
call shower(iqi,e,xi,yi,zi,ui,vi,wi,iri,wti)
end doExample 3. Unpolarized photon source. In a photon transport simulation modeling linear polarization, an unpolarized photon source is automatically generated by setting:uf(1)=0.0
vf(1)=0.0352

wf(1)=0.0
inside the shower call loop.

B.4.9

Output of Results (Step 9)This step is self-explanatory, and is included only for the sake of completeness.

B.5

Specifications for HOWFAR

EGS calls user code HOWFAR when it reaches the point at which it has determined, because of stepsize specifications and/or interaction probabilities, that it would like to transport the top particle
on the stack a straight line distance USTEP in the current media. All of the parameters of theparticle are available to the user via COMMON/STACK/ as described earlier. The user controls thetransport upon return to EGS by altering one or more of the following variables: USTEP, IDISC,IRNEW, and DNEAR(NP). Except for DNEAR (which is in COMMON/STACK/), these are available to theuser via COMMON/EPCONT/. The ways in which these variables may be changed and the way EGS willinterpret these changes is discussed in detail below. (Note, flow diagrams for subroutines ELECTR
and PHOTON have been included in Appendix A of SLAC-R-730/KEK-2005-8 for the user who
requires a more complete understanding of what actually takes place during particle transport.)

IDISC If the user decides that the current particle should be discarded, then IDISC must be set
nonzero (the usual convention is to set IDISC=1).
A positive value for IDISC will cause the particle to be discarded immediately. A negative value
for IDISC will cause EGS to discard the particle when it completes the transport. EGS initializes
IDISC to zero, and if left zero no user requested discard will take place. For example, the easiest
way to define an infinite, homogeneous medium is with the HOWFAR routine:
subroutine howfar
returnend
In this case, particle transport will continue to take place until energy cutoffs are reached. However,a common procedure is to set IDISC=1 whenever the particle reaches a discard region, e.g.outside
the problem geometry.

353

USTEP and IRNEW If immediate discard has not been requested, then the HOWFAR should checkto see whether transport by distance USTEP will cause a region boundary to be crossed. If noboundary will be crossed, then USTEP and IRNEW may be left as they are. If a boundary will becrossed, then USTEP should be set to the distance to the boundary from the current position along
the current direction, and IRNEW should be set to the region index of the region on the other sideof the boundary. For sophisticated geometries, this is the most complex part of the user code.

DNEAR(NP) The setting of DNEAR(NP) by the user is optional. However, in many situations a significant gain in efficiency will result by defining DNEAR(NP) in HOWFAR. It is obvious that distance toboundary calculations are computationally expensive and should be avoided whenever possible. For
electrons traveling in regions in which their step sizes are much smaller than the region dimensions,interrogation of the problem geometry at each electron step can greatly slow the simulation. In
order to avoid this inefficiency, each particle has stored on the stack a variable called DNEAR(NP),which is used by EGS to hold a lower bound on the distance from the particle’s current position
to the nearest region boundary. This variable is used by EGS in the following ways:
1. DNEAR for the incident particle is initialized to zero.
2. Whenever a particle is actually moved (by a straight line distance TVSTEP) the path lengthtransported is deducted from the DNEAR for the particle.
3. Whenever a particle interacts, the DNEAR values for the product particles are set from the
DNEAR value of the parent particle.
4. When EGS has decided it would like to transport the current particle by a distance USTEP(which will be the distance to the next interaction), subroutine HOWFAR will be called to get
the user’s permission to go that far only if USTEP is larger than DNEAR. It is this feature which
permits EGS to avoid potentially cumbersome geometry computations whenever possible.
In summary, to take advantage of these efficiency features, the user should set DNEAR(NP) equal tothe perpendicular distance to the nearest region boundary from the particle’s current position. If it
is easier for the user to compute some quick lower bound on the actual nearest distance, this could
be used to set DNEAR with time savings depending on how close the lower bound is to the actual
nearest distance on the average. It should be understood, however, that if the boundary separations
are smaller than the mean step size, subroutine HOWFAR will still be called and the overall efficiency
will decrease as a result of having to perform the DNEAR calculation so many times. Finally, if themedium for a region is vacuum, the user need not bother computing DNEAR, as EGS will always
transport to the next boundary in only one step in this case.

B.5.1

Sample HOWFAR User Code

Consider, as an example of how to write a HOWFAR subprogram, the three region geometry in B.2.
A particle is shown in Region 2 with coordinates (X,Y,Z) and direction cosines (U,V,W). We will
354

|
|
| Region | Region
|
|
1
|
2
|
3
|
|
| (X,Y,Z) |
|
x
|
|
.
|
Vacuum |. | Air at NTP|. |O---------.+----------------> Z
|
|
|
|.
|
Iron
| .
|
| .
|
|
.
|
|
.
|
|
.
|
|
.
|
|
x
|
|
(U,V,W)||||-->| ZTHICK |<-|||V
RegionX(Y into paper)
Figure B.2: A three-region geometry for a HOWFAR example code.

355

assume that the slab of thickness ZTHICK is semi-infinite (x and y-directions), and that particles
are immediately discarded whenever they go into Region 1 or Region 3. The following HOWFAR code
correctly models this geometry:subroutine howfar
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/passit/zthick
real*8 zthick
real*8 deltaz
integer irnxt! Local variables

if (ir(np).ne.2) thenidisc = 1
returnend ifdnear(np) = dmin1(z(np),zthick-z(np))!----------------------------------! Particle going parallel to planes!----------------------------------if(w(np).eq.0) return!-------------------------------------------------------! Check forward plane first since shower heading that way! most of the time!-------------------------------------------------------if (w(np).gt.0.0) thendeltaz=(zthick-z(np))/w(np)irnxt=3!----------------------------------------------------------! Otherwise, particle must be heading in backward direction.
!----------------------------------------------------------elsedeltaz=-z(np)/w(np)irnxt=1
end if356

if (deltaz.le.ustep) thenustep=deltaz
irnew=irnxt
end ifreturnend
Note that a number of auxiliary geometry subprograms are distributed with the EGS5 Code
System in order to make it easier to write HOWFAR. For example, subroutine PLAN2P could have beenused in place of several lines above and the program would have been easier to read. Example user
codes which employ several of the auxiliary geometry subprograms are described in Chapter 4 of
SLAC-R-730/KEK-2005-8.

B.6

Specifications for AUSGAB

The user subroutine AUSGAB is called at more than 40 places inside various EGS5 subroutines withthe statement:
call ausgab(iarg)
The argument IARG indicates the situation under which AUSGAB is being called. IARG can takeon 31 values starting from zero (i.e., IARG=0 through IARG=30), although only the first five arecalled in the default version of EGS. The remaining 26 IARG situations must be “switched on” via
specification of the array IAUSFL. The 5 values of IARG which are turned on by default and thecorresponding situations in which they initiate calls to user code AUSGAB are given in Table B.18.
The above IARG values are the ones required in the majority of situations in which EGS5 is used
to simulate electromagnetic cascade shower development. In particular, IARG =0 is useful whenevertrack lengths are being calculated or when charged particle ionization loss is being scored. The
large number of situations which initiate calls to AUSGAB for various IARG values allows the user
to extract information about EGS5 simulations without making changes to the EGS code. The
user controls when AUSGAB is to be called by specifying in the user code values of the integer flag
array, IAUSFL(J), for J=1 through 31. IAUSFL(J) takes on values of 1 or 0 depending on whether
AUSGAB is called or not, respectively. For J=1 through 5, which corresponds to IARG of 0 through 4,
IAUSFL(J) is set to 1 by default, and AUSGAB is always called for the situations listed in Table B.18
For the remaining values of J, corresponding to IARG =5 through 31, IAUSFL(J) is set to 0 by
default, and the user must modify IAUSFL(J) in order to initiate any desired AUSGAB calls. The
value for IARG and the corresponding situations for this upper set of IARG values are shown in
Table B.19.
357

Table B.18: IARG values program status for default AUSGAB calls.
IARG
0
1

2
3
4

Situation
Particle is going to be transported by distance TVSTEP.
Particle is going to be discarded because its energy is below the cutoff ECUT (forcharged particles) or PCUT (for photons)—but its energy is larger than the corresponding PEGS cutoff AE or AP, respectively.
Particle is going to be discarded because its energy is below both ECUT and AE (orPCUT and AP).
Particle is going to be discarded because the user requested it (in HOWFAR usually).
Part of particle energy is deposited due to the binding energy. This situation occurs
in one of the following 3 cases:
1. A photoelectric interaction has occurred and the difference in the electron
binding energy and the secondary particle (X-ray or Auger electron) energy isdeposited.
2. Compton interaction has occurred and the electron binding energy is deposited
locally. This is enabled only when the Doppler-broadening option is turnedon.
3. The K-shell EII has occurred and the difference between the electron binding
energy and the secondary particle (K-X ray) energy is deposited. This isenabled only when the EII option is turned on.

358

Note that the code statuses for IARG values from 0 to 3 and from 5 to 24 are the identicalto those found in EGS4. A slight modification has been made in EGS5 for IARG of 4, and thesituations in which IARG values of 25 through 30 initiate a call to AUSGAB are newly added in EGS5.
As an example of how to write an AUSGAB subprogram, consider the previous three region
geometry of Figure B.2. Suppose that we wish to score only photons that emanate from Region 2
into Region 3. The AUSGAB subprogram that will accomplish this is given below (in this example
we print out the stack variables plus IARG).

subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_stack.f’
integer iarg
! Arguments
if(iarg.eq.3.and.iq(np).eq.0.and.ir(np).eq.3) thenwrite(6,1000)e(np),x(np),y(np),z(np),u(np),v(np),w(np),1
iq(np),ir(np),iarg
end if1000 format(7g15.7,3i5)
returnend

359

Table B.19: IARG values, IAUSFL indices, and program status for AUSGAB calls.
IARG
5
6

IAUSFL
6
7

7
8

8
9

9
10

10
11

11
12

12
13

13
14
15

14
15
16

16
17

17
18

18
19

19
20

20
21
22
23
24
25
26
27
28
29
30

21
22
23
24
25
26
27
28
29
30
31

Situation
Particle has been transported by distance TVSTEP.
A bremsstrahlung interaction is to occur and a call to BREMS is about to bemade in ELECTR.
Returned to ELECTR after a call to BREMS was made.
A Møller interaction is to occur and a call to MOLLER is about to be made in
ELECTR.
Returned to ELECTR after a call to MOLLER was made.
A Bhabha interaction is to occur and a call to BHABHA is about to be made in
ELECTR.
Returned to ELECTR after a call to BHABHA was made.
An in-flight annihilation of the positron is to occur and a call to ANNIH is
about to be made in ELECTR.
Returned to ELECTR after a call to ANNIH was made.
A positron has annihilated at rest.
A pair production interaction is to occur and a call to PAIR is about to bemade in PHOTON.
Returned to PHOTON after a call to PAIR was made.
A Compton interaction is to occur and a call to COMPT is about to be made in
PHOTON.
Returned to PHOTON after a call to COMPT was made.
A photoelectric interaction is to occur and a call to PHOTO is about to be madein PHOTON.
Returned to PHOTON after a call to PHOTO was made (assuming NP is non-zero).
Subroutine UPHI was just entered.
Subroutine UPHI was just exited.
A coherent (Rayleigh) interaction is about to occur.
A coherent (Rayleigh) interaction has just occurred.
An EII interaction is about to occur.
Returned to MOLLER after a call to EII was made.
An energy hinge is about to occur in ELECTR.
An energy hinge has just occurred in ELECTR.
A multiple-scattering hinge is about to occur in ELECTR.
A multiple-scattering hinge has just occurred in ELECTR.

360

B.7

UCSAMPL5 — An Example of a “Complete” EGS5 User
Code

The following user code, called UCSAMPL5, simulates electro-magnetic cascade showers initiatedby 1 GeV electrons that are incident (normally) on a 3 cm, semi-infinite slab of iron. The upstream
region of the slab is vacuum and the downstream region is air at NTP. A particle is discarded
whenever it leaves the slab (on either side), or whenever its total energy falls below a preset
cutoff energy of 100 MeV. (Note that the medium assigned to Region 3 is really not importantin this example, and was included solely for purposes of illustration.) Some of the stack variable
information E(NP), Z(NP), W(NP), IQ(NP), IR(NP), plus the IARG value, is printed out on theprinter (first 15 lines only) for photons reaching Region 3.
A total of 10 cases of incident electrons are run and the total energy fraction for each region issummed and printed out at the end of the run for an energy balance check.
The UCSAMPL5 user code is given below.
!***********************************************************************!
!
**************!
**!
* ucsampl5.f *!
**!
**************!
! A complete example of a EGS5 user code, using a simple plane! geometry. For SLAC-R-730/KEK Report 2005-8.
!
! This user code corresponds to ucsampl4.mor for egs4.
! The following shows the geometry!***********************************************************************!
*!
------------------------------------------------*!
1-Dimensional Plane Z Geometry (ucsampl5 example)*!
------------------------------------------------*!
*!
Y (X into page)
*
!
^
*
!
|
*
!
|
|
*
!
| Fe |
Air
*
!
|
|
*
!
|
|
*
!
1 GeV
|
|
*
!
==========>+----+------------------------> Z*!
electron
0
3.0
*
!
*

361

!***********************************************************************!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12!----------------------------------------------------------------------!------------------------------- main code ----------------------------!----------------------------------------------------------------------!----------------------------------------------------------------------!
Step 1. Initialization!----------------------------------------------------------------------implicit none!
!
!

-----------EGS5 COMMONs-----------include ’include/egs5_h.f’! Main EGS "header" fileinclude ’include/egs5_bounds.f’include ’include/egs5_edge.f’include ’include/egs5_media.f’
include ’include/egs5_misc.f’include ’include/egs5_stack.f’
include ’include/egs5_thresh.f’include ’include/egs5_useful.f’
include ’include/egs5_usersc.f’include ’include/egs5_userxt.f’include ’include/randomm.f’!
!
!

---------------------Auxiliary-code COMMONs---------------------include ’auxcommons/lines.f’common/passit/zthick
real*8 zthickcommon/totals/esum(3)real*8 esum*

real*8 ei,ekin,etot,totke,xi,yi,zi,
ui,vi,wi,wti
real tarray(2)real t0,t1,timecpu,tt
real etime
integer i,idinc,iqi,iri,j,ncases
character*24 medarr(2)!
!
!

! Arguments! Local variables---------Open files----------

362

open(UNIT= 6,FILE=’egs5job.out’,STATUS=’unknown’)!
!

====================
call counters_out(0)====================!----------------------------------------------------------------------! Step 2: pegs5-call!----------------------------------------------------------------------!
==============
call block_set! Initialize some general variables!
==============!
!
!

--------------------------------define media before calling PEGS5--------------------------------nmed=2
medarr(1)=’FE-RAYLEIGH
medarr(2)=’AIR AT NTP’’

do j=1,nmed
do i=1,24
media(i,j)=medarr(j)(i:i)
end doend do

!
!
!
100
!
!

chard(1) = 3.0
chard(2) = 3.0-----------------------------Run PEGS5 before calling HATCH-----------------------------write(6,100)
FORMAT(’ PEGS5-call comes next’)=============
call pegs5=============!----------------------------------------------------------------------! Step 3: Pre-hatch-call-initialization!----------------------------------------------------------------------med(1)=0
med(2)=1
med(3)=2!
!
!

---------------------------------Set of option flag for region 2-3
1: on, 0: off

363

!

---------------------------------nreg=3
do i=2,nregecut(i)=100.0
pcut(i)=100.0
iphter(i) = 0
iedgfl(i) = 0
iauger(i) = 0
iraylr(i) = 0
lpolar(i) = 0
incohr(i) = 0
iprofr(i) = 0
impacr(i) = 0
end do! egs cut off energy for electrons! egs cut off energy for photons! Switches for PE-angle sampling! K & L-edge fluorescence! K & L-Auger! Rayleigh scattering! Linearly-polarized photon scattering! S/Z rejection! Doppler broadening! Electron impact ionization

!
!
!
!

-------------------------------------------------------Random number seeds. Must be defined before call hatch.
ins (1- 2^31)-------------------------------------------------------inseed=1
luxlev=1!

=============
call rluxinit=============!

! Initialize the Ranlux random-number generator!----------------------------------------------------------------------! Step 4: Determination-of-incident-particle-parameters!----------------------------------------------------------------------iqi=-1
xi=0.0yi=0.0zi=0.0ui=0.0vi=0.0wi=1.0
iri=2
wti=1.0
ncases=1000
idinc=-1
ei=1000.D0
ekin=ei+iqi*RM!----------------------------------------------------------------------! Step 5:
hatch-call!----------------------------------------------------------------------! Total energy of incident source particle must be defined before hatch! Define posible maximum total energy of electron before hatch
if (iqi.ne.0) thenemaxe = ei
! charged particle

364

else
emaxe = ei + RM
end if
!
!
!

130
!
!
!
!
!

! photon-----------------------------Open files (before HATCH call)-----------------------------open(UNIT=KMPI,FILE=’pgs5job.pegs5dat’,STATUS=’old’)open(UNIT=KMPO,FILE=’egs5job.dummy’,STATUS=’unknown’)
write(6,130)
FORMAT(/,’ HATCH-call comes next’,/)==========
call hatch==========-----------------------------Close files (after HATCH call)-----------------------------close(UNIT=KMPI)
close(UNIT=KMPO)! ---------------------------------------------------------! Print various data associated with each media (not region)! ---------------------------------------------------------write(6,140)
140
FORMAT(/,’ Quantities associated with each MEDIA:’)do j=1,nmed
write(6,150) (media(i,j),i=1,24)
150
FORMAT(/,1X,24A1)
write(6,160) rhom(j),rlcm(j)160
FORMAT(5X,’ rho=’,G15.7,’ g/cu.cm
rlc=’,G15.7,’ cm’)write(6,170) ae(j),ue(j)170
FORMAT(5X,’ ae=’,G15.7,’ MeV
ue=’,G15.7,’ MeV’)
write(6,180) ap(j),up(j)180
FORMAT(5X,’ ap=’,G15.7,’ MeV
up=’,G15.7,’ MeV’,/)end do!----------------------------------------------------------------------! Step 6: Initialization-for-howfar!----------------------------------------------------------------------zthick=3.0!
plate is 3 cm thick!----------------------------------------------------------------------! Step 7: Initialization-for-ausgab!----------------------------------------------------------------------do i=1,nregesum(i)=0.D0
end do365

nlines=0
nwrite=15!----------------------------------------------------------------------! Step 8: Shower-call!----------------------------------------------------------------------tt=etime(tarray)t0=tarray(1)190

write(6,190)
format(/,’ Shower Results:’,///,7X,’e’,14X,’z’,14X,’w’,10X,
1’iq’,3X,’ir’,2X,’iarg’,/)do i=1,ncases

200

if (nlines.lt.nwrite) thenwrite(6,200) i,ei,zi,wi,iqi,iri,idincformat(i2,3G15.7,3I5)
nlines=nlines+1
end ifuf(1)=0.0
vf(1)=0.0
wf(1)=0.0! Needed if lpolar(i)=1

call shower(iqi,ei,xi,yi,zi,ui,vi,wi,iri,wti)
end dott=etime(tarray)t1=tarray(1)210timecpu=t1-t0
write(6,210) timecpu
format(/,’ Elapsed Time (sec)=’,1PE12.5)!----------------------------------------------------------------------! Step 9: Output-of-results!----------------------------------------------------------------------totke=ncases*ekin
write(6,220) ei,zthick,ncases
220
format(//,’ Incident total energy of electron=’,F12.1,’ MeV’,/, ’*Iron slab thickness=’,F6.3,’ cm’,/, ’ Number of cases in run=’,I7,*//,’ Energy deposition summary:’,/)
etot=0.D0
do i=1,nreg
etot=etot+esum(i)esum(i)=esum(i)/totke
write(6,230) i, esum(i)

366

230

240
!
!
!

format(’ Fraction in region ’,I3,’=’,F10.7)
end doetot=etot/totke
write(6,240) etot
FORMAT(//,’ Total energy fraction in run=’,G15.7,/,*’
Which should be close to unity’)----------Close files----------close(UNIT=6)

stop
end
!-------------------------last line of main code-----------------------!-------------------------------ausgab.f-------------------------------! Version:
050701-1615! Reference: SLAC-R-730, KEK-2005-8 (Appendix 2)!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------! A simple AUSGAB to:!
!
1) Score energy deposition!
2) Print out stack information!
3) Print out particle transport information (if switch is turned on)!
! ---------------------------------------------------------------------subroutine ausgab(iarg)
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’
include ’auxcommons/lines.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/totals/esum(3)real*8 esum
integer iarg!
!
!

! Arguments---------------------Add deposition energy---------------------esum(ir(np))=esum(ir(np)) + edep

367

!
!
!

---------------------------------------------------------------Print out stack information (for limited number cases and lines)----------------------------------------------------------------

1240

if (nlines.lt.nwrite) thenwrite(6,1240) e(np),z(np),w(np),iq(np),ir(np),iarg
FORMAT(3G15.7,3I5)
nlines=nlines+1
end ifreturnend!--------------------------last line of ausgab.f-----------------------!-------------------------------howfar.f-------------------------------! Version:
050701-1615! Reference: SLAC-R-730, KEK-2005-8 (Appendix 2)!----------------------------------------------------------------------!23456789|123456789|123456789|123456789|123456789|123456789|123456789|12! ---------------------------------------------------------------------! Required (geometry) subroutine for use with the EGS5 Code System! ---------------------------------------------------------------------! This is a 1-dimensional plane geometry.
! ---------------------------------------------------------------------subroutine howfar
implicit none
include ’include/egs5_h.f’
include ’include/egs5_epcont.f’include ’include/egs5_stack.f’! Main EGS "header" file
! COMMONs required by EGS5 codecommon/passit/zthick
real*8 zthick
real*8 deltaz
integer irnxt! Local variables

if (ir(np).ne.2) thenidisc = 1
returnend ifdnear(np) = dmin1(z(np),zthick-z(np))!----------------------------------! Particle going parallel to planes!----------------------------------if(w(np).eq.0) return368

!-------------------------------------------------------! Check forward plane first since shower heading that way! most of the time!-------------------------------------------------------if (w(np).gt.0.0) thendeltaz=(zthick-z(np))/w(np)irnxt=3!----------------------------------------------------------! Otherwise, particle must be heading in backward direction.
!----------------------------------------------------------elsedeltaz=-z(np)/w(np)irnxt=1
end ifif (deltaz.le.ustep) thenustep=deltaz
irnew=irnxt
end ifreturnend!--------------------------last line of howfar.f------------------------

369

Appendix C

PEGS USER MANUAL
Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANAlex F. Bielajew and Scott J. Wilderman
Department of Nuclear Engineering and Radiological Sciences
The University of Michigan
2355 Bonisteel Boulevard
Ann Arbor, MI 48109, USA
Walter R. Nelson
Department Associate in the Radiation Physics Group (retired)
Radiation Protection Department
Stanford Linear Accelerator Center
2575 Sand Hill Road Menlo Park, CA 94025, USA
This PEGS User Manual is Appendix C of a document called
SLAC-R-730/KEK-2005-8, which can be obtained from the SLAC and KEK web sites.

370

C.1

Introduction

PEGS (Preprocessor for EGS) is a set of FORTRAN subprograms which generate material datafor use with the EGS5 Monte Carlo shower code and which also provide utilities for researchers
studying electro-magnetic interactions. The active operations of PEGS are functionals; that is,they are operations whose arguments are functions (the functions related to physics interactions).
Included among these operations are:• Fitting of functions by means of piecewise linear fits.
• Production of printed plots of selected functions.
• Evaluation of functions at selected points.
• Comparison of functions with sampled spectra.
Associated with these active functionals are other operations:• Selection of material to which the functions refer.
• Selection of energy cutoffs for fits.
• Printing of fitted data.
This manual describes the full general functionality of PEGS. Users interested only in preparing
data sets for EGS5 can proceed directly to section C.3.

C.2

Structural Organization of PEGS

Beginning with version 5 of the EGS code system, MORTRAN is no longer supported, and PEGS
is no longer a stand-alone program, but rather a subroutine to be called by EGS user code. The
main PEGS subroutine (called PEGS5), however is essentially the same as the MAIN program inprevious versions of PEGS, and EGS and PEGS are still quite distinct in EGS5.
The full PEGS5 system consists of almost 7000 of FORTRAN source code that make up over
100 subprograms, plus an additional 610 lines of FORTRAN in 40 files containing COMMON blocks
accessed by the various PEGS subprograms through include statements. New to version 5 ofPEGS, all floating point variables are declared as double precision.
Despite its size, PEGS has a simple structure. Figures C.1 and C.2 below contain a schematic
flowchart of main subroutine (PEGS5) of PEGS. After the once-only initializations, an option loop371

is entered. On each pass through this loop, an option is read (option names are four charactersand are read as 4A1), numeric control parameters are read (using the FORTRAN NAMELIST I/O
extension), and then the option name is looked up in the option table. If an option name is notfound, the job is aborted. When an option name is found, the appropriate code is executed and theprogram returns to the beginning of the option loop. Normal exit from the loop is by selection of
the STOP option or detection of an End-of-File condition on the input file. Detailed descriptionsof the use of the options are contained in section C.3 of this manual.
Figures C.3 and C.4 show some of the subprogram relationships of PEGS1 . Boxed items in
the figures are subprograms, and option names (identified by “:” delimiters, as in “:CALL:”) areused to show which subprograms correspond to which options. The general structure of PEGS
is clearly evident. Subprograms which compute physical quantities are accessed directly by the
PWLF option, but when the utility options TEST, PLTN, PLTI, HPLT, or CALL are invoked,
physics routines are referenced using the function FI, the so-called “function multiplexer.” Thefunction multiplexer FI has five arguments. The first argument (I) tells which physical function to
invoke, and the other four arguments (X1, X2, X3, X4) are used as needed as arguments for thecalled function. Calls to FI return values returned to FI by the called function specified by I. The
full list of physical functions which can be called by FI is given in Figure C.4.
It should also be noted that there are relationships between the PEGS functions shown in
Figure C.3 which are not indicated there. We show the most complicated of these in Figures C.5
and C.6 (Bremsstrahlung Related Functions) and in Figures C.7 and C.8 (Pair Production Related
Functions). One reason for the complexity of these particular subprograms is that higher level formsof the relevant cross sections must be obtained by numerical integration of the more differential
forms.
Tables C.1 and C.2 list the subroutines used in PEGS and provide brief descriptions of theirfunctions. All of the subroutines in Table C.2 are new to PEGS5 and deal with either the calculationof the Goudsmit-Saunderson multiple scattering distribution or the determination of electron energy
loss steps. Note that in the current implementation, when the Goudsmit-Saunderson distribution
is requested, the routines which prepared the data are actually called by EGS during the set up
operations performed in HATCH. We nevertheless refer to these new subprograms as part of PEGS,
as they do involve pre-processing of data.
Tables C.3 through C.6 list the FORTRAN FUNCTIONS used in PEGS along with their mathematical symbols and definitions if appropriate. The names of most of the functions have beenchosen in a mnemonic way. The first three or four letters suggest the process being considered.
The last letter designates the form of the cross section (Z for element, M for mixture, and R for
run-time mixture). The next to last letter describes either the particular form of the cross section(such as D for differential, T for total or R for range-integrated), or it indicates that only the secondary energy is to vary, with other data being passed through a common. The letter F is used insuch cases and the data in common is initialized using the corresponding function that has a next
to last letter of D. If the function word begins with an I through N (i.e., the FORTRAN integer
1

Subprograms involving the low energy physics and the multiple scattering models new to PEGS with the releaseof EGS5 are not included in the diagrams.

372

+------+| PEGS |+------+| |+-------------+
initialize | +------>| OPTION LOOP |<---++--------------++-------------+
A
|
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|
V
V
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+------------------------++-----------------+ ||Compute Physical &||Read Option| || Mathematical Constants ||Name (4A1)| |+------------------------+
|
&
| |
|
|
Read Control | |||Parameters| |V| (NAMELIST/INP/) | |+------------------------++-----------------+ || Read Pair Production, ||||Photoelectric, and||||Rayleigh Data from|||
! PGS5PEPR and PGS5FORM |||+------------------------+||
V|+------+
Yes+--------------+|| Stop |<------------ | End of File? ||+------++--------------+
|
|
|
| No
|
V
|
+-----------------------++---------------+|| Illegal Option---Stop |<----- | Select Option ||+-----------------------++---------------+
|
|
|
V
|
+<------------------------------------ +
|
|
|
V
|
(to next figure)(from next figure)
Figure C.1: Flowchart of the PEGS5 subprogram of PEGS, part 1.

373

(from previous figure)(to previous figure)|A|+-----------------------+|+-->:ELEM: | Set Up Element Medium | -------------------> ||+-----------------------+||+-----------------------+|+-->:MIXT: | Set Up Mixture Medium | -------------------> ||+-----------------------+||+------------------------+|+-->:COMP: | Set Up Compound Medium | ------------------> ||+------------------------+||+------------------------+|+-->:ENER: | Set Energy Cutoffs and | ------------------> ||| Compute Thresholds|||+------------------------+||+---------------------+|+-->:PLTN: | Plot Named Function | ---------------------> ||+---------------------+||+-----------------------+|+-->:PLTI: | Plot Indexed Function | -------------------> ||+-----------------------+||+-----------------------+|+-->:HPLT: | Histogram Theoretical | -------------------> ||| vs Sampled Spectrum |||+-----------------------+||+-------------------------+|+-->:CALL: | Evaluate Named Function | -----------------> ||+-------------------------+||+-----------------------------+|+-->:TEST: | Plot Functions To Be Fitted | -------------> ||+-----------------------------+||+----------------------+|+-->:PWLF: | Piecewise Linear Fit | --------------------> ||+----------------------+||+------------------------+|+-->:DECK: | Print File of Material | ------------------> +|| Dependent Data||+------------------------++-->:STOP:
Figure C.2: Flowchart of the PEGS5 subprogram of PEGS, part 2.

374

+------------+| BLOCK DATA |+------------++------+| MAIN |+------+|+ --- + - + ----- + ----- + ----- + ----- + ---- + --- +
|
|
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|
|
|
|
|
:PWLF: :DECK: :TEST: :HPLT: :CALL: :ENER:||||:PLTN:||+------+ ||+---+ :PLTI: +-----+||PMDCON| |
|
|LAY|
|
|HPLT1||+------+ |
|
+---+
|
+-----+|||+----+||:ELEM:+---+----+|PLOT|||:MIXT:||+----+||:COMP: +-----+ +-----+|||||EBIND| |PWLF1|+-----> | <--- +|+-----+ +-----+|+ ----- + ----- +
|
|
|
|
|
+----+|+---+ +------+ +------+ |QFIT|(to next figure)|MIX| |SPINIT| |DIFFER| +----++---+ +------+ +------+|+ - + --- + --------------- +|||+-----++-----++-----+|EFUNS||GFUNS||RFUNS|+-----++-----++-----+|||+-----+----+-----+--+---+---+ |+------++-----+|||||| +---|PHOTTE||AINTP|+------+ | +------+ | +------+ | |+------++-----+|SPTOTP| | |ANIHTM| | |BHABTM| | |+------++------+ | +------+ | +------+ | +---|COMPTM|||| |+------++------+ | +------+ | +------+ | |+------+|SPTOTE|-+ |AMOLTM|-+ |BREMTM|-+ +---|PAIRTU|+------++------++------+|+------+|+------++---|COHETM|+------+
Figure C.3: Subprogram relationships in PEGS, part 1.

375

(from previous figure)||+ ---------------------- +|FI|| "Function Multiplexer" |+ ---------------------- +|+ ------------------------------------- +
|
|
|
ALIN
APRIM
COMPFM
PAIRTE||ALINI
BHABDMCOMPRM
PAIRTM||
BHABFM
COMPTM
PAIRTR||ADFMOLBHABRMCRATIOPAIRTU||ADIMOLBHABTMEBIND
PAIRTZ||ADDMOL
BREMDR
EBR1
PBR1
|
|
DLOG
BREMFR
EDEDX
PBR2
|
|
DEXP
BREMDZ
ESIG
PDEDX
|
|
AREC
BRMSDZFCOULCPHOTTZ||ALKE
BREMFZ
GBR1
PHOTTE||ALKEI
BRMSFZ
GBR2
PSIG
|
|
AMOLDM
BREMRR
GMFP
SPIONE||AMOLFM
BREMRM
PAIRDRSPIONP||AMOLRM
BREMRZPAIRFRSPTOTE||AMOLTM
BREMTM
PAIRDZSPTOTP||ANIHDM
BREMTRPAIRFZTMXB||
ANIHFM
BRMSRM
PAIRRM
TMXS||
ANIHRM
BRMSRZPAIRRR
TMXDE2||ANIHTM
BRMSTM
PAIRRZXSIF||
COHETM
EEITM||
COHETZ||COMPDM|+ ------------------------------------- +Figure C.4: Subprogram relationships in PEGS, part 2.

376

+------++------+|BREMTM|---------->|BREMRM|+------++------+|
V+------++------+ initialize+------+| QD |<------------- |BREMRZ|-------------> |BREMDZ|+------++------+
BREMFZ+------+||
V
V+------++------++------++------+|DCADRE|-->|BREMFZ|--->|BRMSFZ|<------------- |BRMSDZ|+------++------++------++------+
A
A |
|
| |
+------++---------------+| +-->|APRIM ||| |+------++------++------+
initialize| ||DCADRE|+----|BRMSRZ|---------------> + |+------++------+|+------+
BRMSFZ+-->| XSIF |A|A|+------+||||+------+|+------+|+------+| QD |<--+|BRMSRM|+-->|FCOULC|+------++------++------+A|+------++------++------+|SPTOTE|-----------> |BRMSTM|<----------- |SPTOTP|+------++------++------+||
V
V+------++------++------+|SPIONE|-----------> |SPIONB|<----------- |SPIONP|+------++------++------+
Figure C.5: Bremsstrahlung related functions—most accurate form (used to produce the total crosssections and stopping power).

377

+------+|BREMTR|+------+|
V+------+|BREMRR|+------+|
initialize V+ <---------+--------> +| BREMFR|
V
V+------++------+|BREMDR|| QD |+------++------+|||
V|+------+||DCADRE||+------+|+------+|+ ----> |BREMFR| <---- ++------+
Figure C.6: Bremsstrahlung related functions—with run-time approximations (for comparison withsampled spectra).

378

+------+|PAIRTU|+------+| |+ <------------------- + + -------------------> +||VV+------++------+|PAIRTM||PAIRTE|+------++------+||
V
V+------++------+|PAIRRM||PAIRTZ|+------++------+||
V
V+------+
initialize+------+|PAIRRZ|-------------> +|AINTP |+------+
PAIRFZ|+------+||
V
V+------++------++------+| QD ||PAIRDZ|-----+----> | XSIF |+------++------+|+------+|||+------+||+----> |FCOULC|
V
V+------++------++------+|DCADRE|----------> |PAIRFZ|+------++------+
Figure C.7: Pair production related functions—most accurate form (used to produce the total crosssections and stopping power).

379

+------+|PAIRTR|+------+|
V+------+|PAIRRR|+------+|
initialize V+ <-------- + -------> +| PAIRFR|
V
V+------++------+|PAIRDR|| QD |+------++------+|||
V|+------+||DCADRE||+------+|+------+|+ ----> |PAIRFR| <---- ++------+
Figure C.8: Pair production related functions—with run-time approximations (for comparison With
sampled spectra).

380

Name
DIFFER
EFUNS
GFUNS
HPLT1
LAY
MIX
PLOT
PMDCONPWLF1RFUNSSPINITSFUNSCFUNS3
CFUNS4EIIFUNS
MAKEK1

Table C.1: Subroutines in PEGS.
Description
Determines the various parameters needed for bremsstrahlung and pair production energy sampling.
Subprogram to compute electron functions to be fit in a way that avoids
repetition.
Subprogram to compute photon functions to be fit in a way that avoids
repetition.
Creates line printer plot comparisons of EGS-sampled data (UCTESTSR
User Code) and theoretical functions of PEGS.
Subprogram to produce a file of material dependent data (for subsequent
use by EGS).
Computes Z-dependent parameters that reside in COMMON/MOLVAR/.
Subprogram to plot a given function (referenced by number).
Determines the physical, mathematical, and derived constants in a verymnemonic way.
Subprogram to piecewise linearly fit up to 10 functions simultaneously on
an interval (XL,XU).
Subprogram to compute Rayleigh scattering functions to be fit in a way thatavoids repetition.
Initializes stopping power functions for a particular medium.
Subroutine to compute incoherent scattering functions to be fit.
Subroutine to compute c.d.f. of shell-wise Compton profile to be fit.
Subroutine to compute shell-wise Compton profile to be fit.
Subroutine to compute EII/Møller ratio to be fit.
Prepares tables of initial scattering strength to be fit.

381

Name
DCSLOADDCSSTORDCSTAB
ELASTINO
ELINITESTEPLIM

FINDI
FITMS
GAULEG
GSCOEF
GSDIST
INIGRDINTEG
LEGENP
PRELASTINO
SPLINEWMSFIT

Table C.2: Goudsmit-Saunderson-related subroutines in PEGS.
Description
Loads molecular elastic differential cross sections tables for multi-material
PEGS runs.
Stores molecular elastic differential cross sections tables for multi-material
PEGS runs.
Computes tables of the molecular elastic differential cross sections for electrons and positrons.
Prepares cumulative Goudsmit-Saunderson multiple scattering distributions
on problem dependent energy and scattering strength grids.
Reads differential elastic scattering cross section data for electrons andpositrons.
Computes the maximum electron energy hinge step (in units of fractionalenergy loss) which assures a given error tolerance (currently fixed at 0.1%) in
the total energy loss and total scattering power accumulated over the step.
Does table look-up.
Prepares special fit to Goudsmit-Saunderson multiple scattering data for fastsampling in EGS.
Provides abscissas and weights for Gauss-Legendre quadrature.
Computes Goudsmit-Saunderson transport coefficients.
Computes Goudsmit-Saunderson multiple scattering probability density
functions.
Initializes energy and scattering strength grid tables.
Integrates cubic spline data.
Computes Legendre polynomials.
Writes multiple scattering grid parameters for later consistency checks.
Computes cubic splines.
Writes fitted Goudsmit-Saunderson multiple scattering data parameters.

382

convention) the word is prefixed with the letter A. A few examples are given below:
AMOLDM
BREMDR
BREMRM
BRMSTM
PAIRRRPAIRTZis the differential Møller cross section for a mixture of elements.
is the differential bremsstrahlung cross section for a run-time mixture ofelements.
is the bremsstrahlung cross section, integrated over some energy range, fora mixture of elements.
is the soft bremsstrahlung total cross section for a mixture of elements.
is the pair production cross section, integrated over some energy range, fora run-time mixture of elements.
is the total cross section for pair production for an element.

Note that this method of naming is not strictly adhered to, however. For example, SPIONE is theionization stopping power for an electron, PBR1 and PBR2 are positron branching ratios, and GMFP
is the gamma-ray mean free path.

C.3

PEGS Options and Input Specifications

C.3.1

Interrelations Between Options

Figure C.9 illustrates the logical relationship between options of PEGS. For example, in order tobe able to use the PLTN option, one of the material specification options (ELEM, MIXT, COMP)
must have already been processed. The PWLF option requires that both the ENER option andone of the material specification options precede it. To use the DECK option, it is sufficient tohave validly invoked the PWLF option. The STOP option is seen to be independent of the others.
In the following sections we give for each option its function, parameters which control it, theformat of the data lines in a PEGS input file needed to invoke it, and an explanation of the routines(if any) that are used to implement it. The data lines for a given option are named with the first partof their name being the option name, and the last part the line number. For example, “MIXT2”is the name of the second line needed for the MIXT option. The information is summarized inTables C.7 through C.18. The single line referred to as being read by NAMELIST may in fact beseveral lines, provided that the proper convention for continuing NAMELIST lines is followed. Oncethe first line (indicating the option) has been read in, however, the second line (i.e., NAMELIST/INP/)
must follow (see examples at the end of the next section). It should also be noted that different
compilers may require different formats for NAMELIST data.

383

Table C.3: Functions in PEGS, part 1.
Name
AFFACT
AINTP
ALKE
ALKEI
ALIN
ALINI
ADFMOLADIMOLADDMOLAMOLDMAMOLFMAMOLRMAMOLTM
ANIHDM
ANIHFM
ANIHRM
ANIHTMAPRIM
AREC
BHABDM
BHABFM
BHABRM
BHABTM

Description
Atomic form factor (squared) for an element or mixture of elements.
Linear or log interpolation function.
Log of kinetic energy (LOG(E-RM)), used as a cumulative distribution function for fits and plots.
Inverse of ALKE (=EXP(X)+RM).
Linear cumulative distribution function for plots (ALIN(X)=X).
Inverse of ALIN (=same as ALIN). Used as inverse cumulative distributionfunction in plots.
Approximate cumulative distribution function for Møller and Bhabha crosssections (ADFMOL(E)=-1/(E-RM)).
Inverse of ADFMOL.
Derivative of ADFMOL.
Møller differential cross section for a mixture of elements.
One argument form of AMOLDM.
Møller cross section, integrated over some energy range, for a mixture ofelements.
Møller total cross section for a mixture of elements.
Annihilation differential cross section for a mixture of elements.
One argument form of ANIHDM.
Annihilation cross section, integrated over some energy range, for a mixtureof elements.
Annihilation total cross section for a mixture of elements.
Empirical correction factor in bremsstrahlung cross section.
Reciprocal function (=derivative of ALOG(X)). Used as probability density
function in log plots (AREC(X)=1/X).
Bhabha differential cross section for a mixture of elements.
One argument form of BHABDM.
Bhabha cross section, integrated over some energy range, for a mixture ofelements.
Bhabha total cross section for a mixture of elements.

384

Table C.4: Functions in PEGS, part 2.
Name
BREMDR
BREMFR
BREMDZBREMFZBREMRM
BREMRR
BREMRZBREMTM
BREMTR
BRMSDZBRMSFZBRMSRM
BRMSRZBRMSTM
COHETM
COHETZCOMPDM
COMPFMCOMPRM
COMPTMCPRFILCRATIOCSDARESIGDCADRE
DCSEL
DCSN
EBIND
EBR1
EDEDX

Description
Bremsstrahlung differential cross section for a run-time mixture of elements.
One argument form of BREMDR.
Bremsstrahlung differential cross section for an element.
One argument form of BREMDZ.
Bremsstrahlung cross section, integrated over some energy range, for a mixture of elements.
Bremsstrahlung cross section, integrated over some energy range, for a runtime mixture of elements.
Bremsstrahlung cross section, integrated over some energy range, for anelement.
Bremsstrahlung total cross section for a mixture of elements.
Bremsstrahlung total cross section for a run-time mixture of elements.
Soft bremsstrahlung differential cross section for an element.
One argument form of BRMSDZ.
Soft bremsstrahlung cross section, integrated over some energy range, for a
mixture of elements.
Soft bremsstrahlung cross section integrated over some energy range, for anelement.
Soft bremsstrahlung total cross section for a mixture of elements.
Coherent (Rayleigh) total cross section for a mixture of elements.
Coherent (Rayleigh) total cross section for an element.
Compton differential cross section for a mixture of elements.
One argument form for COMPDM.
Compton cross section, integrated over some energy range, for a mixture ofelements.
Compton total cross section for a mixture of elements.
Shell-wise Compton profiles.
Coherent (Rayleigh) cross section ratio.
Returns previously computed and stored values of CSDA range for electronsand positrons.
Determines the total electron(-) interaction cross section (probability perradiation length).
Quadrature routine to integrate f(x) between a and b using cautious
Romberg extrapolation.
Computes differential elastic scattering cross sections from data which has
been fitted by cubic splines.
Computes arbitrary Goudsmit-Saunderson elastic scattering transport coefficients from cross section data which has been fitted by cubic splines.
Function to get an average photo-electric binding energy.
Function to determine the electron(-) branching ratio (brem/total).
Evaluates SPTOTE with cutoff energies of AE and AP.
385

Table C.5: Functions in PEGS, part 3.
Name
EIITM
ESTEPMAXFCOULCFI
G1E
G1EDEDX
G1PDEDX
GBR1
GBR2
GMFP
K1E
IFUNT
PAIRDR
PAIRDZPAIRFRPAIRFZPAIRRM
PAIRRRPAIRRZPAIRTEPAIRTM
PAIRTRPAIRTUPAIRTZ

Description
K shell EII cross section
Returns previously computed and stored values of maximum electron energy
hinge steps.
Coulomb correction term in pair production and bremsstrahlung cross sections.
Function multiplexer.
Computes scatter powering for positrons and electrons from either splines ofpartial wave cross sections or analytical form of Molière cross section.
Computes product of first elastic transport cross section and inverse stopping
power for electrons.
Computes product of first elastic transport cross section and inverse stopping
power for positrons.
Function to determine the gamma-ray branching ratio (pair/total).
Functiontodeterminethegamma-raybranchingratio((pair+Compton)/total).
Function to determine the gamma-ray mean free path.
Computes scattering strength for electrons and positrons from spline data.
Given PEGS function name, it looks it up name in table and returns thefunction index. Used by options that specify functions by name.
Pair production differential cross section for a run-time mixture of elements.
Pair production differential cross section for an element.
One argument form of PAIRDR.
One argument form of PAIRDZ.
Pair production cross section, integrated over some energy range, for a mixture of elements.
Pair production cross section, integrated over some energy range, for a runtime mixture of elements.
Pair production cross section, integrated over some energy range, for anelement.
Empirical total pair production production cross section for a mixture(=SUM(PZ(I)*PAIRTZ(Z(I))).
Pair production total cross section for a mixture of elements, obtained bynumerical integration of differential cross section.
Pair production total cross section for a run-time mixture of elements.
Pair production total cross section actually used. Same as PAIRTE for
primary energy less than 50 MeV; otherwise, same as PAIRTM.
Computes contribution to empirical pair production total cross section for
an element assuming one atom per molecule. It is obtained by log-linear
interpolation of Storm-Israel data.

386

Table C.6: Functions in PEGS, part 4.
Name
PBR1
PBR2
PDEDX
PHOTTEPHOTTZ
PSIG
QD
QFIT
SUMGA
SPIONBSPIONESPIONPSPTOTESPTOTP
TMXB
TMXS
TMXDE2XSIF
ZTBL

Description
Function to determine the positron branching ratio (brem/total).
Function to determine the positron branching ratio ((brem+Bhabha)/total).
Evaluates SPTOTP with cutoff energies of AE and AP.
Determines the proper mix of PHOTTZ’s for a mixture.
Determines the interpolated total photoelectric cross section from tabulated
data.
Determines the total positron interaction cross section (probability per radiation length).
Driver function for DCADRE, the numerical integration routine.
Utility logical function for the piecewise linear fit subroutine, PWLF1. It
returns .TRUE. if a given partition gives a good fit.
Integrates a function using a 20-point Gauss quadrature method with anadaptive bipartition scheme.
Does the work for SPIONE and SPIONP. One argument tells whether tocompute stopping power for electron or positron.
Calculates the stopping power due to ionization for electrons(-).
Calculates the stopping power due to ionization for positrons.
Calculates the total stopping power (ionization plus soft bremsstrahlung) for
electrons(-) for specified cutoffs.
Calculates the total stopping power (ionization plus soft bremsstrahlung) for
positrons for specified cutoffs.
Determines the maximum total step length consistent with Bethe’s criterion.
Determines the minimum of TMXB and 10 radiation lengths.
Included for possible future modification purposes (TMXDE2 = TMXB / (E**2* BETA**4)). It might be easier to fit this quantity than to fit TMXB andthen apply the denominator in EGS at run-time.
Function to account for bremsstrahlung and pair production in the field of
the atomic electrons.
Given the atomic symbol for an element, it returns the atomic number.

387

+------++------++------++------+|:ELEM:||:MIXT:||:COMP:||:ENER:|+------++------++------++------+
|
|
|
|
|
V
|
|
+ ------> OR <-------- +||V+---------------------------> AND ----+
|
|
|
V
|
|
+ -------+----+----+------- +
|
|
|
|
|
|
|
|
V
V
V
V
|
|
+------+ +------+ +------+ +------+ +------+|||:PLTN:| |:PLTI:| |:HPLT:| |:CALL:| |:TEST:|<---+|+------+ +------+ +------+ +------+ +------+
V+------+|:PWLF:|+------+|
V+------++------+|:STOP:||:DECK:|+------++------+
Figure C.9: Logical relationship between the options of PEGS.

C.3.2

The ELEM, MIXT, COMP Options

The purposes of the ELEMent, MIXTure, and COMPound options are to specify the materials
used by the PEGS functions. The parameters needed to specify a material are its density (RHO),
the number of different kinds of atoms (NE), and, for each different kind of atom, its atomic number(Z(I)), its atomic weight (WA(I)), and its proportion either by number (PZ(I)) for a compound or
by weight (RHOZ(I)) for a mixture. PEGS has tables for the atomic symbol (ASYMT(1:100)) andthe atomic weight (WATBL(1:100)) for elements I=1 through I=100, so the type of atom is specified
by giving its atomic symbol (ASYM(I)). PEGS also has a table of the densities of the elements(RHOTBL(1:100)).
The ELEMent option is used if the material being used has only one type of atom. In this case
PEGS defines that NE=1, takes the density from its internal data table, sets PZ(1)=1, and deduces
Z(1) and WA(1) from ASYM(1). Thus the atomic symbol (ASYM(1)) is the only information thatthe user need supply. If either RHO or WA(I) are specified by the user, they override the table values
in PEGS. This allows the different atoms to be non-standard isotopes and/or allows the overall
density to be adjusted to the experimental state.
388

Table C.7: ELEM option input data lines in PEGS, part 1.
Line and format
ELEM1 (4A1)
ELEM2 NAMELIST

Variables
OPT(1:4)
RHO
WA(1)
IRAYL

IUNRST

IAPRIMISSB
EPSTFLFUDGEMSIBOUND

INCOH

ICPROF

Comments‘ELEM’. Means “select material that is an element.”
Optional. If given, this overrides the PEGS default
density (g/cm3 ) for the element.
Optional. Atomic weight of element. If given, thisoverrides the PEGS default.
Optional. Set to 1 (or 2, which requires additionaldata, see below) to model Rayleigh scattering. (default=0, ignore).
Optional. For printing stopping powers (default = 0,
restricted stopping powers, for EGS runs).
1 unrestricted collision only
2 unrestricted collision and radiative
3 unrestricted collision, restricted radiative. 4 restricted collision, unrestricted radiative
5 unrestricted radiative only
6 restricted radiative only
7 restricted collision only
Optional. For correction to bremsstrahlung cross section. Default (=1), normalizes integrated cross section to ICRU-37 radiative stopping power; set to 0
uses Motz et al.empirical; set to 2 uses no correction.
Optional. Set to 1 to use user-supplied density effectparameters (see below).
Optional. Set to 1 to use user-supplied density effectcorrections (see below).
Optional. Constant in Z(Z + ξM S ) which describes
the contribution of soft electron collisions to multiple
scattering (default = 1.0).
Optional. Set to 1 to use bound total Compton crosssection (default = 0, use free Compton cross section).
Optional. Set to 1 to use incoherent scattering function to model electron binding effect on Compton
scattering angles (default = 0, ignore).
Optional. Set to 3,4,-3 or -4 (see below) to use shellwise Compton profiles to get Compton scattering energies. (default = 0, ignore).

389

Table C.8: ELEM option input data lines in PEGS, part 2.
Line and format
ELEM2 (cont) NAMELIST

Variables
IMPACT

NLEG0

EFRACHEFRACL

ELEM3 (24A1,6X,24A1)

MEDIUM
IDSTRN

ELEM4 (24(A2,1X))

ASYM(1)

Comments
Optional. For selecting electron impact ionization
cross section (default = 0, ignore):
1 Casnati’s formula
2 Kolbenstvedt’s rev. formula
3 Kolbenstvedt’s formula
4 Jakoby’s formula
5 Gryziński’s formula
6 Gryziński’s relativistic. formula
Optional. Number of Legendre coefficients in computing Goudsmit-Saunderson distribution (default =1000).
Optional and not recommended. Fractional energy
loss over a multiple scattering step at E = UE (default= 0.05).
Optional and not recommended. Fractional energy
loss over a multiple scattering step at E = AE (default= 0.20).
Identifier assigned to data set to be produced.
Optional.
Identifier of media name of desired
Sternheimer-Seltzer-Berger density effect coefficientsin PEGS (see Table 2.3 of SLAC-R-730/KEK-2005-8
for a list of default media supplied with the EGS5 distribution.) If not specified, MEDIUM(1:24) is used.
Atomic symbol for element.

390

Table C.9: COMP option input data lines in PEGS.
Line and Format
COMP1 (4A1)

Variables
OPT(1:4)

COMP2 NAMELISTNE
RHO(PZ(I),I=1,NE)
GASP(WA(I),I=1,NE)

COMP3 (24A1,6X,24A1)
COMP4 (24(A2,IX))IRAYLIUNRST
IAPRIMISSB
EPSTFLFUDGEMSIBOUND
INCOH
ICPROF
IMPACT
NELG0
EFRACHEFRACLMEDIUM
IDSTRN(ASYM(I),I=1,NE)

391

Comments‘COMP’. Means “select material that is acompound.”Number of elements in compound.
Density (g/cm3 ) of compound (at NTP forgases).
Relative numbers of atoms in compound.
Optional. Defines state of compound:zero (default) for solid or liquid, otherwise
value gives gas pressure (atm).
Optional. May be used to override default
atomic weights (e.g., to allow for special
isotopes).
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 3.
Same as for ELEM, line 3.
Atomic symbols for the atoms in the compound. Duplicates are allowed if several
isotopes of the same element are present,
or may be required for diatomic molecules(e.g. nitrogen gas).

Table C.10: MIXT option input data lines in PEGS.
Line and Format
MIXT1 (4A1)

Variables
OPT(1:4)

MIXT2 NAMELISTNERHO(RHOZ(I),I=1,NE)
GASP(WA(I),I=1,NE)

MIXT3 (24A1,6X,24A1)
MIXT4 (24(A2,1X))IRAYL
IUNRST
IAPRIMISSB
EPSTFLFUDGEMSIBOUND
INCOH
ICPROF
IMPACT
NELG0
EFRACHEFRACLMEDIUM
IDSTRN(ASYM(I),I=1,NE)

Comments‘MIXT’. Means “select material that is a
mixture.”Number of elements in mixture.
Density (g/cm3 ) of mixture (at NTP forgases).
Relative amount of atom in mixture (byweight).
Optional. Defines state of mixture: zero(default) for solid or liquid, otherwise
value gives gas pressure (atm).
Optional. May be used to override default
atomic weights.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 2.
Same as for ELEM, line 3.
Same as for ELEM, line 3.
Same as for COMP, line 4.

392

Table C.11: ENER option input data lines in PEGS. Note that if the user supplies negative values
for the energy limits above, the absolute values given will be interpreted as being given in units ofthe electron rest mass energy. Thus, AE=-1 is equivalent to AE=0.511 MeV.
Line and Format
ENER1 (4A1)
ENER2 NAMELIST

Variables
OPT(1:4)
AE
UE
AP
UP

Comments‘ENER’. Means “select energy limits.”
Lower cutoff energy (total) for charged particle transport (MeV).
Upper limit energy (total) for charged particle transport (MeV).
Lower cutoff energy for photon transport (MeV).
Upper limit energy for photon transport (MeV).

The COMPound option is used when there is more than one different kind of atom and it isdesired to give the proportions by relative number of atoms (PZ(I)). The only required data is NE,
ASYM(I), and PZ(I) (for I=1,NE). Optionally, any of the values of WA(I) taken from the default
PEGS tables can be overridden if specified by the user.
The MIXTure option is similar to the COMPound option except that the relative atomic proportions are given by weight (RHOZ(I)) rather than by number. When the PZ(I) values havebeen specified, PEGS obtains values of RHOZ(I) using RHOZ(I)=PZ(I)*WA(I). Otherwise, PZ(I)values are obtained from PZ(I)=RHOZ(I)/WA(I). The absolute normalization of the PZ(I) andRHOZ(I) values is not important because of the way the quantities are used. For example, themacroscopic cross sections contain factors like PZ(I)/SUM(PZ(I)*WA(I)) where the denominatoris the molecular weight.
Users may supply values for the various parameters used in calculating the density effect correction by setting the flag ISSB to 1 and specifying the six parameters, AFACT, SK, X0, X1, CBAR,
and IEV) in the NAMELIST/INP input in ELEM, MIXT, or COMP. Note that if one only wants tooverride IEV, one must still input all six parameters.
Alternatively, users may request values of the density effect correction taken from the NBS(now NIST) database EPSTAR, in which case collision stopping powers calculated by PEGS willcorrespond with values from the ICRU-37 standard. To invoke this option, the flag EPSTFL in
NAMELIST/INP/ should be set to 1, and the appropriate data file from either the elements or thecompounds subdirectory of the density corrections directory of the EGS5 distribution data
files should be copied or linked to a file called “epstar.dat” in the working user code directory. Notethat when this option is employed and multiple materials are being processed, all of the appropriate
data files from density corrections must be concatenated onto epstar.dat in the order in which
the materials are referenced in the PEGS input file.
Rayleigh (coherent) scattering data will be appended to the normal PEGS output data if the393Table C.12: PWLF option input data lines in PEGS. Note: The PWLF parameters above (seesection C.3.4 for longer descriptions) are optional and may be overridden by the user. The default
values (assigned in BLOCK DATA) are indicated above, and all of the parameters are elements of both
NAMELIST/PWLFNM and NAMELIST/INP.
Line and Format
Variables
Comments
PWLF1 (4A1)
OPT(1:4)‘PWLF’. Means “perform piecewise linearfit.”
PWLF2 NAMELIST EPE/0.01/
Electron EP parameter.
EPG/0.01/
Gamma EP parameter.
EPR/0.01/
Rayleigh scattering EP parameter.
EPSF/0.01/
Incoherent scattering function EP parameter.
EPCP/0.03/
Compton profile EP parameter.
ZTHRE(1:8)/8*0./
Electron ZTHR parameter.
ZTHRG(1:3)/0.,.1,0./ Gamma ZTHR parameter.
ZTHRS(1:40)/40*0./
Incoherent scattering function ZTHR parameter.
ZTHRR(1:40)/40*0./
Rayleigh scattering ZTHR parameter.
ZTHRC(1:200)/200*0./ Compton profile ZTHR parameter.
ZEPE(1:8)/8*0./
Electron ZEP parameter.
ZEPG(1:3)/0.,.01,0./ Gamma ZEP parameter.
ZEPS(1:40)/40*0./
Incoherent scattering function ZEP parameter.
ZEPR(1:40)/40*0./
Rayleigh scattering ZEP parameter.
ZEPC(1:200)/200*0./Compton profile ZEP parameter.
NIPE/20/
Electron NIP parameter.
NIPG/20/
Gamma NIP parameter.
NIPS/20/
Incoherent scattering function NIP parameter.
NIPR/20/
Rayleigh scattering NIP parameter.
NIPC/20/
Compton profile NIP parameter.
NALE/150/
Electron NIMX parameter.
NALG/1000/
Gamma NIMX parameter.
NALR/100/
Rayleigh scattering NIMX parameter.
NALS/100/
Incoherent scattering function NIMX parameter.
NALC/2000/
Compton profile NIMX parameter.

394

Table C.13: DECK option input data lines in PEGS.
Line and Format
DECK1 (4A1)

Variables
OPT(1:4)

DECK2

NAMELIST

Comments‘DECK’. Means “print fitted data and other usefulparameters.”
No parameters, but &INP and &END tags must bepresent.

Table C.14: TEST option input data lines in PEGS.
Line and Format
TEST1 (4A1)

Variables
OPT(1:4)

TEST2 NAMELIST

NPTS

Comments‘TEST’. Means “Prepare data for plotting the fitted
functions.”
Optional. Number of points to plot per function (Default=50).

Table C.15: CALL option input data lines in PEGS.
Line and Format
CALL1 (4A1)

Variables
OPT(1:4)

CALL2 NAMELISTCALL3 (6A1)

XP(1:4)
NAME(1:6)

Comments‘CALL’. Means “Call the designated function andprint value.”
Values for up to four arguments of the function.
Name of function to be evaluated.

Table C.16: PLTI option input data lines in PEGS.
Line and Format
PLTI1 (4A1)

Variables
OPT(1:4)

PLTI2 NAMELIST

IFUN
XP(1:4)
IV
VLO
VHI
NPTS
IDF

Comments’PLTI’. Means “Plot function given its index and theindex of the distribution function.”
The index of the function to be plotted.
Values for the static arguments (parameters).
Variable telling which argument is to be varied (e.g.,
IV=2 means plot function vs. its second argument).
Lower limit for argument being varied.
Upper limit for argument being varied.
Number of points to plot.
Index of distribution function used to select independent variable.

395

Table C.17: PLTN option input data lines in PEGS.
Line and Format
PLTN1 (4A1)
PLTN2 NAMELISTPLTN3 (2(6A1))

Variables
OPT(1:4)
XP(1:4), IV,
VLO, VHI, NPTS,
IDF, MP
NAME(1:6)
IDFNAM(1:6)

Comments’PLTN’. Means “Plot the named function.”
Same as PLTI2.

Name (6 characters) of function to be plotted.
Name of distribution function to be used.

Table C.18: HPLT option input data lines in PEGS.
Line and Format
HPLT1 (4A1)

Variables
OPT(1:4)

Comments’HPLT’. Means “Plot histogram to compare thesampled spectrum with the range-integrated andthe differential theoretical values.”
HPLT2 NAMELISTEITest particle total energy (MeV).
ISUB
Variable telling which function is being tested:
1=PAIR 2=COMPT 3=BREMS 4=MOLLER
5=BHABHA 6=ANNIH 7=MSCAT (not implemented).
HPLT3: FORMAT(’ TEST DATA FOR ROUTINE=’,12A1,’,#SAMPLES=’,I10,’,NBINS=’,I5)
NAMESB(1:12) Name of subroutine tested.
NTIMES
Number of samples.
NBINS
Number of histogram bins.
HPLT4: FORMAT(’ IQI=’,I2,’,RNLO,RNHI=’,2F12.8,’,IRNFLG=’,I2)
IQI
Charge of test particle.
RNLO,RNHI
Lower and upper limits to random number preceding call to test function.
IRNFLG
Non-zero means to “apply above limits to preceding random number to test for correlation.” Zerovalue means “don’t do this.”
HPLT5...etc, (9I8) NH(1:NBINS)
Sampled data (from User Code UCTESTSR).

396

IRAYL flag is set to 1 or 2 in NAMELIST/INP/. An IRAYL value of 2 indicates that interference effects
in coherent scattering are to be modeled, the user must supply pre-computed values of the coherent
scattering cross section and coherent scattering form factors for the material in question. This isdone by copying or linking the appropriate data files to files named “ics.dat” and ‘iff.dat” in theworking user code directory. Sample data files in the appropriate format for several materials areincluded in the int coherent cs and int form factor directories of the EGS5 distribution data
set. As with the use of the EPSTFL flag, when multiple materials are being used all of the relevantmaterial data files must be concatenated together in the proper order. Note that an option IRAYL=3
exists in which PEGS will print the Form Factors themselves, rather than cumulative distributions
for sampling in EGS runs, when the user wishes to study this data. Note that EGS will abort if it
encounters a PEGS data file prepared with IRAYL=3.
Altering the value of the NAMELIST/INP/ flag IUNRST causes different versions of the stopping
power to be computed and printed in place of the default required by EGS (i.e., the sum of therestricted collision and radiative stopping powers). This functionality is most useful in extracting
data sets from PEGS-only user codes for comparison with publish results, and should not be
invoked when preparing data files for EGS runs. The available options and their effects are given
in Table C.7.
Users may specify the way in which PEGS normalizes bremsstrahlung cross sections by altering
the value of the NAMELIST/INP/ flag IAPRIM, which has a default value of 1 (see Chapter 2 of
SLAC-R-730/KEK-2005-8 for a complete discussion). Since radiative stopping powers calculated
by default in PEGS comply with ICRU standards, changing IAPRIM is not recommended except
during the development of and testing of new bremsstrahlung cross sections.
Users may input values of the empirical constant which is to be used to represent the contributionof soft electron collisions to multiple nuclear elastic scattering distributions by specifying the valueof the parameter FUDGEMS in NAMELIST/INP/. The default of FUDGEMS is 1, which corresponds toapproximating angular deflection due to electron scattering by replacing the Z 2 term in the nuclear
elastic cross section with Z(Z +1). See Chapter 2 of SLAC-R-730/KEK-2005-8 for a full discussion.
When the parameter IBOUND is set to 1 in NAMELIST/INP/, the effect of electron binding on the
total Compton scattering cross section is taken into account, as described in Chapter 2 of SLACR-730/KEK-2005-8. In the default case (IBOUND = 0), atomic electrons are considered to be free
and at rest for the purpose of modeling Compton scattering.
The effects of electron binding on the angular distribution of Compton scattered photons aremodeled when, in conjunction with setting IBOUND to 1, the user also sets the parameter INCOH to
be 1 in NAMELIST/INP/.
The effects of atomic electron motion on the energy distribution of photons Compton scatteredthrough given angles (this is often referred to as “Doppler broadening”) can be explicitly treatedby specifying the value of the parameter ICPROF. Currently, ICPROF may be set to only 3 (in whichcase the user must supply values of shell-by-shell Compton profiles) or -3, in which case shellwise Compton profile data is read from data files provided with the EGS5 distribution. When
397

ICPROF is set to 3, the user must copy or link the input profile data to a file named “scp.dat” inthe working user code directory. If multiple materials are being processed, data for all materials
must concatenated to scp.dat in the order in which the materials are specified in the PEGS input
file. In PEGS-only user codes designed to generate physics data for examination and study, users
may request that PEGS output raw Compton profile data by specifying ICPROF to be 4 (for user
supplied profile data) or -4 (for PEGS generated profile data). In almost all instances, users wishing
to model Compton Doppler broadening should use ICPROF=-3. Note that EGS requires that IBOUND
and INCOH be set to 1 when modeling of Doppler broadening is requested, and that EGS will not
accept PEGS data sets prepared with ICPROF values of 4 or -4.
If electron impact ionization is to be explicitly treated, the user must specify which cross section
is to be used by defining a non-zero value for the parameter IMPACT in NAMELIST/INP. The list ofavailable cross sections and the values of IMPACT needed to invoke them is given in Table C.8. The
user is referred to Chapter 2 of SLAC-R-730/KEK-2005-8 for a discussion of the derivations andlimitations of the various cross sections.
Electron and positron multiple scattering step sizes to be taken in EGS run may be specified by
the user when the parameters EFRACH and EFRACL are defined in NAMELIST/INP in a PEGS input file.
Including these EGS-related quantities in PEGS arises from the need to compute electron multiple
scattering step sizes in terms of scattering strength for the new EGS5 transport mechanics. The
parameters EFRACH and EFRACL refer to the fractional energy loss which occurs over the step atenergies corresponding to UE and AE respectively. See Chapter 2 of SLAC-R-730/KEK-2005-8 for a
detailed description of multiple scattering step-size selection. Note that it is strongly recommended
that users not use EFRACH and EFRACL to control electron step sizes, but instead specify problem“characteristic dimensions,” as described in the EGS User Manual and Chapter 2 of SLAC-R730/KEK-2005-8.
In addition to physically specifying the material being used, a unique name must be supplied(MEDIUM(1:24)) for identification purposes. This name is included in the output file when the
DECK option is selected. One useful feature of this implementation is that different names canbe used for data sets created with the same materials but different parameters. For example, one
might produce PEGS output using a particular material but different energy limits (or fit tolerances,density effect parameters, etc.), with separate identification names for each (e.g., FE1, FE2, etc.).
The quantity IDSTRN(1:24) is used to identify the Sternheimer-Seltzer-Berger density effectparameters that are tabulated in BLOCK DATA (see Chapter 2 of SLAC-R-730/KEK-2005-8 for acomplete list of identifier names and a general discussion of the density effect). If IDSTRN(1) isblank, then IDSTRN is given the same value as MEDIUM. If this name is not identifiable with anyof those in BLOCK DATA, the Sternheimer density effect scheme is replaced with a general formulaby Sternheimer and Peierls. Note that the options for computing stopping powers and the density
effect invoked by either ISSB or EPSTFL take precedence over the specification of IDSTRN.
After reading the input data for these options, subroutine MIX is called in order to compute theZ-related parameters that reside in COMMON/MOLVAR/, subroutine SPINIT is called to initialize thestopping power routines for this material, and subroutine DIFFER is called to compute run-time398

parameters for the pair production and bremsstrahlung sampling routines.
The following are examples of sets of input data lines that can be used with the ELEM, MIXT,
and COMP options:

1. Material - Element is Iron with defaults taken.

Line
ELEM1
ELEM2
ELEM3
ELEM4

Column123456789112345678921234567893123456789412345678..etc.
ELEM&INP &END
IRON
FE
FE

2. Material - Element is Helium-3 with the density and atomic weight overridden by user.

Line
ELEM1
ELEM2
ELEM3
ELEM4

Column123456789112345678921234567893123456789412345678..etc.
ELEM&INP RHO=1.E-2,WA(1)=3 &END
HELIUM-3
HE
HE

3. Material - Compound is sodium iodide, and IDSTRN(1:24) defaults to MEDIUM(1:24).

Line
COMP1
COMP2
COMP3
COMP4

Column123456789112345678921234567893123456789412345678..etc.
COMP&INP NE=2,RHO=3.667,PZ(1)=1,PZ(2)=1 &END
NAI
NA I

4. Material - Compound is polystyrene scintillator (e.g., PILOT-B or NE-102A) with data taken
from: “Particle Properties Data Booklet, April 1982” (Physics Letters 111B, April 1982).
Sternheimer-Peierls default.

Line
COMP1
COMP2
COMP3
COMP4

Column123456789112345678921234567893123456789412345678..etc.
COMP&INP NE=2,RHO=1.032,PZ(1)=1,PZ(2)=1.1 &END
POLYSTYRENE SCINTILLATOR
C H

5. Material - Mixture is lead glass, consisting of five specified elements (and 1 per cent of the
trace elements unspecified). Sternheimer-Peierls default.
399

Line
MIXT1
MIXT2
MIXT3
MIXT4

Column123456789112345678921234567893123456789412345678..etc.
MIXT&INP NE=5,RHO=3.61,RHOZ=41.8,21.0,29.0,5.0,2.2 &END
LEAD GLASSPB SI O K NA

6. Material - Mixture is U-235, U-238, and carbon (not a real material). Sternheimer-Peierls
default.

Line
MIXT1
MIXT2
MIXT3
MIXT4

Column123456789112345678921234567893123456789412345678..etc.
MIXT&INP NE=3,RHO=16,WA=235,238,RHOZ=50,30,10 &END
JUNK
U U C

7. Material - Element is Iron with shell-wise Compton profile.

Line
ELEM1
ELEM2
ELEM3
ELEM4

Column123456789112345678921234567893123456789412345678..etc.
ELEM&INP IBOUND=1,INCOH=1,ICPROF=-3 &END
IRON
FE
FE

8. Material - Element is Iron with EII cross section by Casnati’s formula.

Line
ELEM1
ELEM2
ELEM3
ELEM4

C.3.3

Column123456789112345678921234567893123456789412345678..etc.
ELEM&INP IMPACT=1 &END
IRON
FE
FE

The ENER Option

The ENERgy option is used to define the electron and photon energy intervals over which it isdesired to transport particles, and hence, over which fits to total cross sections and branchingratios must be made. The electron energy interval is (AE,UE) and the photon interval is (AP,UP).
If any of these parameter is entered by the user with a negative value, it is multiplied by -0.511 MeV;
that is, the absolute magnitude of the input energy is assumed to be in units of the electron rest
mass energy. The quantities TE=AE-RM, TET2=2*TE, and TEM=TE/RM, as well as the bremsstrahlung
and Møller thresholds (RM+AP and AE+TM, respectively), are then computed and printed out.
400

The following are examples of sets of input data lines that can be used with the ENER option:
1. Electron and photon cutoff energies are 1.5 MeV and 10 keV, respectively. The upper energy
limit for both is set at 100 GeV. (Note: All energies are in MeV and are total energies).

Line
ENER1
ENER2

Column123456789112345678921234567893123456789412345678..etc.
ENER&INP AE=1.5,UE=100000.,AP=0.01,UP=100000. &END

2. Same as above, except AE=3*RM.

Line
ENER1
ENER2

C.3.4

Column123456789112345678921234567893123456789412345678..etc.
ENER&INP AE=-3,UE=100000.,AP=0.01,UP=100000. &END

The PWLF Option

The PieceWise Linear Fit option performs a simultaneous piecewise linear fit (vs. ln(E − RM ))of ten or twelve electron functions over the energy interval (AE,UE) and a simultaneous piecewise
linear fit (vs. ln E) of three or four photon functions over the energy interval (AP,UP). Eachsimultaneous fit over several functions is accomplished by a single call to subroutine PWLF1, once
for the electrons and once for the photons. (By simultaneous fit we mean that the same energy subintervals are used for all of the functions of a set.)
The PWLF1 subroutine is an executive routine that calls the function QFIT. Function QFIT tries
to perform a fit to the vector function by doing a linear fit with a given number of subintervals.
It returns the value .TRUE. if the fit satisfies all tolerances and .FALSE. otherwise. Subroutine
PWLF1 starts out doubling the number of subintervals until a successful fit is found. Additional calls
to QFIT are then made to determine the minimum number of subintervals needed to give a goodfit. Sometimes, because of discontinuities in the functions being fitted, a fit satisfying the specified
tolerances cannot be obtained within the constraints of the number of subintervals allowed by thearray sizes of EGS. When this happens, PEGS prints out the warning message (for example):
NUMBER OF ALLOCATED INTERVALS(= 150) WAS INSUFFICIENTTO GET MAXIMUM RELATIVE ERROR LESS THAN
0.01
Even in this case a fit is produced which is sufficient most of the time. Suppose NFUN is the numberof components to the vector function F(IFUN,E(J)) (where IFUN runs from 1 to NFUN) and E(J) isa sequence of points (with J from 1 to NI) covering the interval being fitted. The number of points
NI must be about ten times the number of fit intervals NINT so that the fit will be well tested in the401interiors of the intervals. If FEXACT(IFUN,J) and FFIT(IFUN,J) are the exact and fitted values of
the IFUNth component at E(J), then the flow of the logical function QFIT may be shown as follows:
LOGICAL FUNCTION QFIT(NINT)(commons and declarations)
QFIT=.TRUE.
REM=0.0
NI=10*NINT
DO J=1,NI
DO IFUN=1,NFUN
AER=ABS(FEXACT(IFUN,J)-FFIT(IFUN,J))
AF=ABS(FEXACT(IFUN,J))
IF(AF.GE.ZTHR(IFUN)) THEN
IF(AF.NE.0.0) REM=AMAX1(REM,AER/AF)
ELSE
IF(AER.GT.ZEP(IFUN)) QFIT=.FALSE.
END DOEND DOQFIT=QFIT.AND.REM.LE.EP
RETURN
END
Thus we see that EP is the largest allowed relative error for those points where the absolute computed
value is above ZTHR(IFUN), and ZEP(IFUN) is the largest allowed absolute error for those points
where the absolute computed value is less than ZTHR(IFUN).
Other features of the QFIT routine include provisions for aligning a subinterval boundary at a
specified point in the overall interval (in case the fitted function has a discontinuous slope such asat the pair production or Møller thresholds), and computing fit parameters in bins flanking themain interval to guard against truncation errors in sub-interval index computations.
The net result of the fit is the determination of coefficients AX, BX, AF(IFUN,J), and BF(IFUN,J)
such thatFVALUE(E)=AF(IFUN,INTERV)*XFUN(E) + BF(IFUN,INTERV)
is the value of the IFUNth function, and where
INTERV=INT(AX*XFUN(E) + BX).
XFUN is called the distribution function and is ln(E − RM ) for electrons and ln(E) for photons.
402

Note that the coding of EGS is designed to allow a “mapped PWLF” in which we have AX, BX,
AF(IFUN,J), BF(IFUN,J), and M(I), such that when I=INT(AX*XFUN(E)+BX) and J=M(I), thenFVALUE(E)=AF(IFUN,J)*XFUN(E) + BF(IFUN,J)
for the IFUNth function. This kind of fit has the advantage that it yields a better fit with asmaller amount of stored data than the default fit computed by PWFL1. Such a scheme has neverbeen implemented in PEGS, however, despite that more data than necessary is currently used in
describing the functions at higher energies where they vary quite smoothly.
The following is an example of the input data lines that can be used with the PWLF option:

Line
PWLF1
PWLF2

Column123456789112345678921234567893123456789412345678..etc.
PWLF&INP &END

C.3.5

The DECK Option

The DECK option, with the aid of subroutine LAY, prints the data needed to specify the current
material, the energy intervals specified, various computed molecular parameters (e.g., the radiation
length), the run-time parameters for pair production and bremsstrahlung, and the fit data producedby the PWLF option. In other words, DECK prints anything that might be of use to EGS in
simulating showers or to users in analysis routines. Subroutines LAY (in PEGS) and HATCH (in
EGS) are a matched pair in that HATCH reads what LAY writes (PEGS “lays” and EGS “hatches”).
Thus when users alter LAY to write additional material data to PEGS output files, HATCH must be
be updated appropriately.
DECK should be invoked when either ELEM, MIXT, or COMP and ENER and PWLF havebeen run for the current material and before any of these have been executed for the next material(see Figure C.9). The following is an example of the input data lines that can be used with the
DECK option:

Line
DECK1
DECK2

Column123456789112345678921234567893123456789412345678..etc.
DECK&INP &END

403

C.3.6

The TEST Option

The TEST option is used as an easy way to obtain data for plots of some of the functions (notthe fits) that the PWLF option fits. Output is printed to a file “pgs5job.pegs5plot” in the working user code directory, and includes sequential sets of data from PEGS functions ESIG, PSIG,EDEDX, PDEDX, EBR1, PBR1, PBR2, TMXS, TMXDE2, GMFP, GBR1, GBR2 and CRATIO. Thus the
TEST option provides yet another useful PEGS tool for examining and studying raw physics data.
The following is an example of the input data lines that can be used with the TEST option:

Line
TEST1
TEST2

Column123456789112345678921234567893123456789412345678..etc.
TEST&INP NPTS=50 &END

C.3.7

The CALL Option

The CALL option is used whenever one desires to have PEGS evaluate a particular function andprint out the results. The following is an example of the input data lines that can be used with
the CALL option in order to test for discontinuities in GMFP (Gamma Mean Free Path) near 50
MeV. (Note: In this example we have included the (necessary) ELEM option data lines for Lead).

Line
ELEM1
ELEM2
ELEM3
ELEM4
CALL1
CALL2
CALL3
CALL1
CALL2
CALL3

Column123456789112345678921234567893123456789412345678..etc.
ELEM&INP &END
PB
PB
CALL
&INP XP(1)=49.99 &END
GMFP
CALL
&INP XP(1)=50.01 &END
GMFP

The resulting output from a user code accessing PEGS would be:
OPT = CALL
Function call:
OPT = CALL
Function call:

1.95522= GMFPOF

49.9900

1.97485= GMFPOF

50.0100404

IDFNAM
ALIN
DLOG
ALKE
ADFMOL

C.3.8

Table C.19: Distribution functions available with PLTI and PLTN.
PurposeLinear plot.
Natural log plot.
Natural log of electron kinetic energy plot.
Approximation to Møller and Bhabha distributions

The PLTI and PLTN Options

The PLTI and PLTN options may be used to obtain printer plots of any of the functions in the
PEGS function table. These options are somewhat analogous to the TEST option in that they callthe same PEGS output routine (subroutine PLOT) and write output data in the same format to thesame file (‘pgs5job.pegs5plot’ in the working user code directory). However, the information which
can be extracted using the TEST option is restricted to the energy-dependent default functions
over the ranges from AE to UE and AP to UP, while PLTI and PLTN can access arbitrary PEGS
functions and produce physics data over arbitrary ranges. The PLTI option is rather primitive inthat the functions involved must be specified by number, so we shall instead concentrate on the
PLTN option in which the functions are specified by name.
Consider the function BRMSRZ(Z,E,K1,K2), which is the soft bremsstrahlung cross section (for anelectron of total energy energy E and element Z) integrated over the photon energy range (K1,K2).
Suppose we would like to see a plot of BRMSRZ(2,E,0.0,1.5) for values of E from 5 to 100 MeV.
Also assume we want the data points evenly spaced in ln(E). Then (see Table C.17) the function
name we would use is BRMSRZ, the distribution function name to be specified is IDFNAM=ALOG, thestatic arguments needed are XP(1)=2., XP(3)=0.0, XP(4)=1.5, the independent variable to beaccessed is the second argument (i.e., IV=2), and its limits are VLO=5.0 and VHI=100.0. If we want
100 points on the data set we specific NPTS=100. The input data lines necessary to accomplish this
are:

Line
PLTN1
PLTN2
PLTN3

Column123456789112345678921234567893123456789412345678..etc.
PLTN&INP XP(1)=2.,XP(3)=0.0,XP(4)=1.5,IV=2,VLO=5.,
VHI=100.,NPTS=100 &END
BRMSRZALOG

Distribution functions that are available are listed and described in Table C.19.

405

C.3.9The HPLT Option

The Histogram PLoT option is designed to be used in conjunction with UCTESTSR (User Codeto TEST Sampling Routine), which is provided with the EGS5 distribution. This option provides
a way to test the EGS methods used to sample differential distributions functions by looking
for discrepancies between actual differential physics data and histograms of sampled data during
simulations.
Suppose that a probability density function, P DF (X) (see Chapter 2 of SLAC-R-730/KEK2005-8), is to be sampled by EGS (note: P DF (X) will have other static arguments which we
ignore for this discussion). Let CDF (X) be the cumulative distribution function associated withP DF (X). If P DF (X) drops sharply with increasing X, we will not get many samples in thebins with large X unless we make the bins themselves larger in such regions. We accomplish
this by finding another p.d.f.and c.d.f., P DG(X) and CDG(X), respectively, such that P DG(X)approximates P DF (X). If we want N bins, we then pick the X(I) such that
CDG(X(I+1))-CDG(X(I))=(CDG(X(N+1))-CDG(X(I)))/N.
For all I this implies that
X(I)=CDGI((CDG(X(N+1))-CDG(X(1)))*I/(N+1) + CDG(X(1)))
where CDGI is the inverse function of CDG. Thus, if P DG(X) is a reasonable approximation,
the histogram bins at large X should have the same order of magnitude of counts as those at lower
X. The function CDG(X) is called the “distribution function” in the context of the HPLT option.
CDG(X), CDGI(X), and P DG(X) are used by the UCTESTSR user code and HPLT1 subroutine
in PEGS. If we let SP DF (X) and SCDF (X) be the “sampled” data, and P DF (X) and CDF (X)be the theoretical data, then the routine HPLT1 can be summarized by the pseudo-code:
DO I=1,N
PLOT((SCDF(X(I+1))-SCDF(X(I)))/(CDG(X(I+1))-CDG(X(I))))
PLOT((CDF(X(I+1))-CDF(X(I)))/(CDG(X(I+1))-CDG(X(I))))
DO....X(I) at 10 points in the interval (X(I),X(I+1))
PLOT(d(SCDF)/d(CDG)=PDF(X)/PDG(X))
END DO
RETURNEND
All of the input data lines for a specified HPLT option are printed directly by UCTESTSR tostandard output. This data can then be used in a subsequent run of PEGS to produce line-printerstyle plots of the actual and sample functions . Users should refer to the listing for UCTESTSR andcomments therein for a better understanding of the HPLT option (see also Chapter 6 of SLAC-210).
406

C.4

Concluding Remarks

In the previous sections we have seen the various uses for PEGS. We summarize by giving theoption sequences most generally used:

1. Minimal options for a material data set creation (for later use by EGS):
1.
2.
3.
4.

ELEM (or MIXT, or COMP)
ENER
PWLF
DECK

2. Same as above, with default data dumps of functions for which the PWLF option provides
fits:
1.
2.
3.
4.
5.

ELEM (or MIXT, or COMP)
ENER
TEST
PWLF
DECK

3. Comparison of theoretical and sampled distributions by means of the HPLT option:
1.

2.

ELEM (or MIXT, or COMP)
Note: Data lines should agree with those usedwith the UCTESTSR run that generated the HPLT input.
HPLT
Note: Output data from UCTESTSR run.

4. Selective plotting of various functions:
1.
2.
3.
4.
5.
6.

ELEM (or MIXT, or COMP) - for material 1
PLTN - for function 1
PLTN - for function 2,....etc.
ELEM (or MIXT, or COMP) - for material 2,....etc.
PLTN - for function 1
PLTN - for function 2,....etc.

407

Appendix D

EGS5 INSTALLATION GUIDE
Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANAlex F. Bielajew and Scott J. Wilderman
Department of Nuclear Engineering and Radiological Sciences
The University of Michigan
2355 Bonisteel Boulevard
Ann Arbor, MI 48109, USA
Walter R. Nelson
Department Associate in the Radiation Physics Group (retired)
Radiation Protection Department
Stanford Linear Accelerator Center
2575 Sand Hill Road Menlo Park, CA 94025, USA
This EGS5 Installation Guide is Appendix D of a document called
SLAC-R-730/KEK-2005-8, which can be obtained from the SLAC and KEK web sites.

408

D.1

Installation of EGS5

The current version of the EGS5 code system run only on Unix-based operating system. On
Windows platforms, this release of EGS5 can be run on top of Cygwin or any other Linux environment emulator for PCs. Copies of Cygwin, along with setup programs, installation instructions, users guides, and other documentation, can be obtained for free at the Cygwin website,
http://www.cygwin.com.
The EGS5 system is distributed as a compressed (with gzip) Unix “tar” archive, egs5.tar.gz.
It is trivially “installed” by typing:% gunzip -c egs5.tar.gz | tar vxf from the command line in any desired directory into which a user places egs5.tar.gz (to somewhat
mimic the organizational structure of EGS4, this command should be run from the user’s top level
directory). When this command is executed, a directory called egs5 will be created, under which
the subdirectories containing all of the EGS5 FORTRAN, data, tutorial, and sample problem files
will be placed.
Although it employs some non-standard extensions to FORTRAN-77 the basic source code in
EGS5 avoids the use of all system-dependent utilities, and all that is required to run EGS5 on any
Unix-like platform is a FORTRAN compiler1 . Further, all of the tutorial, sample, and auxiliary
user codes provided in the EGS5 distribution should run without modification if the FORTRAN
library routines FDATE and ETIME are available. Likewise, the sample shell scripts for controllingthe compilation and execution of EGS5 jobs will run after minimal modification on the part of the
user.

D.2

Sample Scripts for Running EGS5The EGS5 distribution comes with two shell scripts which can be used (with minimal modification)
to compile, run, and re-run generic EGS5 user codes. The five steps which are required to use thesample script “egs5run” are listed below. Example implementations follow.
1. A copy of egs5run (or a symbolic link to a global copy of egs5run) must be placed in thesubdirectory containing the user code.
2. In the egs5run file, the user must specify the directory in which the main EGS5 distribution
was installed. This is done by setting the shell script variable BASKET to the name of the1

The one system-dependent FORTRAN implementation issue which an EGS5 user may encounter is with the
NAMELIST extension. Different compilers may require different leading characters in the end-of-list tag, usually either&END or /END.

409

egs5 directory (preferably using the full Unix pathname). BASKET is analogous to HEN HOUSE
of EGS4, in that all of the basic EGS5 system resides in this single place.
3. Also in egs5run, the user must specify the operating system type under which the current
computation is to be performed by defining the variable MY MACHINE. This is required primarily
so that egs5run can select the appropriate compiler options.
4. If the user’s operating system is not among the defaults included with the egs5run script, theuser will have to modify the section of egs5run in which the name of the FORTRAN compiler
and the relevant compiler options are set. An example is given below.
5. Lastly, and also in egs5run, the user must select the level of optimization to be used, if any,
during compilation, by specifying the value of the variable OPT LEVEL.
The preamble to egs5run contains some examples of how the variables described above might
be set in egs5run:
#
#
#
#
#
#
#
#
#
#
#

BASKET=/home/wrn/egs5/
MY_MACHINE=LinuxOPT_LEVEL=O
BASKET=/home/Ralph/egs5/
MY_MACHINE=Cygwin-LinuxOPT_LEVEL=O2
BASKET=/afs/slac.stanford.edu/g/rp/egs5
MY_MACHINE=sparc
OPT_LEVEL=

The first example corresponds to a user with login id “wrn” on machine running Linux, with EGS5
installed from his top level (home) directory. The second example is representative of what a PC
user named “Ralph” might see if he were running Cygwin, again with EGS5 installed from his top
level directory. The final example is typical of what a user accessing a globally available version of
EGS5 across an afs file system might see.
Note that the variable MY MACHINE can be defined as anything the user wishes, provided thatcompiler options are included for that OS name later in egs5run. The version of egs5run in thecurrent EGS5 distribution has defined compiler options for four operating systems: Linux, CygwinLinux, sparc and Digital-Unix. If EGS5 is to be used on some other Unix-like platform, the user
must edit egs5run to specify the name of the FORTRAN compiler and any compiler options which
are necessary. This can be done quite simply by changing the 5 instances of user defined * withthe appropriate parameters in the section of egs5run shown below:
elif test "$MY_MACHINE" = "user_defined_machine"
410

then
COMPILER="user_defined_compiler"
DEBUG="user_defined_debug_flags"
CFLAGS="user_defined_compilation_flags"
if test "$OPT_LEVEL" = ""then
OPTIMIZED=""
else
OPTIMIZED="-$OPT_LEVEL user_defined_optimizations"fi

Running egs5run: There are five options for executing egs5run, invoked by specifying a single
command line argument, as shown below:
Command line Action of egs5run
egs5runCompile user code and execute.
egs5run comp Compile user code but do not execute.
egs5run pegs Compile pegs-only user code and execute.
egs5run db
Compile user code for debug (does not execute).
egs5run cl
Clear out files (and links) and exit egs5run.
When egs5run is executed in normal mode (i.e., with no argument), the following sequence ofactions is initiated:
1. Various files and links created in previous runs (if any) are deleted.
2. The specified system name MY MACHINE and compiler are echoed.
3. The user is asked to key-in the name of the FORTRAN file containing the user code for thisrun. The file name provided by the user must have a .f extension. Is the file is not found,the script will exit.
4. A file named egs5job.f is created using the specified user code file, and all files with .f
extensions in the following directories:
user auxcode
user defined auxiliary subroutines, if present.
$BASKET/egs
EGS related subroutines.
$BASKET/pegs
subroutines related to PEGS.
$BASKET/auxcode auxiliary subroutines provided with EGS5.
Note that file extensions should not be keyed-in when specifying file names.
5. Symbolic links are created pointing to the following directories containing files of COMMON
blocks included by the various source code files:$BASKET/include
COMMON block files for EGS.
$BASKET/auxcommons
COMMON block files for EGS5 auxiliary subroutines.
$BASKET/pegscommons COMMON block files for PEGS.

411

6. The user is then asked to key-in the name of the EGS input file for this run. Note that for
many applications, all of the necessary problem-specific data can be specified in the user code
itself, and so an input problem data file need not be used. When this is the case, the user
may simply enter a carriage return. If an EGS problem data file is to be used, it must havea .data extension. If the data file has the same name as the user code file (apart from theextensions .f and .data, of course), this can be specified by keying in a carriage return. If
the user specified file exists, it is copied to file called “egs5job.inp.” If the file does not exist,
egs5run will create an empty (dummy) file called egs5job.inp, which is opened as unit 4 for
input in many EGS5 user codes.
7. The user is next asked to key-in the name of the input data file used by PEGS. This file
must have a .inp extension. A carriage return issued in response to this question will cause
egs5run to look for .inp files named similarly to first the user code and then the data file. If
any of the files exist, a symbolic link named “pgs5job.pegs5inp” will be created, as this fileis opened by default on unit 25 in PEGS. If no appropriate .inp file is found, the script willexit.
8. The egs5job.f will then be compiled, with executable named “egs5job.exe.”
9. If egs5job.exe does not exist because of compilation errors, the script will exit.
10. Next, a symbolic link to the data directory, $BASKET/data, will be created.
11. The user will then be asked if this user code requires input to be keyed-in from the terminal(on unit 5, standard input).
12. egs5job.exe will be executed either in the foreground (if interactive input was specified) orin the background. If the job is executed in the background, standard output (unit 6) isredirected to a file named “egs5job.log.” Note that the user may elect to explicitly open unit
6 in the user code, if desired.
When egs5run is executed with one of the three arguments, “cl,” “db,” or “comp,” the script
follows the basic flow described above except that it exits either immediately after cleaning out old
links (with the “cl” option) or just prior to executing the compiled code (with the “db” or “comp”
arguments). When egs5run is executed with “pegs” as an argument, only the pegs files and the
BLOCK DATA FORTRAN source files from egs are included with the user code when egs5job.f iscreated. With that exception, egs5run with “pegs” as an argument runs exactly as it does with noargument. Note that because none of the main physics routines of EGS are included in egs5job.f,user codes based on PEGS only need not contain either HOWFAR or AUSGAB subroutines.

Example transcript from egs5run Below is a transcript of an egs5run session using the tutorial
problem in tutor1. (Note that input keyed-in by the user is denoted by <-------- USER INPUT.)% egs5run============================
412

egs5run script has started============================
working directory is /afs/slac.stanford.edu/g/rp/egs5/tutorcodes/tutor1-----------------------------------------------------------Erasing files (and links) from previous runs (if they exist)-----------------------------------------------------------OS_TYPE = sparc
Your Compiler is f77--------------------------------------Enter name of User Code(file extension must be ’.f’)--------------------------------------tutor1<-------- USER_INPUT-----------------------------------------Enter name of READ(4) data file
(file extension must be ’.data’)(<CR> for same file name as User Code)-----------------------------------------<-------- USER_INPUT (<CR>)--> Empty file created as egs5job.inp-----------------------------------------Enter name of UNIT(25) (pegs input file)(file extension must be ’.inp’)(<CR> for same file name as data fileor same file name as User Code)-----------------------------------------<-------- USER_INPUT (<CR>)--> tutor1.inp linked to pgs5job.pegs5inp-----------------------------------Compiling (with optimization of O2)---------------------------------------------------------------------------------Does this user code read from the terminal?
(Enter 1 for yes, anything else for no)----------------------------------------------0<-------- USER_INPUT*******************************************************
413

* User code tutor1.f has been compiled and is starting ********************************************************
Running egs5job.exe in background========================
egs5run script has ended========================

Running run5again: If a user wishes to run previously compiled user code with different input
data on either unit 4 (EGS problem data) or unit 25 (PEGS material data) or both, the script
run5again can be executed instead of egs5run. The run5again script takes no arguments, and does
all of what egs5run does except that it does not construct or compile egs5job.f. run5again will
execute either an existing egs5job.exe program or any previously compiled egs5job.f file has beenrenamed and has a .exe file extension. When run5again (which requires no set up on the part of
the user) is executed, the following actions are performed:
1. Symbolic link pgs5job.pegs5inp and file egs5job.inp from previous runs are deleted.
2. The user is asked to key-in the EGS data file name to be copied to egs5job.inp and openedon unit 4, as in egs5run, above.
3. The user is asked to key-in the name of the PEGS data file to be linked to pgs5job.pegs5inp
and opened on unit 25, as with egs5run.
4. The user is then asked to specify the name of an existing executable (with file extension .exe
for this run. If a carriage return is entered, run5again will look for an existing egs5job.exe
executable to use. If neither the specified executable nor egs5job.exe exist, the script willexit.
5. The user will finally be asked if this user code requires input that is to be keyed-in (to unit
5, standard input).
6. Either the specified executable or egs5job.exe will be executed, either in the foreground (ifinteractive input was specified) or in the background. If the job is executed in the background,standard output (unit 6) is redirected to a file named “egs5job.log,” as with egs5run.

Example transcript from run5again

Below is a sample transcript from a run5again job:% run5again============================
run5again script has started============================
414

working directory is /home/user/egs5/tutorcodes/tutor1-----------------------------------------Enter name of READ(4) data file
(file extension must be ’.data’)-----------------------------------------tutor1<-------- USER_INPUT--> Empty file created as egs5job.inp-----------------------------------------Enter name of UNIT(25) (pegs input file)(file extension must be ’.inp’)(<CR> for same file name as data file)-----------------------------------------<-------- USER_INPUT (<CR>)--> d4file used, tutor1.inp linked to pgs5job.pegs5inp-----------------------------------Enter name of the executable(file extension must be ’.exe’)(<CR> to use egs5job.exe)-----------------------------------<-------- USER_INPUT (<CR>)----------------------------------------------Does this user code read from the terminal?
(Enter 1 for yes, anything else for no)----------------------------------------------0<-------- USER_INPUT********************************************** Previously compiled user code is starting **********************************************
Running egs5job.exe in background==========================
run5again script has ended==========================

415

Appendix E

CONTENTS OF THE EGS5
DISTRIBUTION
Hideo Hirayama and Yoshihito Namito
Radiation Science Center
Advanced Research Laboratory
High Energy Accelerator Research Organization (KEK)
1-1 Oho Tsukuba-shi Ibaraki-ken 305-0801 JAPANAlex F. Bielajew and Scott J. Wilderman
Department of Nuclear Engineering and Radiological Sciences
The University of Michigan
2355 Bonisteel Boulevard
Ann Arbor, MI 48109, USA
Walter R. Nelson
Department Associate in the Radiation Physics Group (retired)
Radiation Protection Department
Stanford Linear Accelerator Center
2575 Sand Hill Road Menlo Park, CA 94025, USA
This EGS5 Distribution Listing is Appendix E of a document called
SLAC-R-730/KEK-2005-8, which can be obtained from the SLAC and KEK web sites.

416

The EGS5 distribution archive contains six main groups of files: documentation; EGS FORTRAN source codes; PEGS FORTRAN source codes; material data files; example and tutorial
run-time scripts and user codes (including FORTRAN source codes, input files and sample output); and FORTRAN source codes for auxiliary functions which users may find useful.

E.1

Documentation

Documentation has been included within the EGS5 distribution for two reasons. First, the mostheavily used parts of SLAC-R-730/KEK-2005-8 are expected to be the user manuals, and so .pdf
versions have been placed in the archive so that they can be accessed (or printed) at the discretionand convenience of the user. Second, it is anticipated that when changes and additions are madeto the EGS5 code base, the user manuals will be updated appropriately, while SLAC-R-730/KEK2005-8 may not be. Documentation files includes with the EGS5 distribution are:egs5 user manual.pdf
pegs user manual.pdf
installation guide.pdf
distribution contents.pdf
Writing HOWFAR.pdf

Appendix B of SLAC-R-730/KEK-2005-8
Appendix C of SLAC-R-730/KEK-2005-8
Appendix D of SLAC-R-730/KEK-2005-8
Appendix E of SLAC-R-730/KEK-2005-8
Presentation describing construction of user code subroutine
HOWFAR

These files can be found in the subdirectory docs of the under the main egs5 directory in the
distribution. In addition, a complete copy of SLAC-R-730/KEK-2005-8 is included in the mainegs5 directory, as the .pdf file slac730.pdf.

E.2

EGS-Related FORTRAN Source Files

There are two types of files which are part of the EGS Monte Carlo shower simulation package.
As described in the EGS5 User Manual (Appendix B of SLAC-R-730/KEK-2005-8), some of theutility of MORTRAN macro substitutions used in EGS4 has been retained in EGS5 by having
all FORTRAN COMMON blocks be defined in the subprograms of the source code through include
statements which reference files containing just the FORTRAN listing of each COMMON block. This
makes changing variables and parameters in any COMMON block a global process, similar to what
could be done with a MORTRAN macro in EGS4. Each EGS5 COMMON block is thus containedin a unique file which is named after the COMMON block as in egs5 media.f for the file containing
the declarations for COMMON block MEDIA. All of the FORTRAN PARAMETERS used to the specify
array dimensions in the various COMMON blocks have been collected in a single “header” file calledegs5 h.f, and all of the EGS5 COMMON block files have been placed in a directory called include
under the main egs5 directory. These files contain nothing other than the declarations of thevariables in each COMMON, along with some documentation.
417

All of the actual FORTRAN source code files used in simulating showers have been collected in a
subdirectory of egs5 called egs. Each file contains the source code for one EGS5 subroutine (somesource code files actually contain more than one subroutine, when groups of subprograms are very
closely related), and the naming convention used with the COMMON block files in include has been
followed. Descriptions of the functionality of the various subprograms of egs can be found both in
Chapter 2 of SLAC-R-730/KEK-2005-8 and in the EGS5 User Manual. List below are the namesof all of the files in egs and include, which are subdirectories of egs5 in the EGS5 distribution.

EGS FORTRAN source files:
counters_out.f
egs5_annih.fegs5_aphi.f
egs5_bhabha.fegs5_block_data.f
egs5_block_data_atom.f
egs5_block_set.f
egs5_brems.f
egs5_collis.f
egs5_compt.f
egs5_edgbin.f

egs5_eii.f
egs5_electr.f
egs5_hardx.f
egs5_hatch.f
egs5_kauger.f
egs5_kshell.f
egs5_kxray.f
egs5_lauger.f
egs5_lshell.f
egs5_lxray.f
egs5_moller.f

egs5_mscat.f
egs5_pair.f
egs5_photo.f
egs5_photon.f
egs5_raylei.f
egs5_rk1.f
egs5_rmsfit.f
egs5_shower.f
egs5_uphi.f
randomset.f
rluxinit.f

EGS “included” COMMON block files:
counters.f
egs5_bcomp.f
egs5_bounds.f
egs5_brempr.f
egs5_cdcsep.f
egs5_cdcspl.f
egs5_coefgs.f
egs5_csplcf.f

egs5_edge.f
egs5_eiicom.f
egs5_elecin.f
egs5_epcont.f
egs5_h.fegs5_media.f
egs5_misc.f
egs5_ms.f

egs5_mscon.fegs5_mults.f
egs5_photin.f
egs5_scpw.f
egs5_stack.f
egs5_thresh.f
egs5_uphiin.f
egs5_uphiot.f

egs5_useful.f
egs5_userpr.f
egs5_usersc.f
egs5_uservr.f
egs5_userxt.f
randomm.f

Also included in the egs subdirectory is the EGS5 copyright file, COPYRIGHT, in a version
suitable for inclusion with FORTRAN user codes.

E.3

PEGS-Related FORTRAN Source Files

As with EGS, there are two types of files comprising PEGS, those containing actual FORTRAN
source code (in a subdirectory of egs5 called pegs) and those containing just COMMON block declara418

tions (in a subdirectory called pegscommons). While all of the COMMON blocks in PEGS are foundin unique files (named after the COMMON block with no prefix) all of the subroutines and functions
previously a part of PEGS4 are contained in one file pegs5.f. All of the other source code files
in the pegs subdirectory are new to EGS5 and are either required for calculating the scatteringstrength and scattering power data needed for the new electron transport mechanics of EGS5, orare part of the implementation of the new multiple scattering distribution, both of which are described in Chapter 2 of SLAC-R-730/KEK-2005-8. The full lists of the PEGS-related files foundin pegs and pegscommons, which are both subdirectories of egs5 in the EGS5 distribution, aregiven below.

PEGS FORTRAN source files:
csdar.fdcsel.fdcsn.fdcsstor.f
dcstab.felastino.f

elinit.f
esteplim.f
estepmax.f
findi.ffitms.f
g1dedx.f

g1e.f
gauleg.f
gscoef.f
gsdist.finigrd.f
integ.f

k1e.f
sumga.flegenp.f
wmsfit.fmakek1.f
pegs5.f
prelastino.f
spline.f

PEGS “included” COMMON block files:
bcom.fbremp2.f
cohcom.f
cpcom.f
dbrpr.f
dcsstr.fdercon.f
eimpact.f

E.4

elemtb.f
elmtbc.fepstar.ffuncs.ffuncsc.f
k1spl.flamolm.flanihm.flbhabm.f
lbremr.f
lbremz.f
lcompm.flegacy.f
lpairr.f
lpairz.f
lspion.f

mimsd.f
mixdat.fmolvar.f
mscom.f
mxdatc.f
phpair.f
pmcons.f
pwlfin.fradlen.frngspl.f
rslts.f
scpspl.fsfcom.f
spcomc.f
spcomm.fthres2.fMaterial Data FilesThe EGS5 distribution contains six primary data files plus five subdirectories containing material
dependent data files, as described below. All files and directories are found in the subdirectory dataof egs5. More complete descriptions of the data contained in the files can be found in Chapter 2of SLAC-R-730/KEK-2005-8.

Data files:

The main material data files in data are briefly described below:419

aprime.data
bcomp.dat
incoh.dat
K1.datpgs5form.dat
pgs5phtx.dat

Data for empirical bremsstrahlung correction.
Bound total Compton cross section data (σbC ) for elements Z = 1 ∼ 100.
Incoherent scattering function data (S(x, Z)) for all elements Z = 1 ∼100.
Tables of optimal initial scattering strengths as a function of geometry
region size at various energies for various reference materials.
Rayleigh scattering form factor data (F (x, Z)), identical to
pgs4form.dat from PEGS4.
Photo-electric cross section data, pair-production cross section data (forphoton energies less than 50 MeV), and Rayleigh scattering cross section
data from PHOTX for all elements Z = 1 ∼ 100.

Data subdirectories:

Data subdirectories of data are briefly described as follows:

dcslibdensity corrections

int coherent cs
int form factor
shellwise Compton profile

Differential nuclear elastic scattering cross sections forelectrons and positrons for elements Z = 1 ∼ 95.
Explicit values of the density effect correction to stopping
power which can be used to reproduce ICRU-37 values.
Placed in two subdirectories, elements and compounds.
Coherent scattering cross sections (with interference effects modeled) for selected materials.
Form factors (including interference effects) for selectedmaterials.
Shellwise Compton profiles for elements Z = 1 ∼ 100.

Files in dcslib: Files containing elemental partial-wave differential elastic scattering cross section
data for both electrons and positrons are included in the EGS5 distribution in subdirectory dcslibof data. A straight-forward convention has been used to name these data files, which are listed
below
eeldx001.tabeeldx002.tabeeldx003.tab
eeldx004.tabeeldx005.tabeeldx006.tabeeldx007.tabeeldx008.tab
eeldx009.tab
eeldx010.tab
eeldx011.tabeeldx012.tabeeldx013.tab

eeldx039.tab
eeldx040.tab
eeldx041.tabeeldx042.tabeeldx043.tab
eeldx044.tabeeldx045.tabeeldx046.tabeeldx047.tabeeldx048.tab
eeldx049.tab
eeldx050.tab
eeldx051.tabeeldx077.tabeeldx078.tab
eeldx079.tab
eeldx080.tab
eeldx081.tabeeldx082.tabeeldx083.tab
eeldx084.tabeeldx085.tabeeldx086.tabeeldx087.tabeeldx088.tab
eeldx089.tab

peldx020.tab
peldx021.tab
peldx022.tab
peldx023.tab
peldx024.tab
peldx025.tab
peldx026.tab
peldx027.tab
peldx028.tab
peldx029.tab
peldx030.tab
peldx031.tab
peldx032.tab

420

peldx058.tab
peldx059.tab
peldx060.tab
peldx061.tab
peldx062.tab
peldx063.tab
peldx064.tab
peldx065.tab
peldx066.tab
peldx067.tab
peldx068.tab
peldx069.tab
peldx070.tab

eeldx014.tabeeldx015.tabeeldx016.tabeeldx017.tabeeldx018.tab
eeldx019.tab
eeldx020.tab
eeldx021.tabeeldx022.tabeeldx023.tab
eeldx024.tabeeldx025.tabeeldx026.tabeeldx027.tabeeldx028.tab
eeldx029.tab
eeldx030.tab
eeldx031.tabeeldx032.tabeeldx033.tab
eeldx034.tabeeldx035.tabeeldx036.tabeeldx037.tabeeldx038.tab

eeldx052.tabeeldx053.tab
eeldx054.tabeeldx055.tabeeldx056.tabeeldx057.tabeeldx058.tab
eeldx059.tab
eeldx060.tab
eeldx061.tabeeldx062.tabeeldx063.tab
eeldx064.tabeeldx065.tabeeldx066.tabeeldx067.tabeeldx068.tab
eeldx069.tab
eeldx070.tab
eeldx071.tabeeldx072.tabeeldx073.tab
eeldx074.tabeeldx075.tabeeldx076.tabeeldx090.tab
eeldx091.tabeeldx092.tabeeldx093.tab
eeldx094.tabeeldx095.tab
peldx001.tab
peldx002.tab
peldx003.tab
peldx004.tab
peldx005.tab
peldx006.tab
peldx007.tab
peldx008.tab
peldx009.tab
peldx010.tab
peldx011.tab
peldx012.tab
peldx013.tab
peldx014.tab
peldx015.tab
peldx016.tab
peldx017.tab
peldx018.tab
peldx019.tab

peldx033.tab
peldx034.tab
peldx035.tab
peldx036.tab
peldx037.tab
peldx038.tab
peldx039.tab
peldx040.tab
peldx041.tab
peldx042.tab
peldx043.tab
peldx044.tab
peldx045.tab
peldx046.tab
peldx047.tab
peldx048.tab
peldx049.tab
peldx050.tab
peldx051.tab
peldx052.tab
peldx053.tab
peldx054.tab
peldx055.tab
peldx056.tab
peldx057.tab

peldx071.tab
peldx072.tab
peldx073.tab
peldx074.tab
peldx075.tab
peldx076.tab
peldx077.tab
peldx078.tab
peldx079.tab
peldx080.tab
peldx081.tab
peldx082.tab
peldx083.tab
peldx084.tab
peldx085.tab
peldx086.tab
peldx087.tab
peldx088.tab
peldx089.tab
peldx090.tab
peldx091.tab
peldx092.tab
peldx093.tab
peldx094.tab
peldx095.tab

Files in density corrections: To access values of collision stopping powers derived from ICRU
Report 37, the user must provide density effect values explicitly for each material. Files containing
the appropriate data for 450 elements and compounds are provided in the EGS5 distribution in
subdirectories elements and compounds of data subdirectory density corrections. Files are
titled using a unique and explicit name for each material along with a .density extension. The fulllist of elemental material files (without the .density extension) is given below. A list of the 345 files
containing data for compounds can be found in a README file provided in density corrections.
actiniumaluminium
aluminumamericium
antimonyargon
arsenic
astatinebariumberkelium

germanium
gold
hafnium
heliumholmiumhydrogenhydrogen_liquid
indiumiodineiridium
421

protactinium
radiumradonrheniumrhodium
rubidium
ruthenium
samariumscandium
seleniumberyllium
bismuthboron
brominecadmium
calcium
californium
carbon_graphite_1.700g_cm3
carbon_graphite_2.000g_cm3
carbon_graphite_2.265g_cm3
ceriumcesiumchlorinechromiumcobaltcoppercuriumdysprosium
einsteinium
erbiumeuropium
fermium
fluorine
francium
gadolinium
galliumiron
kryptonlanthanum
leadlithium
lutetiummagnesium
manganese
mercurymolybdenum
neodymiumneon
neptunium
nickelniobium
nitrogenosmiumoxygenpalladium
phosphorus
platinumplutonium
poloniumpotassium
praseodymium
promethium

siliconsilversodiumstrontium
sulfursulfur_1.92g_cm3
sulfur_2.07g_cm3
tantalumtechnetium
tellurium
terbiumthallium
thoriumthuliumtintitaniumtungstenuranium
vanadiumxenon
ytterbium
yttriumzinczirconium

Files in int coherent cs: For computations in which coherent scattering is modeled with interference effects included, the user must provide the interference coherent scattering cross section
data. Included in subdirectory int coherent cs of data in the EGS5 distribution are files of coherent scattering cross section data with interference effects for seven materials: water, PMMA,
fat, muscle, kidney, liver and blood. The data files are provided are named ics followed by the
first letter of the material, as below.
ics_b.dat
ics_f.dat

ics_k.dat
ics_l.dat

ics_m.dat
ics_p.dat

ics_w.dat

Files in int form factor: When coherent scattering interference effects are modeled, the user
must also provide the appropriate form factors. Sample data files containing form factors withinterference effects included are provided in subdirectory int form factor of data of the EGS
distribution for the same seven materials in int coherent cs. The naming convention analogous
to that described above is employed, and the exact EGS5 file names are listed below.
422

iff_b.dat
iff_f.dat

iff_k.dat
iff_l.dat

iff_m.dat
iff_p.dat

iff_w.dat

Files in shellwise Compton profile: The elemental shellwise Compton profile data files included in the EGS5 distribution in subdirectory shellwise Compton profile of data are named
as follows. This data is used in modeling Compton scattering from bound atomic electrons on ashell-by-shell basis. The naming convention employed is intuitive.
z001.datz002.datz003.dat
z004.dat
z005.dat
z006.dat
z007.datz008.datz009.dat
z010.dat
z011.datz012.datz013.datz014.dat
z015.dat
z016.dat
z017.datz018.dat

E.5

z019.dat
z020.dat
z021.datz022.datz023.datz024.datz025.dat
z026.datz027.datz028.datz029.dat
z030.dat
z031.datz032.datz033.datz034.dat
z035.dat
z036.dat

z036n.dat
z037.dat
z038.datz039.dat
z040.dat
z041.datz042.datz043.datz044.datz045.dat
z046.datz047.datz048.datz049.dat
z050.datz051.datz052.datz053.datz054.datz055.datz056.datz057.datz058.datz059.dat
z060.dat
z061.datz062.datz063.datz064.dat
z065.dat
z066.dat
z067.datz068.datz069.datz070.datz071.datz072.datz073.datz074.datz075.datz076.datz077.datz078.datz079.dat
z080.dat
z081.datz082.datz083.dat
z084.dat
z085.dat
z086.dat
z087.datz088.datz089.dat

z090.dat
z091.datz092.datz093.datz094.datz095.dat
z096.datz097.datz098.datz099.dat
z100.datz101.datz102.dat

Sample User Codes and Run Scripts

The EGS5 distributions includes three sets of example user codes and two sample shell scripts forsetting up and executing EGS5 simulations, as described below:
egs5runSample script for compiling and executing user code.
run5again
Sample script for rerunning previously compiled user code.
tutorcodes
Directory containing a series of sample problems, each in a unique subdirectory with FORTRAN source codes, input files, and output files forthe step-by-step tutorials on using EGS5 found in Chapter 3 of SLAC-R730/KEK-2005-8.
samplecodes Directory containing subdirectories with advanced user codes and input files
for example problems described in Chapter 4 of SLAC-R-730/KEK-2005-8.
extra ucodes Directory containing subdirectories with additional sample user codes.

423

The tutorial problems and advanced user codes are discussed in detail in Chapters 3 and 4 of
SLAC-R-730/KEK-2005-8, respectively, while the programs (based on PEGS) included as “extra
codes” are described briefly in Appendix C. Listed below are the input, source, and sample output
files included in the EGS5 distribution in each of these subdirectories.

Files in subdirectories under egs5/samplecodes:
uc_lp/uc_lp.f
uc_lp/uc_lp.inp
uc_lp/uc_wlp.out
uc_lp/uc_wolp.out
uccyl/uccyl.f
uccyl/uccyl.inp

uccyl/uccyl.out
ucbend/ucbend.f
ucbend/ucbend.inp
ucbend/ucbend.log
ucbend/ucbend.pic
ucsampl5/ucsampl5.f

ucsampl5/ucsampl5.inp
ucsampl5/ucsampl5.out
ucsampcg/ucsampcg.data
ucsampcg/ucsampcg.f
ucsampcg/ucsampcg.inp
ucsampcg/ucsampcg.out

Files in subdirectories under egs5/tutorcodes:
tutor1/tutor1.ftutor1/tutor1.inp
tutor1/tutor1.out
tutor2/tutor2.ftutor2/tutor2.inp
tutor2/tutor2.out
tutor3/tutor3.ftutor3/tutor3.inp
tutor3/tutor3.out

tutor4/tutor4.ftutor4/tutor4.inp
tutor4/tutor4.out
tutor5/tutor5.ftutor5/tutor5.inp
tutor5/tutor5.out
tutor6/tutor6.ftutor6/tutor6.inp
tutor6/tutor6.out

tutor7/tutor7.ftutor7/tutor7.inp
tutor7/tutor7_w.out
tutor7/tutor7_wo.out
tutor7/tutor7_wo_db.out
tutor8/tutor8.ftutor8/tutor8.inp
tutor8/tutor8.out

Files in subdirectories under egs5/extra ucodes:
uc_examin/uc_examin.f
uc_examin/uc_examin.inp
uc_examin/uc_examin.out
uc_examin/uc_e_mfps_AL.xvgr
uc_examin/uc_ph_mfp_AL.xvgr

E.6

ucpegs/ucpegs.f
ucpegs/ucpegs.inp
ucpegs/ucpegs.pegs5dat
uctestsr/uctestsr.f
uctestsr/uctestsr.inp

uctestsr/uctestsr.out
uctestsr/hplt1.inp
uctestsr/hplt1.pegs5lst

Auxiliary Subprogram FORTRAN Source Files

The final set of files included with the EGS5 distribution are the “auxiliary” FORTRAN source
codes (in subdirectory auxcode of egs5) and files containing the COMMON blocks (in subdirectory424auxcommons) associated with these codes. The auxiliary codes contain functions and subroutinesthat are useful in performing input, geometry setup, distance-to-boundary, and scoring computations for many generic problems, and are referenced by some of the advanced user codes provided
with the EGS5 distribution. Detailed descriptions of the functionality of the subprograms can befound in the comments in the source code and in Chapter 3 of SLAC-R-730/KEK-2005-8. Lists ofthe files provided with the distribution are given below.

Auxiliary FORTRAN source files found in auxcode:
cg_related.f
chgtr.fcone.fcone2.f
cone21.fcyl2.fcylndr.f
decod_xyz.f
decodeir.f
ecnsv1.fedistr.ffintrn.ffinval.f
geomout.f
ntally.fplan2p.f
plan2x.fplane1.f
plotxyz.f
rdistr.f

sph2.fsphere.f
sphtrn.fswatch.fxyzbound.f“Included” auxiliary COMMON block files found in auxcommons:
aux_h.f
condta.fcyldta.f
dataconst_common.fedata.fetaly1.fetaly2.fetaly3.fgeom_common.fgeortz.fgeorz.fgeoxyz.f

instuf.flines.f
nfac.fntaly1.f
pladta.f
rdata.f425

sphdta.f
trndta.fvoxel.fwatch.f

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Index“shower book”, 37
AE, 28, 338, 343, 358, 393
AINTP, 125, 133
AMOLDM, 67
AMOLRM, 67
AMOLTM, 67
analog Monte Carlo, 207, 209
ANIHDM, 70
ANIHRM, 70
ANIHTM, 70
ANNIH, 360
annihilation, 28, 61, 69
AP, 28, 338, 343, 393
APRIM, 40
AUSGAB, 31, 126, 208, 329, 330, 332, 350, 351,
357–360
average shower behavior, 21, 26
azimuthal angle (see secondary angle), 54

electron field, 41
polar angle, 332, 346
BRMSDZ, 44
BRMSFZ, 44

CALL (PEGS option), 372, 395, 404
cgview, 212, 215
CHARD, 341
charged particle transport, 28
coherent scattering (see Rayleigh scattering), 133
COHETM, 133
COHETZ, 133
collision loss, 20, 21
combinatorial geometry (see geometry, combinatorial), 204
COMP (PEGS option), 388, 391, 393, 399
COMPDM, 64
composition sampling, 24, 44
COMPRM, 64
COMPT, 360
BHABDM, 69
COMPTM, 64
BHABHA, 360
Compton profile, 135, 335, 336, 343, 389, 391,
Bhabha scattering, 41, 61, 66, 68
392, 397, 400
radiative, 41Compton scattering, 20, 21, 27, 61, 62
BHABRM, 69
binding effects, 134, 389, 391, 392, 397
BHABTM, 69
Doppler broadening, 134, 192, 335, 336, 343,
biasing, 208, 209
389, 391, 392, 397, 400
BLOCK SET, 31, 329, 340
conditional density function, 23
Blunck-Leisegang modification, 86
conditional distribution function, 23
Boltzmann transport equation, 26
correction, 28, 37, 40, 42, 43, 52, 85
Born approximation, 37
Coulomb correction, 37, 40, 52
branching ratio, 29, 30
Coulomb scattering, 20, 28
BREMDR, 53
critical energy, 20
BREMDZ, 44, 53
cross section, 27–30
BREMS, 360
cumulative distribution function, 22, 30
bremsstrahlung, 20, 21, 28, 37, 207, 389, 391, cutoff (bremsstrahlung), 37, 38
392, 397
cutoff energy, 26, 28, 30, 31, 38, 54, 86
soft, 28438

CYLNDR, 204
DECK (PEGS option), 395, 398, 403
decomposition (see composition sampling), 24
delta-ray, 20, 28
densitymaterial, 33, 345
scaling, 345
density effect, 83–85, 389–393
density function, 29, 30
DIFFER, 54, 399
direct sampling, 24, 27
distribution function, 27, 29
see also cumulative distribution function, 22
DNEAR, 337, 346, 353, 354
DUNIT, 31, 336, 344, 349
ECNSV1, 205
ECUT, 342, 343, 345, 358
EDEP, 205, 334
efficiency
simulation, 26, 30
EGS3, 1, 75, 87, 125
egs5run, 409
EII, 360
ELECTR, 208, 345, 346, 353, 360
electron binding, 28
electron density, 37, 38
electron impact ionization, 142, 334–336, 343,
390–392, 398, 400
electron transport, 28
ELEM (PEGS option), 388–390, 399
Elwert factor, 43
EMAXE, 339, 342
ENER (PEGS option), 393, 400
energy loss, 20, 28, 73
step-size, 345
energy sampling, 43, 54
energy straggling, 86
EPCONT, 208
ESAVE, 339, 346
ESTEPR, 339, 345
event (see history), 26
event counter, 205, 341
excitation, 20, 21, 28

Faraday cup, 210
FCOULC, 41
Feynman diagram, 37
fluorescence, 204, 333, 344
form factor, 40, 133, 134
FUDGEMS, 389, 391, 392, 397
geometry, 31
combinatorial, 204, 213, 214
multi-cylinder, 204
multi-slab, 204
Goudsmit-Saunderson multiple scattering, 96, 335,
345, 382
Hartree, 40
HATCH, 31, 329, 330, 332, 335, 336, 342–345,
348–350high frequency limit (bremsstrahlung), 43
history, 26, 30, 31
HOWFAR, 31, 205, 210, 211, 213, 329, 330, 332,
350, 353–356, 358
HPLT (PEGS option), 372, 396, 406
IAPRIM, 389, 391, 392, 397
IARG, 357–361
IAUSFL, 208, 360
IDISC, 334, 353
importance sampling, 205, 207, 209
incoherent scattering function, 136, 335, 336, 343,
389, 391, 392, 397
inelastic scattering, 28
infrared catastrophe, 28, 38
interaction probability, 27
ionization, 20, 21, 28, 142, 210
IPHTER, 344
IRNEW, 354
IROLD, 206
IRSPLT, 206
joint density function, 22, 29, 30
joint distribution function, 22
K-edge, 125, 126, 344
K1HSCL, 345
K1LSCL, 345
Klein-Nishina formula, 62439

subroutine descriptions, 381, 382
PEGS3, 42
PEGS5, 31, 329, 330, 332, 335, 341, 342
PHOTO, 125, 360
photoelectric effect, 20, 27, 125, 133, 344
PHOTON, 125, 208, 353, 360
Møller scattering, 41, 61, 66
photon transport, 27
radiative, 41
photoneutron, 21
magnetic field transport, 204, 209
PHOTTE, 125
MAIN, 31, 329, 332, 335, 336, 339–341, 343, 348– PHOTTZ, 125351
PLAN2P, 204
marginal density function, 25
PLANE1, 204
mass absorption coefficient, 21
PLTI (PEGS option), 372, 395, 405
materialization (see pair production), 20
PLTN (PEGS option), 372, 396, 405
mean free path, 27
PMDCON, 33
media data, 31
polar angle (see secondary angle), 54
MIX, 33
polarization, 38
mixed sampling, 25(see also density effect), 37
MIXT (PEGS option), 388, 392, 393, 399
polarized photons
Molière multiple scattering, 87
scattering, 137, 336, 344, 351
Bethe condition, 92, 94, 336, 345
preprocessor code (see PEGS), 30
reduced angle, 88
pressure correction factor, 85
validity, 92, 94, 96, 336, 345
probability density function
MOLLER, 360(see also density function), 22
MORSE-CG, 204, 214
probability theory, 21
multiple scattering, 20, 28, 54, 86, 210, 336, 397 PWLF (PEGS option), 372, 394, 401
see also Goudsmit-Saunderson, 96
radiation integral, 41
see also Molière multiple scattering, 87
step-size dependence, 113, 117, 160, 335, 336, radiation length, 31, 33, 42
radiation loss, 20, 21
341, 345, 390–392, 398
random hinge, 100, 109
NAMELIST, 30
moments, 103
NP, 360
random number generator, 337, 347, 348
NTALLY, 205
restart, 348
random variable, 22, 27, 29
PAIR, 360
range rejection, 345
pair production, 20, 21, 27, 37, 207, 346
Rayleigh scattering, 27, 133, 335, 336, 343, 389,
electron field, 41
391, 392, 397
polar angle, 332
rejection sampling, 24, 25, 44
threshold, 53
restricted stopping power, 74, 389, 391, 392, 397
PAIRDR, 53
RHOR, 345
PAIRDZ, 53
RLUXDAT, 337, 347, 348
particle trajectories, 210
run5again, 414
PCUT, 342, 343, 358
PEGS, 28, 30, 31, 41
sampling theory, 21
function descriptions, 384–387
scattering, 20
Landau distribution, 86
lateral spread, 21
leading particle biasing, 204, 207
Lorentz force, 209
LPM effect, 28, 37, 38

440

scattering power, 104
scattering strength, 104
scoring, 31, 205, 208
screening, 28, 39–41
screening factor, 43
secondary angle, 61, 62, 65, 67, 69, 72, 133, 344,
346
photoelectrons, 339
secondary energy, 62, 64, 65, 67, 69, 72
see also energy sampling, 54
SHOWER, 31, 329, 330, 332, 350, 351
shower, 20, 21, 26, 28, 30
single scattering tail, 93
SLAC, 210
SLAC-265, 1
soft electron scattering, 28
soft interaction, 28
SPINIT, 85
SPIONB, 85
splitting, 204–206
bremsstrahlung, 332, 347
EII x-rays, 194
EII x-rays, 334, 346
stack, 26, 29, 53, 69, 206
stopping power, 73

UCBEND, 209
UCSAMPCG, 214
UCSAMPL5, 214, 361
UE, 338, 393
uniform distribution, 23
units, 33
distance, 31, 336, 344
energy, 31, 33, 39
UP, 338, 393
UPHI, 62, 65, 67, 72, 360
USTEP, 211, 334, 353, 354
variance, 30, 205
variance reduction, 26, 204–207, 209, 347
weighting, 208
XSIF, 42TEST (PEGS option), 372, 395, 404
Thomas-Fermi, 40
time (see efficiency, simulation), 30
TMSTEP, 334
TMXSET, 336, 345
transport mechanics, 100, 109
triplet production, 41
TSTEP, 334
tutorial programs
tutor1, 145tutor2, 155tutor3, 157
tutor4, 160
tutor5, 171
tutor6, 179
tutor7, 182
tutor8, 194
TVSTEP, 334
UC LP, 208
441