PENELOPE 2003 - OECD Nuclear Energy Agency

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6 Chapter 1. Monte Carlo simulation. Basic concepts which takes values from −1 to 1. Notice that cov(x, x) = var(x). When the variables x and y are independent, i.e. when p(x, y) = p x (x) p y (y), we have Moreover, for independent variables, cov(x, y) = 0 and var(x + y) = var(x) + var(y). (1.28) var{a 1 x + a 2 y} = a 2 1 var(x) + a 2 2 var(y). (1.29) 1.2 Random sampling methods The first component of a Monte Carlo calculation is the numerical sampling of random variables with specified PDFs. In this section we describe different techniques to generate random values of a variable x distributed in the interval (x min , x max ) according to a given PDF p(x). We concentrate on the simple case of single-variable distributions, since random sampling from multivariate distributions can always be reduced to singlevariable sampling (see below). A more detailed description of sampling methods can be found in the textbooks of Rubinstein (1981) and Kalos and Whitlock (1986). 1.2.1 Random number generator In general, random sampling algorithms are based on the use of random numbers ξ uniformly distributed in the interval (0,1). These random numbers can be easily generated on the computer (see e.g. Kalos and Whitlock, 1986; James, 1990). Among the “good” random number generators currently available, the simplest ones are the so-called multiplicative congruential generators (Press and Teukolsky, 1992). A popular example of this kind of generator is the following, R n = 7 5 R n−1 (mod 2 31 − 1), ξ n = R n /(2 31 − 1), (1.30) which produces a sequence of random numbers ξ n uniformly distributed in (0,1) from a given “seed” R 0 (< 2 31 −1). Actually, the generated sequence is not truly random, since it is obtained from a deterministic algorithm (the term “pseudo-random” would be more appropriate), but it is very unlikely that the subtle correlations between the values in the sequence have an appreciable effect on the simulation results. The generator (1.30) is known to have good random properties (Press and Teukolsky, 1992). However, the sequence is periodic, with a period of the order of 10 9 . With present-day computational facilities, this value is not large enough to prevent re-initiation in a single simulation run. An excellent critical review of random number generators has been published by James (1990), where he recommends using algorithms that are more sophisticated than simple congruential ones. The generator implemented in the fortran77 function RAND (table 1.1) is due to L’Ecuyer (1988); it produces 32-bit floating point numbers uniformly distributed in the open interval between zero and one. Its period is of the order of 10 18 , which is virtually infinite for practical simulations.
1.2. Random sampling methods 7 Table 1.1: Fortran77 random number generator. C ********************************************************************* C FUNCTION RAND C ********************************************************************* FUNCTION RAND(DUMMY) C C This is an adapted version of subroutine RANECU written by F. James C (Comput. Phys. Commun. 60 (1990) 329-344), which has been modified to C give a single random number at each call. C C The ’seeds’ ISEED1 and ISEED2 must be initialized in the main program C and transferred through the named common block /RSEED/. C IMPLICIT DOUBLE PRECISION (A-H,O-Z), INTEGER*4 (I-N) PARAMETER (USCALE=1.0D0/2.0D0**31) COMMON/RSEED/ISEED1,ISEED2 C I1=ISEED1/53668 ISEED1=40014*(ISEED1-I1*53668)-I1*12211 IF(ISEED1.LT.0) ISEED1=ISEED1+2147483563 C I2=ISEED2/52774 ISEED2=40692*(ISEED2-I2*52774)-I2*3791 IF(ISEED2.LT.0) ISEED2=ISEED2+2147483399 C IZ=ISEED1-ISEED2 IF(IZ.LT.1) IZ=IZ+2147483562 RAND=IZ*USCALE C RETURN END 1.2.2 Inverse transform method The cumulative distribution function of p(x), eq. (1.6), is a non-decreasing function of x and, therefore, it has an inverse function P −1 (ξ). The transformation ξ = P(x) defines a new random variable that takes values in the interval (0,1), see fig. 1.1. Owing to the correspondence between x and ξ values, the PDF of ξ, p ξ (ξ), and that of x, p(x), are related by p ξ (ξ) dξ = p(x) dx. Hence, ( ) −1 ( ) −1 dξ dP(x) p ξ (ξ) = p(x) = p(x) = 1, (1.31) dx dx that is, ξ is distributed uniformly in the interval (0,1). Now it is clear that if ξ is a random number, the variable x defined by x = P −1 (ξ) is randomly distributed in the interval (x min , x max ) with PDF p(x) (see fig. 1.1). This provides a practical method of generating random values of x using a generator of random numbers uniformly distributed in (0,1). The randomness of x is guaranteed by
Page 1 and 2: Data Bank ISBN 92-64-02145-0 PENELO
Page 3 and 4: FOREWORD The OECD/NEA Data Bank was
Page 5 and 6: TABLE OF CONTENTS Foreword ........
Page 7 and 8: 4.3 Combined scattering and energy
Page 9 and 10: PREFACE Radiation transport in matt
Page 11 and 12: The present version of PENELOPE is
Page 13 and 14: Chapter 1 Monte Carlo simulation. B
Page 15 and 16: 1.1. Elements of probability theory
Page 17: 1.1. Elements of probability theory
Page 21 and 22: 1.2. Random sampling methods 9 Nume
Page 23 and 24: 1.2. Random sampling methods 11 whe
Page 25 and 26: 1.2. Random sampling methods 13 can
Page 27 and 28: 1.2. Random sampling methods 15 mus
Page 29 and 30: 1.2. Random sampling methods 17 the
Page 31 and 32: 1.3. Monte Carlo integration 19 Con
Page 33 and 34: 1.4. Simulation of radiation transp
Page 37 and 38: ¡ ¢ 1.4. Simulation of radiation
Page 41 and 42: 1.5. Statistical averages and uncer
Page 43 and 44: 1.6. Variance reduction 31 also imp
Page 45 and 46: 1.6. Variance reduction 33 weight a
Page 47 and 48: Chapter 2 Photon interactions In th
Page 49 and 50: 2.1. Coherent (Rayleigh) scattering
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Page 53 and 54: 2.2. Photoelectric effect 41 ➤E E
Page 55 and 56: F 2.2. Photoelectric effect 43 1E+7
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2.4. Electron-positron pair product
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2.5. Attenuation coefficients 63 di
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2.6. Atomic relaxation 65 2.6 Atomi
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2.6. Atomic relaxation 67 two vacan
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Chapter 3 Electron and positron int
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3.1. Elastic collisions 71 nucleus
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3.1. Elastic collisions 73 1 E - 1
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ρ ρ ρ λ λ λ ρ ρ ρ λ λ λ
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3.1. Elastic collisions 77 becomes
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B ' B ' 3.1. Elastic collisions 79
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3.2. Inelastic collisions 81 where
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3.2. Inelastic collisions 83 global
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3.2. Inelastic collisions 85 plays
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3.2. Inelastic collisions 87 optica
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3.2. Inelastic collisions 89 The DC
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3.2. Inelastic collisions 91 In the
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3.2. Inelastic collisions 93 where
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c b c b 3.2. Inelastic collisions 9
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A l A u 3.2. Inelastic collisions 9
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3.2. Inelastic collisions 99 After
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3.2. Inelastic collisions 101 Table
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3.2. Inelastic collisions 103 In re
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3.2. Inelastic collisions 105 1Ε+4
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3.3. Bremsstrahlung emission 107 an
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3.3. Bremsstrahlung emission 109 1
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3.3. Bremsstrahlung emission 111 1
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3.3. Bremsstrahlung emission 113 wh
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3.3. Bremsstrahlung emission 115 Al
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3.3. Bremsstrahlung emission 117 Th
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3.4. Positron annihilation 119 (198
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3.4. Positron annihilation 121 It i
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Chapter 4 Electron/positron transpo
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4.1. Elastic scattering 125 Lewis (
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4.1. Elastic scattering 127 ⎡ ⎛
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4.1. Elastic scattering 129 simulat
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4.1. Elastic scattering 131 The sim
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4.2. Soft energy losses 133 soft in
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4.2. Soft energy losses 135 deviati
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4.2. Soft energy losses 137 scatter
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4.3. Combined scattering and energy
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4.4. Generation of random tracks 14
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Chapter 5 Constructive quadric geom
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5.1. Rotations and translations 155
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5.2. Quadric surfaces 157 Given a f
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5.2. Quadric surfaces 159 Table 5.1
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5.3. Constructive quadric geometry
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5.4. Geometry definition file 163 1
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5.4. Geometry definition file 165
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5.5. The subroutine package pengeom
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5.5. The subroutine package pengeom
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5.6. Debugging and viewing the geom
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5.7. A short tutorial 173 MODULE (
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5.7. A short tutorial 175 Writing a
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Chapter 6 Structure and operation o
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6.1. penelope 179 and/or numbers in
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6.1. penelope 181 1986). The format
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6.1. penelope 183 The connection of
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6.1. penelope 185 that has to be lo
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G y y y y y 6.1. penelope 187 CALL
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6.1. penelope 189 It is the respons
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6.2. Examples of MAIN programs 191
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6.3. Selecting the simulation param
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! 6.3. Selecting the simulation par
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6.5. Installation 205 radiation is
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6.5. Installation 207 To get the ex
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Appendix A Collision kinematics To
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A.1. Two-body reactions 211 Clearly
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A.2. Inelastic collisions of charge
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m A.2. Inelastic collisions of char
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Appendix B Numerical tools B.1 Cubi
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B.1. Cubic spline interpolation 219
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B.2. Numerical quadrature 221 B.2 N
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Appendix C Electron/positron transp
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C.1. Tracking particles in vacuum.
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C.1. Tracking particles in vacuum.
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C.2. Exact tracking in homogeneous
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C.2. Exact tracking in homogeneous
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Bibliography Abramowitz M. and I.A.
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Bibliography 235 Briesmeister J.F.
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Bibliography 237 James F. (1990),
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Bibliography 239 Reimer L. and E.R.
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Bibliography 241 Yates A.C. (1968),
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