1. Radiation Transport Calculation by Monte Carlo Method
H. Hirayama, Y. Namito
KEK, High Energy Accelerator Research Organization

2. Monte Carlo Method
A mathematical method to solve problem using random number is called as “Monte Carlo Method”
Named by J. von Neumann S. M. Ulam
Accordingly, generation of random number is the most important technique in Monte Carlo method.

3. Method to generate random number
Utilize dice or roulette
Very slow
Utilize table of random number
Statistical nature is well studied.
Need to hold total number necessary as data.
Not most speedy way of generating random number.
Utilize physical phenomena such as disintegration of radioisotope.
Cumbersome in converting to numbers.
Have problem in stability and reproducibility.

4. Pseudorandom number
Select initial seed random number R0 , adequately. Generate next random number by recursion equation Rn+1= f(Rn) .
The rest after divided by m is treated as next random number.
The total number of integer less than m is m. Then, pseudorandom number has finite periodic length.
Good pseudorandom number,
Can be generated QUICKLY.
Have long periodic length.
Have reproducibility.
Have good statistical nature.
A pseudorandom number can be obtained by dividing the generated pseudorandom number by m.

5. Pseudorandom number by linear congruential method
Linear congruential method introduced by D. H. Lehmer is most widely used; Rn+1=mod(aRn+b,m)
mod(aRn+b,m) is a rest when aRn+b is divided by m.
a, b and m are positive integer. m is a maximum integer which compiler can handle.
Name a b m
RANDU
65539
231
SLAC RAN1
69069
SLAC RAN6

6. Generation of pseudorandom number by pocket calculator
Generate 10 pseudorandom number by setting R0=3, a=5, b=0 , and m=16.
Certify that the same pseudorandom number is produced.
What is a cycle length?
Generate pseudorandom number again from a different R0 .
Excel is NOT recommended to use.

7. n Rn Rn*5 Rn+1=mod(Rn*a,m) 1 2 3 R0=3 15 15/16=0・・・Rest 15 R1=15
R0=3 15
15/16=0・・・Rest 15 R1=15 1 2 3
mod(Rn*a,m)： The rest when Rn*a is divided by m.

8. n Rn Rn*5 Rn+1=mod(Rn*a,m) 1 2 3 4 R0=3 15 15/16=0・・・Rest 15 R1=15
R0=3 15
15/16=0・・・Rest 15 R1=15 1
R1=15 75
75/16=4・・・Rest 11 R2=11 2
R2=11 55
55/16=3・・・Rest 7　R3=7 3
R3=7 35
35/16=2・・・Rest 3　R4=3 4
R4=3

9. Calculation p using random number
Choose ten pairs of random numbers (,) from left to right at an arbitrary place in Table 1 (made using SLAC RAN6).
Count number of pair which satisfy following condition.
AREA= p /4
p= 4 x (number of pairs which satisfy the condition.)/(Total number of pairs)

10. How to use Table 1

11. Trial No ξ η R
R≦1 1
0.896
0.618
1.088 2
0.759
0.690
1.026 3
0.251
0.094
0.268 ○ 4
0.371
0.148
0.399 5
0.492
0.519
0.715 6
0.789
0.567
0.972 7
0.397
0.179
0.435 8
0.576
0.341
0669 9
0.517
0.583
0.779 10
0.909
0.380
0.985
A=8
A/10=0.8
(A/10)*4=3.2

12. Reference Web page About random number
Dice, Calculation of π using random number
Pseudorandom number and Monte Carlo method
Changing of value of π when trial number is increased.
Newton August Issue, Page 29.

13. Other way of generating random number
Marasaglia-Zaman random number
G. Masaglia and A. Zaman, “A New Class of Random Number Generator”, Annals of Applied Probability 1(1991)
Long period length –2144 ~1043
A little bit cumbersome for controlling random number
Run on any 32-bit computer
RANLUX random number
F. James, “A Fortran implementation of the high-quality pseudorandom number generators”, Comp. Phys. Comm. 79 (1994)
Period length is
By using seed number of 1-231, independent series of random number can be produced. No overlap of them is expected.
Utilized in egs5 as a default random number generator.

14. Sampling from a discrete probability distribution
Let’s write the physical processes which are independent and rebellion each other as x1, x2,......,xn . And let’s assume that their probability as p1, p2, , pn. (For example, photoelectric effect, Compton scattering, and pair production in interaction of photon with matter.
Let’s write a random number which distributes between 0 and 1 uniformly as . We choose an event xi when following condition is satisfied.
F(xi) is called as Cumulative distribution function.

15. Introduction of sampling from a discrete probability distribution (1)
Example）　Sample interaction using a random number from a distribution of, Photoelectric effect：３０%, Compton scattering :５０%, Pair production: 20%

16. Introduction of sampling from a disrete probability distribution (2)
Random
Probability of Compton scattering 
”Cumulative distribution function” or “Adding up calculation”.

17. Example of sampling from a discrete probability distribution.
Let’s write probability of photoelectric effect, Compton scattering, and pair production as Pphoto, PCompt , Ppair.
　　　Pphoto +PCompt + Ppair =1
Photoelectric effect is sampled when
Compton scattering is sampled when
Pair production is sampled when

18. Sampling from a continuous probability distribution
Let’s write a physical process emerge in a region of x and x+dx with a probability of f(x)dx [axb]. This f(x) is called as probability density function (PDF).
Cumulative distribution function (CDF:F(x))
Let’s write a random number which distribute between 0 and 1 uniformly as . Sampling procedure is written as,
　　x is obtained as ,
This x can be obtained by direct calculation if this equation is analytically solved. This is called as “Direct sampling method”.

19. F(x) 1  x a x b

20. Example of direct sampling – Calculation of flight path length
Let’s write an interaction probability of one incident particle per unit distance as St . The probability that first interaction occur between l and l+dl is,
l: Flight path length
l: Mean free path
h: Random number (1-h and h are equivalent. )

22. Infinite medium
Photon
Incident condition：Energy, position, direction e0, x0, y0, z0, u0, v0, w0

23. Sample distance l toward interaction point l=-ln()/
Coordinate after movement
x=x0+u0l, y=y0+v0l, z=z0+w0l l
Initial condition：Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

24. Sample kind of interaction
Photoelectric effect：a, Compton scattering : b, Pair production ：ｃ
　<=a/(a+b+c): Photoelectric effect
a/(a+b+c)< <=(a+b)/(a+b+c): Compton scattering
>(a+b)/(a+b+c): Pair production 　
x=x0+u0l, y=y0+v0l, z=z0+w0l l
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

25. Trace of generated particle Electron
Sample energy and direction of each particle.
Photon
x=x0+u0l, y=y0+v0l, z=z0+w0l l
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

26. d>l： Move toward interaction point d<=l：Move by a distance of d
Same material: Additional move by a distance of l-d
Different material ：Sample interaction point again
Region boundary
Interaction point
x=x0+u0l, y=y0+v0l, z=z0+w0l l d
Calculate straight path length toward boundary (d) .
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

27. Record information
　　Particle moves：Energy deposition
　　　　　　　　　　Path length
　　　　　　　　　　Boundary crossing
　　Photoelectric effect：Energy deposition
　　Below cut off energy
　　Stop tracing (Ex: Out of boudary)
Region boundary
Calculate straight distance (d) toward boundary l d
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

28. Sample distance toward interaction point Condensed History Technique
As the mean free path of elastic scattering of electron and positron is nm to mm range, direct treatment of this process is unrealistic from the point of calculation efficiency.
Sample distance toward interaction point
Condensed History Technique
Divide a distance toward major interaction into many fine steps, evaluate changing of direction and position and route distance using multiple scattering model.
　　l=-ln()/ l7 l8 l x x l5 x x l6
Charged particle looses part of its energy while moving via ionization or excitation. l4 x l3 x x l2 11
Energy deposition at each step is,
　Route distance x stopping power (dE/dx)
Electron or positron
Initial condition ：Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0

29. Transport calculation by hand calculation
Use random numbers in Table 1 (Made by SLAC RAN6）
Can start at arbitrary point and proceed to arbitrary direction in a series. (No jump, no change direction) when use random number.
Effective digit is 3 in calculation results

30. How to use random number in Table 1

31. Photon transport calculation 1 by hand calculation (Fig.1)
As is shown in Fig.1, there is a material A of 50 cm thickness.
Let’s assume that a photon of 0.5 MeV incident to material A vertically from left side
Let’s assume that mean free path is 20 cm.
Let’s assume that ratio of photoelectric effect and Compton scattering as 1:1 .
Let’s assume that photon energy and direction are not changed after Compton scattering.
Example 1
Initial random number: l=-20.0 x ln(0.234)=29.0
29.0(cm)<50.0(cm)
Next random number:0.208 (<0.5) – Photo electric effect （Terminate）
Example 2
Next random number: l=-20.0 x ln(0.906)=1.97
1.97(cm)<50.0(cm)
Next random number :0.716 (>0.5) – Compton scattering
Next random number： l=-20.0 x ln(0.996)=0.0802
0.0802(cm)< (cm)

32. Single layer No. d(cm) Random number l(cm) d>l dl Photo. Compt.
Exp.1
50.0
0.234
29.0 *
0.208
Exp.2
0.906
1.97
0.716
48.03
0.996
0.0802
0.600
47.95
0.183
34.0
0.868
13.95
0.351
20.9

34. Photon transport calculation by hand 2 (Fig.2)
As is shown in Fig. 2, there are material B of 20 cm thickness after material A of 30 cm thickness.
Let’s assume that a photon of 0.5MeV incident onto material A vertically from left side.
The mean free path and ratio of photoelectric effect to Compton in material A is the same as previous problem.
Let’s assume that mean free path in material B to be 3 cm.
Let’s assume that Photo to Compton ration in material B to be 3:1 .
Like previous example, photon energy and direction are assumed to be un-changed after Compton scattering.

35. Example of calculation
Used random number in table.1
Initial random number : l=-20.0 x ln(0.329)=22.2
22.2(cm)<30.0(cm)
Next random number (>0.5) – Compton scatt.
Next random number: l=-20.0 x ln(0.234)=29.0
29.0(cm)> (cm)
Move toward boundary between A and B (30.0cm)
Next random number : l=-3.0 x ln(0.281)=3.80
3.80(cm)<20.0(cm)

36. Medium A
No.
d(cm)
Random number
l(cm)
d>l
dl
Random
number
Photo.
Compt
Exp.1
30.0
0.329
22.2 *
0.612
7.8
0.234
29.0
Medium B
d(cm)
Random number
l(cm)
d>l
dl
Random
number
Photo.
Compt
20.00
0.281
3.80 *
0.906
16.20
0.716
1.00
0.996
15.20
0.600
1.53
0.183

38. Complex but realistic calculation of photon
Geometry is Al slab of 10 cm thickness. Trace photon trajectory in following assumptions.
Incident photon energy is 0.5 MeV.
Photon scattering angle in Compton scattering is in the unit of 90 degree and scattering probability toward each angle is the same. This is independent of photon energy.
Scattered photon energy is calculated as,

39. Complex but realistic calculation of photon
Scattering azimuth angle is 0 degree and 180 degree with the same probability (1:1) . (Compton scattering occurs in X-Z plane. Left side from propagation direction is defined as 0 degree.)
Read mfp and branching ratio of interaction from Figs. 4 and 5, respectively.
Photon cut off energy is 0.05 MeV.

43. Electron trajectory without secondary particle
Geometry is Al slab of 1 mm thickness as Fig.7.
1.0 MeV electron incident vertically from left side.　
Ignore correction of route distance due to multiple scattering.
Ignore production of secondary particles, such as  ray and bremsstrahlung photon.
Electron step size is 0.01 cm regardless of electron energy.

44. Electron trajectory without secondary particle
Changing of electron direction due to multiple scattering is in unit of 90 degree with the same probability. This is independent of electron energy.
0°-- 1/3, 90°-- 1/3, 180°-- 1/3
Azimuth angle after multiple scattering is either 0 degree and 180 degree with the same probability. 0º and 180º are left side and right side of propagation.
Energy loss due to inelastic scattering is 0.04 MeV per 0.01 cm. This is independent of electron energy.
Electron cut off energy is MeV .

47. Example of random walk A magazine, Newton, Page 42-43, Aug (2009).
Random walk in 1, 2, and 3 dimension is shown as an example of irregular motion.
Step size is common. The different feature of motion due to difference of dimension is shown.

48. Record of modification
English version was made on 16Jan2013 from Japanese version of
Subscript number in inequality in Page 14 was reduced by *
* File name was mot changed as this is a fix of typo.