radiation transport calculation by monte carlo method h. hirayama, y. namito kek, high energy...
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Radiation Transport Calculation by Monte Carlo Method
H. Hirayama, Y. Namito
KEK, High Energy Accelerator Research Organization
2009.8.8
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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.
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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.
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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.
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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 mRANDU 65539 0 231
SLAC RAN1 69069 0 231
SLAC RAN6 663608491 0 231
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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.
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n Rn Rn*5 Rn+1=mod(Rn*a,m)
0 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.
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n Rn Rn*5 Rn+1=mod(Rn*a,m)
0 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
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Calculation 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.
0.122 R
AREA= /4
x (number of pairs which satisfy the condition.)/(Total number of pairs)
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How to use Table 1
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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
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Reference Web page
• About random number– http://www.nikonet.or.jp/spring/sanae/MathTopic/mo
ntecalro/montecalro.htm– Dice, Calculation of π using random number
• Pseudorandom number and Monte Carlo method – http://www.sm.rim.or.jp/~shishido/pie.html– Changing of value of π when trial number is
increased.
• Newton 2009. August Issue, Page 29.
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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)462-480.– 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) 111-114.
– Period length is 10 171
– 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.
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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.
1)(1
n
iin pxF
i
jji
i
jji pxFpxF
1
1
11 )()(
F(xi) is called as Cumulative distribution function.
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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%
0
0.1
0.2
0.3
0.4
0.5
0.6
Photoelectric Compton Pair
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Introduction of sampling from a disrete probability distribution (2)
”Cumulative distribution function” or “Adding up calculation”.
0
0.2
0.4
0.6
0.8
1
1.2
Photoelectric Compton Pair
Rando
m Probability
of Compton scattering
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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
,photoP,Comptphotophoto PPP
, Comptphoto PP
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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 [axb]. 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”.
)1)(( dxxfb
a
ii
x
adxxfxF )()(
)(1 Fx
ii
x
adxxfxF )()(
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F(x)
x0
1
a b
x
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Example of direct sampling – Calculation of flight path length
• Let’s write an interaction probability of one incident particle per unit distance as t . The probability that first interaction occur between l and l+dl is,
)ln(
)1ln()1ln(1
1)()(
)(
10 1
l
l
edllplP
dledllp
t
ll
tl
t
t
lFlight path length
: Mean free path
Random number and are equivalent.
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Photon
Incident condition : Energy, position, direction e0, x0, y0, z0, u0, v0, w0
Infinite medium
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Initial condition : Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
Sample distance l toward interaction point
l=-ln()/
Coordinate after movement
x=x0+u0l, y=y0+v0l, z=z0+w0l
l
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Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
l
x=x0+u0l, y=y0+v0l, z=z0+w0l
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
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l
Photon
Electron
Sample energy and direction of each particle.
Trace of generated particle
x=x0+u0l, y=y0+v0l, z=z0+w0l
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
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l
Interaction point
x=x0+u0l, y=y0+v0l, z=z0+w0l
Region boundary
d
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
Calculate straight path length toward boundary (d) .
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
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l
Region boundary
d
Calculate straight distance (d) toward boundary
Record information
Particle moves : Energy deposition
Path length
Boundary crossing
Photoelectric effect : Energy deposition
Below cut off energy
Stop tracing (Ex: Out of boudary)
Initial Condition: Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
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Electron or positron
Initial condition : Energy, position, and direction
e0, x0, y0, z0, u0, v0, w0
l=-ln()/
l
x x
x
x x
x x
Charged particle looses part of its energy while moving via ionization or excitation.
As the mean free path of elastic scattering of electron and positron is nm to m range, direct treatment of this process is unrealistic from the point of calculation efficiency.
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.
Energy deposition at each step is,
Route distance x stopping power (dE/dx)
11
l2
l3
l4
l5l6
l7 l8
Sample distance toward interaction point
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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
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How to use random number in Table 1
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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:0.234 -- 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:0.906 -- 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 : 0.996 -- l=-20.0 x ln(0.996)=0.0802– 0.0802(cm)<50.0-1.97(cm)
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Single layer
No. d(cm) Random
number
l(cm) d>l dl Random
number
Photo.
Compt.
Exp.1 50.0 0.234 29.0 * 0.208 *
Exp.2 50.0 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 *
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0cm 10cm 20cm 30cm 40cm 50cm
Exp. 1
Exp. 2P
C C C
Fig. 1 Trajectories for a single layer
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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.
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Example of calculation
• Used random number in table.1– Initial random number :0.329 -- l=-20.0 x
ln(0.329)=22.2– 22.2(cm)<30.0(cm)– Next random number 0.612 (>0.5) – Compton scatt.– Next random number:0.234 --l=-20.0 x ln(0.234)=29.0– 29.0(cm)>30.0-22.2(cm)– Move toward boundary between A and B (30.0cm)– Next random number :0.281 --l=-3.0 x ln(0.281)=3.80– 3.80(cm)<20.0(cm)
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Medium A
No.
d(cm) Random number
l(cm) d>l dl 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 dl 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 *
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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,
)cos1(511.0
1 0
0
E
EE
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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.
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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.
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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 0.01 MeV .
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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.
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Record of modification• English version was made on 16Jan2013 from
Japanese version of 2009.8.8. • Subscript number in inequality in Page 14 was
reduced by 1. 2013.1.17*