1. Monte Carlo Simulation Techniques
Pravata K Mohanty
Tata Institute of Fundamental Research, Mumbai
Winter School on Astroparticle Physics, Bose Institute, Darjeeling,
December 2009

2. Deterministic Process
For a given input, the outcome can be exactly determined. s
Example: X1 X2 X3 Y1 Y2 Y3
f(x) = x2

3. Stochastic Process s Atmosphere Outcome can’t be exactly determined
Example: Interaction of primary cosmic rays in the atmosphere and development of EAS s
E E E3
E1 = E2 = E3 = E
Atmosphere
N N N3
N1 = N2 = N3

4. Stochastic Process s P + P Q. what are the life time of the muons? +- 0 + -   s
Q. what are the life time of the muons?

5. Probability and Random numbers
Number of muons survive after time t
N (t) = N(0) e –t/
Number of decays after time t
Ndecay(t)= N(0) - N(t)= N(0) – N(0) e -t/
Decay probability P = N decay(t) / N(0)
= 1 – e -t/
Or t = -  ln(1-P), < P <1
Replace P with R, where you call R as a random
number and 0<R<1
t = -  ln(1-R) s

6. Muon decay s

7. Monte Carlo simulation
The average behaviour of the process obtained from the measurements i.e. the current knowledge about the process
Ex N (t) = N(0) e –t/
Convert it to a probability distribution and use random numbers for probability to generate the variates. s

8. Random Numbers True random numbers: Obtained from natural processes
Pseudo random numbers:
generated by computers using some algorithm.
Example: Linear Congruential Generator
X n+1 = a X n + c mod m
m is the period, X0 is the SEED
m and c should be relatively prime
In C++, m = 232, a=214013, c =

9. How to generate Random Numbers
In C or C++, you can generate random numbers like this
for (int i=0; i<10000; i++) {
r = rand(); //Here rand() is the random
} //number generator

10. Value of  using random numbers
Area of Circle/ Area of Square =  (a/2)2 /a2 = /4
a/2
Manual Method:
1. Randomly throw pebbles inside the square
2. Count the number of pebbles inside the circle
3. Take ratio of the number of pebbles inside
circle to the total.
-a/2
(0,0)
a/2
Using Computer
Generate points with x and y coordinates
uniformly inside the square of side a
x = -a/2 + a/2*R1 R1 and R2 are Random Numbers
y = -a/2 + a/2**R2
2. count the number of points inside the circle
r = (x2 + y2 ) < a/2
-a/2

11. History of Monte Carlo Simulation
The name "Monte Carlo" was popularized by Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, among others; the name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble. The use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino.
Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method to calculate the properties of the newly-discovered neutron. Monte Carlo methods were central to the simulations required for the Manhattan Project, though were severely limited by the computational tools at the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research.

12. Applications of Monte Carlo
Monte Carlo method is used in almost every field of science, mathematics to economics
Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms.
The Monte Carlo method is widely used in statistical physics
In experimental particle physics, these methods are used for designing detectors, understanding their behavior and comparing experimental data to theory.

13. Designing a plastic scintillator detector using Monte Carlo

14. Design Goals High photon yield Good spatial uniformity Good timing
Low Cost
Ease for fabrication

15. Plastic scintillator detector
For total internal reflection,
sin  > sin c
c = 38.7o

16. Meridional and Skew Ray Mode
In skew ray mode
the incident angle
changes at every
reflection

17. Important steps of MC Production and propagation of muons
Generation and propagation of photons inside scintillator considering the various losses
Propagation of photons in WLS fiber
Collection of photon and convert then to photo electrons

18. Generating Cosmic Ray Muons
Angular distribution of muons
dN/d  cos2
 can be randomly chosen from this distribution
The probability of getting a muon between 0 to  is
P =  2 cos2 cos sin d d
602cos2 cos sin d d
= (1 – cos4)/0.9375
Hence  = cos-1( P)1/4

19. Continued P  R, R is the random number
As we know the value of probability between is between 0 to 1, this can be selected using a uniform random number between 0 to 1
P  R, R is the random number
As  distribution is uniform between 0 to 2 
Probability P =  / 2 
or  = 2P = 2R
Then distribute the muons uniformly over the surface of the scintillator.
X = Xmin + ( Xmax – Xmin)*R1
Y = Ymin + (Ymax – Ymin)*R2

20. Energy loss calculation
Though mean energy loss remains fairly constant over a large energy range above minimum ionizing energy (Bethe-Block formula), however there is fluctuation in the energy loss around the mean. The fluctuation of energy loss is described by Landau distribution.
Landau distribution gives distribution of a universal
parameter called , which is independent of the
material and particle velocity.
The relation between energy loss ΔE and  is
ΔE =  [ + ln( x109 2 )/(1- 2 )Z2+1 - 2 - E ]
Where  = (0.1536/ 2) (Z /A) S
S is mass density of the material
The probability distribution of  is used from ROOT
Mathematical function LandauI

21. Photon Production Number of photons produced
N = ΔE Δl / ,  -> Energy required to produce a single photon
Δl -> Incremental path length
 = 100 eV, Navg = 20,000 /cm for vertical muons
Generate  and  for each photon randomly from an isotropic distribution
Track the photon and for every reflection check for the critical angle condition
sinI > sinc
c = 39o 

22. Attenuation loss of photons
Scintillator is not fully transparent to the blue wave length photons because of self-absorption in POPOP
The attenuation formula is
I = Io exp(-x/), Here  = Mean attenuation length
I/Io = exp(-x/ )
P = 1 - I/Io = 1 - exp(-x/ ), x = -  *ln(1-P) = -  *ln(1-R)
Determine the path length of each photon a priori and compare with total path length traversed at each reflection.

23. Diffuse Reflection Lambert’s cosine law dI/d  cos
Hence probability of photons reflected between 0 to 
P =  cos d
/2 cos d
Hence P = sin,  = sin -1 R and  = 2R

24. Photon propagation in WLS fiber
Core n0
Inner clad n1
Outer clad n2

25. Conversion of photons to Photo-electrons
PMT converts the photons to photo-electrons. The conversion efficiency depends on the quantum efficiency of the PMT.

26. Simulation Inputs

27. Photon statistics Number of photon collected at PMT = 208
No of photons produced ,000
Fraction of photons escaped %
Fraction of photons lost due to %
attenuation in scintillator
Fraction of photons captured by %
WLS fiber
Trapping Efficiency in Fiber = 14% of the captured photons
Number of photon collected at PMT = 208
Collection efficiency = 0.45%

28. Photo electron yield and timing comparison
parallel
parallel
matrix
matrix  

29. Photo–electron yield with Number of fibers
Ne  Nfib
Number of Fibers

30. The simulation code
This is a single C++ program of ~ 1000 lines of cde
The code can be compiled by g++ or C++ command
Easy to modify the inputs
Any one interested to use this can contact

31. Summary
The good agreement of simulation with measurement would allow us to design and optimize detector in future by doing simulation prior to the actual construction which would save lot of time and cost

32. THANKS

33. Plastic scintillator detector

34. Photon statistics With Tyvek reflector No reflector
No of photons produced , ,000
Fraction of photons escaped % %
Fraction of photons lost due to % %
attenuation in scintillator
Fraction of photons captured by % %
WLS fiber
Trapping Efficiency in Fiber: For skew rays = 14 % (Real case)
For meridional rays = 11%
Number of photon collected at PMT = 208
Collection efficiency = 0.45%

35. Photo electron yield and timing comparison
parallel
parallel
matrix  

36. Photo electron yield and timing comparison
parallel
parallel
matrix
matrix  