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Monte Carlo Simulation Techniques

Pravata K Mohanty Tata Institute of Fundamental Research, Mumbai. Monte Carlo Simulation Techniques. s. Winter School on Astroparticle Physics, Bose Institute, Darjeeling, 14 - 22 December 2009. Deterministic Process. For a given input, the outcome can be exactly determined. s.

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Monte Carlo Simulation Techniques

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  1. Pravata K Mohanty Tata Institute of Fundamental Research, Mumbai Monte Carlo Simulation Techniques s Winter School on Astroparticle Physics, Bose Institute, Darjeeling, 14 - 22 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 Atmosphere N1 N2 N3 Outcome can’t be exactly determined Example: Interaction of primary cosmic rays in the atmosphere and development of EAS s E1 E2 E3 E1 = E2 = E3 = E N1 = N2 = N3

  4. Stochastic Process P + P 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), 0 < 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 • Truerandom numbers: Obtained from natural processes • Pseudo random numbers: generated by computersusing 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 = 2531011

  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. Applicationsof 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 s 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(1-0.9375P)1/4

  19. Continued 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(5.596707 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. Outer clad n2 Inner clad n1 Core n0 Photon propagation in WLS fiber

  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 No of photons produced 46,000 Fraction of photons escaped 25% Fraction of photons lost due to 60% attenuation in scintillator Fraction of photons captured by 15% 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 pkm@tifr.res.in

  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 s

  34. Photon statistics With Tyvek reflector No reflector No of photons produced 46,000 46,000 Fraction of photons escaped 25% 75% Fraction of photons lost due to 60% 20% attenuation in scintillator Fraction of photons captured by 15% 5% 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  

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