Monte Carlo 1

Monte Carlo simulation ofphoton and electron transport

Francesc Salvat

First Barcelona Techno WeekCourse on semiconductor detectorsICCUB, 11-15th July 2016

Monte Carlo 2

Simulations performed with the code system PENELOPE, an acronym for "PENtration and Energy LOss of Positrons and Electrons"

A general-purpose Monte Carlo simulation code system with- Realistic, well defined interaction models- Fast and accurate random sampling algorithms- Efficient tools for tracking particles through complex geometries (constructive quadric geometry)

- Complementary tools: variance reduction, transport in electromagnetic fields, tabulation of macroscopic interaction parameters, ...

Distributed by the OECD/Nuclear Energy Agency Data Bank (Paris) and the RSICC (Oak Ridge National Laboratory) More than 1,500 copies distributed

References:

F. Salvat, PENELOPE-2014: A Code System for Monte Carlo Simulation of Electron andPhoton Transport. OECD NEA Data Bank/NSC DOC(2011)/5 (OECD Nuclear EnergyAgency, Issy-les-Moulineaux, 2011)http://www.oecd-nea.org/lists/penelope.html

F. Salvat and J.M. Fernández-Varea, "Overview of physical interaction models for photonand electron transport used in Monte Carlo codes", Metrologia 46 (2009) S112–S138

Introduction

Monte Carlo 3Photonuclear absorption is neglected

Photon interactions

Monte Carlo 4Electron rest energy: keV

Electron and positron interactions

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>>>>>>>>> TABLES

The program TABLES reads a material data file and generates a number ofascii files with relevant energy-dependent interaction data (total crosssections, mean free paths, stopping powers and radiation yields ofelectrons and positrons, ranges, ...). [...]

In principle, TABLES gets the interaction properties of a material fromits PENELOPE material data file. [...] If that file does not exist,TABLES builds it by calling the PENELOPE routines; in this case the userhas to provide some information on the material (chemical composition,density, ...), which is entered from the keyboard. For the 280 materialslisted in the file 'material_list.txt', the user only needs to enter theidentification number of the material. [...]

The output file 'tables.dat' is generated by the PENELOPE routines andcontains most of the quantities used in the Monte Carlo simulations. Theoutput files with the extension '.tab' contain sets of relatedquantities, which are described in the heading comments of each file. Tovisualize the contents of output files, we use the plotting softwaregnuplot, which is small in size, available for various platforms(including Windows and Linux) and free (distribution sites are listed atthe gnuplot central site, http://www.gnuplot.info). When gnuplot isinstalled on the computer, the provided script 'fname.gnu' plots thecontents of the output file with the same filename, 'fname.tab'.

Program TABLES

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1.0E-7

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

1.0E+1

1.0E+2

1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6 1.0E+7 1.0E+8 1.0E+9

MFP

*rho

(g/

cm**

2)

Energy (eV)

Electron mean free paths (MFP)

Elastic

Inelastic

Bremsstrahlung

Inner shell ion.

Total

Screenshot of TABLES (gold, Z = 79)

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1.0E+2

1.0E+3

1.0E+4

1.0E+5

1.0E+6

1.0E+7

1.0E+8

1.0E+9

1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6 1.0E+7 1.0E+8 1.0E+9

STP

[eV/

(g/c

m**

2)]

Energy (eV)

Electron mass stopping powers (STP)

Collision

Radiative

Total

Screenshot of TABLES (gold, Z = 79)

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1.0E-8

1.0E-6

1.0E-4

1.0E-2

1.0E+0

1.0E+2

1.0E+4

1.0E+6

1.0E+2 1.0E+3 1.0E+4 1.0E+5 1.0E+6 1.0E+7 1.0E+8 1.0E+9

mu/

rho

(cm

**2/

g)

Energy (eV)

Photon mass attenuation coefficients (mu/rho)

Rayleigh

Compton

Photoabsorption

Pair production

Total

Screenshot of TABLES (gold, Z = 79)

Monte Carlo 9

ξ = random number, U(0,1)

and set , i.e., x isthe solution of the sampling eq.

Consider the cumulative distribution function

Example: exponential distribution,

RITA (Rational Inverse Transform with Aliasing): optimal generic algorithm for sampling arbitrary single-variate distributions (discrete or continuous)

Monte Carlo: Numerical solution methods based on the use of random numbers

Random sampling

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Scattering model: Two interaction mechanisms, A and B, with DCSs

and

Total cross sections:

Path length to the next interaction:

Kind of interaction:

Effect of each interaction:

Practical detailed simulation

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Reliability depends on : 1) accuracy of adopted DCSs2) validity of the "trajectory model"

(de Broglie wavelength, λdB, much less than the inter-atomic spacing) For electrons and positrons

mat. 1 mat. 2vacuum

sθ, φ

En, dn

rn

B

E2, d2

E1, d1r2

r3E3, d3

rn+1

ss

s

AA

B

r1

^

^^

^

W

Monte Carlo 12

>>>>>>>>> SHOWER

The program SHOWER generates electron-photon showers within a slab ofone of the 280 materials listed in the file 'material_list.txt', and ofany material whose definition file has been previously generated byrunning the program TABLES. SHOWER displays the generated showersprojected on the computer screen plane. The program is self-explanatory,and requires only a small amount of information from the user, which isentered from the keyboard in response to prompts from the program.Electron, photon and positron tracks are displayed in different colorsand intensities that vary with the energy of the particle. The maximumnumber of showers that can be plotted in a single shot is limited to 50,because the screen may become too cluttered.

Once on the graphical screen, the view plane can be rotated about thehorizontal screen axis by typing 'r' and the rotation angle in degrees;the screen plane can also be rotated progressively, by 15 deg steps, bypressing the 'enter' key repeatedly. Entering the single-charactercommand 'n' erases the screen and displays a new bunch of showers.Observation of single showers projected on a revolving plane gives atruly three-dimensional perspective of the transport process.

Program SHOWER

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Screenshot of SHOWER (10 MeV electrons in water)

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Statistical uncertainties: All Monte Carlo calculations are equivalent to integrals

Monte Carlo estimators:

Simulation result:

- Central limit theorem ⇒ uncertainty interval includes the exact value with 99.9 % probability

- Usually, simulation is slow (but results come with the associated uncertainties!)

In radiation transport studies, p(x) is unknown. The simulation of individual showers can be regarded as a sampling procedure of the random variable (shower) x

Example: Energy deposited within the sensitive volume of the detector

Statistical uncertainties

Monte Carlo 15

x

z

y

source

r

(x0, y0, z0)

E0

Eθ

t

φ α

Simulation geometry: A material cylinder and a point source of mono-energetic radiation, partially collimated.

Program PENCYL of the PENELOPE code systemDelivers very detailed information on the transport process

Direction vectors and polar coordinates

Monte Carlo simulation code MCtracks

x

y

z

θ

φ

d̂

Monte Carlo 16

TITLE Point source and a homogeneous cylinder..

GSTART >>>>>>>> Beginning of the geometry definition list.LAYER 0 0.1 [Z-lower and Z-higher]CYLIND 1 0 0.1 [Material, R-inner and R-outer]GEND > Source definition.

SKPAR 2 [Primary particles: 1=electron, 2=photon, 3=positron]SENERG 40e3 [Initial energy (monoenergetic sources only)]SPOSIT 0 0 0 [Coordinates of the source center]SCONE 0 0 5 [Conical beam; angles in deg]

>>>>>>>> Material data and simulation parameters.MFNAME Si.mat [Material file, up to 20 chars]MSIMPA 1e3 1e3 1e3 0.05 0.05 1e3 1e3 [EABS(1:3),C1,C2,WCC,WCR]

>>>>>>>> Counter array dimensions and pdf ranges.NBE 0 0 100 [Energy window and no. of bins]NBANGL 45 18 [No. of bins for the angles THETA and PHI]

>>>>>>>> Energy-deposition detectors (up to 25).ENDETC 0 0 250 [Energy window and no. of bins]EDBODY 1 1 [Active cylinder]

>>>>>>>> Dose and charge distributions.DOSE2D 1 1 100 50 [Active body (KL,KC), nos. of bins in Z and R]

>>>>>>>> Job properties.RESUME dump.dat [Resume from this dump file, 20 chars]DUMPTO dump.dat [Generate this dump file, 20 chars]DUMPP 60 [Dumping period, in sec]NSIMSH 1e9 [Desired number of simulated showers]END [Ends the reading of input data]

Input file formats (tracks-Si.in)

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MCtracks results

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MCtracks results

Monte Carlo 19

MCtracks results