MC principles 1

Principles of Monte Carlo calculations

Monte Carlo

MC principles 2

Numerical methods based on the use of random numbers

Able to solve deterministic problems (Monte Carlo)and problems with random components (simulation)

MC/simulation algorithms are fairly “natural”

Formally, a Monte Carlo calculation is equivalent to a set of integrations

Basic ingredient: Random number generator (rng)delivers r.v. ξ uniformly distributed in (0,1)

Simple example: congruential rng

Note: the ξ values are not truly random (pseudo-random)... but the computer does not know that

A good random number generator

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C *********************************************************************C FUNCTION RANDC *********************************************************************

FUNCTION RAND(DUMMY) ! To prevent changes by optimizerCC This is an adapted version of subroutine RANECU written by F. JamesC (Comput. Phys. Commun. 60 (1990) 329-344), which has been modifiedC to give a single random number at each call.C

IMPLICIT DOUBLE PRECISION (A-H,O-Z), INTEGER*4 (I-N)PARAMETER (USCALE=1.0D0/2.0D0**31)COMMON/RSEED/ISEED1,ISEED2 ! Define the ‘state’ of the generator

CI1=ISEED1/53668ISEED1=40014*(ISEED1-I1*53668)-I1*12211IF(ISEED1.LT.0) ISEED1=ISEED1+2147483563

CI2=ISEED2/52774ISEED2=40692*(ISEED2-I2*52774)-I2*3791IF(ISEED2.LT.0) ISEED2=ISEED2+2147483399

CIZ=ISEED1-ISEED2IF(IZ.LT.1) IZ=IZ+2147483562RAND=IZ*USCALE

CRETURNEND

Random sampling

Cumulative distribution function of x [or of p(x)]

⇔

0.0

0.2

0.4

0.6

0.8

1.0P(x)

p(x)

x

Generic problem: generate values of a random variable x with a givenprobability distribution function (pdf) p(x). Typically as a transformof one (or several) random numbers

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Increases monotonically from 0 to 1

Discrete random variables, xi withpoint probability pi , represented as

Inverse transform method

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Random values of x are generated by means of the sampling formula

0.0

0.2

0.4

0.6

0.8

1.0P(x)

p(x)

x

ξ

that gives sampled x values with pdf

the method does work

Example: exponential distribution

Sampling of discrete random variables

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1 0.06252 0.18753 0.50004 1.0000

xx

x

x

0

1

0.5

1

2

3

4

ξ

Lookup table

Generic sampling algorithms

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Discrete r.v.: Walker's aliasing method. Optimal: the sampling "speed" isindependent of the size (number of elements) of the sample space

Continuous r.v.: Piecewise Rational Inverse Transform with Aliasing (RITA)Adaptive interpolation; very accurate

Other sampling techniques:-- Rejection method-- Composition method-- Metropolis-Hastings for high-

dimensionality pdf's (not used in radiation transport)

Example: Isotropic radiation source

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Isotropic source: Emits particles with directions uniformly distributed on the unit sphere

Direction of motion defined as a unit vector

Sampling the initial direction of aparticle from the source:

Alternative method: introduce the variable

Monte Carlo integration

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Formally, all Monte Carlo (MC) calculations are equivalent to integrations

We introduce randomness by considering x as a r.v. with a pdf p(x) that 1) vanishes outside the interval (a,b) and 2) p(x) > 0. We write

with

Monte Carlo estimator: Sample a large number N of values xi from p(x)and compute

NOTE: the variance of f(x), , can be estimated similarly

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Multiple runs of the MC program (using different seeds of the rng) will give different results: is a random variable with mean and variance

The central limit theorem implies that follows a normal distribution

with standard deviation

The uncertainty interval contains the true value of the integral with probability 68.3% if n=1, 95.4% if n =2, 99.7% if n =3 (3σ rule)

Monte Carlo is more efficient than conventional (trapezoidal rule) integration when the dimension of the integration domain is >4

The efficiency of the integration algorithm isdetermined by the adopted pdf p(x) ==>variance reduction techniques

Radiation transport

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Basic problem: Given a radiation source in a material structure, determine the radiation flux and the space distribution of depositedenergy (particle penetration and slowing down, secondary particles)

Notice: 1) the interaction events are stochastic and so is the transportprocess

2) the problem involves multiple variables:- kind of particle- position coordinates (3) - energy (1) - direction of motion (2)

a problem well suited for Monte Carlo simulation

1954: E. Hayward and J. Hubbell, first simulation of photon transport 1963: M. Berger, general strategies for charged particles

Radiation transport

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Basic assumptions / simplifications:

-- The medium is homogeneous, isotropic and amorphous with known composition and density (random scattering medium)

atoms or molecules per unit volume

-- Collisions (interactions) are with single atoms (or molecules)Not valid at low energies (diffraction and coherence effects)

-- All physics is contained in the atomic cross sections (CS)

-- Interactions "localise" the particles (as in a cloud chamber)

-- Individual particle histories are generated as a succession of "free flights and collisions" (trajectory model)

Interactions of photons

MC principles 13electron rest energy

Interactions of electrons and positrons

MC principles 14electron rest energy

Differential cross section

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z

y

x

T

θ

dΩ, dW

φ

d, E

d0, E —W

jinc

ˆ

ˆ

W = energy transfer

Total cross section:

σ (an area) measures the ''interaction probability''Represents the "effective transverse area" of the target

σ may be very different from the "geometrical" x-section

Distribution of free flight lengths

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σ

J J — dJ

dJ = J Nσ ds

ds

N

Late

rally

hom

ogen

eous

bea

m The number of particles that interact equals the number of those that would hit any of the spheres of transverse "area" σ

The interaction probability per unitpath length is (inverse mean free path, IMFP)

The most probable path length to the next interaction is s=0 (!)

Mean free path to the next interaction:

Practical detailed simulation

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

and

Total cross sections:

PDF of the path length to the next event:

Kind of interaction:

Effect of each interaction:

Angular deflections

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z

xy

θ

φ

dn = (u,v,w)

dn+1 = (u0,v0,w0)

rn+1ˆˆ

ˆ

ˆ

ˆˆ

Note: valid also for polarized radiation (Stokes parameters)

A toy model for electron transport

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Non-relativistic, physically motivated, fully analytical simulation

Elastic collisions (A): (Wentzel, screened Rutherford, DCS)

Inelastic collisions (B): (restricted Thomson DCS, binding)

Consistent with Bethe’s stopping power:

Practical detailed simulation

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mat. 1 mat. 2vacuum

sθ, φ

En, dn

rn

B

E2, d2

E1, d1r2

r3E3, d3

rn+1

ss

s

AA

B

r1

^

^^

^

W

Reliability depends on:1) Accuracy of the adopted DCSs2) Validity of the trajectory model ( ),

Generation of random trajectories

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1) Set the initial state of the particle (laboratory frame):

3) Sample the length of the free flight:

4) Move to the new position:

5) Sample the type of interaction:

6) Simulate the interaction from the corresponding DCS, i.e., sample theenergy loss W and the scattering angles θ and φ

7) New energy and direction of motion:

8) Check for absorption and interface crossings. If still "alive" go to 2

2) Determine the total cross sections and mean free path at this energy

a b means that the value a is replaced by b

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10 photons, no electrons

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10 electrons, no photons

What causes the problem?

MC principles 24

Al (Z=13)20 keV electrons

1 MeV electrons in gold undergo about 20,000 elastic collisions before slowing down to rest, and a similar number of inelastic collisions (!) ... but most of them are very ‘soft’

Possible simulation strategies

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Detailed (analogue) simulation, interaction by interaction+ Nominally exact− Doable only for low-E, thin media− Requires very large data bases (interpolation is not a problem)

Class I (condensed) simulation, complete grouping+ Works for high energies and/or thick media− Difficulties to describe space displacements − Interface crossings require specific actions− Difficult to incorporate purely numerical interaction models− Usually applied only to elastic scattering, not easy for inelastic cols.

and bremms.

Class II (mixed) simulation + Hard events are described "exactly" from their DCSs+ Elastic, inelastic and bremsstrahlung are “tuned” independently+ Flexible (from detailed to class I)− Slow when cutoffs are too small

Energy straggling

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From the transport equation:

with

(moments of the energy loss in a single interaction)

Simplification: small path lengths, accumulated energy loss

Moments of the energy-loss distribution:

Moments of the energy-loss distribution

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Stopping power:

S(E) represents the average energy loss per unit path length

Energy straggling parameter:

2(E) represents the increase of variance per unit path length

Continuous slowing down approximation (CSDA)

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Heavy charged particles undergo many interactions with small W (the central limit theorem is applicable)

Valid only if (not for electrons!)

Particles lose energy continuously at the rate given by the stopping power S(E)

CSDA range: average path length to rest

Energy loss ∆s after a path length s: exact (within CSDA)

Energy loss distribution:

The energy-loss distribution of Landau (1944)

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Assumptions:L1. No bremss (as in other straggling theories!), only collision losesL2. Short path lengths,L3. Thomson cross section for hard events,

L4. L5. Stopping power for soft events evaluated from the Bethe theory

with

Landau solved the transport equation using the Laplace transform, whichrequires setting early in the calculation

Note: the second moment (straggling) of soft interactions is neglected

The energy-loss distribution of Landau (1944)

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Landau distribution:

where

Beyond Landau...

MC principles 31

Blunk and Leisegang (1950) correction: includes second moment of soft interactions

(convolution of Landau and a normal distribution with zero mean and variance )

Vavilov (1957): accounts for the finite value of Wmax (removes L4)

Bichsel and Saxon (1975): Vavilov's theory modified by including the second moment of soft interactions

Numerical solution of transport eq.: finite difference method withrealistic cross sections (including bremss!)

Numerical straggling vs. Monte Carlo

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Numerical straggling vs. Monte Carlo

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Multiple (elastic) scattering

MC principles 34

Single-scattering distribution: axial symmetry

Wentzel (1927) model, screened Rutherford

First Born approximation:

Allows analytical calculations, basis of the Molière (1948) theory

ICRU 77 database. Electrons and positrons, Z = 1— 99 (relativistic,Dirac, partial-wave expansion method)

Legendre expansion

with

Goudsmit-Saunderson (1940) theory

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Folding theorem. Initial direction along the z axis

- 1 collision:

- 2 collisions:

- n collisions:

Collisions in a path length s (Poisson):

Goudsmit-Saunderson distribution: exact angular distribution

Goudsmit-Saunderson (1940) theory

MC principles 36

... after little rearrangement

with

transport cross sections

λ are the transport (mean) free paths

Moments:

"scattering power"

Molière (1948) theory: GS for the Wentzel DCS, with mathematical approximations (Fernández-Varea et al., 1993)

Goudsmit-Saunderson (1940) theory

MC principles 37

(ICRU 77 DCSs)

Multiple scattering with energy loss. Lewis (1950)

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Energy loss from the CSDA:

All energy-dependent quantities can be regarded as functions of sExample: average number of collision in s

Transport equation:

with the DIMFP

Lewis' solution method: expansion in spherical harmonics, gives the angular distribution and low-order spatial moments

where

Multiple scattering with energy loss. Lewis (1950)

MC principles 39

Angular distribution:

with

Reduces to the GS distribution when the energy loss ∆s is small

Moments:

Practical calculations: (Negreanu et al., 2005; ICRU Report 77)

- ICRU 77 DCSs (Dirac partial-wave method)

- ICRU 37 stopping powers

Lewis (1950) theory

MC principles 40

(ICRU 77 DCSs)

Lewis (1950) theory

MC principles 41

Spatial moments:

Results: Kawrakow and Bielajew (1998)

Completely determined by λ1(E) and λ2(E), independent of λ(E)