Bielajew A.F. Fundamentals of the Monte Carlo method for neutral and charged particle transport

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8.4. OBTAINING µ FROM MICROSCOPIC CROSS SECTIONS 91

92 CHAPTER 8. TRANSPORT IN MEDIA, INTERACTION MODELS

which we have been discussing in the previous sections.

The cross section, σ can also depend on the energy, E, of the incoming particle. In this case

we may write the cross section diﬀerential in energy as σ(E). The symbol σ is reserved for

total cross section. If there is a functional dependence it is assumed to be diﬀerential in that

quantity.

Once the particle interacts, the scattered particle can have a diﬀerent energy, E

and direc-

tion, Θ, Φ relative to the initial z directions. The cross section can be diﬀerential in these

parameters as well. Thus,

σ(E)=

σ(E,E

dΩ σ(E,Θ, Φ) =

dΩ σ(E,E

, Θ, Φ) , (8.29)

relates the cross section to its diﬀerential counterparts. The interaction coeﬃcient µ can

have similar diﬀerential counterparts.

Now, consider a cross section diﬀerential both in scattered energy and angle. We may write

this as:

µ(E,E

, Θ, Φ) = µ(E)p(E,E

, Θ, Φ) , (8.30)

where p(E,E

, Θ, Φ) is a normalized probability distribution function, which describes the

scattering of particle with energy E into energy E

and angles Θ, Φ. We are now ready to

specify the ﬁnal ingredients in particle transport.

Given a particle with energy E and cross section σ(E,E

, Θ, Φ):

• Form its interaction coeﬃcient,

µ(E)=n

dΩ σ(E,E

, Θ, Φ) . (8.31)

• Usually the number density n is not tabulated but it can be determined from:

n =

. (8.32)

where ρ is the mass density, A is the atomic weight (g/mol) and N

is Avogadro’s

number, the number of atoms per mole (6.0221367(36) × 10

mol

−1

) [C. 98]

• Assuming µ(E) does not depend on position, provide the distance to the interaction

point employing (now familiar) sampling techniques:

s = −

µ(E)

log(1 − r) , (8.33)

where r is a uniformly distributed random number between 0 and 1. Provide this s and

execute the transport step ~x = ~x

+ ~µs. (Do not confuse the direction cosine vector ~µ

with the interaction coeﬃcient µ!)

The latest listing of physical constants and many more good things are available from the Particle Data

Group’s web page: http://pdg.lbl.gov/pdg.html

8.5. COMPOUNDS AND MIXTURES 93

• Form the marginal probability distribution function in E

m(E,E

dΩ p(E,E

, Θ, Φ) , (8.34)

and sample E

• Form the conditional probability distribution function in Θ, Φ:

p(E,Θ, Φ|E

p(E,E

, Θ, Φ)

m(E,E

)

, (8.35)

sample from it and provide [Θ, Φ] to the rotation mechanism that will obtain the new

direction after the scattering.

8.5 Compounds and mixtures

Often our scattering media are made up of compounds (e.g. water, H

O) or homogeneous

mixtures (e.g. air, made up of nitrogen, oxygen, water vapor and a little bit of Argon). How

do we form interaction coeﬃcients for these?

We can employ Equations 8.27 and 8.28 and extend them to include partial fractions, n

atoms each with its own cross section σ

µ(E)=

. (8.36)

Often, however, compounds are speciﬁed by fractional weights, w

, the fraction of mass due

to one atomic species compared to the total. From Equation 8.36 and 8.32:

µ(E)=

˜ρ

. =

˜ρ

= ρ

, (8.37)

µ(E)

. (8.38)

where ρ

is the “normal” density of atomic component i (very often it is µ

/ρ

, called the

mass attenuation coeﬃcient, that is provided in numerical tabulations), ˜ρ

is the “actual”

mass density of the atomic species in the compound, ρ is the mass density of the compound

and the fractional weight of atomic component i in the compound is w

=˜ρ

/ρ.

94 CHAPTER 8. TRANSPORT IN MEDIA, INTERACTION MODELS

8.6 Branching ratios

Sometimes the interaction coeﬃcient represents the compound process:

µ(E)=

(E) . (8.39)

For example, a photon may interact through the photoelectric, coherent (Rayleigh), inco-

herent (Compton) or pair production processes. Thus, from Equation 8.5 we have:

p(z)=

(z)=

(E)e

−zµ(E)

(E)p

(z) , (8.40)

where we assume (a non-essential approximation) that the µ

(E)’s do not depend on position.

Thus the fractional probability for interaction channel i is:

(z)

p(z)

(E)

µ(E)

. (8.41)

In other words, the interaction channel is decided upon the relative attenuation coeﬃcients

at the location of the interaction, not on how the particle arrived at that point.

This is one of the few examples of discrete probability distributions and we emphasize this

by writing an uppercase P for this probability. The sampling of the branching ratio takes

the form of the following recipe:

1. Choose a random number r.Ifr ≤ µ

(E)/µ(E), interact via interaction channel 1.

Otherwise,

2. if r ≤ [µ

(E)+µ

(E)]/µ(E), interact via interaction channel 2. Otherwise,

3. if r ≤ [µ

(E)+µ

(E)]/µ(E), interact via interaction channel 3. Other-

wise...and so on.

The sampling must stop because

(E)/µ(E) and a random number must always satisfy

one of the conditions along the way.

Note that this technique can be applied directly to sampling from the individual atomic

species in compounds and mixtures. The fractional probabilities can be calculated according

to the discussion of the last section and the sampling procedure above adopted.

8.7 Other pathlength schemes

For complete generality it should be mentioned that there are other schemes for selecting

pathlengths. We shall see this in the later chapter on electron transport. In many applica-

tions, electron transport steps are provided in a pre-prescribed, non-stochastic way, such as

8.8. MODEL INTERACTIONS 95

the pathlength for which a certain average energy loss would be obtained, or a pathlength

that produces a given amount of angular deﬂection. Deeper discussion would invoke a long

discussion that we will defer until later, after we have built up all the basic of the Monte

Carlo technique that we will need to solve a broad class of problems.

8.8 Model interactions

In this section we will introduce several model interactions and employ them to study some

basic characteristics of Monte Carlo results.

8.8.1 Isotropic scattering

The scattering of low to medium-energy neutrons (before resonances begin to appear) is

nearly isotropic in the center-of-mass. Therefore the scattering of low to medium-energy

neutrons from heavy nuclei is very nearly isotropic. Isotropic scattering is sampled from the

distribution function:

p(Θ, Φ) dΘ dΦ =

4π

sin Θ dΘ dΦ . (8.42)

The sampling procedure is:

cosΘ=1− 2r

Φ=2πr

(8.43)

where the r

are uniform random numbers between 0 and 1.

8.8.2 Semi-isotropic or P

scattering

The ﬁrst order correction to isotropic scattering of low to medium-energy neutrons, a cor-

rection that becomes increasingly important for scattering from lighter nuclei is the semi-

isotropic cross angular distribution:

p(Θ, Φ) dΘ dΦ =

4π

(1 + a cosΘ)sinΘdΘdΦ |a|≤1 . (8.44)

The sampling procedure is:

cos Θ =

2 − a − 4r

1 − a(2 − a − 4r

)

Φ=2πr

(8.45)

where the r

are uniform random numbers between 0 and 1.

96 CHAPTER 8. TRANSPORT IN MEDIA, INTERACTION MODELS

8.8.3 Rutherfordian scattering

A reasonable approximation to the elastic scattering of charged particles is the Rutherfordian

distribution:

p(Θ, Φ) dΘ dΦ =

a(2 + a)

4π

sin Θ dΘ dΦ

(1 − cos Θ + a)

0 ≤ a<∞ . (8.46)

The sampling procedure is:

cosΘ=1− 2a

1 − r

a +2r

Φ=2πr

(8.47)

where the r

are uniform random numbers between 0 and 1. An interesting feature f the

Rutherfordian distribution is that it becomes the isotropic distribution in the limit a −→ ∞

and the “no-scattering” distribution δ(1 −cos Θ) in the limit a −→ 0.

8.8.4 Rutherfordian scattering—small angle form

One approximation that is made to the Rutherfordian distribution in the limit of small a is

to make the small-angle approximation to Θ as well. The distribution then takes the form:

p(Θ, Φ) dΘ dΦ =

ΘdΘdΦ

(Θ

+2a)

0 ≤ a<∞ , 0 ≤ Θ < ∞ . (8.48)

The sampling procedure is:

Θ=

2ar

1 − r

Φ=2πr

(8.49)

If Θ >πis sampled in the ﬁrst step, it is rejected and the distribution in Θ is resampled.

One also has to take care that the denominator in the

√

does not become numerically zero

through some quirk of the random number generator.

Bibliography

[C. 98] C. Caso et al. The 1998 Review of Particle Physics. European Physical Journal,

C3:1, 1998.

Problems

The following are paper exercises, not computer exercises.

1. Give a derivation of the sampling procedure in Equation 8.45 for the semi-isotropic

probability distribution given in Equation 8.44 is the isotropic distribution.

2. Prove that the a −→ ∞ limit of the Rutherfordian distribution given in Equation 8.46

is the isotropic distribution.

3. Prove that the a −→ 0 limit of the Rutherfordian distribution is the δ-function

δ(1 − cos Θ).

4. Consider the small-angle form of the Rutherfordian distribution:

p(Θ, Φ) dΘ dΦ = N

ΘdΘdΦ

(Θ

+2a)

0 ≤ a<∞ , 0 ≤ Θ <π, (8.50)

where we now exclude the non-physical regime, Θ >π. What is the normalization

factor, N? Develop the sampling procedure for this distribution.

98 BIBLIOGRAPHY

Chapter 9

Lewis theory

This chapter contains a discussion of Lewis theory [Lew50], which provides us with a very

powerful way to evaluate the quality of our Monte Carlo algorithms. Lewis’ theory provides

us with scattering-model-independent exact predictions of spatial moments (e.g. hzi, hx

i···), angular moments (e.g. hcos θi, hsin θ

cos φ

···), and their correlations (e.g. hz cos θi,

hx sin θ cos φi···). It is one of the most remarkable (and often ignored) developments in

transport theory.

Particle transport of the form we consider in this course is described by the linear Boltzmann

transport equation:

∂

∂s

+ ~µ ·

∇ +Σ(E)

ψ(~x, ~µ, s)=

4π

dµ

Σ(~µ · ~µ

,E)ψ(~x, ~µ

,s) . (9.1)

where ~x is the position, ~µ is a unit vector indicating the direction of the electron, E is the

energy of the electron and s is the pathlength. Σ

(~µ · ~µ

,E) is the macroscopic diﬀerential

scattering cross section,

Σ(E)=

4π

dµ

Σ(~µ · ~µ

,E) (9.2)

is the total macroscopic cross section

(probability per unit length) and ψ(~x, ~µ, E, s)d~x dµ dE

is the probability of there being an electron in d~x about ~x,indµ about ~µ and in dE about

E at after having been transported a pathlength s. The boundary condition to be applied

is:

ψ(~x, ~µ, E, 0) = δ(~x)δ(ˆz − ~µ)δ(E

− E) , (9.3)

where we have assumed that at the start of transport the particle is at the origin and pointed

in the z-direction. (ˆz is a unit vector pointing along the z-axis.) The energy of the particle

at the start of the transport process is E

There is a slight departure in notation from the previous chapter. The macroscopic cross section was

called µ there. In this chapter the appearence of µ and ~µ together in the same equation seem too awkward

to be employed.

100 CHAPTER 9. LEWIS THEORY

This formulation is a very powerful and comprehensive description of particle scattering. It

can encompass transport models where the drift between collision points happens with no

loss of energy. In this case:

s = vt , (9.4)

where v is the velocity and t is the time, or it can allow for any non-stochastic functional

connection between pathlength and energy. One important energy-loss model is the contin-

uous slowing down (CSD) approximation (ir CSDA) which connects pathlength and energy

through the relation:

s = vt =

L(E

)

, (9.5)

where L(E) is called the “stopping power” and describes, for example, how electrons slow

down in media. The cross section still depends on E which may be calculated from Equa-

tion 9.5.

Except under very special circumstances (such as no scattering in simple geometries), Equa-

tion 9.5 can not be solved analytically by any means and so the Monte Carlo method is

adopted as a practical approach. However, Lewis [Lew50] has presented a “formal” solution

to Equation9.1 from which all the spatial-angular moments can be derived. We sketch below

and derive the major results.

9.1 The formal solution

Assume that ψ can be written as an expansion in spherical harmonics,

ψ(~x, ~µ, s)=

(~x, s)Y

(~µ) , (9.6)

one ﬁnds that

∂

∂s

+ κ

(~x, s)=−

λµ

∇ψ

λµ

(~x, s) ·

λµ

, (9.7)

where

(E)=

2π

dΦ

dΘ sin Θ Σ(cos Θ,E)[1 − P

(cos Θ)] , (9.8)

and

λµ

4π

dµY

∗

(~µ) ~µY

λµ

(~µ) . (9.9)

It has been assumed that the scattering law depends only on Θ and not Φ, that is, it is

azimuthally symmetric. The boundary condition in the parametrization is:

(~x, 0) = δ

δ(~x)Y

(0) =

2l +1

4π

δ(~x) . (9.10)