NE582 Monte Carlo
Spring semester 1998
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NE582 Monte Carlo
Spring semester 1998
|
Lesson 5 - Techniques for Common PDFsWe will wrap up our study of techniques for choosing random numbers according to PDFs with a quick overview of the choices that we will be normally making in the course of following a particle in a Monte Carlo transport calculation.The lifecycle decisions that we will look at are:
Particle initial positionDecisions about the initial position of a particle is usually at multidimensional parameter determination based on a given position distribution over volume, .
The mathematical approach to this is to define this function in terms of
an appropriate coordinate system and then independently choose random numbers
in each of the dimensions according to that dimension's "part" of the total
distribution. This is best demonstrated by example, so we shall look
at the three typical spatial coordinate systems: Cartesian, cylindrical,
and spherical.
Cartesian coordinate systemThe classic shape in Cartesian coordinate system is a right parallelpiped (i.e., 3D rectangle) in (x,y,z) with upper and lower limits of and 
in x, 
and 
in y, and 
and 
in z:
A differential volume element would be defined by:
If we want to pick a point with a flat distribution (i.e., each volume element equally likely), then the total distribution would be:
with
then we must have:
This means, of course, that each of the dimensions x, y, and z would be chosen according to constant distributions over their respective ranges, giving us:
where Cylindrical coordinate systemFor a cylinder of radius
and z limits
and :
The volume element is:
where
Use of these PDFs over the ranges of the dimensions would result in the following equations for choosing the position variables.
In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into:
Spherical coordinate systemFor a sphere of radius
with dimensions r (distance from origin), 
(polar angle), and 
(azimuthal angle):
The volume element is:
The distributions for each of the dimensions, then are:
Use of these PDFs over the ranges of the dimensions would result in the following equations for choosing the position variables.
In the Cartesian coordinate system (that most Monte Carlo codes run in) these would be translated into:
Choosing an initial point from multiple sourcesFor a situation in which source particles are chosen from multiple source (possibly of various shapes, sizes, and source rate density), the user should apply a probability mixing strategy whereby:
The point within the chosen source is picked using the appropriate shape's equations from above. Non-uniform spatial distributionsOne additional consideration is what should be done if the spatial source distribution is not uniform. In this case, the PDFs for the individual dimensions would be multiplied by the non-uniform distribution.
Particle initial directionThe choice of direction is based on probabilities on ,
which is a differential element of solid angle on the surface of a unit
sphere:
with the value:
where we note that the specification of the polar axis to be the z axis
in this figure is completely arbitrary. The polar axis can
be oriented in any direction that the analyst desires. If we define
where the minus sign is present because This gives us a dimensional PDFs of:
Since
Generally, Monte Carlo methods require directions in the form of direction cosines, which would be:
Particle initial energyGenerally, choice of the initial particle energy is based on either a continuous, discrete, or multigroup source spectrum.If the source distribution is continuous, the particular distribution
(in units of If the source distribution is discrete - which is common for
photon production from decay of radioactive sources - the data is usually
in the form of particular particle energies coupled with the yield
(i.e., the percentage of decay events that produce a gamma of the particular
energy). In this situation, the yield values for the various possible
emitted energies serve as the probabilities ( Similarly, if the source distribution is in multigroup form,
the individual source contribution in a given group is the integrated source
over the group (and therefore is not a distribution in "per unit energy"
units). Therefore, the individual group source values are exactly
analogous to discrete yields, so would be used as the Expected distance to next collisionFor an infinite material with a known total cross section, ,
the probability distribution for collision inside a differential path element
dx a distance x from the previous collision is:
Therefore the PDF is:
which is already normalized over the range The associated CDF is:
which inverts to give us the formula:
In terms of the optical path length,
(which corresponds to the number of mean free paths traveled) we can use:
Actually since
Type of collisionOnce a collision is known to have occurred, the choice of reaction type is based on the reaction macroscopic cross sections at the particle energy. Generally, the reactions of interests are scattering, fission, and capture, with relative probabilities based on ratios with the total cross section:
which gives us probabilities of:
We make the choice between reaction types by using these probabilities
as a discrete distribution.
Outcome of scattering eventThe outcome of a scattering event by a particle with initial energy E is given (formally) by the double differential distribution:
where M = material and the primed variables are associated with the particle after the collision. Mathematically, this is handled as described in the text for multi-dimensional
distributions -- we reduce it to a one dimensional choice by integrating
over the other dimension. Typically, for continuous energy representations
(like MCNP),it is the outgoing energy that is eliminated to give us a distribution
over the cosine of the deflection angle,
Once this distribution is sampled to give us a particular deflection
cosine (which can be translated into a polar angle of deflection), we would
choose an azimuthal angle uniformly from 0 to Knowing this value, then the outgoing energy is then chosen from:
It should be noted that for elastic scattering events (and inelastic scattering from known nuclear levels) there is a unique relationship among scattering deflection angle, initial energy, and final energy. This situation would, of course, reduce this last distribution to a single value (times a Dirac delta). For multigroup energy representations in which the angular dependence of the group-to-group scattering is represented by a Legendre expansion in deflection angle, the energy group of the outgoing distribution is generally determined first -- using the 0th order Legendre cross sections -- and then the direction is chosen from the angular expansion of the chosen groups, using the particular group-to-group Legendre coefficients. |
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Return to Course Outline © 2006 by Ronald E. Pevey. All rights reserved. |
is the 
pseudo-random number.
.
The distributions for each of the dimensions, then are:
?
,
this becomes:
decreases as 
increases.
from 0 to 
,
the resulting equations for the variables are:
or 
)
must be dealt with in the usual ways -- either by a direct approach (if
the distribution can be integrated and inverted) or with a rejection method.
)
used for choosing from a discrete distribution.
.
,
defined by:
is "as
random" as 
,
it is traditional to simplify this to:
,
:
.
From this, we can determine the outgoing direction 
.
