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7. Mathematical Formulations
7.1 Formulations of Collision Probabilities
We present in this section the formulations to evaluate the collision probabilities for a
multi-region cell expressed by either of three one-dimensional coordinate systems (plane, sphere and
cylinder) or two-dimensional cylindrical coordinates. They are expressed in a form suitable to apply a
numerical scheme “Ray-Trace” method developed by Tsuchihashi 33) .
Description of this chapter will be started by the derivation of the linear equation to solve the
Boltzmann equation by the collision probability method (CPM) then followed by the formulations of
collision probabilities in various coordinate systems, in which care is paid to evaluate the surface
problem and also the directional probabilities to yield the anisotropic diffusion coefficients.
For the slab geometry, the formulation of the ordinary collision probabilities expressed by the
E i3 function has been given by Honeck 34) starting from the plane transport kernel by E i1 function. Here,
we shall start from the basic point kernel, operate first, the double integration along a neutron path
(line), finally, achieve the angular integration to yield the general form of the directional collision
probabilities expressed by the integral exponential function E in .
For the one-dimensional cylinder, the formulation derived by Takahashi 35) needs that the angular
integration to scan the collision regions be repeated for each source region. The similar formulation by
Honeck used in the THERMOS code 34) approximates the attenuation of neutron emitted from a source
region by that from the mid-point of the source region. This approximation could be covered by
sub-dividing the system into so many concentric regions. The formulation given in this chapter is a
special case of the general two-dimensional cylinder. For an annular geometry, no integration over the
azimuthal angle is needed. The main difference between the present formulation and the formers’ is in
the sequence and the coordinates for the integration. Owing to the variable transformation, the angular
integration appearing in the formers’ is replaced by the integration over the volume element d ρ
where ρ is the distance from the center to one of the parallel lines drawn across the system. The
number of lines drawn for the integration is quite small compared with that of the THERMOS code to
have the same measure of accuracy, partly because no repeated integration by source region is required,
partly because the analytic integration along the line in the source region does not need so many
sub-division of the system.
The formulation for collision probabilities in the spherical system which can be sub-divided into
an arbitrary number of spherical shells has been given by Tsuchihashi & Gotoh 36) in the course of
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