If in a system the sources are isotropic and only the isotropic component of the flux is required, it is possible to write the transport equation in a simpler form. Let us treat the monoenergetic time-independent case.

Let us consider a neutron starting from r’ in the direction ft. The probability that it reaches the point r is given by

= exp -[t(x)]

where

t(x)= X,(r — x’fl) dx’

JO

is the optical distance (probability of having a reaction along the path x), and X, is the total cross-section. The neutron flux in r is given by

ф(г, П) = exp [-r(x)]Q(r-xft, ft) dx

where Q is the source representing fission or scattering. If the sources can be assumed to be independent of ft the equation can be integrated over the angle. Analytical expressions are only possible in particular cases, but in general one obtains an expression of the form

(4.53)

In terms of collision density фХ, we have:

(4.54)

The kernel K(r — r’|) is then the probability of neutron penetration from point r’ to point r. K(]r — r’|)2,(r) is the probability of a neutron starting from point r’ to suffer its first collision in r.

In the case of a multi-group calculation eqn. (4.54) holds for every group, and the source Q,(r) is given for each group / by

Q,(r) = Sj(r) + d>,(r)2s(r) = Si(r)+ ф,(г)Х,(г)с

where Xs IX, = c is the number of secondary neutrons per collision and S,(r) represents the fission and slowing-down source and is therefore dependent on the flux of all groups. It is always possible to calculate numerically the kernels K(r — r’|).

These methods are usually called “collision probability” methods because the number of neutrons colliding in one region is calculated from the number of neutrons produced or scattered in other regions multiplied by the probability that they will make their next collision in the region considered. This implies a discretization of the reactor space in a certain number of regions.

In general collision probabilities depend not only on geometry and composition of the regions considered but also on the spatial flux distribution within each region.

What makes these methods attractive is the fact that, in practice, if the regions are small compared with the mean free path, the collision probabilities are rather independent from the flux distribution so that they can be calculated assuming flat flux in each region. The above-mentioned assumption of isotropic sources refers both to fission and scattered neutrons.

This last condition is not always satisfied, but is sufficiently accurate for graphite because of its relatively high atomic weight. Besides in the thermal energy range crystal bindings and thermal motion of the graphite atoms tend to give a more isotropic scattering. In a more general energy-dependent case we have

This expression is sometimes known as Peierl’s equation. Here we can distinguish a transport kernel

K(r-r’,E)

and a scattering kernel

2,(r£’->£)•

Energy is then discretized in G groups and space in N regions obtaining the following expression:

= 2 PS f Qi*v, + v, 2 slrvl

i=l L к=1 J

where 2,8 = total cross-section region j group g,

Vj = volume region j,

Фf = flux in group g region j

2ЇГ8 = scattering kernel discretized for region i, group к to g,

Q8 = source region і group g,

Pfi = collision probability: probability that a neutron of group g originating in region і makes its next collision in region j.

We use here a superscript to indicate the energy group in order not to generate confusion with the various other subscripts.

In this equation the transport kernel K(r-r’,E) has been discretized in the collision probabilities P%. This implies the assumption that the collision probabilities are not dependent on the flux distribution within each region and can be calculated with the flat flux approximation.

One can then obtain for P% the following expression:

the exponential factor 5.‘(s)ds

0

gives the probability that the neutron does not suffer collisions along the path s. From (4.57) follows the reciprocity relation

SiaV, PS = SSVjPS. (4.58)

In an infinite system

N

2 Pn = і

j-i

because a neutron born in region і will certainly make its next collision system regions, if leakage is excluded. The probabilities P8 are normally calculated by numerical integration of eqn. (4.57). Monte Carlo methods can be used for special cases. Once these collision probabilities are known it is possible to calculate the fluxes ФіК by means of eqn. (4.56). Usually iterative methods are used, e. g. inserting a trial
solution for the fluxes on the right-hand side of eqn. (4.56) and obtaining an improved solution on the left-hand side (power iteration method).

A typical example of this method is given by the THERMOS"21 code where only the thermal energy range is considered in the geometry of a reactor cell. A unit slowing — down source is assumed over the cell.

Another example is given by the PIJ code"31 which allows an r — в geometry upon which can be superimposed rather general distributions of cylindrical “rods” themselves subdivided in r and в. A typical HTR block representation for the PIJ code is given in Fig.

4.2. "4)

Collision probability methods are also used to calculate control rods (e. g. MINOTAUR code.’15’

Some extension of the collision probability method has been developed in order to treat anisotropic sources and scattering,"4’ but this is not very important in graphite­moderated systems. Approximate methods are sometimes used to calculate collision probabilities. For cylindrical geometry the widely used Bonalumi method approximates the fate of neutrons beyond the second boundary crossing."6’

Sometimes for spectrum calculations the reactor volume is discretized in very big regions (MULTICELL method,"7’ and the flat flux approximation is no longer valid: corrections based upon diffusion theory have been developed for this purpose."8’

Collision probability methods are best suited to cases where the required accuracy is not very high and approximate methods can be used to calculate collision probabilities. If a high accuracy is demanded an S„ representation may be more advantageous."4’19’