The escape probability P0 can be easily calculated for isolated lumps in the two extreme cases of very large or very small mean free path, under the assumption of a flat source of neutrons in the region. If the body is large compared with the mean free path (black limit)

Vo volume, So surface (see ref. 3, p. 117). For very large mean free path obviously Po-»l.

For intermediate situations the following rational approximation has been proposed

or defining the escape cross-section 2e = l//0

The values given by this approximation are systematically too low, up to 20% in the neighborhood of /02,o = 1.

In general for isolated lumps (that is with a spacing of several mean free paths) P0 is given as a function of /02,o (see, for example, Case, de Hoffmann, Placzek<l3) where these functions are given for various simple geometries). If the fuel lumps are closely packed the so-called Dancoff correction has to be employed.

The Po to be used in (7.14) has in this case to be substituted by Pо which is given by the escape probability P0 of an isolated lump multiplied by the probability that the neutron escaped from the lump has its next collision in the moderator region.

This correction was first introduced by Dancoff and Ginsburg.<l4> Its exact calculation can only be performed with Monte Carlo methods. A useful approximation can be obtained as shown in Fig. 7.1.<l5>

Let us consider the neutron trajectory of Fig. 7.1. The probability of survival without collision on the leg Ri is

Г, =

The escape probability from one lump is given by

Ро = г=-(1-Г„) (7.22)

where l0 = mean chord length,

X, o = total cross-section,

Го = average of Г0 over all possible chords of the lump.

The probability of making the next collision in the moderator is

P’o = ї4-{(1 — Г„Ж1 — Г,) + Г, Г2(1 — Гз) + • • ]}• (7.23)

I oZlo

The averaging is over all possible trajectories from the surface of the first lump.

In eqn. (7.23) the terms linear in the Г, are the most important ones if the lumps are big compared with the mean free paths. For the higher terms the average of the products is then replaced by the product of the averages. This will also be a good approximation for small dimensions. We obtain then

p. 1 (1-ГоХі-Г,)

/оХіо 1 — FoFi

here we have assumed that in a lattice geometry the averages of Г, for fuel lumps (even i) are all equal to Г0 and those for the moderator (odd i) are all equal to Г,. Introducing the Dancoff factor

С = Г, = e

and using for P0 the expression (7.22) we obtain

p’ = P ________ * ~ Є______

0 °1 -(1-/oS, oPo)C-

Using the rational approximation for Po

, 1 2, 0 1 + lo2,o 2, + 2ю 2,(1-C) SI 2ю + 2,(1 — C) 2u> + 2,

we have

(7.25) (7.26)

where 2, is defined as

here the Dancoff correction appears as a shadowing of the lump surface due to the adjacent lumps. A rational approximation can also be found for C.

Since (1 — C) is the probability that a neutron entering the moderator collides in it, C is the moderator escape probability Pi. In the rational approximation we have [see eqn. (7.21)]

(7.27)

with

(7.28)

The accuracy of this last expression is not very high.

The Wigner rational approximation for P0 does not give the correct behaviour for grey fuel lumps.

An improvement has been proposed by Bell introducing the so-called Bell factor a:

a 2,

a 2, — H 2to

a is not constant but depends on the geometry and on the scattering cross-section of the moderator mixed with the fuel.<l6) The values of this factor for various geometries are given, for example, in ref. 17.