For the purposes of designing or operating a reactor the direction in which the neutrons are travelling is usually of little interest and there is an incentive to adopt a simpler calculation method that eliminates the two independent scalar variables Й. This is the treatment known as “diffusion theory”.

The basis of diffusion theory is to assume that ф is either isotropic or at most linearly anisotropic. (This is equivalent to P1 transport theory, with all but the first two terms on the right-hand side of equation (1.6) being ignored.) The transport equation is integrated over all directions and various approximations are made to obtain an energy-dependent diffusion equation in terms of the neutron flux ф, independent of direction, defined by

ф(г, E) = ф(г, E, to)dto, (1.7)

4n

and the neutron current

J(r, E) = &ф(і, E, to)dto. (1.8)

4n

The assumption that the flux is at most linearly anisotropic is accurate except in a strongly absorbing medium or where the properties of the medium change substantially over distances comparable to the mean free path of the neutrons. For many fast reactor calculations these limitations are not particularly important. Fast neutron cross­sections are usually small and mean free paths are typically 0.1 m or longer. The nuclear properties of fuel, coolant and structure are very different but because the dimensions of individual fuel elements and structural members are usually only a few millimetres, over distances comparable with the mean free path large regions of the reactor can be treated as homogeneous and diffusion theory can be used. Even in control rods the capture cross-sections for fast neutrons are low enough for diffusion theory to be a good approximation for many purposes. The worst inaccuracies arise at the edge of the core, and deep in the neutron shielding that surrounds the reactor.

There are some types of reactors for which transport theory has to be used, such as the small experimental fast reactors of the 1950s and 1960s that had cores with dimensions comparable with the mean free path, and also experimental reactors assembled from various materials in the form of thin plates, typically 50 mm square, arranged in arrays comparable in size with the mean free paths.

The great advantage of diffusion theory over transport theory is that, because for a steady-state calculation it deals with only four independent scalar variables than six, it makes smaller demands on computing power. For this reason up to the first decade of the 21st century most reactor design and operational flux calculations made use of diffusion theory. However in recent years the capacity of computers has increased to the extent that transport theory calculations have become more feasible and cheaper and therefore more widely used.

The rest of this discussion of fast reactor physics is presented in terms of diffusion theory because its formulation is much simpler than that of transport theory. Nevertheless it should be borne in mind that the same considerations apply in the latter, but the algebra is heavier.