Neutron flow theory

neutron age theory

An approximate description of the slowing down of neutrons resulting from their elastic scattering on the nuclei of a medium. It may be used to determine the spatial distribution of neutrons with different energies only in media not containing light nuclei (the permitted mass numbers are $M\gg2$, i.e. hydrogen and deuterium, for example, are not permitted). The assumption which underlies neutron flow theory is that the slowing down neutrons lose their energy continuously rather than discretely, and that the real behaviour of a large number of individual neutrons is replaced by certain averages; this is legitimate only if, at any given moment after the emergence of the neutrons from their source, their energy scattering is small. On this assumption it is possible to introduce the neutron flow function as an independent variable and employ the neutron flow approximation to obtain the neutron flow equation (or neutron age equation)

$$\nabla^2q=\frac{\partial q}{\partial\tau}$$

in the diffusion approximation; the form of the equation is the same as that of the heat equation. The neutron flow equation is a parabolic partial differential equation which describes the slowing down of the neutrons. The unknown function $q$ is the slowing down density, i.e. the number of neutrons in a unit volume which, during their slowing down, cross a given energy value in unit time. The neutron flow function, which acts as independent variable $\tau$, is understood to mean the symbolic age of the neutrons, which is equal to the time of slowing down of the neutrons (the chronological age) multiplied by the coefficient of diffusion averaged over the slowing down time. The square root of the age of thermal neutrons is said to be the slowing down length of a thermal neutron. The physical sense of the concept of "age" is one-sixth of the mean square of the distance by which the neutron is displaced from the moment of its emergence from a point source (age $\tau=0$) to the moment under study, corresponding to the age $\tau$. The age approximation, which, together with the diffusion approximation, yields the neutron flow equation, is represented by the transition from accurate energy operators which describe the slowing down of the neutron to approximate operators.

The spatial distribution of the neutrons, produced by their diffusion in the course of slowing down, determines the probability of neutron losses in a nuclear reactor, and affects its critical dimensions.

References

[1]	S. Glasstone, "The elements of nuclear reactor theory" , v. Nostrand , New York (1952)

References

[a1]	A.M. Weinberg, E.P. Wigner, "The physical theory of neutron chain reactors" , Univ. Chicago Press (1958)
[a2]	A.D. Galanin, "Thermal reactor theory" , Pergamon (1960) (Translated from Russian)
[a3]	M.M.R. Williams, "The slowing down and thermalization of neutrons" , North-Holland (1966)