A variational principle for the Milne problem linear extrapolation length

Abstract

A simple bilinear functional F is introduced on behalf of the Milne subcritical problem with replication parameter 0⩽c⩽1. This functional depends upon two arguments, respectively intended to be the neutron flux and its adjoint, and is stationary about the true solution pair where, in addition, it vanishes. The stationarity and null value can then be united as a basis for the demand that F continue to vanish even when flux and adjoint are both approximated by just the two modes from the discrete eigenvalue spectrum, a representation akin to what is known as the asymptotic portion of the neutron flux, and one which is clearly incapable of matching interface boundary conditions. The stationarity of F, however, renders it tolerant of such boundary defect, as a result of which one can expect the persisting null demand, F=0, to yield the best possible value for the ratio of the two discrete mode amplitudes.We go on to implement this program, and find as its outcome that the optimum amplitude ratio is determined as one preferred solution of a simple quadratic equation. With that solution in hand, it is an easy step then to a computation of the linear extrapolation length λ. We follow through with a numerical embodiment of these ideas, obtaining the discrete, real and positive eigenvalue ν0 on the run via a Newton–Raphson tangent encroachment root hunt. With sufficient start-up care the Newton–Raphson root hunt proves here to be exceedingly rapid, and it, together with the quadratic underpinning, provides for λ a string of values that differ by less than 0.5% from those found in the classic compendium on neutron transport from the pens of Case, de Hoffmann, and Placzek. In particular, we are able to bypass in this way, and with quite elementary tools indeed, a known canonical machinery of far greater weight and sophistication, be it based upon the Wiener–Hopf method, or else upon flux decomposition along both discrete and singular eigenfunction modes. To our way of thinking, such a simple alternative is aesthetically pleasing in its own right, and both provides a measure of confirmation to, and is itself checked by, the more formidable apparatus.