A variational principle for the Milne problem linear extrapolation length

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.