Surface Tension of Nuclear Matter and the Enumeration of Eigenfunctions of an Enclosed Particle

Abstract

THE enumeration of eigenfunctions for a bounded continuum to a closer approximation than the Rayleigh asymptotic expression has, in recent years, largely because of its significance in problems of room-acoustics, received considerable attention¹. The number of eigenfunctions with their wave-numbers equal to or less than K is given by where the value of θ depends on the boundary condition which the eigenfunctions (U) have to satisfy; θ = 1 when U vanishes at the boundary, and θ = +1 when n (n being the normal to the surface) vanishes at the boundary. We could contemplate other possible boundary conditions for which θ could take values between 1 and +1. V is the volume of the enclosure, S its surface area, and L has the dimension of length, which for a rectangular enclosure is the sum of the edge-lengths. That the volume term is independent of the shape of the enclosure was shown by Weyl (1911); for the surface term this is assumed, but a demonstration is lacking. The L-term is still more uncertain. (The order term O(K) is probably of order less than K^3/2.) The effect of retaining the surface term in equation (1) on Stefan's law of radiation and on Debye's theory of specific heat was dealt with by one of us (D. S. K.) and has also been recently treated by Brager and Schuchowitzky². They have further derived, by applying (1) to a completely degenerate electron gas, the following expression (assuming one free electron per atom) for the surface tension of metals, where A is the atomic weight and (gm. cm.³) the density of the metal. It is interesting to observe that the above equation, but with the value of α reduced by a factor of about ½, was obtained on certain assumptions from the electron theory of metals by Gogate and Kothari, jun.³, who found the equation in reasonably good accord with experiment; the experimental values of σ give α as about 2.2 × 10⁴.