Commentary on a conjecture of Shoenberg’s concerning the de Haas—van Alphen effect

Abstract

A thermodynamical proof is given of Shoenberg's conjecture that the oscillatory magnetization M of a metal exhibiting the de Haas-van Alphen effect may be correctly obtained by substituting B for H in the standard expression for M(H), derived on the assumption that B and H are not significantly different. When the oscillations are strong this leads to a many-valued function M(H). It is shown that a resolution of the ambiguities by a dubious thermodynamical argument leads to the wrong answer, and that eddy currents induced in the metal by changes of M cause the observed magnetization to follow a well-defined single-valued curve. Shoenberg's observations of harmonics generated in strong signals by this process are analysed quantitatively, for which purpose the influence of eddy currents is examined in some detail and earlier treatments are shown to be seriously in error. The problem involves discussion of a non-linear diffusion equation, for which solutions are given in the limiting cases of very weak and very strong de Haas-van Alphen signals. It is concluded that various apparent anomalies seen by Shoenberg and Gold are capable in principle of explanation along the lines suggested, but both the observations and the mathematical theory are too incomplete at present to allow a full explanation to be given.