The Fermi Surfaces of Copper, Silver and Gold I. The de Haas-van Alphen Effect

Following the discovery of the effect in copper, silver and gold by the impulsive high-field method, a study of the variation of the oscillatory frequency (which is proportional to the extremal area of cross-section of the Fermi surface by planes normal to the field) with the direction of the magnetic field relative to the crystal axes has made possible a detailed determination of the Fermi surfaces of these metals. As well as high-frequency oscillations associated with major ('belly') sections of the Fermi surface, low frequency oscillations are observed for the field in the [111] direction, which can be associated with the 'necks' in which the Fermi surface makes contact with the hexagonal faces of the Brillouin zone, as suggested by Pippard. The general topology of Pippard's model is confirmed by the existence of medium frequencies for the field along [100] and [110], which can be associated with 'hole' sections having the shape of a 'rosette' and a 'dog's bone', respectively. By making the high-frequency belly oscillations beat with the oscillations from a reference specimen, it has proved possible to measure the small variations of the belly frequency with field direction, which amount to only a few parts per cent, with an accuracy of a few parts per cent. These variations and the neck frequency have been fitted by Roaf to an analytical representation of the Fermi surface in the form of a Fourier series for each metal as explained in the accompanying paper, and the calculated surfaces are found to give the correct frequency for every direction studied. The surfaces have been chosen to have a volume exactly half that of the Brillouin zone, and the fact that the absolute values of the frequencies predicted by Roaf's formulae agree within experimental error with the observed frequencies confirms the validity of the assumption that these metals have exactly one electron per atom. It shows also that electron interaction effects do not modify appreciably the theoretical relation between frequency and area which was derived by Onsager without taking such effects into account. From the variation of the amplitude of the oscillations with temperatures and field, information is deduced about cyclotron masses and electron relaxation times. The absolute amplitude of the oscillations is found to agree in order of magnitude with theoretical prediction, but to explain the very strong harmonic content of the oscillations, it is necessary to invoke a new mechanism. This is the frequency modulation of the applied field caused by the oscillations of the magnetization, the contribution of which to the local field is appreciable when the differential susceptibility is, as here, comparable to 1/4π , but is neglected in the theory as usually developed. It is shown that this frequency modulation effect can account in order of magnitude for the harmonic content and is probably also responsible for certain anomalies observed in the field and temperature variation. The paper concludes with a discussion of a number of effects which can appreciably reduce the observed amplitude. These are field inhomogeneity, specimen imperfections such as bending and random substructure, inhomogeneity of field due to eddy currents, and various possible heating effects which can raise the temperature of the specimen above that of the bath. In many of the experiments resonant amplification is used and the rapid passage through resonance ('gliding tone' effect) which is relevant to various of the amplitude problems, is discussed in an appendix.