Introduction

During the last 25 years de Haas–van Alphen studies have led to a spectacular advance in our knowledge of the Fermi surface of metals and (to a lesser extent) of the differential properties of the surfaces of constant energy in the vicinity of the FS. This progress has been made possible partly by parallel developments in the theoretical understanding of band structures, but perhaps more significantly by advances in technology. These advances, in the production of high magnetic fields and low temperatures, in electronic techniques and data processing and in the growing of purer and more perfect single crystals, are still continuing and hopefully will continue to be exploited to extend our knowledge still further.

By far the greatest effort has gone into measurements of dHvA frequencies F with a view to the determination of the FS of metals through the Onsager relation and by now the FS of nearly all the metallic elements and of many intermetallic compounds have in fact been determined. The level of determination achieved however, varies both in the degree of certainty with which the qualitative nature of the surface has been established and in the precision of the quantitative specification of the surface. At best, the qualitative nature of the surface (i.e. the number and shapes of the separate sheets) is reliably known and the dimensions of the various sheets determined with a precision of order 1 in 10.

It is just over 50 years ago that an oscillatory magnetic field dependence was first observed in the electrical resistance of bismuth by Shubnikov and de Haas and in the magnetization by de Haas and van Alphen. It was not long before Peierls showed how these effects could be understood in principle and, indeed, Landau had implicitly predicted oscillatory behaviour even before the experimental discovery, but the effects remained somewhat of a scientific curiosity for upward of 20 years. It was only in the 1950s with the observation of magnetic oscillations in many metals other than bismuth and the advent of improved theoretical understanding that it began to be realized that the effect was not only an aesthetically pleasing curiosity but potentially a powerful tool for understanding the electronic structure of metals.

During the following 20 years, exploitation of this possibility became somewhat of a ‘band wagon’ and with ever improving experimental and theoretical techniques, an immense amount of detailed information about the ‘Fermiology’ of individual metals has emerged. More recently the pace has slackened, though there are still many loose ends and unsolved problems, and during a sabbatical half year at the University of Waterloo in 1977, I felt the time was ripe for a new comprehensive review and somewhat light-heartedly embarked on it little thinking it would be five years before it was ready for the Press.

Systems of more than one phase

When a system consists of more than one phase, each phase may be considered as a separate system within the whole. The thermodynamic parameters of the whole system may then be constructed out of those of the component phases. If the interaction between the phases were restricted to energy exchange (flow of heat and performance of work), then application of thermodynamics to the whole system would not lead to any essentially new results. However, if we allow new degrees of freedom within the system, such as mass transport between phases or chemical reaction between constituents, the conditions for thermodynamic equilibrium (derived in section 7.4) do lead to new results which are related to the restrictions which equilibrium places on the new degrees of freedom. In this chapter, we shall restrict ourselves to considering systems whose chemical composition is uniform (for example, systems of one component) but in which more than one phase is present. For simplicity, we shall again develop the general results for a system subjected to work by hydrostatic pressure only.

The condition for equilibrium between phases

Let us first consider a one-component system of two phases maintained at constant pressure and temperature (Fig. 10.1). This might be a liquid in contact with its vapour. If we ignore any possible surface effects at the interface, both temperature and pressure will be uniform throughout.

Origins of thermodynamics

The increase of mechanization during the nineteenth century involved the construction of machines, such as the steam engine, for the conversion of heat energy into mechanical power. It was from the study of these heat engines that thermodynamics grew. The initial development was rapid. By 1900, the subject was firmly established, and although its application had at first been restricted to thermal engineering, its laws were soon recognized to be of such great generality as to be useful and important in many other branches of science also. Broadly speaking, thermodynamics is applicable to all processes in which temperature or heat play an important part. In physics, it provides a way of understanding phenomena as different as thermal radiation on the one hand and the low temperature properties of paramagnetic salts on the other. It supplies the basic theory of chemical reactions and underlies much of chemical engineering. It is applicable not only to steam engines but to refrigeration and rocketry.

With this very wide range of application, it is possible to adopt various terms of references within which to develop the subject. We shall choose examples which are primarily of interest to the physicist. The fundamental structure of the subject, however, is little affected by the applications one has in mind. This is because the basic theory can be developed in a precise and self-contained way with much of the rigour of a mathematical argument.