Following the discussion of electron states in the previous chapter, the focus here will be the magnetic properties of liquid metals, both near to their melting points and along the coexistence curve towards the critical point.

In discussing the magnetism of normal metals, it has been traditional to separate the magnetic susceptibility into three contributions:

1. Core diamagnetism

2. Pauli spin susceptibility

3. Conduction electron orbital diamagnetism

Available evidence is that (1) can be dealt with adequately and thus it will be treated only in passing. The interest in (2) turns out, at least for simple s-p metals, to focus on the role of electron-electron interactions in enhancing the independent electron Pauli susceptibility. Contribution (3), as will be seen, is also of interest from the point of view of neutron scattering studies on a solid metal like Ni. The discussion will also embrace nuclear magnetic resonance and, in particular, the Knight shift in liquid metals.

Following this, which is centred on liquid metals just above their melting points, some consideration will also be given to the magnetic properties of expanded liquid metals. Here both phenomenology and also a treatment that leans on heavy Fermion theory (see Appendix 11.1) will be developed in order to interpret the properties of expanded liquid Cs, in particular, taken up toward the critical point along the coexistence curve.

Spin susceptibility of normal liquid metals

The most elementary theory of the spin susceptibility of a liquid metal follows that given in the early days of metal physics theory by Pauli.

In this chapter, the focus is the study of the properties of liquid metals at high pressure and temperature, with special emphasis on shock-wave studies. Naturally, theory can be invoked to bring the experimental observations into contact with what has been learned about liquid metals in the earlier chapters. Then, in the concluding chapter, some emphasis is placed on the relevance of such properties of the lightest and most abundant elements, H and He, in considering the giant planets Jupiter and Saturn.

In a shock-wave experiment, one measures the shock and particle velocities; from these, the pressure and density of the final state can be obtained directly. For some materials, it is also possible to measure temperature and optical properties. But, in the main, detailed information about atomic and molecular processes must come from theoretical studies.

In Section 15.2 the fundamental relations of shock physics are introduced. This is followed by a discussion of some specific results on hot expanded metals: data necessarily limited by large binding energies and high values of the critical constants.

Shock compression

Here some of the essential physics of shock compression are summarized, following the account of Ross (1985). A shock wave is a disturbance propagating at supersonic speed in a material, preceded by an extremely rapid rise in pressure, density, and temperature. It seems natural, at first sight, to associate shock waves with explosions and other uncontrolled and irreversible processes. Although shock waves are irreversible, the process is well understood and can be controlled to produce a desired response.

Although this volume is basically about metallic conductors, it is relevant to a discussion of electron states in disordered materials to give here some general background, plus relevant references, to the theory of localization of noninteracting electrons in a random potential.

The pioneering paper in this area was that of Anderson (1958) on the absence of diffusion in certain random lattices. But there was subsequently a lot of effort on this problem of noninteracting electrons in a static disordered potential before there was agreement on, at least, the answers to some important questions in this general area.

One might, at first sight, think it fruitful to compare the theory of electrons in disordered systems with the Bloch wave theory of the behaviour of electrons in a regular lattice. As Thouless (1979) emphasizes in his survey article, the theory of electrons in disordered systems is much more closely analogous to and owes much more to the theory of critical phenomena (see Chapter 9). In a sense then, developments in electron states in disordered systems awaited the synthesis of the theory of critical behaviour.

Following Thouless (1979), it is useful to group approaches into a number of areas, the first category being perturbative methods.

Perturbative methods

Anderson's original paper (1958) was based on the application of perturbation theory to a system that was strongly localized by a lot of disorder. In some ways, it resembles the Ursell-Mayer approach to phase transitions by examining the convergence of a high temperature perturbation series.

So far, one has assumed that the electron states in liquid metals can be described by the nearly free electron approximation. It seems unlikely that this can be true in the presence of strong electron-ion scattering, such as exists in (1) rare earth metals, (2) transition metals, and (3) even sp metals such as the heavy alkalis, as these are taken up along the coexistence curve toward the critical point (Chapman and March, 1988). Therefore, in this chapter, it is important to summarize a basic approach to the calculation of electron states in a framework that transcends the nearly free electron approximation.

Electron states in simple s-p metals

Following Edwards (1958, 1962; see also Cusack, 1963; 1987), Ballentine (1966; 1975) has employed the Green function formalism to make perturbation calculations of the density of states N(E) for Al and Zn, thereby allowing him to discuss the range of its validity and limitations and motivating more satisfactory calculations for Al and Bi.

As discussed more specifically later in this chapter, in order to study the electronic structure of a liquid metal, represented by a model of independent electrons interacting with a disordered array of ions, one should start from a fixed configuration of ions and then average over the ensemble of all possible arrangements of the ions. The Green function formalism as set up by Edwards (1958; 1962) allows the calculation of ensemble averages of physical quantities without ever calculating the wave functions for a specific ionic array. The summary of the Green function procedure below follows closely the account of Ballentine (1966; 1975).