The only matter tensor having a rank as high as 6 that appears in Chapters 1 and 2 is the so-called ‘third-order elastic constant’ tensor of type Ts(6). (See Section 1–6 and Table 1–1.) This tensor couples a quantity Y, which is a thermodynamic tension, ti, of type Ts(2) to a quantity X, which is the symmetric product of Lagrangian strains, ηiηi, of type Ts(4). (Here, both ti and ηj are second-rank symmetric tensors whose six components are written in single-index hypervector notation.) The resulting matter tensor K is then a Ts(6) tensor, symmetric in the interchange of all the indices. The first objective of this chapter will be to obtain the independent components of such a Ts(6) tensor. This will require us to go beyond the material contained in the S-C-T tables (Appendix E).

Relation betweenTs(2) andTs(4)

We wish to consider the usual relation: Y = KX, in which Y is a Ts(2) tensor that transforms as a six-vector, and X is a Ts(4) tensor which transforms as the 21 symmetric products of six-vectors, αiαj. (Here we use the notation α for a Ts(2) tensor as in the S-C-T tables and Eq. (4–4).) In terms of the single-index quantities Y and X, K then become a two-index (6 × 21) matrix.

A crystal is made up of a regular arrangement of atoms in a pattern that repeats itself in all three spatial dimensions. This structural feature has the practical result that most properties of crystals are anisotropic, that is, different values are obtained for the property when measured along the different directions of a crystal. This anisotropy distinguishes crystals from non-crystalline materials (glasses) or from random polycrystalline aggregates, both of which show isotropic properties.

The purpose of this book is to show how to analyze the anisotropic properties of crystals in terms of the (tensor) nature of the properties and of the symmetry of the crystals. In the first two chapters, we focus on the nature of the various properties with which we will be dealing. We will see how the tensor character of a property helps to define its variation with orientation. Questions of crystal symmetry will then be dealt with, starting from Chapter 3.

Definition of crystal properties

For a crystal, regarded as a thermodynamic system (i.e. in equilibrium with its surroundings), any physical property can be defined by a relation between two measurable quantities. For example, the property of crystal density is simply the ratio of mass to volume, both measurable quantities; similarly, elastic compliance is the ratio of mechanical strain to stress. Often, one of the measurable quantities can be regarded as a generalized ‘forces’ and the other as a response to that force.

k-states in disordered metals

The simplest amorphous metals are probably the liquid non-transition metals such as liquid sodium or liquid zinc. The simplicity arises partly because no d-electrons are involved and partly because the liquids are of a single component whereas by contrast all metallic glasses involve at least two components. So let us see how far we can understand the electron transport of these amorphous metals before we tackle systems with the additional complication of two or more components.

As soon as we confront the problem of electron transport in highly disordered systems like liquids or glasses several questions spring to mind. How useful is the concept of a k-state when we are so far from having translational symmetry? How valid is the concept of a Fermi surface? The answers depend, not surprisingly, on the degree of scattering involved. Thus it is not just the degree of disorder involved but also the strength of the individual scattering processes. If the mean free path of the electrons is l and the electron wavelength at the Fermi level is λ we require that l be much greater than λ. More commonly we choose the almost equivalent condition kFl ≫ 1 where kF = 2π/λ. In a highly disordered system the mean free path tends to be only weakly temperature dependent so that the condition itself is essentially temperature independent. Experience with other systems of limited mean free path, for example, calculations on concentrated random binary crystalline alloys which give results in accord with experiment, suggests that a Fermi surface and its associated k-vectors for the conduction electrons are useful and satisfactory concepts provided that kFl≫ 1.

The source of electrical resistance

In trying to understand the electron transport properties of metallic glasses – properties such as electrical conductivity, Hall coefficient and thermopower – we shall start by using conventional theories that have been successful in accounting for the corresponding properties in crystalline metals and alloys and see how far these theories are successful in describing the properties of metallic glasses. I will explain what I mean by ‘conventional’ theories as we go along.

The starting point in understanding the electrical conductivity, σ, or resistivity, ρ(= 1/σ) of metals is the fact that the de Broglie waves which represent the conduction electrons can propagate without attenuation through a perfectly periodic lattice, such as that formed by the positive ions of an ideally pure and perfect crystalline metal at absolute zero. There is thus no electrical resistivity. More strictly, the electrons are scattered by the ions but only coherently as in Bragg reflections from the lattice planes. Such coherent scattering alters the way the electrons respond to applied electric and magnetic fields but does not cause electrical resistance. Such resistance comes about through random, incoherent scattering of the electron waves; this occurs only when the periodicity of the lattice and its associated potential is upset.

This means that, if you now add to your pure and perfect crystal chemical impurities randomly distributed, they will disturb the perfect periodicity and cause resistance to the flow of the electric current. Likewise, physical imperfections such as vacancies, dislocations or grain boundaries will produce electrical resistance even at the absolute zero. These imperfections and chemical impurities upset the perfect periodicity and so scatter the electrons that carry the electric current.

The purpose of this book is to explain in physical terms the many striking electrical properties of disordered metals or alloys, in particular metallic glasses. The main theme is that one central idea can explain many of the otherwise puzzling behaviour of these metals, particularly at low temperatures and in a magnetic field. That idea is that electrons in such metals do not travel ballistically between comparatively rare scattering events but diffuse through the metal. These new effects are not large but they are so universal in high-resistivity metals, so diverse and qualitatively so different from anything to be expected in metals where the electrons have a long mean free path, that they cry out for an explanation.

The book is not a critical research review; the motivation is mainly to explain. In interpreting theory there are always the dangers of overinterpretation, misinterpretation and failure to interpret and I do not expect to have escaped these completely. Nonetheless, our new understanding of disordered metals and alloys constitutes a substantial addition to conventional Boltzmann theory and deserves to be more widely known and appreciated.

The book is aimed at those who know little of the subject such as students starting work in this field or those outside the field who wish to know of developments in it. There is no attempt at rigorous derivations; the aim is to present the physics as clearly as possible so that readers can think about the subject for themselves and be able to apply their thinking in new contexts.

The results that we have just obtained are rather formal but do give us some important insights into the nature of electrical conduction in metals. We see from equation (3.35) that this conductivity depends entirely on the properties of the conduction electrons at the Fermi level. Moreover, the properties involved are of two distinct kinds: the first kind relates to the dynamics of the electrons, as represented by the distribution of electron velocities over the Fermi surface. The second kind represented by τ is concerned with scattering, our theme in this chapter.

As we have already noted, scattering in a metal arises from anything that upsets the periodicity of the potential: disorder of the ionic positions, which is paramount in metallic glasses; random changes in chemical composition, which are of great importance in random alloys; impurities, physical imperfections, thermal vibrations, random magnetic perturbations and so on. Let us therefore see how the scattering from some of these can be treated.

In this chapter we deal in section 4.1 with some basic ideas about scattering theory; in 4.2 there is a very brief discussion of Fourier transforms because they appear so frequently in scattering problems; in 4.3 we look at the influence of scattering angle on resistivity; in 4.4 at the effect of the Pauli exclusion principle on scattering; in 4.5 we consider electron screening in metals because the mobile electrons can markedly alter the scattering potential by electrical screening of the scatterer; and in 4.6 the pseudopotential because this has been used very successfully to represent the scattering potential in some simple amorphous metals.

Definitions

The origin of the thermoelectric effects is very simple. They arise because an electric current in a conductor carries not only charge but also heat. Consequently when an electric current flows through the junction of one conductor with another, although the charge flow is exactly matched, there is in general a mismatch in the associated heat flow; the difference is made manifest as the Peltier heat. If the current flows through a conductor in which there is a temperature gradient the heat shows up as the Thomson heat which is the heat that must be added to or subtracted from the conductor to maintain the temperature gradient unchanged; the electric current behaves as if it were a fluid with a heat capacity (either positive or negative). The third manifestation of thermoelectricity is the Seebeck effect which is the inverse of the other two. In this a heat current is established by means of a temperature gradient and this produces an electric current. However this cannot be done with a single material since in such a closed circuit the current induced in one part would cancel that in the other. Instead two materials are needed; moreover it is more convenient to measure not the circulating current that results but the emf that arises when the electrical circuit is broken. More explicitly, if conductor A is connected to conductor B at its two ends and the two junctions are maintained at different temperatures, an emf appears in the circuit.