Functional integration is one of the most powerful methods of modern theoretical physics. The functional integration approach to systems with an infinite number of degrees of freedom turns out to be very suitable for the introduction and formulation of the diagram perturbation theory in quantum field theory and statistical physics. This approach is simpler than that using an operator method.

The application of functional integrals in statistical physics allows one to derive numerous interesting results more quickly than when using other methods. The theories of phase transitions of the second kind, superfluidity, superconductivity, lasers, plasma, the Kondo effect, the Ising model – a list of problems which is far from complete, for which the application of the functional integration method appears to be very fruitful. In some of the problems, it allows us to give a rigorous proof of results obtained by other methods. If there is a possibility of an exact solution, the functional integration method gives a simple way to obtain it. In problems far from being exactly solvable (for example, the general theory of phase transitions), the application of functional integrals helps to build up a qualitative picture of the phenomenon and to develop the approximative methods of calculations.

Functional integrals are especially useful for the description of collective excitations, such as plasma oscillations in the system of particles with Coulomb interaction, quantum vortices and long-wave phonons in superfluidity and superconductivity, collective modes in 3He-type systems and in 3He−4He mixtures.

In this section we outline the functional integral approach to the mathematically rigorous theory of the so-called model Hamiltonians system such as the Bardeen–Cooper–Schriffer (BCS) model in superconductivity (Bardeen, Cooper & Schriffer, 1957) or the Dicke model of superradiation. The problem of developing a rigorous theory of such models was put forward by Bogoliubov, Zubarev & Tserkovnikov (1960). This problem was solved for the BCS model by Bogoliubov (1960), who developed the so-called approximation Hamiltonian method. This method may be applied to the Dicke model as well. A rigorous theory of the Dicke model was suggested by Hepp & Lieb (1973). They have shown that there exists an exact solution of the model in the thermodynamic limit. The superradiation phase transition in this model was also described by these authors.

The approximation Hamiltonian method was applied to the Dicke model by Bogoliubov Jr (1974). Rigorous results for free energy and boson averages were obtained (see Bogoliubov Jr et al., 1981).

Functional integral methods were also applied to the Dicke model. Moshchinsky & Fedianin (1977) obtained asymptotics of Z/Z0, where Z is the partition function of the model and Z0 is the partition function of the corresponding free system. Kirianov and Yarunin (1980) investigated the Bose excitation spectrum of the system below the phase transition point in the superradiation state.

In this section we will prove the asymptotics of Z/Z0 for the Dicke model with a single mode of the radiation field (Popov & Fedotov, 1982).

The interaction of a superconductive system with electromagnetic field results in some interesting effects. One of the most remarkable is the Meissner effect, the consequence of which is that a sufficiently weak magnetic field is pushed out of the superconductor. There are two possible scenarios when the magnetic field is increased:

(1) The field penetrates the superconductor, changing it into the ordinary state. Such superconductors are called ‘superconductors of the first kind’.

(2) What emerges in the system is the so-called ‘mixed state’. During this process the field partially penetrates the superconductor in which a periodic lattice of quantum vortices is formed. This effect is called ‘superconductivity of the second kind’, and the superconductors exhibiting this type of superconductivity are called the second kind superconductors.

The creation of the quantum vortex lattice was predicted by Abrikosov (1952) in the framework of the phenomenological superconductivity theory of Ginsburg & Landau (1950). Formulation of the microscopic superconductivity theory allowed the derivation of the Ginsburg–Landau equations microscopically and the understanding of the meaning of the parameters appearing in the equations. In particular, it turned out that the effective charge of the phenomenological theory is equal to twice the electron charge, i.e. that it corresponds to the charge of a Cooper pair.

In this section we shall apply the functional integration method to some of the problems of superconductivity of the second kind. First, we shall show that the homogeneous superconducting state cannot exist in a constant homogeneous magnetic field.

Collective electron and localised moment models

Discussions about the magnetic structures of metals and alloys have traditionally involved consideration of two basic models which represent the extremes of electron localisation. The localised moment model assumes that the unpaired d or f electrons that give rise to a magnetic moment are confined to the atoms concerned. As such, Fe atoms in Cu should have the same integral moment atom as Fe atoms in Ni, for example. Below the spin ordering Curie temperature Tc the spins are spontaneously aligned, producing a net magnetic moment. Above Tc the thermal energy destroys the cooperative spin alignment resulting in zero net magnetic moment. On the other hand, the collective electron model (also known as the band or itinerant electron model) assumes that these electrons are completely delocalised and exist in narrow bands throughout the material, although the spin density need not be uniformly distributed. Below the Curie temperature the bands are exchange split, the resultant imbalance between the number of up and down spins leading to the observed magnetic moment. Above Tc there should be no splitting and so no moment.

The 3d electrons of metals of the first transition series definitely form narrow bands and their orbital angular momentum is effectively quenched by the crystal field giving a spin-only magnetic moment. These materials do not have integral moments and in fact usually display a moment per atom that varies smoothly with composition.

We now want to consider the electrical resistivity of simple metallic alloys in terms of the pseudopotential formalism developed in Chapter 4 for the scattering potential and using the methods developed in Chapter 2 to describe the structure of the alloys of interest. In most cases we will assume that this scattering process can be adequately described in terms of the relaxation time approximation discussed in Chapter 1. Some of the alternative methods which are applicable to non-simple metals are discussed in Chapter 6, although it is noted that these are equally suited to the study of simple metals as evidenced by some of the examples discussed in that chapter.

The alloy features that can be considered within this framework include atomic correlations and their associated static atomic displacements, vacancies and interstitials and their associated static atomic displacements and thermally induced displacements. Of these, the effects of atomic correlations, vacancies, interstitials and their associated displacement fields are dealt with directly in terms of the deviation lattice described in Chapter 4. This is because the defects are adequately described in terms of the site occupation parameters (or their averages) and the scattering potentials can be expressed in terms of a deviation from the average potential. In the case of atomic correlations this is just the difference in the atomic form factors wA(r) — wB(r).

The previous chapter was concerned with the electrical resistivity of simple metals and alloys. The result was a quantitative understanding of much of the behaviour associated with dilute and concentrated simplemetal alloys in terms of the electronic scattering processes and microscope phase distribution. Unfortunately there has been less progress in the understanding of non-simple metals and alloys. This is not especially due to any lack of a basic understanding of the processes concerned, but rather to the complicated nature of the techniques that are currently employed. In the case of simple metals the analysis was enormously simplified by the introduction of a weak scattering formalism. This was manifest at many different stages of the calculation: the approximate solution of the Boltzmann equation, the use of perturbation theory, the assumption of nearly free electrons and linear screening, for example. In alloys containing transition metals, however, the distinction between the valence and core states is no longer so clearcut. The d-states in the transition metals cannot simply be bundled into the core region and incorporated into some smoothly varying lattice pseudopotential. The wavefunctions associated with these states have significant amplitude outside the atomic core region and play an important role either by overlapping with those from adjacent atoms to form a narrow d-band or via the effects of hybridisation with the sp conduction bands. As a further consequence the Fermi surface may be quite irregular and have some parts that are s-like and others that are d-like.

Introduction

Understanding the physical processes that determine the electrical resistivity of a concentrated metallic alloy is a daunting task because of the large number of possible contributions that could be involved. In addition to conduction electron scattering from thermally induced atomic displacements (which may depend upon concentration and degree of atomic and magnetic order) there will be other direct contributions from atomic and magnetic disorder, strain and band structure effects. The magnitude of such effects will be influenced by the homogeneity of the microstructure and will depend specifically upon whether the spatial extent or ‘scale ’ of the inhomogeneity is greater or less than the conduction electron mean free path length.

The purpose of this first chapter is to introduce in a general way the relationship between the electrical resistivity and conduction electron scattering and band structure effects. It will be assumed that the reader is familiar with the fundamental concepts of electron waves in solids which have been very adequately considered in a variety of other texts (Ashcroft & Mermin 1976; Coles & Caplin 1976; Harrison 1970; Kittel 1976; Mott & Jones 1936; Blatt 1968; Ziman 1960, 1969, 1972). Other topics which are not specifically considered in detail in this text but which have been considered elsewhere include the electrical properties of pure metals (Meaden 1966; Wiser 1982; Pawlek & Rogalla 1966; Bass 1984; van Vucht et a,. 1985), galvanomagnetic effects (Hurd 1974; Jan 1957).

In this final chapter, three other aspects of the resistivity of metals and alloys will be discussed. As with many of the topics included in this book, each of these could easily justify a complete text in its own right and so the discussions here will be of necessity rather brief. It is hoped, however, that this introduction to these topics, all of which are still in a state of development, will serve to assist the interested reader in pursuing the matters in more detail.

Resistivity at the critical point

Some general comments

As noted in Chapter 2, a second order phase transition is characterised by a long range order parameter that continuously decays to zero as the critical transition temperature Tc is approached. Any of the physical properties of such a system which depend upon the derivative of S (or M) with respect to temperature, dS / dT (or dM//dT), such as the specific heat or temperature coefficient of resistivity dp/dT, will diverge at Tc. However, this gives a static picture of the transition which is somewhat misleading. Near Tc, small fluctuations in temperature produce fluctuations in the correlation parameter that are large in both magnitude and spatial extent. Since the resistivity is proportional to the magnitude of such correlations (the subject of Chapters 5 and 6), it is expected that the detailed behaviour at the critical point will be determined by these fluctuations.

In order to proceed with calculation of the electrical resistivity of a concentrated alloy it is necessary to find some suitable description of the atomic configuration.

Dilute and concentrated alloys

We start by defining a dilute alloy as one in which the solute atoms are in sufficiently small concentration that they have no effect on the electronic structure or phonon spectrum of the bulk alloy, although localised perturbations may result. In this regime the impurity atoms will act as independent scattering centres for the conduction electrons so that the effect of N impurity atoms is just N times that of a single impurity. This will only be possible if the solute atoms are widely separated (and hence randomly distributed), implying no direct or indirect interaction between them. In real alloys this composition range is usually restricted to less than 1 or 2 % solute concentration.

At higher solute concentrations the electronic structure or phonon spectrum of the bulk alloy will start to suffer perturbations, and the solute atoms may no longer be randomly distributed. Such alloys will be described as concentrated alloys and are the main concern of this text.