In this chapter we summarize some background information concerning molecular collisions, dipoles and radiation, spectroscopy, and statistical mechanics that will be needed later. This Chapter should be skipped in a first reading. It is hoped that a reader who comes back to this Chapter later with specific questions will find the answers here — or, at least, some useful reference for further study.

Intermolecular potentials

The ideal gas law, Eq. 1.1 with B = C = … = 0, may be derived with the assumption of non-interacting ‘point particles’. While in the case of rarefied gases at high temperatures this assumption is successful in that it predicts the relationship between pressure, density and temperature of a gas in close agreement with actual measurement, it was clear that important features of gaseous matter, such as condensation, the incompressibility of liquids and solids, etc., could not be modeled on that basis. As early as in 1857, Clausius argued convincingly that intermolecular forces must be repulsive at short range and attractive at long range. When in 1873 van der Waals developed his famous equation of state, a significant improvement over the ideal gas law, he assumed a repulsion like that of hard spheres at near range, and attraction at a more distant range.

Introduction

In Chapters 6–10 we have dealt with the general structure of the linear response, kinetic theory and random-walk approaches to the calculation of the phenomenological coefficients and to the dielectric and anelastic response functions. We gave some straightforward examples of approximate results that can be obtained from the general expressions. The mathematical inter-relations between the three approaches demonstrated in those chapters allow the use of common techniques for the evaluation of the expressions for the transport coefficients of particular models.

In the present chapter, which, like Chapters 7–10, is concerned with dilute alloys and solid solutions, we first consider these techniques and then go on to present some results and applications of those results. It divides therefore into three more or less distinct parts. Techniques are the subject of §§11.2 and 11.3, the resulting transport coefficients are the subject of §§11.4–11.6 while various applications are reviewed in §§11.7–11.10.

In Chapters 7 and 8 general expressions were derived for the phenomenological coefficients and response functions from kinetic and linear response theories while consistent expressions for diffusion coefficients were obtained from random walk theory in Chapter 10. The techniques for the evaluation of these expressions are reviewed here in §§11.2 and 11.3. In the first of these sections the techniques are limited to calculations for which, in the terminology introduced in Chapter 10, only one type of jump occurs in the formal analysis. Important examples of such calculations are furnished by evaluations of the three independent phenomenological coefficients LAA, LAB, LBB for dilute binary alloys of cubic structure with transport by single vacancies, simple interstitials or dumb-bell interstitials.

Introduction

In the preceding Chapters 6–12 we have dealt with the general structure of the various statistical theories of atomic transport in solids and with the relations between them. However, in obtaining analytical results for specific classes of model the emphasis has been mostly on dilute alloys and solid solutions containing only low concentrations of defects. The theory of such systems is made easier because we need to retain only the lower-order terms in defect and solute concentrations. In this chapter we turn to theories of so-called lattice-gas models which give physically simplified representations of both concentrated alloys and systems containing high concentrations of defects. In such systems the concentrations of the components are not useful expansion parameters. As a result, and as is usual in statistical mechanics, an accurate theory is much more difficult to formulate in these circumstances.

The rather broad range of systems of current interest includes (i) ideal or random alloys and solid solutions (such as were discussed in §§5.6 and 10.8), (ii) alloys displaying short-range or long-range order and (iii) various systems where the number of mobile atoms is significantly less than the number of sites available to them. In this third group these sites are commonly interstitial sites within a structure formed by relatively immobile atoms. Examples are provided by the β-aluminas and other fast ion conductors (Laskar and Chandra, 1989), metals containing interstitial hydrogen or deuterium atoms and various other interstitial solid solutions (e.g. the tungsten ‘bronzes’; Cox, 1992). We shall refer to the sublattice of interstitial sites in these systems simply as the lattice.

Introduction

In Chapter 1 we briefly introduced the extended macroscopic equations of atomic transport that are provided by non-equilibrium thermodynamics. In this chapter and the next we expand this description, in this chapter in general terms and then in the next by a series of applications. The objectives are (i) to show how to use the description provided by non-equilibrium thermodynamics and (ii) to focus attention on the corresponding transport coefficients (denoted by Lij) and the expressions of various practical transport coefficients in terms of them. By working in terms of these basic L-coefficients we obtain a better understanding of the relations among the various practical coefficients and we provide a more sharply defined objective for the statistical atomic theories to be developed later.

The subject of non-equilibrium thermodynamics is now supported by a considerable body of statistical mechanical theory. Since our interest in this book lies primarily with the use of this formalism and with the calculation of the quantities appearing in it, it is not appropriate to go into this body of general theory widely. For this purpose we refer the reader to the books by de Groot and Mazur (1962) and by Haase (1969) for wide-ranging treatments, especially of the use of the macroscopic formulation, and to Kreuzer (1981) for a more recent account which emphasizes the general statistical mechanical foundations of this formalism. Here, a less extensive review will be given of those aspects pertinent to our interest in atomic transport in solids.

As we have already pointed out in §1.4 the immediately obvious features of the formalism are threefold: …

Introduction

The theory presented in the last chapter has already provided important general insight into atomic transport coefficients, as well as specific results which define routes to the evaluation of these coefficients for particular systems, both dilute and concentrated. We could therefore go straight to these evaluations. However, there are good reasons for delaying the matter. One is that there are alternative approaches to dilute systems which convey particular insights into the processes of atomic transport and which define additional quantities of experimental interest, most notably the so-called correlation factor (Chapters 8 and 10). Two of the most important of these alternative approaches are the kinetic and random-walk theories: the former are the subject of this and the next chapter while the latter are dealt with in Chapters 9–10. Although these theories are very different in appearance we shall show that in the end they can lead to the same expressions for measurable quantities in terms of specific atomic features of the system under study. Ultimately therefore they may use the same techniques for the evaluation of these expressions (as we shall show in Chapter 11 especially).

Although we shall henceforth be much more concerned with the physical details of the systems of interest, both approaches relate rather closely to what we have presented in Chapter 6. We shall take kinetic theory first because it offers a rather direct representation of many of the basic equations of Chapter 6 together with a number of easily obtained approximate results for particular systems. At the same time some of its notions are useful in the development of the random-walk theories.

In any area of physical science there are stages in its historical development at which it becomes possible and desirable to pull its theory together into a more coherent and unified whole. Grand examples from physics spring immediately to mind – Maxwell's electro-magnetic theory, Dirac's synthesis of wave-mechanics and matrix mechanics, the Salam–Weinberg electro-weak theory. Yet this process of unification takes place on every scale. One example of immediate concern to us in this book here is afforded by the emergence of the thermodynamics of irreversible processes, where the coming together of separate strands of theory is made plainly visible in the short book by Denbigh (1951). In connection with imperfections in crystal lattices the Pocono Manor Symposium held in 1950 (Shockley et al. 1952) might be said to mark another such stage of synthesis. At any rate a great growth in the understanding of the properties of crystal imperfections occurred in the years following and this carried along a corresponding achievement in understanding atomic transport and diffusion phenomena in atomic terms. Around 1980, in the course of a small Workshop held at the International Centre for Theoretical Physics, Trieste we concluded from several observations that such a stage of synthesis could soon be reached in the study of atomic transport processes in solids. First of all, despite the great growth in detailed knowledge in the area, the theory, or rather theories, which were used to relate point defects to measurable quantities (e.g. transport coefficients of various sorts) were recognizably still largely in the moulds established considerably earlier in the 1950s and 1960s, notwithstanding the advances in purely numerical techniques such as molecular dynamics and Monte Carlo simulations.

Introduction

In the previous chapter we considered the evaluation of the macroscopic transport coefficients for models of substances displaying small degrees of disorder. In this chapter we turn to the evaluation of the effects of atomic movements upon the relaxation processes arising in nuclear magnetic resonance, as defined in §1.7 in Chapter 1. Our reason for devoting a whole chapter to this subject is that nuclear magnetic relaxation is the most widely applicable nuclear technique for studying atomic migration in solids after the direct application of radioactive tracers. Furthermore, it is still extending in various new directions, most notably for our purposes by the use of nuclear methods of alignment and detection with unstable nuclei, especially β-emitters (see e.g. Ackermann, Heitjans and Stöckmann (1983), Heitjans (1986) and Heitjans, Faber and Schirmer (1991)).

Except for self-diffusion coefficients determined by pulsed field gradient methods, the task which this subject presents to theory is rather different from that tackled in the previous chapter and it presents different mathematical problems, some of which derive from the need to deal with a two-particle correlation function (cf. (1.7.4)), while others derive from the nature of the physical interaction between the particles. For the most part the interaction we are concerned with is the nuclear magnetic dipole–dipole interaction, since this is always present (so long as the nuclear spin I > 0, i.e. so long as n.m.r. is possible). We shall also give some attention to the interaction between internal electric-field gradients and nuclear electric-quadrupole moments (I > 1/2). The dependence of these interactions upon relative position and orientation introduces mathematical complications, …

Introduction

In this and following chapters we shall consider those theories which are based on microscopic models of atomic movements and which allow us to obtain expressions for the macroscopic phenomenological coefficients introduced in the preceding two chapters. The principle on which all these particular models are founded has already been introduced in §§2.3 and 3.11 and is that the motion of atoms (or ions) of the solid may be divided into (i) thermal vibrations about defined lattice sites and (ii) displacements or ‘jumps’ from one such site to another, the mean time of stay of an atom on any one site being many times both the lattice vibration period and the time of flight between sites. The justification of this principle is provided widely in solid state physics. In general terms it derives from the strong cohesion of solids and the associated strong binding of atoms to their assigned lattice sites. This leads to well-defined phonon spectra such as have now been observed, especially by neutron scattering methods, in a wide range of solids. Theories of solid structures also show that there are generally large energy barriers standing in the way of the displacement of atoms from one site to another, even in the presence of imperfections in the structure, with the consequence that such displacements must be thermally activated and will occur relatively infrequently compared to the period of vibrational motions. Thus even the highest diffusion rates (say D ∼ 10−9m2 s−1) correspond to a mean time of stay τ ∼ 10−11 s, i.e. 100 times longer than the period of the lattice vibrations. It is more difficult to obtain information about the time …

Introduction

In the last two chapters we have studied kinetic theories of relaxation and diffusion as specific representations of the general master equation (6.2.1). Such analyses allowed us to obtain insight into a number of aspects of these processes, especially in dilute systems, e.g. the L-coefficients, the way correlations in atomic movements enter into diffusion coefficients, the relation of diffusion to relaxation rates and so on. In this chapter and the next we turn to another well-established body of theory, namely the theory of random walks. This too can be presented in the context of the general analysis of Chapter 6 and additional insights obtained.

The basic model or system which is analysed in the mathematical theory of discrete random walks is that of a particle (or ‘walker’) which moves in a series of random jumps or ‘steps’ from one lattice site to another. It can be used to represent physical systems such as interstitial atoms (e.g. C in α-Fe) or point defects moving through crystal lattices under the influence of thermal activation, as long as the concentrations of these species are low enough that their movements do not interfere with one another. The mathematical theory of such random walks has received considerable attention and is well recorded in many books and articles (recent examples include Barber and Ninham, 1970 and Haus and Kehr, 1987). For this reason it would be superfluous (and impractical) to go over all the same ground again here. Nevertheless there are various results which are useful in the theory of atomic transport either directly (e.g. in the evaluation of transport coefficients accordings to eqns.

Introduction

In the previous chapter we surveyed the ideas of point defects which form the physical basis for the description of diffusion and other atomic transport phenomena. In the present chapter we shall describe the application of equilibrium statistical mechanics to these physical models under conditions of thermodynamic equilibrium. This will allow us to obtain the contributions of thermally created point defects to thermodynamic quantities, e.g. lattice expansions, specific heat, etc. But there is a wider interest in such calculations, as will be fully apparent in later chapters. For, although the processes of atomic transport involve systems not in overall thermodynamic equilibrium, the theory of these processes nevertheless assumes the existence of local thermodynamic equilibrium; i.e. of equilibrium in regions small compared to the size of the entire macroscopic specimen but large enough to allow a thermodynamic description locally. There are two aspects to this, one concerns the chemical potentials μi and the other the transport coefficients, Lij (cf. §1.4). Statistical thermodynamics allows us to calculate the chemical potentials of the various components, i, of any particular model system as functions of the intensive thermodynamic variables. Gradients of these chemical potentials then give the thermodynamic forces Xi which generate the fluxes of atoms, defects, etc. under non-equilibrium conditions. For the transport coefficients we shall need the equilibrium concentrations of point defects and certain related quantities. Statistical thermodynamics again will provide these for particular models as a function of thermodynamic variables (e.g. P, T) without regard to the mechanism by which this equilibrium is attained (although, of course, it is implicit that the density of defect sinks and sources is adequate).

Introduction

The processes of the migration of atoms through solids enter into a great range of other phenomena of concern to solid state physics and chemistry, metallurgy and materials science. The nature of the concern varies from field to field, but in all cases the mobility of atoms manifests itself in many ways and contributes to many other phenomena. The study of this mobility and its physical and chemical manifestations is thus a fundamental part of solid state science, and one which now has a substantial history. Like much else in this science it is for the most part concerned with crystalline solids, although it also includes much which pertains also to glasses and polymers. Unfortunately, it is not possible to treat crystalline and non-crystalline solids together in any depth. The present work, which is concerned with the fundamentals of these atomic transport processes, is thus limited to crystalline solids, although occasionally we can ‘look over the fence’ and see implications for non-crystalline substances. Even with the study of crystalline solids there is a further important sub-division to be made, and that is between atomic movements in good crystal, where the arrangement of atoms is essentially as one expects from the crystal-lattice structure, and in regions where this arrangement is disturbed, as at surfaces, at the boundaries between adjoining grains or crystallites, along dislocation lines, etc. The atomic arrangements in these regions are not by any means without structure, but the structures in question are diverse and far from easy to determine experimentally. Since the basis of much of what we have to say is the assumption that there is a known and regular …

Introduction

In the preceding chapter we reviewed a number of properties of crystalline solids which demonstrate the movement and migration of atoms through these solids. The atomic theory of these properties is the subject of this book, and in this chapter we review the relevant basic mechanisms. Fundamental to these are questions of structure, both the ideal lattice structure of the perfect crystal, as one would determine it by X-ray or neutron diffraction, and those local modifications of this crystalline arrangement of atoms, called imperfections or defects, which facilitate this movement of atoms through the body of the crystal. Mostly we shall be dealing with systems in which the fraction of atoms contained in imperfect regions of the crystal is small, i.e. we are dealing with nearly perfect crystals. Notable exceptions, which we shall consider to varying degrees in this and later chapters, include order–disorder alloys, fast ion conductors and concentrated solutions of hydrogen in metals.

There are two important and interesting general facts about these crystal imperfections. This first is that there is not an arbitrary number of distinct types, but just a few. This is true for topological reasons even if we represent the solid as a continuum, when we would classify the elementary imperfections as point defects, dislocations (linear) and surfaces (two-dimensional). In a crystal lattice there are others, such as stacking faults and grain boundaries, for which there are no analogues in a continuum, but the total number of distinct types remains small. The second general fact is that the same classification of imperfections is useful irrespective of the type of bonding between the atoms of the solid.

Introduction

In this chapter we show how to use some aspects of the random-walk theory to describe the macroscopic diffusion of solute atoms (and isotopic atoms) caused by point defects. In doing so we have to handle solute movements and defect movements together, rather than the random walk of just one type of ‘walker’ or ‘particle’, as in the preceding chapter. This presents a more complex problem; nevertheless the approach we shall describe has provided a major part of the subject for many years. It was started by Bardeen and Herring (1952) who drew attention to the fact that the movement of solute atoms by the action of point defects was such that the directions of successive jumps of the solute atom were necessarily correlated with one another for the reasons already presented in Chapters 2 and 5. In Chapter 8 we showed how these correlated movements are handled in kinetic theories. However, it is the calculation of these correlation effects by random-walk theory which has engaged the attention of theoreticians to a considerable extent. Although the consequence of these effects for isotopic self-diffusion may be to introduce only a simple, numerical factor into the expression for the diffusion coefficient, the consequences for solute diffusion can be qualitatively significant, particularly when the motion of the defect is substantially affected in the vicinity of the solute. Under these same conditions qualitatively significant correlations can also appear in the self-diffusion of the solvent. The calculation of these correlation effects is thus an important part of this chapter.

Introduction

In the previous chapter we showed how the thermally activated movements of defects of low symmetry (e.g. solute–defect pairs) may give rise to dielectric and anelastic relaxation processes. To do so we employed a particular example of the master equation (already introduced in Chapter 6) to represent the movement of these defects among the possible configurations and orientations open to them. These systems were, by assumption, spatially uniform, but it is natural to seek to extend such a theoretical approach to spatially non-uniform systems in which necessarily there will be diffusion processes taking place. Such extensions are the subject of this chapter. We call them kinetic theories because in them we are concerned with the changes in time of the distributions of the solute atoms, defects, solute–defect pairs, etc. in space and among available configurations. We shall use such approaches to find expressions for the diffusion coefficients and the more general transport coefficients of non-equilibrium thermodynamics for dilute alloys and solid solutions. Both interstitial and substitutional solid solutions are considered. In doing so we shall give quantitative expression to the correlation effects anticipated in atomic migration coefficients when defect mechanisms are active (cf. §§2.5.3 and 5.5). The treatments are mostly limited to isothermal systems, although they can be extended to systems in a thermal gradient.

Kinetic theories come in a variety of forms, depending on the characteristics of the systems of interest and on the physical quantities to be represented (e.g. transport coefficients, quasi-elastic scattering functions, Mössbauer cross-sections, etc.). As elsewhere, the aim is to achieve a useful level of generality and this is achieved for systems dilute in both defects and solute atoms.

Introduction

The development in the previous chapter has necessarily been fairly abstract. In particular, although it is clear that the flux equations, when taken in conjunction with the corresponding continuity equations, in principle allow us to calculate the course of particular diffusion processes, we have not been able to give many indications of the physical characteristics of the L-coefficients appearing in these flux equations. This is, of course, to be expected at this stage since much of what follows in later chapters is about the theory of these quantities themselves. For the moment therefore they remain phenomenological coefficients, functions of thermodynamic variables (T, concentrations and stress) and governed by Onsager's theorem. It is helpful therefore at this stage to look at the relations between these L-coefficients and more familiar phenomenological coefficients adequate in special situations and which have been studied experimentally, e.g. chemical and isotopic diffusion coefficients, electrical ionic mobilities and ionic conductivity. Throughout we shall keep our assumptions about the nature of the systems being considered fairly broad. In particular, we shall not go out of our way to deal explicitly with dilute solid solutions, even though these are enormously important in fundamental studies. The more detailed results which are obtained for such systems are considered later after we have presented the atomic theory of the L-coefficients and evaluated them for particular models.

We begin by considering one of the simplest applications of the formalism, namely that to interstitial solid solutions in metals (§5.2). The simplicity of these systems allows us to describe several distinct phenomena with a minimum of mathematical manipulation.