The Griffith study usefully identifies two distinct stages in crack evolution, initiation and propagation. Of these, initiation is by far the less amenable to systematic analysis, governed as it invariably is by complex (and often illdefined) local nucleation forces that describe the flaw state. Accordingly, we defer investigation of crack initiation to chapter 9. A crack is deemed to have entered the propagation stage when it has outgrown the zone of influence of its nucleating forces. The term ‘propagation’ is not necessarily to imply departure from an equilibrium state: indeed, for the present we shall concern ourselves exclusively with equilibrium crack propagation. Usually (although not always), a single ‘well-developed’ crack, by relieving the stress field on neighbouring nucleation centres, propagates from a ‘dominant flaw’ at the expense of its potential competitors. In the construction of experimental test specimens for studying propagation mechanics such a well-developed crack may be artificially induced, e.g. by machining a surface notch. This pervasive notion of a well-developed crack, taken in conjunction with the fundamental Griffith energy-balance concept, provides us with the starting point for a powerful analytical tool called fracture mechanics, the many facets of which will become manifest in the remaining chapters.

The formulation of fracture mechanics began with Irwin and his associates round about 1950. The impetus for the development of this discipline originally came from the increasing demand for more reliable safety criteria in engineering design.

Introduction

In chapter 1 we introduced the s-d and the Anderson models as the basic models for magnetic impurities in simple metallic hosts. In the succeeding chapters we have outlined techniques for predicting the static and dynamic behaviour of these models over most of the possible parameter regimes. These techniques give results which are either exact or which are within well controlled approximations (for a detailed summary of these results see appendix K). We know from the comparisons between theory and experiment which we have made so far, that the physical picture which emerges is in broad agreement with the experimental observations. The parameter regime of primary interest is the Kondo regime where there is local moment behaviour at high temperatures or high magnetic fields with a Curie law susceptibility. This undergoes a broad transition or crossover at temperatures of the order TK (in weak or zero fields) to Fermi liquid behaviour and a Pauli susceptibility at low temperatures. The lnT contributions calculated by Kondo explain the resistance minimum, and the Fermi liquid theory gives the power law behaviour of the resistance at very low temperatures. The narrow peak in the one electron density of states, the Kondo resonance, qualitatively explains the basic trends in the change in thermodynamic behaviour with temperature, such as the peak in the specific heat.

I welcome the opportunity that the new edition of this book gives for me to correct a number of minor slips and omissions that escaped my notice in the original text. I am grateful to the help given me by Dr Jan von Delft in compiling this list of corrections.

I have also used this occasion to add a short section in the form of an Addendum to cover some recent developments. The subject of strongly correlated electron systems continues to be a very active field of research so within the short space at my disposal I could only briefly mention some results that are particularly related to the topics covered in the original edition. However, I have also included references to some more recent review articles where a fuller discussion of these topics can be found, as well as references to other related work.

Introduction

In the last chapter we introduced some new techniques of calculation which, though approximate, gave asymptotically exact results in the limit N → ∞. This is a chapter of advanced topics in which we develop these techniques further, particularly with a view to calculating dynamic quantities and response functions. We first of all extend the perturbational scheme of section 7.2 to a self-consistent scheme which takes into account all non-crossing diagrams, and known generally as the Non-Crossing Approximation (NCA). This is used to calculate dynamic response functions at finite temperatures. It has been used extensively for making predictions to compare with experiment for anomalous rare earth systems. Following this we also reconsider the mean field slave boson approach. The aim here is to go beyond the mean field theory and take some account of fluctuations. It is possible to generalize the approach so that the corrections to the mean field can be systematically treated in a 1/N expansion. We shall also develop approximations so that the full spectrum of the one particle Green's function can be calculated, and not just the quasi-particle contribution. Finally we discuss briefly yet another formulation of a 1/N expansion, the variational approach. This has been developed for the calculation of the spectral density of Green's functions and response functions for comparison with several types of photoemission experiments, and also for one electron absorption spectra. It has been used extensively in the interpretation of data on Ce and other rare earth compounds.

This book charts the progress in the theory of magnetic impurities since the late 50s, from the early developments leading to the Kondo impurity problem and its solution to the challenging problems posed by the recent work on heavy fermions and the high temperature superconductors. The first eight chapters cover, largely in chronological order, the techniques which have been developed to deal with single impurity problems. Some of these techniques, such as Green's functions, Feynman diagrams and perturbation theory, are covered in standard many-body texts (for example, Fetter and Walecka, 1971: Abrikosov, Gorkov and Dzyaloshinski, 1975: Mahan, 1990). Others may be less familiar so for these techniques I have included general introductory sections in the relevant chapters, and some appendices with further details, in order to make the text as self-contained as possible. The aim has been to make the book readable at two levels. At the higher level I have tried to present the general development of ideas, the emphasis being on the results of the theory and the general physical picture that emerges. The equations at this level are included to make it clear how these results are obtained. I have tried to make it readable also at a second more detailed level by including enough information in the text and appendices so that one can work from one equation to the next.

Introduction

Our main concern so far has been with theories of a single magnetic impurity in a simple host metal. These theories have been developed over a period of more than thirty years and most of the models put forward for explaining these systems are now well understood. The concepts for understanding their behaviour in terms of scaling trajectories, fixed points, spin compensation, quasi-particles, Fermi liquid theory and Kondo resonances have been developed and there are exact solutions for the thermodynamic behaviour of many of the models as well as good approximations for their dynamic response (for the reader who has skipped the details of chapters 4–8 the important single impurity results are summarized in appendix K). As we saw in the previous chapter there is still much to be done in the way of detailed predictions for specific systems to compare with experiment. There is also the possibility of new types of experiments to probe the theory further (for example, the possibility of probing the Kondo resonance by resonant tunnelling has recently been discussed by Hershfield, Davies and Wilkins, 1991). These may throw up new puzzles to be answered. The general belief, however, is that if these occur they will be relatively minor and will not require a fundamental reworking of the theory. With these provisos it may be claimed the single impurity problem has been ‘solved’.