The electrical resistance or resistivity of a conducting solid can be experimentally determined without much difficulty and for many years it has been used as a research tool to investigate various microstructural and physical phenomena. However, unlike conventional diffraction methods which are capable of mapping out scattered intensities in twoor three-dimensional space, an electrical resistivity measurement gives only a single value (at any fixed temperature and structural state) representing an average over all directions of conduction electron scattering. As there is no means of performing the back-transform from this single point, the analysis of resistivity data in terms of microstructure must incorporate calculations of conduction electron scattering based on some model of the structure or microstructure concerned. With the refinement and greater availability of more direct methods, particularly X-ray, neutron or electron diffraction, transmission and analytical electron microscopy (especially atom-probe field ion microscopy which allows an atom-by-atom picture of a material to be established) there has been a declining utilisation of such an indirect method in microstructural investigations. Nevertheless, while a resistivity study may require support from some other technique to allow an unambiguous interpretation of the results, there are many cases where such studies still have particular value, either by virtue of their simplicity or lack of alternative techniques. These include studies of defects, pre-precipitation processes, short and long range ordering or phase separation (particularly with respect to transformation kinetics) and determination of critical transformation compositions and/or temperatures.

The structure of magnetic material was discussed in some detail in Chapter 3 and we now want to consider how this magnetic structure effects the motion of conduction electrons. However, the purpose of this chapter is not so much to give a complete theory of the resistivity of magnetic materials but, rather, to alert the reader to some of the complications that can arise when the alloy being investigated is magnetic or nearly magnetic. We will thus concentrate mainly on those aspects of the resistivity of magnetic materials that have a dependence upon composition and atomic correlations. Furthermore, as much of the theory of the magnetic state is still being developed, some of the analysis will of necessity be qualitative rather than quantitative.

Magnetic materials with long range magnetic order

Overview

As noted in Section 3.1, there are still some aspects of the nature of the magnetic state at finite temperatures in transition metals that are uncertain. As such we cannot as yet give a comprehensive discussion about the variation of resistivity with temperature and the effects of alloying. References to some of the more general theories of the resistivity of magnetic materials are given in Table 7.1. We can, however, give a qualitative description by adopting the view that a clear separation is possible between the conduction and magnetic electrons and that they are scattered by deviations from perfect ordering of the magnetic moments.

Introduction

The purpose of this chapter is to review the application of Mössbauer spectroscopy to the study of dynamics. The main emphasis will be on the new areas in which significant advances have been made in recent years, such as the study of diffusive processes, the influence of motion on lineshape, time-dependent studies and dynamics at phase transitions. Rather less attention will be devoted to areas such as f-factors in solids, the Goldanskii–Karyagin effect and temperature shift, which have been studied more extensively in the past. The trends towards future areas for this research will also be considered, together with an evaluation and comparison between Mössbauer spectroscopy and other methods for investigating these phenomena.

No attempt will be made to develop the formal theory of dynamics in this chapter. General physical arguments will be employed to describe the phenomena and the main theoretical results to be used in interpreting the experimental data will be summarised. Representative experimental data and results will be presented in the various sections.

The phenomenon of the recoil-free resonant absorption of gamma rays is the result of the special dynamics of nuclei in solids. This fact was recognised by Mössbauer, who interpreted his apparently anomalous observations in terms of lattice dynamics of a similar nature to those associated with neutron scattering (Mössbauer, 1958). He found that the recoil momentum of the recoiling nucleus is shared by many nuclei of the crystal and thus the recoil energy is extremely small.

Introduction

In this chapter the various kinds of time-dependent phenomena that can be studied by Mössbauer spectroscopy will be reviewed and the relationship between the data obtained by Mössbauer spectroscopy and that derived from other techniques will be assessed.

In many discussions on time-dependent or dynamical effects in Mössbauer spectroscopy the essential physical ideas get obscured by complicated theoretical formalism. The ideas, however, are quite simple and are capable of being appreciated at a qualitative level before embarking on detailed calculations. The present chapter is an attempt to fulfil this need and bring out the important concepts at a non-specialist level. Where necessary, the reader may refer to the original sources cited for more elaborate treatments. For these same reasons, only a few simple models have been chosen, which can be analysed without a detailed mathematical treatment.

As briefly outlined in Chapter 1, the Mössbauer effect concerns the resonant absorption of a gamma ray, of frequency ω0 and wavevector k, by a nucleus. If the absorbing nucleus is located at the position r, the electromagnetic field of the gamma ray at the nucleus can be represented by

where ΓN is the natural linewidth of the decaying nucleus in its excited state, which is given by 2π/τN, where τN is the mean lifetime of the nuclear excited state. In general, in this chapter, linewidths are expressed in terms of frequency, which must be multiplied by ħ, the Planck constant divided by 2π, in order to convert to an energy.

Introduction

The creation of a chemical bond involves the sharing of two or more electrons between a pair of atoms. If the atoms are identical, as in a homonuclear diatomic molecule, there is little change in the total electron density on each atom, but there may be a slight redistribution of electrons between the valence orbitals to achieve a more satisfactory hybridisation. If the atoms differ, the bond will be more or less polar, depending on the disparity in electronegativities: one atom will gain electron density at the expense of the other. This effect is at its most extreme in the formation of an ionic bond. When an atom forms more than one bond, considerable rehybridisation may take place, with or without gain or loss of electron density by the electronegativity effect. An examination of the distribution of electron density about a particular atom can therefore give considerable information about the bonds involving that atom.

Mössbauer spectroscopy is well suited to such a study, since two major parameters of a spectrum can be related directly to the populations and changes in population of the valence shell orbitals. The isomer shift relates to the total electron density on the atom, and the quadrupole splitting reflects any asymmetry in the distribution of electron density. Thus, each parameter is capable of giving information about bonding: from their combination, it is often possible to make quite detailed analyses.

Many techniques have developed in recent years which allow experimental scientists to extend their studies in new directions. An awareness of the scope, strengths, potential, and limitations of a particular technique is essential if it is to be utilised to the full. The aim of this book is to review the unique contribution that Mössbauer spectroscopy can make to the study of the bonding, structural, magnetic, time-dependent, and dynamical properties of various systems. Particular emphasis is given to the types of information which Mössbauer spectroscopy provides, as well as to a comparison with the information in these areas which can be obtained using other techniques.

In order to achieve the objectives outlined above, each of the main chapters relates to one aspect of the information which can be obtained by Mössbauer spectroscopy, and these chapters have been written by scientists with considerable experience in these different areas. We are particularly indebted to the authors of the individual chapters for embracing our overall philosophy for this book and the need to produce a coherent whole, while still reflecting their personal enthusiasm for their own particular area of interests, as well as the differences inherent in these different aspects.

One of the striking features of the growth of Mössbauer spectroscopy, following the discovery by Rudolf Mössbauer in 1957 of the effect which bears his name, was the rapidity with which the potential of the new technique was recognised and developed.

Introduction

The aim of this section is to begin with the most general and qualitative picture of the interactions of a Mössbauer nucleus with its environment. By developing and quantifying the treatment of these interactions, we seek to establish connections between the components of the environment and the energies of the nuclear levels and thus to build up an understanding of the features of the environment that shape the Mössbauer spectrum.

Classes of interaction

A schematic picture of the interactions that affect the energy levels of a Mössbauer nucleus is shown in Figure 4.1. In this representation it is seen that the Mössbauer nucleus senses its environment directly through atomic–nuclear interactions [1], solid–nuclear interactions [3] and the external field–nuclear interaction [5]. It is indirectly sensitive to the solid–atom interaction [2] and the external field–atom interaction [4], which are felt through their effect on the nucleus' own atom.

It is usual in magnetic solids for the main influence on the Mössbauer spectrum to be felt through the interactions [1] and [2]. That is, the nucleus senses its own atom and the state of the solid via the effect of the solid on that atom.

Introduction

The opening chapters of this book have shown how the hyperfine interactions which give rise to the Mössbauer spectrum are a means by which the electronic environment of the nucleus may be examined. The preceding chapter in particular has shown how the Mössbauer parameters can be related to various aspects of chemical bonding and the geometrical properties of compounds. In this chapter the application of Mössbauer spectroscopy in the examination of electronic, molecular, and lattice structure will be considered. Given the close connection between electronic structure and chemical bonding, which has been covered in Chapter 2, this chapter will devote particular attention to molecular and lattice structure, which are both properties amenable to investigation by Mössbauer spectroscopy.

The interpretation of the Mössbauer spectrum obtained from a molecular solid is highly dependent on the particular situation. For example, in some cases two inequivalent Mössbauer atoms may be involved in the molecular complex and this can provide a means by which the Mössbauer data may be more comprehensively interpreted. However, in the majority of compounds studied by Mössbauer spectroscopy only one Mössbauer atom is present and, while acknowledging the advantages of having such a sensitive probe of the immediate local environment, there are often problems in making conclusions concerning the number or disposition of the atoms or groups around the Mössbauer atom.

Introduction

Mössbauer spectroscopy has now established itself as a technique which can provide valuable information in many areas of science. This book sets out to consider the unique contribution of Mössbauer spectroscopy in terms of the specific information it can give to the study of the bonding, structural, magnetic, time-dependent and dynamical properties of various systems. Accordingly, each of the following chapters is devoted to a detailed consideration of one fundamental aspect of the total information that can be provided by Mössbauer spectroscopy. It is intended that by adopting this approach the book will provide an overview and appraisal of the technique that will be of value both to the experienced practitioner who has perhaps worked in one particular area of the technique, and also to the newcomer or student who needs to see the technique in its proper context. It is also hoped that those working in other areas of science may find this book a helpful guide in relating information obtained from Mössbauer spectroscopy with that derived from other techniques and also in evaluating the potential value of Mössbauer spectroscopy for providing information relevant to their particular problems and systems.

Introduction

Further exciting developments in rate equations are possible. Some of these more advanced uses and techniques are touched on in this chapter, singling out laser devices which will find applications in communications. The statistical information that can be provided by rate equations has been one theme in this book and is developed further to demonstrate how the output of a single mode injection laser changes from a chaotic distribution to a Poisson distribution as the drive current into the laser is increased.

The injection laser normally has several modes. It is useful then to show how rate equations can handle such multimode problems. In particular this section emphasises the importance of spontaneous emission in determining mode amplitudes.

Rate equations, as interpreted here, have been concerned with rates of change of energy, momentum, quanta, charge and so on. These equations have all removed any information of the phase of quantum or electromagnetic waves. In phenomena where phase is important more detailed discussions using full quantum or electromagnetic theories are usually required. To demonstrate the importance of phase and also to demonstrate how the rate equation approach can sometimes be modified to include phase, the ‘mode locked’ laser is discussed briefly. This topic follows on naturally from the multimode rate equations because in a mode locked laser there are many optical modes at equally spaced frequency intervals but with their amplitudes locked to zero phase at one time. The resultant output from such mode locked lasers can be a train of exceptionally short pulses, down to subpicosecond durations with nanosecond repetition rates.