Introduction

The nature of silica under pressure is a textbook example of the intersection of condensed-matter physics and mineralogy [1]. Silica is of obvious importance in mineralogy, as SiO2 is abundant in the Earth's crust and plays a major role in the deep interior, both as a product of chemical reactions and as an important secondary phase. From the point of view of condensed-matter physics, SiO2 presents an important system for investigating pressure-induced polymorphism, providing examples of first-order reconstructive transitions [2], displacive (soft-mode-driven) transitions [3], pressure-induced amorphization [4, 5], and polymorphism and polyamorphism [4–6]. Silica is also of great technological interest in both its crystalline and glassy forms.

Introduction

We discuss work that we have carried out on geophysically important phenomena, namely (1) the equations of state of MgSiO3 perovskite and (2) diffusion in MgO (see Fig. 3.4.1).

1. To model the composition of the lower mantle it is necessary to obtain accurate thermoelastic parameters that are used in equations of state of the component minerals. Experiments at these extreme pressure and temperature conditions are difficult and can lead to large uncertainties in some of the thermoelastic constants. Recent reports reveal that there are discrepancies in the data obtained by various experimental studies of silicate perovskite, the most abundant mineral in the Earth. Both the x-ray-diffraction diamond-anvil-cell measurements of Mao et al. [1] and the multianvil high-pressure experiments of Wang et al. [2] are in agreement in their measurement of compressibility and the associated Birch–Murnaghan equation of state, but they differ for thevalues of the Griineisen and the Anderson–Griineisen parameters inferred. Moreover, the thermodynamic analyses of Anderson and Masuda [3] and the spectroscopic measurements of Chopelas [4] have yielded thermoelastic data that are in good agreement with the measurements of Wang et al.

2. […]

Introduction

Planetary interiors represent a unique environment in the universe in which the behavior of condensed matter presents a considerable challenge. The nature and the evolution of planetary interiors, even that of our own Earth, are complex, poorly understood, and difficult to predict with current theoretical understanding. In contrast, we have a much better understanding in many ways of the interiors of distant stars. For example, we are able to calculate the structures and evolutionary history of stars with some certainty, an exercise that is not yet possible for the Earth.

Introduction

Much attention has been paid to the following phase changes of Earth interiors under high pressure and temperature; (1) phase transformation (transition, melt, amorphization); (2) decomposition, (exsolution, dissociation, phase separation); (3) chemical reaction (solid–solid, solid–molten salt); (4) electronic changes, (electronic excitation, charge transfer, high–low spin state). Many of these changes are associated with lattice instabilities accompanied by changing pressure and/or temperature conditions.

Pressure-induced phase transformation including amorphization is a significant subject not only for phase transition but also for high-pressure studies of industrial materials. Pressure-induced amorphizations of H2O (Ih) and Sio2 were previously regarded as thermodynamically metastable phases, which correspond to the supercooled liquid phase at room temperature [1, 2]; since these discoveries, several crystalline substances have been found to transform into amorphous states under compression at sufficiently low temperatures (Table 4.3.1). Some retain the amorphous state on the release of pressure to ambient, (i.e., showing an irreversible behaviour).

The energy spectrum of an isolated magnetic ion in a crystal carries information about the magnetic ion itself, its crystalline host and the interaction between these two components of the system. Crystal field theory comprises a range of techniques for extracting as much of this information as possible from observed spectra. The aim is to express the information in a form that can be used to predict the energy spectra of related systems.

Magnetic ions in crystals have many useful physical properties. In particular, there is an ongoing search for new types of laser crystal [Kam95, Kam96] and new magnetic materials. In order to design the new systems required for specific applications, it is necessary to be able to predict energy level structures and transition intensities for any magnetic ion in any crystalline environment. The techniques described in this book go a considerable way towards the achievement of this goal.

The term ‘crystal field theory’ has been applied to describe two quite different approaches. One of these, the so-called phenomenological approach, which involves the use of linear parametrized operator expressions to fit experimental results, provides a highly successful predictive tool. The alternative, so-called ab initio approach, in which energy levels and transition intensities are calculated from first principles, has proved far less useful. This book is mainly concerned with showing how the phenomenological approach can be used to obtain information about the physical properties of magnetic ions in crystals directly from observed spectra.

As was pointed out in Chapter 2, a considerable number of high-precision optical absorption and fluorescence spectra for lanthanide ions in ionic host crystals have been gathered since 1960. Crystal field fits, such as those described in Chapter 4, to the 100+ lowest-lying energy levels in lanthanide ions are generally good. However, fits to some particular ‘problem’ multiplets are invariably poor, e.g. the 3K8 multiplet of Ho3+, the 1D2 multiplet of Pr3+, and the 2H11/2 multiplet of Nd3+. In addition, some of the variations in the values of fitted parameters across the lanthanide series cannot be understood in terms of differences in ion size, or in terms of the ionic dependence of site distortions.

Problems with the one-electron crystal field parametrization also occur in fitting the energy levels of actinide and 3d transition metal ions. However, for these ions, the problems are not so easy to characterize as they are in the case of lanthanides. For this reason, while the main focus of this chapter is on the analysis of specific inadequacies of the one-electron model of lanthanide crystal fields, the conceptual aspects of this analysis should be understood as being relevant to all magnetic ion spectra.

Some of the difficulties found in fitting one-electron crystal field parameters can be associated with the use of inaccurate free-ion basis states resulting from the use of an inadequate parametrization of the free-ion Hamiltonian.

This appendix describes the structure of the QBASIC programs and data files employed in the main text. In a few cases complete printouts of the programs have been provided for the reader's convenience. All programs and data files can be downloaded from the Cambridge University Press web site:

http://www.cup.cam.ac.uk/physics

or from the web site maintained by Dr M. F. Reid:

http://www.phys.canterbury.ac.nz/crystalfield

The QBASIC programs available from these sites are as follows.

1. (i) THREEJ.BAS, which calculates values of 3j symbols. Its use is described in Section 3.1.1. The subprogram, which forms the core of this program, is listed in Section A2.1.1.

2. (ii) REDMAT.BAS, which is used to determine J-basis reduced matrix elements of the tensor operators C(k) from LS-basis reduced matrix elements, such as those tabulated by Nielson and Koster [NK63]. The use of this program in calculating matrix elements is described in Section 3.1.1. It includes the subprogram SIXj, listed in Section A2.1.2.

3. (iii) ENGYLVL.BAS, which is used to calculate single J multiplet energy levels and eigenfunctions from crystal field parameters. This program is discussed at length in Chapter 3. A specific example of its use is given in Section 3.1.3.

4. (iv) ENGYFIT.BAS, which is used to fit crystal field parameters to sets of energy levels of a single J multiplet. This program is discussed at length in Chapter 3. A specific example of its use is given in Section 3.2.

5. (v) CORFACW.BAS, which determines combined coordination factors (Wybourne normalization) from input coordination angles. An example of the use of this program is given in Section 5.3.2.1.

6. […]

Our aim in producing this book is to make a wide range of up-to-date techniques for the analysis of crystal field splittings accessible to non-specialists and specialists alike. All of these techniques are based on the phenomenological approach, in which the aims are to

1. (i) parametrize observed crystal field splittings using models which provide the maximum predictive power,

2. (ii) test model assumptions directly by means of the quality of parameter fits and the accuracy of their predictions,

3. (iii) use model parameters as a convenient interface between first principles calculations and experiment.

It is expected that readers will be mainly interested in the first two of these aims. Hence relatively little space is devoted to the description of first principles calculations. While most of the book is concerned with practical applications of the phenomenological approach, some space is given to providing a critical appraisal of relationships between the this approach and other approaches to crystal field theory.

A series of examples, backed with a set of QBASIC programs, is provided to enable readers to rapidly attain proficiency in the basic techniques of phenomenological analysis. Those who wish to embark on more sophisticated analyses are directed to other available program packages. In addition to the procedures covered in the QBASIC programs, the book covers three topics which have barely been touched upon elsewhere.

Calculations of crystal field parameters from first principles depend upon a knowledge of the crystal structure in the immediate neighbourhood of the magnetic ion. The calculated crystal field contributions for single ligands (see Chapter 1) can then be combined to estimate values of the crystal field parameters for real systems. Most calculations of this type [New71] have been based on the assumption that the combination process is purely additive, i.e. that the ‘superposition principle’ holds. At the present time, however, the superposition principle is mostly employed in the analysis of experimentally determined crystal field parameters. This phenomenological application is known as the ‘superposition model’.

This chapter, taken together with some of the QBASIC programs described in Appendix 2, is designed to provide a comprehensive ‘Do It Yourself’ kit, supporting various applications of the superposition model in the calculation and interpretation of phenomenological crystal field parameters. The mathematical manipulations are relatively simple; some can even be carried out by hand using tables provided in this chapter.

Section 5.1 summarizes the physical assumptions that underlie the superposition model. It also gives the basic equations that describe the model and lists the various ways in which the model can be applied. A comprehensive survey of the empirical values of the intrinsic, or single-ligand, crystal field parameters is given in Section 5.2. Practical means of simplifying superposition model analyses, together with some examples of the use of the model to estimate empirical crystal field parameters from the intrinsic parameters, are described in Section 5.3.

In order to interpret the information obtained from magnetic ion spectra using parametrized crystal field models it is necessary to have a qualitative understanding of the physical mechanisms through which the crystalline environment induces energy level splittings. The discussion of mechanisms given in this chapter should be seen in this light. It is not intended to provide a practical basis for quantitative ab initio calculations of crystal field splittings. Instead, it provides a conceptual description of the various mechanisms that contribute to crystal field splittings.

The first three sections give a qualitative description of the most important mechanisms which contribute to crystal field splittings. These comprise the electrostatic, charge penetration, screening, exchange, overlap and covalency contributions. Sections 1.4 and 1.5 express these contributions in terms of a simple algebraic formalism related to the ‘tight-binding’ model in solid state physics. Several other formal approaches to crystal field theory, including the phenomenological approach, are described in the subsequent sections. Numerical results for a particular system, based on the formalism developed in Sections 1.4 and 1.5, are given in Section 1.9. The significance of these results to the phenomenological methods used in this book is summarized in Section 1.10.

The crystal field as a perturbation of free-ion open-shell states

Crystals can be regarded as assemblies of free ions, bound together in several possible ways.

The intrinsic physical properties of the crystal field Hamiltonian have led over the years to various formats and conventions being used in the literature to express crystal field parameters. Hence comparison of crystal field parameter sets from various sources often requires several manipulations and conversions. To facilitate the computations involved in this process, the user-friendly computer package ‘CST’ (Conversion, Standardization and Transformation) has been developed [RAM98, RAM97]. The package is useful for various general manipulations of the format of the experimental zero-field splitting parameters as well as the crystal field ones. Its capabilities include CONVERSIONS: unit conversions – between several units most often used for crystal field and zero-field splitting parameters; normalization (or notation) conversions – between several major normalizations for crystal field and zero-field splitting parameters; STANDARDIZATION of orthorhombic, monoclinic and triclinic crystal field and zero-field splitting parameters; and TRANSFORMATIONS of crystal field and zero-field splitting parameters into an arbitrary axis system, including the rotation invariants.

In this appendix the structure and capabilities of the CST package are briefly presented. In keeping with the overall approach of a book designed as a ‘Do-It-Yourself’ text, the emphasis is put on the practical implementations of the package, illustrated using examples of experimental data taken from recent crystal field literature. A detailed description of the CST modules is provided in the Manual [RAM97], whereas examples of applications to zero-field splitting parameters are dealt with in [RAM97, Rud97, RM99].