Crystal field theory is one of several chemical bonding models and one that is applicable solely to the transition metal and lanthanide elements. The theory, which utilizes thermodynamic data obtained from absorption bands in the visible and near-infrared regions of the electromagnetic spectrum, has met with widespread applications and successful interpretations of diverse physical and chemical properties of elements of the first transition series. These elements comprise scandium, titanium, vanadium, chromium, manganese, iron, cobalt, nickel and copper. The position of the first transition series in the periodic table is shown in fig. 1.1. Transition elements constitute almost forty weight percent, or eighteen atom percent, of the Earth (Appendix 1) and occur in most minerals in the Crust, Mantle and Core. As a result, there are many aspects of transition metal geochemistry that are amenable to interpretation by crystal field theory.

Cosmic abundance data for the transition elements (Appendix 1) show that each metal of the first transition series is several orders of magnitude more abundant than all of the metals of the second (Y–Ag) and third (Hf–Au) transition series, as well as the rare earth or lanthanide series (La–Lu). Iron, Fe, is by far the predominant transition element, followed by Ni, Cr and Mn. Crustal abundance data also show high concentrations of Fe relative to the other first-series transition elements on the Earth's surface.

Introduction

Earlier chapters have demonstrated that spectral features of most rock-forming minerals in the visible to near-infrared region originate from the presence of transition elements in their crystal structures. Iron and titanium have higher crustal abundances on terrestrial planets relative to other transition elements and, consequently, are expected to contribute significantly to the reflectance spectra of planetary surfaces. Spectral profiles of sunlight reflected from planetary surfaces, when correlated with measured optical spectra of rock-forming minerals, may be used to detect the presence of individual transition metal ions, to identify constituent minerals, and to determine modal mineralogies of regoliths on terrestrial planets. The origin and applications of such remotesensed reflectance spectra measured through Earth-based telescopes are described in this chapter.

Chemical composition of the terrestrial planets

Properties of the terrestrial planets that are central to this chapter are summarized in table 10.1 and information about element abundances is contained in Appendix 1. The crustal abundance data for the Earth indicate the presence of relatively high concentrations of Fe, and to a lesser extent Ti, compared to other first-series transition elements.

Introduction

In the previous chapter it was shown how electrostatic fields produced by anions or negative ends of dipolar ligands belonging to coordination sites in a crystal structure split the 3d orbitals of a transition metal ion into two or more energy levels. The magnitude of these energy separations, or crystal field splittings, depend on the valence of the transition metal ion and the symmetry, type and distances of ligands surrounding the cation. The statement was made in 2.8 that separations between the 3d orbital energy levels may be evaluated from measurements of absorption spectra in the visible to near-infrared region. The origins of such crystal field spectra, also termed d–d spectra and optical spectra, are described in this chapter. Later chapters focus on measurements and applications of crystal field spectra of transition metal-bearing minerals.

Units in absorption spectra

When light is passed through a compound or mineral containing a transition metal ion, it is found that certain wavelengths are absorbed, often leading to coloured transmitted light. One cause of such absorption of light is the excitation of electrons between the split 3d orbital energy levels. Measurements of the intensity of light incident on and transmitted through the transition metalbearing phase produces data for plotting an absorption spectrum.

This book arose from a series of lectures given at the Universities of Cambridge and Oxford during the Spring of 1966. The lectures were based on material compiled by the author between 1961 and 1965 and submitted in a Ph.D. dissertation to the University of California at Berkeley. At the time crystal field theory had become well established in chemical literature as a successful model for interpreting certain aspects of transition metal chemistry.

For many years the geochemical distribution of transition elements had been difficult to rationalize by conventional principles based on ion size and bond type criteria. In 1959 Dr R. J. P. Williams, a chemist at Oxford conversant with the data from the Skaergaard intrusion, was able to present an explanation based on crystal field theory of the fractionation patterns of transition elements during crystallization of basaltic magma. This development led to the author's studying, under the supervision of Professor W. S. Fyfe, other mineralogical and geochemical data of the transition elements which might be successfully interpreted by crystal field theory.

In addition to suggesting applications of crystal field theory to geology, this book reviews the literature on absorption spectral measurements of silicate minerals and determinations of cation distributions in mineral structures. Many of these data have not been published previously. Spectral measurements on minerals have revealed many advantages of silicates as substrates for fundamental chemical studies. First, crystal structures of most rock-forming silicates are known with moderate to high degrees of precision. Secondly, minerals provide a range of coordination symmetries many of which are not readily available to the synthetic inorganic chemist.

Introduction

One outcome of interpreting transition metal geochemistry by crystal field theory is that the theory has enabled some of the basic concepts of geochemistry to be critically evaluated and defined more rigorously. In earlier chapters, crystal field theory was used to explain why some transition elements deviate from periodic crystal chemical and thermodynamic trends shown by other cations with similar charges and ionic radii. In this chapter, criteria for interpreting trace element geochemistry are examined. Examples are highlighted where fractionation patterns applicable to many elements sometimes deviate for transition metal ions. Crystal field effects are shown to be dominant factors influencing the distributions of several of the transition elements in crustal processes during the petrogenesis of igneous, sedimentary and metamorphic rocks.

Trace elements

The classification of chemical elements into major and minor or trace element categories is somewhat arbitrary. Thermodynamically, a minor element may be defined as one that is partitioned between coexisting phases in compliance with laws of dilute solutions, such as Henry's law, eq. (7.2b). In geochemical parlance, however, trace elements are usually categorized on the basis of abundance data. In this context, the mineral, rock or environment containing the chemical elements must be defined as well as the concentration boundary separating a major and trace element.

In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function Gαβ(Q, ω) appearing as a distinct component of the the dynamic structure factor S(Q, ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S(Q, ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between Gαβ(Q, ω) and S(Q, ω) in a Bose-condensed fluid (n0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through Tλ, although these excitations have very temperature-dependent weights in Gαβ(Q, ω) and S(Q, ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about Gαβ(Q, ω) from inelastic neutron-scattering data on S(Q, ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid.

After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4He.

In this book, we have developed the theory of the excitation spectrum of superfluid 4He in which the Bose condensate plays the central role. In Chapter 5, we showed how a Bose broken symmetry inevitably leads to a mixing of the single-particle and density fluctuations. Combining the general results of the dielectric formalism with recent high-resolution neutron-scattering data over a wide range of wavevectors, energies and temperatures, we were led in Chapter 7 to a new interpretation of the well known phonon–maxon–roton dispersion curve. In Section 12.1, we briefly recapitulate this new scenario and discuss how it developed from preceding theoretical work. We also review earlier studies which had independently suggested that rotons were in fact atomic-like single-particle excitations, quite different from the long-wavelength phonons. In addition, we address the question of what Feynman's work says about the nature of rotons.

The most important topic which has not been covered in this book is superfluid 3He–4He mixtures. The appropriate dielectric formulation has been developed by Talbot and Griffin (1984c), as we briefly summarize in Section 12.2. Much work remains to be done in using these formal results in a detailed analysis of experimental S(Q, ω) data, even at the level of Section 7.2 in the case of pure 4He.

Finally, in Section 12.3, we list some specific topics where further theoretical and experimental work would be useful. This list, which brings together suggestions scattered throughout the book, also acts as a convenient summary of our major themes.

The major goal of the present book is to outline the field-theoretic analysis of the dynamical behaviour of a Bose-condensed fluid that has developed since the late 1950's. While we often use the weakly interacting dilute Bose gas (WIDBG) for illustrative purposes, the emphasis is on the dynamical properties of a specific Bose-condensed liquid, superfluid 4He. We attempt to develop a coherent picture of the excitations in liquid 4He which is consistent with, and rooted in, an underlying Bose broken symmetry. Recent high-resolution neutron-scattering studies in conjunction with new theoretical studies have led to considerable progress and it seems appropriate to summarize the current situation. The only other systematic account of superfluid 4He as a Bose-condensed liquid is the classic monograph by Nozières and Pines (1964, 1990).

The phenomenon of Bose condensation plays a central role in many different areas of modern condensed matter physics (Anderson, 1984). Historically it was first studied in an attempt to understand the unusual properties of superfluid 4He (London, 1938a). In a generalized sense, however, it also underlies much of the physics involved in superconductivity in metals and the superfluidity of liquid 4He, in which Cooper pairs play the role of the Bosons (see, for example, Leggett, 1975 and Nozières, 1983). In recent years, there has been increased research on the possibility of creating a Bose-condensed gas, involving such exotic composite Bosons as excitons in optically excited semiconductors, spin-polarized atomic Hydrogen, and positronium atoms.

In this chapter, we use the formalism developed in Sections 3.2, 5.1 and 5.4 to discuss various correlation functions in the long-wavelength, low-frequency limit. It is important that the microscopic theory based on a Bose broken symmetry used to describe the high-frequency excitations probed by neutrons also explains the low-frequency behaviour which characterizes superfluidity. In Section 6.1, we show how the generalized Ward identities given in Section 5.1 lead in a simple way to several rigorous zero-frequency sum rules. We discuss the structure of the low-frequency, long-wavelength response functions and make contact with the two-fluid description of Landau. In Section 6.2, we discuss the structure of the correlation functions in the hydrodynamic region, as given by the two-fluid equations of Landau (see Khalatnikov, 1965). While the hydrodynamic region of S(Q, ω) is difficult to probe by thermal neutron scattering, it can be studied by inelastic Brillouin light scattering (for excellent reviews, see Stephen, 1976; Greytak, 1978).

In Section 6.3, starting from the Gavoret–Nozières formalism summarized in Section 5.4, we review GN's explicit calculation of the phonon spectrum of Gαβ and χnn (at T=0). We comment on the significance of the infrared divergences in the Q, ω → 0 limit first noted by Gavoret and Nozières (1964) and clarified in later work, by Nepomnyashchii and Nepomnyashchii (1978), Popov and Serendniakov (1979), and Nepomnyashchii (1983). We also discuss the relation between first and second sound which occurs in the hydrodynamic region and the phonons which arise in the collisionless region.