Hardcover
-
PICK UP IN STORECheck Availability at Nearby Stores
Available within 2 business hours
Related collections and offers
Overview
The power of the method is further demonstrated by showing that it can be extended to include lattice vibration and conduction by electron hopping such as probably occurs in high-Tc superconductors. Thus Stevens shows how an apparently successful ad hoc method of the past can be replaced by a much more soundly based one that not only incorporates all the previous successes but appears to open the way to extensions far outside the scope of the previously available methods. So far only some of these have been explored. The book should therefore be of great interest to all physicists and chemists concerned with understanding the special properties of molecules and solids that are imposed by the presence of magnetic ions.
Originally published in 1997.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Product Details
| ISBN-13: | 9780691636801 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 04/19/2016 |
| Series: | Princeton Legacy Library , #351 |
| Pages: | 268 |
| Product dimensions: | 6.20(w) x 9.30(h) x 0.80(d) |
Read an Excerpt
Magnetic Ions in Crystals
By K. W. H. Stevens
PRINCETON UNIVERSITY PRESS
Copyright © 1997 Princeton University PressAll rights reserved.
ISBN: 978-0-691-02693-0
CHAPTER 1
Introduction
§1.1 Microscopic Magnetism
From a microscopic point of view magnetic properties arise from two sources, the translational motion of charged particles, particularly electrons, and the magnetic moments associated with the spins of electrons and nuclei. The steps that eventually led to this picture began in the early 1900s, when Zeeman studied the changes that occur in the optical spectra of free ions in the presence of external magnetic fields. It was still a long road though, through classical physics, Bohr theory, quantum theory, and the introduction of a wide range of experimental techniques, before the present understanding of the properties of magnetic ions was established. It was initially assumed that spectral lines were in some way associated with harmonic oscillations, but in spite of a good deal of analysis and ingenuity little progress was made until the Bohr theory came along, in 1915. This substituted transitions between energy levels for harmonic-oscillator-like resonances. Rapid progress then ensued, at least in interpreting the spectroscopy of the simplest ions, but gradually this too slowed down as it became increasingly difficult to explain the observations on more complicated ions [see Van Vleck (1926) for a detailed account of the position just before the advent of quantum mechanics]. Quantum mechanics, in the late 1920s, opened a new avenue, one which was quickly recognized as being more promising and which could be rapidly developed in its wave-mechanical form using mathematical techniques familiar from the Bohr theory. Within just a few years, enough progress had been made for Condon and Shortley to begin writing, in 1931, a detailed account of the theory of atomic spectra. The book, which appeared in 1935 (with a later revision, Condon and Shortley, 1967), is still one of the standard texts on atomic spectroscopy.
§ 1.2 Ions in Solids
With the theory of the isolated ions in a satisfactory form, attention turned to using it to explain the spectroscopic properties of molecules and the macroscopic properties of liquids and solids. In the context of magnetism, it was known that iron was a ferromagnet, that many metals had temperature-independent paramagnetic susceptibilities, that many ionic crystals were diamagnetic, and that others had susceptibilities that obeyed either the Curie or the Curie-Weiss law, χ = C/(T + Δ), where C and Δ (= 0 in the Curie law) are constants and T is the absolute temperature. (In paramagnets the moment is in the direction of the applied field, while in diamagnets it is in the opposite direction. For each, the susceptibility is defined as the ratio of the magnetic moment to the magnetic field in the limit when the field tends to zero.) The paramagnetism of the normal metals was fairly soon explained (in the 1930s) with the introduction of a simple form of band theory, but the ferromagnetism of iron, cobalt, and nickel remained a problem for much longer.
For the ionic magnetic crystals the direction to go had already been indicated, for even before the advent of the Bohr theory an explanation of their magnetic properties had been given. This assumed that the ions contained microscopic circulating currents, each of which would give rise to a magnetic moment. Thus in some crystals there might be "permanent" microscopic magnetic moments which in zero magnetic field would be randomly arranged and so cancel, whereas in others the internal moments would be such that there were no permanent moments to be canceled. An external magnetic field could be expected to induce a precessional motion of all the currents, which would produce a diamagnetic moment in both types of ion. However, in the ions with permanent moments it would also induce a partial alignment, and it was the thermal distribution over the associated energies that added a temperature-dependent paramagnetism. [Although the theory was worked out in some detail and seemed to be satisfactory it was later realized that it was unsound. See Chapters 4 and 5 of Van Vleck (1932), which, incidentally, seems to have been the first book to apply quantum mechanics to solid state problems.]
The advent of the quantum theories led to the concept of electron shells, and when this was supplemented by the idea that no two electrons could have the same quantum numbers (originally three in number, but increased to four when electron spin was discovered), the theory led to an understanding of the magnetic properties of the free atoms and ions and an understanding of the periodic table of the elements. In the lowest-energy arrangement, the shells are filled progressively in energy so that in some atoms and ions there are just the right number of electrons to fill all the occupied shells. Between these, there will be others in which the outer shells, usually just one, are partially filled. Those atoms or ions with filled shells have no orbital and no spin moment so they can be identified with the entities which give diamagnetism. Those with partially filled outer shells usually have all the inner shells filled (the rare earths are an exception), so diamagnetism can be associated with their inner shells and paramagnetism with their partially filled outer shells. The chemical properties follow when the free atom picture is supplemented by the idea that in compounds electrons have moved from one atom to another to produce ions with closed-shell structures.
It was also known that in many of the iron and rare earth group compounds the ions do not have closed-shell structures, and it was therefore not surprising that they showed properties typical of those expected from microscopic permanent magnetic moments. There was, however, a problem, particularly for the iron group compounds, that the moments of the ions in a crystal, obtained from susceptibility measurements, were usually quite different from those of the same ion in free space, obtained from the Zeeman effect. (Less was known about the susceptibilities of rare earth compounds. Over the temperature ranges that had been used the agreement seemed much better. This did not last when measurements were made at lower temperatures.)
The step which led to the resolution of the problem was taken in the early 1930s, by Van Vleck, who introduced what is known as the crystal field. Each magnetic ion was regarded as being subject to an electric field due to its ionic nonmagnetic neighbors, and the theoretical problem became one of including this interaction in the Hamiltonian of the free ion, which already included a number of interactions of differing magnitudes and which had been treated by a sequence of approximations. An important consideration was that large perturbations needed to be treated before smaller ones, so it was necessary to estimate the magnitude of the crystal field perturbation, incorporate it in the right place in the perturbation sequence, and work out the consequences. It is not necessary to go into any details at this stage, but two remarks can be made. The first is that although the theory is an extension of the isolated ion theory, much of the detail of that theory is not needed, for magnetism is a property which, in the context of solids, is confined to a restricted thermal range. So in the theory the only energy levels of interest are those in a range which extends to not much more than kT above the ground level, where k is Boltzmann's constant and T is the absolute temperature. The other remark is that the theory appeared to be very successful.
In the late 1940s the technique of electron paramagnetic resonance (EPR) was introduced. In due course it was used to study the low-lying energy levels of isolated magnetic ions, impurities in an otherwise nonmagnetic host. (Nonmagnetic ions are ions with closed-shell structures. As they typically show a small temperature-independent diamagnetism they are not, strictly, nonmagnetic. Nevertheless, they are so described to distinguish them from the ions that have partially filled shells of electrons, which typically show much larger paramagnetic susceptibilities.) Until then, crystal field theory had been used to explain thermal properties, which invariably involved taking averages over energy level distributions and examining how they change under the influence of external perturbations, such as the application of a magnetic field. The spectroscopic nature of EPR gives energy differences, so thermal considerations enter only in the intensities of the resonance lines. EPR (which tends to be used interchangeably with ESR), therefore provided a much more sensitive test of the validity of crystal field theory.
The results are almost always expressed using the concept of spin-Hamiltonians, which is apt to give the impression that they depend on crystal field theory. In fact some of them, such as those for covalently bonded ions, have come from a theory which is much more like that used for molecules, and for a few others, particularly S-state ions, it is not obvious where they have come from.
Nevertheless, in the early days of EPR it seemed that it was the crystal field that provided the explanation of the observations, and even now the way of describing the underlying theory usually involves a reference to crystal fields, with, perhaps, a few remarks that modifications may be needed (e.g., delocalization of orbitals and/or covalency). It is, therefore, of interest to recall that at about the same time that crystal field theory was being actively developed, band theory was coming into prominence as the way in which to describe periodic systems. It would have been possible to claim that there was no conflict between the two because there was no common ground, for crystal field theory was being verified only for isolated magnetic ions in otherwise diamagnetic crystals. This would clearly have been unconvincing because there was no doubt that it was expected that it could be applied to fully concentrated, and therefore periodic, magnetic crystals (with a few additional interactions such as magnetic dipole and exchange interactions between the ions). The exponents of band theory believed that their way of proceeding would eventually lead to a better explanation of the resonance results, and that it was incorrect to use a model in which the electrons are allocated to specific sites rather than placed in delocalized orbitals that extend throughout the crystal. The weakness of this argument was the lack of evidence that it could come anywhere near crystal field theory in explaining the resonance results.
The second criticism was that the electrons on the magnetic ions were being distinguished from the electrons on the neighboring ions, which were simply regarded as producing a large part of the crystal field. A fundamental principle, that electrons are indistinguishable, was therefore being violated.
Of these two criticisms, it is now established that the first is not a flaw. The Hamiltonian of a periodic arrangement of ions is a complicated expression, and it is certainly an approximation to replace the detailed interactions between a large number of charges by an assumption that each electron moves in a periodic potential. Doing so inevitably leads to a description in terms of bands. But the approximation is not necessarily a good one. The second criticism is one that needs to be taken seriously, though it was not until the 1970s that a satisfactory way out of the difficulty was found.
An alternative to crystal field theory was proposed by Van Vleck (1935) at almost the same time that he was putting forward the concept of a crystal field. It was based on the idea of regarding the magnetic ion and its neighbors as a unit in which the electrons move in orbitals that extend over the whole complex. It is therefore rather similar to the picture used in band theory, the main difference being that the effective potential is not periodic. It has been extensively developed, and is usually referred to as ligand field theory. It runs into much the same difficulty as occurs in band theory, the step of replacing energies in a fictitious electrostatic field by the actual Coulomb interactions. There are a number of other theories, an example being the superposition model, which endeavors to relate the spin-Hamiltonian parameters for an ion with many neighbors to those that would be found for the same ion with just one neighbor. (As the purpose of this book is to describe a way in which the problems with these theories can be circumvented, there seems little need to describe them in detail here. They do, though, form an important part of the development of the understanding of electron spin resonance. It has therefore seemed only right to give the reader an entry into what is an extensive literature, through a selection of references which should provide an entry into the whole field. These can be found at the end of this chapter.)
The new method relies a good deal on symmetry arguments and perturbation theory, as does crystal field theory, so instead of going immediately to its description it has seemed better to introduce it through crystal field theory. It then becomes possible to make comparisons of the two methods which, it is hoped, will reveal the rather subtle differences between the two and show the logical advantages of the new method. As there are a number of theoretical points that form a common background to both theories, the rest of this chapter will be devoted to a discussion of these.
§1.3 The Choice of Hamiltonian
Until early in the twentieth century, dynamics was based entirely on Newtonian mechanics, and its inadequacies only became apparent when particles of atomic dimensions came to be studied. Two revolutionary changes then occurred, the discoveries of quantum mechanics and relativity. For most solid state purposes, relativistic considerations are of minor importance and most theoretical work therefore uses nonrelativistic quantum mechanics. This practice will be followed here. It produces many simplifications while still leaving a number of questions open, one being that of how to choose the generic Hamiltonian. A reader coming new to solid state theory might suppose that there is no such problem, because Hamiltonians are frequently written down with no justifications except, possibly, that they have been used in previous work. Turning to the elementary texts on quantum mechanics for guidance is unlikely to be much help, for they are usually concerned with the exact mathematical treatments of a relatively small number of model Hamiltonians, which have been obtained from classical mechanics, with no mention of their relevance to solid state physics.
From one point of view this is surprising, for quantum theory might be expected to be quite different from classical mechanics. But from the purely practical point of view one can see why it was done initially, for what other choice was there? Elements of this approach still remain, but so much has now been learnt that it is well established that while the main "classical energies of a solid are indeed present, as operators, there should also be a number of other operators, which have been shown to be present in the Hamiltonians of isolated atoms and ions. There is also the nonclassical requirement that the wave functions of many-electron systems must be antisymmetric with respect to interchanges of electrons.
It is therefore possible to begin writing down a generic Hamiltonian which should be adequate for all solids. Indeed, for many purposes it is probably sufficient to write down just the main terms, assuming the nuclei are fixed, which are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
to represent the kinetic energy of the electrons, i being a label that runs over all the electrons,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
which represents their Coulomb energy in the field of the nuclei, I being a label for nuclei, and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
which represents the energy associated with the mutual Coulomb repulsion between the electrons. If the nuclei are not fixed there should be extra terms to represent their kinetic energies and their mutual electrostatic energies.
Unfortunately, no exact eigenvalues, eigenstates, and solutions for the Schrödinger equation are known for any Hamiltonian of the above form for a system that contains more than two particles. This is probably why it is not usually stressed that the great diversity of properties shown by solids all result from what is basically the same Hamiltonian, it being necessary only to vary the number of nuclei, their charges, and the number of electrons.
The diversity is more commonly accounted for by the variety of approximations made in dealing with the generic Hamiltonian, which is one of the reasons why only the above energies have been given specifically, for there is usually enough difficulty in dealing with these without including any more, particularly as there is no virtue in making approximations unless progress can then be made. This is where the standard problems of the textbooks come in useful, for a common aim is to approximate so as to reduce the problem to one that is standard and soluble.
(Continues...)
Excerpted from Magnetic Ions in Crystals by K. W. H. Stevens. Copyright © 1997 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
Table of Contents
Preface Chapter 1 — Introduction Chapter 2 — Group Theory Chapter 3 — Perturbation Theory Chapter 4 — Crystal Field Theory Chapter 5 — Beyond Crystal Field Theory Chapter 6 — Second Quantization Chapter 7 — From Generic to Spin-Hamiltonian Chapter 8 — The Interactions between Ions Chapter 9 — Cooperative Systems Chapter 10 — Conductors Chapter 11 — Nuclear Symmetry Index