Magnons (or spin waves) are the low-lying excitations that occur in ordered magnetic materials. The concept of spin waves was introduced by Bloch (1930), who envisaged some of the spins as deviating slightly from their ground-state alignment and this disturbance propagating with a wave-like behaviour through the solid. Because the spins are properly described by quantum-mechanical operators, the spin waves are also quantised with the basic quantum being referred to as the magnon (by analogy with the phonon for quantised lattice vibrations). Magnons can be studied through their contribution to thermodynamic properties (e.g. specific heat) or more directly by techniques such as light scattering, neutron scattering and magnetic resonance.

We begin this chapter by giving an introductory account of the theoretical models required to describe magnons in simple ferromagnets and antiferromagnets. Using the Heisenberg model we then present the theory of bulk magnons in infinite ferromagnets, as a preliminary to generalising the theory to semi-infinite ferromagnets and to ferromagnetic films. In the last two cases we show that surface magnons may occur localised near the surface(s) and that bulk-magnon properties are modified. After extending the theory to bulk magnons and surface magnons in Heisenberg antiferromagnets, we give a review of experimental results for surface magnons in Heisenberg magnetic materials.

Preamble

In this chapter we shall describe the application of some of the ideas presented in Chapters 1 and 2 to the dilution refrigerator. This device has been of crucial importance for the development of low temperature physics since the mid-sixties when it was first demonstrated (Hall et al. (1966)) to be a practical proposition. Several sources have provided the material for this chapter, and may be consulted by the reader wishing to pursue the matter in great depth. These sources importantly include two research monographs by Lounasmaa (1974) and by Betts (1976), both of which need to be updated, a much-used conference review article by Frossati (1978) and its sequel by Vermeulen and Frossati (1987), and a privately circulated manual by Sagan (1981). There are also useful articles by Wheatley et al (1968) and (1971), Niinikoski (1976), Frossati et al (1977), Lounasmaa (1979) and Bradley et al (1982). It would I think be generally acknowledged that Frossati is the master in this field, and his article is the main inspiration for this chapter. The aim is briefly to cover the functions of the various components, and to offer a guide to design considerations and to the sort of performance which can be achieved at present. In the early days dilution refrigerators were homemade but most users now buy them commercially in much the same way as consumers buy domestic refrigerators, though at considerably greater expense. The main supplier at present is the Oxford Instrument Company based in the UK.

Evaporation cooling

It is useful to begin by having in mind a simple image of an evaporation cooler as shown in Figure 3.1.

Preamble

The dilution refrigerator, to be dealt with in Chapter 3, can only be fully understood in terms of the properties of the two helium isotopes 3He and 4He and their mixtures. The purpose of this chapter is very briefly to review the relevant properties indicating, where necessary, sources giving more details. A useful general reference is by Wilks and Betts (1987) which also serves as an introduction to the research literature. As in the whole of this book, the reader is expected to derive at least as much information from the figures as from the minimal text.

Phase diagrams

The phase diagrams of 4He, 3He, and their mixtures are given below as Figures 2.1 and 2.2 respectively. Particular points of interest are discussed in the rubric and text following each figure.

Liquid 4He can be solidified by pressures above about 25 bar at temperatures below about 1 K. Higher pressures are needed at higher temperatures but below 1 K the melting pressure is nearly constant. The liquid phase is superfluid when the pressure is below the melting pressure and when in addition the temperature is below the lambda line (labelled in the figure) which runs from 2.17 K at p = 0 bar to 1.76 K at p = pmelt ≍ 30 bar. The superfluid phase is called He-II to distinguish it from normal He-I.

Liquid 3He has a pronounced minimum in its melting pressure of pmelt ≍ 29 bar at T ≍ 0.32 K and at lower temperatures pmelt rises to about 34 bar as T-→0 K. Normal (that is, non superfluid) liquid 3He persists down to a few millikelvin where it is well-described as a Fermi liquid.

The origin of this volume was an invitation I received from Dr Marek Finger of the Charles University, Prague, and the Joint Institute for Nuclear Research in Dubna, Russia, to give four lectures on low temperature methods at an international summer school on hyperfine interactions and physics with oriented nuclei organised at a chateau in Bechyně in the Czechoslovakian countryside, in September 1985. The topic of the summer school was something I knew little about, but low temperature physics is my metier and the preparation of the lectures was frankly not a large task, particularly in view of the fact that I was already the author of Refrigeration and thermometry below one kelvin (Sussex University Press, 1976). I decided to use a minimum of prose, produced in the usual garish colours, together with a large number of diagrams from various sources converted into transparencies. I would depend on my knowledge of the subject matter to talk through the transparencies in an unscripted way. It took me four days to think through the content and prepare the material. All the lectures were given on 3 September 1985. The organising committee originally had no intention of publishing proceedings but many participants expressed their desire to have the lectures and contributions presented in written form. My heart sank at the thought of converting my bundle of transparencies into something which could fairly be described as a camera-ready manuscript, but I agreed to try. It was like trying to turn a movie into a novella. I worked spasmodically on it, missing all of a series of extended deadlines until eventually the editors gave up on me and the proceedings appeared without my contribution.

Preamble

The theoretical bases of thermometry are discussed in Chapter 1. It is rare for experimentalists to use methods which relate directly to a Carnot cycle, though there are certainly users of secondary methods which have respectable theoretical support of another sort (e.g. the Boltzmann factor in 60Co y-ray anisotropy). Some properties of some materials have useful dependences on temperature in the desired range (e.g. the electrical resistance of carbon composites) but are best described by polynomial fits rather than fundamental theory. The fact is that there are many ways of measuring temperatures in the millikelvin range and experimentalists make choices which best fit their needs. The extents to which these techniques give temperatures which are accurate (i.e. equal to the true Kelvin temperatures) or consistent when compared with temperatures determined in other ways are often not fully certain. Users frequently rest their cases on widely accepted results obtained and published by respected workers in the past (e.g. for the melting pressure of 3He (Greywall (1985)), or they take strength from demonstrations by others that two methods do in fact agree within acceptable limits. Examples of comparisons include platinum NMR versus nuclear orientation (Berglund, Collan et al. (1972)), platinum NMR versus CLMN susceptibility (Alvesalo et al (1980)), noise versus 60Co y-ray anisotropy (Soulen and Marshak (1980)), and 3He melting pressure versus CLMN susceptibility (Parpia et al. (1985)). Disputes can and do arise about whether small temperature-dependent effects are truly properties of the system being studied or whether they are artifacts of an imperfect scale. A recent example of this has been the debate about whether the low temperature heat capacity of liquid 3He does or does not contain a term proportional to T3 ln T.

Diffusion patterns and single crystal growth rates

Gels are obviously permeable, but the fact that convection currents are suppressed, can easily be demonstrated. With an ordinary microscope it is possible to verify that particles have streaming motion in the ungelled solution but none after gelling. With a laser-ultramicroscope arrangement of the kind described by Vand et al (1966), this demonstration can be extended to smaller particles, e.g. down to about 600 Å and even below, depending on the wavelength and intensity of the laser light. Such tests do not rule out the possibility of convection currents on a submicroscopic scale, but it is implausible to believe that these play any major role.

In the absence of convection, the only mechanism available for the supply of solute to the growing crystal or a Liesegang Ring is diffusion. The complete solute diffusion pattern can evidently be very complicated, and attempts to analyze it commit us to several layers of simplification. The choices we make in this must depend, in turn, on the nature of the situation envisaged. One such situation might involve prominent Liesegang Rings, and that will be discussed in Chapter 5. Another might be represented by a small crystal, growing far from anywhere (and, in particular, far from any other crystal) in a large amount of gel. One might then consider that the solute super-saturation ϕ∞ at large (lateral) distances from the crystal remains unchanged during growth. This would be so, if the total amount of matter in the crystal were small compared with that in the gel, an assumption which is specially appropriate for the initial stages of crystal growth.

Introduction

It has long been appreciated that advances in solid state science depend critically on the availability of single crystal specimens. As a result, an enormous amount of labor and care has been lavished on the development of growth techniques. In terms of crystal size, purity, and perfection, the achievements of the modern crystal grower are remarkable indeed, and vast sections of industry now depend on his products. So do the research workers whose preoccupation is with new materials, no matter whether these are under investigation for practical reasons or because a knowledge of their properties might throw new light on our understanding of solids in general.

In one way or another, a very large number of materials has already been grown as single crystals, some with relative ease and others only after long and painstaking research. Nevertheless, there are still many substances which have defied the whole array of modern techniques and which, accordingly, have never been seen in single crystal form. Others, though grown by conventional methods, have never been obtained in the required size or degree of perfection. All these constitute a challenge and an opportunity, not only for the professional crystal grower but, as it happens, also for the talented amateur. New and unusual methods of growing crystals are therefore of wide interest; and if the crystals are by themselves beautiful, as they often are, there is no reason why this interest should be confined to professional scientists.

Crystal growth in gels has been a subject of intense interest for many years, but the process has remained a laboratory research activity. Certainly, it has not (or, shall we delicately say, not yet) given rise to a new industry, but it is nevertheless recognized as useful. In contrast, it would until quite recently have been necessary to admit that Liesegang Ring formation, for all its aesthetic merits, has yet to find its practical niche. One would have had to argue (as, indeed, one still does) that the study of this beautiful phenomenon is instructive in a general way, and it is its own justification. However, a new possibility has emerged which changes this position, namely that of designing anisotropic ceramics with Liesegang microstructures. This is a potentially potent technology but, of course, outside the immediate context of ‘crystal growth in gels.’

R. E. Liesegang

A tribute, first, to the discoverer who gave this set of phenomena his name. Somehow the name Liesegang has an ancient ring, and it is tempting to believe that its owner was one of the earliest chemical pioneers. In fact, Raphael Eduard Liesegang (born 1869) was as much a physicist as he was a chemist, though it was never easy to classify him into any professional niche. He lived until 1947, and did many things in his lifetime that would make a modern scientist recoil in awe. Thus, for instance, he wrote one of the earliest papers on the possibility of television (in 1891, and before the discovery of electrons), was active as a bacteriologist, contributed to chromosome theory and to the beginnings of paper chromatography, not forgetting the properties of aerosols and gelatins, the origins of silicosis, the mechanism of the photographic process in black-and-white and color (1889).