In the early 1970's, attention was drawn to the remarkable similarity between the excitation spectra exhibited by S(Q, ω) in solid 4He and superfluid 4He at low temperatures (Werthamer, 1972; Horner, 1972a; Glyde, 1974), as shown very dramatically in the theoretical results of Figs. 11.1 and 11.2. While various suggestions have been made as to the origin of this similarity, it remains an unresolved and intriguing problem. In this brief chapter, we compare the theoretical description of excitations in a quantum solid with those of a Bose-condensed liquid. While we review the key ideas, we assume that the reader has some familiarity with an introductory account of quantum crystals. (The modern theory of excitations in quantum crystals was essentially completed in the early 1970's. For background and a more detailed discussion of solid 4He than we give in this chapter, we recommend the review by Glyde, 1976.)

In both condensed phases, it is important to distinguish clearly between the elementary excitations and the density fluctuations. We argue that the phonons in solid 4He are the natural analogue of the single-particle excitations in liquid 4He. In Section 11.1, defining the phonons as the poles of the displacement correlation function, we briefly review theories which start with the self-consistent harmonic (SCH) approximation or something similar. In Section 11.2, we discuss the relation between the displacement–displacement and the density–density correlation functions in solid 4He.

The well known Landau theory of the low-temperature properties of superfluid 4He starts from a weakly interacting gas of phonons and rotons. This theory is very successful but it is essentially phenomenological since it makes no reference to the Bose condensate. The core of this book is a discussion of the modern microscopic theory of Bose-condensed systems based on finite-temperature Green's function techniques (the dielectric formalism). My emphasis is on developing the language and concepts of this formalism in a way that brings out the essential physics. This book is the first general account of the progress made in the last two decades toward understanding the excitations in superfluid 4He specifically within the framework of a Bose-condensed liquid. I hope it will be a guide and stimulus to a new generation of experimentalists and theorists studying superfluid 4He. The book should also be of interest to a much wider audience, since the phenomenon of Bose condensation, with its associated macroscopic quantum effects, plays a central role in modern condensed matter physics (Anderson, 1984).

The goal of this book is two-fold: (a) to summarize the field-theoretic analysis of a Bose-condensed fluid and (b) to use this formalism to understand the nature of the excitations in superfluid 4He. I emphasize how a Bose broken symmetry inevitably leads to certain characteristic features in the structure of various correlation functions, the most spectacular being the phenomenon of superfluidity. A major theme is the way in which a Bose condensate mixes the elementary excitation and density fluctuation spectra.

In this chapter, we review the high-resolution neutron-scattering data for the dynamic structure factor S(Q, ω) and suggest an interpretation within a unified picture of the excitations in liquid 4He consistent with the ideas of Chapter 5. We argue that the phonons (0.1 ≲ Q ≲ 0.7 Å–1) in the collisionless region and rotons (Q ∼ 1.9 Å–1) are really two separate branches of the density fluctuation spectrum in the superfluid phase which are hybridized by the condensate. The low-wavevector phonon is interpreted as a zero sound collective density fluctuation while the large-Q maxon–roton is interpreted as a strongly renormalized single-particle excitation. In the intermediate-wavevector region 0.8 ≲ Q ≲ 1.2 Å–1, we argue that there is evidence that both excitation branches, a sharp single-particle (or atomic-like) maxon excitation and a broad high-energy zero sound phonon, are observed in S(Q, ω). Within this scenario, the appearance of the sharp maxon-roton resonance (0.8 ≲ Q ≲ 2.4 Å–1) in S(Q, ω) below the superfluid transition temperature Tλ is direct dynamical evidence for the Bose broken symmetry and the associated Bose condensate in superfluid 4He.

In Section 7.1, we review the neutron-scattering intensity data for small, intermediate and large wavevectors. In Section 7.2, these results are interpreted starting from the assumption that superfluid He is a Bose-condensed liquid. The condensate inevitably leads to a mixing of the single-particle spectrum described by Gαβ(Q, ω) and the density fluctuation spectrum described by S(Q, ω).

At many points in this book, we have mentioned the high-frequency scattering intensity which appears in the S(Q, ω) data. This high-frequency component (see Fig. 1.6) is usually identified with the spectrum of two excitations (with total momentum Q) and is thus referred to as the multiphonon or multiparticle component. In addition to inelastic neutron scattering, this two-excitation spectrum can be more directly probed by inelastic Raman light scattering, but only at Q = 0. In this chapter, we briefly review the microscopic theory of such pair excitations and how they show up in both neutron and Raman scattering cross-sections.

Raman light scattering in superfluid 4He has been extensively studied both theoretically and experimentally, especially with regard to the possible formation of bound states involving roton–roton, roton–maxon and maxon–maxon pairs (Ruvalds and Zawadowski, 1970; Iwamoto, 1970). High-resolution Raman experiments over a wide range of pressure and temperature are reviewed by Greytak (1978) and more recently by Ohbayashi (1989, 1991). An excellent theoretical introduction at a phenomenological level is given by Stephen (1976). Our emphasis will be on the role of the Bose broken symmetry.

In earlier chapters, we have carefully distinguished the single-particle Green's function G1(Q, ω) (which may be a 2 × 2 matrix) and the density-response function χnn(Q, ω). The latter gives the dynamic structure factor measured by neutron scattering. In the present chapter, we introduce several additional correlation functions which are needed to describe the pair-excitation spectrum and Raman scattering.

The slab model

The simplest way to model the behaviour of a composite containing continuous, aligned fibres is to treat it as if it were composed of two slabs bonded together, one of the matrix and the other of the reinforcement, with the relative thickness of the latter in proportion to the volume fraction of the fibres (designated as f). The response of this ‘composite slab’ to external loads can be predicted quite easily, but its behaviour will closely mirror that of the real composite only under certain conditions (Fig. 2.1).

Over the last 30 years or so, metal matrix composites have emerged as an important class of materials. During this period, a very substantial research effort has been directed towards an improved understanding of their potential and limitations, invoking principles of physical metallurgy, stress analysis and processing science. This book is intended as an introduction to the field, covering various aspects of the structure, behaviour and usage of these materials. It is designed primarily for scientists and technologists, but the content is also suitable for final year degree course students of materials science or engineering and for postgraduate students in these disciplines.

The structure of the book is designed to allow several different modes of usage. Chapters 2 and 3 provide a background to stress analysis techniques used to describe the mechanical behaviour of MMCs. In these chapters we have aimed to introduce the concepts pictorially, while the details are discussed in the main text. The finer points of these treatments are relevant to those with a keen interest in composite mechanics, but they are not essential for use of the rest of the book. The following four chapters then form a core description of the load-bearing behaviour. Chapters 4 and 5 cover the basic deformation mechanisms and characteristics, over a range of temperature. A chapter is then devoted to various aspects of the interface between matrix and reinforcement. This is relevant to several areas, particularly the fracture behaviour outlined in Chapter 7.

Engineering properties of MMCs

Stiffness enhancement

Potential for the enhancement of stiffness, and specific stiffness, is one of the most attractive features of MMCs. Stiffness is a critical design parameter for many engineering components, as the avoidance of excessive elastic deflection in service is commonly the overriding consideration, and the incentive to achieve even a modest increase is often very high indeed. This is the case for many rotating parts, support members, structural bodywork, etc., for which metals offer essential combinations of toughness, formability, environmental stability and strength. However, with very few exceptions (Al–Li being the prime example), there is no scope for increasing the stiffness of a metal by minor additives or microstructural control.

The significance of interfacial bond strength

In the previous two chapters it has become clear that many important phenomena can take place at the matrix/reinforcement interface. For polymer-based composites, although the chemistry involved may be complex, the objectives in terms of interfacial properties are often the rather straightforward ones of a high bond strength (to transfer load efficiently to the fibres) and a good resistance to environmental attack. In designing ceramic composites, on the other hand, the aim is usually to make the interface very weak, as the prime concern is in promoting energy dissipation at the interface so as to raise the toughness. For MMCs, a strong bond is usually desirable, but there may be instances where inelastic processes at the interface can be beneficial.

Thermal and electrical conduction

There are many applications in which the high electrical and/or thermal conductivities of metals are exploited. A problem commonly arises in situations where this needs to be combined with good mechanical properties, in that conventional strengthening by alloying leads to sharp reductions in these conductivities.

A considerable body of mechanical test data for discontinuously reinforced MMCs is now available, although some of these results have been obtained with rather poor quality material. However, study of data such as those for Al/SiC summarised in Table 4.1, reveals some systematic trends:

• the incorporation of reinforcement improves both yield stress (0.2% proof stress) and ultimate tensile stress (UTS)

• whiskers provide more effective reinforcement than particles

• yield stress rises with increasing volume fraction; UTS is not always similarly affected

• for whisker-reinforced composites, increases in yield strength are often much greater in compression than in tension

• for whisker-reinforced composites, increases in tensile yield strength are greater transverse to the whisker alignment than parallel to it

• there is a wide scatter in experimental results

Dislocation structure and behaviour

Plastic deformation in the matrix of a composite is never completely homogeneous. The reinforcement interrupts flow, giving rise to distinctive dislocation structures. As well as having immediate implications for the flow properties of the matrix, these structures also indirectly influence flow behaviour via changes in the precipitation and aging response.

The influence of thermal stress on dislocation structure

In many MMCs, thermal stresses can give rise to dislocation densities which are 10–100 times greater than those for comparable unreinforced alloys. The conditions under which the creation of thermally stimulated dislocations is energetically favourable were first studied by Ashby and Johnson, who found that, for incoherent particles, the critical misfit decreases with increasing particle size. For short fibres and particles, the simplest mechanism of stress relief is to punch out a dislocation loop (a disc of vacancies or interstitials) into the matrix (Fig. 10.1(a,b)). This mechanism, illustrated schematically in Fig. 4.16, is well understood in terms of a relaxation in the local stress field.

Types of MMC and general microstructural features

The term metal matrix composite (MMC) encompasses a wide range of scales and microstructures. Common to them all is a contiguous metallic matrix. The reinforcing constituent is normally a ceramic, although occasionally a refractory metal is preferred. The composite microstructures may be subdivided, as depicted in Fig. 1.1, according to whether the reinforcement is in the form of continuous fibres, short fibres or particles. Further distinctions may be drawn on the basis of fibre diameter and orientation distribution. Before looking at particular systems in detail, it is helpful to identify issues relating to the microstructure of the final product. A simplified overview is given in Table 1.1 of the implications for composite performance of the main microstructural features. Whereas some of these microstructural parameters are readily pre-specified, others can be very difficult to control.

Thermal stresses and strains

Differential thermal contraction stresses

As is displayed in Fig. 5.1, metals generally have larger thermal expansion coefficients (α) than ceramics. Since fabrication of MMCs almost inevitably involves consolidation at a relatively high temperature, it is not surprising that they often contain significant differential thermal contraction stresses at ambient temperatures (e.g. see Fig. 5.17). Assuming the material to be effectively stress-free at some (high) temperature, Tesf, the stress state at a lower temperature can be envisaged as arising from the fitting of an oversized inclusion into an undersized hole in the matrix. The misfit strain is then simply Δα ΔT, where ΔT = Tesf - T0, the ambient temperature.