Introduction

Granular materials are ubiquitous throughout nature. From the beauty of sand dunes and the rings of Saturn to the destructive power of snow avalanches and mudslides, the flow of ice floes, and the manner in which plate tectonics determine much of the morphology of the Earth [1–4]. These phenomena arise from the interplay between structural and dynamical properties that result in the collective behavior of a vast number of smaller, distinct entities that we call “grains.” From a technological point of view, granular materials play a dominant role in numerous industries, such as mining, agriculture, civil engineering, pharmaceuticals manufacturing, and ceramic component design. Even apparently the most mundane of activities from coffee bean bag filling at the grocery store to pouring salt onto our dinner plates at night involve various aspects of granular matter mechanics that continue to puzzle us. It is estimated that particulate media are second only to water as the most manipulated material for human usage [4], amounting to trillions of dollars per annum in the US alone. The importance of granular materials to our daily lives cannot be overstated.

In the previous chapter we have considered the properties of strongly correlated electrons. The systems we mostly had in mind were the compounds of transition metal and maybe rare earth elements with partially filled inner d or f shells. We have discussed only the correlated electrons themselves, the prototype model being the Hubbard model (12.1).

When turning to real materials, several extra factors missing in the model (12.1) are important. One of them is the possible influence of orbital degrees of freedom, especially in cases with orbital degeneracy, treated in Section 12.9.

In many situations there is yet another very important factor. There may exist in a system, besides correlated electrons, also electrons of other bands, e.g. electrons in wide conduction bands, responsible for ordinary metallic conductivity. Such is for instance the situation for magnetic impurities in metals, or in the concentrated systems like rare earth metals and compounds in which localized f electrons coexist with the metallic electrons in broad spd bands. The interplay between localized, or, better, strongly correlated electrons and itinerant electrons of the wide bands can lead to a number of very interesting consequences; these will be discussed in this chapter.

Localized magnetic moments in metals

When we put transition metal impurities in ordinary metals (e.g. Mn or Fe in Cu, Au), the result may be two-fold. In certain cases the impurities retain their magnetic moment, but in others they lose it.

Foreword

There are many good books describing the foundations and basics of solid state physics, such as Introduction to Solid State Physics by C. Kittel (2004) or, on a somewhat higher level, Solid State Physics by N. W. Ashcroft and N. D. Mermin (1976). However there is a definite lack of books of a more advanced level which would describe the modern problems of solid state physics (including some theoretical methods) on a level accessible for an average graduate student or a young research worker, including experimentalists.

Usually there exists a rather wide gap between such books written for theoreticians and those for a wider audience. As a result many notions which are widely used nowadays andwhich determine ‘the face’ of modern solid state physics remain ‘hidden’ and are not even mentioned in the available literature for non-specialists.

The aim of the present book is to try to fill this gap by describing the basic notions of present-day condensed matter physics in a way understandable for an average physicist who is going to specialize in both experimental and theoretical solid state physics, and more generally for everyone who is going to be introduced to the exciting world of modern condensed matter physics – a subject very much alive and constantly producing new surprises.

In writing this book I tried to follow a unifying concept throughout. This concept, which is explained in more detail below, may be formulated as the connection between an order in a system and elementary excitations in it.

The state of different condensed matter systems is characrerized by different quantities: density, symmetry of a crystal, magnetization, electric polarization, etc. Many such states can have a certain ordering. Different types of ordering can be characterized by order parameters.

Examples of order parameters are, for instance: for ferromagnets – the magnetization M; for ferroelectrics – the polarization P; for structural phase transitions – the distortion uαβ, etc. Typically the system is disordered at high temperatures, and certain types of ordering may appear with decreasing temperature. This is clear already from the general expressions for thermodynamic functions, see Chapter 1: at finite temperatures the state of the system is chosen by the condition of the minimum of the corresponding thermodynamic potential, the Helmholtz free energy (1.8) or the Gibbs free energy (1.10), and from those expressions it is clear that with increasing temperature it is favourable to have the highest entropy possible, i.e. a disordered state. But some types of ordering are usually established at lower temperatures, where the entropy does not play such an important role, and the minimum of the energy is reached by establishing that ordering.

The general order parameter η depends on temperature, and in principle also on other external parameters – pressure, magnetic field, etc. Typical cases of the dependence of the order parameter on temperature are shown in Fig. 2.1.

Introduction

The modeling and simulation of granular materials is important to our understanding of their behavior and the wealth of phenomena they exhibit [1–3]. Many phenomena and practical applications, such as the design of industrial processes, remain out of reach of traditional simulation methods due to the large numbers of grains involved. A liter of fine sand, mgrain ≈ 0.1 mg, may contain 107 grains. An industrial process such as mixing or hopper flow can easily involve 101 to 103 liters. Geophysical processes, such as sand dunes and earthquake faults, involve even larger numbers of grains. Traditional simulation techniques are currently unable to deal with such large numbers of grains; however, cellular automata models are able to simulate larger numbers of grains for longer times and show promise in the simulation of large, real-world granular flows.

Molecular dynamics, where Newton's laws are applied to individual grains and the resulting motion is determined from the forces, has made dramatic progress in the past two decades through improved techniques and more powerful computers. Molecular dynamics is particularly effective for gases of simple grains interacting through hard-sphere collisions. In principle, even the physics of complex grain interactions can be included. However, as the complexity of the interactions increases or the number of grains increases, the computational demands increase as well. As a result, molecular dynamics is primarily used for relatively small numbers of grains (102–105), whereas real granular flows often contain many more.

As we have seen in the previous chapters, interactions, if they are not too strong, preserve many features of the electronic system which are present in the noninteracting case (Fermi gas, Chapter 7). In general, however, the interactions are not at all small: for instance in typical metals, rs ∼ 2–3, and not rs « 1 as was implicitly assumed in Chapter 9 and which was actually the condition for the applicability of perturbation theory used there. Nevertheless we know that the description of normal metals using the concepts developed for the free Fermi gas or Fermi systems with weak interactions (such asDrude theory, for example) is very successful.

An explanation of the success of the conventional theory of metals, and the generalization of the corresponding description to a more general situation, was given by Landau in his theory of Fermi liquids. This theory is very important conceptually, although in the usual metals there are only few special effects which indeed require this treatment. However, there exist also systems (3He, or rare earth systems with mixed valence and heavy fermions) for which this approach is really vital. Also the emerging new field of non-Fermi-liquid metallic systems requires first an understanding of what is the normal Fermi liquid.

The foundations of the Fermi-liquid theory

Themain assumptions of the Landau Fermi liquid theory are completely in linewith our general approach.