Introduction

In this chapter we develop a general formalism to describe the scattering of waves by a large system of particles and show that the scattering pattern relates directly to the structural arrangement of the particles. We develop this formalism using the specific example of light waves, composed of oscillating electromagnetic fields. But, in principle, the waves could represent any wave-like entity including matter waves such as traveling electrons or neutrons. The characteristic scattering pattern is known as the static structure factor, and it results from the collective interference of waves scattered by particles in the system. This interference is sensitive to the relative separation between the particles, and the static structure factor is shown to be just a spatial Fourier transform of the particle structure as it is represented by the density–density correlation function.

The dipole field

All condensed matter is constructed of atoms that contain nuclei and electrons. The nuclei reside at the atom center and the electrons, while bound up in the atom, orbit about the nucleus at a relatively large distance under the attraction of a Coulomb force. In considering the interaction of an atom with an external electric field, we know that both the electron and the nucleus experience opposing forces owing to their opposite charge. However, because neutrons and protons are about two thousand times more massive than the electron, we can largely disregard any disturbances in the location of the nucleus and instead focus on the motion of electrons alone.

Introduction

Up to now, we have considered only those inherent microscopic dynamics in a material that are present at equilibrium and are driven by the thermal energy content of the material itself. Here, in our last chapter dealing with dynamics, we consider instead the macroscopic, bulk dynamics of materials in non-equilibrium situations where an external force is applied or removed. Examples include the stretching or bending of a solid that results from application of a mechanical force, or the polarization of a dielectric material resulting when an external electric field is applied.

Several common features emerge in the response of a material to an external force or field. In all cases, there is some aspect of elasticity by which application of the force results in the storage of potential energy, that is returned when the force is removed. In all cases, this storage of energy is accompanied by some element of viscous drag or damping by which a portion of the work done during the deformation is lost in the form of heat. Like friction, this damping is a microscopic feature inherent in the thermodynamic fluctuations, and the energy lost during the deformation is returned to the same thermal bath from which it was derived. In fact, we will show that an important theorem exists, known as the fluctuation–dissipation theorem, which relates the macroscopic dissipation of energy in these bulk, non-equilibrium, processes directly to the inherent microscopic fluctuations present at equilibrium.

Purpose and motivation

This textbook was designed to accompany a one-semester, undergraduate course that itself is a hybridization of conventional solid state physics and “softer” condensed matter physics.

Why the hybridization? Conventional (crystalline) solid state physics has been pretty much understood since the 1960s at a time when non-crystalline physics was still a fledgling endeavour. Some 50 years later, many of the foundational themes in condensed matter (scaling, random walks, percolation) have now matured and I believe the time is ripe for both subjects to be taught as one. Moreover, for those of us teaching at smaller liberal arts institutions like my own, the merging of these two subjects into one, better accommodates a tight curriculum that is already heavily laden with required coursework outside the physics discipline.

Why the textbook? For some years now I have taught a one-semester course, originally listed as “solid state physics”, which evolved through each biannual reincarnation into a course that now incorporates many significant condensed matter themes, as well as the conventional solid state content. In past offerings of the course, a conventional solid state textbook was adopted (Kittel’s Introduction to Solid State Physics) and students were provided with handouts for the remaining material. This worked poorly. Invariably, the notation and style of the handouts clashed with that of the textbook and the disjointed presentation of the subject matter was not only annoying to students, but a source of unnecessary confusion. Students were left with the impression that solid state and condensed matter were two largely unrelated topics being crammed into a single course. Frustrated, I opted to spend a portion of a recent sabbatical assembling all of the material into a single document that would better convey the continuity of these two fields by threading both together into a seamless narrative.

Introduction

We often think of crystals as the gemstones we give to a loved one, but most metals (e.g. copper, aluminum, iron) that we encounter daily are common crystals too. In this chapter, we will examine the structure of crystalline matter in which particles are arranged in a repeating pattern that extends over very long distances. This long-range order is formally described by identifying small local groupings of particles, known as a basis set, that are identically affixed to the sites of a regularly repeating space lattice. As it happens, most crystals found in nature assume one of a limited set of special space lattices known as Bravais lattices. These lattices are special by virtue of their unique symmetry properties wherein only discrete translations and rotations allow the lattice to appear unchanged. Chief among these Bravais lattices are the cubic and hexagonal lattice structures that appear most frequently in nature. We focus extra attention on both to provide a useful introduction to coordination properties and packing fractions.

Crystal lattice

Crystals have a decided advantage because of the inherent repeating pattern present in their structure. In an ideal (perfect) crystal, this repeating pattern extends indefinitely. However, for real crystals found in nature, the pattern is often interrupted by imperfections known as defects that can include vacancies, in which a single particle is missing, and dislocations in which the repeating pattern is offset. These defects are important for some crystal properties, but for now we restrict ourselves to only ideal structures. Besides, even in real crystals large regions containing substantial numbers of particles exist in which a perfectly repeating pattern is maintained.

SCATTERING

Most of the light that enters our eyes has been scattered and when we see objects we see them because of the diffuse scattering of light they produce. Even the sky is blue because of how it scatters sunlight. But scattering is also an important mechanism for observing very small objects. As a classic example, recall how Lord Rutherford unveiled the internal structure of the atom by studying the scattering pattern of alpha particles directed at gold atoms. The abnormally large number of particles backscattered by these gold atoms pointed to the existence of a small, but very dense, center which we now refer to as the nucleus.

In the next chapter, we develop the basic framework for the scattering of waves by condensed matter by looking at how electromagnetic waves scatter from the electrons contained in the particles. Although this is strictly relevant only for the scattering of X-rays and visible light, much of the formalism that develops will apply equally to other waves, including particle waves (electrons or neutrons) that interact with things other than electrons. In the following chapter (Chapter 6), we look at how X-rays scatter from crystals. There we will find scattering that is reminiscent of how visible light is scattered by a diffraction grating in that the scattered radiation exits as a set of discrete beams. This discrete (Bragg) diffraction is contrasted in Chapter 7 by the continuous pattern of scattering produced by glasses or liquids.

Introduction

In the last chapter, we took a brash and somewhat unrealistic approach to treating the motion of electrons in a crystal. Although we know that the electron travels through a periodic potential caused by the regular arrangement of ion cores, we disregarded this “bumpy terrain” and considered instead only the barest consequences of the electron being trapped in the crystal “box” as a whole. In spite of its simplicity, this free electron model provided insightful explanations, not only for the origin of the small electronic contribution to specific heat and the temperature dependence of the electrical resistivity, but also for a host of emission phenomena, including the photoelectric effect.

However, the free electron model fails to provide any insight into additional questions regarding electrical conduction, such as (1) the anomaly of positive Hall coefficients that would imply positive charge carriers, and (2) the peculiar pattern of conductors, insulators and semiconductors that is found in the periodic table. In this chapter, we examine the nearly free electron model as a natural extension in which a weak, periodic potential is introduced. As a direct consequence of this addition, the continuum of electron energies in our free electron model now becomes separated into bands of allowed electron energy, separated by disallowed energy gaps. This separation of the electron energy into bands and gaps is key to understanding the division of materials into conductors, insulators and semiconductors, as well as providing a natural interpretation for the positive Hall coefficients.

Introduction

Percolation theory refers to properties of a simple experiment in which random events produce features common to second-order transitions; namely a continuously developing order parameter and self-similar, critical-like fluctuations. The model itself is quite simple, yet as we will see, it has been used extensively to interpret many phenomena found in nature, including not only the conditions under which liquids percolate through sand (from which the theory obtains its name), but also the manner in which stars form in spiral galaxies.

In this chapter, we investigate the percolation process in some rigorous detail to demonstrate how percolation clusters develop in a self-similar, power law manner near the percolation threshold. We also take this opportunity to introduce both the finite-sized scaling and renormalization techniques. Both of these techniques exploit the inherent self-similarity to gain insight into the critical exponents that characterize a second-order phase transition, and will prove useful to us in the next chapter.

The percolation scenario

At the heart of percolation theory is the question of how long-range connections develop through a random process. Consider a geometrical lattice of some arbitrary dimension such as the two-dimensional networks of pipes shown in the form of a square lattice in Fig. 16.1a. Here, the pipes are fully connected and fluid is free to flow from one edge of the network to the other. Suppose we now insert valves throughout this arrangement of pipes in one or the other of two ways. In the first instance, which corresponds to bond percolation, the valves are placed inside the pipes (i.e. inside the “bonds” between intersections), as shown in Fig. 16.1b. In the alternate case, referred to as site percolation, the valves are placed at the intersection of the pipes. Again, when all the valves are opened, the network is fully connected and fluid can flow readily from one side to the other. But, if all the valves are closed, the network is fully unconnected and fluid is unable to flow anywhere.

Introduction

In the last chapter, we investigated the dynamics of liquids whose particles are free to wander about due to the reasonably weak level of inter-particle bonding. In a solid (crystal or glass), bonding between particles is stronger and the translational motion of the particles is arrested. Nevertheless, these “solid” particles continue to move and execute small, localized vibrations about a fixed point in space. In this chapter and the next, we investigate the nature of this vibrational motion and its impact on the thermal properties of a solid. Here we begin by considering a simple model of masses connected by ideal springs to demonstrate how vibrations of individual atoms are, in reality, a consequence of propagating waves traveling through the crystal lattice. In order to connect these waves with the quantum mechanical perspective of each atom behaving as a quantized harmonic oscillator, we find ourselves introducing the concept of a quantum of elastic wave, known as a phonon.

An important outcome of our development of quantized elastic waves is a growing appreciation for a special region of reciprocal space known as the Brillouin zone, which is populated by all the wave vectors, K, corresponding to allowed phonon waves in the crystal. For phonons whose K matches the edge of this zone, significant Bragg scattering results, to produce two equivalent standing wave patterns separated by an energy gap. We will revisit the Brillouin zone often in the chapters to come, and we will begin to appreciate the significance of this boundary for the motion of all waves that attempt to travel within a crystal.

Introduction

We have now witnessed the similar patterns associated with second-order phase transitions in both fluid and magnetic systems. These patterns include laws of corresponding states and similarity in critical exponents, that govern how properties evolve near the transition point. Furthermore, the Landau theory provides a framework for understanding the commonality of these second-order phase transitions, in terms of similarity in the functional dependence of the free energy on an appropriately chosen order parameter, and a simple expansion that can be performed near the critical point. In this chapter, we examine yet another significant phase transition found in condensed matter: the transition of a material to a state of virtually infinite conductivity or superconductivity. Here the transition involves the sticking together of two electrons into a boson-like, superconducting charge carrier known as a Cooper pair, and we again find evidence of a second-order transition consistent with mean field theory.

Superconducting phenomena

Discovery

In 1908, H. K. Onnes perfected the technique for cooling helium gas to its condensation point and soon after began using this new technology to investigate the properties of various elements at ultra low temperatures. In one instance, Onnes was curious about the ultimate demise of the resistivity of an electronic conductor. As we saw in Chapter 12, the resistivity of most conductors decreases linearly with temperature at high temperatures, due to the scattering of electrons by lattice phonons, but approaches a limiting value at low temperatures, associated with a mean free path determined by macroscopic imperfections of the crystal lattice. In a series of studies, Onnes measured the resistance of gold and platinum and observed an approach to a limiting resistance at low temperatures. In an effort to eliminate the effects of imperfections, he extended the study in 1911 to include mercury, which at that time could be refined to a highly pure form. The results of this study, shown in Fig. 18.1, are quite dramatic. A roughly linear temperature dependence was observed above about 4.2 K, which decreased abruptly to an immeasurably small resistance at lower temperatures. On reheating, the resistance was identically retraced, and Onnes concluded that mercury had undergone a unique phase transition to a new state characterized by virtually zero resistance – a “superconducting” phase.