Much of what we observe in nature is either time- or frequency-dependent. In this chapter, we will introduce language to describe time- and frequency-dependent phenomena in condensed matter systems near thermal equilibrium. We will focus on dynamic correlations and on linear response to time-dependent external fields that are described by time-dependent generalizations of correlation functions and susceptibilities introduced in Chapters 2 and 3. These functions, whose definitions are detailed in Sec. 7.1, contain information about the nature of dynamical modes. To understand how and why, we will consider linear response in damped harmonic oscillators in Sees. 7.2 and 7.3, and in systems whose dynamics are controlled by diffusion in Sec. 7.4. These examples show that complex poles in a complex, frequency-dependent response function determine the frequency and damping of system modes. Furthermore, the imaginary part of this response function is a measure of the rate of dissipation of energy of external forces.

A knowledge of phenomenological equations of motion in the presence of external forces is sufficient to determine dynamical response functions. The calculation of dynamical correlation functions in dissipative systems requires either a detailed treatment of many degrees of freedom or some phenomenological model for how thermal equilibrium is approached. In Sec. 7.5, we follow the latter approach and introduce Langevin theory, in which thermal equilibrium is maintained by interactions with random forces with well prescribed statistical properties. Frequency-dependent correlation functions for a diffusing particle and a damped harmonic oscillator are proportional to the imaginary part of a response function.

In the preceding several chapters, we have seen that the order established below a phase transition breaks the symmetry of the disordered phase. In many cases, the broken symmetry is continuous. For example, the vector order parameter m of the ferromagnetic phase breaks the continuous rotational symmetry of the paramagnetic phase, the tensor order parameter Qij of the nematic phase breaks the rotational symmetry of the isotropic fluid phase, and the set of complex order parameters ρG of the solid phase breaks the translational symmetry of the isotropic liquid. In these cases, there are an infinite number of equivalent ordered phases that can be transformed one into the other by changing a continuous variable θ. If rotational symmetry is broken, θ specifies the angle (or angles) giving the direction of the order parameter; if translational symmetry is broken, θ specifies the origin of a coordinate system. Uniform changes in θ do not change the free energy. Spatially non-uniform changes in θ, however, do. In the absence of evidence to the contrary, one expects the free energy density f to have an analytic expansion in gradients of θ. Thus we expect a term in f that is proportional to (∇θ)2 for θ varying slowly in space. We refer to this as the elastic free energy, fel, since it produces a restoring force against distortion, and we will refer to θ as an elastic or hydrodynamic variable.

In Chapter 9, we studied topological defects in ordered systems with a broken continuous symmetry. In this chapter, we will study fundamental defects in systems with discrete symmetry such as the Ising model. These defects are surfaces, of one dimension less than the dimension of space, that separate regions of equal free energy but with different values of the order parameter. They are variously called domain walls, kinks, solitons, discommensurations, or simply walls, depending on the particular system and context. They can also be regarded as interfaces, such as, for example, the interface separating coexisting liquid and gas phases. They play an important, if not dominant, role in determining the physical properties of systems with discrete symmetry.

We begin this chapter (Sec. 10.1) with a number of examples of walls. In Sec. 10.2, we study the continuum mean-field theory for kinks and solitons. Then, in Sec. 10.3, we discuss in some detail the Frenkel-Kontorowa model for atoms adsorbed on a periodic substrate. This will introduce in a natural way a lattice of interacting kinks (called discommensurations in this case) to describe the incommensurate phase of adsorbed monolayers. After investigating the properties of interacting kinks at zero temperature, we will in Sec. 10.4 study thermal fluctuations of walls in dimensions greater than one and show that structureless walls in three dimensions or less have divergent height fluctuations that render them macroscopically rough.

Mean-field theory, presented in the preceding chapter, correctly describes the qualitative features of most phase transitions and, in some cases, the quantitative features. Since mean-field theory replaces the actual configurations of the local variables (e.g. spins) by their average value, it neglects the effects of fluctuations about this mean. These fluctuations may or may not be important. The more spins that interact with a particular test spin, the more the test spin sees an effective average or mean field. If the test spin interacts with two neighbors, the averaging is minimal and the fluctuations are large and important. The number of spins producing the effective field increases with the range of the interaction and with the dimension. Thus we will find that mean-field theory is a good approximation in high dimensions but fails to provide a quantitatively correct description of second-order critical points in low dimensions. This chapter will be devoted to the study of second-order phase transitions when mean-field theory is not a good approximation.

We will begin by considering fluctuations of the order parameter about its spatially uniform mean-field value. We will find that for spatial dimensions below an upper critical dimension dc (typically dc = 4), fluctuations always become important and invalidate mean-field theory for temperatures sufficiently close to Tc. This will motivate a generalization of Landau theory that incorporates spatial fluctuations about a mean-field free energy minimum.

Thermodynamics provides a description of the equilibrium states of systems with many degrees of freedom. It focuses on a small number of macroscopic degrees of freedom, such as internal energy, temperature, number density, or magnetization, needed to characterize a homogeneous equilibrium state. In systems with a broken continuous symmetry, thermodynamics can be extended to include slowly varying elastic degrees of freedom and to provide descriptions of spatially nonuniform states produced by boundary conditions or external fields. Since the wavelengths of the elastic distortions are long compared to any microscopic length, the departure from ideal homogeneous equilibrium is small. In this chapter, we will develop equations governing dynamical disturbances in which the departure from ideal homogeneous equilibrium of each point in space is small at all times.

Conserved and broken-symmetry variables

Thermodynamic equilibrium is produced and maintained by collisions between particles or elementary excitations that occur at a characteristic time interval τ. In classical fluids, τ is of order 10−10 to 10−14 seconds. In low-temperature solids or in quantum liquids, τ can be quite large, diverging as some inverse power of the temperature T. The mean distance λ between collisions (mean free path) of particles or excitations is a characteristic velocity v times τ. In fluids, v is determined by the kinetic energy, v ~ (T/m)½, where m is a mass. In solids, v is typically a sound velocity. Imagine now a disturbance from the ideal equilibrium state that varies periodically in time and space with frequency ω and wave number q.

In Chapter 6, we studied the states of systems with broken continuous symmetry in which the slowly varying elastic variables described distortions from a spatially constant ground state configuration. These distortions arose from the imposition of boundary conditions, from external fields, or from thermal fluctuations. In this chapter, we will consider a class of defects, called topological defects, in systems with broken continuous symmetry. A topological defect is in general characterized by some core region (e.g., a point or a line) where order is destroyed and a far field region where an elastic variable changes slowly in space. Like an electric point charge, it has the property that its presence can be determined by measurements of an appropriate field on any surface enclosing its core. Topological defects have different names depending on the symmetry that is broken and the particular system in question. In superfluid helium and xy-models, they are called vortices; in periodic crystals, dislocations; and in nematic liquid crystals, disclinations.

Topological defects play an important role in determining properties of real materials. For example, they are responsible to a large degree for the mechanical properties of metals like steel. They are particularly important in two dimensions, where they play a pivotal role in the transition from low-temperature phases characterized by a non-vanishing rigidity to a high-temperature disordered phase.

This chapter begins (Sec. 9.1) with a discussion of how topological defects are characterized and a brief introduction to the concepts of homotopy theory.

The use and understanding of matter in its condensed (liquid or solid) state have gone hand in hand with the advances of civilization and technology since the first use of primitive tools. So important has the control of condensed matter been to man that historical ages – the Stone Age, the Bronze Age, the Iron Age – have often been named after the material dominating the technology of the time. Serious scientific study of condensed matter began shortly after the Newtonian revolution. By the end of the nineteenth century, the foundations of our understanding of the macroscopic properties of matter were firmly in place. Thermodynamics, hydrodynamics and elasticity together provided an essentially complete description of the static and dynamic properties of gases, liquids and solids at length scales long compared to molecular lengths. These theories remain valid today. By the early and mid-twentieth century, new ideas, most notably quantum mechanics and new experimental probes, such as scattering and optical spectroscopy, had been introduced. These established the atomic nature of matter and opened the door for investigations and understanding of condensed matter at the microscopic level. The study of quantum properties of solids began in the 1920s and continues today in what we might term “conventional solid state physics”. This field includes accomplishments ranging from electronic band theory, which explains metals, insulators and semiconductors, to the theory of superconductivity and the quantum Hall effect.

In the preceding two chapters, we have discussed various types of order that can occur in nature and how the ordering process can be quantified by the introduction of order parameters. We also developed a formalism for dealing with the thermodynamics of ordered states. In this chapter, we will use mean-field theory to study phase transitions and the properties of various ordered phases. Mean-field theory is an approximation for the thermodynamic properties of a system based on treating the order parameter as spatially constant. It is a useful description if spatial fluctuations are not important. It becomes an exact theory only when the range of interactions becomes infinite. It, nevertheless, makes quantitatively correct predictions about some aspects of phase transitions (e.g. critical exponents) in high spatial dimensions where each particle or spin has many nearest neighbors, and it makes qualitatively correct predictions in physical dimensions. Mean-field theory has the enormous advantage of being mathematically simple, and it is almost invariably the first approach taken to predict phase diagrams and properties of new experimental systems.

Before proceeding, let us review some simple facts about phase transitions. At high temperatures, there is no order, and the order parameter 〈φ〉 is zero. At a critical temperature, Tc, order sets in so that, for temperatures below Tc, 〈φ〉 is nonzero. If 〈φ〉 rises continuously from zero, as shown in Fig. 4.0.1a, the transition is second order.

Condensed matter physics

Imagine that we knew all of the fundamental laws of nature, understood them completely, and could identify all of the elementary particles. Would we be able to explain all physical phenomena with this knowledge? We could do a good job of predicting how a single particle moves in an applied potential, and we could equally well predict the motion of two interacting particles (by separating center of mass and interparticle coordinates). But there are only a few problems involving three particles that we could solve exactly. The phenomena we commonly observe involve not two or three but of order 1027 particles (e.g., in a liter of water); there is little hope of finding an analytical solution for the motion of all of these particles. Moreover, it is not clear that such a solution, even if it existed, would be useful. We cannot possibly observe the motion of each of 1027 particles. We can, however, observe macroscopic variables, such as particle density, momentum density, or magnetization, and measure their fluctuations and response to external fields. It is these observables that characterize and distinguish the many different thermodynamically stable phases of matter: liquids flow, solids are rigid; some matter is transparent, other matter is colored; there are insulators, metals and semiconductors, and so on.

Condensed matter physics provides a framework for describing and determining what happens to large groups of particles when they interact via presumably well known forces.

Large collections of particles can condense into an almost limitless variety of equilibrium and nonequilibrium structures. These structures can be characterized by the average positions of the particles and by the interparticle spatial correlations. Periodic solids, with their regular arrangements of particles, are more ordered and have lower symmetry than fluids with their random arrangements of particles in thermal motion. There are a number of equilibrium thermodynamic phases that have higher symmetry than periodic solids but lower symmetry than fluids. Typically interacting particles at low density and/or high temperature form a gaseous phase characterized by minimal interparticle correlations. As temperature is lowered or density increased, a liquid with strong local correlations but with the same symmetries as a gas can form. Upon further cooling, various lower-symmetry phases may form. At the lowest temperatures, the equilibrium phase of most systems of particles is a highly ordered low-symmetry crystalline solid. Nonequilibrium structures such as aggregates can have unusual symmetries not found in equilibrium structures.

In this chapter, we will investigate some of the prevalent structures found in nature and develop a language to describe their order and symmetry. We will also study how these structures can be probed with current experimental methods. Though tools such as scanning force and tunneling microscopes can now provide direct images of charge and particle density, at least near surfaces, most information about bulk structure, especially at the angstrom scale, is obtained via scattering of neutrons, electrons, or photons.

Introduction

It is easily seen from either the Sayre equation (3.52) or the tangent formula (3.60) that direct methods are likely to be much more powerful in phase extension and refinement than in ab initio phasing, since nothing can be known about the left-hand side of either the Sayre equation or the tangent formula without first putting into the right-hand side at least a small number of starting phases. One of the reasons why multi-solution procedures are so successful in practice is that they provide the possibility of having some trial sets with an initial pattern of phases able to converge to the correct point in the multi-dimensional phase space. On the other hand, if the phases of a sufficiently large number of reflections can be estimated in advance then direct methods will work even more efficiently. This gives the possibility of combining direct methods with other methods to tackle the phase problem in a number of special cases.

Fragment development

In the analysis of complex crystal structures, it is often the case that a fragment instead of the complete structure is first obtained. Hence fragment development plays an important role in crystal structure determination.

Recycling methods

Fourier synthesis with partial-structure phases has been a very efficient approach to obtaining the complete structure, especially when this is associated with weighting functions (Woolfson, 1956; Sim, 1960). A reciprocal-space alternative is a phase extension and refinement procedure based on partial structure information.

Multiple-beam scattering

A description of multiple-beam scattering

In fig. 1.12 there is shown a graphical means of determining the condition under which a particular diffracted beam will be produced. If the reciprocal lattice point at position s is the only one that touches the sphere of reflection then the associated diffracted beam will be the only one to occur – except of course for the straight-through diffracted beam, corresponding to the point O, which always occurs. Such a situation is described as two-beam diffraction. It is also possible to have more than two reciprocal lattice points on the surface of the reflecting sphere and in fig. 8.1 the points O, P and Q lie on the surface corresponding to the reciprocal-lattice vectors 0, h and k. This would be a case of three-beam diffraction. Four- and more beam diffraction is also possible but here we shall be restricting our attention to the three-beam case.

The line CQ in fig. 8.1 gives the direction of the k diffracted beam. Let us suppose that a beam of radiation was incident on the crystal from that direction. The crystal would still be in the same orientation, as would be the reciprocal lattice which is rigidly attached to the crystal, and the sphere of reflection would be displaced to put the new origin point, O, on the old point Q. It is now clear that after the displacement the reciprocal lattice point h – k will now fall at the same point as P and hence that this diffracted beam would occur.

When von Laue and his assistants produced their first smudgy X-ray diffraction photographs in Munich in 1912 they could not have known of the developments that would follow and the impact that these would have on such a wide range of science. Structural crystallography, the ability to find the arrangement of atoms inside crystals, has advanced over the years both theoretically and experimentally. Technical advances, such as the development of computers both for control of instruments and for complex calculations, and also the advent of high power synchrotron X-ray sources have all played their part.

In this book we bring together all the methods that have been and are being used to solve crystal structures. We broadly divide these methods into two main classes, non-physical and physical methods. In the first category we place those methods that depend on a single set of diffraction data produced by the normal Thomson scattering from the individual atoms. The Patterson methods and direct methods described in chapters 2 and 3 respectively are non-physical methods. In chapter 4 the basic principles are explained for two physical methods – isomorphous replacement, which combines the data from two or more related compounds to obtain phase information, and anomalous scattering, which uses data at wavelengths for which some of the atoms scatter anomalously, i.e. with an amplitude and phase differing from that given by the Thomson process. In chapter 5 the method of isomorphous replacement is explored in much greater depth and in chapter 6 the same is done for anomalous scattering.