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.

Some atomic bound states have simple structure in the sense that a straightforward calculation obtains correct energy levels. In some cases optical oscillator strengths probe further detail. Collision theory has reached the stage where experimental observables for electron collisions involving such states can be calculated within experimental error. Observables whose calculation is sensitive to structure details constitute a probe for structure which verifies the details in more-difficult cases.

Scattering experiments are usually not very sensitive to structure. On the other hand the differential cross section for ionisation in a kinematic region where the plane-wave impulse approximation is valid gives a direct representation (10.31) of the structure of simple targets in the form of the momentum-space orbital of a target electron.

Electron momentum spectroscopy (McCarthy and Weigold, 1991) is based on ionisation experiments at incident energies of the order of 1000 eV, where the plane-wave impulse approximation is roughly valid. The differential cross section is measured for each ion state over a range of ion recoil momentum p from about 0 to 2.5 a.u. Noncoplanar-symmetric kinematics is the usual mode. In such experiments the distorted-wave impulse approximation turns out to be a sufficiently-refined theory. Checks of this based on a generally-valid sum rule will be described.

The reaction depends as much on the observed state |f〉 of the residual ion as on the ground state |0〉 of the target. Not only the single-particle structure but electron correlations in each state are sensitively probed in different circumstances.

Electron–atom collisions that ionise the target provide a very interesting diversity of phenomena. The reason for this is that a three-body final state allows a wide range of kinematic regions to be investigated. Different kinematic regions depend sensitively on different aspects of the description of the collision.

Up to now there has been no calculation of differential cross sections by a method that is generally valid. We use a formulation due to Konovalov (1993). Understanding of ionisation has advanced by an iterative process involving experiments and calculations that emphasise different aspects of the reaction. Kinematic regions have been found that are completely understood in the sense that absolute differential cross sections in detailed agreement with experiment can be calculated. These form the basis of a structure probe, electron momentum spectroscopy, that is extremely sensitive to one-electron and electron-correlation properties of the target ground state and observed states of the residual ion. It forms a test of unprecedented scope and sensitivity for structure calculations that is described in chapter 11.

Other kinematic regions require a complete description of the collision, which may be facilitated by including the boundary condition for the three charged particles in the final state. This is nontrivial because there is no separation distance at which the Coulomb forces in the three-body system are strictly negligible. The pioneering experiments of Ehrhardt et al. (1969) are of this type.

An accurate description of ionisation channels is essential in a theory of scattering, even to low-lying discrete states at low incident energy. The first test of such a description is provided by the total ionisation cross section and asymmetry.

In the last chapter we discussed how our understanding of electron impact excitation of atoms has substantially improved in recent years. Sophisticated experimental techniques are available for revealing sensitive details of the collision process, in addition to providing accurate and reliable differential and total cross section data. These details include the shape and inherent angular momentum of the excited atoms after the scattering process, measured as a function of the scattering angle and incident energy. These studies have provided stringent tests of current scattering theories, particularly at intermediate energies and backward angles.

In conventional collision experiments the strong Coulomb interaction generally masks the much weaker relativistic spin-dependent interactions. The role of the spin-dependent interactions, such as the exchange and spin–orbit interactions, has also been clarified by sophisticated measurements with spin-polarised electrons and/or spin-polarised targets, sometimes employing spin analysis after the collision process (Kessler, 1985, 1991; Hanne, 1983).

Such measurements were first applied with considerable success to elastic scattering. Indeed one was able to discuss experiments which would determine all the theoretically calculable amplitudes (Bederson, 1970). For inelastic processes, such measurements necessitate the simultaneous application of spin selection techniques and the alignment and orientation measurements discussed in the previous chapter. The experiments have become feasible with the advancement of experimental techniques. The first successful differential electron impact excitation study with spin-polarised electrons and alignment and orientation measurements was performed by Goeke et al. (1983) for the e–Hg case.

To understand an electron–atom collision means to be able to calculate correctly the T-matrix elements for excitations from a completely-specified entrance channel to a completely-specified exit channel. Quantities that can be observed experimentally depend on bilinear combinations of T-matrix elements. For example the differential cross section (6.55) is given by the absolute squares of T-matrix elements summed and averaged over magnetic quantum numbers that are not observed in the final and initial states respectively. This chapter is concerned with differential and total cross sections and with quantities related to selected magnetic substates of the atom.

In the study of electron–atom collisions there has been a constant emphasis on increasing the state selectivity of the particles in both the initial and final states. Thus while total cross section measurements define the initial kinetic energies, measurements of differential cross sections as a function of angle give additional information on the final momentum states of the separating particles. Added state selectivity is obtained through the use of spin-polarised electrons, or spin-polarised atoms, and with spin analysis of scattered particles (see e.g. Kessler (1985)). The great progress that has been made with the use of spin-selected particles will be discussed in the next chapter. Much of the progress that has been made has been for the case of elastic scattering.

Collisional alignment and orientation

The status for inelastic collisions is a little less satisfactory than for elastic collisions. Collisional excitation of atoms involves excited states with several magnetic substates.

The background for the details of multichannel scattering calculations has now been established. We consider methods based on the integral-equation formulation of chapter 6. These momentum-space methods have proved accurate at all energies in a sufficient variety of situations to justify the belief that they can be generally applied. In some situations sufficient accuracy is achieved without resorting to the full power of the integral-equation solution. The methods used in these situations are distorted-wave methods. Their relationship to the full solutions will be examined in a simplified illustrative case. A brief outline will be given of alternative methods based on a coordinate-space formulation of the multichannel problem.

There are two characteristic difficulties of multichannel many-fermion problems. The first is that computational methods can of course directly address only a finite number of channels whereas the physical problem has an infinite number of discrete channels and the ionisation continuum. The second is that the electrons are identical so that the formulation in terms of one-electron states must be explicitly antisymmetric in the position (or momentum) and spin coordinates.

We first show how to set up the problem within the framework of formal scattering theory using antisymmetric products of one-electron states. The problem is then formulated in terms of the calculation of reduced T-matrix elements relating the absolute values of initial and final momenta in different angular-momentum states. This depends on a knowledge of the corresponding potential matrix elements, whose calculation we treat in detail. We then show how the target continuum is accounted for in the scattering formalism.

We have considered the measurement of observables in electron–atom collisions and the description of the structure of the target and residual atomic states. We are now in a position to develop the formal theory of the reaction mechanism. Our understanding of potential scattering serves as a useful example of the concepts involved.

Reactions are understood in terms of channels. A channel is a quantum state of the projectile–target system when the projectile and target are so far apart that they do not interact. It is specified by the incident energy and spin projection of the projectile and the quantum state of the N-electron target, which may be bound or ionised.

The reaction mechanism is studied by considering targets whose description is simple and, at least from the spectroscopic point of view, believable within an accuracy appropriate to the scattering experiment. Hydrogen is the obvious example, although experiments are difficult because of the need to make the atomic target by dissociating molecules. Sodium is a target for which a large quantity of experimental data is available and whose structure can be quite well described for the lower-energy states. When the reaction mechanism is sufficiently understood the reaction may be used as a probe for the structure of more-complicated target or residual systems.

Formulation of the problem

Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v.