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Structures ant! Vibrational Properties of Solicls INTRODUCTION Matter in a solid state consists of many nuclei and electrons that form a structure in space. Knowledge of this structure is essential for understanding the physical properties of the solid, for example, whether it is a metal, a semiconductor, or an insulator or whether magnetic order can be produced by the electronic interactions. Vibra- tions of the nuclei around their average positions produce excited states of the solid structure. Since the nuclei have much heavier masses than the electrons, their characteristic vibrational frequencies, ~10'3 s-l, are much lower than the frequencies of ~10~5 s-l typical of many electronic excitations. These low-frequency vibrations are ubiquitous aspects of all solids: they propagate, and in so doing carry heat and information; they are important in the thermodynamics of solids; they are always present to absorb or scatter such experimental probes as electromagnetic radiation and neutrons, as well as other excitations in a solid such as electronic and magnetic excitations; and they lead to important electronic ordering effects such as superconductivity. The vibrational properties of many solids can be understood on the basis of the harmonic approximation, in which the force acting on a given nucleus is assumed to be a linear function of the displacements of that nucleus and of the other nuclei from their average positions. The problem can be solved exactly in this approximation. The quantized 58

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STRUCTURES AND VIBRATIONAL PROPERTIES OF SOLIDS 59 units of vibration are called phonons, and the quantum of energy is the vibrational frequency ~ times Planck's constant h. The nature of the vibrations is closely related to the structure of the solid: in crystals they form collective propagating excitations whose frequencies are de- scribed by dispersion curves ~ = make, where k is the wave vector associated with the excitation (the direction of k is the direction of propagation of the excitation, and the magnitude of k is 2~/A, where A is the wavelength of the excitation); in disordered systems they are sensitive to the topology of the structure of the solid and to the local order. The addition of small enharmonic forces leads to finite lifetimes and scattering of these phonons. In some systems the enharmonic forces are large, however. The dynamics of the nuclei in strongly enharmonic systems may be qualitatively different from the behavior of simple oscillators. There can be stable nonlinear excitations termed solitons, interesting statistical mechanics of thermally excited interact- ing vibrational states, and phase transitions to structures of different symmetry. Because this field is extensive and closely related to other topics, many of its aspects are considered in separate chapters, in particular, critical phenomena at phase transitions, structures of surfaces and interfaces, defects in crystals, and properties of particular classes of solids. THEORETICAL CALCULATIONS The primary goal in the theory of the structures of solids is to understand both why different types of solids form and how the resultant structures control the properties of solids. This is a many- body problem involving ~1023 electrons and nuclei. One of the highlights of research during the past decade is the progress toward a unified theoretical understanding of the combined many-electron/ many-nucleus problem. Indeed, predictions of the structures and vibrational excitations of solids are currently a crucial test of our understanding of the ground state of the electronic system. Since the mid-1970s, there has been a qualitative change in the ability to predict structures and related properties of solids a priori without using any information from experiments. This rapid development has been made possible both by the increase in power and availability of computers and by the formulation of new ways to treat the quantum many-body problems. Of these developments in the treatment of the electronic system discussed in Chapter 1, it is the density functional approach to electronic exchange and correlation that is the basis for the

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60 A DECADE OF CONDENSED-MATTER PHYSICS recent progress in accurate first-principles calculations of a wide range of structural and vibrational properties of solids. Other techniques, such as the many-body Monte Carlo quantum methods, make it possible to study the simplest cases in great depth. Among the primary achievements of such calculations are the phase diagrams of many elements and compounds as functions of pressure. Recent results include the structures of transition metals, semiconduc- tor-metal transitions, graphite and diamond structures of carbon, and many other crystals. An exemplary case is hydrogen, which is ex- pected to transform from an insulating molecular system to a metallic solid at high pressure. This is illustrated in Figure 2.1, which gives the total energy versus average proton separation, found from quantum Monte Carlo calculations. Similar results are found from perturbation -0.70 -0.75 _ _ -0.90 _ -0.80 -0.85 -0.95 -1.00 -1.05 -1.10 -1.15 -1.20 $ \ \\ \1 \; \ Molecular 4;\ ~ FCC crystal (Pa3) \\\ why Metallic FCC crystal Metallic static I ~ - on '] .0 1.2 1.4 1.6 1.8 2.0 2.2 rS FIGURE 2.1 Ground-state energy of hydrogen as a function of the average proton separation a, in units of the Bohr radius aO calculated by an approximate Monte Carlo simulation of the many-body fermion problem. The solid curve gives the energies for molecular and monatomic metallic phases. The dashed curves show the effect of fixing the protons, i.e., eliminating their zero-point motion, in the metallic phase. The results indicate a first-order transition from the molecular to the metallic phase near the crossing point at a ~ 1.35aO. (Courtesy of D. M. Ceperly and B. J. Adler, Lawrence Laboratory.)

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STR UCTURES AND VIBRA TIONAL PROPERTIES OF SOLIDS 61 theory and density functional calculations that have also considered different metallic structures. The salient result is that hydrogen will become a metal at pressures -2 mbar, which may be achieved in diamond anvil cells in the near future. One of the currently interesting developments is the emergence of a unified theoretical approach to structures, vibrations, and electron- phonon interactions. This is made possible by density functional calculations for crystals with atoms displaced from their equilibrium positions to determine small energy differences, forces acting on individual atoms, and the macroscopic stress. From the restoring forces and stresses, the vibrational properties can be obtained with no input from experiment. Calculations to date include complete phonon dispersion curves Arks, the pressure dependence of phonon frequen- cies and other enharmonic coefficients, and anomalous soft phonon modes. Results of calculations carried out so far agree with experi- ments to within a few percent and predict other properties not known experimentally. MEASUREMENTS OF STRUCTURES AND PHONON SPECTRA The basis of experimental measurements of the structures and dynamics of condensed matter is the absorption or the scattering of particles whose momentum and energy can be measured. The average structure is measured by the intensity of scattering as a function of the difference between the momenta of the incoming and outgoing parti- cles. Dynamical information can be obtained by measurement of the energy lost or gained by the particles. Conservation of energy requires that the excitation that is created (or destroyed) has energy equal to either (1) the energy of a particle that is absorbed (or emitted) or (2) the difference between the incoming and outgoing energies of a particle that is scattered inelastically. Experimental probes used in current investigations of structures and dynamics include x rays, photons, neutrons, electrons, atoms, and ions. Experiments using electrons and atoms are particularly suited for studies of surfaces and are discussed in Chapter 7. Inelastic neutron scattering is a powerful technique for the study of the dynamics of atoms in condensed matter (see Appendix E). The spectrometers needed to resolve the energies of the neutrons, devel- oped in the 1950s and 1960s, continue to provide an extensive body of knowledge on phonons and other excitations in condensed matter. Recently neutron scattering has provided the crucial short-wavelength probe for exploration of the challenging problems associated with

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62 A DECADE OF CONDENSED-MA TTER PHYSICS phase transitions, anomalous phonon dispersion curves due to strong electron-phonon interactions, and dynamics of nonlinear systems. There have been two major advances in neutron-scattering methods recently. One is the development of a neutron spin-echo spectrometer, which can measure energy transfers as small as a few microelectron volts. This resolution makes it possible to determine the dynamics of low-frequency, quasi-elastic phonons and the intrinsic lifetimes of phonons. The second is the advent of spallation sources, which are described in Appendix E. These sources produce neutrons with large usable ranges of momentum and energy that can provide increased spatial resolution and measurements of high-energy phonons, particu- larly those involving light atoms such as hydrogen. Experiments using x-ray scattering and absorption have become much more powerful because of recent advances in the production of intense, tunable x rays from synchrotron sources. The increase in angular resolution and intensity has made possible new experiments. One is the study of melting of two-dimensional systems of rare-gas atoms, described in Chapter 3. Another is the first measurement of the phase of the scattered x rays. This advance offers the possibility of yielding powerful new information on structures but is controversial at present. The pulsed nature of synchrotron radiation has been utilized to measure the rapid melting and recrystallization on nanosecond time scales that occur in pulsed-laser annealing. There has been an enor mous increase in the number of measurements of extended x-ray absorption fine structure (EXAFSJ spectra, which are being used to determine the local environment of a given type of atom. The most important results have been obtained for alloys, disordered solids, ionic conductors, and liquids, where EXAFS provides detailed infor- mation on the correlation functions of different atoms. The interaction of light with solids provides many of the most useful and versatile techniques for studying the dynamics of condensed matter. Although the range of momenta that can be studied by this technique is limited compared with that of neutron scattering, the absorption and scattering of photons have much greater resolution, dynamic range, and sensitivity than is possible with neutron scattering. Furthermore, because light couples to phonons primarily through the . . . . . ... . . . . . . ~ . electron~c polar~zab~l~ty, these experiments provide unique information on linear and nonlinear interactions of electrons, photons, and phonons. The modern era of light scattering began in 1962 when lasers were first introduced as monochromatic sources of light. Since that time, Raman scattering has become the most widely applied technique to

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STRUCTURES AND VIBRATIONAL PROPERTIES OF SOLIDS 63 determine vibrations in solids. In recent work, for example, scattering from tiny crystals under the extreme pressures that are generated in diamond anvil pressure cells is giving much new information on the nature of matter at compressed density. The use of optical interference enhancements has made possible detection of the vibrational spectra of molecules adsorbed on surfaces at submonolayer densities and of crystalline compounds formed in very thin (~20 A) layers at interfaces between different solids. In addition, the use of intense laser beams and optical nonlinearities leads to new effects, such as coherent stimulated Raman scattering and hyper-Raman scattering involving several pho- tons. The former has made possible lasing at Raman frequencies in op- tical fibers. The latter leads to different selection rules, so that vibrations can be detected that are not observable by ordinary Raman scattering. Inelastic scattering of light with small frequency shifts <10~0 Hz, often termed Brillouin scattering, has expanded greatly, aided by development of highly selective multiple-pass interferometers. Among the recent accomplishments of this technique are measurements of acoustic vibrations in metals through inelastic reflection caused by dynamical rippling of the surface. Low-frequency scattering also plays a crucial role in investigations of nonlinear systems, including such problems as the detection of tunneling modes in glasses, ionic motion in superionic conductors, large increases in quasi-elastic scattering near phase transitions, and dynamics of incommensurate structures described later. The nature of the coupling of electrons and phonons can be studied by resonance scattering, in which selected electronic states are en- hanced by their resonance with the light frequency. Because the extreme resonance conditions occur at energies where the light is absorbed, understanding the phenomenon has required the develop- ment of theoretical tools to deal with the difficult problems of nonequilibrium excited states coupled to the stochastically fluctuating environment. This has been applied particularly to investigate impurity states coupled to the lattice and the scattering mechanisms for elec- trons and holes in semiconductors. Infrared light can be used to study optically active phonons through reflectivity and absorption. As in the scattering experiments, the advent of infrared lasers has made possible new experimental areas, and many recent advances have been in the areas of low-frequency measurements. This is one of the powerful tools for studying ionic conductors, amorphous metals, and the coupled electron-phonon sys- tem in semiconductors. Each of these experimental tools for determining structures and

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64 A DECADE OF CONDENSED-MATTER PHYSICS dynamics has an important role in exploring the properties of solids and the physics of condensed matter. Some of the highlights and opportu- nities made possible by these techniques are mentioned below. PHONON TRANSPORT In the area of phonon transport varied aspects of phonons as elementary excitations of condensed matter are explored: the spectrum of energies, the velocities of propagation, scattering and decay of phonons, and their interactions with defects and other excitations. Before 1965, phonon transport was almost always studied by measur- ing the temperature dependence of the thermal conductivity. This yields a transport coefficient that is an average over different scattering processes due to anharmonicity, defects, and surfaces, weighted by the equilibrium distribution of phonons. In contrast, new techniques for generation and detection of high-frequency phonons have made possi- ble the direct study of phonon properties, selected by their frequency, velocity, and direction of propagation, in frequency ranges extending to >1 THz (10~2 sob. The initial experiments used heat pulses and measurement of the time of flight of phonons from heater to detector. They could resolve individual phonon modes, which propagate ballistically with their respective group velocities, as well as diffusive heat transport resulting from multiply scattered phonons. Important results included the ob- servations of second sound, the propagation of temperature waves in solids, and the propagation of solitons. The latter are well-defined excitations of a nonlinear lattice. This work was a stimulus for interest in nonlinear problems in other areas. There are several new methods of energy-selective generation and detection of high-frequency phonons. These include phonon-assisted tunneling, optical techniques, and time-of-flight selection of high- frequency phonons using the dispersion of velocities. Superconducting tunnel junctions bonded to the sample surface can selectively study phonons with energies up to the superconducting gap of ~0.5 THz. Optical techniques utilizing visible lasers can be used in many trans- parent solids to generate and detect phonons through coupling to sharp impurity states. State-of-the-art techniques of pulsing and focusing visible lasers make possible complete studies with simultaneous spec- tral, spatial, and temporal resolution. Also, phonons can be generated by infrared lasers using surface piezoelectric effects. This approach has the potential of creating phonons with phase coherence limited only by surface roughness.

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STR UCTURES AND VIBRA TIONA ~ PROPERTIES OF SOLIDS 65 Transport of energy in different phonon modes has been shown to vary enormously. In particular, low-frequency transverse phonons can often propagate over large distances, and their weak scattering mech- anisms can be studied in detail. Perhaps the most dramatic experimen- tal consequences of the long lifetime of certain acoustic phonons are the phenomena of phonon imaging and focusing caused by anisotropy in the velocity of propagation. For example, phonons produced by a heater at one point on a sample can be focused along particular crystallographic directions and can propagate ballistically for distances of ~1 cm under readily achievable conditions. An example of this striking anisotropic transport of energy in germanium is shown in Figure 2.2. Other developments include the study of anisotropic phonon winds and their effect on the shape of electron-hole droplets in semiconduc- tors; measurement of the frequency dependence of scattering by defects such as donors and acceptors in semiconductors; stimulated directional emission of phonons; demonstration of phonon mirrors created by superlattices of semiconductors; measurement of lifetimes of optic phonons in the picosecond range; generation and study of high-frequency surface phonons; and observation of anomalous trans- port in glasses at low temperatures due to coupling to low-frequency tunneling modes. ELECTRON-PHONON INTERACTIONS The interactions of phonons with photons' electrons, magnons, and excitors are indispensable ingredients in understanding the physical properties of solids. In cases of weak coupling, the phonons cause scattering, which is an important limitation on the mean free path of electrons, i.e., on the conductivity of metals and the mobility of carriers in semiconductors. Since the electrons also affect the phonon frequencies, the same interactions can be manifested in anomalous dispersion of the phonon frequencies and in phase transitions such as superconductivity and structural transitions. There can also be nonlinear solutions for localized electronic states coupled to atomic displacements. The best known recent example is the formation of fractionally charged solitons in conducting polymers (Chapter 101. For reasons such as these, electron-phonon interactions are of great importance in solid-state physics, and there is a growing interest in studying and utilizing the consequences of these interactions. The transition metals and their compounds are the focus of much of the activity in this area because the electron-phonon interactions are

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66 A DECADE OF CONDENSED-MATTER PHYSICS FIGURE 2.2 Phonon focusing. The bright areas represent heat energy propagating to the surfaces of a germanium crystal produced by a pulse of heat at a point on the back surface of the crystal. The phenomenon is caused by intense channeling of heat flux along certain crystal directions. (Courtesy of J. P. Wolfe University of lllinois.) thought to be responsible for high-temperature superconductivity in compounds like V3si and NbC, as well as for phonon softening and displacive phase transitions. A striking example of experimental and theoretical work is the m-phase transition, in which the bee structure is unstable to displacements of planes of atoms perpendicular to the (1 1 1) axis. The dynamics of this transition in Zr have been studied by neutron scattering, which has detected an anomalously low phonon frequency shown in Figure 2.3 and an increase in intensity of the central-peak scattering at zero frequency at the wavelength corre- sponding to the periodicity of the m-phase. Theoretical density

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STRUCTURES AND VIBRATIONAL PROPERTIES OF SOLIDS 67 functional calculations have determined an entire curve for the energy as a function of the positions of the planes, giving the low phonon frequency, two stable solutions in the bee and m-phase structures for Zr, and insight into why the effects are greatly reduced in the neighboring elements Nb and Mo. The electron-phonon interactions in transition-metal compounds have also made possible a new class of experiments involving light scattering, normally not observable in metals. The same interactions that cause the phonon anomalies also give rise to coupling to the light through the electrons. For example, NbSe2 distorts into an incommen- surate structure (discussed below) owing to the electron-phonon cou- pling, and the dynamics of the atomic displacements have been 8 7 6 5 I , 1 , 1 i 1 , I 1 1 o 0.0 0.2 0.4 11 Mo 0 Nb O bee Zr ll L [111] ~1 I '7 I? 0.6 0.8 1.0 FIGURE 2.3 Phonon dispersion curves for the longitudinal (111) branch measured by inelastic neutron experiments on Mo, Nb, and the high-temperature (1400 K) bcc phase of Zr near the m-phase transition. (Courtesy of C. Stassis and B. N. Harmon, Iowa State University.)

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68 A DECADE OF CONDENSED-MATTER PHYSICS detected in the light-scattering spectrum presented in Figure 2.4. Perhaps the most striking observation is the new peaks at low temperatures, interpreted as electronic excitations across the super- conducting gap. These results have led to new theoretical and experi- mental work to understand the basic phenomena involved and the role of the interactions in superconductivity and other properties. Other areas in which electron-phonon interactions play a crucial role are inelastic electron tunneling and a new experimental technique termed point-contact spectroscopy. The use of tunneling spectroscopy in superconductors to determine phonon densities of states, weighted by electron-phonon couplings, is now well established. Recent ad- vances in making tunnel junctions of superior quality have made possible tunneling in transition metals, high-temperature superconduc- tors, and magnetic superconductors. Such measurements on magnetic superconductors show the disappearance of the superconducting en- ergy gap as the magnetic transition is approached. In the high- temperature superconductors, e.g., Nb3Sn, tunneling results indicate 35 30 20 2H NbSe2 TICK Aig , i 1 1 1 1 -!~ 0 20 40 60 0 20 40 60 RAMAN SHI FT (cm~l ) FIGURE 2.4 Raman spectrum of 2H NbSe2 at low temperature. The peaks at ~40 cm-' are amplitude modes of the incommensurate structure and those at ~20 cm-' are excitations of the electrons across the superconducting gap. (Courtesy of M. V. Klein, University of Illinois.)

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STRUCTURES AND VIBRATIONAL PROPERTIES OF SOLIDS 69 that, of all the phonons, those of low frequency are most effective in promoting superconductivity. Point-contact spectroscopy involves measuring the current-voltage relation for a current of electrons through a metallic point. If the dimensions of the point are smaller than the electronic mean free path, electrons can be accelerated to the energy eV, where V is the voltage drop. Measurement of the current as a function of V gives direct information on the energy dependence of the scattering mechanisms. At present, theoretical work is attempting to derive the relations to the underlying phonon properties. One advantage of this technique is that it can be applied to many materials and is not restricted to superconductors. DISORDERED SOLIDS AND INCOMMENSURATE PHASES A growing area of research is concerned with disordered solids that present intellectual challenges, unique phenomena, and extensive applications. One class of disordered materials is the amorphous or glassy solids, which have no long-range order. The atomic structures of glasses, nevertheless, have characteristic types of short-range order, e.g., favored coordination numbers and angular arrangements of the nearest neighbors associated with specific types of bonding. For example, in vitreous silica the oxygen atoms have twofold coordination and the silicon atoms have fourfold tetrahedral coordination, whereas in amorphous metals the coordination number is higher, ~8-12. Ex- perimental information on the short-range order is obtained by diffrac- tion of x rays, neutrons, and electrons and by EXAFS, which deter- mine angle-averaged radial distributions of the probability of finding neighboring atoms. These measurements cannot determine the three- dimensional structure uniquely, but they provide stringent conditions on models of the structure. Research in this area has increased dramatically in recent years owing to the availability of synchrotron facilities as intense, tunable, collimated x-ray sources and the advent of spallation facilities as sources of higher-energy neutrons, which can give improved spatial resolution. The intellectual challenges that have highlighted recent research in this area are concerned with the ways that groups of atoms with short-range order can be connected together to build space-filling rigid structures with no long-range order. An interesting contribution to the theory of such structures is the demonstration that small sets of regular polyhedra can be packed to generate nonperiodic, disordered struc- tures that fill three-dimensional space. There are, however, many

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70 A DECADE OF CONDENSED-MATTER PHYSICS degrees of freedom to consider in a physical glass, and there is much controversy and continuing research on the thermodynamics of the glass transition and the nature of the structures formed. The vibrational excitations are especially pertinent to the studies of disordered structures because they depend sensitively on both the short-range order and the connectivity or topology of the structure. Theoretical studies of vibrational properties of strongly coupled disor- dered networks, especially with topological disorder, have led to new perspectives on excitations in disordered systems. Experimental mea- surements of vibration frequencies in glasses, together with the im- proved theoretical understanding, have motivated new explorations of the topology of glasses, such as silica. Another aspect of the dynamics is the existence of low-frequency modes, which appear to occur universally in disordered systems. These nonlinear excitations dominate many low-frequency aspects of glasses, e.g., low-temperature heat capacity, thermal transport, elec- trical resistivity, and dielectric loss. Although they are thought to involve finite displacements of atoms by tunneling or thermal hopping, the microscopic origins of these modes are unknown. A different class of disordered solids are crystals in which there is intrinsic disorder. The two areas of most current interest are ionic conductors and plastic crystals. Crystals called superionic conductors contain large densities of ions that can diffuse with rates comparable with those of ions in liquids. For example, in the high-temperature phase of AgI the I ions form a solid bcc lattice in which the Ag ions are as mobile as in the melt. Studies of these materials have been stimulated by their technological applications. The term plastic crystal denotes crystals containing molecules that are orientationally disor- dered. The low-frequency reorientations that these molecules can undergo are strongly coupled enharmonic motions, which lead to unusual mechanical properties of these solids. For ionic conductors, plastic crystals, and other dynamically disordered systems, the basic questions are: Why do such crystals form, and how do the ions or molecules move? Investigations on a microscopic scale currently utilize x-ray and neutron scattering, EXAFS, nuclear magnetic reso- nance, light scattering, high-frequency conductivity, and theoretical work on these highly enharmonic, nonlinear problems. An exciting class of structures is one in which there are simulta- neously two incommensurate periodicities coexisting in the same solid. Such a structure is not periodic because there is no translation that is equal to integral numbers of primitive translations of both periodicities. However, each periodicity can be separately observed in a scattering

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STRUCTURES AND VIBRATIONAL PROPERTIES OF SOLIDS 71 experiment. Such structures were known for some time (e.g., the spin density wave in chromium), but only in the last decade have they taken their place in the field of phase transitions and their symmetries and dynamics studied extensively. Several types of incommensurate solids have been found. In one, which has been discussed in Chapter 1, the electron-phonon interaction stabilizes a distortion with the Fermi wave vector kF, which is incommensurate with the lattice periodicity. Examples include chain compounds like TTF-TCNQ and layered metals like NbSe2. A different mechanism that can occur in either metals or insulators is a zero phonon frequency at an incommensurate wave vector k, which can be caused by simple combinations of interatomic forces. This is a soft mode that leads to a phase transition, as happens in K2SeO4 and ThBr4. Another type of incommensurate structure results from the coexist- ence of interpenetrating lattices with different periodicities. An exam- ple is Hg2.72(AsF6) in which the mercury atoms form linear metallic chains with an average spacing that is incommensurate with that of the AsF6 lattice. The vibrational states of incommensurate systems differ from those of ordinary crystals in fascinating ways. In particular, since it requires no energy to slide or change uniformly the relative phase of one periodicity relative to the other, there may be phason excitations with zero frequency at infinite wavelength. In the harmonic approximation, there is a phason dispersion curve with frequency linear in wave vector k at small k, in addition to the ordinary sound modes present in all solids. There has been a widespread search for these modes leading to their observation in ThBr4 and Hg2.72(AsF6) by neutron scattering. The difficulty in observing these modes at longer wavelength and lower frequency, e.g., in light-scattering experiments, appears now to be understood in terms of a fundamental difference between phasons and true acoustic modes. The latter become more precisely defined prop- agating modes as the frequency decreases, whereas the phasons are greatly modified by anharmonicity and become overdamped at low frequencies. The strongly nonlinear character of phason modes leads to domainlike descriptions of incommensurate phase transitions like those described below. PHASE TRANSITIONS AND NONLINEAR EXCITATIONS Phase transitions that involve a change in the structure of a solid are among the archetypal examples of this general phenomenon. There are two paradigms for structural transitions order-disorder and displa

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72 A DECADE OF CONDENSED-MATTER PHYSICS civet The former is a change in degree of disorder present in the structure. The latter involves displacement of atoms from sites of high symmetry to ones of lower symmetry. Each paradigm is illustrated in the previous two sections by recent work on disordered crystals, incommensurate phases, and structural transitions, such as the transition in Zr. These and other phase transitions, e.g., ferroelectric- ity, continue to provide major conceptual challenges and phenomena with technological applications. Research on nonlinear excitations involving finite displacements of atoms has become a stimulating area of physics. Although exact solutions to simple nonlinear models and phenomena like solitary waves have been known for many years, a veritable explosion in the study of such excitations has occurred in condensed-matter physics since the mid-1970s. An impetus to this work was the progress in understanding displacive phase transitions, where studies of the dy- namics revealed domain wall-type solutions that cannot be represented by perturbation expansions in the displacements of the atoms from their equilibrium positions. The dynamics of such systems consist not only of spatially extended, small-amplitude phonons, but also of spatially compact, large-amplitude excitations, often referred to as solitons. Although this has developed into an exciting new subfield, there is still controversy over how these excitations affect the thermo- dynamics of phase transitions. Many stimulating developments in nonlinear dynamics have been made in the context of quasi-one-dimensional systems. A particularly interesting case is the conducting polymer polyacetylene (CH)X, whose properties are striking consequences of the electron-phonon interac- tion. They are described in detail in Chapter 10. The general ideas underlying such excitations have widespread ramifications in physics and are discussed in Chapters 1, 3, 4, and 11. OPPORTUNITIES The ability to carry out theoretical calculations that predict the structures and vibrational properties of solids is expanding rapidly and will play a major role in future work. Because the calculations can be done accurately for real solids, there is emerging a new relation between theory and experiment and a more unified understanding of structural, vibrational, and electronic properties of matter. The poten- tial of future work is to develop new ideas and methods for excited states and nonzero temperatures, to make simple models that describe the essential points, and to gain greater insight into the nature of condensed matter.

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S TR UC TURKS A ND VIBRA TI ONA L PR OPER TIES OF S OLI DS 7 3 New experiments on structures and dynamics of solids can be made possible by improved synchrotron sources of x rays and by high-flux steady-state or pulsed sources of neutrons. Exciting possibilities include direct determination of structures using the phases of scattered x rays, measurements of fast transient structures, and improved energy resolution that can enable inelastic scattering of x rays to measure dynamics of atoms and electrons. High fluxes of neutrons would enable measurements to be made with greater resolution and on the small samples often crucial for forefront research. Pulsed spallation sources will permit inelastic scattering at high energies, e.g., at energies comparable with those of the vibrations of hydrogen atoms. Current and future innovations in light scattering, such as femtosec- ond pulses and resolution of small frequency shifts, will make possible experiments on new materials, conditions, and time scales. important contributions will likely occur for fast-reaction kinetics, properties of surfaces and interfaces, phase transitions, nonlinear excitations, and novel superconductors, for example. High pressures achievable in diamond anvil cells open many possi- bilities for understanding why structures form and creating states of matter never before accomplished in a laboratory, such as metallic hydrogen. Future areas of research in phonon transport will likely include increased emphasis on lower-dimensional systems, superlattices, nonlinear lattices, transport of phonons through interfaces, phonon dynamics in the subpicosecond range, and coherent excitations. The most important need for future work is the development of simple, sensitive tunable generators and detectors of phonons to extend measurements to wider classes of materials. This work will also have an impact on other areas of condensed-matter science, such as heat transport in small fast electronic devices, transfer of energy in pulsed- laser annealing, and steps toward development of a phonon laser. The structures of glasses and other solids with disorder are at present only partially understood, and there is much controversy concerning the degree to which spatial order extends to intermediate ranges. New information and ideas are needed to understand such basic features of the structures of disordered solids. The microscopic origins of the low-frequency modes that occur almost universally in disordered systems are~vestigation of these modes by many different,~chniques will be an important area of future research in disordered systems. Thee theoretical and experimental study of solitons and other nonlinear phenomena is an exciting area of research with many ~f~damental questions to be answered, such as the stability of solitons -

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74 A DECADE OF CONDENSED-MA TTER PH YSICS to small displacements, their role in phase transitions, effects of quantum fluctuations on them, and the nature of fractionally charged excitations. There are many possibilities for entirely new nonlinear phenomena in physical systems that may-be realized through imagina- tive ideas and novel synthesis of materials. Synthesis of new materials will likely provide unforeseen structures and phenomena as stimulating as those of the recent past, such as organic conductors and superconductors, incommensurate structures, and lower-dimensional systems. The creation of man-made artificial structures, such as semiconductor superlattices, is just beginning to reveal the range of new possibilities. Studies of structure and vibra- tions will certainly continue to probe phenomena of intrinsic interest as well as to provide keys to understanding the nature of new materials.