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OCR for page 58
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
OCR for page 59
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
OCR for page 60
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.)
OCR for page 61
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
OCR for page 62
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
OCR for page 64
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
OCR for page 66
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.
Representative terms from entire chapter:
vibrational properties
