Encyclopedia Dictionary of Condensed Matter Physics

2562 Pages·2005·78.82 MB·English

by  G. Bassani

Editors
Franco Bassani, Scuola Normale Superiore, Pisa, Italy
Gerald L Liedl, Purdue University, West Lafayette, IN, USA
Peter Wyder, Grenoble High Magnetic Field Laboratory, Grenoble, France
Editorial Advisory Board
Vladimir Agranovich, Russian Academy of Sciences, Moscow, Russia
Angelo Bifone, GlaxoSmithKline Research Centre, Verona, Italy
Riccardo Broglia, Universita degli Studi di Milano, Milano, Italy
Kikuo Cho, Osaka University, Osaka, Japan
Ge!rard Chouteau, CNRS and MPI-FKF, Grenoble, France
Roberto Colella, Purdue University, West Lafayette, IN, USA
Pulak Dutta, Northwestern University, Evanston, IL, USA
Leo Esaki, Shibaura Institute of Technology, Japan
Jaap Franse, Universiteit van Amsterdam, Amsterdam, The Netherlands
Alexander Gerber, Tel Aviv University, Tel Aviv, Israel
Ron Gibala, University of Michigan, Ann Arbor, MI, USA
Guiseppe Grosso, Universita" di Pisa, Pisa, Italy
Jurgen M Honig, Purdue University, West Lafayette, IN, USA
Massimo Inguscio, Dipartmento di Fisica e L.E.N.S., Firenze, Italy
A G M Jansen, Institut Max Planck, Grenoble, France
Th W J Janssen, Katholieke Universiteit Nijmegen, Nijmegen, The Netherlands
Giorgio Margaritondo, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland
Emmanuel Rimini, Universita di Catania, Catania, Italy
Robin D Rogers, The University of Alabama, Tuscaloosa, AL, USA
John Singleton, Los Alamos National Laboratory, Los Alamos, NM, USA
Carl H Zweben, Devon, PA, USA
INTRODUCTION
Physics is the paradigm of all scientific knowledge. Over the centuries it has evolved to a complexity that has
resulted in a separation into various subfields, always connected with one another and very difficult to single
out.FreemanDyson,inhisbeautifulbook‘InfiniteinAllDirections’,distinguishestwoaspectsofphysicsand
twotypes ofphysicists: the unifiersandthe diversifiers. The unifiers lookforthe most general lawsof nature,
likethe universal attraction betweenmassesandelectriccharges,the lawsofmotion,relativityprinciples,the
simplest elementary particles, the unification of all forces, symmetry violation and so on. The diversifiers
considertheimmensevarietyofnaturalphenomena,infiniteintheirextension,trytoexplainthemonthebasis
ofknowngeneralprinciples,andgeneratenewphenomenaanddevicesthatdonotexistinnature.Evenatthe
beginningofmodernscienceGalileoGalilei,besidesstudyingthelawsofmotionandlayingdowntheprinciple
of relativity, was interested in the phenomenon of fluorescence and disproved the theories put forward at his
time. He was both a unifier and a diversifier. The full explanation of fluorescence had to await the advent of
quantum mechanics, as did the explanation of other basic phenomena like electrical conductivity and
spectroscopy.
The past century witnessed an explosive expansion in both aspects of physics. Relativity and quantum
mechanics were discovered and the greatest of the unifiers, Albert Einstein, became convinced that all reality
couldbecomprehendedwithasimplesetofequations.Ontheotherhandawiderangeofcomplexphenomena
was explained and numerous new phenomena were discovered. One of the great diversifiers, John Bardeen,
explained superconductivity and invented the transistor.
In physics today we encounter complex phenomena in the behavior of both natural and artificial complex
systems,inmatterconstitutedbymanyparticlessuchasinteractingatoms,incrystals,inclassicalandquantum
fluids as well as in semiconductors and nanostructured materials. Furthermore, the complexity of biological
matterandbiologicalphenomenaarenowmajorareasofstudyaswellasclimatepredictiononaglobalscale.
Allofthishasevolvedintowhatwenowcall‘‘condensedmatterphysics’’.Thisisamorecomprehensiveterm
than‘‘solidstatephysics’’fromwhich,whentheelectronicpropertiesofcrystalsbegantobeunderstoodinthe
thirties, it originated in some way. Condensed matter physics also includes aspects of atomic physics,
particularly when the atoms are manipulated, as in Bose–Einstein condensation. It is now the largest part of
physics and it is where the greatest number of physicists work. Furthermore, it is enhanced through its
connectionswithtechnologyandindustry.Incondensedmatterphysicsnewphenomena,newdevices,andnew
principles,suchasthe quantum Halleffect,are constantlyemerging.Forthis reasonwe thinkthat condensed
matterisnowtheliveliestsubfieldofphysics,andhavedecidedtoaddressitinthepresentEncyclopedia.Our
focus is to provide some definitive articles for graduate students who need a guide through this impenetrable
forest,researcherswhowantabroaderviewintosubjectsrelatedtotheirown,engineerswhoareinterestedin
emergingandnewtechnologiestogetherwithbiologistswhorequireadeeperinsightintothisfascinatingand
complex field that augments theirs.
InthisEncyclopediawehaveselectedkeytopicsinthefieldofcondensedmatterphysics,providedhistorical
backgroundtosomeofthemajorareasanddirectedthereader,throughdetailedreferences,tofurtherreading
resources. Authors were sought from those who have made major contributions and worked actively in the
viii INTRODUCTION
areaofthetopic.Weareawarethatcompletenessinsuchaninfinitedomainisanunattainabledreamandhave
decidedtolimitourefforttoasix-volumeworkcoveringonlythemainaspectsofthefield,notallofthemin
comparable depth.
AsignificantpartoftheEncyclopediaisdevotedtothebasicmethodsofquantummechanics,asappliedto
crystals and other condensed matter. Semiconductors in particular are extensively described because of their
importanceinthemoderninformationhighways.Nanostructuredmaterialsareincludedbecausetheabilityto
produce substances which do not exist in nature offers intriguing opportunities, not least because their
propertiescanbetailoredtoobtainspecificdeviceslikemicrocavitiesforlightconcentration,speciallasers,or
photonicbandgapmaterials.Forthesamereasonsopticalpropertiesaregivenspecialattention.Wehavenot,
however,neglectedfoundationaspectsofthefield(suchasmechanicalproperties)thatarebasicforallmaterial
applications, microscopy which now allows one to see and to manipulate individual atoms, and materials
processing which is necessary to produce new devices and components. Attention is also devoted to the ever-
expandingroleoforganicmaterials,inparticularpolymers.Specificefforthasbeenmadetoincludebiological
materials, which after the discovery of DNA and its properties are now being understood in physical terms.
Neuroscience is also included, in conjunction with biological phenomena and other areas of the field.
Computational physics and mathematical methods are included owing to their expanding role in all of
condensedmatterphysicsandtheirpotentialinnumerousareasofstudyincludingapplicationsinthestudyof
proteinsanddrugdesign.Manyarticlesdealwiththedescriptionofspecificdeviceslikeelectronandpositron
sources, radiation sources, optoelectronic devices, micro and nanoelectronics. Also, articles covering essential
techniques such as optical and electron microscopy, a variety of spectroscopes, x-ray and electron scattering
and nuclear and electron spin resonance have been included to provide a foundation for the characterization
aspect of condensed matter physics.
We are aware of the wealth of topics that have been incompletely treated or left out, but we hope that by
concentrating on the foundation and emerging aspects of the infinite extension of condensed matter physics
these volumes will be generally useful.
Wewishtoacknowledgethefruitfulcollaborationofthemembersofthescientificeditorialboardandofthe
Elsevier editorial staff.
Special thanks are due to Giuseppe Grosso, Giuseppe La Rocca, Keith Bowman, Jurgen Honig, Roberto
Colella, Michael McElfresh, Jaap Franse, and Louis Jansen for their generous help.
Franco Bassani, Peter Wyder, and Gerald L Liedl
Permission Acknowledgments
The following material is reproduced with kind permission of Nature Publishing Group
Figure 12 of Ionic and Mixed Conductivity in Condensed Phases
Figure 1 of Porous Silicon
Figure 5 of Quasicrystals, Electronic Structure of
Figure 2 of Superconductivity: Flux Quantization
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The following material is reproduced with kind permission of the American Association for the
Advancement of Science
Figure 6 of Biomolecules, Scanning Probe Microscopy of
Figure 4 of Excitons: Theory
Figures 5, 6 and 7 of Genetic Algorithms for Biological Systems
Figures 1 and 3 of Rhodopsin and the First Step in Vision
Figures 4 and 5 of Scattering, Inelastic: Electron
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The following material is reproduced with kind permission of Taylor & Francis Ltd
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Figure 2 of Semiconductor Nanostructures
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Table 1 of Crystal Tensors: Applications
Figure 2 of Cyclotron Resonance: Metals
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A
Acoustics: Physical Principles and Applications to Condensed
Matter Physics
J DMaynard,ThePennsylvania StateUniversity, line of liquid helium. For the field of supercon-
UniversityPark, PA,USA ductivity, a paper on acoustic attenuation was in-
&2005,ElsevierLtd.AllRightsReserved. cluded as one of the relatively small number of
selected reprints on superconductivity published by
the American Institute of Physics. Acoustic measure-
ments are among the first performed whenever a
material involving novel physics is discovered. Mod-
Introduction
ern acoustic techniques, discussed below, can probe
For many condensed matter systems, including liq- thepropertiesofsamplesonlyafewhundredmicrons
uidsaswellassolids,acousticmeasurementsprovide in size and nanoscale thin films, and may be utilized
a crucial probe of important and fundamental phys- in practical applications such as micro-electro-me-
icsofthesystem.Inthecaseofsolids,oneofthefirst chanical systems (MEMS). Acoustic measurements
fundamental properties to be determined would be provide significant information about condensed
the atomic structure, defined by the minimum in the matter systems, and their accurate and precise meas-
freeenergywithrespecttothepositionsoftheatoms. urement is certainly important.
Thenextfundamentalcharacteristicofinterestmight
be the curvature of the free energy in the vicinity of
the minimum, and this would be manifest in the
Acoustics in Solids
elastic constants for the material. As derivatives of
the free energy, elastic constants are closely connect- For solids, acoustic phenomena reflect the elastic
edtothermodynamicpropertiesofthematerial;they properties of the material. Interest in elasticity dates
can be related to specific heat, Debye temperature, back to Galileo and other philosophers in the seven-
the Gruneisen parameter, and they can be used to teenth century, who were interested in the static
checktheoreticalmodels.Extensivequantitativecon- equilibriumofbendingbeams.Withthebasicphysics
nections may be made if the elastic constants are introduced by Hooke in 1660, the development of
known as functions of temperature and pressure. the theory of elasticity followed the development of
Acoustic measurements not only probe lattice prop- the necessary mathematics, with contributions from
erties,theyarealsosensitiveprobesoftheenvironm- Euler, Lagrange, Poisson, Green, etc., and the result-
ent in which all interactions take place, and may be ing theory was summarized in the treatise by A E H
used to study electronic and magnetic properties Love in 1927.
(e.g., through magnetostriction effects). As will be Acoustic and elastic properties of solids are quan-
discussedlater,acousticmeasurementsinvolvetensor tifiedinasetofelasticconstants.Theseconstantsare
quantities,andthuscanprobeanisotropicproperties like spring constants, relating forces, and displace-
of crystals. The damping of elastic waves provides ments, and they may be measured with a static tech-
information on anharmonicity and on coupling with nique, in which a displacement is measured as a
electrons and other relaxation mechanisms. One of linear response to a small applied force. However, it
the most important features of acoustic measure- was long ago learned that a better method is to
ments is that they provide a sensitive probe of phase measure an elastic vibration, as found, for example,
transitionsandcriticalphenomena;importantexam- in a propagating sound wave. Most existing com-
ples,inadditiontotheobviousexampleofstructural pletesetsofelasticconstantsformaterialshavebeen
transitions in solids, include the superconducting determined by measuring the time of flight of sound
transition and the superfluid transition. Indeed, one pulses. Recently, a relatively new method, resonant
of the most impressive successes in critical phenom- ultrasound spectroscopy (RUS), is being used. In the
enahasbeentheuseofacousticstostudythelambda RUSmethod,ratherthanmeasuringsoundvelocities,
2 Acoustics:PhysicalPrinciplesandApplicationstoCondensedMatterPhysics
one measures the natural frequencies of elastic vib- in the i-direction is @s =@x, and Newton’s law may
ij j
ration for a number of normal modes of a sample, be written as
and processes these in a computer, along with the
shape and mass of the sample. With a proper con- @s @2c
ij ¼r i ½3(cid:2)
figuration, a single measurement yields enough fre- @x @t2
j
quenciestodeterminealloftheelasticconstantsfora
material (as many as 21 for a crystal with low sym-
where r is the mass density.
metry).Samplesmaybepreparedinawidevarietyof
The symmetric nature of the definitions, and the
shapes,includingrectangular,spherical,etc.,anditis
assumption thattheelasticenergymust bequadratic
not necessary to orient crystalline samples. A com-
in the strains, reduces the number of independent
pelling reason for using RUS has to do with the na-
elementsofc from81to21.Abasicsymmetryhas
ijkl
ture of samples of new materials. Whenever a new
c invariant if the indices are exchanged in the first
ijkl
material is developed, initial single crystal samples
pairorsecondpairofthefoursubscripts(c ¼c ,
jikl ijkl
are often relatively small, perhaps on the order of a
etc.); thus a reduced system of indices may be used:
fraction of a millimeter in size. Also, with new 11-1,22-2,33-3,23-4,13-5,12-6,sothat
developments in nanotechnology and the possibility c -c . The reduced system is used when tabulat-
ijkl mn
ofapplicationsinthemicroelectronicsindustry,there
ingvaluesofelasticconstants;however,thefullfour-
is a great interest in systems which are very small in
indextensormustbeusedincalculations.Additional
one or more dimensions, such as thin films and one-
symmetries of a particular crystal group will reduce
dimensional wires. For such small systems, pulse
the number of independent elastic constants further
measurements are difficult, if not impossible, but
below 21; for example, orthorhombic crystals have
RUS methods may be readily used.
nine independent elastic constants, cubic crystals
have three, and isotropic solids have only two.
Physical Principlesfor Acoustics in Solids
Later, the case of an isotropic elastic solid will be
To begin a theory for acoustics in solids, one may useful for the purposes of illustration. In this case,
imagine a spring, extended with some initial tension, one has c ¼c ¼c , c ¼c ¼c , c ¼c ¼
11 22 33 44 55 66 12 13
andconsidertwopointsatpositionsx,andxþdx.If c ¼ c (cid:4)2c , and all other elements of the
23 11 44
one applies an additional local tension, or stress, s, elastic tensor are zero. The two independent elastic
then the spring stretches and the two points are dis- constants may be taken as c and c , but other
11 44
placed by c(x), and cðxþdxÞ respectively. The sep- combinations, such as Young’s modulus Y ¼
aration between the twopointswill have changed by c ð3c (cid:4)4c Þ=ðc (cid:4)c Þ and the bulk modulus
44 11 44 11 44
dc, and the fractional change in the separation, de- B¼c (cid:4)4c =3 are also used. The bulk modulus
11 44
fined asthe strain, ise(cid:1)dc=dx. Hooke’s law for the appears in an important thermodynamic identity
springtakestheforms¼ce,wherecisaone-dimen- involving g, the ratio of the specific heat at constant
sionalelasticconstant.Forathree-dimensionalelastic pressure c to that at constant volume, c :
p v
solid,onemayuseindices(i,j,etc.)whichcantakeon
the values 1, 2, and 3, referring to the x, y, and z c Tb2B
g¼ p ¼1þ ½4(cid:2)
coordinate directions, and generalize the strain to c rc
v p
1(cid:1)@c @c(cid:2)
e (cid:1) iþ j ½1(cid:2) Here T is the temperature and b is the thermal ex-
ij 2 @x @x
j i pansion coefficient (TEC).
The symmetric form of e avoids pure rotations,
ij Anharmonic Effects
which do not involve stress. The stress is generalized
to s , a force per unit area acting on a surface ele- The basic formulation of acoustics in solids involves
ij
ment, where the first index refers to the coordinate theexpansionofenergyminimaaboutequilibriumto
directionofacomponentoftheforce,andthesecond second order, or equivalently, assuming a harmonic
index refers to the coordinate direction of the unit potential, quadratic in strain. However, there are a
normal to the surface element. Hooke’s law becomes numberofeffectswhichrequiregoingbeyondsecond
order. Some effects are related to exceeding ‘‘small
s ¼c e ½2(cid:2) displacements’’ from equilibrium, such as in quan-
ij ijkl kl
tum solids with large zero-point motion, and solids
where c is the 3(cid:3)3(cid:3)3(cid:3)3 (81 element) elastic at high temperatures (near melting) where thermal
ijkl
tensor, and where a summation over repeated indices motionsarelarge.Othereffectsoccurinequilibrium
is implied. For a small volume element, the net force atnormalorlowtemperatures;theseincludethermal
Acoustics:PhysicalPrinciplesandApplicationstoCondensedMatterPhysics 3
expansion, lattice thermal conductivity, and acoustic wave solutions can superimpose and pass through
dissipation. oneanotherwithnoeffect.Ananharmonicpotential
The relationship between anharmonic effects and allows sound waves (lattice vibrations) to interact
thermal expansion is worth discussing. As tempera- andscatterfromoneanother,permittingthetransfer
tureisincreased,theamplitudeofatomicoscillations of energy from an ordered to a disordered form
increases, or equivalently the occupation of higher (acoustic dissipation), and allowing a change in dis-
quantized energy levels increases. If the potential tribution functions in passing from one location to
energy was exactly quadratic in displacements, then another (lattice thermal conductivity).
the center of oscillation, or the expectation value of Anharmonic effects may be probed with acoustic
displacement,wouldremainthesame.Withthesame experiments by measuring the changes in the elastic
average positions for the atoms, the system would constants as the sample is subjected to increasing
not expand with increasing temperature, and the uniaxial or uniform hydrostatic pressure. The coef-
TEC b would be zero. From the thermodynamic ficients which relate the changes to the pressure are
identity in eqn [4], one would also have c ¼c . On referred to as ‘‘third-order elastic constants.’’ How
p v
theotherhand,whenpotentialsareanharmonic,one elastic constants themselves are determined with
may have (in the typical case) a stiffer repulsion at acoustic measurements is discussed next.
short interatomic distances, and weaker attractive
forces at largerdistances. The result is that at higher
energy levels, the ‘‘center’’ position between classical Determining ElasticProperties Experimentally
turning points moves to larger distances, and the
Todeterminethenatureofsoundpropagationinsol-
system thermally expands. This situation is illustrat-
ids, one must solve eqns [1] through [3] with some
ed in Figure 1.
specified boundary conditions. Because of the tensor
Anharmonicityandthermalexpansioncanalsobe
natureoftheequations,therelation betweenparticle
readily related to nonlinear acoustics in fluids. For
displacement and the direction of wave propagation
fluidacoustics,nonlineareffectsareproportionaltoa
is quite complicated. To tackle the complexity and
dimensionless second order parameter ðr=vÞð@v=@rÞ,
make a connection between ultrasound measure-
where v is the sound speed, and the derivative is at
mentsandtheelasticconstants,twoapproachesmay
constantentropy.Forgases,thisparameteris(g(cid:4)1),
be taken. The first approach, used in conventional
which by the thermodynamic identity in eqn [4], is
pulse ultrasound, is to note that if one had a sample
proportional to the thermal expansion coefficient b.
with a large (infinite) plane surface which is perpen-
Thus the absence of nonlinear acoustic effects coin-
dicular toone ofthe principleaxesof theelasticten-
cides with a vanishing thermal expansion.
sor, andif a planewave could belaunched from that
That an anharmonic potential results in acoustic
surface, then the tensor equations would uncouple,
dissipation and lattice thermal conductivity may be
and a longitudinal wave or one of two transverse
understoodbynotingthatwithaharmonicpotential,
wavescouldpropagateindependently.Inthiscase,for
onegetsalinearsecond-orderwaveequation,whose
each wave, the relationship between the sound velo-
city and the independent elastic constants is fairly
straightforward.Whilethedeterminationoftheprin-
ciple axes and the relationships between the three
sound speeds and the relevant elastic constants may
be done analytically, the manipulations are compli-
catedandmustbedoneonacase-by-casebasis;there
y
g is no elucidating general formula. The simplest case
er Atomic position
n ofanisotropicelastic solidwillbepresentedherefor
e
al purposes of illustration. In this case, Newton’s law
enti maybewrittenintermsofthetwoindependentelas-
ot
P tic constants, c and c :
11 44
d2w
r ¼c r2w(cid:4)c =(cid:3)ð=(cid:3)wÞ ½5(cid:2)
dt2 11 44
¼c r2wþðc (cid:4)c Þ=ð=.wÞ ½6(cid:2)
44 11 44
Figure1 Illustrationoftherelationshipbetweenanonquadratic
wherethetwoequationsarerelatedbyanidentityfor
potential energy curve (and nonlinear acoustics) and the phe-
nomenonofthermalexpansion. the = operator. If one has =(cid:3)w¼0, then the first
4 Acoustics:PhysicalPrinciplesandApplicationstoCondensedMatterPhysics
equation becomes a simple wave equation for a ExperimentalMethodsforAcousticMeasurements
longitudinalwavewithspeedv ¼pfficffiffiffiffiffiffi=ffiffirffiffiffi,andifone in Solids
l 11
has =(cid:5) w¼0, then the second equation becomes a
Acoustic measurements with the pulse method are
simplewaveequationfortransversewaveswithspeed
fairly straightforward; emphasis is on careful bond-
pffiffiffiffiffiffiffiffiffiffiffiffi
v ¼ c =r.
t 44 ingoftransducerstosamplesand theuseofsuitable
Although the pulse ultrasound method has been
high-frequency pulse electronics. The RUS method
used extensively in the past, it has a number of di-
is less well known, and can be briefly described as
sadvantages,includingproblemswithtransducerring-
follows.
ing,beamdiffraction,andside-wallscattering,andthe
In a general RUS measurement, the natural fre-
inconvenience that the sample must be recut, repol-
quencies of a sample with stress-free boundary con-
ished, and reattached to a transducer if one wants
ditions are determined by measuring the resonance
more than the three elastic constants accessible with
frequenciesofthesamplewhenheld(lightly,withno
one measurement. The second approach to determi-
bonding agents, at two positions on the sample sur-
ning elastic constants avoids all of the disadvantages.
face) between two transducers. One transducer acts
Thesecondapproachistheonedescribedearlieras
as a drive to excite vibrations in the sample at a
RUS, which involves the use of a computer to nu-
tunable frequency, and the second measures the
merically solve the elastic constants given a set of
amplitude (and possibly the phase) of the response
measurednaturalfrequenciesforasolidwithagiven
ofthesample;asthefrequencyofthedriveisswept,
shape and boundary conditions (usually stress-free
asequenceofresonancepeaksmayberecorded.The
conditions). The computer processing involves sol-
positions of the peaks will determine the natural
ving a ‘‘forward problem’’ (finding the natural fre-
frequencies f (and hence the elastic constants), and
n
quencies in terms of the elastic constants) first and
the quality factors (Q’s, given by f divided by the
n
then inverting. Unlike the conventional pulse ultra-
full width of a peak at its half-power points) will
sound approach, the forward problem does not
provide information about the dissipation of elastic
provide a simple relationship between the modes of
energy.
vibrationandtheelasticconstants;thedisplacements
RUSmayalsobeusedtomeasurethepropertiesof
in the various modes involve all of the elastic con-
thin films on a substrate, to determine the effects of
stants in a complicated manner, and a numerical
induced strain from lattice mismatch, etc. In this
computation is required to sort it all out.
case, the natural frequencies of the substrate alone
The forward problem may be posed as the min-
are measured, then the same sample is again meas-
imization of a Lagrangian L given by
ured with the film in place. From the shifts in the
naturalfrequencies,thepropertiesofthefilmmaybe
ZZZ
L¼ ðro2cc (cid:4)c e e ÞdV ½7(cid:2) determined.
i i ijkl ij kl
A simple apparatus for making RUS measure-
ments is illustrated in Figure 2. In the illustration, a
The minimization is accomplished numerically with
rectangular parallelepiped sample is supported by
a Rayleigh–Ritz method, and the results yield dis-
transducers at diametrically opposite corners. Cor-
crete resonance frequencies, f ¼2po , given the
n n ners are used for contact because they provide elas-
elasticconstants,c .FortheRUStechnique,whatis
ijkl tically weak coupling to the transducers, greatly
needed is the inverse. In most cases, there will be
reducing loading, and because the corners are al-
more measured frequencies than independent elastic
ways elastically active (i.e., they are never nodes),
constants; so what is required is to find a set of in-
and thus can be used to couple to all of the normal
dependent elastic constants which best fits the meas-
modes of vibration.
ured frequencies, usually in a least squares sense.
Furthermore, when there are more measured fre-
quencies than independent elastic constants, then
Acoustics in Fluids
other parameters may be varied in order to best fit
the measured frequencies. Such parameters may in- The thermo-hydrodynamic state of a fluid may be
clude the shape and dimensions of the sample (al- specified with five fields, which may be taken as the
though one known length is necessary), and the massdensityr(r,t),thepressurep(r,t),andthemean
orientation of the crystallographic axes relative to flow velocity u(r, t). The five equations needed to
the faces of the sample. In any case, it is not neces- determine the five fields are conservation of mass,
sary that crystallographic axes be oriented with re- Newton’slawforthemotionofthecenterofmassof
specttofacesofasample,althoughcomputationsare afluidelement(threecomponents),andconservation
greatly simplified if they are oriented. ofenergyformotionaboutthecenterofmass.These