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1/13182 Los Alamos Science Number 21 1993

Crystals and Ultrasoundold-fashioned materials science

Albert Migliori and Zachary Fisk

Stone Age, Bronze Age, Iron Age,

the age of exploration and thesearch for gold, the industrial rev-

olution and the steel that made it possi-ble, and, of course, the present age of electronicstotally dependent on tinychunks of ultrapure, single-crystal sili-conit seems that entire chapters inhuman history are strongly connectedto certain special materials that enabledmassive bursts of what has come to becalled progress.

But the pursuit of human progress isnot the driving force behind physicistsstudying new materials today. We havemuch simpler motivations, more akin tothe curiosity of the scientist who fourthousand years ago first noticed thebrilliant luster revealed by scratchingthe surface of the glob of bronze pulledfrom a state-of-the-art, charcoal-driven,high-temperature, controlled-atmos-phere furnace. That scientist had, byserendipity, produced some sort of veryimperfect alloy. The appearance andthe properties of even that poor excuse

for bronze were so remarkable that

many, many others spent lifetimes ad-vancing available technology to im-prove that material to the point wherethe fully developed and vastly im-proved copper alloys were to the firstbronze nugget as that nugget was to thestone it replaced.

Just as important as the productionof high-performance bronzes were theparallel developments necessary to un-derstand completely the properties of such alloys. And so goes the story formany other materials. The entire busi-ness of materials science, from the firstdiscovery of a new substance to the fulldevelopment of its properties, is atremendously attractive intellectualpuzzle of sufficient complexity to inter-est almost anyone. Moreover, the in-credible importance of new materials,the qualitative changes in human lifethat they produce, their intrinsic physi-cal attractiveness, their appeal to the in-tellect, and our total inability to foreseehow important their impact will be

make the study of new materials excit-

ing, risky, difficult, and what seemsbest described as fun.

Essential to the richness of the fieldis an understanding of the laws of physics and chemistry that enable de-tection and realization of new proper-ties of matter. But despite having itsroots in basic science, condensed-mat-ter physics is viewed as a kind of ap-plied science: After all, we know all of the fundamental physical laws that de-scribe the individual atoms in solids.The interesting fact, however, is thatthe collective properties of many atomsassembled together are too complex tobe predicted from first principles. Forexample, the sharp transition from non-magnetic to magnetic behavior in ironat 1043 kelvins cannot be predictedfrom a calculation of the properties of an assemblage of a few or even a fewhundred iron atoms. Such calculationsindicate no abrupt transition to ferro-magnetism but only a gradual changefrom weakly magnetic to more strongly


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magnetic. Nobel laureate P. W. Ander-son has most clearly laid out this idea inhis article More is Different: Hemakes the point that phenomena arisingas more and more particles interact arenot simple extrapolations of the behav-ior of a few particles. Further, we can-not, nor do we want to, solve the mostbasic equations for the many-atom sys-tem because those equations are toocomplex and provide too much informa-

tion. No one cares where a particularatom is at a particular time in the mag-netic powder of a cassette tape. What isimportant is the overall magnetizationof millions of atoms. Thus most of theuseful theoretical descriptions of solidsrely on statistical averages to predictmeasurable quantities. There is, then, atheoretical gap between our knowledgeof the basic laws for each atom and ourability to predict what large numbers of interacting atoms do.

In the last few years some attempts

have been made to control macroscopicmaterial properties by constructing arti-ficial materials through the depositionof sequential layers of carefully con-trolled composition and thickness. At-tempts are being made also to producenanostructures, materials assembledatom by atom through such processesas molecular-beam epitaxy and chemi-cal or vapor deposition. Both ap-proaches produce very small, but notquite microscopic, building blocks, orrepeated units. Such designed materi-als are applicable to the production of practical devices as well as the study of quantum physics, and they have proper-ties that are roughly predictable fromtheory. Still, the theoretical tools need-ed to predict their material propertiesnecessarily include statistics andmacroscopic approximations.

Recent work on materials by de-sign contrasts strongly with the work of what one could call old-fashionedmaterials scientists, who work semi-

empirically and produce structures thatare largely only what nature allows;that is, the materials are microscopical-ly homogeneous and theoretically in-tractable. The payoff from this old-fashioned work is the discovery of completely new phenomena, whichoften arise when the number of atomsis large (on the order of 10 18 ).

A particular innovation that arosefrom the large-number effect coupled

with old-fashioned intuition was theinadvertent discovery by K. Mueller in1987 that certain cuprates (namely,those copper-oxide compounds that aredoped with transition-metal and otherimpurities) are high-temperature super-conductors, that is, they become super-conducting at temperatures well aboveabsolute zero (above 30 kelvins).Mueller knew that those cuprates arequite unusual solids. He also knew thatin many materials, if a smaller atom isdeliberately substituted for a larger

atom at the center of each unit cell inthe crystal lattice, then when the solidcools the smaller atom and its cloud of electrons would not have a stable rest-ing spot in the center of its symmetriccage. As a result it must move to someother position in the unit cell and spon-taneously break the crystal symmetry.The effect (called a Jahn-Teller insta-bility) is driven by large symmetric ar-rays of atoms and is only poorly under-stood in detail (large numbers + in-tuition = mysterious phenomena).Muellers brilliant conjecture was that,in electrical conductors, the distortionof the crystal lattice resulting from theJahn-Teller instability would producenew material properties provided theenergy associated with the lattice dis-tortion is comparable to one of the en-ergy scales of the conduction electrons.As it turned out, the marvelous super-conducting properties of the cuprateswere not attributable to Muellers con-

jecture, but nevertheless his very cre-

ative idea led to their discovery.The unpredictable effects found

when large numbers of various atomsare assembled thus provide both moti-vation and justification for the empiri-cally based search for new physics andchemistry through the study of new ma-terials. Fortunately, the production of any single new material is often accom-plished by a very few scientists work-ing with a small budget and a limited

collection of inexpensive equipment.The demonstration of superconductivi-ty in heavy-fermion compounds by F.Steglich at Universitt zu Kln, and thediscovery of the high-temperature su-perconductors by K. Meller at IBMZurich are typical examples of small,successful efforts. However, underly-ing all of the apparently small efforts isa powerful information and technologybase easily accessible to those workingwithin the umbrella of a large researchlaboratory or university. That access is

crucial; without it small groups of sci-entists, driven by whatever odd notionsmotivate them, could not succeed.

The Remarkable Effectsof Impurities

Also important to success is a clearfocus on some particular set of materialproperties. Many groups today are fo-cused on modifying the electronic andmagnetic properties of solids by dopingthem with impurities and studying the ef-fects of those impurities as the solids arecooled to low temperatures. Often thepresence of impurities causes the solidsto undergo a drastic change in somephysical property as they cool, from con-ducting to superconducting, from non-magnetic to magnetic, from paraelectricto ferroelectric, and so on. These drasticchanges are called phase transitions.

It is a curious finding that someseemingly intrinsic properties of solids

1831993 Number 21 Los Alamos Science

Crystals and Ultrasound


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owe their existence tosmall amounts of im-purities. For example,off-the-shelf tungstenmetal is typically hardand brittle. Whenpains are taken to re-move the various traceimpurities dissolved innominally pure tung-sten, it becomes quite

soft. The change inhardness is so pro-nounced that the purityof a tungsten samplecan be reliably esti-mated with an ordinaryDremel grinding tool.The very pure metal issofter than annealedcopper, and its x-raypattern becomesblurred, no doubt be-cause of the large de-

formations that easilyoccur in its soft condi-tion. Perhaps a moresurprising case is thatof salt (NaCl). UsualNaCl contains a smallamount of hydroxl,OH . When great careis taken to prepareOH -free NaCl, the re-sulting solid can be spread like butteron bread.

The most important example ever inwhich the effects of impurities were un-derstood and then exploited was in thedevelopment of transistors. In the earlydecades of this century, work byThompson, Drude, Pauli, Fermi, Dirac,and Sommerfeld led from the discoveryof the electron to an understanding of the special quantum physics governingconduction processes in pure and im-pure (doped) semiconductors.

This revolution in understanding theelectronic properties of solids had, by

the late 1930s and early 1940s, germi-nated into an idea in the minds of JohnBardeen, William Shockley, and WalterBrattain at AT&T Bell Laboratories.They realized that the intrinsic proper-ties of very pure (but still doped) semi-conductors would allow the electricalconductivities of these materials to becontrolled and varied by externally ap-plied voltage, just as the flow of elec-trons is in a vacuum tube. Thereforesuch materials might serve as replace-ments for the reed relays in telephoneswitchboards. In order to obtain thiseffect, they knew they needed to pro-

duce single crystalsof germanium andsilicon with the im-purity content keptto the unprecedentedlow level of 1 partper millionandthey used everypiece of technologyavailable to reachtheir goal.

Along the way,when state-of-the-artwas inadequate,their team developednew methods of growing and charac-terizing crystals,such as zone-refine-ment, a simplemethod for remov-ing impurities whilea crystal is growing.In the end they were

able to make thefirst transistors. Inthe process of suc-ceeding, they signif-icantly advanced thetheory and methodsof solid-statephysics. Not onlydid they predict theneed for purity so

high as to seem unnecessary, unachiev-able, and up until then unmeasurable,they also developed methods to pro-duce crystals of the desired purity, de-veloped characterization schemes,grew the crystals, and then producedpractical devices whose function wastotally reliant on the remaining andcarefully controlled impurities. Theirodyssey from the purest of basic re-search to the stupendous discovery of one of the most important practical de-vices ever was one of the greatestachievements of twentieth centuryphysics.

184 Los Alamos Science Number 21 1993


Figure 1. Materials Science at WorkThe production of new materials with exotic properties leads to the devel-opment of new tools to study them, the use of those tools to advance theunderstanding of the new materials and the application of the new materi-als, the new tools, and the new science in the commercial sector. Here weelaborate on this paradigm with examples from our own work.

Ne w Ph ys ic s

No vel e lec tronic,

s truc tura l, and

magne tic phenomena

N e w

s u p e r c o n d u c t o r

s

N e w m a g n e t s

N e w t e c h n i q u e s

f o r

n o n d e s t r u c t i v e

t e s t i n g

N e w T o o l s R e s o n a n t u l t r a s o u n d s p e c t r o s c o p y

N e w M a t e r i a l s

S y s t ems w i t h hi ghl y c o r r e l a t ed e l ec t r o ns suc h as hi gh T c sup er c o nd uc t o r s and heav y -f er mi o n sup er c o nd uc t o r s

T e c h n o l o g y

T r a n s f e r


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Second-Order Phase Transitionsand the Measurement of

Elastic Properties

We have been interested in produc-ing and understanding materials thatundergo second-order phase transitions.In some metals the transition from thenormal conducting phase to the super-conducting phase is second order, asare many other structural, magnetic,

and electric phase transitions. In thistype of phase transition, the symmetryand material properties of the solidchange gradually rather than abruptly(as in first-order phase transitions), butthe changes nevertheless begin at avery distinct temperature called thecritical temperature, T c. The solid is ina more symmetric phase above T c thanbelow.

In a crudely parallel way the intro-duction of various amounts of impuri-ties can also break the symmetry of a

solid, at least locally. The combinationof changes in temperature and controlof impurity levels can thus be used totune, in more or less continuous ways,important physical properties inmetastable materials. Therefore, thestudy of such systems is widespread inmaterials science. It involves manytools, requires science, intuition, andluck, and it produces returns in bothfundamental physics and very practicalapplications.

The tale we are about to tell con-cerns the development of a very oldtechnique into a powerful and essential-ly new one for determining the changesin the elastic stiffness of crystals, par-ticularly as they undergo second-orderphase transitions. This now maturetechnique, called resonant ultrasoundspectroscopy, has led to a simple proce-dure for making what once was a verydifficult measurement. Our work illus-trates the synergism among the produc-tion of new materials, the invention of

new tools to study them, the subsequentuse of those tools to advance thephysics of other materials, and the ap-plication of the tools, the science, andthe materials to industrial applications.In other words, our story illustrates a

paradigm of how materials science ac-tually works (Figure 1).

Measuring the elastic stiffness, orelasticity, of a solid as a function of temperature has been a traditional andvery important technique for studying



Discontinuities at Second-Order Phase Transitions

Three thermodynamic quantities, the specic heat C P , the thermal-expansioncoefcient , and the elastic-stiffness tensor cij , can be discontinuous at thecritical temperature T c of a second-order phase transition. Each is proportionalto a second derivative of G , the change in the Gibbs free energy per unitmass across the boundary separating two phases. Although G is always zeroin any phase transition, in a second-order phase transition the rst derivativesof G are always continuous and the rst sign of any discontinuities occurs inthe second derivative of G .

The relationships between G and C P , , and cij are particularly simpleto write down for a liquid because the elastic-stiffness tensor has only onecomponent, namely, the bulk modulus, B .

In terms of temperature T , pressure P , and volume V , the thermodynamicrelations are

@2 G@P 2

=@ V@P

= 1B

; B = bulk modulus

@2 G@T2

= @ S@T

= C PT

; C P = specic heat

@2 G@P @T

=@ V

@T= : = thermal-expansion

coefcient

V is the change in volume per unit mass across the phase boundary and Sis the change in entropy per unit mass across the phase boundary. Although G , V , and S are zero and the phase transition is continuous, B , C P , and can exhibit discontinuities at the critical temperature T c .

The relationships for a solid are more complicated and are usually written interms of the stress tensor ij and the strain tensor kl instead of P and V .Further, the bulk modulus B becomes the elastic-stiffness tensor cijkl , which

relates the stress to the strain in terms of the fundamental relation

ij =X

kl

cijkl kl :


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second-order phase transitions in mate-rials doped with impurities. Naturehelps us with the measurement becausethe elasticity of a solid is discontinuousat T c; that is, it jumps to a differentvalue when that solid begins to undergoa second-order phase transition. The

jump is perhaps surprising because, aswe mentioned above, the solid exhibitsno obvious microscopic changes at thecritical temperature. The atoms do not

suddenly change position, magnetismand ferroelectricity do not suddenly ap-pear, and a conductor is not suddenlyable to carry super-large currents.However, three thermodynamic quanti-tiesthe elastic stiffness, the specificheat, and the thermal-expansion coeffi-cientdo exhibit discontinuities at T cand those abrupt changes can be mea-sured. The accompanying box definesthese three quantities in terms of varia-tions with respect to temperature andpressure of the change in the Gibbs free

energy per unit mass across the bound-ary between the two phases.

Discontinuities are a boon to the ex-perimentalist because they are often themost unambiguous of measured quanti-ties. Of the three discontinuous quanti-ties mentioned, elastic stiffness is par-ticularly informative because in a solidthis quantity is a fourth-rank tensor,c ijkl , with 81 components. The elastic-stiffness tensor relates the stresses(forces) to the strains (displacements)in the solid through the fundamental re-lation

where ij is the stress tensor and kl isthe strain tensor. This relation betweenstress and strain is simply the general-ization of Hookes law, F = - kx, whereF is the force developed in a spring if itis stretched a distance x. Thus the elas-tic-stiffness tensor c ijkl , in analogy withthe spring constant k , describes the

stiffness of the bonds holding the solidtogether. Although c ijkl has 81 compo-nents (or moduli), in crystalline struc-tures that do not produce external mag-netic fields, the tensor has at most 21independent components. Consequent-ly the elastic stiffness tensor is com-monly written as cij , where i and j runfrom 1 to 6, for pragmatic rather thanmathematical reasons.

It is also important to realize that the

stiffness of a metallic solid comes notonly from the chemical bonds that holdthe ions together but also from the de-generacy pressure of the Fermi sea of electrons. Thus a great deal of informa-tion about the crystal is contained in acomplete description of its elasticity.Because of this wealth of information,if one knows the crystal structure andcan measure the changes in the elastic-stiffness tensor as a function of temper-ature through a phase transition, then itis possible to infer the detailed changes

in the crystal lattice or in the electronicstructures that are driving the phasetransition. Few other measurementscan reveal as much of the physics of phase transitions. Consequently wewould like to be able to make this mea-surement in high- T c superconductors,heavy-fermion superconductors, andother exotic materials exhibiting sec-ond-order phase transitions.

The Need for a NewMeasurement Technique

Unfortunately, nature, while permit-ting us to produce single crystals of high-temperature superconductors,heavy-fermion superconductors, andother materials in variously doped ver-sions, has often restricted the dimen-sions of the crystals that can be easilygrown to the millimeter range. Thatsize is entirely adequate for many mea-surements. But for the crucial determi-

nation of elastic stiffness, a thermody-namic measurement that provided thefirst verification of the BCS theory of ordinary low-temperature superconduc-tivity, millimeter-sized crystals are adisaster.

What we describe below is a solu-tion to this measurement problem. Itinvolves measuring the resonances (ornatural frequencies of vibration) of acrystal and often makes measurement

of the elastic-stiffness tensor more triv-ial than measurement of the resistivitytensor.

Prior to development of this the newresonance approach, elastic moduliwere determined mostly from measure-ments of the speeds of longitudinal andshear sound waves along different di-rections in a sample. Figures 2 showsthe simple relationships among thesesound speeds and the elastic moduli fora cubic crystal. For a large chunk of isotropic material such as ordinary

glass, there are only two independentsound speeds, one for longitudinalwaves in which the atoms vibrate alongthe direction of the sound wave (theseare the waves we hear), and shearwaves, in which the atoms vibrate in adirection perpendicular to the directionof the sound waves. A simple measure-ment of the pulse-echo time of eachtype of sound is easy to do and yieldssound speeds for glass accurately andcompletely. As shown in Figure 2, thepulse-echo time is the time for a shortsound pulse to travel the distance fromone face of the sample to the oppositeface and back again.

In applying the pulse-echo techniqueto single crystals of more exotic materi-als, the first catch is that the speed of sound in nearly every solid that inter-ests us is a blazing 5 millimeters permicrosecond or so (Mach 15 in jetfighter units). The second catch is thatwe need to use very-high-frequency ul-trasound (several hundred megahertz or



ij =X

kl

cijkl kl :


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higher) to measure the sound speed insamples whose largest dimension isabout a millimeter. But often ultrason-ic attenuation increases to prohibitivelevels at such frequencies.

Lets say that we want to measurethe speed of sound in a 1-centimetercube (big!) of La 2CuO 4. Above 530kelvins this cuprate has a tetragonal

structure (all three crystal axes of theunit cell are at right angles, two axesare equal in length, and one is longer orshorter). La 2CuO 4 is an interestingmaterial because when doped withstrontium it becomes a high-tempera-ture superconductor and its doped ver-sions undergo a structural phase transi-tion as they cool, from a tetragonal to

an orthorhombic structure (see Figure3). Therefore, were interested in thevariation in sound speed (or equivalent-ly, elastic-stiffness tensor) as a functionof temperature through the structuralphase transition. In our 1-centimetersample, a single cycle (wavelength) of 50-megahertz sound is 0.1 millimeterlong. To obtain a good signal-to-noise



(a) Longitudinal and Shear Sound Waves

In longitudinal sound waves atoms vibratealong the direction of wave propagation. Inshear sound waves atoms vibrate perpen-dicularly to the direction of wave propagation.

(b) Sound Speeds and Elastic Moduli

The elastic-stiffness tensor of a cubic crystalhas three indepedent elastic moduli ( c 11 , c 12 ,and c 44 ) and therefore there are three soundspeeds in a cubic crystal. The elasticmodulus c 11 is determined from the speed v l of the longitudinal sound wave shown atright. The two shear moduli c 12 and c 44 aredetermined from two speeds v s1 and v s2 ofthe two shear waves shown at right.

(c) Pulse-Echo Measurements of Sound Speed

Conventional measurements of the elastic stiffness tensor c ij are made bymeasuring the sound speeds of longitudinal and shear sound waves alongvarious directions in a large crystal. In the example shown here a shortpulse of longitudinal sound is generated by a transducer on one crystalface; the sound pulse travels as a narrow beam through the crystal andbounces back from the opposite face to produce an echo that is picked upby the transducer. The measured time between the inititation of the pulseand the echo (called the pulse-echo time, echo ) is equal to 2 L / v l , where L is the distance between the two crystal faces and v l is the longitudinalsound speed. Thus the sound speed can be calculated directly fromecho . Further, if the density of the crystal, , is known, the measurementalso provids a direct determination of c 11 , one component of the elastic-stiffness tensor. By placing the transducer at various locations on thecyrstal and using both longitudinal and shear sound waves, all thecomponents of the elastic-stiffness tensor can be determined.

z

x

y

Longitudinal sound wave

Shear sound wave

Shear sound wave

Direction of sound

propagation

Direction of

vibrational motion

Longitudinal wave

First shear wave

Second shear wave

v l =c 11

v s1 =c 44

v s2 =c 11 c 12

2

Orientation in cubic crystal

Sound speed

( is the crystal density)

v l =2L

echo

Sound speed

L

Soundpulse

Transducer

Figure 2. Sound Speeds, Elastic Moduli, and Pulse-Echo Measurements


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(a) Tetragonal Structure of La 2CuO 4 above T c

The unit cell of La 2CuO 4 above the criticaltemperature, T c = 525 kelvins, is tetragonal;that is, the axes are at right angles, the x and y axes are of equal length, and the z axis islonger. Note that the oxygen atoms form anoctahedron. Each oxygen atom at the apex ofthe octahedron sits in a double-well potential,

V (shown here in red). Thermal motions of theapical oxygen atoms are sufficiently large thatthe equilibrium position of each is at the centerof its potential well.

(b) Orthorhombic Structure of La 2CuO 4 below T c

At temperatures below the critical temperature,each apical oxygen atom falls into one or theother minima of its double-well potential sothat the octahedron of oxygen atoms in eachunit cell has a static tilt. Since octahedra inadjacent cells tilt in opposite directions, two ofthe old unit cells form the unit cell of the newstructure. Thus the crystal now has anorthorhombic structure (all three axes of theunit cell are unequal in length).

(c) Top View of Phase Transition fromTetragonal to Orthorhombic Structures ofLa2CuO 4

c 44

c 66z

x y

Shearwaves

T > T c T > T c

T < T c

T < T c

T < T c

E n e r g y

V

V

Position of apical oxygen atom

y

x

T > T c

As the temperature decreases belowT c and oxygen octahedra develop

permanent tilts, shear forces developand change the square array of copperatoms to a rhombus. Here thedistortion is exaggerated for thepurposes of illustration. Theorypredicts that the shear modulus c 66 ,which characterizes shear forces in thex-y plane, will shift abruptly during thephase transition whereas the shearmodulus c 44 , which characterizesshear forces in the z-y and z-x planes,will remain unaffected by the phasetransition.

Figure 3. Structural Phase Transition of La 2CuO 4

Oxygen

Lanthanum

Copper

Orthorhombic unit cell

Tetragonalunit cell


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ratio, we send in a pulse consisting of many cycles of sound and bounce thepulse from the inside walls of the crys-tal. Such a pulse might last, then, 0.2microseconds and be 1 millimeter long,10 percent of the width of the sample.

To determine the relative change insound speed to 1 part per million, a notunreasonable goal, we would need totime the pulse echo to an accuracy of 2picoseconds, corresponding to about

1/1000 of a wavelength of sound. Allof this is not wildly difficult to do forthe frequencies appropriate to big sam-ples. But for a 1-millimeter crystal wemust increase the frequency, and there-fore the timing accuracy, by a factor of 10. At 500 megahertz and 0.2-picosec-ond timing accuracy, things get tough.Even worse, the orthorhombic phase of La 2CuO 4, whether pure or doped, hasnine independent elastic moduli andtherefore nine different sound speeds,each requiring a separate measurement.

And each measurement requires that asmall transducer be glued to the sampleand that it not fall off as we cool thesample from room temperature down toa temperature well below the criticaltemperature. The pulse echo is an in-termittent signal, and the signal caneasily become so greatly attenuated thatbarely one echo can be detected.

The Development of ResonantUltrasound Spectroscopy

Direct measurement of the pulse-echo time (or sound speed) to deter-mine elastic stiffness would seem to benearly hopeless for small crystals. Butusually, when nature makes the mea-surement of time tough, the measure-ment of frequency is much easier. Infact, if we apply continuous sound to asolid sample, the solid will resonate, orring just like a bell, provided the ap-plied sound frequency is one of the

solids natural vibrational frequencies.(Application of this technique to long,thin rods is perhaps hundreds of yearsold!) Because the resonant frequenciesare related to the pulse-echo times, wecan measure resonant frequencies in-stead of pulse-echo times to determineelastic-stiffness tensors.

We measure the frequency of a reso-nance by driving the sample continu-ously with ultrasound and slowlychanging the frequency of the sounduntil the sample suddenly starts to res-onate (1-millimeter samples resonate at

1 megahertz or so, a wonderfully lowfrequency). The resonating sample actslike a natural amplifier, greatly increas-ing the amplitude of the vibrations (afactor of 10,000 is not uncommon). Wenow need only measure frequency. Wecan take as long as we like to do it, andthe naturally amplified signal is presentduring the entire time we are doing themeasurement. Therefore, a resonancemeasurement can easily have a signal-to-noise ratio that is a million timeshigher than a measurement of the echotime of short pulses of sound.



Figure 4. Distortions of a Cube-Shaped Sample on ResonanceEach figure above is an example of the distortions that a cubical object undergoeswhen it is driven at a natural (resonant) vibrational frequency. At each resonant fre-quency the vibrational motion is dependent on complicated linear combinations of allthe elastic moduli as well as the exact shape of the object. Because of this complexi-ty, Rayleigh, Love, and others were unable to compute the resonances of such short,fat objects from their elastic moduli. With the advent of big computers and someclever algorithms, such computations are now easily done.


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You might, at this point, wonderwhy anyone would have used echo-times rather than resonant frequenciesto determine the components of theelastic-stiffness tensor. The catch isthat the resonant frequencies are hardto interpret because they have a some-what complicated relationship to theelastic moduli.

Over a hundred years ago JohnWilliam Strutt, Baron Rayleigh, at-

tempted to calculate the resonant fre-quencies of cubes, short cylinders, andother short fat objects from knownelastic moduli. Unlike the correspond-ing calculation for thin rods and plates,this wonderfully tantalizing and seem-ingly simple problem stymied the bril-liant Rayleigh, who finally concludedthat the problem . . . has for the mostpart, resisted attack. Figure 4 illus-trates the origin of the complexity:When a cube resonates, it exhibits sig-nificant distortion typically involving

all the elastic moduli in complicatedlinear combinations. A. E. H. Love,Willis Lamb, and others were alsostymied by this problem. But in the1960s Orson Anderson of ColumbiaUniversity and his postdoc, Harold De-marest, hit on a fast, accurate numericalalgorithm for obtaining the solution.The algorithm requires computationsonly at the surface of the object andachieves an accuracy much greater thanthe relatively crude finite-elementtechniques, which compute throughoutthe volume of a sample and are the onlyalternate method. With this fast, accu-rate algorithm Anderson and hiscoworkers were able to make the firstfully interpreted resonant ultrasoundmeasurements on large, high qualitymineral crystals. That is, they wereable to match measured resonance fre-quencies to predicted ones, but theyhad big crystals, big signals, and almostperfectly known answers for the elasticmoduli before they started. In contrast,

our small and completely uncharacter-ized samples could not be so easily at-tacked, so we had to develop new hard-ware and refined analysis procedures.

To measure the resonant frequenciesof a 1-millimeter object, we had better(1) not disturb the samples resonanceswith the measuring device and (2) notgenerate resonances in the measuringsystem that might confuse the issue.To not change what we wish to measure

in the process of measuring it requiresthat our transducers make extremelyweak contact with the sample. So if our sample is approximately cubical,we lightly contact its corners withtransducers, using no glue or any othercoupling medium, and apply just agram or so of force (pardon the unit).

With that light contact we lose a fac-tor of 1000 in our signal-to-noise ratio.We must also drive the sample lightlyso as not to destroy it. Fortunately, thenatural amplification at resonance re-

covers most of what is lost with a gen-tle drive. The resonances generated inthe measurement apparatus are anothermatter. We are not ourselves smallenough to make transducers muchsmaller than 1 millimeter or so; there-fore, most transducers would ring in

just the same frequency range as thesample and spoil the signal. The wayaround this problem is to make thetransducers of a composite structureconsisting mostly of single-crystal dia-mond. Diamond, with a sound velocityof 17 millimeters per microsecond(Mach 50!) has such a high soundspeed that its resonance frequencies aremuch higher than those of the crystalsof comparable size that we wish tomeasure.

When we hooked all the pieces to-gether and connected them to electron-ics specially designed (with the help of Thomas Bell) to maximize the signal-to-noise ratio, John Sarrad, a studentworking with us on his thesis research,

was able to measure all of the lowerfifty or so resonant frequencies of acrystal with a largest dimension of 1millimeter or so. The entire experimentfits on the top of a desk and costs lessthan a car. Figure 5 shows the mea-surement apparatus and a typical reso-nance spectrum.

Analysis of Resonant

Ultrasound Data

To our surprise, the algorithm tocompute resonant frequencies from elas-tic moduli and our simple technique tomeasure accurately the resonances of auseful sample were not quite enough toproduce a robust technique for determin-ing elastic moduli from measured reso-nances. (Note that this is the inverse of the problem solved by Orson Anderson.)In real life the measured resonant fre-quencies have some errors relative to the

resonant frequencies of a perfect sam-ple. The errors arise from many sourcesincluding anomalies in the geometry of the sample (such as chips and roundededges) and transducer loading effects.Typically, the resonances of the mathe-matical model of the solid and thosemeasured for the real object differ in fre-quency by 0.1 percent, more or less.Further, one or two resonances out of the thirty or forty that exist in a givenfrequency range may be missing fromour data because some resonances, byaccident, may have no component of vi-brational motion along the driving direc-tion of the transducer. The experimentaldata are fed into a computer programthat tries to find a set of elastic moduliconsistent with the measured reso-nances, the sample dimensions, and thesymmetry of the samples crystal lattice.If one or two resonances are missingfrom the measured resonance spectrum,the computer will invent an answer thatis not even wrong.




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We have not yet developed a com-puter program for guessing at the ab-sence of a resonance; however, thoseguesses can be made by looking at thedata and relying on the intuition thatcomes from experience. After severaliterations and applications of WilliamM. Visschers pioneering computer al-gorithms, we usually do guess whichresonance is missing, and then the com-puter instantly determines the elastic

moduli of the sample to an accuracy of 0.05 percent or better. The algorithmsalso determine the true dimensions of the sample to an accuracy as high as0.01 micron. The results are the mostaccurate and complete measurements todate of the elastic-stiffness tensor, andthe many elastic moduli are obtainedsimultaneously from a single measure-ment of the resonance spectrum.

Our ability to infer the existence of missing resonances in our data was crit-ically dependent on our being able to

produce some samples that were perfectsmall single crystals and then usingtheir known perfection to refine our ex-periment for less perfect crystals. ThisMcLuhanesque bootstrapping of the de-velopment of a measurement techniqueby using known properties of near-per-fect materials is not uncommon and isoften the only safe route. Without thecollaboration among crystal growers,instrument developers, and theorists, theresonant ultrasound spectrometer couldnot have been developed to its full po-tential. Without the instrument our per-fect small single crystals would have re-mained strangers to the ultrasound andtheir elastic properties would still be amystery. Although the project has in-volved many types of expertise, the sizeof the overall effort has remained small.The required components of the effort,however, were not predictable when webegan developing the instrument. Thusthe work really needed to be done at anational laboratory, which could serve

as a technology supermarket with alarge and available stock from which toselect just the right things.

Applications of the ResonantUltrasound Spectrometer

With our resonant ultrasound spec-trometer and a wonderful collection of

crystals doped with various impurities,we, our postdoc, and our students havebeen able to make steady progress inexpanding our knowledge of second-order phase transitions. Our resonantultrasound measurements on pure anddoped La 2CuO 4 single crystals neartheir structural phase transition fromtetragonal to orthorhombic are particu-larly interesting.



Electronicsignal

generator

Ulrasonic transducers

SampleElectronicdetector

Shown here are theresonances between 0.4

and 1.1 megahertz. Theresonance lines areextremely sharp as shownby the enlargement of atypical line. In fact, theresonant frequency isdetermined to an accuracyof 1 part per million.

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1Frequency (megahertz)

A m p

l i t u

d e

( v o

l t s

)

0.0

0.2

0.895 0.897

30 hertzwide

(b) Resonance Spectum of La 1.86 Sr 0.14 CuO 4 at T = 297.2 kelvins

(a) Measurement Technique

The photograph shows a sample ofLa1.86 Sr 0.14 CuO 4 held between two trans-ducers, one of which drives the sampleover a continuous range of frequencies inthe megahertz range. The apparatus isplaced in a cryostat, which can cool thesample to the desired temperatures.While the temperature is held fixed, thefrequency of the driver is gradually andautomatically changed by an electronicsignal generator until the sampleresonates and thereby amplifies theapplied signal. The amplified signal ispicked up by the second transducer andrecorded automatically by a speciallydesigned electronic detector. Thefrequency of the driver then continues tochange again until another resonantfrequency is reached. All the resonantfrequencies of the sample are thus

recorded during a single experimentalrun. The entire resonance spectrum ismeasured in several seconds. To study asecond-order phase transition in thesample, the experiment is repeated atvarious temperatures between roomtemperature and the critical temperatureof the phase transition.

Figure 5. Ultrasound Measurements of La 1.86 Sr 0.14 CuO 4


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When we started the measurement,the structural phase transition wasthought to be well understood. Asshown in Figure 3a, at high tempera-tures each apical oxygen atom in theunit cell sits in the center of a double-well potential (a potential with twominima). More precisely, fast thermalmotions cause each oxygen atom to vi-brate so that it appears to fill the avail-able space symmetrically. Near thecritical temperature of the structuralphase transition, the energy of thermalvibrations is just about equal to theheight of the bump (local maximum) inthe center of the potential well. Thusthe thermal motion is now too weak tokeep the atom buzzing in the center

(the atom can no longer make it overthe bump of the potential well). As thesolid cools further, the atom graduallydrops into one of the two small wells atthe bottom of the potential. Thus thearrangement of atoms is altered; that is,the solid experiences a structural phasetransition.

Figure 3b shows the rearrangement.Each oxygen atom in the unit cell un-dergoes a displacement such that theoctahedron formed by the oxygenatoms develops a permanent tilt. Be-cause the octahedra in neighboring unitcells develop tilts in alternate direc-tions, the two neighboring unit cells areno longer identical. In fact, the unitcell of the new structure is now com-

posed of two of the old unit cells andhas orthorhombic symmetry.

In Figure 3c, a projection of thecrystal lattice onto the x- y plane showsthat as the oxygen octahedra developalternating tilts during the phase transi-tion, they pull apart the corners of theunit cells and shear the square array of atoms into a rhombus (or diamondshape). Thus, according to theory, theshear modulus c66 should drop abruptlyas the temperature drops below the crit-ical temperature, whereas the shearmodulus c44 should remain unaffected.In fact, very general symmetry argu-ments can be used to predict exactlyhow each of the six independent elasticmoduli of the tetragonal structure



0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 c 4 4

( 1 0 1 2 d y n e s p e r s q u a r e c e n

t i m e

t e r )

220 240 260 280 300Temperature (kelvins)

(a) Variation of c 44 with Temperature

Theoreticalprediction

(b) Variation of c 66 with Temperature

0.6

0.5

0.4

0.3

0.2

0.1

0.0220 240 260 280 300

Temperature (kelvins)

Theoreticalprediction

c 6 6

( 1 0 1 2 d y n e s p e r s q u a r e c e n

t i m e

t e r )

c 44

resonancec 66

resonance

Figure 6. Surprising Results from Resonant Ultrasound MeasurementsTo study the transition of the superconducting cuprate La 1.86 Sr 0.14 CuO 4 from a tetragonal structure to an orthorhombic structure,we measured the elastic moduli of an approximately cubical, strontium-doped sample as a function of temperature using the reso-nant ultrasound technique. Resonance spectra were taken at various temperatures from room temperature down to the criticaltemperature of the sample, 223 kelvins. The values of the elastic moduli at the various temperatures were determined from theresonance spectra by using a computer program discussed in the main text. As explained in Figure 4 and the main text, theorypredicts that the shear modulus c 44 should remain constant through the transition, whereas the shear modulus c 66 should remain

constant but then undergo an abrupt change at (or slightly above) T c. Above are shown the results from the resonant ultrasoundmeasurements. (a) shows that the value of c 44 remains constant, as expected, and (b) shows that the value of c 66 begins de-creasing at a temperature roughly 80 kelvins above T c and continues to decrease as the temperature decreases to T c. The depar-ture from theoretical predictions (shown in red) indicates that this second-order phase transition is less well understood than wasthought. The dotted and dashed lines of the cube inset in each graph indicate peak distortions at opposite ends of the vibrationalcycle for each resonance.


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should change as a function of tempera-ture near the critical temperature.

Much to our surprise, the behaviorof the elastic moduli as a function of temperature determined from our reso-nant ultrasound measurements differedmarkedly from theoretical predictions.Rather than changing abruptly at thecritical temperature, the shear modulusc66 changes smoothly over a tempera-ture drop of 80 kelvins or so (Figure 6).

Because our simultaneous measure-ments of the six elastic moduli are ex-tremely precise, there can be no doubtas to the validity of our results. Whatwe are left with is a clear indicationthat this structural phase transition isnot as well understood theoretically aswas thought. Moreover, measurementson pure La 2CuO 4 as well as samplesdoped with oxygen, barium, and stron-tium showed similar results. Whetherthese results are related to the mysteriesof high-temperature cuprate supercon-

ductors remains to be seen.Our results for the La 2Cu0 4 system

are typical of the surprises we find withresonant ultrasound measurements.The surprises provide motivation forimproving our understanding of thebasic physics, which in turn enables usto see the way to make new materialsthat have desirable structural, magnetic,or other properties. Thus, the physicswe uncover has the promise that it willeventually apply directly to the real en-gineering aspects of materials.

A Breakthrough inNondestructive Testing

Because of our justifiable confi-dence in the accuracy and precision of resonant ultrasound measurements,when the measured resonance spectrumof a sample of known structure cannotbe made to fit the mathematical modelto within 1 percent or so, as happened

with a few brittle samples we attemptedto study, we were able to conclude thatthe samples were not the perfect cube-like chunks that the microscope re-vealed but were, in fact, cracked. Thecracks altered the resonance spectrum,and therefore, the data could not be fitto any object shaped like the sample.From this simple effect, discovered ac-cidentally during our research on sin-gle-crystal samples, we developed newnondestructive testing approaches thatsubsequently received a 1991 RD100award and a 1993 Federal LaboratoryConsortium Award for Excellence inTechnology Transfer. In particular, to-gether with Raymond D. Dixon and

others we have shown how certainanomalies in the resonance spectrumcan be used to detect cracks and otherflaws in small, precision objects includ-ing aluminum plates, ball bearings,high-strength permanent magnets forlightweight motors, and more.

One test for cracks involves identi-fying the presence of second harmonicsin the resonance spectrum under dryconditions and the absence of thosesecond harmonics under wet condi-tions. We, together with George W.Rhodes, discovered that a crack pro-duces second harmonics when dry butnot when filled with fluid because thefluid prevents the crack from banging



757.50 765.00750.00 757.50 765.00750.00Frequency (kilohertz) Frequency (kilohertz)

(a) Resonance Line in Spectrum of PerfectBall Bearing

(b) Split Resonance Line in Spectrum ofScratched Ball Bearing

Figure 7. Fast, Accurate Detection of Flaws in Ball BearingsThe resonant ultrasound technique is accurate enough to detect tiny flaws in high-pre-cision objects. The figure compares two resonance lines, one from a perfect siliconnitride ball bearing and the other from a similar ball bearing with a tiny surfacescratch. The flaw causes the single line to split into two lines separated in frequencyby 824 parts per million. The resonant ultrasound measurement can detect errors insphericity as small as 0.005 microns in seconds, whereas commonly used optical in-

spections typically require one hour. The resonant ultrasound spectrometer is nowbeing readied to test these newly developed ceramic ball bearings, which can with-stand very high temperatures and acidic environments and are compatible with dry lu-bricants. A resonant ultrasound spectrometer specifically designed to test ball bear-ings is now available from Quatro Corporation as a result of a collaborative technolo-gy-transfer effort between that company and Los Alamos National Laboratory (seeFigure 8).


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shut when mechanically excited on res-onance. Ceramic (Si 3N4) ball bearings,which can withstand very high temper-atures, are being developed for use innaval and military aircraft operations.Accurate resonance measurements of these precision objects can reveal tinyinternal and surface flaws. As shown

in Figure 7, these flaws shift and break the symmetry of the resonances of theball bearings. As a result, 0.005-mi-cron errors in sphericity of a ball bear-ing can be detected in seconds, insteadof the hour needed with present opticalstudies. These nondestructive testingtechniques, fathered by basic research

in materials science at the Laboratory,have been transferred to private indus-try (Figure 8). Thus new jobs are nowbeing created in Albuquerque as a di-rect result of the technology we devel-oped here.

Further Reading

William M. Visscher, Albert Migliori, ThomasM. Bell, and Robert A. Reinert. 1991. On thenormal modes of free vibration of inhomoge-neous and anisotropic elastic objects. Journal of the Acoustical Society of America 90: 2154 .

A. Migliori, J. L. Sarrao, William M. Visscher, T.M. Bell, Ming Lei, Z. Fisk, and R. G. Leisure.1993. Resonant ultrasound spectroscopic tech-niques for measurement of the elastic moduli of solids. Physica B 183: 124.


Figure 8. Resonant Ultrasound Spectrometers on the MarketThe photograph shows a commercial resonant ultrasound spectrometer complete withcomputer, receiver, transducers, and sample. This system, produced by Quatro Cor-

poration, is designed to perform nondestructive testing of ball bearings and othersmall objects. After having devised applications of resonant ultrasound technology tothe detection of flaws in precision objects, we sought through a public advertisementa commercial company that could develop the technology for a wide variety of appli-cations. Quatro came to us in 1991 and in 1992 obtained a license to develop, manu-facture, and market systems for nondestructive testing based on our resonant ultra-sound measurement techniques and computer software. The systems under devel-opment by Quatro will perform nondestructive inspections of metal, ceramic,composite, and rigid plastic parts in a high-volume manufacturing environment. Qua-tro is presently working with numerous clients to design custom-engineered systemsthat meet the particular needs of each application. The Quatro system for testing ballbearings is being adapted to handle thousands of ball bearings per hour and is sensi-tive to geometry errors as small as 2 parts per million. Another system tests the in-tegrity of oxygen sensors. The commercial production of research electronics, hard-ware, and software by Quatro has, in turn, assisted us in expanding research effortsand government applications of resonant ultrasound spectroscopy..

Albert Migliori (left) received his B.S. inphysics from Carnegie-Mellow University andhis M.S. and Ph.D. in physics from the Universi-ty of Illinois. He came to Los Alamos as a Labo-ratory Postdoctoral Fellow, remained as a Na-tional Science Foundation Postdoctoral Fellow,and then accepted his current position as a Staff Member in the Condensed Matter and ThermalPhysics Group. He was honored by the Labora-tory in 1989 with a Distinguished PerformanceAward. He has authored numerous publicationsand been awarded forteen patents.

Zachery Fisk was educated at Harvard Universi-ty and the University of California, Dan Diego.He received his Ph.D. in physics at the latter in1969. A postdoctoral year at Imperial College,London, was followed by a year as AssistantProfessor of Physics at the University of Chica-go. He returned to UCSD, becoming ResearchPhysicist and Adjunct Professor of Physics be-fore joining the Laboratory in 1981 as a staff member in the Physical Metallurgy Group of what is now the Materials Science and Technolo-gy Division. His research interests include thelow-temperature electrical a nd magnetic proper-ties of metals and the growth of single crystals of these materials. Fisk is a Laboratory Fellow.

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