Crystals and Ultrasound Old-Fashioned Materials Science

Crystals and Ultrasound old-fashioned materials science

Albert Migliori and Zachary Fisk

tone Age, Bronze Age, Iron Age, for bronze were so remarkable that make the study of new materials excit- the age of exploration and the many, many others spent lifetimes ad- ing, risky, difficult, and what seems Ssearch for gold, the industrial rev- vancing available technology to im- best described as fun. olution and the steel that made it possi- prove that material to the point where Essential to the richness of the field ble, and, of course, the present age of the fully developed and vastly im- is an understanding of the laws of electronics—totally dependent on tiny proved copper alloys were to the first physics and chemistry that enable de- chunks of ultrapure, single-crystal sili- bronze nugget as that nugget was to the tection and realization of new proper- con—it seems that entire chapters in stone it replaced. ties of matter. But despite having its human history are strongly connected Just as important as the production roots in basic science, condensed-mat- to certain special materials that enabled of high-performance bronzes were the ter physics is viewed as a kind of ap- massive bursts of what has come to be parallel developments necessary to un- plied science: After all, we know all of called progress. derstand completely the properties of the fundamental physical laws that de- But the pursuit of human progress is such alloys. And so goes the story for scribe the individual atoms in solids. not the driving force behind physicists many other materials. The entire busi- The interesting fact, however, is that studying new materials today. We have ness of materials science, from the first the collective properties of many atoms much simpler motivations, more akin to discovery of a new substance to the full assembled together are too complex to the curiosity of the scientist who four development of its properties, is a be predicted from first principles. For thousand years ago first noticed the tremendously attractive intellectual example, the sharp transition from non- brilliant luster revealed by scratching puzzle of sufficient complexity to inter- magnetic to magnetic behavior in iron the surface of the glob of bronze pulled est almost anyone. Moreover, the in- at 1043 kelvins cannot be predicted from a state-of-the-art, charcoal-driven, credible importance of new materials, from a calculation of the properties of high-temperature, controlled-atmos- the qualitative changes in human life an assemblage of a few or even a few phere furnace. That scientist had, by that they produce, their intrinsic physi- hundred iron atoms. Such calculations serendipity, produced some sort of very cal attractiveness, their appeal to the in- indicate no abrupt transition to ferro- imperfect alloy. The appearance and tellect, and our total inability to foresee magnetism but only a gradual change the properties of even that poor excuse how important their impact will be from weakly magnetic to more strongly

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Crystals and Ultrasound

magnetic. Nobel laureate P. W. Ander- empirically and produce structures that ative idea led to their discovery. son has most clearly laid out this idea in are largely only what nature allows; The unpredictable effects found his article “More is Different”: He that is, the materials are microscopical- when large numbers of various atoms makes the point that phenomena arising ly homogeneous and theoretically in- are assembled thus provide both moti- as more and more particles interact are tractable. The payoff from this old- vation and justification for the empiri- not simple extrapolations of the behav- fashioned work is the discovery of cally based search for new physics and ior of a few particles. Further, we can- completely new phenomena, which chemistry through the study of new ma- not, nor do we want to, solve the most often arise when the number of atoms terials. Fortunately, the production of basic equations for the many-atom sys- is large (on the order of 1018). any single new material is often accom- tem because those equations are too A particular innovation that arose plished by a very few scientists work- complex and provide too much informa- from the “large-number effect” coupled ing with a small budget and a limited tion. No one cares where a particular with “old-fashioned” intuition was the collection of inexpensive equipment. atom is at a particular time in the mag- inadvertent discovery by K. Mueller in The demonstration of superconductivi- netic powder of a cassette tape. What is 1987 that certain cuprates (namely, ty in heavy-fermion compounds by F. important is the overall magnetization those copper-oxide compounds that are Steglich at Universität zu Köln, and the of millions of atoms. Thus most of the doped with transition-metal and other discovery of the high-temperature su- useful theoretical descriptions of solids impurities) are high-temperature super- perconductors by K. Müeller at IBM rely on statistical averages to predict conductors, that is, they become super- Zurich are typical examples of small, measurable quantities. There is, then, a conducting at temperatures well above successful efforts. However, underly- theoretical gap between our knowledge absolute zero (above 30 kelvins). ing all of the apparently small efforts is of the basic laws for each atom and our Mueller knew that those cuprates are a powerful information and technology ability to predict what large numbers of quite unusual solids. He also knew that base easily accessible to those working interacting atoms do. in many materials, if a smaller atom is within the umbrella of a large research In the last few years some attempts deliberately substituted for a larger laboratory or university. That access is have been made to control macroscopic atom at the center of each unit cell in crucial; without it small groups of sci- material properties by constructing arti- the crystal lattice, then when the solid entists, driven by whatever odd notions ficial materials through the deposition cools the smaller atom and its cloud of motivate them, could not succeed. of sequential layers of carefully con- electrons would not have a stable rest- trolled composition and thickness. At- ing spot in the center of its symmetric tempts are being made also to produce cage. As a result it must move to some The Remarkable Effects nanostructures, materials assembled other position in the unit cell and spon- of Impurities atom by atom through such processes taneously break the crystal symmetry. as molecular-beam epitaxy and chemi- The effect (called a Jahn-Teller insta- Also important to success is a clear cal or vapor deposition. Both ap- bility) is driven by large symmetric ar- focus on some particular set of material proaches produce very small, but not rays of atoms and is only poorly under- properties. Many groups today are fo- quite microscopic, building blocks, or stood in detail (“large numbers” + “in- cused on modifying the electronic and repeated units. Such designed materi- tuition” = “mysterious phenomena”). magnetic properties of solids by doping als are applicable to the production of Mueller’s brilliant conjecture was that, them with impurities and studying the ef- practical devices as well as the study of in electrical conductors, the distortion fects of those impurities as the solids are quantum physics, and they have proper- of the crystal lattice resulting from the cooled to low temperatures. Often the ties that are roughly predictable from Jahn-Teller instability would produce presence of impurities causes the solids theory. Still, the theoretical tools need- new material properties provided the to undergo a drastic change in some ed to predict their material properties energy associated with the lattice dis- physical property as they cool, from con- necessarily include statistics and tortion is comparable to one of the en- ducting to superconducting, from non- macroscopic approximations. ergy scales of the conduction electrons. magnetic to magnetic, from paraelectric Recent work on “materials by de- As it turned out, the marvelous super- to ferroelectric, and so on. These drastic sign” contrasts strongly with the work conducting properties of the cuprates changes are called phase transitions. of what one could call “old-fashioned” were not attributable to Mueller’s con- It is a curious finding that some materials scientists, who work semi- jecture, but nevertheless his very cre- seemingly intrinsic properties of solids

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owe their existence to duce single crystals small amounts of im- New Materials of germanium and purities. For example, silicon with the im- Systems with highly off-the-shelf tungsten correlated electrons such as purity content kept high metal is typically hard Tc superconductors to the unprecedented and brittle. When and heavy-fermion low level of 1 part superconductors pains are taken to re- per million—and move the various trace they used every impurities dissolved in piece of technology nominally pure tung- available to reach sten, it becomes quite their goal. soft. The change in Along the way, hardness is so pro- New when state-of-the-art nounced that the purity Physics was inadequate, of a tungsten sample their team developed Novel electronic, can be reliably esti- structural, and new methods of mated with an ordinary magnetic phenomena growing and charac- Dremel grinding tool. terizing crystals, Technology New Tools The very pure metal is Transfer such as zone-refine- softer than annealed New Resonant ment, a simple copper, and its x-ray ultrasound method for remov- superconductors spectroscopy pattern becomes New magnets ing impurities while blurred, no doubt be- New techniques for a crystal is growing. nondestructive cause of the large de- testing In the end they were formations that easily able to make the occur in its soft condi- first transistors. In tion. Perhaps a more the process of suc- surprising case is that Figure 1. Materials Science at Work ceeding, they signif- Migliori fig1 of salt (NaCl). Usual The production of new materials with3/11/93 exotic properties leads to the devel- icantly advanced the NaCl contains a small opment of new tools to study them, the use of those tools to advance the theory and methods amount of hydroxl, understanding of the new materials and the application of the new materi- of solid-state − OH . When great care als, the new tools, and the new science in the commercial sector. Here we physics. Not only is taken to prepare elaborate on this paradigm with examples from our own work. did they predict the OH−-free NaCl, the re- need for purity so sulting solid can be spread like butter the late 1930s and early 1940s, germi- high as to seem unnecessary, unachiev- on bread. nated into an idea in the minds of John able, and up until then unmeasurable, The most important example ever in Bardeen, William Shockley, and Walter they also developed methods to pro- which the effects of impurities were un- Brattain at AT&T Bell Laboratories. duce crystals of the desired purity, de- derstood and then exploited was in the They realized that the intrinsic proper- veloped characterization schemes, development of transistors. In the early ties of very pure (but still doped) semi- grew the crystals, and then produced decades of this century, work by conductors would allow the electrical practical devices whose function was Thompson, Drude, Pauli, Fermi, Dirac, conductivities of these materials to be totally reliant on the remaining and and Sommerfeld led from the discovery controlled and varied by externally ap- carefully controlled impurities. Their of the electron to an understanding of plied voltage, just as the flow of elec- odyssey from the purest of basic re- the special quantum physics governing trons is in a vacuum tube. Therefore search to the stupendous discovery of conduction processes in pure and im- such materials might serve as replace- one of the most important practical de- pure (doped) semiconductors. ments for the reed relays in telephone vices ever was one of the greatest This revolution in understanding the switchboards. In order to obtain this achievements of twentieth century electronic properties of solids had, by effect, they knew they needed to pro- physics.

184 Los Alamos Science Number 21 1993 Crystals and Ultrasound

Second-Order Phase Transitions and the Measurement of Discontinuities at Second-Order Phase Transitions Elastic Properties Three thermodynamic quantities, the speci®c heat C P , the thermal-expansion We have been interested in produc- coef®cient ®, and the elastic-stiffness tensor cij, can be discontinuous at the ing and understanding materials that critical temperature Tc of a second-order phase transition. Each is proportional undergo second-order phase transitions. to a second derivative of ¢ G , the change in the Gibbs free energy per unit In some metals the transition from the mass across the boundary separating two phases. Although ¢ G is always zero normal conducting phase to the super- in any phase transition, in a second-order phase transition the ®rst derivatives conducting phase is second order, as of ¢ G are always continuous and the ®rst sign of any discontinuities occurs in are many other structural, magnetic, the second derivative of ¢ G . and electric phase transitions. In this type of phase transition, the symmetry The relationships between ¢ G and C P , ®, and cij are particularly simple and material properties of the solid to write down for a liquid because the elastic-stiffness tensor has only one change gradually rather than abruptly component, namely, the bulk modulus, B . (as in first-order phase transitions), but the changes nevertheless begin at a In terms of temperature T, pressure P , and volume V , the thermodynamic very distinct temperature called the relations are critical temperature, T . The solid is in c @2¢ G @¢ V 1 a more symmetric phase above Tc than = = ¡ ; B = bulk modulus below. @P 2 @P B In a crudely parallel way the intro- 2 duction of various amounts of impuri- @ ¢ G @¢ S C P = ¡ = ¡ ;C P = speci®c heat ties can also break the symmetry of a @T 2 @T T solid, at least locally. The combination of changes in temperature and control @2¢ G @¢ V = = ®: ® = thermal-expansion of impurity levels can thus be used to @P @T @T tune, in more or less continuous ways, coef®cient important physical properties in metastable materials. Therefore, the ¢ V is the change in volume per unit mass across the phase boundary and ¢ S study of such systems is widespread in is the change in entropy per unit mass across the phase boundary. Although materials science. It involves many ¢ G , ¢ V , and ¢ S are zero and the phase transition is continuous, B , C P , and tools, requires science, intuition, and ® can exhibit discontinuities at the critical temperature Tc. luck, and it produces returns in both fundamental physics and very practical The relationships for a solid are more complicated and are usually written in applications. terms of the stress tensor ij and the strain tensor ²kl instead of P and V . The tale we are about to tell con- Further, the bulk modulus B becomes the elastic-stiffness tensor cijkl, which cerns the development of a very old relates the stress to the strain in terms of the fundamental relation technique into a powerful and essential- X ly new one for determining the changes ij = cijkl²kl: in the elastic stiffness of crystals, par- kl ticularly as they undergo second-order phase transitions. This now mature technique, called resonant ultrasound new tools to study them, the subsequent paradigm of how materials science ac- spectroscopy, has led to a simple proce- use of those tools to advance the tually works (Figure 1). dure for making what once was a very physics of other materials, and the ap- Measuring the elastic stiffness, or difficult measurement. Our work illus- plication of the tools, the science, and elasticity, of a solid as a function of trates the synergism among the produc- the materials to industrial applications. temperature has been a traditional and tion of new materials, the invention of In other words, our story illustrates a very important technique for studying

1993 Number 21 Los Alamos Science 185 Crystals and Ultrasound

second-order phase transitions in mate- stiffness of the bonds holding the solid nation of elastic stiffness, a thermody- rials doped with impurities. Nature together. Although cijkl has 81 compo- namic measurement that provided the helps us with the measurement because nents (or moduli), in crystalline struc- first verification of the BCS theory of the elasticity of a solid is discontinuous tures that do not produce external mag- ordinary low-temperature superconduc- at Tc; that is, it jumps to a different netic fields, the tensor has at most 21 tivity, millimeter-sized crystals are a value when that solid begins to undergo independent components. Consequent- disaster. a second-order phase transition. The ly the elastic stiffness tensor is com- What we describe below is a solu- jump is perhaps surprising because, as monly written as cij, where i and j run tion to this measurement problem. It we mentioned above, the solid exhibits from 1 to 6, for pragmatic rather than involves measuring the resonances (or no obvious microscopic changes at the mathematical reasons. natural frequencies of vibration) of a critical temperature. The atoms do not It is also important to realize that the crystal and often makes measurement suddenly change position, magnetism stiffness of a metallic solid comes not of the elastic-stiffness tensor more triv- and ferroelectricity do not suddenly ap- only from the chemical bonds that hold ial than measurement of the resistivity pear, and a conductor is not suddenly the ions together but also from the de- tensor. able to carry super-large currents. generacy pressure of the Fermi sea of Prior to development of this the new However, three thermodynamic quanti- electrons. Thus a great deal of informa- resonance approach, elastic moduli ties—the elastic stiffness, the specific tion about the crystal is contained in a were determined mostly from measure- heat, and the thermal-expansion coeffi- complete description of its elasticity. ments of the speeds of longitudinal and cient—do exhibit discontinuities at Tc Because of this wealth of information, shear sound waves along different di- and those abrupt changes can be mea- if one knows the crystal structure and rections in a sample. Figures 2 shows sured. The accompanying box defines can measure the changes in the elastic- the simple relationships among these these three quantities in terms of varia- stiffness tensor as a function of temper- sound speeds and the elastic moduli for tions with respect to temperature and ature through a phase transition, then it a cubic crystal. For a large chunk of pressure of the change in the Gibbs free is possible to infer the detailed changes isotropic material such as ordinary energy per unit mass across the bound- in the crystal lattice or in the electronic glass, there are only two independent ary between the two phases. structures that are driving the phase sound speeds, one for longitudinal Discontinuities are a boon to the ex- transition. Few other measurements waves in which the atoms vibrate along perimentalist because they are often the can reveal as much of the physics of the direction of the sound wave (these most unambiguous of measured quanti- phase transitions. Consequently we are the waves we hear), and shear ties. Of the three discontinuous quanti- would like to be able to make this mea- waves, in which the atoms vibrate in a ties mentioned, elastic stiffness is par- surement in high-Tc superconductors, direction perpendicular to the direction ticularly informative because in a solid heavy-fermion superconductors, and of the sound waves. A simple measure- this quantity is a fourth-rank tensor, other exotic materials exhibiting sec- ment of the pulse-echo time of each cijkl, with 81 components. The elastic- ond-order phase transitions. type of sound is easy to do and yields stiffness tensor relates the stresses sound speeds for glass accurately and (forces) to the strains (displacements) completely. As shown in Figure 2, the in the solid through the fundamental re- The Need for a New pulse-echo time is the time for a short lation Measurement Technique sound pulse to travel the distance from X one face of the sample to the opposite ij = cijkl²kl: Unfortunately, nature, while permit- face and back again. kl ting us to produce single crystals of In applying the pulse-echo technique σ ε where ij is the stress tensor and kl is high-temperature superconductors, to single crystals of more exotic materi- the strain tensor. This relation between heavy-fermion superconductors, and als, the first catch is that the speed of stress and strain is simply the general- other materials in variously doped ver- sound in nearly every solid that inter- ization of Hooke’s law, F = -kx, where sions, has often restricted the dimen- ests us is a blazing 5 millimeters per F is the force developed in a spring if it sions of the crystals that can be easily microsecond or so (Mach 15 in jet is stretched a distance x. Thus the elas- grown to the millimeter range. That fighter units). The second catch is that tic-stiffness tensor cijkl, in analogy with size is entirely adequate for many mea- we need to use very-high-frequency ul- the spring constant k, describes the surements. But for the crucial determi- trasound (several hundred megahertz or

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

(a) Longitudinal and Shear Sound Waves Direction of sound Longitudinal sound wave In longitudinal sound waves atoms vibrate propagation along the direction of wave propagation. In Shear sound wave shear sound waves atoms vibrate perpen- dicularly to the direction of wave propagation. Direction of

Shear sound wave vibrational motion

(b) Sound Speeds and Elastic Moduli Orientation in cubic crystal Sound speed The elastic-stiffness tensor of a cubic crystal has three indepedent elastic moduli (c11, c12, and c ) and therefore there are three sound c11 44 Longitudinal wave vl = √ speeds in a cubic crystal. The elastic ρ modulus c11 is determined from the speed vl c of the longitudinal sound wave shown at v = 44 First shear wave s1 √ ρ right. The two shear moduli c12 and c44 are determined from two speeds vs1 and vs2 of z the two shear waves shown at right. c Ð c y v = 11 12 Second shear wave s2 √ 2ρ x (ρ is the crystal density)

Conventional measurements of the elastic stiffness tensor c are made by ij L measuring the sound speeds of longitudinal and shear sound waves along various directions in a large crystal. In the example shown here a short Sound speed pulse of longitudinal sound is generated by a transducer on one crystal Transducer face; the sound pulse travels as a narrow beam through the crystal and Sound 2L pulse v = bounces back from the opposite face to produce an echo that is picked up l τ by the transducer. The measured time between the inititation of the pulse echo τ and the echo (called the pulse-echo time, echo) is equal to 2L/vl, where L is the distance between the two crystal faces and vl is the longitudinal sound speed. Thus the sound speed can be calculated directly from τ ρ echo. Further, if the density of the crystal, , is known, the measurement also provids a direct determination of c11, one component of the elastic- stiffness tensor. By placing the transducer at various locations on the cyrstal and using both longitudinal and shear sound waves, all the components of the elastic-stiffness tensor can be determined. higher) to measure the sound speed in structure (all three crystal axes of the an orthorhombic structure (see Figure samples whose largest dimension is unit cell are at right angles, two axes 3). Therefore, we’re interested in the about a millimeter. But often ultrason- are equal in length, and one is longer or variation in sound speed (or equivalent- ic attenuation increases to prohibitive shorter). La2CuO4 is an interesting ly, elastic-stiffness tensor) as a function levels at such frequencies. material because when doped with of temperature through the structural Let’s say that we want to measure strontium it becomes a high-tempera- phase transition. In our 1-centimeter the speed of sound in a 1-centimeter ture superconductor and its doped ver- sample, a single cycle (wavelength) of cube (big!) of La2CuO4. Above 530 sions undergo a structural phase transi- 50-megahertz sound is 0.1 millimeter kelvins this cuprate has a tetragonal tion as they cool, from a tetragonal to long. To obtain a good signal-to-noise

1993 Number 21 Los Alamos Science 187 Crystals and Ultrasound

Figure 3. Structural Phase Transition of La2CuO4

T > T c T > Tc V (a) Tetragonal Structure of La2CuO4 above Tc

The unit cell of La2CuO4 above the critical temperature, Tc = 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 is Oxygen longer. Note that the oxygen atoms form an T < T c V Lanthanum

octahedron. Each oxygen atom at the apex of Energy the octahedron sits in a double-well potential, V (shown here in red). Thermal motions of the Copper apical oxygen atoms are sufficiently large that the equilibrium position of each is at the center of its potential well. Position of apical oxygen atom Tetragonal unit cell

T < Tc (b) Orthorhombic Structure of La2CuO4 below Tc

At temperatures below the critical temperature, each apical oxygen atom falls into one or the other minima of its double-well potential so that the octahedron of oxygen atoms in each unit cell has a static tilt. Since octahedra in adjacent cells tilt in opposite directions, two of the old unit cells form the unit cell of the new structure. Thus the crystal now has an orthorhombic structure (all three axes of the unit cell are unequal in length).

Orthorhombic unit cell

As the temperature decreases below x Tc and oxygen octahedra develop permanent tilts, shear forces develop and change the square array of copper atoms to a rhombus. Here the distortion is exaggerated for the purposes of illustration. Theory predicts that the shear modulus c66, which characterizes shear forces in the x-y plane, will shift abruptly during the phase transition whereas the shear c modulus c , which characterizes 44 Shear 44 z shear forces in the z-y and z-x planes, waves c66 will remain unaffected by the phase y transition. x

188 Los Alamos Science Number 21 1993 Crystals and Ultrasound

Migliori fig4 3/11/93 ratio, we send in a pulse consisting of many cycles of sound and bounce the pulse from the inside walls of the crystal. Such a pulse might last, then, 0.2 microseconds and be 1 millimeter long, 10 percent of the width of the sample. To determine the relative change in sound speed to 1 part per million, a not unreasonable goal, we would need to time the pulse echo to an accuracy of 2 picoseconds, corresponding to about 1/1000 of a wavelength of sound. All of this is not wildly difficult to do for the frequencies appropriate to big samples. But for a 1-millimeter crystal we must increase the frequency, and therefore 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 La2CuO4, whether pure or doped, has nine independent elastic moduli and therefore nine different sound speeds, each requiring a separate measurement. And each measurement requires that a small transducer be glued to the sample and that it not fall off as we cool the Figure 4. Distortions of a Cube-Shaped Sample on Resonance sample from room temperature down to Each figure above is an example of the distortions that a cubical object undergoes a temperature well below the critical when it is driven at a natural (resonant) vibrational frequency. At each resonant fre- temperature. The pulse echo is an in- quency the vibrational motion is dependent on complicated linear combinations of all termittent signal, and the signal can the elastic moduli as well as the exact shape of the object. Because of this complexi- easily become so greatly attenuated that ty, Rayleigh, Love, and others were unable to compute the resonances of such short, barely one echo can be detected. fat objects from their elastic moduli. With the advent of big computers and some clever algorithms, such computations are now easily done.

The Development of Resonant solid’s natural vibrational frequencies. 1 megahertz or so, a wonderfully low Ultrasound Spectroscopy (Application of this technique to long, frequency). The resonating sample acts thin rods is perhaps hundreds of years like a natural amplifier, greatly increas- Direct measurement of the pulse- old!) Because the resonant frequencies ing the amplitude of the vibrations (a echo time (or sound speed) to deter- are related to the pulse-echo times, we factor of 10,000 is not uncommon). We mine elastic stiffness would seem to be can measure resonant frequencies in- now need only measure frequency. We nearly hopeless for small crystals. But stead of pulse-echo times to determine can take as long as we like to do it, and usually, when nature makes the mea- elastic-stiffness tensors. the naturally amplified signal is present surement of time tough, the measure- We measure the frequency of a reso- during the entire time we are doing the ment of frequency is much easier. In nance by driving the sample continu- measurement. Therefore, a resonance fact, if we apply continuous sound to a ously with ultrasound and slowly measurement can easily have a signal- solid sample, the solid will resonate, or changing the frequency of the sound to-noise ratio that is a million times ring just like a bell, provided the ap- until the sample suddenly starts to res- higher than a measurement of the echo plied sound frequency is one of the onate (1-millimeter samples resonate at time of short pulses of sound.

1993 Number 21 Los Alamos Science 189 Crystals and Ultrasound

You might, at this point, wonder our small and completely uncharacter- was able to measure all of the lower why anyone would have used echo- ized samples could not be so easily at- fifty or so resonant frequencies of a times rather than resonant frequencies tacked, so we had to develop new hard- crystal with a largest dimension of 1 to determine the components of the ware and refined analysis procedures. millimeter or so. The entire experiment elastic-stiffness tensor. The catch is To measure the resonant frequencies fits on the top of a desk and costs less that the resonant frequencies are hard of a 1-millimeter object, we had better than a car. Figure 5 shows the mea- to interpret because they have a some- (1) not disturb the sample’s resonances surement apparatus and a typical reso- what complicated relationship to the with the measuring device and (2) not nance spectrum. elastic moduli. generate resonances in the measuring Over a hundred years ago John system that might confuse the issue. William Strutt, Baron Rayleigh, at- To not change what we wish to measure Analysis of Resonant tempted to calculate the resonant fre- in the process of measuring it requires Ultrasound Data quencies of cubes, short cylinders, and that our transducers make extremely other short fat objects from known weak contact with the sample. So if To our surprise, the algorithm to elastic moduli. Unlike the correspond- our sample is approximately cubical, compute resonant frequencies from elas- ing calculation for thin rods and plates, we lightly contact its corners with tic moduli and our simple technique to this wonderfully tantalizing and seem- transducers, using no glue or any other measure accurately the resonances of a ingly simple problem stymied the bril- coupling medium, and apply just a useful sample were not quite enough to liant Rayleigh, who finally concluded gram or so of force (pardon the unit). produce a robust technique for determin- that “the problem . . . has for the most With that light contact we lose a fac- ing elastic moduli from measured reso- part, resisted attack”. Figure 4 illus- tor of 1000 in our signal-to-noise ratio. nances. (Note that this is the inverse of trates the origin of the complexity: We must also drive the sample lightly the problem solved by Orson Anderson.) When a cube resonates, it exhibits sig- so as not to destroy it. Fortunately, the In real life the measured resonant fre- nificant distortion typically involving natural amplification at resonance re- quencies have some errors relative to the all the elastic moduli in complicated covers most of what is lost with a gen- resonant frequencies of a perfect sam- linear combinations. A. E. H. Love, tle drive. The resonances generated in ple. The errors arise from many sources Willis Lamb, and others were also the measurement apparatus are another including anomalies in the geometry of stymied by this problem. But in the matter. We are not ourselves small the sample (such as chips and rounded 1960s Orson Anderson of Columbia enough to make transducers much edges) and transducer loading effects. University and his postdoc, Harold De- smaller than 1 millimeter or so; there- Typically, the resonances of the mathe- marest, hit on a fast, accurate numerical fore, most transducers would ring in matical model of the solid and those algorithm for obtaining the solution. just the same frequency range as the measured for the real object differ in fre- The algorithm requires computations sample and spoil the signal. The way quency by 0.1 percent, more or less. only at the surface of the object and around this problem is to make the Further, one or two resonances out of achieves an accuracy much greater than transducers of a composite structure the thirty or forty that exist in a given the relatively crude “finite-element” consisting mostly of single-crystal dia- frequency range may be missing from techniques, which compute throughout mond. Diamond, with a sound velocity our data because some resonances, by the volume of a sample and are the only of 17 millimeters per microsecond accident, may have no component of vi- alternate method. With this fast, accu- (Mach 50!) has such a high sound brational motion along the driving direc- rate algorithm Anderson and his speed that its resonance frequencies are tion of the transducer. The experimental coworkers were able to make the first much higher than those of the crystals data are fed into a computer program fully interpreted resonant ultrasound of comparable size that we wish to that tries to find a set of elastic moduli measurements on large, high quality measure. consistent with the measured reso- mineral crystals. That is, they were When we hooked all the pieces to- nances, the sample dimensions, and the able to match measured resonance fre- gether and connected them to electron- symmetry of the sample’s crystal lattice. quencies to predicted ones, but they ics specially designed (with the help of If one or two resonances are missing had big crystals, big signals, and almost Thomas Bell) to maximize the signal- from the measured resonance spectrum, perfectly known answers for the elastic to-noise ratio, John Sarrad, a student the computer will invent an answer that moduli before they started. In contrast, working with us on his thesis research, is “not even wrong.”

190 Los Alamos Science Number 21 1993 Crystals and Ultrasound

Figure 5. Ultrasound Measurements of La1.86Sr0.14CuO4

(a) Measurement Technique

The photograph shows a sample of Electronic Ulrasonic transducers signal Electronic La1.86Sr0.14CuO4 held between two transducers, one of which drives the sample generator Sample detector over a continuous range of frequencies in the megahertz range. The apparatus is placed in a cryostat, which can cool the

sample to the desired temperatures.

We have not yet developed a com- While the temperature is held fixed, the puter program for guessing at the ab- frequency of the driver is gradually and sence of a resonance; however, those automatically changed by an electronic guesses can be made by looking at the signal generator until the sample resonates and thereby amplifies the data and relying on the intuition that applied signal. The amplified signal is comes from experience. After several picked up by the second transducer and iterations and applications of William recorded automatically by a specially M. Visscher’s pioneering computer al- designed electronic detector. The gorithms, we usually do guess which frequency of the driver then continues to change again until another resonant resonance is missing, and then the com- frequency is reached. All the resonant puter instantly determines the elastic frequencies of the sample are thus moduli of the sample to an accuracy of recorded during a single experimental 0.05 percent or better. The algorithms run. The entire resonance spectrum is also determine the true dimensions of measured in several seconds. To study a second-order phase transition in the the sample to an accuracy as high as sample, the experiment is repeated at 0.01 micron. The results are the most various temperatures between room accurate and complete measurements to temperature and the critical temperature date of the elastic-stiffness tensor, and of the phase transition. the many elastic moduli are obtained simultaneously from a single measure- (b) Resonance Spectum of La Sr CuO at T = 297.2 kelvins ment of the resonance spectrum. 1.86 0.14 4 Our ability to infer the existence of 0.2 missing resonances in our data was crit- Shown here are the ically dependent on our being able to resonances between 0.4 produce some samples that were perfect and 1.1 megahertz. The resonance lines are small single crystals and then using extremely sharp as shown their known perfection to refine our ex- by the enlargement of a typical line. In fact, the 30 hertz periment for less perfect crystals. This wide McLuhanesque bootstrapping of the de- resonant frequency is velopment of a measurement technique determined to an accuracy of 1 part per million. by using known properties of near-per- Amplitude (volts) fect materials is not uncommon and is often the only safe route. Without the collaboration among crystal growers, 0.895 0.897 instrument developers, and theorists, the resonant ultrasound spectrometer could 0.0 not have been developed to its full po- 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 tential. Without the instrument our per- Frequency (megahertz) fect small single crystals would have remained strangers to the ultrasound and as a technology supermarket with a crystals doped with various impurities, their elastic properties would still be a large and available stock from which to we, our postdoc, and our students have mystery. Although the project has in- select just the right things. been able to make steady progress in volved many types of expertise, the size expanding our knowledge of second- of the overall effort has remained small. order phase transitions. Our resonant The required components of the effort, Applications of the Resonant ultrasound measurements on pure and however, were not predictable when we Ultrasound Spectrometer doped La2CuO4 single crystals near began developing the instrument. Thus their structural phase transition from the work really needed to be done at a With our resonant ultrasound spec- tetragonal to orthorhombic are particu- national laboratory, which could serve trometer and a wonderful collection of larly interesting.

1993 Number 21 Los Alamos Science 191 Crystals and Ultrasound

(a) Variation of c44 with Temperature (b) Variation of c66 with Temperature

0.7 0.6

0.6 Theoretical 0.5 Theoretical prediction prediction 0.5 0.4 0.4 0.3 0.3 0.2 0.2 dynes per square centimeter) dynes per square centimeter) c44

12 12 c66 0.1 resonance 0.1 resonance (10 (10

44 66 c 0.0 c 0.0 220 240 260 280 300 220 240 260 280 300 Temperature (kelvins) Temperature (kelvins)

Figure 6. Surprising Results from Resonant Ultrasound Measurements To study the transition of the superconducting cuprate La1.86Sr0.14CuO4 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 resonant ultrasound technique. Resonance spectra were taken at various temperatures from room temperature down to the critical temperature of the sample, 223 kelvins. The values of the elastic moduli at the various temperatures were determined from the resonance spectra by using a computer program discussed in the main text. As explained in Figure 4 and the main text, theory predicts that the shear modulus c44 should remain constant through the transition, whereas the shear modulus c66 should remain constant but then undergo an abrupt change at (or slightly above) Tc. Above are shown the results from the resonant ultrasound measurements. (a) shows that the value of c44 remains constant, as expected, and (b) shows that the value of c66 begins de- creasing at a temperature roughly 80 kelvins above Tc and continues to decrease as the temperature decreases to Tc. The depar- ture from theoretical predictions (shown in red) indicates that this second-order phase transition is less well understood than was thought. The dotted and dashed lines of the cube inset in each graph indicate peak distortions at opposite ends of the vibrational cycle for each resonance.

When we started the measurement, (the atom can no longer make it over posed of two of the old unit cells and the structural phase transition was the bump of the potential well). As the has orthorhombic symmetry. thought to be well understood. As solid cools further, the atom gradually In Figure 3c, a projection of the shown in Figure 3a, at high tempera- drops into one of the two small wells at crystal lattice onto the x-y plane shows tures each apical oxygen atom in the the bottom of the potential. Thus the that as the oxygen octahedra develop unit cell sits in the center of a double- arrangement of atoms is altered; that is, alternating tilts during the phase transi- well potential (a potential with two the solid experiences a structural phase tion, they pull apart the corners of the minima). More precisely, fast thermal transition. unit cells and shear the square array of motions cause each oxygen atom to vi- Figure 3b shows the rearrangement. atoms into a rhombus (or diamond brate so that it appears to fill the avail- Each oxygen atom in the unit cell un- shape). Thus, according to theory, the able space symmetrically. Near the dergoes a displacement such that the shear modulus c66 should drop abruptly critical temperature of the structural octahedron formed by the oxygen as the temperature drops below the crit- phase transition, the energy of thermal atoms develops a permanent tilt. Be- ical temperature, whereas the shear vibrations is just about equal to the cause the octahedra in neighboring unit modulus c44 should remain unaffected. height of the bump (local maximum) in cells develop tilts in alternate direc- In fact, very general symmetry argu- the center of the potential well. Thus tions, the two neighboring unit cells are ments can be used to predict exactly the thermal motion is now too weak to no longer identical. In fact, the unit how each of the six independent elastic keep the atom buzzing in the center cell of the new structure is now com- moduli of the tetragonal structure

192 Los Alamos Science Number 21 1993 Crystals and Ultrasound

should change as a function of tempera- (a) Resonance Line in Spectrum of Perfect (b) Split Resonance Line in Spectrum of ture near the critical temperature. Ball Bearing Scratched Ball Bearing Much to our surprise, the behavior of the elastic moduli as a function of temperature determined from our resonant ultrasound measurements differed markedly from theoretical predictions. Rather than changing abruptly at the critical temperature, the shear modulus c66 changes smoothly over a temperature drop of 80 kelvins or so (Figure 6). Because our simultaneous measurements of the six elastic moduli are extremely precise, there can be no doubt as to the validity of our results. What 750.00 757.50 765.00 750.00 757.50 765.00 Frequency (kilohertz) Frequency (kilohertz) we are left with is a clear indication that this structural phase transition is not as well understood theoretically as Figure 7. Fast, Accurate Detection of Flaws in Ball Bearings was thought. Moreover, measurements The resonant ultrasound technique is accurateMigliori fig7 enough to detect tiny flaws in high-pre- 3/11/93 on pure La2CuO4 as well as samples cision objects. The figure compares two resonance lines, one from a perfect silicon doped with oxygen, barium, and stron- nitride ball bearing and the other from a similar ball bearing with a tiny surface tium showed similar results. Whether scratch. The flaw causes the single line to split into two lines separated in frequency these results are related to the mysteries by 824 parts per million. The resonant ultrasound measurement can detect errors in of high-temperature cuprate supercon- sphericity as small as 0.005 microns in seconds, whereas commonly used optical in- ductors remains to be seen. spections typically require one hour. The resonant ultrasound spectrometer is now Our results for the La2Cu04 system being readied to test these newly developed ceramic ball bearings, which can with- are typical of the surprises we find with stand very high temperatures and acidic environments and are compatible with dry lu- resonant ultrasound measurements. bricants. A resonant ultrasound spectrometer specifically designed to test ball bear- The surprises provide motivation for ings is now available from Quatro Corporation as a result of a collaborative technolo- improving our understanding of the gy-transfer effort between that company and Los Alamos National Laboratory (see basic physics, which in turn enables us Figure 8). to see the way to make new materials that have desirable structural, magnetic, with a few brittle samples we attempted others we have shown how certain or other properties. Thus, the physics to study, we were able to conclude that anomalies in the resonance spectrum we uncover has the promise that it will the samples were not the perfect cube- can be used to detect cracks and other eventually apply directly to the real en- like chunks that the microscope re- flaws in small, precision objects includ- gineering aspects of materials. vealed but were, in fact, cracked. The ing aluminum plates, ball bearings, cracks altered the resonance spectrum, high-strength permanent magnets for and therefore, the data could not be fit lightweight motors, and more. A Breakthrough in to any object shaped like the sample. One test for cracks involves identi- Nondestructive Testing From this simple effect, discovered ac- fying the presence of second harmonics cidentally during our research on sin- in the resonance spectrum under dry Because of our justifiable confi- gle-crystal samples, we developed new conditions and the absence of those dence in the accuracy and precision of nondestructive testing approaches that second harmonics under wet condi- resonant ultrasound measurements, subsequently received a 1991 RD100 tions. We, together with George W. when the measured resonance spectrum award and a 1993 Federal Laboratory Rhodes, discovered that a crack pro- of a sample of known structure cannot Consortium Award for Excellence in duces second harmonics when dry but be made to fit the mathematical model Technology Transfer. In particular, to- not when filled with fluid because the to within 1 percent or so, as happened gether with Raymond D. Dixon and fluid prevents the crack from banging

1993 Number 21 Los Alamos Science 193 Crystals and Ultrasound

in materials science at the Laboratory, have been transferred to private indus- try (Figure 8). Thus new jobs are now being created in Albuquerque as a direct result of the technology we developed here.