The use of mechanical resonances to determine the elastic moduli of materials of interest to condensed-matter physics, engineering, materials science and more is a steadily evolving process. With the advent of massive computing capability in an ordinary personal computer, it is now possible to find all the elastic moduli of low-symmetry solids using sophisticated analysis of a set of the lowest resonances. This process, dubbed “resonant ultrasound spectroscopy” or RUS, provides the highest absolute accuracy of any routine elastic modulus measurement technique, and it does this quickly on small samples. RUS has been reviewed extensively elsewhere, but still lacking is a complete description of how to make such measurements with hardware and software easily available to the general science community. In this article, we describe how to implement realistically a useful RUS system.

## INTRODUCTION

The mechanical resonances of solids with low mechanical dissipation are sharp (the $Q=f\u2215\Delta f\u2aa21$, where $f$ is the resonance frequency and $\Delta f$ is the full width at half maximum). When the $Q$ is large, the elastic hysteresis is small and the elastic moduli that determine $f$ are well defined in the sense that they relate stress and strain linearly and uniquely^{1} with little hysteresis. However, the relationship between resonance frequencies and the modulus tensor is not simple, requiring sophisticated computations based on finite element methods, or, for simple shapes such as rectangular parallelepipeds (RP), cylinders, or spheres, Lagrangian-minimization approaches. Until a certain level of computational speed was available to everyone, the deconvolution of such resonances into elastic moduli was limited to Young’s and shear moduli of rods. We contrast this with time-of-flight (pulse-echo) approaches to the measurement of moduli where a pocket calculator, the time of flight of a sound pulse, its crystallographic direction of travel, and the sample length along the sound path are all that is needed to extract *one* modulus. However, even such a simple monocrystalline solid as Fe has three moduli, while orthorhombic materials have nine, and the most general solid has 21.^{2} To extract all three moduli of Fe as a function of, say, magnetic field and temperature using pulse-echo approaches requires a fair amount of work, but it also requires two separate measurements because the crystal must be cut first along 100 and then along 110 to obtain all three moduli (only two because for good samples, transducers are available that can produce both shear and compressional waves). For orthorhombic symmetry, many measurements are needed, and many cuts on the specimen must be made. For each measurement, transducers must be bonded carefully to polished faces. The opportunity for sample damage, transducers falling off, radiation damage, martensite formation, grants expiring, and many other time-dependent effects is correspondingly increased. In addition, it is quite often that one can force a sound pulse through a solid and successfully measure its transit time, but the solid itself may be flawed, the moduli are extremely dissipative, or other effects intrude to make the elastic moduli so measured meaningless with no warning to the investigator.

In contrast to time-of-flight methods (time-domain approaches) resonance methods (frequency-domain approaches) use natural resonances of objects. Until the personal computer (PC) became ubiquitous, resonance methods were limited to the measurement of the Young’s and shear moduli of circular cross-section rods of high aspect ratio. However, with the advent of the PC, the sophistication of the analysis of mechanical resonances blossomed and the technique called resonant ultrasound spectroscopy (RUS) was born.

RUS is qualitatively different from other “ultrasonic” techniques because it is sensitive to all the components of the elastic tensor, “sees” only the true thermodynamic dissipation (a pulse’s echo can have its amplitude reduced by dissipation, errors in parallelism of the faces, and transducer bond absorption). The technique has been applied to samples with the minimum elastic symmetry (triclinic), though it is most often, and most easily, applied to systems of rhombohedral or higher symmetry. Key to the use of this technique is the ability of modern personal computers to perform calculations nearly as rapidly as the supercomputers of recent years. For example, a modern RUS analysis would require approximately $2h$/iteration on the first IBM PC, $10s$ on the first Cray, and $0.8s$ on a $2GHz$ notebook PC.

Because RUS is used so extensively today, many review articles have been written about it that contain the history, details of the complex computational techniques, transducer design and much more.^{3–13} But no publication to date provides a complete description of how to get up and running a modern RUS system based on off-the-shelf electronics and simple, easy-to-fabricate transducers, for the measurement of the elastic modulus tensor of samples of order $1mm$ cube. Our purpose here is to provide a description of just how to do this. The description consists of (a) computations, (b) electronics, (c) transducers, and (d) sample preparation.

## COMPUTATIONS

All RUS measurements are performed on samples that to good accuracy are free vibrators. Because a complete analytical solution for the free vibrations of even a RP does not exist, one must rely on approximations. We briefly review here the most useful approach for the experimenter looking for moduli of small samples: the rectangular parallelepiped resonator or RPR. Migliori and Sarrao^{7} give a complete treatment of this. The procedure begins with an object whose volume, $V$, is bounded by its free surface, $S$, a critical point that needs to be remembered when making a measurement. The Lagrangian is given by

where KE is the kinetic energy density

and PE is the potential energy density

Here, $ui$ is the $i$th component of the displacement vector, $\omega $ is the angular frequency from harmonic time dependence, $cijkl$ is a component of the elastic tensor, and $\rho $ is the density. Subscripts $i$, $j$, etc., refer to Cartesian coordinate directions.

To find the minimum of the Lagrangian, calculate the differential of $L$ as a function of $u$, the arbitrary variation of $u$ in $V$ and on $S$. This yields

Because $ui$ is arbitrary in $V$ and on $S$, both terms in square brackets in Eq. (4) must be zero. The first term yields the $n\xafi$ elastic wave equations, the second, an expression of free-surface boundary conditions; is the unit vector normal to $S$. Thus, the set of $ui$ that satisfies the above conditions are precisely those displacements that correspond to $\omega $ being a normal-mode frequency of the system. The implementation of such an approach involves expansion of the $ui$ in a complete (but not necessarily orthogonal) set of basis functions, substituting that expression into Eq. (1) and finding the stationary points of the Lagrangian which then yields an eigenvalue problem easily solved by common techniques. The stationary points of the Lagrangian are found by solving the eigenvalue problem that results from Eq. (4), namely,

$a$ are the approximations to the motion expanded in a complete basis set, $E$ comes from the kinetic energy term, and $\Gamma $ comes from the elastic energy term. The order of the matrices is $103$ or so for good approximations, that is, for the computation to be more accurate than the measurement.

Equation (5) determines the resonance frequencies from the elastic moduli. To compute elastic moduli from frequencies is not (a) a well-posed problem, (b) has never successfully incorporated certain experimental problems such as modes missed, and (c) requires some experience to help the computer. Migliori^{7} and co-workers have coded this so-called inverse problem enough to enable proper error analysis and some degree of automation. Unfortunately, both experience and good initial guesses of elastic moduli are still important for consistent success in using even these well-tested and frequently used codes to analyze RUS data. The coding and testing of a program to solve this problem is a formidable task. Accordingly, we supply in the associated EPAPS (Ref. 7) file the Fortran 77 source code and executable for the most useful version: the RPR code that fits up to orthorhombic symmetry.

It is possible during measurements that a resonance frequency (mode) will be missed perhaps because the mode has a node at the place where the sample touches a transducer, or because the mode displacement is in a direction for which the transducer is insensitive. If this occurs, the experimenter must be prepared to guess where it might be. This is easier than it sounds, but not quite easy, by running one or two iterations of the code, with rational guesses, often one can see where a missing mode should be. Inserting a missing mode at this point in the input file [see EPAPS (Ref. 7) file] may quickly reduce the error [Eq. (6)] and see below. Our code is based on Levenberg-Marquard minimization and carries out a nonlinear least-squares fit by minimizing the functional defined by the sum of weighted residuals

where $g$ and $f$ are the vectors of measured and calculated frequencies, respectively, and $wi$ is a weighting factor chosen based on the confidence the experimenter has for the measurement of that particular mode frequency. It is customarily set to unity. $N$ has to be chosen large enough to ensure some amount of over determination (i.e., there should be more modes measured than the number of moduli). However, it will be found that many of the lower modes depend on shear only, so that $N$ must be large enough that many non-shear modes are included. Of extreme importance are the residuals [Eq. (6)] and the “chi-square” test.^{7}

For a useful fit, five things must be true.

(1) The fit must be predictive. If a mode is missing, and the code predicts it is there, one should be able to find it by remeasuring, or remounting the sample.

(2) The rms error should be below 0.8%, often below 0.2%, sometimes as low as 0.03%.

(3) The effective curvature of the minimum in elastic-modulus space for each modulus represents a real-world measure of the accuracy of the measurement tested by much experience, and must be acceptable for the purpose intended. The accuracy is very definitely *not* the rms error. The code provides a measure of this curvature by computing the percentage changes in “chi-square” for changes in each modulus. This is the proper approximate estimate for the “1-sigma” uncertainty for that modulus. Because most of the lower modes are shear, the shear moduli are always found more accurately than the others.

(4) There must be many modes that are not pure shear. Typically, one must use 40 modes for orthorhombic symmetry, but maybe only 15 for isotropic.

(5) The $Q$ of the modes must be high, of order a few hundred and up. We have observed a $Q$ of $1.5\xd7106$ for diamond.

## ELECTRONICS

The function of the electronics is to excite mechanical resonances in the specimen and to detect them. Although Weaver^{14} has used the noise spectrum of samples without excitation to measure resonances, and others have used white or pink noise as the excitation source, the most common, and easiest to implement is a system where frequency is swept and response is detected using a phase-sensitive synchronous detector (swept-sine). In fact, in all implementations that we are aware of, frequency is not exactly swept but incremented in small steps, the step size chosen to be much smaller than the width of a resonance. Because RUS relies on free-surface boundary conditions, and because most samples of interest to condensed-matter physics are of order millimeters in size, transducers, to be discussed later, are not attached to the sample. They make weak contact, are nonresonant, and produce small signals in the range from hundreds of kilohertz to several megahertz. The phase of the resonance is useful in separation of nearly overlapping modes. The block diagram for the electronics used in a swept-sine phase-sensitive RUS measurement is shown in Fig. 1. Note that a complete system useful for samples of order $3mm$ cubes, complete with software but lacking phase information and the ability to interface the software with common temperature controllers, a drawback, is available from Quasar International.^{15} The synthesizer (DS345), and phase sensitive detector (844RF) are available from Stanford Research.^{16} If long (more than a meter or so) coaxial connections to the transducers are not required, then no amplifier is needed.

For longer leads, a charge amplifier must be used. This can easily be built from the circuit diagram of Fig. 2. The key to operation of this amplifier is that pin 2 of U9 is held at virtual ground by the CMOS op-amp OPA627. Therefore, the coaxial connection from the receive transducer to P6 has the center conductor held at the same ground potential as the shield so that none of the charge developed by the RUS signal is lost charging the cable capacitance, the charge is pumped to the $50pF$ C29. The gain is independent of frequency above the RC time constant of C29-R28, and is Cx/C29 where Cx is the transducer capacitance. JP1 can be jumpered for low or high frequency operation, useful because the low frequency noise of this circuit is large because of the effect of R28-but this has no effect on the detected signal if a bandwidth-limited phase-sensitive detector is connected to the output, P2 such as the 844RF. R36 is adjusted for little (less than $0.1V$) dc offset at P2. The circuit requires the usual attention to short leads and good ground planes, but produces remarkably good signals.

There is a generic problem with all commercial phase-sensitive detectors (lock-in amplifiers) when the reference frequency is swept, having to do with the rate of sweep and the tendency to “unlock” from the frequency synthesizer input. Care must be taken to prevent this. With the Stanford Research 844RF, an additional problem exists because the internal phase-locked loop tends to switch divide ratio at undocumented, but consistent, values of reference frequency. When this occurs, the 844RF briefly ceases to function and produces a spurious signal. However, the 844RF uses synchronous digitization of the signal, a substantial advantage that makes this amplifier, for RUS applications, superior to others because it can settle faster after a frequency change. The workaround for the spurious signals is implemented in a Labview program available from us^{17} that includes file logging and operation of a temperature controller.^{18} The basic approaches for solving this and the “lock” issue involve (a) checking for “lock” after every frequency step before reading the 844RF (slow, but reliable), (b) programming frequency sweeps until one approaches known frequencies where this problem occurs and then implementing “lock” detect only around those frequencies.

It is useful to note that the time required to sweep through an entire resonance must be more than $1\u2215\Delta f$ or else the specimen cannot respond completely. In addition, by sweeping in steps much smaller than $\Delta f$, the shape of the resonance is revealed, useful for determining the $Q$ and hence the ultrasonic attenuation. Unfortunately, the dissipation does include small effects from energy leaking into the transducers. A measure of this is the highest $Q$ detectable with the actual transducers used for the attenuation measurement. The system described here has an “instrumental” $Q$ of order $1.5\xd7106$. Thus an observed $Q$ of $104$ would be in error by less than 1% because of the dissipation caused by the measurement system.

## TRANSDUCERS

Although magnetic,^{12} quasimagnetic,^{19} and piezo/ferroelectric transducers have all been used for RUS, the most common approach is to use a piezoelectric or ferroelectric transducer. For extreme environments, such as $1800K$, Anderson^{11} used buffer rods. But for most applications, a very simple transducer can be constructed from $LiNbO3$ piezoelectric transducers,^{20} or from ferroelectric films such as polyvinylidine flouride. Note that almost any piezoelectric transducer operated off resonance will do, but that without some damping of the transducer resonances themselves, especially for high-Q transducers, the transducer resonances will intrude. Often, transducer resonances are undocumented by the manufacturer because they consist of torsional and bending modes much lower than the specified resonance. By attaching the transducer to a nonresonant mount, these modes are suppressed. The attachment can simply be any epoxy.

The $LiNbO3$ assembly we use (Fig. 3) is constructed as follows: (a) machine the Vespel^{21} rods to length, bore a clearance hole for the coax, and machine a recess that is about half the transducer thickness and slightly greater in diameter than the transducer; (b) use silver epoxy^{22} to attach the coax^{23} center conductor to the center of one side of the transduce; (c) epoxy the transducer assembly into the recess on the Vespel, ensuring that there is no epoxy on the transducer face away from the coax; (d) evaporate or sputter gold to coat transducer and Vespel. This is the ground connection for the transducer. (e) Use silver epoxy to attach the coax shield onto the gold-coated Vespel.

The fabricated transducers are mounted in a cell for variable temperature applications, or just for room temperature work. Although many designs are usable, in Fig. 4 we show here a graphic representation of one we use in an Oxford Optistat flow cryostat^{24} capable of operation to $500K$. This cell incorporates all that we have learned in making RUS easier. Some important features are (a) gravity provides the contact force with $1g$ or so of mass on top of the movable transducer piston, and (b) a plastic tube (such as a soda straw) in which specimen and both transducers fit loosely, prevents ferromagnetic samples from “escaping” in a magnetic field. The very weak contact between straw, specimen, and transducers appears to be undetectable.

An alternate approach, based on polyvinylidene flouride transducers (PVDF) may be used to measure very small samples, on the order of $0.03-0.04cm$ on all edges, and in this case very small transducers are used. Simple small transducers may be fabricated with thin-film piezoelectric plastic, such as polyvinylidene fluoride (PVDF). Sheets of film as thin as 9 microns are available commercially.^{25} Transducers may be made by beginning with a thin-film sample about two centimeters square. The film is metallized on one side up to a line near the center, and then metallized on the opposite side up to a line about $0.05cm$ beyond the first line, Fig. 5, producing a region where the metallizations overlap in a strip $0.05cm$ wide. A razor blade is then used to cut the film into strips, about $0.05cm$ wide, aligned perpendicular to the region where the metallizations overlap. The result is a strip with metallization extending from the ends of the strip to the region in the center where the metallizations overlap; the overlap region forms a $0.05cm\xd70.05cm$ square capacitor with the 9 micron PVDF film between the plates, and this constitutes the transducer. The strip may be supported under slight tension by its ends (Fig. 6); the metallizations extending from the ends of the strip form the electrical leads to the transducer. The PVDF film transducer can provide very light support with no damage to fragile RUS samples.

## SAMPLE PREPARATION

Although it is possible to find the elastic moduli of a monocrystal RPR with crystallographic axes not parallel to the edges of the specimen,^{26} this is difficult and prone to error. The most useful application of RUS is to a RPR specimen where the edges of the specimen are oriented parallel to the crystallographic axes (except for hexagonal symmetry where only the sixfold axis need be parallel to one edge because the elastic moduli are isotropic in the plane perpendicular to that axis). Usually, Laue diffraction is used to orient the sample in a goniometer, and then one face is polished flat and perpendicular to, say, a 100 direction. The sample is then removed, and a second face parallel to the first is polished. Note that because the wavelength of vibrations in the sample is of order the sample dimensions, a good polish is unnecessary, 600 grit paper will do a fine job on $1mm$ specimens. The sample is remounted and oriented so that a 010 face can be polished. At this point, the normals to all the remaining faces are known for samples as low as orthorhombic in symmetry, and the remaining faces can be polished. Noting that it is important to maintain the faces appropriately parallel and perpendicular, and that the polishing of a face perpendicular to another is not a common operation, we have developed a simple procedure (Fig. 7) to achieve the desired accuracy and, at the same time, support the edges of brittle samples with Crystalbond,^{27} an adhesive implemented by melting it on a hot plate, and shims, to prevent chipping of edges. We find that errors in parallelism and perpendicularity of a few parts in $103$ can be tolerated.

In earlier work, pains were taken to make point contact to the corners of samples polished so they were not cubes. The rational was that the corners were low symmetry points so that it was unlikely that a mode had a node there, and that by making the length, width, and thickness of the sample different, accidental degeneracies, especially in high-symmetry specimens, would be broken, enabling one to see all the modes. Both approaches, as it turns out, are not good ones. The corners are fragile, and it is hard to mount the sample this way and still insert the system into a cryostat without the sample falling off the transducers. Furthermore, the sharp corners tend to wear away both sample and transducer for high-$Q$ samples and long runs. By mounting the sample flat, one avoids these problems, and, because the very small errors in transducer construction render the transducers slightly nonparallel, weak point contact is always achieved, even though now it is almost by accident.

It is also advantageous to make the sample a cube, or if only thin plates are available, a square, if possible. Again, small errors in sample polishing always make the cube or square with edges very slightly different from perfectly equal. These small differences split degeneracies slightly as well, producing new groups of separated modes that are easily identified as broken degeneracies by the code. Under these circumstances, it is quite easy to spot a missing mode that is degenerate because it should be very close to its degenerate pair. The aspect ratio of samples where it is not possible to make a cube is very important. For thin samples, it is better to make the length and width as small as possible, or else many of the lowest, and most important, modes will be pure bending modes with little volume change, making it impossible to obtain $cii$-related modes. We have successfully measured moduli on, for example, $EuB6$ monocrystals $0.19cm$ by $0.039cm$ $0.047cm$, but much beyond this aspect ratio, the diagonal moduli are poorly determined. In Fig. 8 we display resonances obtained as described in this article on a polycrystal specimen of $\alpha -Pu$.

## ACKNOWLEDGMENTS

This research was performed under the auspices of the National Science Foundation, the State of Florida, and the U. S. Department of Energy.

## REFERENCES

**183**,