The mechanical resonant response of a solid depends on its shape, density, elastic moduli and dissipation. We describe here instrumentation and computational methods for acquiring and analyzing the resonant ultrasound spectrum of very small (0.001 cm 3 ) samples as a function of temperature, and provide examples to demonstrate the power of the technique. The information acquired is in some cases comparable to that obtained from other more conventional ultrasonic measurement techniques, but one unique feature of resonant ultrasound spectroscopy (RUS) is that all moduli are determined simultaneously to very high accuracy. Thus in circumstances where high relative or absolute accuracy is required for very small crystalline or other anisotropic samples RUS can provide unique information. RUS is also sensitive to the fundamental symmetry of the object under test so that certain symmetry breaking effects are uniquely observable, and because transducers require neither couplant nor a flat surface, broken fragments of a material can be quickly screened for phase transitions and other temperature-dependent responses.

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The Hamilton's principle approach to the calculation of vibrational modes of elastic objects with free boundaries is exploited to compute the resonance frequencies of a variety of anisotropic elastic objects, including spheres, hemispheres, spheroids, ellipsoids, cylinders, eggs, shells, bells, sandwiches, parallelepipeds, cones, pyramids, prisms, tetrahedra, octahedra, and potatoes. The paramount feature of this calculation, which distinguishes it from previous ones, is the choice of products of powers of the Cartesian coordinates as a basis for expansion of the displacement in a truncated complete set, enabling one to analytically evaluate the required matrix elements for these systems. Because these basis functions are products of powers of x, y, and z, this scheme is called the xyz algorithm. The xyz algorithm allows a general anisotropic elastic tensor with any position dependence and any shape with arbitrary density variation. A number of plots of resonance spectra of families of elastic objects are displayed as functions of relevant parameters, and, to illustrate the versatility of the method, the measured resonant frequencies of a precision machined but irregularly shaped sample of aluminum (called a potato) are compared with its computed normal modes. Applications to materials science and to seismology are mentioned.

Independent heat reservoirs are postulated to interact with the atoms in a harmonic system as a substitute for real nonharmonic forces. The temperature of each reservoir is determined by the condition that it exchange no energy with the system in the steady state. An explicit solution for the covariance matrix is obtained for the case of the linear chain. The thermal conductivity is finite and inversely proportional to the coupling to the reservoirs.A serious deficiency of the harmonic lattice as a model for real solids is that it is not an ergodic system. The state of the lattice can be expressed as a superposition of normal modes (phonons), and the coefficients of the different normal modes (occupation numbers and phases of phonons) do not change with time. Any harmonic lattice that is started at t =0 away from equilibrium will never approach the canonical equilibrium distribution e PH On .the other hand, in the real crystal, phonons can be created and destroyed by anharmonic interactions with other phonons and with the electrons in the solid. Explicit inclusion of these interactions generally makes lattice dynamics problems intractable.The nonharmonic interactions are the sole mechanism by which an isolated system can approach equilibrium.They facilitate the diffusion in phase space and the dissipation which together bring a system toward equilibrium and keep it there. We propose to simulate these nonharmonic interactions by postulating that each atom in the harmonic crystal interacts stochastically with its own thermal reservoir.In the classical or high-temperature case, where the interactions of the lattice with the electrons in the crystal are incoherent, the description of the electron-phonon interaction in this way is undoubtedly accurate. Less intuitive is the validity of the replacement of the anharmonic phonon-phonon interactions by stochastic reservoirs at the atoms; it would be more realistic to allow correlations between different reservoirs, but also much more complicated.A convenient characterization of the reservoirs is by a nonconservative Fokker-Planck force' acting on each atom in the crystal. The Fokker-Planck force is equivalent to an impulsive interaction of the atom with a Maxwellian gas of hard particles that are much lighter than the atoms of the crystal, and which collide with the crystal atom frequently compared to the characteristic time of the lattice. The Fokker-Planck force can be writtenMp.~v. "z z where A. . is the dissipative viscosity parameter, P; = 1/T; with T; equal to k times the temperature of the ith reservoir, and Mv is the momentum of the ith lattice atom. When it operates on a velocity distribution function, Fz is the sum of an attraction toward the origin in velocity space and a diffusive term that tends to flatten the velocity distribution. The constants Xi can, in principle, be related to anharmonic and electron-phonon interactions; to do so would be a complicated task, so we shall simply assume that all the X are equal to a common value that is to be regarded as an...