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.

Experimental, phenomenological, and theoretical analyses are given of the dependence on strain of the ferromagnetic Tc of the colossal magnetoresistance (CMR) rare earth manganese perovskites. It is found that Tc is extremely sensitive to biaxial strain; by implication other physical properties are also. The results indicate that biaxial strain is an important variable which must be considered in the design of devices based on thin films and provide evidence in favor of the relevance of the Jahn–Teller electron-phonon coupling to the CMR phenomenon.

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2Close to optimal doping, the copper oxide superconductors show 'strange metal' behavior 1,2 , suggestive of strong fluctuations associated with a quantum critical point [3][4][5][6] . Such a critical point requires a line of classical phase transitions terminating at zero temperature near optimal doping inside the superconducting 'dome'. The underdoped region of the temperature-doping phase diagram from which superconductivity emerges is referred to as the 'pseudogap' 7-13 because evidence exists for partial gapping of the conduction electrons, but so far there is no compelling thermodynamic evidence as to whether the pseudogap is a distinct phase or a continuous evolution of physical properties on cooling. Here we report that the pseudogap in YBa 2 Cu 3 O 6+δ is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behavior and therefore superconductivity in the cuprates.Resonant ultrasound spectroscopy (RUS) measures the frequencies f n and widths Γ n of the vibrational normal modes of a crystal acting as a free mechanical resonator. The frequencies of the normal modes are determined by density and geometry of the crystal as well as its elastic properties. The elastic component of the temperature evolution of these frequencies, ∆f n (T ), depends on a linear combination of all elastic moduli and reflects changes in the thermodynamic state of the system such as those associated with a phase transi- (Figure 4(a,b)). Causality requires that the maxima in energy absorption are accompanied by elastic stiffening over the same temperature range. This stiffening is observed in addition to the distinct break in slope at T * (Figure 2(b)).The potential for RUS to determine the broken symmetry in the pseudogap phase was limited in this study by the precision with which crystal shape could be controlled, an issue that may be resolvable as sample preparation techniques improve. The pseudogap phase 5 transition is located by our RUS measurements with ±3K uncertainty, improving on the ±30K uncertainty in onset of neutron spin-flip scattering. This clearly separates the onset of magnetic order 8-11 at T * from the onset T K of the Kerr rotation signal 27 and charge order 28 at lower temperature (Figure 3). In our measurements we observe an increase in energy absorption over a broad region near T K (Figure 2(c)), however we do not observe an accompanying thermodynamic signature there. Our observed evolution of the pseudogap phase boundary from underdoped to overdoped establishes the presence of a quantum critical point inside the superconducting dome, suggesting a quantum-critical origin for both the strange metallic behavior and the mechanism of superconducting pairing.

We constructed computer-based simulations of the lattice dynamical properties of plutonium using an electronic structure method, which incorporates correlation effects among the f-shell electrons and calculates phonon spectra at arbitrary wavelengths. Our predicted spectrum for the face-centered cubic delta phase agrees well with experiments in the elastic limit and explains unusually large shear anisotropy of this material. The spectrum of the body-centered cubic phase shows an instability at zero temperature over a broad region of the wave vectors, indicating that this phase is highly anharmonic and can be stabilized at high temperatures by its phonon entropy.

Specific heat of a material is a measure of heat necessary to raise the temperature of a given amount of material, typically a gram or a mol, by 1 Kelvin. Near absolute zero, this bulk thermodynamic quantity is a sensitive probe of the low energy excitations of a complex quantum

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.

High-temperature superconductivity is achieved by doping copper oxide insulators with charge carriers. The density of carriers in conducting materials can be determined from measurements of the Hall voltage--the voltage transverse to the flow of the electrical current that is proportional to an applied magnetic field. In common metals, this proportionality (the Hall coefficient) is robustly temperature independent. This is in marked contrast to the behaviour seen in high-temperature superconductors when in the 'normal' (resistive) state; the departure from expected behaviour is a key signature of the unconventional nature of the normal state, the origin of which remains a central controversy in condensed matter physics. Here we report the evolution of the low-temperature Hall coefficient in the normal state as the carrier density is increased, from the onset of superconductivity and beyond (where superconductivity has been suppressed by a magnetic field). Surprisingly, the Hall coefficient does not vary monotonically with doping but rather exhibits a sharp change at the optimal doping level for superconductivity. This observation supports the idea that two competing ground states underlie the high-temperature superconducting phase.