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.