Resonant ultrasound spectroscopy (RUS) involves the study of the mechanical resonances of solids. The resonant response of a particular object depends on its shape, elastic constants, crystallographic orientation, density, and dissipation. It is possible to obtain the complete elastic constant matrix of relatively low-symmetry materials from a RUS spectrum on a single small sample (<1 mm 3 ). The measurement and the computation of the RUS spectra of solids are reviewed. Several examples of the use of the technique are discussed. Contents 1 Introduction 6001 2 Ultrasonic waves in condensed matter 6002 2.1 Crystalline elasticity 6002 2.2 Plane-wave propagation methods 6003 3 Resonant ultrasound spectroscopy 6005 4 Theoretical basis of RUS 6008 4.1 Computation of the eigenfrequencies 6008 4.1.1 The general computational method. 6009 4.1.2 Use of symmetry to speed the calculations. 6011 4.2 Determination of the sample parameters 6017 4.2.1 Minimization of the error function. 6017 4.2.2 Calculation of the derivatives of the error function. 6017 4.3 Effects of dissipation 6019 5 Applications of RUS 6020 5.1 Phase transitions 6020 5.2 Hydrogen-metal systems 6024 5.3 Other uses 6025 5.4 Outlook 6025 6 Conclusions 6027

Resonant ultrasound spectroscopy has been used to measure the complete set of elastic constants of LiKSO 4 over the temperature range of 200 to 300 K including both the hexagonal room temperature phase and the lower temperature trigonal phase. Large step changes are observed in all the elastic constants, except C 13 , at 213 K on cooling and at 243 K on warming. These step changes are associated with the hexagonal/trigonal crystallographic phase transition. The bulk modulus is approximately 15% higher in the trigonal phase than in the hexagonal phase. The hexagonal-phase elastic constants exhibit very little temperature dependence while the trigonal-phase elastic constants show a stronger dependence on temperature. The transitions are noted to be quite sluggish, taking a few hours to equilibrate. The results are described in terms of a Landau-type free energy expansion using two Ising-like order parameters with strong coupling between the two parameters. Biquadratic coupling between the order parameters and strains is shown to account for the step changes in the elastic constants as well as the temperature dependence in the trigonal phase. Coupling linear in the strains is shown to be insufficient to explain the results.

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We present the results of magnetization measurements on single-crystal holmium using a SQUID magnetometer in the temperature range from 4 to 140 K in magnetic fields up to 5.5 T. In low fields (0.01 T) the magnetization versus temperature data show a spiral to conical transition at Tc=16 K and the Néel temperature at 132 K. In addition, we observe new anomalies in the temperature dependence of the magnetization along the a, b, and c axes at 20, 24, 42, and 98 K. These new anomalies appear at the same temperatures as observed by Bates et al. [J. Phys. C 21, 4125 (1988); 21, 4113 (1988)] in ultrasonic velocity measurements on holmium. These anomalies could be accounted for within the frame work of the ‘‘spin-slip’’ model of Gibbs and co-workers. In the c axis magnetization we observe a splitting of the Néel temperature in magnetic fields greater than 0.5 T. The H-T phase diagrams of the magnetic phases of holmium for fields in three directions (along the a, b, and c axes) are presented.

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