Sphere-resonance method to determine elastic constants of crystal

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Abstract
Theoretical basis is developed for the sphere-resonance method to determine elastic constants of a very small crystal. In this method, a sphere of crystal is formed, and frequencies of its free oscillations are measured by the resonance method. To calculate frequencies of the sphere, we use the Rayleigh–Ritz (variational) method as in the rectangular parallelepiped resonance method developed by Ohno [J. Phys. Earth 24, 355 (1976)]. It is necessary in the Rayleigh–Ritz method to evaluate matrix elements which are quadratic forms in the strain tensor with coupling by the elastic tensor. To do this the elastic tensor is transformed to spherical coordinates, and the contravariant canonical components of the elastic tensor are expanded by the generalized spherical harmonics YNml with nonzero coefficients only for l=0, 2, 4. For seven crystal systems, properties of the matrix elements are examined, and free oscillations are classified. We also obtain selection rules for excitation which are useful for the mode identification.