Mechanical Resonance Dispersion in Quartz at Audio-Frequencies

Measurements of the complex shear compliance (${\mathrm{J}}^{*}=J^{\prime }-i{J}^{\prime \prime }$) of single crystals of quartz and fused quartz at frequencies from 100 to 5000 cps have resulted in the discovery of sharp resonances in the compliance similar to those recently found in polycrystalline metals and crystalline polymers. The number, locations, and magnitudes of the resonances depend on crystal orientation with respect to the applied dynamic stress and vary with temperature, external static stress, and, in some cases, with time. The presence of numerous resonances in fused quartz may result from the existence of regions of long-range order (100 to 200 A) in this material. Analysis of the data on the basis of a generalized stress-strain relation involving a linear combination of strain and its first and second time derivatives gives a close fit to the experimental curves. An explanation of the resonances is suggested by calculations of Fermi, Pasta, and Ulam for nonlinear systems in which no tendency toward equipartition of energy among modes was found. Accordingly it is proposed that (1) crystalline solids with nonlinear forces between atoms do not share their vibrational energy among all of the available modes, but pass energy back and forth among relatively few modes, (2) the frequency of the energy exchange among modes may be low, and in particular, much lower than the frequencies of the lattice vibrations, and (3) the resonance dispersions observed in the dynamic mechanical compliance of quartz and other materials occur at these various acoustic exchange frequencies. This lack of energy equipartition will not necessarily be noticed in specific heat measurements of solids, but can be expected to have some consequences in other areas.