Brillouin Light Scattering from Crystals in the Hydrodynamic Region

With the availability of modern lasers, light scattering can now be used as a probe of the energy, damping, and relative weight of the various hydrodynamic collective modes in anharmonic insulating crystals. We first give a general review of the distinction between hydrodynamic and high-frequency (or dynamic) vibrational modes. We then express the intensity and spectral distribution of scattered light in terms of the Fourier transform of the displacement-displacement correlation function ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}\ensuremath{\Omega})$, which is the spectral weight of the phonon propagator. This follows some work of Loudon, who has discussed first-order Raman (or Brillouin) scattering using standard quantum-mechanical perturbation theory. Next we summarize what can be said about the spectral weight ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}\ensuremath{\Omega})$ in the region of low frequencies (or small energy transfer). We use the model calculations of Kwok and Martin, as well as the standard theory of an elastic medium with a nonlocal form of Fourier's law of heat diffusion. In the case of pure isotropic anharmonic crystals ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\mathrm{K}, \ensuremath{\Omega})$ has resonances corresponding to first-sound (pressure) and second-sound (temperature) waves, in addition to that from transverse or shear elastic waves. Unless fairly restrictive conditions are met, the second-sound wave does not propagate and reduces to the ordinary thermal diffusion mode of Landau and Placzek. The special nature of second sound in He II is discussed in an Appendix.