International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 2.4, pp. 330-331
Section 2.4.4. Brillouin scattering in crystals^{a}Laboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France |
Brillouin scattering occurs when an incident photon at frequency interacts with the crystal to either produce or absorb an acoustic phonon at , while a scattered photon at is simultaneously emitted. Conservation of energy gives where positive corresponds to the anti-Stokes process. Conservation of momentum can be written where Q is the wavevector of the emitted phonon, and , are those of the scattered and incident photons, respectively. One can define unit vectors q in the direction of the wavevectors k by where n and are the appropriate refractive indices, and is the vacuum wavelength of the radiation. Equation (2.4.4.3b) assumes that so that is not appreciably changed in the scattering. The incident and scattered waves have unit polarization vectors and , respectively, and corresponding indices n and . The polarization vectors are the principal directions of vibration derived from the sections of the ellipsoid of indices by planes perpendicular to and , respectively. We assume that the electric vector of the light field E_{opt} is parallel to the displacement D_{opt}. This is exactly true for many cases listed in the tables below. In the other cases (such as skew directions in the orthorhombic group) this assumes that the birefringence is sufficiently small for the effect of the angle between and to be negligible. A full treatment, including this effect, has been given by Nelson et al. (1972).
After substituting (2.4.4.3) in (2.4.4.2), the unit vector in the direction of the phonon wavevector is given by The Brillouin shift is related to the phonon velocity V by Since , from (2.4.4.5) and (2.4.4.3), (2.4.4.4) one finds where is the angle between and .
The power , scattered from the illuminated volume V in a solid angle , where and are measured inside the sample, is given by where is the incident light intensity inside the material, is the appropriate elastic constant for the observed phonon, and the factor results from taking the fluctuation–dissipation theorem in the classical limit for (Hayes & Loudon, 1978). The coupling coefficient M is given by In practice, the incident intensity is defined outside the scattering volume, , and for normal incidence one can write Similarly, the scattered power is observed outside as , and again for normal incidence. Finally, the approximative relation between the scattering solid angle , outside the sample, and the solid angle , in the sample, is Substituting (2.4.4.9a,b,c) in (2.4.4.7), one obtains (Vacher & Boyer, 1972) where the coupling coefficient is In the cases of interest here, the tensor is diagonal, without summation on i, and (2.4.4.11) can be written in the simpler form
References
Hayes, W. & Loudon, R. (1978). Scattering of light by crystals. New York: Wiley.Nelson, D. F., Lazay, P. D. & Lax, M. (1972). Brillouin scattering in anisotropic media: calcite. Phys. Rev. B, 6, 3109–3120.
Vacher, R. & Boyer, L. (1972). Brillouin scattering: a tool for the measurement of elastic and photoelastic constants. Phys. Rev. B, 6, 639–673.