Brillouin scattering in crystals

Vacher, R.; Courtens, E.

doi:10.1107/97809553602060000641

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.4, pp. 330-331

Section 2.4.4. Brillouin scattering in crystals

R. Vacher^a ^* and E. Courtens^a

^a Laboratoire des Verres, Université Montpellier 2, Case 069, Place Eugène Bataillon, 34095 Montpellier CEDEX, France
Correspondence e-mail: rene.vacher@ldv.univ-montp2.fr

2.4.4. Brillouin scattering in crystals

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2.4.4.1. Kinematics

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Brillouin scattering occurs when an incident photon at frequency $[\nu _i]$ interacts with the crystal to either produce or absorb an acoustic phonon at $[\delta \nu]$ , while a scattered photon at $[\nu _s]$ is simultaneously emitted. Conservation of energy gives $[\delta \nu = \nu _s - \nu _i, \eqno (2.4.4.1)]$ where positive $[\delta \nu]$ corresponds to the anti-Stokes process. Conservation of momentum can be written $[{\bf Q}= {\bf k}_s - {\bf k}_i, \eqno (2.4.4.2)]$ where Q is the wavevector of the emitted phonon, and $[{\bf k}{_s}]$ , $[{\bf k}{_i}]$ are those of the scattered and incident photons, respectively. One can define unit vectors q in the direction of the wavevectors k by $[\eqalignno{{\bf k}_i &= 2\pi {\bf q}n/\lambda _0, & (2.4.4.3a) \cr {\bf k}_s &= 2\pi {\bf q}'n'/\lambda _0, &(2.4.4.3b)}]$ where n and [n'] are the appropriate refractive indices, and $[\lambda _0]$ is the vacuum wavelength of the radiation. Equation (2.4.4.3b) assumes that $[\delta \nu \ll \nu _i]$ so that $[\lambda _0]$ is not appreciably changed in the scattering. The incident and scattered waves have unit polarization vectors $[{\bf e}]$ and $[{\bf e}']$ , respectively, and corresponding indices n and [n'] . The polarization vectors are the principal directions of vibration derived from the sections of the ellipsoid of indices by planes perpendicular to $[{\bf q}]$ and $[{\bf q}']$ , 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 $[{\bf E}_{\rm opt}]$ and $[{\bf D}_{\rm opt}]$ 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 $[\hat{\bf Q}= {{n'{\bf q}' - n{\bf q}}\over {\left| {n'{\bf q}' - n{\bf q}}\right|}}. \eqno (2.4.4.4)]$ The Brillouin shift $[\delta \nu]$ is related to the phonon velocity V by $[\delta \nu = VQ/2\pi. \eqno (2.4.4.5)]$ Since $[\nu \lambda _0 = c]$ , from (2.4.4.5) and (2.4.4.3), (2.4.4.4) one finds $[\delta \nu \cong (V/{\lambda _0 }) [{n^2 + ({n'} )^2 - 2nn'\cos \theta } ]^{1/2}, \eqno (2.4.4.6)]$ where $[\theta ]$ is the angle between $[{\bf q}]$ and $[{\bf q}']$ .

2.4.4.2. Scattering cross section

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The power $[{\rm d}P_{\rm in}]$ , scattered from the illuminated volume V in a solid angle $[{\rm d}\Omega _{\rm in}]$ , where $[P_{\rm in}]$ and $[\Omega _{\rm in}]$ are measured inside the sample, is given by $[{{{\rm d}P_{\rm in}}\over {{\rm d}\Omega _{\rm in}}}= V{{k_B T\pi ^2 n'}\over {2n\lambda _0^4 C}}MI_{\rm in}, \eqno (2.4.4.7)]$ where $[I_{\rm in}]$ is the incident light intensity inside the material, $[C = \rho V^2]$ is the appropriate elastic constant for the observed phonon, and the factor [k_B T] results from taking the fluctuation–dissipation theorem in the classical limit for $[h\delta \nu \ll k_B T]$ (Hayes & Loudon, 1978). The coupling coefficient M is given by $[M = | {e_m e'_n \kappa _{mi}\kappa _{nj}p'_{ijk\ell }\hat u_k \hat Q_\ell } |^2. \eqno (2.4.4.8)]$ In practice, the incident intensity is defined outside the scattering volume, $[I_{\rm out}]$ , and for normal incidence one can write $[I_{\rm in} = {{4n}\over { ({n + 1} )^2 }}I_{\rm out}. \eqno (2.4.4.9a)]$ Similarly, the scattered power is observed outside as $[P_{\rm out}]$ , and $[P_{\rm out} = {{4n'}\over { ({n' + 1} )^2 }}P_{\rm in}, \eqno (2.4.4.9b)]$ again for normal incidence. Finally, the approximative relation between the scattering solid angle $[\Omega _{\rm out}]$ , outside the sample, and the solid angle $[\Omega _{\rm in}]$ , in the sample, is $[\Omega _{\rm out} = ({n'} )^2 \Omega _{\rm in}. \eqno (2.4.4.9c)]$ Substituting (2.4.4.9a,b,c) in (2.4.4.7), one obtains (Vacher & Boyer, 1972) $[{{{\rm d}P_{\rm out}}\over {{\rm d}\Omega _{\rm out}}}= {{8\pi ^2 k_B T}\over {\lambda _0^4 }}{{n^4 }\over {({n + 1})^2 }}{{({n'})^4 }\over {({n' + 1})^2 }}\beta VI_{\rm out}, \eqno (2.4.4.10)]$ where the coupling coefficient $[\beta]$ is $[\beta = {1 \over {n^4 ({n'} )^4 }}{{ | {e_m e'_n \kappa _{mi}\kappa _{nj}p'_{ijk\ell }\hat u_k \hat Q_\ell } |^2 }\over C}. \eqno (2.4.4.11)]$ In the cases of interest here, the tensor $[\boldkappa]$ is diagonal, $[\kappa _{ij} = n_i^2 \delta _{ij}]$ without summation on i, and (2.4.4.11) can be written in the simpler form $[\beta = {1 \over {n^4 ({n'})^4 }}{{| {e_i n_i^2 p'_{ijk\ell }\hat u_k \hat Q_\ell e'_j n_j^2 }|^2 }\over C}. \eqno (2.4.4.12)]$

References

Hayes, W. & Loudon, R. (1978). Scattering of light by crystals. New York: Wiley.Google Scholar

Nelson, D. F., Lazay, P. D. & Lax, M. (1972). Brillouin scattering in anisotropic media: calcite. Phys. Rev. B, 6, 3109–3120.Google Scholar

Vacher, R. & Boyer, L. (1972). Brillouin scattering: a tool for the measurement of elastic and photoelastic constants. Phys. Rev. B, 6, 639–673.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 2.4, pp. 330-331