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

International Tables for Crystallography (2013). Vol. D, ch. 2.4, pp. 350-351

Section 2.4.4.2. Scattering cross section

R. Vachera* and E. Courtensa

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

2.4.4.2. 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[link]). 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)[link][link][link] in (2.4.4.7)[link], one obtains (Vacher & Boyer, 1972[link]) [{{{\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)[link] 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.
Vacher, R. & Boyer, L. (1972). Brillouin scattering: a tool for the measurement of elastic and photoelastic constants. Phys. Rev. B, 6, 639–673.








































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