Symmetry properties of the scattering cross section

Gregora, I.

doi:10.1107/97809553602060000640

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. 2.3, pp. 316-317

Section 2.3.3.2. Symmetry properties of the scattering cross section

I. Gregora^a ^*

^a Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: gregora@fzu.cz

2.3.3.2. Symmetry properties of the scattering cross section

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The quantity that controls the symmetry properties of the scattering cross section due to excitation $[Q{_j}({\bf q})]$ is the squared modulus of the corresponding second-order susceptibility (second-rank tensor), contracted with the two polarization vectors of the incident and scattered light: $[\left| {{\bf e}_S \boldchi ^{(j)}({{\bf q},\omega _I, - \omega }){\bf e}_I }\right|^2 \equiv \left| {e_{S\alpha }\chi _{\alpha \beta }^{(j)}({\bf q},\omega _I, - \omega)e_{I\beta }}\right|^2. \eqno (2.3.3.9)]$ The nonlinear susceptibility tensor $[{\boldchi}^{(j)}({\bf q},{\omega}{_I},-{\omega})]$ is usually referred to as the first-order Raman tensor (defined in the literature to within a factor). Before discussing the consequences of the crystal symmetry on the form of the Raman tensor, let us mention two important approximations on which conventional analysis of its symmetry properties is based.

In a general case, the second-order susceptibilities are not necessarily symmetric. However, they fulfil a general symmetry property which follows from the symmetry of the scattering with respect to time inversion. Since the anti-Stokes process can be regarded as a time-inverted Stokes process (exchanging the role of the incident and scattered photons), it can be shown that in non-magnetic materials the susceptibilities obey the relation $[\chi _{\beta \alpha }^{(j)}({- {\bf q},\omega _S, \omega }) = \chi _{\alpha \beta }^{(j)}({{\bf q},\omega _I, - \omega }). \eqno (2.3.3.10)]$

In the quasi-static limit, i.e. if the scattering frequency is negligibly small compared with the incident photon frequency ( $[0\approx {\omega}\ll{\omega}{_I}\approx{\omega}{_S}]$ ), it follows that the susceptibilities of non-magnetic materials become symmetric in the Cartesian indices α, β. This symmetry is very well fulfilled in a great majority of cases. Appreciable antisymmetric contributions are known to occur, e.g. under resonant conditions, where the quasi-static approximation breaks down as the energy of the incident (or scattered) photon approaches those of electronic transitions.

Thus, in the first approximation, we set $[{\omega}]$ equal to zero and remove the time dependence in the phonon amplitudes, treating the normal coordinates as static. Then the nonlinear susceptibilities correspond to susceptibility derivatives, $[\chi _{\alpha \beta }^{(j)}({\bf q},\omega _I, - \omega) \to R_{\alpha \beta }^j ({\bf q}) \equiv \chi _{\alpha \beta }^{(j)}\left({{\bf q},\omega _I }\right) = {{\partial \chi _{\alpha \beta }(\omega _I)}\over {\partial Q_j^* ({\bf q})}}, \eqno (2.3.3.11)]$ where we suppressed the explicit dependence on $[{\omega}{_I}]$ and introduced a simplified notation for the Raman tensor $[{\bf R}{^j}({\bf q})]$ , still keeping the dependence on the scattering wavevector.

In deriving the symmetry properties of the Raman tensor $[{\bf R}{^j}({\bf q})]$ that follow from the crystal lattice symmetry, the main point is thus to determine its transformation properties under the symmetry operation of the crystal space group.

Since the magnitude of the scattering vector $[q\equiv |{\bf q}|]$ is very small compared with the Brillouin-zone dimensions, another conventional approximation is to neglect the q dependence of the susceptibilities. Setting $[{\bf q}\rightarrow 0]$ enables us to analyse the symmetry of the Raman tensor in terms of the factor group $[G{_0}]$ , which is isomorphous to the point group of the crystal lattice. This approach is, again, appropriate for the vast majority of cases. An important exception is, for instance, the scattering by acoustic modes (Brillouin scattering) or scattering by longitudinal plasma waves in semiconductors (plasmons): in these cases the Raman tensor vanishes for $[{\bf q} = 0]$ , since this limit corresponds to a homogeneous displacement of the system. Possible q-dependent effects can be treated by expanding the Raman tensor in powers of q and using compatibility relations between the symmetries at $[{\bf q} = 0]$ and at the full symmetries applicable in the $[{\bf q} \neq 0]$ case.

Let us mention that another notation is sometimes used in the literature for the Raman tensor. Since the square modulus of a second-rank tensor contracted with two vectors can be written as a fourth-rank tensor contracted with four vectors, one can introduce a fourth-rank tensor I^(j), $[I_{\alpha \beta \mu \nu }^{(j)} = {{\partial \chi _{\alpha \beta }^* }\over {\partial Q_j^{}}}{{\partial \chi _{\mu \nu }^{}}\over {\partial Q_j^ * }},]$ so that the scattering cross section of the jth mode is $[{{{\rm d}\sigma ^{(j)}}\over {{\rm d}\Omega \,\,{\rm d}\omega _S }} \approx e_{I\alpha }e_{I\mu }e_{S\beta }e_{S\nu }I_{\alpha \beta \mu \nu }^{(j)}.]$ If there are no antisymmetric components in the susceptibility derivatives, it can be shown that the fourth-rank tensor $[{\bf I}=(I_{\alpha\beta\mu\nu})]$ has at most 21 independent components, as for the elastic constants tensor.

References

International Tables for Crystallography (2006). Vol. D. ch. 2.3, pp. 316-317