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. 314-315

Section 2.3.2.2. 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

2.3.2.2. Cross section

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In the absence of any excitations, the incident field $[{\bf E}{_I}]$ at frequency $[\omega_I]$ induces in the crystal the polarization P, related to the field by the linear dielectric susceptibility tensor $[{\chi}]$ ( $[\varepsilon {_0}]$ is the permittivity of free space): $[{\bf P} = \varepsilon {_0} {\bf \chi}(\omega {_I}) {\bf E}{_I}. \eqno (2.3.2.4)]$ The linear susceptibility $[ \chi(\omega{_I})]$ is understood to be independent of position, depending on the crystal characteristics and on the frequency of the radiation field only. In the realm of nonlinear optics, additional terms of higher order in the fields may be considered; they are expressed through the respective nonlinear susceptibilities .

The effect of the excitations is to modulate the wavefunctions and the energy levels of the medium, and can be represented macroscopically as an additional contribution to the linear susceptibility . Treating this modulation as a perturbation, the resulting contribution to the susceptibility tensor, the so-called transition susceptibility $[{\delta}{ \chi}]$ can be expressed as a Taylor expansion in terms of normal coordinates $[Q{_j}]$ of the excitations: $[\chi \rightarrow \chi + \delta \chi, \hbox{ where }\delta \chi = \textstyle\sum\limits_j {\chi ^{(j)}Q_j + \textstyle\sum\limits_{j,j'}{\chi ^{(j,j')}Q_j Q_{j'} + \ldots}}. \eqno (2.3.2.5)]$ The tensorial coefficients $[\chi ^{(j)},\chi ^{(j,j')},\ldots]$ in this expansion are, in a sense, higher-order susceptibilities and are often referred to as Raman tensors (of the first, second and higher orders). They are obviously related to susceptibility derivatives with respect to the normal coordinates of the excitations. The time-dependent polarization induced by $[{\delta}{\chi}]$ via time dependence of the normal coordinates can be regarded as the source of the inelastically scattered radiation.

The central quantity in the description of Raman scattering is the spectral differential cross section, defined as the relative rate of energy loss from the incident beam (frequency $[\omega{_I}]$ , polarization $[{\bf e}{_I}]$ ) as a result of its scattering (frequency $[\omega{_S}]$ , polarization $[{\bf e}{_S}]$ ) in volume V into a unit solid angle and unit frequency interval. The corresponding formula may be concisely written as (see e.g. Hayes & Loudon, 1978) $[{{{\rm d}^2 \sigma }\over {{\rm d}\Omega \,\,{\rm d}\omega _S }}= {{\omega _S^3 \omega _I V^2 n_S }\over {(4\pi)^2 c^4 n_I }}\left\langle {\left| {{\bf e}_I \delta \chi {\bf e}_S }\right|^2 }\right\rangle _\omega. \eqno (2.3.2.6)]$ The symbol $[\left\langle {\ldots}\right\rangle _\omega ]$ stands for the power spectrum (correlation function) of the transition susceptibility fluctuations. The spectral differential cross section is the quantity that can be directly measured in a Raman scattering experiment by analysing the frequency spectrum of the light scattered into a certain direction. By integrating over frequencies $[\omega{_S}]$ for a particular Raman band and, in addition, over the solid angle, one obtains, respectively, the differential cross section ( $[{\rm d}\sigma/{\rm d}\Omega]$ ) and the total cross section ( $[\sigma_{\rm tot}]$ ): $[{{{\rm d}\sigma }\over {{\rm d}\Omega }} = \int {\left({{{{\rm d}^2 \sigma }\over {{\rm d}\Omega \,\,{\rm d}\omega _S }}}\right)}{\rm d}\omega _S, \quad \sigma _{\rm tot} = \int {\left({{{{\rm d}\sigma }\over {{\rm d}\Omega }}}\right){\rm d}\Omega }.]$ These quantities are useful in comparing the integrated scattered intensity by different excitations.

References

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

International Tables for Crystallography (2006). Vol. D. ch. 2.3, pp. 314-315