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

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

Section 2.3.2. Inelastic light scattering in crystals – basic notions

I. Gregoraa*

aInstitute 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. Inelastic light scattering in crystals – basic notions

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Although quantum concepts must be used in any complete theory of inelastic scattering, basic insight into the problem may be obtained from a semiclassical treatment. In classical terms, the origin of inelastically scattered light in solids should be seen in the modulation of the dielectric susceptibility of a solid by elementary excitations. The exciting light polarizes the solid and the polarization induced via the modulated part of the susceptibility is re-radiated at differently shifted frequencies. Thus inelastic scattering of light by the temporal and spatial fluctuations of the dielectric susceptibility that are induced by elementary excitations provides information about the symmetry and wavevector-dependent frequencies of the excitations themselves as well as about their interaction with electromagnetic waves.

2.3.2.1. Kinematics

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Let us consider the incident electromagnetic radiation, the scattered electromagnetic radiation and the elementary excitation to be described by plane waves. The incident radiation is characterized by frequency [\omega{_I}], wavevector [{\bf k}{_I}] and polarization vector [{\bf e}{_I}]. Likewise, the scattered radiation is characterized by [{\omega}{_S}], [{\bf k}{_S}] and [{\bf e}{_S}]: [{\bf E}_{I,S}({\bf r},t) = E_{I,S}{\bf e}_{I,S}\exp(i{\bf k}_{I,S}{\bf r}-{\omega}t). \eqno (2.3.2.1)]

The scattering process involves the annihilation of the incident photon, the emission or annihilation of one or more quanta of elementary excitations and the emission of a scattered photon. The scattering is characterised by a scattering frequency [{\omega}] (also termed the Raman shift) corresponding to the energy transfer [\hbar \omega] from the radiation field to the crystal, and by a scattering wavevector q corresponding to the respective momentum transfer [\hbar {\bf q}]. Since the energy and momentum must be conserved in the scattering process, we have the conditions[\eqalignno{{\omega}_I - {\omega}_S &= {\omega}, &\cr {\bf k}{_I} - {\bf k}{_S} & = {\bf q}.&(2.3.2.2)}]Strictly speaking, the momentum conservation condition is valid only for sufficiently large, perfectly periodic crystals. It is further assumed that there is no significant absorption of the incident and scattered light beams, so that the wavevectors may be considered real quantities.

Since the photon wavevectors ([{\bf k}{_I}], [{\bf k}{_S}]) and frequencies ([{\omega}{_ I}], [{\omega}{_S}]) are related by the dispersion relation [{\omega}= ck/n], where c is the speed of light in free space and n is the refractive index of the medium at the respective frequency, the energy and wavevector conservation conditions imply for the magnitude of the scattering wavevector q[c^2 q^2 = n_I^2 \omega _I^2 + n_S^2 (\omega _I - \omega)^2 - 2n_I n_S \omega _I (\omega _I - \omega)\cos \varphi, \eqno (2.3.2.3)]where [{\varphi}] is the scattering angle (the angle between [{\bf k}{_I}] and [{\bf k}{_S}]). This relation defines in the ([\omega, q]) plane the region of wavevectors and frequencies accessible to the scattering. This relation is particularly important for scattering by excitations whose frequencies depend markedly on the scattering wavevector (e.g. acoustic phonons, polaritons etc.).

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[link]) [{{{\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.

2.3.2.3. Experimental aspects

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In a scattering experiment on crystals, the choice of the scattering geometry implies setting the propagation directions [{\bf k}{_I}] and [{\bf k}{_S}] and the polarization of the incident and scattered light with respect to the crystallographic axes and defining thus the direction of the scattering wavevector q as well as the particular component (or a combination of components) of the transition susceptibility tensor [{\delta\chi}]. In practice, the incident radiation is almost exclusively produced by a suitable laser source, which yields a monochromatic, polarized narrow beam, with a well defined wavevector [{\bf k}{_I}]. The light scattered in the direction of [{\bf k}{_S}] is collected over a certain finite solid angle [\Delta\Omega]. Its polarization is analysed with a suitable polarization analyser, and the scattered intensity as a function of frequency [\omega{_S}] (or Raman shift [{\omega}]) is analysed using a spectrometer.

To characterize the Raman scattering geometry in a particular experimental arrangement, standard notation for the scattering geometry is often used, giving the orientation of the wavevectors and polarization vectors with respect to a reference Cartesian coordinate system, namely: [{\bf k}{_I}({\bf e}{_I}, {\bf e}{_S}){\bf k}{_S}]. Thus, for example, the symbol [x(zy)z] means that right-angle scattering geometry is used, where the incident beam polarized in the [z=[001]] direction propagates along the [x=[100]] axis, while the scattered beam is collected in the z direction and the polarization analyser is set parallel with the [y = [010]] direction. The measured intensity, being proportional to [|\chi_{zy}|{^2}], gives information on this particular component of the transition susceptibility tensor. By virtue of the momentum conservation, the scattering wavevector q in this case is oriented along the [[10\bar 1]] direction.

In a typical Raman experiment with visible light ([\omega \ll \omega{_I} \approx\omega{_S}]), the magnitudes of the wavevectors [ k{_I}\approx k{_s}= k] are of the order of 105 cm−1, much lower than those of the reciprocal-lattice vectors K ([\approx 10^{8}] cm−1). Consequently, the range of the magnitudes of the scattering wavevectors q accessible by varying the scattering geometry from [{\varphi}= 0]° (forward scattering) to [{\varphi}=180]° (back scattering) is [0 \leq q \leq 2k], i.e. by about three orders of magnitude lower than the usual dimensions of the Brillouin zone. The use of back-scattering geometry is imperative in the case of opaque samples, which show stronger absorption for the exciting (or scattered) light.

It should be noted that the general formula for the spectral differential cross section (2.3.2.5)[link] applies to the situation inside the crystal. Since in real experiments the observer is always outside the crystal, several corrections have to be taken into account. These are in particular due to refraction, reflection and transmission of the incident and scattered light at the interfaces, as well as absorption of light in the crystal. Attention must be paid in the case of anisotropic or gyrotropic crystals, where birefringence or rotation of the polarization direction of both incident and scattered light may occur on their paths through the crystal, between the interfaces and the scattering volume.

We conclude this section by remarking that, owing to the obvious difficulties in taking all the properties of the experimental setup and the corrections into consideration, measurements of absolute Raman intensities tend to be extremely rare. There exist, however, several crystals for which absolute determination of the cross section for particular excitations has been made with reasonable reliability and which may serve as secondary standards.

References

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








































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