International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 2.3, pp. 326327
Section 2.3.6. Higherorder scattering^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic 
In higherorder processes, the scattering involves participation of two or more quanta (j and ) of the elementary excitations. Let us discuss briefly the secondorder scattering by phonons, where the energy and wavevector conservation conditions read The combinations of signs in these equations correspond to four possibilities, in which either both phonons, j and , are created (Stokes process: ), both annihilated (antiStokes process: −−), or one is created and the other annihilated (difference process: , ). If in the Stokes or antiStokes case both excitations are of the same type, , one speaks of overtones. The corresponding terms in the transition susceptibility are the coefficients of a bilinear combination of normal coordinates in the expansion of .
In the quasistatic limit, the transition susceptibilities for the secondorder scattering correspond, again, to the susceptibility derivatives. Thus, the spectral differential cross section for the secondorder scattering (Stokes component) can be formally written as with . In this formula, we have suppressed the universal factors [see (2.3.3.5)] and the explicit expression for the response function (thermal factors). Instead, the delta function (response function in the limit of zero damping) expresses the energyconservation condition.
The wavevector selection rules in the longwavelength limit, with , imply that (the same holds for antiStokes components, while for difference scattering), so the wavevectors themselves need not be small and, in principle, scattering by phonons with all wavevectors from the Brillouin zone can be observed.
Without invoking any symmetry arguments for the Raman activity, such as the restrictions imposed by crystal symmetry on the susceptibility derivatives, it is clear that the intensity of secondorder scattering at a frequency is controlled by the number of those combinations of phonons whose frequencies obey . The quantity determining this number is the combined density of states of phonon pairs, i.e. This function can be calculated provided the dispersion curves of the excitations are known. The density of states is a continuous function and shows features known as the van Hove singularities corresponding to the critical points, where one or more components of the gradient vanish. Most of the critical points occur for wavevectors on the boundary, where the vanishing gradients of the individual dispersion curves are often dictated by the crystal symmetry, but they also occur in those regions of the reciprocal space where both dispersion curves have opposite or equal slopes at the same wavevector q. To a first approximation, the secondorder spectrum is thus essentially continuous, reflecting the twophonon density of states, with peaks and sharp features at frequencies close to the positions of the van Hove singularities. This is to be contrasted with the firstorder scattering, where (in perfect crystals) only single peaks corresponding to longwavelength () phonons occur.
Grouptheoretical arguments may again be invoked in deriving the selection rules that determine the Raman activity of a particular combination of excitations (Birman, 1974). The susceptibility derivative again transforms as a tensor. For a given pair of excitations () and () responsible for the modulation, the combined excitation symmetry is obtained by taking the direct product of the irreducible representations of the space group corresponding to the participating excitations, The representation , unlike , corresponds to a zerowavevector representation of the crystal space group and is therefore equivalent to a (reducible) representation of the crystal point group. It can be decomposed into irreducible components. Raman scattering of the pair is allowed if a Ramanactive representation is contained in this decomposition of or, alternatively, if the product contains the totally symmetric representation .
The selection rules for the secondorder scattering are, in general, far less restrictive than in the firstorder case. For example, it can be shown that for a general wavevector q in the Brillouin zone there are no selection rules on the participation of phonons in the secondorder scattering, since the representations contain all Ramanactive symmetries. In specific crystal structures, however, restrictions occur for the wavevectors corresponding to special symmetry positions (points, lines or planes) in the Brillouin zone. This implies that the selection rules may suppress some of the van Hove singularities in the secondorder spectra.
Morphic effects in secondorder scattering, due to an applied external force F (see Section 2.3.4.1), may be investigated using the same criteria as in firstorder scattering, i.e. decomposing the representation and searching for the matrix form of the corresponding secondorder Raman tensors.
Generalization to third and higherorder processes is obvious.
Concluding this section, we note that in a Ramanscattering experiment, higherorder features in the spectra can in principle be distinguished from firstorder features by different behaviour of the differential scattering cross section with temperature. For example, the respective thermal factors entering the expression for the secondorder scattering cross section are given in Table 2.3.6.1.

References
Birman, J. L. (1974). Theory of crystal space groups and infrared and Raman lattice processes of insulating crystals. Handbuch der Physik, Vol. 25/2b. Berlin, Heidelberg, New York: Springer.