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

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 342-343

Section General remarks

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: General remarks

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Various types of applied forces – in a general sense – can be classified according to symmetry, i.e. according to their transformation properties. Thus a force is characterized as a polar force if it transforms under the symmetry operation of the crystal like a polar tensor of appropriate rank (rank 1: electric field E; rank 2: electric field gradient [\nabla{\bf E}], stress T or strain S). It is an axial force if it transforms like an axial tensor (rank 1: magnetic field H). Here we shall deal briefly with the most important cases within the macroscopic approach of the susceptibility derivatives. We shall treat explicitly the first-order scattering only and neglect, for the moment, q-dependent terms.

In a perturbation approach, the first-order transition susceptibility [\delta\chi] in the presence of an applied force F can be expressed in terms of Raman tensors [{\bf R}{^j}({\bf F})] expanded in powers of F: [\displaylines{\delta\chi ({\bf F}) = \textstyle\sum\limits_j {\bf R}^j ({\bf F})Q{_j },\cr \hbox{ where }{\bf R}{^j}({\bf F}) = {\bf R}{^j}{^0} + {\bf R}{^j}{^F}{\bf F }+ {\textstyle{1 \over 2}}{\bf R}{^j}{^F}{^F}{\bf F}{\bf F} + {\ldots}.\cr\hfill (}]Here, [{\bf R}{^j}{^0} = \chi ^{(j)}(0) = ({{{\partial \chi _{\alpha \beta }}/{\partial Q_j }}})] is the zero-field intrinsic Raman tensor, whereas the tensors [\eqalignno{{\bf R}{^j}{^F}{\bf F} &= \left({{\partial ^2 \chi _{\alpha \beta }}\over{\partial Q_j \partial F_\mu }}\right)F_\mu, &\cr {\bf R}^{jFF}{\bf FF} &= \left({{\partial ^3 \chi _{\alpha \beta }}\over {\partial Q_j \partial F_\mu \partial F_\nu }}\right)F_\mu F_\nu \,\,etc.&(}]are the force-induced Raman tensors of the respective order in the field, associated with the jth normal mode. The scattering cross section for the jth mode becomes proportional to [|{\bf e}{_S}({\bf R}^{j0} + {\bf R}^{jF}{\bf F} + {\textstyle{1 \over 2}}{\bf R}^{jFF}{\bf F}{\bf F} + {\ldots}){\bf e}{_I}|{^2}], which, in general, may modify the polarization selection rules. If, for example, the mode is intrinsically Raman inactive, i.e. [{\bf R}^{j0} = 0] whereas [{\bf R}^{jF} \neq 0], we deal with purely force-induced Raman scattering; its intensity is proportional to [F{^2}] in the first order. Higher-order terms must be investigated if, for symmetry reasons, the first-order terms vanish.

For force-induced Raman activity, in accordance with general rules, invariance again requires that a particular symmetry species [\Gamma(j)] can contribute to the first-order transition susceptibility by terms of order n in the force only if the identity representation is contained in the reducible representation of the nth-order Raman tensor.

An equivalent formulation is that the nth-order tensor-like coefficients in the corresponding force-induced Raman tensor, i.e. [R_{\alpha \beta \mu\ldots\nu }^{jF\ldots F} = \left({{{\partial ^{1 + \,n}\chi _{\alpha \beta }^{}}\over {\partial Q_j \partial F_\mu\ldots\partial F_\nu }}}\right) \hbox{ in the term }{\bf R}^{jF \ldots F}{\bf F}{\ldots}{\bf F},]vanish identically for symmetry reasons unless [[{\Gamma} _{\rm PV} \otimes {\Gamma}_{\rm PV}] \otimes [{\Gamma}({\bf F})]^{n}_{S}\supset{\Gamma}(j)]. Here [[\Gamma({\bf F})]^{n}_{S} =] [[\Gamma({\bf F})] [\otimes] [ \Gamma({\bf F})] [ \otimes] [\ldots \otimes\Gamma({\bf F})]{_S}] is the symmetrized nth power of the representation [\Gamma({\bf F})] according to which the generalized force F transforms under the operation of the point group. The requirement for the symmetrized part is dictated by the interchangeability of the higher-order derivatives with respect to the components of the force. We recall that the first factor representing the susceptibility, [[\Gamma_{\rm PV} \otimes \Gamma_{\rm PV}]], need not be symmetric in general. However, for most purposes (non-resonant conditions, non-magnetic crystals in the absence of a magnetic field) it can be replaced by its symmetrized part [[\Gamma_{\rm PV} \otimes \Gamma_{\rm PV}]{_S}].

Standard group-theoretical methods can be used to determine the force-induced Raman activity in a given order of the field and to derive the matrix form of the corresponding Raman tensors. Before treating several important cases of morphic effects in more detail in the following sections, let us make a few comments.

Beside the force-induced effects on the scattering tensors, there are also the direct morphic effects of the forces on the excitations themselves (possible frequency shifts, lifting of mode degeneracies etc.), which can be investigated by an analogous perturbation treatment, i.e. by expanding the dynamical matrix in powers of F and determining the corresponding force-induced corrections in the respective orders.

The lifting of degeneracies is a typical sign of the fact that the symmetry of the problem is reduced. The extended system crystal + applied force corresponds to a new symmetry group resulting from those symmetry operations that leave the extended system invariant. Consequently, the new normal modes (in the long-wavelength limit) can be formally classified according to the new point group appropriate for the extended system, which qualitatively accounts for the new reduced symmetries and degeneracies.

The force-induced Raman tensors referring to the original crystal symmetry should thus be equivalent to the Raman tensors of the corresponding modes in the new point group via the compatibility relations. The new point-group symmetry of the extended system is often used to investigate Raman-induced activity. It should be noted, however, that this approach generally fails to predict to what order in the force the induced changes in the Raman tensors appear. Such information is usually of prime importance for the scattering experiment, where appropriate setup and detection techniques can be applied to search for a force-induced effect of a particular order. Thus the perturbation method is usually preferable (Anastassakis, 1980[link]).

In the following sections, we shall briefly treat the most important cases in the conventional limit [{\bf q}\to 0] (neglecting for the moment the spatial dispersion).


Anastassakis, E. M. (1980). In Dynamical properties of solids, edited by G. K. Horton & A. A. Maradudin, Vol. 4, pp. 159–375. Amsterdam, New York, Oxford: North-Holland.

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