International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 343344
Section 2.3.4.2. Electricfieldinduced scattering^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic 
Expanding the linear dielectric susceptibility into a Taylor series in the field, we write The coefficients of the fielddependent terms in this expansion are, respectively, third, fourth and higherrank polar tensors; they describe linear, quadratic and higherorder electrooptic effects. The corresponding expansion of the Raman tensor of the jth optic mode is written as .
Since the representation , the coefficients of the linear term in the expansion for , i.e. the thirdrank tensor , transform according to the reducible representation given by the direct product: Firstorder fieldinduced Raman activity (conventional symmetric scattering) is thus obtained by reducing this representation into irreducible components . Higherorder contributions are treated analogously.
It is clear that in centrosymmetric crystals the reduction of a thirdrank polar tensor cannot contain evenparity representations; consequently, electricfieldinduced scattering by evenparity modes is forbidden in the first order (and in all odd orders) in the field. The lowest nonvanishing contributions to the fieldinduced Raman tensors of evenparity modes in these crystals are thus quadratic in E; their form is obtained by reducing the representation of a fourthrank symmetric polar tensor into irreducible components . On the other hand, since the electric field removes the centre of inversion, scattering by oddparity modes becomes allowed in first order in the field but remains forbidden in all even orders. In noncentrosymmetric crystals, parity considerations do not apply.
For completeness, we note that, besides the direct electrooptic contribution to the Raman tensor due to fieldinduced distortion of the electronic states of the atoms in the unit cell, there are two additional mechanisms contributing to the total firstorder change of the dielectric susceptibility in an external electric field E. They come, respectively, from fieldinduced relative displacements of atoms due to fieldinduced excitation of polar optical phonons and from fieldinduced elastic deformation (piezoelectric effect, d being the piezoelectric tensor). In order to separate these contributions, we write formally and get, to first order in the field,
The first term in these equations involves the susceptibility derivative at constant and S. The second term involves the secondorder susceptibility derivatives with respect to the normal coordinates: . Since , where the quantity is the effective charge tensor (2.3.3.4) of the normal mode p, its nonzero contributions are possible only if there are infraredactive optical phonons (for which, in principle, ) in the crystal. The third term is proportional to the fieldinduced elastic strain via the elastooptic tensor and can occur only in piezoelectric crystals.
Example: As an illustration, we derive the matrix form of linear electricfieldinduced Raman tensors (including possible antisymmetric part) in a tetragonal crystal of the 4mm class. The corresponding representation in this class reduces as follows: Suitable sets of symmetrized (s) and antisymmetrized (a) basis functions (thirdorder polynomials) for the representations of the 4mm point group can be easily derived by inspection or using projection operators. The results are given in Table 2.3.4.1. Using these basis functions, one can readily construct the Cartesian form of the linear contributions to the electricfieldinduced Raman tensors for all symmetry species of the class crystals. The tensors are split into symmetric (conventional allowed scattering) and antisymmetric part.
