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. 342345
Section 2.3.4. Morphic effects in Raman scattering^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic 
By morphic effects we understand the effects that arise from a reduction of the symmetry of a system caused by the application of external forces. The relevant consequences of morphic effects for Raman scattering are changes in the selection rules. Applications of external forces may, for instance, render it possible to observe scattering by excitations that are otherwise inactive. Again, grouptheoretical arguments may be applied to obtain the symmetryrestricted component form of the Raman tensors under applied forces.
It should be noted that under external forces in this sense various `builtin' fields can be included, e.g. electric fields or elastic strains typically occurring near the crystal surfaces. Effects of `intrinsic' macroscopic electric fields associated with longwavelength LO polar phonons can be treated on the same footing. Spatialdispersion effects connected with the finiteness of the wavevectors, q or k, may also be included among morphic effects, since they may be regarded as being due to the gradients of the fields (displacement or electric) propagating in the crystal.
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 , 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 firstorder scattering only and neglect, for the moment, qdependent terms.
In a perturbation approach, the firstorder transition susceptibility in the presence of an applied force F can be expressed in terms of Raman tensors expanded in powers of F: Here, is the zerofield intrinsic Raman tensor, whereas the tensors are the forceinduced 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 , which, in general, may modify the polarization selection rules. If, for example, the mode is intrinsically Raman inactive, i.e. whereas , we deal with purely forceinduced Raman scattering; its intensity is proportional to in the first order. Higherorder terms must be investigated if, for symmetry reasons, the firstorder terms vanish.
For forceinduced Raman activity, in accordance with general rules, invariance again requires that a particular symmetry species can contribute to the firstorder transition susceptibility by terms of order n in the force only if the identity representation is contained in the reducible representation of the nthorder Raman tensor.
An equivalent formulation is that the nthorder tensorlike coefficients in the corresponding forceinduced Raman tensor, i.e. vanish identically for symmetry reasons unless . Here is the symmetrized nth power of the representation 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 higherorder derivatives with respect to the components of the force. We recall that the first factor representing the susceptibility, , need not be symmetric in general. However, for most purposes (nonresonant conditions, nonmagnetic crystals in the absence of a magnetic field) it can be replaced by its symmetrized part .
Standard grouptheoretical methods can be used to determine the forceinduced 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 forceinduced 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 forceinduced 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 longwavelength 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 forceinduced 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 pointgroup symmetry of the extended system is often used to investigate Ramaninduced 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 forceinduced effect of a particular order. Thus the perturbation method is usually preferable (Anastassakis, 1980).
In the following sections, we shall briefly treat the most important cases in the conventional limit (neglecting for the moment the spatial dispersion).
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.

In a magnetic field, the dielectric susceptibility tensor of a crystal is known to obey the general relation (Onsager reciprocity theorem for generalized kinetic coefficients)Further, in the absence of absorption, the susceptibility must be Hermitian, i.e. Hence, is neither symmetric nor real. Expanding in the powers of the field, it follows that all terms of the magneticfieldinduced Raman tensor that are of odd powers in H are purely imaginary and antisymmetric in α and β, whereas all terms of even powers in H are real and symmetric.
Let us discuss in more detail the symmetry properties of the firstorder term, which can be written as where the tensor f, referred to as the magnetooptic tensor, is real and purely antisymmetric in the first two indices: The representation of the magnetooptic tensor f may thus be symbolically written as since the antisymmetric part of the product of two polar vectors transforms like an axial vector, and the product of two axial vectors transforms exactly like the product of two polar vectors. Hence, the representation is equivalent to the representation of a general nonsymmetric secondrank tensor and reduces in exactly the same way (2.3.3.14).
We arrive thus at the important conclusion that, to first order in the field, only the modes that normally show intrinsic Raman activity (either symmetric and antisymmetric) can take part in magneticfieldinduced scattering. Moreover, the magneticfieldinduced Raman tensors for these symmetry species must have the same number of components as the general nonsymmetric Raman tensors at zero field.
In order to determine the symmetryrestricted matrix form of the corresponding fieldinduced Raman tensors (linear in H) in Cartesian coordinates, one can use the general method and construct the tensors from the respective (antisymmetric) basis functions. In this case, however, a simpler method can be adopted, which makes use of the transformation properties of the magnetooptic tensor as follows.
From the definition of the tensor f, it is clear that its Cartesian components must have the same symmetry properties as the product . The antisymmetric factor transforms, however, as , where is the fully antisymmetric thirdrank pseudotensor (Levi–Civita tensor). Consequently, must transform in the same way as , which in turn transforms identically to . Therefore, comparison of the matrices corresponding to the irreducible components provides a simple mapping between the components of the Cartesian forms of the linear fieldinduced Raman tensors and the intrinsic Raman tensors . Explicitly, this mapping is given byFor any given symmetry species, this relation can be used to deduce the matrix form of the firstorder fieldinduced Raman tensors from the tensors given in Table 2.3.3.1.
Example: We consider again the 4mm class crystal. The representation of the magnetooptic tensor f in the class reduces as follows: Straightforward application of the mapping mentioned above then gives the following symmetryrestricted matrix forms of contributions to the magneticfieldinduced Raman tensors for all symmetry species of the 4mmclass crystals. The number of independent parameters for each species is the same as in the intrinsic nonsymmetric zerofield Raman tensors: Let us note that the conclusions mentioned above apply, strictly speaking, to nonmagnetic crystals. In magnetic materials in the presence of spontaneous ordering (ferro or antiferromagnetic crystals) the analysis has to be based on magnetic point groups.
Stressinduced Raman scattering is an example of the case when the external `force' is a higherrank tensor. In the case of stress, we deal with a symmetric secondrank tensor. Since symmetric stress (T) and strain (S) tensors have the same symmetry and are uniquely related via the fourthrank elastic stiffness tensor (c),it is immaterial for symmetry purposes whether stress or straininduced effects are considered. The linear straininduced contribution to the susceptibility can be written as so that the respective strain coefficients (conventional symmetric scattering) transform evidently as i.e. they have the same symmetry as the piezooptic or elastooptic tensor. Reducing this representation into irreducible components , we obtain the symmetryrestricted form of the linear straininduced Raman tensors. Evidently, their matrix form is the same as for quadratic electricfieldinduced Raman tensors. In centrosymmetric crystals, straininduced Raman scattering (in any order in the strain) is thus allowed for evenparity modes only.
References
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: NorthHolland.