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. 315322
Section 2.3.3. Firstorder scattering by phonons^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic 
In what follows, we shall be more specific and by underlying excitations we shall explicitly understand lattice vibrations – phonons – although the treatment is also applicable to other types of collective excitations in a crystal.
Let us recall (see Chapter 2.1 ) that atomic displacements in the crystal lattice can be expressed as linear combinations of the normal modes of vibrations – eigenvectors of the dynamical matrix. where l denotes the primitive unit cell, () is the index of the atom (mass ) in the unit cell and N is the number of unit cells in the crystal. The eigenvectors , also called polarization vectors, are normalized vectors describing the displacement pattern of atoms in the reference unit cell in a jth normal mode of vibration () with a wavevector q. The normal modes can be classified according to irreducible representations of the crystal space group and labelled correspondingly. In the case of degeneracy, the branch index j stands, in fact, for two indices: one for the irreducible representation, and the other distinguishing the degenerate partners of the same frequency. The coefficients are called normal coordinates and represent the timedependent amplitudes of the normal modes, vibrating with frequencies . The reality of the displacements and the hermiticity of the dynamical matrix impose the following conditions:In the limit of , three of the vibrational branches correspond to homogeneous displacements of the crystal with vanishing frequency along three independent directions. These branches are termed acoustic. The remaining branches (provided that ) are called optic; they correspond to relative displacement of sublattices with nonzero frequency.
For completeness, we note that in a phenomenological treatment the amplitude of the polarization set up by atomic displacements of the normal modes with wavevector q can be expressed as introducing the () effective charge matrix of the jth normal mode of vibration (per unit cell of volume ), The quantity represents the microscopic effective charge of the th atom in the unit cell. Owing to dynamic contributions of ionic deformability, it is not a scalar but rather a () tensor in general. Hence, in general, the direction of the contribution of a mode to the electrical polarization (2.3.3.3) need not correspond to its mechanical polarization, as given by the mode eigenvectors .
In some cases, the optic modes carrying a nonzero effective charge (socalled polar optic modes) may be classified as transverse (TO) or longitudinal (LO). Whenever applicable, this classification should be understood in the correct sense, i.e. according to the orientation of the associated electric polarization relative to the wavevector q.
In the firstorder scattering by a single excitation of the normal mode () of frequency and wavevector , the energy and wavevector conservation conditions give with the sign corresponding to a Stokes process (one excitation quantum is created) and the − sign to an antiStokes process (one quantum is annihilated).
Let us explicitly consider the Stokes component, described by the term linear in the normal coordinate of the excitation. Inserting the planewave expressions for the quantities into the first term of the expansion (2.3.2.5) for and comparing the terms with common time dependence, we get for the Stokes polarization due to the normal mode ()which corresponds to a plane wave at a frequency of .
Since, in the harmonic approximation, the normal modes () are dynamically independent, i.e. uncorrelated, the cross section for the Stokes component of the firstorder scattering by phonons can be written as a sum over contributions from individual excitations. A summation convention over repeated Cartesian indices is understood throughout this chapter:
Let us first briefly review the last term in this expression, which – together with the universal frequencydependent first factor – essentially determines the shape of the scattered light spectrum, i.e. the frequency and temperature dependence of the spectral differential cross section. It depends exclusively on the fluctuation properties of the excitations participating in the scattering process.
The power spectrum of the fluctuations can be calculated using the linear response theory. The normal coordinates, i.e. excitation amplitudes of the normal modes, satisfy the decoupled equations of motion (with phenomenological damping constant added to take into account the finite lifetime of excitations in real crystals within the quasiharmonic approximation): If a fictitious generalized force with Fourier components is applied to the system, the average of the Fourier components of the excitation amplitude and the force are proportional, where the proportionality coefficient is called a linear response function.
The fluctuation–dissipation theorem (or, more exactly, its quantum version) relates the power spectrum of a fluctuating quantity to the imaginary part of the corresponding response function. The results for the Stokes and antiStokes components of the scattering are, respectively: where is the Bose–Einstein statistical factor which gives the occupation number of phonon states.
The linear response function of the normal coordinates is readily obtained from the equation of motion (N being the number of primitive cells in the crystal): If the damping parameter of the excitations is much smaller than the frequency , as is usually the case for phonons in perfect crystals, this function is closely approximated by a Lorentzian centred at , with a full width at half maximum. Hence, introducing a suitably normalized lineshape function ,we can write for the power spectra (2.3.3.5) of the jth normal mode in the spectral differential cross section for the Stokes and antiStokes case, respectively.
Note that at low temperatures the differential cross section of the antiStokes component in the spectrum becomes vanishingly small, because the mean number of thermally excited phonons vanishes at .
Information about the interaction of photons with individual excitations is contained in the central term in the expression for the cross section (2.3.3.5), i.e. in the nonlinear susceptibility .
The quantity that controls the symmetry properties of the scattering cross section due to excitation is the squared modulus of the corresponding secondorder susceptibility (secondrank tensor), contracted with the two polarization vectors of the incident and scattered light: The nonlinear susceptibility tensor is usually referred to as the firstorder Raman tensor (defined in the literature to within a factor). Before discussing the consequences of the crystal symmetry on the form of the Raman tensor, let us mention two important approximations on which conventional analysis of its symmetry properties is based.
In a general case, the secondorder susceptibilities are not necessarily symmetric. However, they fulfil a general symmetry property which follows from the symmetry of the scattering with respect to time inversion. Since the antiStokes process can be regarded as a timeinverted Stokes process (exchanging the role of the incident and scattered photons), it can be shown that in nonmagnetic materials the susceptibilities obey the relation
In the quasistatic limit, i.e. if the scattering frequency is negligibly small compared with the incident photon frequency (), it follows that the susceptibilities of nonmagnetic materials become symmetric in the Cartesian indices α, β. This symmetry is very well fulfilled in a great majority of cases. Appreciable antisymmetric contributions are known to occur, e.g. under resonant conditions, where the quasistatic approximation breaks down as the energy of the incident (or scattered) photon approaches those of electronic transitions.
Thus, in the first approximation, we set equal to zero and remove the time dependence in the phonon amplitudes, treating the normal coordinates as static. Then the nonlinear susceptibilities correspond to susceptibility derivatives, where we suppressed the explicit dependence on and introduced a simplified notation for the Raman tensor , still keeping the dependence on the scattering wavevector.
In deriving the symmetry properties of the Raman tensor that follow from the crystal lattice symmetry, the main point is thus to determine its transformation properties under the symmetry operation of the crystal space group.
Since the magnitude of the scattering vector is very small compared with the Brillouinzone dimensions, another conventional approximation is to neglect the q dependence of the susceptibilities. Setting enables us to analyse the symmetry of the Raman tensor in terms of the factor group , which is isomorphous to the point group of the crystal lattice. This approach is, again, appropriate for the vast majority of cases. An important exception is, for instance, the scattering by acoustic modes (Brillouin scattering) or scattering by longitudinal plasma waves in semiconductors (plasmons): in these cases the Raman tensor vanishes for , since this limit corresponds to a homogeneous displacement of the system. Possible qdependent effects can be treated by expanding the Raman tensor in powers of q and using compatibility relations between the symmetries at and at the full symmetries applicable in the case.
Let us mention that another notation is sometimes used in the literature for the Raman tensor. Since the square modulus of a secondrank tensor contracted with two vectors can be written as a fourthrank tensor contracted with four vectors, one can introduce a fourthrank tensor I^{(j)}, so that the scattering cross section of the jth mode is If there are no antisymmetric components in the susceptibility derivatives, it can be shown that the fourthrank tensor has at most 21 independent components, as for the elastic constants tensor.
The scattering cross section, being a scalar quantity, must be invariant with respect to all symmetry elements of the space group of the crystal. This invariance has two important consequences: it determines which normal modes (j) can contribute to the scattering (Raman activity of the modes) and it also gives the restrictions on the number of independent components of the Raman tensor (polarization selection rules).
At , the transformation properties of the incident and scattered light are described by the threedimensional polar vector representation of the appropriate point group of the crystal, since the quantities that characterize the light (, , ) are all polar vectors, i.e. firstrank polar tensors (). The transformation properties of a normal mode j must correspond to an irreducible matrix representation of the crystal point group. We recall that in cases of two or threedimensional representations (degeneracy), the index j represents two indices.
In order that a particular normal mode j in a given crystal be Raman active, i.e. symmetryallowed to contribute to the (firstorder) scattering cross section, the necessary condition is that the corresponding irreducible representation must be contained in the decomposition of the direct Kronecker product representation at least once: In this case, the Kronecker product contains the identity representation at least once, so the cross section remains invariant under the transformation of the crystal point group. In the phenomenological formulation, the susceptibility derivatives correspond to third derivatives of a particular potential energy (interaction Hamiltonian), such that the product is the reducible representation of the Raman tensor. If condition (2.3.3.12) holds, then the Raman tensor does not vanish identically and may have at least one independent nonzero component. As the representation is that of a secondrank polar tensor, equivalent formulation of the Raman activity of a normal mode j is that the corresponding normal coordinate must transform like one or more components of a polar tensor. The transition susceptibility transforms accordingly. The task of determining whether a given normal mode j is Raman active or not thus consists of simply decomposing the representation and identifying the irreducible components .
The second consequence of the invariance condition is the imposition of restrictions on the Cartesian components of the Raman tensor for modes allowed to participate in the scattering. By virtue of the properties of the irreducible representations , some components of the corresponding Raman tensor are required to vanish whereas others may have related values. This fact results in anisotropies in the observed cross section depending on the polarization directions of the incident and scattered light, and is usually referred to as polarization selection rules. As the scattering cross section of the excitation (j) is proportional to the scalar quantity one can generally `isolate' a given component of the Raman tensor by suitably arranging the scattering geometry in the experiment, i.e. by choosing the orientation of the wavevectors and and the polarization vectors and with respect to crystallographic axes.
For each normal mode (j) allowed in the scattering, the number of independent components of its Raman tensor is given by the multiplicity coefficients of the irreducible representation in the decomposition where the multiplicity coefficient corresponds to the number of times the given irreducible representation enters the decomposition. If the representation is two or threedimensional, then for each occurrence of in (2.3.3.14) there are two or three degenerate partners (of the same frequency) whose Raman tensors are symmetryrelated.
The matrix form of the Raman tensor corresponding to a given irreducible representation – i.e. symmetry species – can be readily constructed by finding the appropriate bilinear basis functions that transform according to the corresponding irreducible representation . The required number of such independent bases is given by the multiplicity coefficient . Alternatively, one may construct invariant polynomials (transforming as scalars) of order four, i.e. of the same order as the product .
Making allowance for possible antisymmetric scattering, we have not explicitly supposed that the Raman tensor is symmetric. We recall that the derivative (2.3.3.13) is not necessarily symmetric in the α and β indices as long as the fields and correspond to different frequencies (inelastic scattering). However, each secondrank polar tensor (nine components), transforming according to , can be decomposed into a symmetric part (six components), transforming like a symmetric polar tensor , and an antisymmetric part (three components), transforming like an axial vector (for the definition of axial tensors, see Section 1.1.4.5.3 ) according to .
The symmetryrestricted forms of the () Raman tensors corresponding to all Ramanactive symmetry species are summarized in Table 2.3.3.1 (see e.g. Hayes & Loudon, 1978) for each of the 32 crystal symmetry classes. Spectroscopic notation is used for the irreducible representations of the point groups. The symbols (x, y or z) for some Ramanactive symmetry species in the noncentrosymmetric classes indicate that the respective components of polar vectors also transform according to these irreducible representations. Hence the normal coordinates of the phonons of these polar symmetry species (polar phonons) transform in the same way and, consequently, the corresponding component of the effective charge tensor , see (2.3.3.4), is not required by symmetry to vanish. Polar phonons thus may carry a nonzero dipole moment and contribute to the polarization in the crystal, which manifests itself in infrared activity and also in the Raman scattering cross section (see Section 2.3.3.5).

For convenience, the Raman tensors are explicitly split into a symmetric and possible antisymmetric part (upper and lower row of each part of the table, respectively, in each case). The conventional symmetric Raman tensors are appropriate for most cases of practical interest. Besides the resonant conditions mentioned above, there are other exceptions. For instance, there are optical phonons that transform like axial vectors, such as in the case of (or , ) modes in some uniaxial crystal classes, where the Raman tensor is purely antisymmetric. Antisymmetric scattering by these modes may become allowed at finite wavevector q. Antisymmetric Raman tensors are also needed for analysing the symmetry of scattering in magnetic materials (scattering by spin waves – magnons), or nonmagnetic materials under a magnetic field, where the susceptibility itself is essentially nonsymmetric.
We note that the matrix form of the Raman tensors depends on the setting of the Cartesian axes with respect to the crystallographic axes. To avoid ambiguities and apparent disagreement with other sources, we give the results for alternative orientations of the point groups in several cases where different settings of the twofold axes or mirror planes with respect to the Cartesian axes are commonly used. This concerns all monoclinic classes (unique direction parallel to y or z), tetragonal class , trigonal classes 32, 3m and , as well as hexagonal class .
In those point groups that contain the inversion operation, i.e. in the eleven centrosymmetric (nonpolar) crystal classes the irreducible representations are divided into two groups, odd and even, according to the parity. Since secondrank polar tensors must transform according to the even parity representations only, whereas polar vectors transform according to odd parity representations, the selection rules for electric dipole absorption (infrared activity) and for Raman scattering are incompatible. This is often expressed as the mutual exclusion rule or complementarity principle: The excitations in a crystal belonging to a centrosymmetric class cannot be simultaneously active in infrared absorption and in Raman scattering. Let us note, however, that evenparity excitations are not necessarily all Raman active, and that oddparity excitations are not necessarily infrared active.
In the remaining noncentrosymmetric crystal classes, the excitations have no defined parity with respect to inversion and can be, in principle, both Raman and infrared active.
Example: Consider a Raman scattering experiment on a crystal of tetragonal symmetry, class . Ramanactive phonons, allowed in conventional symmetric scattering, are of the symmetry species , , and . (the species admits purely antisymmetric scattering only). Straightforward application of Table 2.3.3.1 makes it possible to determine the polarization selection rules, i.e. to determine which symmetry species will contribute to the scattering cross section in various experimental configurations. Choosing the Cartesian axes , , consistent with the standard setting of the point group, i.e. the fourfold rotation axis , let us further introduce the notation , . Then the contributions to the cross section for different symmetry species can be distinguished by their dependence on the polarization vectors and of the incident and scattered light:
Examples of some special scattering geometries that permit the separation of the contributions of different symmetry species are shown in Table 2.3.3.2 (five distinct configurations are sufficient to determine the five independent parameters a, b, d, e, f of the symmetric Raman tensors ).

If, for some reason, antisymmetric scattering is allowed, possible contribution of the modes should be considered as well. The contribution to cross section from these modes is proportional to , hence it can be distinguished from the contribution of the symmetry species by a suitable choice of the scattering geometry.
Special care is required in treating the scattering by those optical phonons in the 21 noncentrosymmetric (polar) crystal classes (1, 2, m, 222, mm2, 4, , 422, 4mm, , 3, 32, 3m, 6, , 622, 6mm, , 23, 432, ) that are simultaneously infraredactive. Since these polar modes carry a nonzero macroscopic effective charge (2.3.3.4), they contribute to the total polarization in the crystal, hence also to the macroscopic electric field, which in turn leads to a coupling between these modes. The polarization being a polar vector, the modes that contribute have the same symmetry character, i.e. they must also transform like the components of polar vectors.
An important consequence of the macroscopic field associated with polar modes in the crystal is the partial lifting of the degeneracies of the longwavelength () mode frequencies (socalled TO–LO splitting). Since the macroscopic field in the crystal is longitudinal, it must be proportional to the longitudinal component of the polarization. Hence, the equations of motion for all polar modes carrying a nonzero longitudinal polarization (i.e. ) become coupled by the field and, consequently, their frequencies depend on the direction of q. This phenomenon is called directional dispersion and is connected with the fact that in the electrostatic approximation the dynamical matrix with longrange Coulomb forces shows nonanalytic behaviour for . In lattice dynamics, the limit can be treated correctly by taking into account the retardation effects in the range where cq becomes comparable to , i.e. in the crossing region of free photon and optical phonon dispersion curves. As a result, one finds that for small q the true eigenmodes of the system – polaritons – have a mixed phonon–photon character and their frequencies show strong dispersion in the very close vicinity of . Experimentally, this polariton region is partially accessible only in nearforward Raman scattering [see (2.3.2.3)]. For larger scattering wavevectors in the usual rightangle or backscattering geometries, the electrostatic approximation, , is well applicable and the excitations behave like phonons. Owing to the coupling via the longitudinal macroscopic electric field, however, the directional dispersion of these phonon branches remains.
Detailed analysis is complicated in the general case of a lowsymmetry crystal with more polar modes (see e.g. Claus et al., 1975). In crystals with at least orthorhombic symmetry, the principal axes of the susceptibility tensor are fixed by symmetry and for the wavevectors oriented along these principal axes the polar optic modes have purely transverse (TO) or longitudinal (LO) character with respect to the associated polarization. The character of a mode is usually mixed for a general direction of the wavevector.
Strictly speaking, conventional symmetry analysis in terms of irreducible representations of the factor group (point group) of the crystal, though giving a true description of polaritons at , cannot account for the lifting of degeneracies and for the directional dispersion of polar modes. A correct picture of the symmetries and degeneracies is, however, obtained by taking into account the finiteness of the wavevector q and classifying the vibrations according to the irreducible (multiplier) corepresentations of the point group of the wavevector , which is a subgroup of the factor group. Compatibility relations of the representations at can then be used to establish a correspondence between the two approaches.
The oscillating macroscopic field associated with longwavelength LO polar modes acts as another source of modulation of the susceptibility. In addition to the standard atomic displacement contribution connected with the mechanical displacements of atoms, one also has to consider that the transition susceptibility also contains the electrooptic term arising from the distortion of electron shells of atoms in the accompanying macroscopic field E. Separating both contributions, we may write or, in terms of the Raman tensor, where we introduce the notation and for the atomic displacement and electrooptic contributions to the Raman tensor . As usual, stands for the normal coordinate of the jth mode and E for the total macroscopic electric field resulting from the longitudinal polarization of all optic modes. The modes that contribute to E are only LO polar modes; they transform as Cartesian components of polar vectors (x, y, z). Hence the electrooptic term contributes to the Raman cross section only if , i.e. if the mode has at least partially longitudinal character. Hence, not only the frequencies but also the scattering cross sections of the TO and LO components of polar modes belonging to the same symmetry species are, in general, different.
Nevertheless, in view of the fact that the macroscopic electric field associated with LO polar phonons transforms in the same way as its polarization vector, the symmetry properties of both the atomic displacement and the electrooptic contributions to the Raman tensors of polar modes are identical. They correspond to thirdrank polar tensors, which have nonzero components only in piezoelectric crystals. The symmetryrestricted form of these tensors can also be derived from Table 2.3.3.1 by combining the matrices corresponding to the x, y and z components. Note that these may belong to different irreducible representations in lowersymmetry classes (e.g. z cannot mix with x, y), and that in some uniaxial classes the z component is missing completely. Finally, in the noncentrosymmetric class 32 of the cubic system, the Raman tensors of the triply degenerate polar modes () are purely antisymmetric; therefore all components of the piezoelectric tensor also vanish.
Example: To illustrate the salient features of polarmode scattering let us consider a crystal of the class, where of the Ramanactive symmetry species the modes and are polar. According to Table 2.3.3.1, their () Raman tensors are identical to those of the and modes in the preceding example of a class crystal. Owing to the macroscopic electric field, however, here one has to expect directional dispersion of the frequencies of the long wavelength () and E optic phonon modes according to their longitudinal or transverse character. Consequently, in determining the polarization selection rules, account has to be taken of the direction of the phonon wavevector (i.e. the scattering wavevector) q with respect to the crystallographic axes. Since for a general direction of q the modes are coupled by the field, a suitable experimental arrangement permitting the efficient separation of their respective contributions should have the scattering wavevector q oriented along principal directions. At , the phonons are longitudinal () and both E modes () are transverse, remaining degenerate, whereas at or , the phonons become transverse () and the E phonons split into a pair of (, ) modes of different frequencies. The subscripts or explicitly indicate the orientation of the electric dipole moment carried by the mode with respect to the fourfold axis ().
Schematically, the situation (i.e. frequency shifts and splittings) at can be represented by
For a general direction of q, the modes are of a mixed character and their frequencies show directional (angular) dispersion. The overall picture depends on the number of and E phonons present in the given crystal, as well as on their effective charges and on the ordering of their eigenfrequencies. In fact, only the modes remain unaffected by the directional dispersion.
Table 2.3.3.3 gives the corresponding contributions of these modes to the cross section for several representative scattering geometries, where subscripts TO and LO indicate that the components of the total Raman tensor may take on different values for TO and LO modes due to electrooptic contributions in the latter case.

So far, we have not explicitly considered the dependence of the Raman tensor on the magnitude of the scattering wavevector, assuming (the effects of directional dispersion in the case of scattering by polar modes were briefly mentioned in the preceding section). In some cases, however, the Raman tensors vanish in this limit, or qdependent corrections to the scattering may appear. Formally, we may expand the susceptibility in a Taylor series in q. The coefficients in this expansion are higherorder susceptibility derivatives taken at . The symmetryrestricted form of these tensorial coefficients may be determined in the same way as that of the zeroorder term, i.e. by decomposing the reducible representation of the third, fourth and higherorder polar Cartesian tensors into irreducible components . General properties of the qdependent terms can be advantageously discussed in connection with the socalled morphic effects (see Sections 2.3.4 and 2.3.5).
References
Claus, R., Merten, L. & Brandmüller, J. (1975). Light scattering by phononpolaritons. Berlin, Heidelberg, New York: Springer.Hayes, W. & Loudon, R. (1978). Scattering of light by crystals. New York: John Wiley & Sons.