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

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 337-339

Section 2.3.3.3. Raman tensor and selection rules at [{\bf q}\approx 0]

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: gregora@fzu.cz

2.3.3.3. Raman tensor and selection rules at [{\bf q}\approx 0]

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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 [{\bf q}\approx 0], the transformation properties of the incident and scattered light are described by the three-dimensional polar vector representation [\Gamma_{\rm PV}] of the appropriate point group of the crystal, since the quantities that characterize the light ([{\bf E}{_I}], [{\bf E}{_S}], [{\bf P}\ldots]) are all polar vectors, i.e. first-rank polar tensors ([T{_\alpha}]). The transformation properties of a normal mode j must correspond to an irreducible matrix representation [\Gamma(j)] of the crystal point group. We recall that in cases of two- or three-dimensional 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. symmetry-allowed to contribute to the (first-order) scattering cross section, the necessary condition is that the corresponding irreducible representation [\Gamma(j)] must be contained in the decomposition of the direct Kronecker product representation [\Gamma_{\rm PV}\otimes\Gamma_{\rm PV}] at least once: [{\Gamma}_{\rm PV}\otimes{\Gamma}_{\rm PV}\supset{\Gamma}(j). \eqno (2.3.3.12)]In this case, the Kronecker product [\Gamma_{\rm PV}\otimes{\Gamma}_{\rm PV}\otimes{\Gamma}(j)] 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 [\Phi] (interaction Hamiltonian), [R_{\alpha \beta }^j = \left({{{\partial \chi _{\alpha \beta } }\over {\partial Q_j^ * }}}\right)\sim \left({{{\partial ^3 \Phi }\over {\partial E_{I\alpha }\partial E_{S\beta }^ * \partial Q_j^ * }}}\right), \eqno (2.3.3.13)]such that the product [\Gamma_{\rm PV}\otimes{\Gamma}_{\rm PV}\otimes{\Gamma}(j)] is the reducible representation of the Raman tensor. If condition (2.3.3.12)[link] holds, then the Raman tensor [{\bf R}{^j}] does not vanish identically and may have at least one independent nonzero component. As the representation [\Gamma_{\rm PV}\otimes\Gamma_{\rm PV}] is that of a second-rank polar tensor, equivalent formulation of the Raman activity of a normal mode j is that the corresponding normal coordinate [Q{_j}] must transform like one or more components of a polar tensor. The transition susceptibility [{\delta \chi}^{(j)}] transforms accordingly. The task of determining whether a given normal mode j is Raman active or not thus consists of simply decomposing the representation [\Gamma_{\rm PV}\otimes\Gamma_{\rm PV}] and identifying the irreducible components [\Gamma (j)].

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 [\Gamma(j)], 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 [\left| {{\bf e}_S {\bf R}^j {\bf e}_I }\right|^2 \equiv \left| {e_{S\alpha }R_{\alpha \beta }^j e_{I\beta }}\right|^2,]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 [{\bf k}{_I}] and [{\bf k}{_S}] and the polarization vectors [{\bf e}{_I}] and [{\bf e}{_S}] 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 [c^{(j)}] of the irreducible representation [\Gamma(j)] in the decomposition [\Gamma_{\rm PV}\otimes\Gamma_{\rm PV} = c^{(1)}\Gamma(1)\oplus c^{(2)}\Gamma(2)\oplus \ldots, \eqno (2.3.3.14)]where the multiplicity coefficient [c^{(j)}] corresponds to the number of times the given irreducible representation [\Gamma(j)] enters the decomposition. If the representation [\Gamma(j)] is two- or three-dimensional, then for each occurrence of [\Gamma (j)] in (2.3.3.14)[link] there are two or three degenerate partners (of the same frequency) whose Raman tensors are symmetry-related.

The matrix form of the Raman tensor corresponding to a given irreducible representation – i.e. symmetry species[\Gamma(j)] can be readily constructed by finding the appropriate bilinear basis functions that transform according to the corresponding irreducible representation [\Gamma(j)]. The required number of such independent bases is given by the multiplicity coefficient [c^{(j)}]. Alternatively, one may construct invariant polynomials (transforming as scalars) of order four, i.e. of the same order as the product [E_{I\alpha}E_{S\beta}Q{_j}].

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)[link] is not necessarily symmetric in the α and β indices as long as the fields [E_{I\alpha}] and [E_{S\beta}] correspond to different frequencies (inelastic scattering). However, each second-rank polar tensor [T_{\alpha\beta}] (nine components), transforming according to [\Gamma_{\rm PV}\otimes \Gamma_{\rm PV}], can be decomposed into a symmetric part [T'_{\alpha\beta}=T'_{\beta\alpha}] (six components), transforming like a symmetric polar tensor [[\Gamma_{\rm PV}\otimes \Gamma_{\rm PV}]{_S}], and an antisymmetric part [T''_{\alpha\beta}=-T''_{\beta\alpha}] (three components), transforming like an axial vector (for the definition of axial tensors, see Section 1.1.4.5.3[link] ) according to [\Gamma_{\rm AV}=[\Gamma_{\rm PV} \otimes \Gamma_{\rm PV}]{_A}].

The symmetry-restricted forms of the ([3\times 3]) Raman tensors corresponding to all Raman-active symmetry species are summarized in Table 2.3.3.1[link] (see e.g. Hayes & Loudon, 1978[link]) 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 Raman-active 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 [Z_{j\alpha}({\bf q}=0)], see (2.3.3.4)[link], 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[link]).

Table 2.3.3.1| top | pdf |
Symmetry of Raman tensors in the 32 crystal classes

The symbols a, b, c, d, e, f, g, h and i in the matrices stand for arbitrary parameters denoting possible independent nonzero components (in general complex) of the Raman tensors. Upper row: conventional symmetric Raman tensors; lower row: antisymmetric part. Alternative orientations of the point group are distinguished by subscripts at 2 or m in the class symbol indicating the direction of the twofold axis or of the normal to the mirror plane.

Triclinic

  [\pmatrix{ a & d & f \cr d & b & h \cr f & h & c \cr }]
1 [{\rm A}(x,y,z)]
[\bar 1] [{\rm A}{_g}]
  [\pmatrix{. & e & g \cr {- e}&. & i \cr {- g}& {- i}&. \cr }]

Monoclinic, unique axis z

  [{\pmatrix{ a & d &. \cr d & b &. \cr. &. & c \cr }}] [{\pmatrix{. &. & f \cr. &. & h \cr f & h &. \cr }}]
[2_z] [{\rm A}(z)] [{\rm B}(x,y)]
[m_z] [{\rm A'}(x,y)] [{\rm A}''(z)]
[2_z/m] [{\rm A}{_g}] [{\rm B}{_g}]
  [{\pmatrix{. & e &. \cr {- e}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & g \cr. &. & i \cr {- g}& {- i}&. \cr }}]

Monoclinic, unique axis y

  [{\pmatrix{ a & . &d \cr . & b &. \cr d&. & c \cr }}] [{\pmatrix{. &f & . \cr f &. & h \cr . & h &. \cr }}]
[2_y] [{\rm A}(y)] [{\rm B}(x,z)]
[m_y] [{\rm A'}(x,z)] [{\rm A}''(y)]
[2_y/m] [{\rm A}{_g}] [{\rm B}{_g}]
  [{\pmatrix{. & . &e \cr .&. &. \cr{-e} &. &. \cr }}] [{\pmatrix{. &g & . \cr-g &. & i \cr .& {- i}&. \cr }}]

Orthorhombic

  [{\pmatrix{ a &. &. \cr. & b &. \cr. &. & c \cr }}] [{\pmatrix{. & d &. \cr d &. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. &. \cr f &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & h \cr. & h &. \cr }}]
222 [{\rm A}] [{\rm B}{_1}(z)] [{\rm B}{_2}(y)] [{\rm B}{_3}(x)]
[mm2] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm B}{_1}(x)] [{\rm B}{_2}(y)]
[mmm] [{\rm A}{_g}] [{\rm B}_{1g}] [{\rm B}_{2g}] [{\rm B}_{3g}]
    [{\pmatrix{. & e &. \cr {- e}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & g \cr. &. &. \cr {- g}&. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & i \cr. & {- i}&. \cr }}]

Tetragonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{ d & e &. \cr e & {- d}&. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. & h \cr f & h &. \cr }}\quad{\pmatrix{. &. & {- h} \cr. &. & f \cr {- h}& f &. \cr }}]
[4] [{\rm A}(z)] [{\rm B}] [{\rm E}(x,y)]
[\bar 4] [{\rm A}] [{\rm B}(z)] [{\rm E}(x,-y)]
[4/m] [{\rm A}{_g}] [{\rm B}{_g}] [{\rm E}{_g}]
  [{\pmatrix{. & c &. \cr {- c}&. &. \cr. &. &. \cr }}]   [{\pmatrix{. &. & g \cr. &. & i \cr {- g}& {- i}&. \cr }}\quad{\pmatrix{. &. & {- i} \cr. &. & g \cr i & {- g}&. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{ d &. &. \cr. & {- d}&. \cr. &. &. \cr }}] [{\pmatrix{. & e &. \cr e &. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. &. \cr f &. &. \cr }}\quad{\pmatrix{. &. &. \cr. &. & f \cr. & f &. \cr }}]
422 [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm B}{_1}] [{\rm B}{_2}] [{\rm E}(-y,x)]
[4mm] [{\rm A}{_1}(z)] [{\rm A}{_2} ] [{\rm B}{_1}] [{\rm B}{_2}] [{\rm E}(x,y)]
[\bar 42_xm_{xy}] [{\rm A}{_1}] [{\rm A}{_2}] [{\rm B}{_1}] [{\rm B}{_2}(z) ] [{\rm E}(y,x)]
[\bar 4m_x2_{xy}] [{\rm A}{_1}] [{\rm A}{_2}] [{\rm B}{_1}] [{\rm B}{_2}(z) ] [{\rm E}(-x,y)]
[4/mmm] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm B}_{1g}] [{\rm B}_{2g}] [{\rm E}{_g}]
    [{\pmatrix{. & c &. \cr {- c}&. &. \cr. &. &. \cr }}]     [{\pmatrix{. &. & g \cr. &. &. \cr {- g}&. &. \cr }}\quad{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}]

Trigonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{ c & f & e \cr f & {- c}& d \cr e & d & . \cr }}\quad{\pmatrix{ f & {- c}& {- d} \cr {- c}& {- f}& e \cr {- d}& e &. \cr }}]
3 [{\rm A}(z)] [{\rm E}(x,y)]
[\bar 3] [{\rm A}{_g}] [{\rm E}{_g}]
  [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & i \cr. &. & g \cr {- i}& {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. & i \cr g & {- i}&. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{f & . &. \cr . &-f & d \cr. & d &. \cr }}\quad{\pmatrix{ . &-f & {- d} \cr-f & .&. \cr {- d}&. &. \cr }}]
[32_x] [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm E}(x,y)]
[3m_x] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}(y,-x)]
[\bar 3m_x] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm E}_{g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{. & f &. \cr f &. & d \cr. & d &. \cr }}\quad{\pmatrix{ f &.& {- d} \cr. & -f&. \cr {- d}&. &. \cr }}]
[32_y] [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm E}(x,y)]
[3m_y] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}(y,-x)]
[\bar 3m_y] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm E}_{g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]

Hexagonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{. &. & e \cr. &. & d\cr e & d &. \cr }}\quad{\pmatrix{. &. & {- d} \cr. &. & e \cr {- d}& e &. \cr }}] [{\pmatrix{ c & f &. \cr f & {- c}&. \cr. &. &. \cr }}\quad{\pmatrix{ f & {- c}&. \cr {- c}& {- f}&. \cr. &. &. \cr }}]
6 [{\rm A}(z)] [{\rm E}{_1}(x,y)] [{\rm E}{_2}]
[\bar 6] [{\rm A}' ] [{\rm E}''] [{\rm E}'(x,y)]
[6/m] [{\rm A}{_g}] [{\rm E}_{1g}] [{\rm E}_{2g}]
  [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & i \cr. &. & g \cr {- i}& {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. & i \cr g & {- i}&. \cr }}]  

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{. &. &. \cr. &. & d \cr. & d &. \cr }}\quad{\pmatrix{. &. & {- d} \cr. &. &. \cr {- d}&. &. \cr }}] [{\pmatrix{. & f &. \cr f &. &. \cr. &. &. \cr }}\quad{\pmatrix{ f &. &. \cr. & {- f}&. \cr. &. &. \cr }}]
622 [{\rm A}{_1}] [{\rm A}{_2}(z) ] [{\rm E}{_1}(x,y)] [{\rm E}{_2}]
[6mm] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}{_1}(y,-x)] [{\rm E}{_2}]
[\bar 6m_x2_y] [{\rm A}{_1}'] [{\rm A}{_2}' ] [{\rm E}''] [{\rm E}'(x,y)]
[\bar 62_xm_y] [{\rm A}{_1}'] [{\rm A}{_2}' ] [{\rm E}''] [{\rm E}'(y,-x)]
[6/mmm] [{\rm A}_{1g}] [{\rm A}_{2g} ] [{\rm E}_{1g}] [{\rm E}_{2g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]  

Cubic

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & a \cr }}] [{\pmatrix{ b &. &. \cr. & b &. \cr. &. & {- 2b} \cr }}\quad{\pmatrix{ {- \sqrt{3}b}&. &. \cr. & \sqrt{3}b&. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & c \cr. & c &. \cr }}\quad{\pmatrix{. &. & c \cr. &. &. \cr c &. &. \cr }}\quad{\pmatrix{. & c &. \cr c &. &. \cr. &. &. \cr }}]
23 [{\rm A}] [{\rm E}] [{\rm F}(x,y,z)]
[m3] [{\rm A}{_g}] [{\rm E}{_g}] [{\rm F}{_g}]
      [{\pmatrix{. &. &. \cr. &. & d \cr. & {- d}&. \cr }}\quad{\pmatrix{. &. & d \cr. &. &. \cr {- d}&. &. \cr }}\quad{\pmatrix{. & d &. \cr {- d}&. &. \cr. &. &. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & a \cr }}] [{\pmatrix{ b &. &. \cr. & b &. \cr. &. & {- 2b} \cr }}\quad{\pmatrix{ {- \sqrt{3}b}&. &. \cr. & {\sqrt{3}}b &. \cr. &. &. \cr }}]   [{\pmatrix{. &. &. \cr. &. & c \cr. & c &. \cr }}\quad{\pmatrix{. &. & c \cr. &. &. \cr c &. &. \cr }}\quad{\pmatrix{. & c &. \cr c &. &. \cr. &. &. \cr }}]
432 [{\rm A}{_1}] [{\rm E}] [{\rm F}{_1}(x,y,z)] [{\rm F}{_2}]
[\bar 43m] [{\rm A}{_1}] [{\rm E}] [{\rm F}{_1}] [{\rm F}{_2}(x,y,z)]
[m3m] [{\rm A}_{1g}] [{\rm E}_{g}] [{\rm F}_{1g}] [{\rm F}_{2g}]
      [{\pmatrix{. &. &. \cr. &. & d \cr. & {- d}&. \cr }}\quad{\pmatrix{. &. & d \cr. &. &. \cr {- d}&. &. \cr }}\quad{\pmatrix{. & d &. \cr {- d}&. &. \cr. &. &. \cr }}]  

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 [{\rm A}{_2}] (or [{\rm A}_{2g}], [{\rm A}'_2]) 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 non-magnetic 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 [\bar 4 2 m], trigonal classes 32, 3m and [\bar 3 m], as well as hexagonal class [\bar 6 2 m].

References

Hayes, W. & Loudon, R. (1978). Scattering of light by crystals. New York: John Wiley & Sons.








































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