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

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 339-341

Section Centrosymmetric crystals

I. Gregoraa*

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

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In those point groups that contain the inversion operation, i.e. in the eleven centrosymmetric (nonpolar) crystal classes [{\bar 1}, 2/m, mmm, 4/m, 4/mmm, {\bar 3}, {\bar 3m}, 6/m, 6/mmm, m3, m3m,]the irreducible representations are divided into two groups, odd and even, according to the parity. Since second-rank 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 even-parity excitations are not necessarily all Raman active, and that odd-parity 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 [4/mmm]. Raman-active phonons, allowed in conventional symmetric scattering, are of the symmetry species [{\rm A}_{1g}], [{\rm B}_{1g}], [{\rm B}_{2g}] and [{\rm E}{_g}]. (the [{\rm A}_{2g}] species admits purely antisymmetric scattering only). Straightforward application of Table[link] 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 [x \equiv [100]], [y \equiv [010]], [z \equiv [001]] consistent with the standard setting of the [4/mmm] point group, i.e. the fourfold rotation axis [4\parallel{\bf z}], let us further introduce the notation [x' \equiv [110]], [y' \equiv [\bar 110]]. Then the contributions to the cross section for different symmetry species can be distinguished by their dependence on the polarization vectors [{\bf e}{_I}] and [{\bf e}{_S}] of the incident and scattered light:[\eqalign{{\rm A}_{1g}&: |(e{_I}{_x}e{_S}{_x} + e{_I}{_y}e{_S}{_y})a + e{_I}{_z}e{_S}{_z}b|{^2}\cr {\rm B}_{1g}&:|(e{_I}{_x}e{_S}{_x}-e{_I}{_y}e{_S}{_y})d|{^2} \cr {\rm B}_{2g}&: |(e{_I}{_x}e{_S}{_y} + e{_I}{_y}e{_S}{_x}) e|{^2}\cr {\rm E}_{g}&: [(e{_I}{_x}e{_S}{_z} + e{_I}{_z}e{_S}{_x}){^2} + (e{_I}{_y}e{_S}{_z} + e{_I}{_z}e{_S}{_y}){^2}] |f|{^2}.\cr}]

Examples of some special scattering geometries that permit the separation of the contributions of different symmetry species are shown in Table[link] (five distinct configurations are sufficient to determine the five independent parameters a, b, d, e, f of the symmetric Raman tensors).

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Raman selection rules in crystals of the [4/mmm] class

Scattering configurationCross section for symmetry species
Back scatteringRight-angle scattering[{\rm A}_{1g}][{\rm B}_{1g}][{\rm B}_{2g}][{\rm E}{_g}]
[\bar z(xx)z,\bar z(yy)z] [y(xx)z, x(yy)z] [\sim|a|^2] [\sim|d|^2]
[\bar x(zz)x, \bar y(zz)y] [x(zz)y] [\sim|b|^2]
[\bar z(xy)z] [y(xy)x, z(xy)x] [\sim|e|^2]
[\bar y(xz)y, \bar x(yz)x] [y(xz)x, x(yz)y] [\sim|f|^2]
[\bar z(x'x')z] [y'(x'x')z] [\sim|a|^2] [\sim|e|^2]
[\bar z(x'y')z] [y'(x'y')z] [\sim|d|^2]

If, for some reason, antisymmetric scattering is allowed, possible contribution of the [{\rm A}_{2g}] modes should be considered as well. The contribution to cross section from these modes is proportional to [|(e{_I}{_x}e_{Sy}-e{_I}{_y}e{_S}{_x}) c|{^2}], hence it can be distinguished from the contribution of the [{\rm B}_{2g}] symmetry species by a suitable choice of the scattering geometry.

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