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

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 344-345

Section Raman scattering in a magnetic field

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: Raman scattering in a magnetic field

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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)[{\chi}_{\alpha\beta}({\bf H}) = {\chi}_{\beta\alpha}(-{\bf H}). \eqno (]Further, in the absence of absorption, the susceptibility must be Hermitian, i.e. [{\chi}_{\alpha\beta}({\bf H}) = {\chi}_{\beta\alpha}^*({\bf H}). \eqno (]Hence, [{\boldchi}({\bf H})] is neither symmetric nor real. Expanding [{\boldchi}({\bf H})] in the powers of the field, [\chi _{\alpha \beta }({\bf H}) = \chi _{\alpha \beta }(0) + {{\partial \chi _{\alpha \beta }}\over {\partial H_\mu }}H_\mu + {{\partial ^2 \chi _{\alpha \beta }}\over {\partial H_\mu \partial H_\nu }}H_\mu H_\nu + \ldots, \eqno (]it follows that all terms of the magnetic-field-induced 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 first-order term, which can be written as [\Delta \chi _{\alpha \beta }({\bf H}) = i f_{\alpha \beta \mu }H_\mu, \eqno (]where the tensor f, referred to as the magneto-optic tensor, is real and purely antisymmetric in the first two indices: [f_{\alpha \beta \nu} \equiv - i(\partial{\chi}_{\alpha\beta}/\partial H {_\nu}) = - f_{\beta\alpha\nu}.]The representation [\Gamma({\bf f})] of the magneto-optic tensor f may thus be symbolically written as [\eqalignno{\Gamma({\bf f}) &= [\Gamma_{\rm PV} \otimes \Gamma_{\rm PV}]{_A} \otimes \Gamma_{\rm AV} = \Gamma_{\rm AV} \otimes \Gamma_{\rm AV} =\Gamma_{\rm PV} \otimes \Gamma_{\rm PV} &\cr &= \Gamma(T{_\alpha}T{_\beta}), &(}]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 [\Gamma({\bf f})] is equivalent to the representation of a general nonsymmetric second-rank tensor and reduces in exactly the same way ([link]. [\Gamma({\bf f}) = \Gamma_{\rm PV} \otimes \Gamma_{\rm PV} = c^{(1)}\Gamma(1) \oplus c^{(2)}\Gamma(2) \oplus \ldots.]

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 magnetic-field-induced scattering. Moreover, the magnetic-field-induced 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 symmetry-restricted matrix form of the corresponding field-induced 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 magneto-optic tensor as follows.

From the definition of the tensor f, it is clear that its Cartesian components [f_{\alpha\beta\nu}] must have the same symmetry properties as the product [[E{_\alpha}E{_\beta}]{_A}H{_\nu}]. The antisymmetric factor [[E{_\alpha}E{_\beta}]{_A}] transforms, however, as [{\varepsilon}_{\alpha \beta\mu}H{_\mu}], where [{\varepsilon}_{ \alpha \beta \mu}] is the fully antisymmetric third-rank pseudotensor (Levi–Civita tensor). Consequently, [f_{\alpha \beta \nu}] must transform in the same way as [{\varepsilon}_{\alpha \beta \mu}H{_\mu}H{_\nu}], which in turn transforms identically to [{\varepsilon}_{\alpha \beta \mu}E{_\mu}E{_\nu}]. Therefore, comparison of the matrices corresponding to the irreducible components [\Gamma(j)] provides a simple mapping between the components of the Cartesian forms of the linear field-induced Raman tensors [{\bf R}{^j}({\bf H})={\bf R}^{jh}{\bf H}] and the intrinsic Raman tensors [{\bf R}^{j0}]. Explicitly, this mapping is given by[R_{\alpha \beta \nu }^{jH} \equiv {{\partial ^2 \chi _{\alpha \beta }}\over {\partial Q_j \partial H_\nu }} = if_{\alpha \beta \nu }^{(j)}\,\, \leftarrow \,\, i\varepsilon _{\alpha \beta \mu } R_{\mu \nu }^{jo}. \eqno (]For any given symmetry species, this relation can be used to deduce the matrix form of the first-order field-induced Raman tensors from the tensors given in Table[link].

Example: We consider again the 4mm class crystal. The representation [\Gamma({\bf f})] of the magneto-optic tensor f in the [4mm] class reduces as follows: [\Gamma({\bf f}) ={ \Gamma}_{\rm PV} \otimes { \Gamma}_{\rm PV} = 2{\rm A}{_1} \oplus {\rm A}{_2} \oplus {\rm B}{_1} \oplus {\rm B}{_2} \oplus 2{\rm E}.]Straightforward application of the mapping mentioned above then gives the following symmetry-restricted matrix forms of contributions to the magnetic-field-induced Raman tensors [{\bf R}^{jH}{\bf H}] for all symmetry species of the 4mm-class crystals. The number of independent parameters for each species is the same as in the intrinsic nonsymmetric zero-field Raman tensors: [\eqalign{A{_1}: \quad&\pmatrix{. & {ib'H_z }& {- ia'H_y } \cr {- ib'H_z }&. & {ia'H_x } \cr {ia'H_y }& {- ia'H_x }&. \cr } \cr A{_2}:\quad&\pmatrix{. &. & {ic'H_x } \cr. &. & {ic'H_y } \cr {- ic'H_x }& {- ic'H_y }&. \cr }\cr B{_1}:\quad &\pmatrix{. &. & {id'H_y } \cr. &. & {id'H_x } \cr {- id'H_y }& {- id'H_x }&. \cr }\cr B{_2}:\quad&\pmatrix{. &. & {- ie'H_x } \cr. &. & {ie'H_y } \cr {ie'H_x }& {- ieH_y }&. \cr }\cr E:\quad &\pmatrix{. & {ig'H_x }&. \cr {- ig'H_x }&. & {if'H_z } \cr. & {- if'H_z }&. \cr }\cr &\pmatrix{. & {ig'H_y }& {- if'H_z } \cr {- ig'H_y }&. &. \cr {if'H_z }&. &. \cr }.}]Let us note that the conclusions mentioned above apply, strictly speaking, to non-magnetic crystals. In magnetic materials in the presence of spontaneous ordering (ferro- or antiferromagnetic crystals) the analysis has to be based on magnetic point groups.

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