The mathematical structure of the Raman tensor

>The symmetry constrained Raman tensor

The symmetry constraint the Raman tensor is:

$\begin{array}{c}\{\begin{array}{rl}& R:\sum _{mn}{R}_{im}{R}_{jn}{\alpha }_{mn}^{(k)}=\sum _{p=1}^d{\mathrm{\Gamma }}_{pk}^{(r)}(R){\alpha }_{ij}^{(p)},\\ & R^{\prime }:\sum _{mn}{R}_{im}{R}_{jn}{\alpha }_{mn}^{(k)}=\sum _{p=1}^d{\mathrm{\Gamma }}_{pk}^{(r)}(R){\alpha }_{ji}^{(p)}.\end{array}\end{array}$$\left\{ \begin{align} & R:\sum\limits_{mn}{{{R}_{im}}}{{R}_{jn}}\alpha _{mn}^{(k)}=\sum\limits_{p=1}^{d}{\Gamma _{pk}^{(r)}(R)\alpha _{ij}^{(p)}}, \\ & R':\sum\limits_{mn}{{{R}_{im}}}{{R}_{jn}}\alpha _{mn}^{(k)}=\sum\limits_{p=1}^{d}{\Gamma _{pk}^{(r)}(R)\alpha _{ji}^{(p)}.} \\ \end{align} \right.$ $R$$R$ and $R^{\prime }$$R'$ is the unitary and anti-unitary operation of a specific magnetic point group, respectively. $R_{ij}$$R_{ij}$ is the elements of the operation $R$$R$ or $R^{\prime }$$R'$. Noticing that the time-reversal operator $T$$T$ exchanges tensor indices ($ij$$ij$↔$ji$$ji$) in the second equation.

> The mathematical structure of the Raman tensors

• For more information see The group theory of Raman effect in magnetic materials Rui-Chun Xiao et. al.

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