Raman Tensors in All Magnetic Groups

Notes:

• The symmetric part $\left[{\alpha }_{ij}^S\right]$$\left[\alpha_{i j}^S\right]$ is time-reversal even ($T$$T$-even), and satisfies ${\alpha }^S(-\omega )={[{\alpha }^S(\omega )]}^{\ast }$${{\mathbf{\alpha }}^{S}}(-\omega )={{\left[ {{\mathbf{\alpha }}^{S}}(\omega ) \right]}^{*}}$. While the antisymmetric part $\left[{\alpha }_{ij}^A\right]$$\left[\alpha_{i j}^A\right]$ is time-reversal odd ($T$$T$-odd), and satisfies ${\alpha }^A(-\omega )={[-{\alpha }^A(\omega )]}^{\ast }$${{\mathbf{\alpha }}^{A}}(-\omega )={{\left[ -{{\mathbf{\alpha }}^{A}}(\omega ) \right]}^{*}}$. Consequently, the $T$$T$-even ${\alpha }_{ij}^S(0)$$\alpha _{ij}^{S}(0)$ may have a nonzero real component at zero frequency limit. Whereas ${\alpha }^A(0)$${{\alpha }^{A}}(0)$ vanishes, indicating the ${\alpha }^A$${{\alpha }^{A}}$ belongs a non-equilibrium tensor.

• The Raman coefficients ${\alpha }_{ij}(\omega )$$\alpha_{ij}(\omega)$ possess both real and imaginary parts, analogous to the dielectric function, and satisfy the Kramers-Kronig relations.

• The second row of each table presents the symmetric components of the Raman tensor, and the Raman tensor elements are denoted with the letter $\alpha$$\alpha$. While the remaining rows correspond to the antisymmetric components, and the Raman tensor elements are written as the letter "A".

• Since the black-and-white MPG $M=S+R^{\prime }(G-S)$$M=S+R' (G-S)$ has the unitary ($R$$R$) and anti-unitary ($R^{\prime }$$R'$) elements, we assign them the values +1 and $-1$$-1$, respectively. In this way, the MPG $M$$M$ can be regarded as a 1D real irrep ${\chi }_{mag}$${{\chi}_{mag}}$ of the isomorphic nonmagnetic point group $G$$G$, as listed in the first column of each table.

• For double degenerate $E$$E$ modes and triply degenerate $T$$T$ modes, three indices are used (e.g., ${\alpha }_{112}$$\alpha_{112}$ and $A_{212}$$A_{212}$), where the first index denotes the number of the irreducible representation, and the last two are the Raman tensor subscripts.

• For the MPGs ${42}^{\prime }{2}^{\prime }$$42'2'$, $4m^{\prime }{m}^{\prime }$$4m'm'$, $\overline{4}{2}^{\prime }{m}^{\prime }$$\bar{4}2' m'$, $4/m{m}^{\prime }{m}^{\prime }$$4/mm' m'$, ${32}^{\prime }$$32'$, $3m^{\prime }$$3m'$, $\overline{6}{m}^{\prime }{2}^{\prime }$$\bar{6}m'2'$, $\overline{3}{m}^{\prime }$$\bar{3}m'$, ${62}^{\prime }{2}^{\prime }$$62'2'$, $6m^{\prime }{m}^{\prime }$$6m'm'$, $6/m{m}^{\prime }{m}^{\prime }$$6/mm'm'$, ${\overline{4}}^{\prime }3m^{\prime }$$\bar{4}'3m'$, $4^{\prime }{32}^{\prime }$$4'32'$, and $m\overline{3}{m}^{\prime }$$m\bar{3}m'$, the $E$$E$ modes undergo splitting, forming the chiral phonons (such as Co${}_3$$_3$Sn${}_2$$_2$S${}_2$$_2$, and we indicate them by “[S]” after the MPG names.

• The lattice character directions of point groups are given here.

• The mathematical structures of the Raman tensors are given here.

• You can also download the pdf version.

• Reference: The group theory of Raman effect in magnetic materials Rui-Chun Xiao et. al.

Useful tools on Raman study

• FINDSYM (to find the magnetic groups of materials)

• Structure IR Raman and Hyper-Raman Activity

• Character Tables for Point Groups used in Chemistry.

Other symmetry webs of mine

• Classification of SHG effect in magnetically ordered materials

• Symmetry-adapted tensors constrained by magnetic and nonmagnetic groups

• Ruichun/TensorSymmetryPackage

(c) The authors Institute of Quantum Materials and Physics, Institute of Physical Science and Information Technology and Information Materials, Anhui University, Hefei 230601, China (安徽大学 物质科学与信息技术研究院 量子材料与物理研究所) Contact: xiaoruichun@ahu.edu.cn (Rui-Chun Xiao 肖瑞春)

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