Raman Tensor Calculator

Notes

The goal of this calculator is to compute the contribution from each irreducible representations (irreps) given a set of scattering configurations in a Raman scattering experiment. In each experiment, the result depends on 3 things, what is the crystal symmetry, how the incident and scattered light polarization is oriented with respect to the crystalline axes. Raman tensors relate these 3 parameters to the scattering cross-section, i.e. I = |es† · RT(θ) · M · R(θ) · ei|², where M is the Raman tensor for a given irrep. Each irrep has one or more partner tensors (e.g. E modes have 2 partners, T modes have 3). R(θ) = rotation matrix around N_s by azimuth angle θ. For degenerate modes (E, T): intensity = sum over all partner tensors. The crystal symmetry dictates the forms of tensors M, the vectors ei and es are the polarization vectors of the incident and scattered light, respectively.

The Raman tensors used here were tabulated in Cardona's Inelastic Light Scattering in Solids Vol. 2 Ch. 2. The forms of Raman tensors depend only on the point groups of the crystal lattice, and not on the space groups. Moreover, some point groups share the same irreps that are Raman active [1]. For example, Oh, Td and O groups all have 4 Raman active irreps, i.e. A1g, Eg, T1g and T2g of the Oh group, and A1, E, T1 and T2 for Td/O groups. The addition of "g" in the irreps of Oh group simply implies the existence of inversion symmetry, and does not change the forms of the Raman tensors of the two groups.

Phonon mode — why the antisymmetric part is not allowed. The tensors above are the general forms, which permit an antisymmetric part. For first-order, non-resonant phonon Raman scattering that part is forbidden: the scattering amplitude is the derivative of the polarizability with respect to the phonon coordinate, ∂χ_ij/∂Q, and χ is a symmetric tensor, so the Raman tensor must satisfy M = MT. Two things follow. Elements that sit at transposed positions are forced equal — for D4h the Eg tensor [[0,0,f],[0,0,0],[g,0,0]] requires f = g — and any irrep whose tensor is purely antisymmetric drops out of the phonon spectrum entirely: A2g in D4h, T1g in Oh, A2g in D3d, A2 in D6h, and the c parameter inside the A/Ag irreps of C4h, C6 and C3. Under these constraints the forms reduce to exactly the symmetric tensors tabulated on the Bilbao Crystallographic Server (Raman and Hyper-Raman Tensors). That is what the Phonon switch next to the point-group selector does, and it is on by default. Turn it off to recover the general Cardona forms, which are the relevant ones for resonant and electronic Raman scattering and for magnetic (spin-flip) excitations, where an antisymmetric component can appear.

For each point group and scattering plane, there exist a set of light polarization combinations that allows clean isolation of contributions from each irreps. For example, backscattering from the (001) plane of a sample with D4h point group symmetry couples to A1g, A2g, B1g and B2g irreps. For a single scattering geometry, such as XX [2], it will couple to both A1g and B1g irreps. But if we acquire data from XX, XY, X'X', X'Y', RR, and RL, then we can isolate individual irreps, e.g., A1g = (XX+X'X'-RL)/2, and similarly for other irreps.

To use this calculator, follow these steps.
1, select the relevant point group symmetry.
2, define the crystal surface normal vector, Ns. This is only important if you want to study the azimuthal dependence. Otherwise just input any random vector.
3, define the incident/scatter light polarization vector, ei/es, for each experimental configuration being performed. This calculates the composition of matrix elements in the defined configuration. For example, in the XX scattering geometry with X=(100) from crystal with Oh symmetry gives a^2+4b^2, that is A1g+4Eg1.

[1] These point groups may have other irreps that are not Raman active, which we will disregard here.
[2] The first index refers to the polarization of the incident photon. In most cases, the result can be greatly simplified if the linear polarization is aligned with one of the crystal axes, such as (100) or (110). In this example here, X is aligned with (100), Y is naturally perpendicular to it, that is (010). We then denote (110) as X', and (1-10) as Y', and R/L are circularly polarized.

Webpage maintained by Sean Kung at the University of British Columbia.