1 i 0 (right circular),
1 -i 0 (left circular), 1+2i 0 0, etc.
Compact notation also works: 100, 001, 1i0.
All vectors are automatically normalized to unit length
(e.g. 1 1 0 → [1,1,0]/√2).
| Irrep | Symbolic (current geometry) | I = |es† · RT(θ) · M · R(θ) · ei|² | Relative intensity | % |
|---|
Raman intensity vs. rotation angle θ (−180° to 180°). Dashed vertical line = current θ. All other parameters fixed at current values.
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 Ns 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.
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.