The Cartesian rank two representation of spherical tensors with an application to Raman scattering

The concept of Cartesian representations of lower rank spherical tensors is reviewed. To facilitate the introduction of the latter, basis vectors and tensors that transform according to the laws for spherical tensors are introduced, as well as a fast projection of a rank two Cartesian tensor onto its spherical components. From this the rotational invariants of the tensors are deduced. The method is applied to the problem of orientational averaging of the Raman polarizability tensor. General, complex, expressions for the depolarization ratio and the reversal ratio are derived. The entire treatment is carried out in a classical context.