INVARIANT REPRESENTATIONS OF FINITE ROTATION MATRICES AND SOME APPLICATIONS

Standard representations of finite rotation matrices (FRM) are defined by expressions that are wedded to two particular coordinate frames su ch as, e.g., are involved in the definition of Euler angles. We presen t here representations for the FRM in invariant tensor forms comprised of vectors defined in a space-fixed coordinate frame K. Three explici t expressions for a FRM are presented: First, in terms of tensor produ cts of the spherical or Cartesian basis vectors of frame K, second in a differential form containing the tensor products of gradient operato rs, and, third, as the superposition of so-called ''minimal'' bipolar harmonics depending on any pair of unit vectors a, b connected with a frame K. Based on these results for the FRM, the transformation rule f or an irreducible tensor set under space rotations may be written in t erms of bipolar harmonics. Our results are especially useful for analy zing angular distributions in atomic processes involving a precise acc ounting of all effects of photon and target polarizations. Four exampl es are considered as illustrations of the techniques presented. First, an invariant representation of the photon polarization tensor is foun d in terms of linear and circular polarization degrees of the photon b eam. Second, an invariant decomposition of tripolar harmonics of secon d rank in terms of very simple, rank 2 tensors is presented. Third, a convenient parametrization is proposed for the polarization state mult ipoles of a polarized atomic target. Fourth, a simple invariant formul a is derived for the angular distribution of polarized photons resulti ng from electric dipole photon emission by an arbitrary polarized atom .