Studies in the application of group theory
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis presents several developments in the mathematical tool of symmetry, group theory. In addition to studies calculating transformation coefficients in the Racah-Wigner calculus, we present several applications of such coefficients to physical systems. We investigate the calculation of transformations between particular symmetric group bases. We discuss symmetric group bases adapted to subgroup factors of the form Sa x Sb. We call these particular bases split bases. A special case of transforming from standard to split bases is considered. We generalise that result and describe a simple method for relating permuted bases. We present the block-selective conjecture for calculating transformations between general standard and split bases. We intensively examine a particular neglected multiplicity case, as part of obtaining algebraic solutions for transformations from the standard basis to the split basis adapted to Sn-₃ x S₃ . We prove that the Littlewood-Richardson rule does not fix the choice of multiplicity separation. We use continuous groups to study the double delta function model of correlation crystal fields. We obtain an explicit expression for the transformation coefficients which relate terms in the model to physical operators. This allows us to improve the understanding of why only some terms contribute in this model. In the last major study, we use point groups to study magneto-optical effects (Kerr rotation) in chromium trihalides. We discuss how the Racah-Wigner algebra is used by the computer programme RACAH to calculate spin-orbit coupling coefficients. Those coefficients are then used in the analysis of the reflectance spectra of chromium trihalides. We provide support for the recently proposed reassignment of the transitions contributing to the Kerr rotation of chromium tribromide. Several minor investigations complement the major studies. Possible lines of further investigation are discussed where appropriate.