This thesis is concerned with the mechanics of transforming between observational data of lanthanide ion transition intensities and the parameters which model the observed intensities. The accuracy of intensity measurements and the underlying accuracy of the model used to predict the observables is investigated. Investigation is made of the effect of uncertainties for the fitting of transition intensities and rotatory strengths. In this thesis it is shown that the parameterisation of the transition intensity operator using the Reid-Richardson Aλtp intensity parameters is inherently, multi-valued. When the superposition model is not explicitly assumed,, distinct sets of Aλtp parameters can exist which produce identical calculated spectra. In this thesis, a new parameterisation was developed which allows the easy identification of the multiple solutions of the electric-dipole transition parameters. It is possible to transform between the different Aλtp solutions by a matrix transformation. Additionally, this new parameterisation allows the different polarisations that can be observed in the symmetry to be treated separately. For example, in most high symmetry systems it is possible to write a parameterisation for the π polarisation and fit the resulting parameters independently of the parameters for the σ polarisation.
With only a single polarisation for the intensities or circular dichroism it is not possible to determine all of the Aλtp parameters, both σ and π are required. Published analyses of circular dichroism are repeated using the new parameterisation scheme showing the effect of this polarisation dependence, and the improvement over previous work. The intensities for low symmetry systems, or the π polarised intensities for most symmetries, can be written as a linear matrix equation. The possible use of Stark effect splittings to determine the intensity parameters is investigated. The use of the Stark effect in the investigation of transition intensities appears to have been previously unexplored by other researchers. It is found that such Stark splittings are expected to be resolvable and have magnitudes of the order of 0.1cm-¹. It may be possible to utilise splittings of hyperfine lines to determine parameters which describe the transition intensities. Intensity parameters are presented for a variety of systems that have been analysed for the first time in this thesis. Intensity parameters for the C4v symmetry centre of Pr³⁺:CaF₂ and Pr³⁺:SrF₂ systems are presented. The vibronic transitions for the Cs₂NaEr³⁺Cl₆ system were analysed and the vibronic intensity parameters were determined. Re-analysis of the previous results was carried out uising the new parameterisation and a new set of intensity parameters is presented. The level of statistical significance that can be assigned to distinct parameter sets is examined. It is found that quite small variations in the χ² value for fitting intensities can be resolved. Using Monte-Carlo methods it is possible to derive levels of confidence for the distinguishability of the resolveability of two minima into distinct sets.