The ring of integers is a very interesting ring, it has the amazing property that each of its elements may be expressed uniquely, up to order, as a product of prime elements. Unfortunately, not every ring possesses this property for its elements. The work of mathematicians like Kummer and Dedekind lead to the study of a special type of ring, which we now call a Dedekind domain, where even though unique prime factorization of elements may fail, the ideals of a Dedekind domain still enjoy the property of unique prime factorization into a product of prime ideals, up to order of the factors. This thesis seeks to establish the unique prime ideal factorization of ideals in a special type of Dedekind domain: the ring of algebraic integers of an imaginary quadratic number field.

<p>Atoms, molecules and clusters all constitute building blocks of macroscopic matter. Therefore, understanding the electronic and geometrical properties of such systems is the key to understanding the properties of solid state objects.</p><p>In this thesis, some atomic, molecular and cluster systems (clusters of O₂, CH₃Br, Ar/O₂, Ar/Xe and Ar/Kr; dimers of Na; Na and K atoms) have been investigated using synchrotron radiation, and in the two last instances, laser light. We have performed x-ray photoelectron spectroscopy (XPS) on all of these systems. We have also applied ultraviolet photoelectron spectroscopy (UPS), resonant Auger spectroscopy (RAS) and near-edge x-ray absorption spectroscopy (NEXAFS) to study many of the systems. Calculations using <i>ab initio</i> methods, namely density functional theory (DFT) and Møller-Plesset perturbation theory (MP), were employed for electronic structure calculations. The geometrical structure was studied using a combination of <i>ab initio</i> and molecular dynamics (MD) methods.</p><p>Results on the dissociation behavior of CH₃Br and O₂ molecules in clusters are presented. The dissociation of the Na₂ molecule has been characterized and the molecular field splitting of the Na 2<i>p</i> level in the dimer has been measured. The molecular field splitting of the CH₃Br 3<i>d</i> level has been measured and the structure of CH₃Br clusters has been determined to be similar to the structure of the bulk solid. The diffusion behavior of O₂, Kr and Xe on large Ar clusters, as a function of doping rate, has been investigated. The shake-down process has been observed from excited states of Na and K. Laser excited Na atoms have been shown to be magnetically aligned. The shake-down process was used to characterize the origin of various final states that can be observed in the spectrum of ground-state K.</p>

Atoms, molecules and clusters all constitute building blocks of macroscopic matter. Therefore, understanding the electronic and geometrical properties of such systems is the key to understanding the properties of solid state objects. In this thesis, some atomic, molecular and cluster systems (clusters of O2, CH3Br, Ar/O2, Ar/Xe and Ar/Kr; dimers of Na; Na and K atoms) have been investigated using synchrotron radiation, and in the two last instances, laser light. We have performed x-ray photoelectron spectroscopy (XPS) on all of these systems. We have also applied ultraviolet photoelectron spectroscopy (UPS), resonant Auger spectroscopy (RAS) and near-edge x-ray absorption spectroscopy (NEXAFS) to study many of the systems. Calculations using ab initio methods, namely density functional theory (DFT) and Møller-Plesset perturbation theory (MP), were employed for electronic structure calculations. The geometrical structure was studied using a combination of ab initio and molecular dynamics (MD) methods. Results on the dissociation behavior of CH3Br and O2 molecules in clusters are presented. The dissociation of the Na2 molecule has been characterized and the molecular field splitting of the Na 2p level in the dimer has been measured. The molecular field splitting of the CH3Br 3d level has been measured and the structure of CH3Br clusters has been determined to be similar to the structure of the bulk solid. The diffusion behavior of O2, Kr and Xe on large Ar clusters, as a function of doping rate, has been investigated. The shake-down process has been observed from excited states of Na and K. Laser excited Na atoms have been shown to be magnetically aligned. The shake-down process was used to characterize the origin of various final states that can be observed in the spectrum of ground-state K.