The Longitudinal Collective Instabilities of Nonlinear Hamiltonian Systems in a Circular Accelerator.

The understanding of collective instabilities of high intensity beams is essential to modern accelerators. In this dissertation, the subject of longitudinal collective instabilities of a single bunched beam in a circular accelerator, for which the Hamiltonian used to describe particle motion is nonlinear, is systematically studied. The analysis described in this work is applied to study the newly discovered instability, the longitudinal head-tail instability, and to substantially improve upon the study of the well known longitudinal mode-coupling instability. It is necessary to adopt a nonlinear Hamiltonian to precisely describe the motion of a particle in an accelerator. Traditionally, the perturbation theory is used to study the instabilities. However, the conventional formalism is based on the linear model and cannot be directly applied to nonlinear systems. In this work, a technique is developed to treat the instabilities of nonlinear systems. This technique involves a canonical transformation from the conventional variables to a new set of generalized variables. The new set of variables consist of the Hamiltonian itself, which serves as the new action variable, and another conjugate variable. This technique is applied to a system with a nonlinear momentum compaction factor to study the newly discovered longitudinal head-tail instability. The growth rate of the instability seems to agree quantitatively with experimental observations. In a linear system, the particles in a bunched beam move in a parabolic potential well. A distortion of the potential well, due to the electromagnetic fields of the beam, also contributes nonlinear terms to the otherwise linear Hamiltonian. This so-called potential well distortion effect is studied, and given as a possible cause of an apparent discrepancy between experimental observations and conventional calculations of the longitudinal mode -coupling instability. The results from the perturbation theory in this work, which includes the potential well distortion, are comparable with the experimental observations. One of the key inputs to study of the collective instabilities is a detailed knowledge of the impedance, a function describing the conductive environment quantitatively. The algorithm commonly used to calculate the impedance is studied in this work. An intrinsic numerical error, which leads to a fictitious high frequency impedance, is discovered by applying the algorithm to an intuitive case. This study is also included in this dissertation.