UBC Theses and Dissertations
Field tolerances associated with some resonances in the triumf cyclotron Bolduc, Jean Louis
This thesis is concerned with tolerances for magnetic field imperfections in the TRIUMF cyclotron set by the betatron oscillation resonances v[formula omitted] = 1.0, v[formula omitted] – v[formula omitted] = 1.0 and v[formula omitted] = 1.5. These resonances, encountered during acceleration, can lead to undesirable growth in the amplitudes of the betatron oscillations. We first derive equations of motion that take into account non-linear terms and field imperfections, and show how resonance conditions may occur. These conditions were simulated in our orbit codes and numerical calculations were made to determine the tolerances they impose on the magnetic field. We have made a detailed investigation of the effect on the behaviour of the beam of first harmonic bumps at radii less than 150 in. The first harmonic tolerance of 0.1 G to produce an increase in the radial amplitude of 0.1 in. is in agreement with analytical calculations. We have also shown that this tolerance, too small to be seen in the magnetic field survey, may be achieved by suitable adjustments in the harmonic coil settings. Tolerances on the second harmonic imperfection are also presented. To determine the tolerances set by the coupled resonance v[formula omitted] – v[formula omitted] = 1.0, we have simulated a first harmonic twist in the median plane. The results show that, for high current poor resolution experiments, the magnitude of the twist on entering the resonance is of no importance, provided the amplitudes of the radial and vertical betatron oscillations are not much larger than the estimate of 0.2 in. For high resolution experiments, the tolerances on the slope of the twist are of the order of a few mrad. Analytical estimates of these tolerances are also presented. Finally, we have determined that the v[formula omitted] = 1.5 resonance sets an upper limit to the gradient of the third harmonic of about 0.2 G/in., which produces a 20% increase in the radial betatron amplitude.