Exact analytic solution of the envelope equations for a matched quadrupole-focused beam in the low space charge limit

The Kapchinskij-Vladimirskij equations describe the evolution of the beam envelopes in a periodic system of quadrupole focusing cells and are widely used to help predict the performance of such systems. Being nonlinear, they are usually solved by numerical integration. There have been numerous papers describing approximate solutions with varying degrees of accuracy. We have found an exact solution for a matched beam in the limit of zero space charge. The model is FODO with a full occupancy, piecewise-constant focusing function. Our explicit result for the envelope a(z) is exact for phase advances up to 180deg and all other values except multiples of 180deg. The peak envelope size is minimized at 90deg. The higher stable bands require large, very accurate, field strengths while producing significantly larger envelope excursions.