Applications of matrix optics to acceptance studies in low-beta ion linacs

Four-gap superconducting resonators have been developed at Argonne for use in the low-beta positive ion injector (PII) for ATLAS. These structures have been used successfully for ion velocities as low as 0.007c with q/m=0.1. First order matrix optics and linear theory for the phase space transformations in accelerating systems are applied to the PII linac which includes low-beta heavy ion RF resonators and magnetic solenoids. These provide a new method to match initial phase space ellipses when used with higher order transfer maps or ray-tracing calculations. And also we present a quantitative measure of nonlinearities using the concept of RMS emittances and deviations of phase space coordinates between linear and nonlinear transformations. As a byproduct a way of identifying the dominant source of nonlinearity of system is indicated