Stable Particle Motion in a Linear Accelerator with Solenoid Focusing

We derived the equation governing stable particle motion in a linear ion accelerator containing discrete rf and either discrete or continuous solenoid focusing. We found for discrete solenoid focusing that cos ¿ = (1 + d¿) cos ¿/2 + (l¿/¿ - d¿/21 - ¿¿d2/41) sin ¿/2 ¿ = 1/f and l + 2d = ß¿, where ¿, ¿, f, l, and d are the phase advance per cell, precession angle in the solenoid, focal length of the rf lens, length of the solenoid in one cell, and the drift distance between the center of the rf gap and the effective edge of the solenoid. The relation for a continuous solenoid is found by setting d equal to zero. The boundaries of the stability region for ¿ vs ¿ with fixed l and d are obtained when cos ¿ = +1.