Acceleration and bunching by a gap

The relativistic equations of axial motion of a charged particle in an RF electric field inside a gap are analysed. The solution to these equations can be expressed as the incomplete elliptic integrals of the first and third kinds. The approximate solutions based on the average particle velocity in the gap are also found. The bunching by a gap is analysed by considering the phase /spl phisub e/ after the drift as a function of the entrance phase /spl phisub i/:/spl phisub e/=/spl phisub i/+/spl phisub t/+/spl phisub d/, /spl phisub t/ is the transit angle in the gap, /spl phisub d/ drift angle. Starting from the general condition for bunching d/spl phisub ed/spl phisub ispl les/1 it is shown that bunching begins with the input phase /spl phisub ipm/ corresponding to the minimum of the momentum P/sub m/ at the exit of the gap and ends at phase /spl phisub ipM/ when the momentum attains its maximum p/sub M/. The width of the bunched phases /spl Deltaspl phisub e/ is analysed as a function of different parameters according to the imposed conditions.< >