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 φ_e after the drift as a function of the entrance phase φ_i:φ_e=φ _i+φ_t+φ_d, φ_t is the transit angle in the gap, φ_d drift angle. Starting from the general condition for bunching dφ_e/dφ_i ⩽1 it is shown that bunching begins with the input phase φ _ipm corresponding to the minimum of the momentum P_m at the exit of the gap and ends at phase φ_ipM when the momentum attains its maximum p_M. The width of the bunched phases Δφ_e is analysed as a function of different parameters according to the imposed conditions