Second-order Transit Time Factors for a Two Gap Resonator

The HIE-ISOLDE linac at CERN will operate independently phased quarter-wave resonators (QWRs) in order to accelerate radioactive ion beams (RIBs), with mass to charge states in the range 2.5 < A/q < 4.5, from 1.2 MeV/u up to an energy of at least 10 MeV/u. The low-β version of the QWR will also be used to decelerate beams below 1.2 MeV/u. The combination of low velocity and high gradient results in a significant change of the ion velocity and a breakdown of the first-order approximation commonly used to calculate the energy gain in accelerating cavities. The first-order transit-time factor for two gaps is briefly reviewed before higher-order transit-time factors are derived and the energy gain expressed, taking into account the variation in velocity, to second-order. The formalism of J.R. Delayen, introduced in [1], is used throughout.