Synchrobetatron coupling effects in alternating phase focusing

In the alternating phase focusing linac with a symmetric synchronous phase sequence, the lowest order resonance (σ_l -2σ_t=0) due to the synchrobetatron coupling occurs naturally, causing significant emittance transfer between the longitudinal and transverse motions. A model for the coupled motion including the amplitude dependent tune shift is proposed. Two approximate invariants are derived. Results from computer simulations of a bunched beam yield good agreement with the formula for the invariant derived from the theory. We provide, moreover, a way to move the parameters away, from the lowest order resonance, and thus lower the emittance exchange due to the resonance