Crossing of resonance in a relativistic electron ring with discrete and continuous eigenvalue spectrum

Abstract

The Vlasov formalism is used to calculate the first and second order perturbations of a ring of relativistic electrons after crossing the betatron resonance v r = 1. The presence of coherent self-fields may result in separate crossing of the coherent and incoherent integral resonances associated with the discrete and continuous eigenvalues. The quantity χ giving the ratio of the spread in the single particle frequencies to the coherent frequency shift characterizes the excitation of the continuous spectrum, the eigenmodes of which are similar to the singular von Kampen modes in the theory of plasma oscillations. Compared with an existing approach the present derivation is valid for arbitrary values of χ and is selfconsistent in that it correctly includes the dynamics of the variable giving rise to the finite spread in v r . The increase in beam size resulting from crossing of the incoherent integral resonance is compared with that due to crossing of the half-integral resonance, for which an improved formula is derived with the Vlasov formalism.