Kinetic description of the beam relaxation caused by incoherent synchrotron radiation in high-energy electron storage rings

Abstract

The asymptotic beam-envelope matrix for an ideal electron-positron storage ring is shown to be the unique fixed point of a mapping, derived from the linearized Fokker-Planck equation, which includes the integrated effect of radiation damping and quantum excitation over one ring revolution. The corresponding beam distribution has azimuthal periodicity and is Gaussian in all the phase-space variables. In the limit of weak dissipation and far from linear resonances, this fixed point can be approximated by a linear combination of three periodic Twiss matrices, associated with the normal modes of the system. To first order in the relative energy loss per turn, the coefficients of the linear combination are constant along the ring and coincide with the equilibrium beam emittances. A general formula is derived for the equilibrium emittances, which reduces to well-known expressions in terms of radiation integrals for a storage ring with no horizontal-vertical coupling and with vanishing dispersion in the RF-cavities. In case of linear resonance, it is shown that two further “generalized equilibrium emittances” are required to specify the beam-envelope matrix. It is also shown that, in the limit of weak dissipation, the asymptotic beam-envelope matrix can be derived from a variational principle.