Equilibrium beam distribution and quantum lifetime in the presence of a single nonlinear resonance

In the proximity of a nonlinear resonance ν≈m/n, the beam distribution in a storage ring is distorted depending on how close is the resonance and how strong is the resonance strength. In the 1-dimensional case, it is well known that the particle motion near the resonance can be described in a smooth approximation by a Hamiltonian of the form (ν-m/n)J+D_ν(J)+f₁(φ,J), where (φ,J) are the phase space angle and action variables, D_ν is the detuning function, and f₁ is an oscillating resonance term. In a proton storage ring, the equilibrium beam distribution is readily solved to be any function exclusively of the Hamiltonian. For an electron beam, this is not true and the equilibrium distribution is more complicated. This paper solves the Fokker-Planck equation near a single resonance for an electron beam in a storage ring. The result is then applied to obtain the quantum lifetime of an electron beam in the presence of this resonance. Resonances due to multipole fields and due to the beam-beam force are considered as examples.