Equilibrium Longitudinal Distribution for Localized Regularized Inductive Wake

We use the Gaussian approximation to confirm that, as noted for Haissinski’s equation, a steady state solution for the longitudinal phase space distribution function always exists if a physically regularized inductive wake is used. INTRODUCTION In a recent paper [1] we have shown that assuming a localized wake and the Gaussian approximation for the longitudinal beam distribution function one can understand the nature of the stationary solutions for the inductive wake, by comparison between the resulting map and the Haissinski equation, which rules the (less realistic) case of a uniformly distributed wake. In particular we showed that the non-existence of solutions of Haissinski’s equation when the inductive wake strength exceeds a certain threshold [2] corresponds to the onset of chaos in the map evolving the moments of the beam distribution from turn to turn. In this paper we use the same map to confirm that, as noted in [2] for Haissinski’s equation, a steady state solution for the longitudinal phase space distribution function always exists if a physically regularized inductive wake is used. THE MOMENT MAPPING The longitudinal beam dynamics in electron storage rings can be described by the stochastic equations of motion for a single particle (Langevin equations). Introducing the canonical variables: x1 = longitudinal displacement natural bunch length , x2 = relative energy spread natural energy spread , and integrating the Langevin equations over one turn, we obtain the following stochastic mapping: ( x1 x2 )′ = U ( x1 Λx2 + r̂ √ 1− Λ2 − φ(x1) ) , where X ′ = (x ′ 1, x ′ 2) is X = (x1, x2) after one turn. Here U is the rotation matrix: U = ( cosμ sinμ − sinμ cosμ ) , (1) μ = 2πνs, νs being the synchrotron tune, Λ = exp(−2/T ), T being the synchrotron damping time measured in units of the revolution period, r̂ is a Gaussian random variable with < r̂ >= 0 and < r̂ >= 1. The wake force φ(x1) is represented by: φ(x1) = Qtot σ0E0 ∫ ∞ 0 ρ(x− u)W (u)du. (2) where E0 is the nominal beam energy, σ0 is the nominal relative energy spread (σ0E0 is the natural energy spread), W (x) is the wake potential and ρ(x) is the charge density normalized to one. Note that synchrotron oscillations have been linearized, and radiation is localized at one point of the ring [3]. The above stochastic mapping is equivalent to an infinite hierarchy of deterministic mappings in the following statistical quantities: x̄i =< xi >, σij =< (xi − x̄j)(xj − x̄j) >, and so on, which are the moments of the distribution function ψ( x), < ∗ > indicating an average over all particles. Our main assumption is that the distribution function in phase space is always a Gaussian, even in the presence of a wake force: ψ(x1, x2) = exp[12 ∑2 i,j σ −1 i,j (xi − x̄i)(xj − x̄j)] 2π √ detσ . (3) We consider a purely inductive wake function