Numerical solution of the coupled Haiessinski equations for the equilibrium state of an arbitrary bunch train in an electron storage ring

摘要

We study the effects of the beam loadings in the main and harmonic cavities and short-range wakefields on the bunch lengthening performance, in an electron storage ring. The compensation of the main cavity beam loading is modeled by adjusting the RF generator voltage so that the total voltage from the main cavity for each bunch can get as close as possible to the desired effective accelerating voltage under the ideal main cavity assumption. A system of coupled Haiessinski equations, which determine the bunch positions and profiles in the equilibrium state for arbitrary beam-fill patterns, are derived and solved numerically by a relaxation iteration. The proposed method has advantages of being a simple algorithm, displaying effectiveness as well as good convergence. The parameters of the planned Advanced Light Source upgrade (ALS-U) storage ring are used to test the method and good agreement is obtained comparing with particle tracking simulations.