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Register a new userEquilibrium of an Arbitrary Bunch Train in the Presence of Multiple Resonator Wake Fields
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Authors
Robert Warnock
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A higher harmonic cavity (HHC), used to cause bunch lengthening for an increase in the Touschek lifetime, is a feature of several fourth generation synchrotron light sources. The desired bunch lengthening is complicated by the presence of required gaps in the bunch train. In a recent paper the author and Venturini studied the effect of various fill patterns by calculating the charge densities in the equilibrium state, through coupled Haissinski equations. We assumed that the only collective force was from the beam loading (wake field) of the harmonic cavity in its lowest mode. The present paper improves the notation and organization of the equations so as to allow an easy inclusion of multiple resonator wake fields. This allows one to study the effects of beam loading of the main accelerating cavity, higher order modes of the cavities, and short range geometric wakes represented by low-$Q$ resonators. As an example these effects are explored for ALS-U. The compensation of the induced voltage in the main cavity, achieved in practice by a feedback system, is modeled by adjustment of the generator voltage through a new iterative scheme. Except in the case of a complete fill, the compensated main cavity beam loading has a substantial effect on the bunch profiles and the Touschek lifetimes. A $Q=6$ resonator, approximating the effect of a realistic short range wake, is also consequential for the bunch forms.
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This paper continues the work of two previous treatments of bunch lengthening by a passive harmonic cavity in an electron storage ring. Such cavities, intended to reduce the effect of Touschek scattering, are a feature of fourth generation synchrotron light sources. The charge densities in the equilibrium state are given by solutions of coupled Haissinski equations, which are nonlinear integral equations. If the only wake fields are from cavity resonators, the unknowns can be the Fourier transforms of bunch densities at the resonator frequencies. The solution scheme based on this choice of unknowns proved to be deficient at the design current when multiple resonators were included. Here we return to the conventional formulation of Haissinski equations in coordinate space, the unknowns being charge densities at mesh points on a fine grid. This system would be awkward to solve by the Newton method used previously, because the Jacobian matrix is very large. Here a new solution is described, which is both Jacobian-free and much simpler. It is based on an elementary fixed point iteration, accelerated by Andersons method. The scheme is notably fast and robust, accommodating even the case of extreme over-stretching at current far beyond the design value. The Anderson method is promising for many problems in accelerator theory and beyond, since it is quite simple and can be used to attack all kinds of nonlinear and linear integral and differential equations. Results are presented for ALS-U, with updated design parameters. The model includes harmonic and main r.f. cavities, compensation of beam loading of the main cavity by adjustment of the generator voltage, and a realistic short range wake field (rather than the broad-band resonator wake invoked previously).
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