Abstract
In typical numerical simulations, the space-charge force is calculated by slicing a beam into many longitudinal segments and by solving the two-dimensional Poisson equation in each segment. This method neglects longitudinal leakage of the space-charge force to nearby segments owing to its longitudinal spread over 1/γ. By contrast, the space-charge impedance, which is the Fourier transform of the wake function, is typically calculated directly in the frequency-domain. So long as we follow these approaches, the longitudinal leakage effect of the wake function will remain to be unclear. In the present report, the space-charge wake function is calculated directly in the time domain by solving the three-dimensional Poisson equation for a longitudinally Gaussian beam. We find that the leakage effect is insignificant for a bunch that is considerably longer than the chamber radius so long as the segment length satisfies a certain condition. We present a criterion for how finely a bunch should be sliced so that the two-dimensional slicing approach can provide a good approximation of the three-dimensional exact solution.
Highlights
We present a criterion for how finely a bunch should be sliced so that the two-dimensional slicing approach can provide a good approximation of the three-dimensional exact solution
The space-charge wake function is conventionally described by the δ-function, assuming that the imaginary part of the impedance is constant over frequency
This may be a good approximation for an ultra-relativistic beam, but because the space-charge force is more important in the case of a non-relativistic beam, we need more rigorous descriptions of the space-charge wake functions for any magnitude of beam energy
Summary
For ρ < rb, where we use the formula:. λ2σz[2 2] cos λz.
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