Rigorous formulation of space-charge wake function and impedance by solving the three-dimensional Poisson equation
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 l...
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| Autores principales: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
Nature Portfolio
2018
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/af5083b9063347019724fa276082aca4 |
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| Sumario: | 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. |
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