Wake fields and impedance of space charge in Cartesian coordinate system

TL;DR

This paper develops a method to calculate wake fields and impedance caused by space charge effects in a rectangular vacuum chamber using eigenfunction expansions, aiding 3D modeling of space-charge phenomena.

Contribution

It introduces a Green-function based approach in Cartesian coordinates to compute wake fields and impedance for space-charge effects, extending existing theories.

Findings

Derived Green-function forms for wake functions and impedance.

Validated the method through comparisons with existing theories.

Enabled calculation of wake fields for various particle distributions.

Abstract

In Cartesian coordinate system, the fields generated by a point charge moving parallel to the axis of a rectangular vacuum chamber can be formulated in terms of eigenfunctions of the rectangular waveguide using the mode expansion method. In combination with the conventional impedance theory, the Green-function forms of the wake functions and impedance for space-charge effects can be obtained, and are found to be functions of the positions of both the source and test particles. Using the Green's functions calculated for a point charge, the wake fields and impedance of a beam with various distributions can be calculated, and should be useful to model the three-dimensional space-charge effects. This paper summarizes our findings and also show comparisons to the existing theories.