Wake fields and impedance of space charge in Cartesian coordinate system

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. I. INTRODUCTIONSpace charge is an important source of collective effects and can exert a strong impact on the machine performance of modern particle accelerators with low-energy but high-intensity beams, or with high-energy but high peak-current beams. To simulate the beam dynamics with space charge, many tracking codes use self-consistent models based on particle-in-cell (PIC) method (for typical examples, see Refs. [1, 2]). On the other hand, many other codes use non-self-consistent models based on space charge impedance with given beam distribution (for an example, see Ref. [3]).The analytic theories of space charge wake functions and impedance have usually been derived in the cylindrical coordinate system with round vacuum chambers [4][5][6]. With the wake potential expanded in terms of cylindrical coordinates, usually the leading terms of longitudinal monopole and the transverse dipole are of important concern for evaluations of space charge effects [7]. In recent years, efforts have been made to extend the theories to cover various cases such as non-round chambers or non-uniform beam distributions. In Ref.[8], the longitudinal space charge impedance of a round uniform beam was studied in details in the presence of parallel-plates or rectangular chambers. In Refs. [9,10], the authors looked into the case of chambers with elliptic geometry for which the Mathieu functions can be used to express the electromagnetic fields.

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