Space charge fields in azimuthally symmetric beams: integrated Green's function approach

TL;DR

This paper presents a comprehensive analytical solution for space charge electromagnetic fields in azimuthally symmetric relativistic beams within a round pipe, enabling efficient numerical modeling across all current variation scales.

Contribution

It introduces an integrated Green's function approach that provides a full-range solution for space charge fields in symmetric beams, simplifying and accelerating computations.

Findings

Derived a compact analytical expression for the Green's function.

Validated the approach for constant transverse charge density.

Enabled efficient numerical evaluation of space charge effects.

Abstract

Electromagnetic fields induced by the space charge in relativistic beams play an important role in Accelerator Physics. They lead to emittance growth, slice energy change, and the microbunching instability. Typically, these effects are modeled numerically since simple description exists only in the limits of large- or small-scale current variations. In this paper we consider an axially symmetric charged beam inside a round pipe and find the solution of the space charge problem that is valid in the full range of current variations. We express the solution for the field components in terms of Green's functions, which are fully determined by just a single function. We then find that this function is an on-axis potential from a charged disk in a round pipe, with transverse charge density $\rho_\perp(r)$, and it has a compact analytical expression. We finally provide an integrated Green's function based approach for efficient numerical evaluation in the case when the transverse charge density stays the same along the beam.