Improved Numerical Method for Solution of the Ha\"issinski Equation

TL;DR

This paper introduces a robust and efficient numerical method using matrix Newton's iteration to solve the Ha"issinski equation, enabling accurate modeling of electron beam equilibrium states with improved convergence and normalization.

Contribution

The paper presents a novel matrix Newton's iteration approach for solving the Ha"issinski equation, overcoming previous limitations and simplifying the solution process.

Findings

Convergence is highly robust even at high currents.

Method automatically normalizes the solution.

Applicable to various realistic wake potentials.

Abstract

The longitudinal charge density of an electron beam in its equilibrium state is given by the solution of the Ha\"issinski equation, which provides a stationary solution of the Vlasov-Fokker-Planck equation. The physical input is the longitudinal wake potential. We formulate the Ha\"issinski equation as a nonlinear integral equation with the normalization integral stated as a functional of the solution. This equation can be solved in a simple way by the matrix version of Newtons's iteration, beginning with the Gaussian as a first guess. We illustrate for several quasi-realistic wake potentials. Convergence is extremely robust, even at currents much higher than nominal for the storage rings considered. The method overcomes limitations of earlier procedures, and provides the convenience of automatic normalization of the solution.