3D Theory of Microscopic Instabilities Driven by Space-Charge Forces

TL;DR

This paper develops a comprehensive 3D theoretical framework for understanding microscopic instabilities caused by space-charge forces in particle beams, providing conditions for its applicability and simplified models.

Contribution

It introduces a rigorous 3D theory for space-charge driven instabilities and derives a linear integral equation, with criteria for reduction to simpler differential equations.

Findings

Provides a condition for the theory's applicability to arbitrary accelerator systems.

Derives a linear integral equation describing the instability.

Identifies when the integral equation simplifies to a second order differential equation.

Abstract

Microscopic, or short-wavelength, instabilities are known for drastic reduction of the beam quality and strong amplification of the noise in a beam. Space charge and coherent synchrotron radiation are known to be the leading causes for such instabilities. In this paper we present rigorous 3D theory of such instabilities driven by the space-charge forces. We define the condition when our theory is applicable for an arbitrary accelerator system with 3D coupling. Finally, we derive a linear integral equation describing such instability and identify conditions when it can be reduced to an ordinary second order differential equation.