The evolution of the phase space density of particle beams in external fields

TL;DR

This paper presents a generalized proof of the Robinson theorem, describing how particle beam phase space density evolves in external fields, applicable to various conditions including accelerators and ion cooling.

Contribution

A simplified, unified proof of the Robinson theorem that extends its applicability to arbitrary external fields, averaged beam fields, and various frictional forces.

Findings

Robinson theorem proven for arbitrary external fields

Limits of theorem's applicability in ion cooling scenarios

Includes particle accelerators as a special case

Abstract

The evolution of the phase space density of particle beams in external fields is found proceeding from the continuity equation in the six-dimensional (6D) phase space (mu-space). The Robinson theorem, which includes the Liouville theorem as a special case, was proved in a more simple and consistent alternative way valid for arbitrary external fields, averaged fields of the beam (self-generated electro-magnetic fields except intrabeam scattering) and arbitrary frictional forces (linear, nonlinear). It includes particle accelerators as a special case. The limits of the applicability of the Robinson theorem in case of cooling of excited ions having a finite living time are presented.