On the Evolution of Ion Bunch Profile in the Presence of Longitudinal Coherent Electron Cooling

TL;DR

This paper analyzes how ion bunch profiles evolve under longitudinal coherent electron cooling, providing analytical solutions without diffusion and numerical solutions with diffusion, validated against macro-particle tracking.

Contribution

It offers the first analytical solutions for ion bunch profile evolution in the absence of diffusion and numerical methods for cases with diffusion, validated by simulations.

Findings

Analytical solutions for ion profile evolution without diffusion.

Numerical solutions for cases with diffusion.

Good agreement with macro-particle tracking results.

Abstract

In the presence of longitudinal coherent electron cooling, the evolution of the line-density profile of a circulating ion bunch can be described by the 1-D Fokker-Planck equation. We show that, in the absence of diffusion, the 1-D equation can be solved analytically for certain dependence of cooling force on the synchrotron amplitude. For more general cases with arbitrary diffusion, we solved the 1-D Fokker-Planck equation numerically and the numerical solutions have been compared with results from macro-particle tracking.