# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1905.03373