Charged Particle Motion in a Plasma: Electron-Ion Energy Partition

TL;DR

This paper calculates the energy distribution between electrons and ions when a charged particle slows down in a plasma, providing precise results including cases with different electron and ion temperatures relevant for nuclear fusion.

Contribution

It introduces a detailed Fokker-Planck approach to accurately determine energy partitions, including sub-leading order terms, for plasmas with unequal electron and ion temperatures.

Findings

Derived new formulas for energy fractions in plasmas with different temperatures.

Confirmed agreement with previous results for equal temperature cases.

Identified a new relaxation mechanism that equilibrates electron and ion temperatures.

Abstract

A charged particle traversing a plasma loses its energy to both plasma electrons and ions. We compute the energy partition, the fractions $E_e/E_0$ and $E_\smI/E_0$ of the initial energy $E_0$ of this `impurity particle' that are deposited into the electrons and ions when it has slowed down into an equilibrium distribution that we shall determine. We use a well-defined Fokker-Planck equation for the phase space distribution of the charged impurity particles in a weakly to moderately coupled plasma. The Fokker-Planck equation holds to first sub-leading order in the dimensionless plasma coupling constant, which means we compute to order $n\ln n$ (leading) and $n$ (sub-leading) in the plasma density $n$. Previously, the order $n$ terms had been estimated, not calculated. Since the charged particle does not come to rest, the energy loss obtained by an integration of a $dE/dx$ has an ambiguity of order of the plasma temperature. Our Fokker-Planck formulation provides an unambiguous, precise definition of the energy fractions. For equal electron and ion temperatures, we find that our precise results agree well with a fit obtained by Fraley, Linnebur, Mason, and Morse. The case with differing electron and ion temperatures, a case of great importance for nuclear fusion, will be investigated in detail in the present paper. The energy partitions for this general case, partitions that have not been obtained before, will be presented. We find that now the proper solution of the Fokker-Planck equation yields a quasi-static equilibrium distribution to which fast particles relax that has neither the electron nor the ion temperature. This "schizophrenic" final ensemble of slowed particles gives a new mechanism to bring the electron and ion temperatures together. The rate at which this new mechanism brings the electrons and ions in the plasma into thermal equilibrium will be computed.