Analytic expressions for electron-ion temperature equilibration rates from the Lenard-Balescu equation

In this work, we elucidate the mathematical structure of the integral that arises when computing the electron-ion temperature equilibration time for a homogeneous weakly coupled plasma from the Lenard-Balescu equation. With some minor approximations, we derive an analytic formula, requiring no input Coulomb logarithm, for the equilibration rate that is valid for moderate electron-ion temperature ratios and arbitrary electron degeneracy. For large temperature ratios, we derive the necessary correction to account for the coupled-mode effect, which can be evaluated very efficiently using ordinary Gaussian quadrature.

Figure 1

Top panel: plot of the piece Eq. (69) of the integrand Eq. (33) for $y=2.1$ and $\alpha =0$; bottom panel: same as left but with $\alpha =1.8\times {10}^{-6}$. Retaining even a small $\alpha$ is crucial in obtaining the correct ion-acoustic peak height.

Figure 2

The function $Q\left(x\right)$ defined in Eq. (86). This is constant if $\alpha =0$, but here $\alpha =1.8\times {10}^{-6}$ and $Q\left(x\right)$ is constant for $x\lesssim 3$ but then drops to zero. This function is a part of the integrand in Eq. (83) and provides convergence in the coupled mode regime.

Figure 3

Equilibration rates for hydrogen at density $n_i=n_e={10}^{26}{\mathrm{cm}}^{-3}$ and $T_i=0.8T_e$; solid green: Eq. (63); dashed red: Brysk, Eq. (C2), with the GMS6 Coulomb logarithm; dotted blue: Brysk with prescription Eqs. (92, 93, 94, 95, 96, 97, 98); circles: numerical integration of Eq. (1). Fermi energy: $E_F=786.60$ eV.

Figure 4

Equilibration rates calculated for hydrogen at $n_i=n_e={10}^{26}{\mathrm{cm}}^{-3}$ and $T_e=500\mathrm{eV}$ over a range of ion temperatures; solid green: Eq. (63); dashed red: Brysk, Eq. (C2), with the GMS6 Coulomb logarithm; dotted blue: Brysk with prescription Eqs. (92, 93, 94, 95, 96, 97, 98); circles: numerical integration of Eq. (1). Fermi energy: $E_F=786.60$ eV.

Figure 5

Equilibration rates calculated for argon ($Z=18$) at $n_i={10}^{25}{\mathrm{cm}}^{-3}$ and $T_i=1.1T_e\mathrm{eV}$ over a range of ion temperatures; solid green: Eq. (63); dashed red: Brysk, Eq. (C2), with the GMS6 Coulomb logarithm; dotted blue: Brysk with prescription Eqs. (92, 93, 94, 95, 96, 97, 98); circles: numerical integration of Eq. (1). Fermi energy: $E_F=1162.48$ eV.

Figure 6

Equilibration rates calculated for hydrogen at $n_i=n_e={10}^{25}{\mathrm{cm}}^{-3}$ and $T_e=100\mathrm{eV}$ over a range of ion temperatures. Black x: direct integration of Eq. (1); red squares: Eq. (63); green circles: Eq. (63)+Eq. (64). Fermi energy: $E_F=169.25$ eV.