Analytic electrical-conductivity tensor of a nondegenerate Lorentz plasma

Abstract

We have developed explicit quantum-mechanical expressions for the conductivity and resistivity tensors of a Lorentz plasma in a magnetic field. The expressions are based on a solution to the Boltzmann equation that is exact when the electric field is weak, the electron-Fermi-degeneracy parameter $\Theta \gg 1,$ and the electron-ion Coulomb-coupling parameter $\Gamma /Z\ll 1.$ (Γ is the ion-ion coupling parameter and Z is the ion charge state.) Assuming a screened $1/r$ electron-ion scattering potential, we calculate the Coulomb logarithm in the second Born approximation. The ratio of the term obtained in the second approximation to that obtained in the first is used to define the parameter regime over which the calculation is valid. We find that the accuracy of the approximation is determined by $\Gamma /Z$ and not simply the temperature, and that a quantum-mechanical description can be required at temperatures orders of magnitude less than assumed by Spitzer [Physics of Fully Ionized Gases (Wiley, New York, 1962)]. When the magnetic field $\mathbf{B}=0,$ the conductivity is identical to the Spitzer result except the Coulomb logarithm $\ln {\Lambda }_1=\left(\ln {\chi }_1-\frac{1}{2}\right)+[{(2\mathrm{Ze}}^2/\lambda m_e{v}_{e1\mathrm{}}^2)\left(\ln {\chi }_1-\ln 2^{4/3}\right)],$ where ${\chi }_1\equiv {2m}_e{v}_{e1\mathrm{}}\lambda /ħ,$ $m_e$ is the electron mass, $v_{e1\mathrm{}}\equiv {(7k}_B{T/m}_e{)}^{1/2},$ $k_B$ is the Boltzmann constant, T is the temperature, λ is the screening length, ħ is Planck’s constant divided by 2π, and e is the absolute value of the electron charge. When the plasma Debye length ${\lambda }_D$ is greater than the ion-sphere radius a, we assume $\lambda ={\lambda }_D;$ otherwise we set $\lambda =a.\mathrm{}$ The $\mathbf{B}=0$ conductivity is consistent with measurements when $Z\gtrsim 1,$ $\Theta \gtrsim 2,$ and $\Gamma /Z\lesssim 1,$ and in this parameter regime appears to be more accurate than previous analytic models. The minimum value of $\ln {\Lambda }_1$ when $Z>~1,$ $\Theta >~2,$ and $\Gamma /Z<~1$ is 1.9. The expression obtained for the resistivity tensor $\left(\mathbf{B}\ne 0\right)$ predicts that ${\eta }_{\perp }/{\eta }_{\parallel }$ (where ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ are the resistivities perpendicular and parallel to the magnetic field) can be as much as 40% less than previous analytic calculations. The results are applied to an idealized 17-MA z pinch at stagnation.