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>>1, and the electron-ion Coulomb-coupling parameter Gamma/Z<<1. (Gamma 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 B=0, the conductivity is identical to the Spitzer result except the Coulomb logarithm ln Lambda(1)=(ln chi(1)-1 / 2)+[(2Ze(2)/lambdam(e)v(2)(e1))(ln chi(1)-ln 2(4/3))], where chi(1) identical with 2m(e)v(e1)lambda/ variant Planck's over 2pi, m(e) is the electron mass, v(e1) identical with (7k(B)T/m(e))(1/2), k(B) is the Boltzmann constant, T is the temperature, lambda is the screening length, variant Planck's over 2pi is Planck's constant divided by 2pi, 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. The B=0 conductivity is consistent with measurements when Z greater, similar 1, Theta greater, similar 2, and Gamma/Z less, similar 1, and in this parameter regime appears to be more accurate than previous analytic models. The minimum value of ln Lambda(1) when Z> or =1, Theta> or =2, and Gamma/Z< or =1 is 1.9. The expression obtained for the resistivity tensor (B not equal 0) predicts that eta( perpendicular )/eta( parallel ) (where eta( perpendicular ) 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.

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