Low-frequency plasma conductivity in the average-atom approximation

Low-frequency properties of a plasma are examined within the average-atom approximation, which presumes that scattering of a conducting electron on each atom takes place independently of other atoms. The relaxation time $\tau$ distinguishes a high-frequency region $\omega \tau >1$, where the single-atom approximation is applicable explicitly, from extreme low frequencies $\omega \tau <1$, where, naively, the single-atom approximation is invalid. A proposed generalization of the formalism, which takes into account many-atom collisions, is found to be accurate in all frequency regions, from $\omega =0$ to $\omega \tau >1$, reproducing the Ziman formula in the static limit, results based on the Kubo-Greenwood formula for high frequencies and satisfying the conductivity sum rule precisely. The correspondence between physical processes leading to the conventional Ohm’s law and the infrared properties of QED is discussed. The suggested average-atom approach to frequency-dependent conductivity is illustrated by numerical calculations for an aluminum plasma in the temperature range $2–10\mathrm{eV}$.

Figure 1

Two Feynman diagrams represent the amplitude, which describes electron scattering with absorption of a quantum of the electromagnetic field, which has frequency $\omega$ and small wave vector that leaves the electron momentum unchanged. The solid lines show the electron propagation, the dashed line the quantum of the electromagnetic field, and the solid circles the elastic scattering process. In diagrams (a) and (b) the lines that represent the electromagnetic quantum are inserted into the outer legs, which makes these diagrams infrared singular, $\propto 1∕\omega$, when $\omega \to 0$. Other possible diagrams have no such singularity.

Figure 2

The conductivity of an aluminum plasma at $T=5\mathrm{eV}$. Solid line, the Kubo-Greenwood formula, Eq. (2.10). Dashed line, simplified Kubo-Greenwood formula, Eq. (2.15), which predicts a second-order pole $\sigma \left(\omega \right)\propto 1∕{\omega }^2$. Dotted line, results from Ref. 10, which used an interpolating procedure to extend the results of the Kubo-Greenwood approach to the static approximation described by the Ziman formula, Eq. (3.1). Dot-dashed line, prediction of Eq. (4.8).

Figure 3

The conductivity of an aluminum plasma at different temperatures. Calculations are based on Eq. (4.8). The parameters used in this formula are evaluated with the help of the approach of Ref. 10.