Effect of Partially Screened Nuclei on Fast-Electron Dynamics

We analyze the dynamics of fast electrons in plasmas containing partially ionized impurity atoms, where the screening effect of bound electrons must be included. We derive analytical expressions for the deflection and slowing-down frequencies, and show that they are increased significantly compared to the results obtained with complete screening, already at subrelativistic electron energies. Furthermore, we show that the modifications to the deflection and slowing down frequencies are of equal importance in describing the runaway current evolution. Our results greatly affect fast-electron dynamics and have important implications, e.g., for the efficacy of mitigation strategies for runaway electrons in tokamak devices, and energy loss during relativistic breakdown in atmospheric discharges.

Figure 1

(a) The deflection frequency and (b) the slowing-down frequency as a function of the incoming-electron momentum, normalized to the completely screened collision frequencies. The models employed here (TF-DFT and Bethe-like) are plotted in black, while the full DFT model and the approximate RP model are shown in green. Note in (a) the lines overlay almost exactly. A pure argon plasma [of either ${\mathrm{Ar}}^{+}$ (dotted line) or ${\mathrm{Ar}}^{2+}$ (dash-dotted line)], with $T=10\mathrm{eV}$ and $n_{\mathrm{Ar}}=10^{20}{\mathrm{m}}^{-3}$ was assumed, giving $\ln {\mathrm{\Lambda }}_0^{+}=9.9$ and $\ln {\mathrm{\Lambda }}_0^{2+}=10.3$.

Figure 2

Contours of the distribution after 25 ms of collisional deceleration from an initial beamlike state (dotted black line), with and without screening effects. The limit of complete screening for both ${\nu }_D^{ei}$ and ${\nu }_S^{ee}$ (dashed green line) is shown, together with the TF-DFT model for ${\nu }_D^{ei}$ but unmodified ${\nu }_S^{ee}$ (dash-dotted blue line), and with both the TF-DFT and Bethe-like models (solid black line). The contours ${\log }_{10}(F)=-16.5$ and $-18$ are shown, where $F=(2\pi m_{e}T{)}^{3/2}{f}_e/n_e$, with (a) $B=0\mathrm{T}$ and (b) $B=4\mathrm{T}$. A parallel cut through the distribution of (a) is shown in (c). Parameters: $T=10\mathrm{eV}$, $n_{\mathrm{H}}=10^{20}$ ${\mathrm{m}}^{-3}$ and ${\mathrm{Ar}}^{+}$ with density $n_{\mathrm{Ar}}=n_{\mathrm{H}}$, $E=2E_c$.

Figure 3

Decay of the runaway-electron current as a function of argon density and time. The full model, with both the TF-DFT and Bethe-like contributions (solid black line), as well as the ${\nu }_D^{ei}$ (dash-dotted blue line) and ${\nu }_S^{ee}$ (dotted red line) models separately, are shown. These are compared to the RP model for inelastic collisions (dotted green line) and the limit of complete screening (dashed green line). The initial distribution of Fig. 2 was used, assuming a H plasma with $n_{\mathrm{H}}=10^{20}{\mathrm{m}}^{-3}$, and ${\mathrm{Ar}}^{+}$ impurities with $n_{\mathrm{Ar}}=10n_{\mathrm{H}}$; the light bands show the range of results obtained by varying the argon density such that $n_{\mathrm{Ar}}\in [0.5n_{\mathrm{H}},100n_{\mathrm{H}}]$. $T=10\mathrm{eV}$, $E=2E_c$, and $B=2\mathrm{T}$.