Electric-field-dependent bimodal distribution functions for electrons in argon, xenon, and krypton owing to the Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections 

This paper considers solutions of the linear Fokker-Planck equation for the steady velocity distribution functions of electrons dilutely dispersed in an excess of either argon, xenon, or krypton. Owing to the large Ramsauer-Townsend minima in the electron-atom momentum-transfer cross sections for these atoms, the steady electron distribution, often referred to as the Davydov distribution, exhibits a bimodal distribution which varies dramatically with the strength of the external electric field. This effect provides a unique verification of the location and shape of the Ramsauer-Townsend minima in the cross sections. It is anticipated that current experimental techniques that probe the details of nonequilibrium electron distributions will verify the results reported in this paper. In addition, these nonequilibrium, non-Maxwellian distributions cannot be rationalized in terms of either the Gibbs-Boltzmann entropy nor the Tsallis nonextensive entropy.

Figure 1

Momentum transfer cross sections $\sigma \left(\epsilon \right)$ in Å${}^2$ vs energy $\epsilon$ in electron volts for electron-argon collisions. The nearly coincident curves denoted by P (solid line) and HO (dashed line) are the cross sections reported by Pitchford et al. [35] and Haddad and O'Malley [31], respectively. The solid curve with square symbols denoted by KK is the momentum-transfer cross section reported by Kitajami et al. [37] whereas the solid curve with circles denoted by HCC is the cross section reported by Hunter, Carter, and Christophorou [34]. The log-log plot enhances the Ramsauer-Townsend minimum in the momentum-transfer cross section. The symbols are used to identify the momentum-transfer cross section and are not actual data.

Figure 2

Momentum transfer cross sections $\sigma \left(\epsilon \right)$ in Å${}^2$ vs energy $\epsilon$ in electron volts for electron-krypton collisions as reported by Mozumder [30] (MOZ: solid line with circles), Koizumi, Shirakawa, and Ogawa [32] (KSO: solid line with squares), England and Elford [33] (EE: solid line), and Hunter, Carter, and Christophorou [34] (HCC: dotted line). The log-log plot enhances the Ramsauer-Townsend minimum in the momentum-transfer cross section. The symbols are used to identify the momentum-transfer cross section and are not actual data.

Figure 3

Momentum transfer cross sections $\sigma \left(\epsilon \right)$ in Å${}^2$ vs energy $\epsilon$ in electron volts for electron-xenon collisions. The solid curve with circles is the momentum-transfer cross section reported by Mozumder [30]. The solid curve with squares is the momentum-transfer cross section reported by Koizumi, Shirakawa, and Ogawa [32]. The heavy solid curve denoted by HCC is the momentum-transfer cross section reported by Hunter et al. [34]. The dotted line denoted by KK is the cross section reported by Kurokawa et al. [37]. The log-log plot enhances the Ramsauer-Townsend minimum in the momentum-transfer cross section. The symbols are used to identify the momentum-transfer cross section and are not actual data.

Figure 4

Bimodal distributions of electrons in argon for different electric field strengths $E/n, x=\sqrt{m{v}^2/2k_B{T}_b}, E/n$ in townsend, $\text{Td}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K. The $e$-Ar momentum-transfer cross section is the analytic fit reported in Ref. [26] as given by Eq. (7).

Figure 5

Bimodal distributions of electrons in Kr for different electric field strengths $E/n$ with the momentum-transfer cross section reported by Hunter, Carter, and Christophoru [34]; $x=\sqrt{m{v}^2/2k_B{T}_b}, E/n$ in townsend, $\text{Td}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K.

Figure 6

Bimodal distributions of electrons in Xe for different electric field strengths with the momentum-transfer cross section reported by Hunter, Carter, and Christophoru [34]; $x=\sqrt{m{v}^2/2k_B{T}_b}, E/n$ in townsend, $\text{Td}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K.

Figure 7

Upper graph: Bimodal distributions of electrons in Kr, $E/n=0.025$ Tn, with the momentum-transfer cross sections reported by England and Efford [33] (dashed line) and the momentum-transfer cross section reported by Hunter, Carter, and Christophorou [34] (solid line); $x=v/v_{\text{th}}$ with $v_{\text{th}}=\sqrt{2k_B{T}_b/m}, E/n$ in townsend, $\text{Tn}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K. Lower graph: Bimodal distributions of electrons in Xe, $E/n=0.062$ Tn, with the momentum-transfer cross sections reported by England and Efford [33] (dashed line) and the momentum-transfer cross section reported by Hunter, Carter, and Christophorou [34] (solid line); $x=v/v_{\text{th}}$ with $v_{\text{th}}=\sqrt{2k_B{T}_b/m}, E/n$ in townsend, $\text{Tn}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K.

Figure 8

Upper graph: The solid line is the bimodal distribution of electrons in Kr, $E/n=0.015$ Tn, with the momentum-transfer cross sections reported by Hunter, Carter, and Christophorou [34]; $x=v/v_{\text{th}}$ with $v_{\text{th}}=\sqrt{2k_B{T}_b/m}, E/n$ in townsend, $\text{Tn}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K. The solid circles identify the outermost extrema in the bimodal distribution. Lower graph: Analysis of the bimodal distribution based on Eq. (11). The solid curve is ${\widehat{\sigma }}^2\left(x\right)$ with the Ramsauer-Townsend minimum and the dashed curve is as noted.

Figure 9

Upper graph: The solid line is the bimodal distribution of electrons in Xe, $E/n=0.055$ Tn, with the momentum-transfer cross sections reported by Hunter, Carter, and Christophorou [34]; $x=v/v_{\text{th}}$ with $v_{\text{th}}=\sqrt{2k_B{T}_b/m}, E/n$ in townsend, $\text{Tn}={10}^{-17}$ V/${\mathrm{cm}}^2$; $T_b=290$ K. The solid circles identify the outermost extrema in the bimodal distribution. Lower graph: Analysis of the bimodal distribution based on Eq. (11). The solid curve is ${\widehat{\sigma }}^2\left(x\right)$ with the Ramsauer-Townsend minimum and the dashed curve is as noted.