Cluster virial expansion of the equation of state for hydrogen plasma with $e{\text{-H}}_2$ contributions

The equation of state of partially ionized hydrogen plasma is considered with special focus on the contribution of the $e\text{−}{\mathrm{H}}_2$ interaction. Traditional semiempirical concepts such as the excluded volume are improved using microscopic approaches to treat the $e\text{−}{\mathrm{H}}_2$ problem. Within a cluster virial expansion, the Beth-Uhlenbeck formula is applied to infer the contribution of bound and scattering states to the temperature-dependent second virial coefficient. The scattering states are calculated using the phase expansion method for the polarization interaction that incorporates experimental data for the $e\text{−}{\mathrm{H}}_2$ scattering cross section. We present results for the scattering phase shifts, differential scattering cross sections, and the second virial coefficient due to the $e\text{−}{\mathrm{H}}_2$ interaction. The influence of this interaction on the composition of the partially ionized hydrogen plasma is confined to the parameter range where both the ${\mathrm{H}}_2$ and the free-electron components are abundant.

Figure 1

Composition of hydrogen plasma at $T=15000$ K using mass action laws [10]. Particle-density fractions of electrons, hydrogen atoms, hydrogen molecules, negative hydrogen ions, and molecular ions are shown. Interaction between electrons and bound states are not taken into account.

Figure 2

Cross section for $e{\text{-H}}_2$ collision due to different scattering channels: $Q_{\mathrm{elas}}$, elastic scattering; $Q_{\mathrm{diss}}$, impact dissociation; $Q_{\mathrm{rot}}$, rotational excitations; $Q_{\mathrm{vibr}}$, vibrational excitations; and $Q_{b^3{\Sigma }_{\mathrm{u}}^{+}}$, electronic excitations, adapted from Ref. [25].

Figure 3

The $s$-wave scattering phase shifts for $e\text{−}{\mathrm{H}}_2$. The $R$-matrix data [39] are compared with the results of the present calculation on the basis of the polarization models (7) and (6).

Figure 4

The $p$-wave scattering phase shifts for $e\text{−}{\mathrm{H}}_2$. The $R$-matrix data [39] are compared with the results of the present calculation on the basis of the polarization model (7).

Figure 5

The $d$-wave scattering phase shifts for $e\text{−}{\mathrm{H}}_2$. The $R$-matrix data [39] are compared with the results of the present calculation on the basis of the polarization model (7).

Figure 6

Differential cross sections for $e\text{−}{\mathrm{H}}_2$ at incident energies 1 eV: solid line, calculation with the polarization model (7); diamonds, experimental data from Ref. [28]; circles, experimental data from Ref. [47].

Figure 7

Differential cross sections for $e\text{−}{\mathrm{H}}_2$ at incident energies of 3.4 eV: solid line, calculation with the polarization model (7); diamonds, experimental data from Ref. [28]; circles, theoretical data from Ref. [39].

Figure 8

Differential cross sections for $e\text{−}{\mathrm{H}}_2$ at incident energies of 4.5 eV: solid line, calculation with the polarization model (7); diamonds, experimental data from Ref. [28].

Figure 9

Differential cross sections for $e\text{−}{\mathrm{H}}_2$ at incident energies of 8 eV: solid line, calculation with the polarization model (7); diamonds, experimental data from Ref. [28]; circles, experimental data from Ref. [47]; dashed line, present calculation with the polarization model (6).

Figure 10

Free-electron-density fraction ${\alpha }_e$ at $T=15000$ K. The solid line shows the result of the set of equations (12). The dashed line is the result of the set of equations (12) neglecting interaction terms $\Delta {\mu }_{e\text{−}\mathrm{H}}^{\mathrm{sc}}$ and $\Delta {\mu }_{e\text{−}{\mathrm{H}}_2}^{\mathrm{sc}}$.

Figure 11

Composition of hydrogen plasma at $T=15000$ K taking into account $e$, H, ${\text{H}}_2,{\text{H}}^{-}$, and ${{\text{H}}_2}^{-}$ constituents. The solid line shows the result of the set of equations (12). The dashed line is the result of the set of equations (12) neglecting interaction terms $\Delta {\mu }_{e\text{−}\mathrm{H}}^{\mathrm{sc}}$ and $\Delta {\mu }_{e\text{−}{\mathrm{H}}_2}^{\mathrm{sc}}$.