Dissociative recombination of electrons with diatomic molecular cations above dissociation threshold: Application to ${{\mathbf{H}}_{\mathbf{2}}}^{+}$ and ${\mathbf{HD}}^{+}$

Our approach to the dissociative recombination and competitive processes based on the multichannel quantum defect theory is extended to the full account of the dissociative excitation, including the electronic excitation of the molecular ion. Compared to other existing modelings, ours relies on a simpler and less-time-consuming discretization of the vibrational continua of the target ion and to a more accurate account of the Rydberg-valence interactions via a second-order solution of the Lippman-Schwinger equation. A thorough study of the competition among the dissociative recombination, vibrational excitation, and dissociative excitation is performed, including a detailed analysis of the two different mechanisms governing the ion dissociation. The application of our method to the high-energy electron collisions with ${{\mathrm{H}}_2}^{+}$ and ${\mathrm{HD}}^{+}$ ions results in a cross section in good agreement with the best previous modeling and with the most recent measurements performed in the Test Storage Ring of the Max-Planck-Institut für Kernphysik in Heidelberg.

Figure 1

${\mathrm{HD}}^{+}-\mathrm{HD}$ states relevant for the electron-${\mathrm{HD}}^{+}$ reactive collisions. Potential-energy curves for dotted curve (blue): ${\mathrm{HD}}^{+}{}^2{\Sigma }_g^{+}\left(1s{\sigma }_g\right)$; dashed curve (orange) ${\mathrm{HD}}^{+}{}^2{\Sigma }_u^{+}\left(2p{\sigma }_u\right)$; solid line (red): ${\mathrm{HD}}^{**}{}^1{\Sigma }_g^{+}\left(2p{\sigma }_u^2\right)$ dissociative states. The vibrational levels relying on the ground and excited cores are represented by horizontal thin lines, colored blue and orange, respectively.

Figure 2

Direct dissociative recombination of the ground-state ${\mathrm{HD}}^{+}$ ion with a $d(l=2)$ electron into the lowest ${}^1{\Sigma }_g^{+}{\left(2p{\sigma }_u\right)}^2{\mathrm{HD}}^{**}$ state. Dotted black curve: only $v^{+}=v_i^{+}=0$ included in the calculation. Dashed (blue) curve: all bound vibrational levels ($v^{+}=0–20$) are included. Solid (red) curve: bound vibrational levels and discretized vibrational levels from the continuum of the ground-state ion (dissociative excitation of the first kind) are included.

Figure 3

DE1 and DE2 cross sections for the ${\mathrm{HD}}^{+}$ ground-state ion on impact with electrons. The contribution of the lowest dissociative state of the ${}^1{\Sigma }_g^{+}$ symmetry and of the $l=2$ partial wave of the Rydberg electron is illustrated, the calculations having been performed in the second order of the reaction matrix. The solid (green) line stands for the total DE cross section, whereas, the dashed black and dotted red lines illustrate the the DE1 and DE2 cross sections, respectively.

Figure 4

Dissociative recombination of the ground state of the ${\mathrm{HD}}^{+}$ ion, global cross sections. Dashed blue (solid red) curves: second-order calculations with dissociative excitation neglected (accounted). Black dot-dashed curve: computation of Takagi [15]. Circles: measurements of Tanabe et al. [12]. Triangles: measurements representing the Heidelberg data in Al-Khalili et al. [24].

Figure 5

Dissociative recombination of the ground-state ${\mathrm{HD}}^{+}$ ion, contributions from each symmetry (indicated in the panel) to the global DR cross section.

Figure 6

Dissociative recombination of the ground-state ${\mathrm{HD}}^{+}$ ion, contributions of each pair of atomic states resulting from dissociation. “$\mathrm{H}(n=1)+\mathrm{D}(n=2,3,4,...)$” stands for “$\mathrm{D}(n=1)+\mathrm{H}(n=2,3,4,...)$” too.

Figure 7

Dissociative recombination of the ground-state ${\mathrm{HD}}^{+}$ ion, relative atomic $\mathrm{H}\left(n\right)$, or $\mathrm{D}\left(n\right)$ final-state yields.

Figure 8

Dissociative recombination—black curves—and dissociative excitation—lighter (green) curves—of the ${{\mathrm{H}}_2}^{+}$ molecular ion. Thin (thick) solid curves: first-order (second-order) calculations. Dashed curves: theoretical results of Takagi [15].