Metastable states of D${}_2{}^{-}$ observed by foil-induced Coulomb explosion imaging

In recent years it was demonstrated that the negative hydrogen molecular ions H${}_2{}^{-}$ and D${}_2{}^{-}$ exist in long-lived states with lifetimes exceeding 1 $\mu$s (H${}_2{}^{-}$) and 1 ms (D${}_2{}^{-}$). These metastable ions exhibit very large internuclear distances and possess high angular momenta, which stabilizes them against autodetachment. Here we present the results of a foil-induced Coulomb explosion imaging experiment that allows for the measurement of the rovibrational wave functions of the metastable D${}_2{}^{-}$ ions. We compare our results to previous predictions from a nonlocal resonance theory. Our measurements do not confirm the discrepancy between experiment and theory that was inferred from previous photofragmentation studies. In fact, we find good agreement between the experiment and calculated wave functions for the most long-lived states with rotational quantum numbers $J=37$ and 38.

Figure 1

Schematic of the CEI beamline. The D${}_2{}^{-}$ beam is defined by two collimators and impinges on a DLC stripping foil that is supported by a nickel mesh. Via a magnetic separation field, the two D${}^{+}$ fragments are selected and guided toward a 3D imaging detector. Between the DLC foil and the detector, the fragments gain a distance of a few centimeters due to the Coulomb explosion process. A fast beam chopper ensures the detection of single-molecule events by blocking the D${}_2{}^{-}$ beam after the first impact at the detector.

Figure 2

Principle of Coulomb explosion imaging. The solid line in the lower panel shows the ${}^2{\Sigma }_u^{+}(J=37)$ potential energy curve of D${}_2{}^{-}$ [13], which leads asymptotically to D${}^{-}$+D. Also plotted is the vibrational ground state together with the corresponding distance distribution $P_{37,0}\left(R\right)$ determined by the square of the rovibrational wave function for $J=37,v=0$ (dashed blue line). Assuming an instantaneous stripping process, the two positively charged deuterons find themselves sitting on the potential energy curve $V_{J=37}^C\left(R\right)$ displayed in the upper panel, which is given by the sum of the Coulomb potential and the centrifugal potential for $J=37$. The potential energy of the two deuterons is then transformed into kinetic energy $E_{\mathrm{kin}}$, which leads to a kinetic energy release distribution $P_{37,0}\left(E_{\mathrm{kin}}\right)$ shown by the dashed blue line.

Figure 3

Comparison of the quantum-mechanical and semiclassical treatment of the Coulomb explosion process. (a) Rovibrational wave function of the ${}^2{\Sigma }_u^{+}(J=37,v=0$) state (blue solid line) and of the ${}^2{\Sigma }_u^{+}(J=37,v=1)$ state (dashed green line) as calculated in the nonlocal resonance framework [13]. (b) Corresponding KER distributions calculated using the quantum-mechanical Coulomb wave functions (lines) and applying the semiclassical approximation (symbols).

Figure 4

Kinetic energy release (KER) distributions measured for D${}_2{}^{-}$ after all cuts. The error bars show the uncertainty resulting from the counting statistics on a 1$\sigma$ level. The uncertainty in the KER is on the order of the size of the solid dots. The left panel (a) shows a fit (solid red line) using the simulated distributions for the three predicted metastable states that are long-lived enough to be detectable. These states are ($J=37$, $v=0$), ($J=37$, $v=1$), and ($J=38,v=0$). For the simulation, the calculated wave functions of Čížek et al. [13] were used as input of the CEI Monte Carlo code (see text for details). The right panel 4b shows a fit (solid red line) using only two wave functions, namely ($J=37$, $v=0$) and ($J=37$, $v=1$). The choice of these two states is motivated by the measurements of Lammich et al. [18], who used the same type of ion source that was used for the present work and did not observe significant population in the ($J=38$, $v=0$) state. The populations of the individual states were used as fit parameters; the resulting populations are given in the insets. The broken lines show the respective simulated KER distributions for the individual wave functions.

Figure 5

Potential energy curves relevant for the photodissociation experiment of D${}_2{}^{-}$ performed by Lammich et al. [18]. Photodetachment of an electron from the ($X{}^2{\Sigma }_u^{+},J=37$) state results in a neutral D${}_2$ molecule being either on the ($b{}^3{\Sigma }_u^{+},J=37$) or ($X{}^1{\Sigma }_g^{+},J=37$) potential energy curve, assuming the rotational angular momentum to be conserved in the detachment process. The dissociation of the D${}_2$ then leads to kinetic energy distributions indicated at the right-hand side of the upper panel.

Figure 6

Comparison of the kinetic energy release distributions calculated using Eq. (A1) (solid lines) to those using the semiclassical approximation (symbols) for photofragmentation of the $(J=37,v=0)$ and $(J=37,v=1)$ state of D${}_2{}^{-}$ via (a) the ($b{}^3{\Sigma }_u^{+},J=37$) and (b) the ($X{}^1{\Sigma }_g^{+},J=37$) channel. The distributions are folded with the experimental energy resolution.

Figure 7

Kinetic energy release distribution observed by Lammich et al. [18] in their photofragmentation experiment of a D${}_2{}^{-}$ beam 10 $\mu$s after the ion production (dots with error bars, redrawn from Fig. 6 of Ref. [18]). The experimental KER distribution is compared to calculated distributions, which are based on the wave functions ${\Psi }_{J,v}\left(R\right)$ of Ref. [13] and on populations $Q_{J,v}$ deduced in Ref. [18] to be $Q_{37,0}:Q_{37,1}:Q_{38,0}=54:45:1$. The theoretical distributions were folded with the experimental energy resolution. (a) The blue (dashed) and red (dashed-dotted) lines show the KER distributions resulting from the dissociation via the ($X{}^1{\Sigma }_g^{+},J^{\prime }$) and ($b{}^3{\Sigma }_u^{+},J^{\prime })$ potential energy curve, respectively, assuming $J^{\prime }=J$. The black solid line represents the sum of the two contributions. (b) Summed KER distributions assuming $J^{\prime }=J-1$ (solid line) and $J^{\prime }=J$ (dashed line). In (a) and (b) the relative intensities of the two dissociation channels were adjusted to $I\left(X{}^1{\Sigma }_g^{+}\right):I\left(b{}^3{\Sigma }_u^{+}\right)=70:30$.