Momentum vector correlations in three-particle fragmentation of triatomic hydrogen

The correlation pattern in the center of mass motion of the three fragments from dissociation of well-defined Rydberg states of ${\mathrm{H}}_3$ and ${\mathrm{D}}_3$ is studied. Dissociation of the molecules is induced by an external electric field. Through a comparison with results obtained in radiative cascading we can show that the correlation pattern is that of the short-lived $2s\hspace{0.5em}{}^2{A}_1^{{}^{\prime }}$ electronic state, of which a tiny amplitude is admixed by the external electric field. A comparison of our results with the predictions by M. Lehner and M. Jungen [J. Phys. B 42, 065101 (2009)] and U. Galster [Phys. Rev. A 81, 032517 (2010)] for predissociation of the $2s\hspace{0.5em}{}^2{A}_1^{{}^{\prime }}$ state is made. We show that the experimental vector correlation maps are direct images of the spatial symmetry of the product of the vibrational wave function and spatial dependence of the nonadiabatic coupling operator.

Figure 1

Experimental setup for ${\text{D}}_3$. The geometrical dimensions of the Stark-field section are given in [7]. For the measurements with ${\text{H}}_3$ a slightly different Stark-field section with bigger electrodes has been used; see Fig.  2 of [8].

Figure 2

Energy levels studied in this work with the example of ${\mathrm{H}}_3$. The state label is $\{v_1,v_2,N,K\}$ where $v_1$ denotes the symmetric stretch mode excitation, $v_2$ the degenerate bending mode, $N$ the total angular momentum disregarding spins, and $K$ its projection along the figure axis.

Figure 3

Kinetic energy release spectra for the three particle decay of ${\mathrm{D}}_3$. The eight traces are measured at different maximal field strengths. They are normalized to measurement time and show the event rate per minute. The appearing peaks are labeled corresponding to $n\ell$(${\nu }_1$,${\nu }_2$).

Figure 4

Spectrum of total kinetic energy release in D(1$s$) $+$ D(1$s$) $+$ D(1$s$) following photoexcitation to the 3$s$$A_1^{\prime }$(0,0,1,0) state of ${\mathrm{D}}_3$ [6]. Dotted lines give the expected energy release for different final state levels. The labels a-j refer to final states populated in cascading transitions; k is the photoexcited state. a: $2{\mathit{sA}}_1^{\prime }$(0,0). b: $2{\mathit{sA}}_1^{\prime }$(0,1). c: $2{\mathit{sA}}_1^{\prime }$(1,0). d: $2{\mathit{sA}}_1^{\prime }$(0,2). e: $2{\mathit{sA}}_1^{\prime }$(1,1). f: $2{\mathit{sA}}_1^{\prime }$(0,3). g: $2{\mathit{sA}}_1^{\prime }$(1,2). h: $2{\mathit{sA}}_1^{\prime }$(2,1). i: $2{\mathit{pA}}_2^{″}$(0,0,2,0). k: $3{\mathit{sA}}_1^{\prime }$(0,0,1,0). j: $3{\mathit{pE}}^{\prime }$(0,0,2/1,1). When $N$ and $K$ cannot be given with certainty the notation is $n\ell \mathcal{S}\hspace{0.5em}(v_1,v_2)$.

Figure 5

On the connection between hyperspherical coordinates and corresponding spatial configurations. The spatial configuration sphere spanned by the three hyperspherical parameters $\vartheta$, $\phi$, and $\rho$ is shown at the left. If $\rho$ is fixed and the sphere surface is projected into the equatorial plane, the Dalitz plot appears, which describes the momentum configurations of the fragments in 3-body decay. One obtains the momentum vectors by drawing arrows from the center of mass of any given triangle to its corners; see Fig. 1 of [2].

Figure 6

Signal and error bars for the segments in two representative Dalitz maps are shown in the lower row, together with the absolute location of the numbered segment in each 60${}^{°}$ slice of the phase space. The signal $s$ gives the relative intensity normalized to the maximal count rate in each Dalitz plot, the total counts in each spectrum being $\approx$$1\times {10}^4$.

Figure 7

Comparison of experimental Dalitz maps for $2s$ states of ${\mathrm{D}}_3$ obtained via cascading (left column) and via the Stark mixing method (right column). The dissociation rate of 2$p$ states at high field is dominated by the $2s$-state amplitude. As a result close similarities appear in the dissociation-configuration pattern with the maps recorded after cascading transitions into the respective $2s$ states. The gray-scale function used can be seen on the upper right side of Fig. 6.

Figure 8

Comparison of Dalitz plots for ${\mathrm{H}}_3$ obtained by cascading into 2$s$ (left column) with those formed by Stark-mixing (right column). At 20 kV/cm the Dalitz plots of the 2$p$ states very much resemble those of the corresponding $2s$ states. The coarse binning into circular segments in Figs. 7 and 8 is chosen to control the statistical error in each bin to a value of less than 10%.

Figure 9

Dalitz maps of $2s$ vibrational states of ${\mathrm{H}}_3$, obtained by Stark-admixing $2s$ character to the metastable $2p$ vibrational states present in our fast neutral beam.

Figure 10

Cuts through the potential surfaces of ${\mathrm{H}}_3$. The center panel shows the effect of a distortion conserving $\rho$ and the $C_{2v}$ symmetry. The panels on the right and left give the energies of the distorted molecule as a function of $\rho$. The vertical lines indicate coordinates common to the center panel (from [5]).

Figure 11

Comparison of experimental Dalitz maps for the $2s(0,0)$ states of ${\mathrm{H}}_3$ and ${\mathrm{D}}_3$ (left column) with the semiclassical results predicted for dissociating molecules initially entering the upper surface (center) and the lower surface (right) [1]. The experimental maps (left columns) show a dominance of obtuse-angle configurations.

Figure 12

Comparison of experimental Dalitz maps for the $2s(0,1)$ states of ${\mathrm{H}}_3$ and ${\mathrm{D}}_3$ (left column) with the semiclassical results predicted for dissociating molecules initially entering the upper surface (center) and the lower surface (right) [1]. The experiment shows a substantial contribution from acute-angle and equilateral-angle contributions.