Permanent dipole transitions remain elusive in HD ⁺  strong-field dissociation

J McKenna A M Sayler, B Gaire, Nora G Johnson, M Zohrabi K D Carnes B D Esry and I Ben-Itzhak

J R Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, KS 66506, USA

E-mail: ibi@phys.ksu.edu

Received 11 May 2009
Published 1 June 2009

1. Introduction

Laser-molecular science is in the midst of an exciting era when new applications continually emerge, such as laser control of chemical reactions [1] and molecular imaging [2, 3]. These applications are born of and continue to benefit from basic research on the interaction of ultrashort intense laser pulses with molecules. For many years, the hydrogen molecular ion has been considered the benchmark system to study for this purpose. The interest in this molecule is fuelled by its elementary structure, composed of two nuclei and one electron, and has resulted in a plethora of theoretical calculations on its intense field dynamics accompanied by numerous experiments. Together these have deepened our understanding of how H ⁺ ₂, and molecules in general, respond to intense laser pulses (see [4, 5] for detailed reviews).

To date, the homonuclear molecules H ⁺ ₂ and D ⁺ ₂ have most commonly been studied. However, HD ⁺  has its own special role as the simplest example of a heteronuclear molecule. The dynamics of heteronuclear molecules in strong fields potentially differ from homonuclear molecules for several reasons, but in particular, because of their permanent electric dipole moment. The permanent dipole moment allows transitions within the same electronic state thereby opening new pathways for dissociation, e.g. transitions in HD ⁺  that involve the direct absorption of an even number of photons. However, despite some theoretical predictions for permanent dipole transitions in small molecules [6, 7], so far there has been very little experimental evidence to support their occurrence. Against this trend, Kiess et al recently renewed hope of observing permanent dipole transitions by reporting the first evidence for direct two-photon dissociation of HD ⁺ , involving its permanent dipole moment [8]. Alas, we find new evidence to the contrary that shows that the small two-photon peak they observed instead comes from the well-known one-photon bond-softening mechanism [9–11] observed also in the homonuclear H ⁺ ₂ molecule (e.g. [12]).

Typically, there are two ways to produce the hydrogen molecular ion for study in intense laser fields: either (i) by laser-induced ionization of neutral H₂, D₂ or HD on the leading edge of the pulse, e.g. [13, 14], or (ii) by electron-impact ionization of H₂ in a separate ion source and then transporting it into the laser focus as an ion beam [15–17]. While these methods are complementary, an ion beam has the additional benefit of allowing the detection of the neutral fragments from dissociation. This, coupled with the coincidence three-dimensional imaging technique developed by the authors [17, 18], enables clear separation of dissociation events (H ⁺ ₂ → H ⁺   +  H) from ionization events (H ⁺ ₂ → H ⁺   +  H ⁺   +  e^–). The application of this method is most vital for the study of HD ⁺  as it also allows distinction of the individual dissociation channels, HD ⁺ → H ⁺  + D from HD ⁺ → H + D ⁺ . Without correlation, energetic D or D ⁺  fragments may not easily be discerned from low energy H or H ⁺  fragments making the channel assignment of kinetic energy release peaks difficult.

As alluded to above, the dynamics of HD ⁺  in an intense laser pulse are in many respects richer than those of H ⁺ ₂ or D ⁺ ₂. The distinguishability of the nuclei (proton and deuteron) means that the symmetry of the molecule under their exchange (that leads to gerade and ungerade states) is absent. One consequence of this nuclear asymmetry is a splitting of the separated-atoms limit of the Born–Oppenheimer potentials. Specifically, the mass of deuterium is larger than hydrogen resulting in a binding energy difference of 3.7 meV. Thus, the HD ⁺  ground (1sσ) and first excited (2pσ) states asymptotically correlate to H ⁺  + D and H + D ⁺ , respectively. In principle, this splitting may lead to a laser-induced asymmetry in the dissociation yields for the two channels similar to that observed for very low energy fragments following proton-impact ionization of HD [19, 20]. In reality, an observable asymmetry is quite improbable at 790 nm as evident from the calculations of Charron et al [21] and the lack of an asymmetry in the recent experimental data of Kiess et al [8]. Succinctly, the 1sσ and 2pσ states of HD ⁺  are strongly coupled around R~ 10 au [22]. Thus, if the dissociation energy is more than a few meV, the population on these states will get mixed [19, 23], thereby removing any dissociation asymmetry (unless controlled in some special way, e.g. using carrier-envelope phase [24, 25] or two-colour mixing [26, 27]).

Another consequence of the nuclear asymmetry is that the centre of mass of HD ⁺  is offset from the centre of charge, giving HD ⁺  a permanent electric-dipole moment. Consequently, permanent dipole transitions within the same electronic state are allowed, giving rise to possible dissociation pathways in HD ⁺  that are absent from H ⁺ ₂ and D ⁺ ₂. The recent Kiess et al [8] study of HD ⁺  used 790 nm, 100 fs pulses and observed a small dissociation peak that coincided in fragment energy with the expected energy for direct two-photon dissociation (arising from a permanent dipole transition). Thus, they concluded that they had found evidence for direct two-photon dissociation, under the assumption that this peak is associated with a D (D ⁺ ) fragment. However, for their HD ⁺  dissociation data they were not able to clearly distinguish H (H ⁺ ) from D (D ⁺ ) and thus based their assignment of peaks solely on the fragment energy—which can be insufficient. By applying our coincidence technique, we unambiguously discern the H + D ⁺  channel from the H ⁺  + D channel as well as separate both dissociation channels from ionization. Therefore, this improved experimental technique should allow us to verify whether direct two-photon dissociation does occur as reported [8] without the need to assume the identity of a fragment. Under similar laser conditions as those used in the previous work, we find no evidence for the direct two-photon pathway. Specifically, our data indicate that the peak assigned previously as two-photon and coming from high energy D (D ⁺ ) fragments is in fact due to low energy H (H ⁺ ) fragments from one-photon bond softening.

2. Experimental method

The details of our experimental set-up have been described elsewhere [17, 18]. In summary, linearly polarized 790 nm, 30 fs pulses from a Ti:sapphire laser system are stretched to 100 fs by adding negative chirp to the pulses using a grating-based compressor. The laser is focused at 90° onto a molecular-ion beam target using an off-axis parabolic mirror. Using an intensity selective scan technique [28, 29], the intensity is varied between 3 × 10¹³ and 5 × 10¹⁴ W cm^–2. The finite size of the ion beam (~0.6 × 0.6 mm²) helps to limit the volume of molecules exposed to the laser field in the laser propagation direction, thus reducing intensity-averaging effects.

The HD ⁺  molecular-ion beam is formed by electron-impact ionization of HD in an electron-cyclotron resonance (ECR) ion source, then extracted, accelerated to 9 keV, mass-selected and collimated. Following interaction with the laser pulses, fragments with several keV laboratory energy from the molecular-ion beam are detected using a time- and position-sensitive hex-anode delay-line detector (80 mm diameter) positioned downstream from the interaction region. The primary ion beam is collected in a Faraday cup (2 mm diameter) that blocks a small region at the centre of the detector. The polarization of the laser is oriented in the plane of the detector to increase the spatial spread of the fragments on the detector. From the measured spatial and temporal information, the 3D momentum vectors of all fragments for each molecule are retrieved. In the interaction region a weak static electric field applied in the direction of ion beam propagation accelerates charged fragments, thus clearly separating the H ⁺ , D ⁺  and neutral fragments by their time of flight. This, in conjunction with the coincidence between fragment hits on the detector, allows all dissociation and ionization channels to be distinguished.

3. Floquet dressed states

To provide context for the discussion of our measured dissociation spectra, we first visit the Floquet light-dressed potential energy curves [4, 30, 31] of H ⁺ ₂ and HD ⁺  at 790 nm displayed side by side in figures 1(a) and (b), respectively. Direct comparison immediately reveals that HD ⁺  has more ways to dissociate. The reason for the extra pathways in HD ⁺  is its permanent dipole moments, which are shown in figure 1(c) along with the 1sσ – 2pσ transition dipole moment present in both H ⁺ ₂ and HD ⁺ . In H ⁺ ₂, the gerade (1sσ_g) and ungerade (2pσ_u) states can only be coupled through the transition dipole moment by the direct absorption (or emission) of an odd number of photons due to dipole selection rules. That is, the absorption of each photon must change the electronic state of the molecule, in this case from 1sσ_g 2pσ_u. In HD ⁺ , however, selection rules allow direct absorption (and emission) of both odd and even numbers of photons between different and the same electronic state(s). Therefore, in addition to the pathways present in H ⁺ ₂, HD ⁺  can dissociate through the crossing marked 2ω in figure 1(b), either along the diabatic (direct) two-photon pathway, |1sσ – 0ω → |2pσ – 2ω, or along the adiabatic (net) one-photon pathway, |1sσ – 0ω → |2pσ – 2ω → |1sσ – 1ω. The diabatic two-photon pathway is the main focus of this work.

4. Results and discussion

From earlier studies of H ⁺ ₂ dissociation, we know that when H ⁺ ₂ absorbs one photon at low intensity (<5 × 10¹³ W cm^–2), vibrational states nearest the 1ω crossing dissociate via bond softening, i.e. v~ 10 for HD ⁺  at 790 nm as shown in figure 1. The associated kinetic energy release (KER) is the energy difference between the vibrational energy and the asymptotic |2pσ – 1ω dissociation limit, e.g. 0.76 eV for v =  10. Therefore, for high-resolution KER measurements, we expect to find a series of KER peaks corresponding to dissociation of different vibrational states through the 1ω crossing.

Figures 2(a) and (b) display the KER-cosθ and KER distributions, respectively, for dissociation of HD ⁺ → H + D ⁺  at 790 nm. The angle θ is the angle between the molecular breakup direction and the laser polarization. Note that as expected, the H ⁺  + D channel is similar to H + D ⁺  and is therefore not shown. As predicted, a sequence of peaks in KER are observed that can be assigned to the v states of HD ⁺ . The vibrational comb for one-photon dissociation, indicating the expected KER for each state, is marked along the top of each panel in figure 2(b).

We note that while these data were recorded with negatively chirped 100 fs pulses, the previous work that we wish to compare with was performed using transform-limited 100 fs pulses [8]. Nonetheless, we repeated several of the measurements using positively chirped pulses and there was no change in the shape of the distributions—only small KER shifts in the peaks were found (e.g. ~20 meV at 3 ×10¹³ W cm^–2). We therefore do not expect significant differences for transform-limited pulses, except that the peaks may become narrower and thus even more pronounced due to the smaller pulse bandwidth.

At 3 × 10¹³ W cm^–2, the largest peak is centred at KER  =  0.76 eV in excellent agreement with the expected value for v =  10. The neighbouring peaks also agree very well with the vibrational comb values. The lowest state that shows substantial dissociation is v =  9, as the gap at the 1ω avoided crossing is only weakly open—this is despite the Floquet adiabatic picture (not shown) indicating that 3 × 10¹³ W cm^–2 should be sufficient to dissociate states lower than v =  9. In general, intensity averaging from the laser focal volume lowers the overall effective intensity. Nevertheless, high v states above the 1ω crossing, up to near the |1sσ – 0ω limit, do dissociate as indicated by the high-KER cut-off value of ~1.6 eV (corresponding to the energy of one photon). The peak structure of the high v states progressively merges as the spacing between v states becomes smaller while their amplitude follows approximately the Franck–Condon population distribution [32, 33].

With increasing peak intensity, dissociation of lower v states becomes more apparent. Also, their angular distribution is more aligned due to the preference for alignment of bond softening. The lowest v state that can physically dissociate through the 1ω crossing by bond softening is v =  6 and may be observed at 5 × 10¹⁴ W cm^–2. However, at this intensity we can also expect to observe some dissociation by above-threshold dissociation (ATD) [9–11]. The ATD adiabatic pathway |1sσ – 0ω → |2pσ – 3ω → |1sσ – 2ω shown in figure 1(b) gives KER ~1.3 eV, which overlaps the high-KER part of the main one-photon peak. Hence, it cannot be easily distinguished, although our observed angular distribution in this KER region does not dramatically differ from lower intensity angular distributions suggesting that any ATD contribution is small. The permanent dipole pathways |1sσ – 0ω → |2pσ – 2ω → |1sσ – 1ω and |1sσ – 0ω → |2pσ – 2ω should give KER values of 0.2 eV and 1.8 eV, respectively. The first pathway, if present, would overlap one-photon dissociation of v =  6, and again cannot be distinguished. The second pathway, the one reported recently as direct two-photon dissociation [8], is clearly not present in our data as no dissociation is observed above 1.6 eV.

To shed further light on this contradiction with the earlier results [8], we plot our dissociation spectra as fragment velocity distributions in figure 2(c) in order to emulate the presentation of the previous data. We note that our coincidence method allows us to separate D ⁺  from H (or D from H ⁺ ) contributions and H from D. The earlier authors did not have this ability and hence they observed only the sum of the two fragment components. This forced them to make some assumptions about the identity of the fragments in order to interpret their data. The peak in question, assigned previously as direct two-photon dissociation, resides between 7000 and 8000 m s^–1. Our data clearly show an H fragment peak in this region at higher intensities that comes from one-photon dissociation of v =  8 by bond softening, and not a high KER D-fragment peak from two-photon dissociation, as suggested previously [8].

For a closer comparison with the earlier work of Kiess et al [8], we replot their neutral fragment data (at 3 × 10¹³ and 2 × 10¹⁴ W cm^–2) alongside ours (at 2 × 10¹⁴ W cm^–2) in figures 3(b) and (a), respectively. The reason it was argued in [8] that the peak circled, and marked `?', in figure 3(b) comes from the two-photon pathway is because it coincides with the expected position for D two-photon dissociation of v =  6—see the vibrational comb tick. As their measurement was not able to separate the D from H components, it was not clear which, or both, mechanisms were responsible for the peak, i.e. one-photon H(v =  8) or two-photon D(v =  6). Our coincidence measurement in figure 3(a) decisively indicates that the peak arises from one-photon H(v =  8) dissociation and that there are no D fragments in this velocity region, hence contesting the earlier assignment of this peak.

In addition to the question about the direct two-photon dissociation, it is interesting to note that the spectra shown in figures 3(a) and (b) look quite different for the two experiments. Particularly, the data of [8] tend to peak for lower v states than in our experiment. This suggests a higher effective intensity in the experiment of [8] as it causes more bond softening to allow the lower v states to dissociate. The fact that the data of [8] do peak for lower v, specifically D(v =  8) at 3 × 10¹³ W cm^–2, lends further credence to the conclusion that if there is a large D(v =  8) peak, there should be an equivalent H(v =  8) peak at 7000–8000 m s^–1.

Several factors can cause the difference in effective intensity of the two experiments. For example, the spectra of [8] are plotted for a narrower angular cut (±4°) along the laser polarization than in our spectra (±18°). This tends to enhance the higher intensity features that show a stronger degree of alignment—the alignment may be geometric or dynamic. We plot an ±18° cut to retain reasonable statistics. However, we have also tried a narrower cut and our spectra do not change dramatically indicating that this is not the biggest factor—for a narrower cut, the contribution from the lower v states simply becomes slightly enhanced as they are more aligned. Another, seemingly more important, difference was that the ion beam used in [8] was collimated to a much smaller size (25 × 250 µm²) than that used here (600 × 600 µm²). This reduces focal-volume intensity averaging again giving more weight to higher intensity features. Furthermore, the size of the laser focus in the two experiments can differ depending on focusing conditions. The comparison in figure 3, therefore, highlights the important impact of the different intensity profile over the interaction volume (approximately 1D in [8] and 2D in our measurements) on the measured KER spectrum due to intensity averaging.

5. Summary

In summary, we find no evidence for the role of permanent dipole transitions in the strong-field dissociation of HD ⁺  using 790 nm, 100 fs pulses in the intensity interval 3 × 10¹³–5 × 10¹⁴ W cm^–2. Our data show that dissociation is dominated by one-photon bond softening with clear vibrational structures that agree well with expected dissociation energies. This is in disagreement with the recent work of Kiess et al [8] that reports evidence for direct two-photon dissociation via the permanent-dipole initiated pathway |1sσ – 0ω → |2pσ – 2ω. Our data strongly suggest that the peak they observed was incorrectly assigned due to the lack of coincidence information, and instead it is in accord with one-photon dissociation of v =  8. This work highlights the value of being able to distinguish among all fragments (H ⁺ , D ⁺ , H and D) in the coincidence measurement allowing dissociation and ionization channels to be unambiguously separated.

Acknowledgments

The authors gratefully acknowledge Professor Z Chang and his group members for providing the laser-beam and Dr C Fehrenbach for assistance with the ion beam. This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy.

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