Imaging resonances in low-energy NO-He inelastic collisions
Sjoerd N. Vogels, Jolijn Onvlee, Simon Chefdeville, Ad van der Avoird, Gerrit C. Groenenboom, Sebastiaan Y.T. van de Meerakker
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Imaging resonances in low-energy NO-He inelasticcollisions
Sjoerd N. Vogels ∗ , Jolijn Onvlee ∗ , Simon Chefdeville, Ad van der Avoird,Gerrit C. Groenenboom ∗∗ , Sebastiaan Y.T. van de Meerakker ∗∗ Radboud University, Institute for Molecules and MaterialsHeijendaalseweg 135, 6525 AJ Nijmegen, the Netherlands ∗ Who contributed equally to this work; ∗∗ To whom correspondence should be addressed;E-mail: [email protected], [email protected]
In molecular collisions, resonances occur at specific energies where the collid-ing particles temporarily form quasi-bound complexes, resulting in rapid vari-ations in the energy dependence of scattering cross sections. Experimentally, ithas proven challenging to observe such scattering resonances, especially in dif-ferential cross sections. We report the observation of resonance fingerprints inthe state-to-state differential cross sections for inelastic NO-He collisions in the13 to 19 cm − energy range with 0.3 cm − resolution. The observed structureswere in excellent agreement with quantum scattering calculations. They wereanalyzed by separating the resonance contributions to the differential crosssections from the background through a partitioning of the multichannel scat-tering matrix. This revealed the partial wave composition of the resonances,and their evolution during the collision. ℓ ( ).Unraveling the contribution of each individual partial wave to a collision cross section wouldprovide the ultimate information that can be retrieved from any collision event. Experimentally,however, it is impossible to select a single partial wave from the pre-collision conditions, and tostudy how the interaction transforms it into post-collision properties. The number of contribut-ing partial waves depends on the de Broglie wavelengths of the particles; observable quantitiessuch as scattering cross sections therefore necessarily represent the quantum mechanical su-perposition of many partial waves ( ). Classically, this can be compared to the unavoidableaveraging over all possible impact parameters of the two colliding particles ( ).At low temperatures, special conditions exist where a single partial wave can dominate thescattering process, mitigating this fundamental obstacle. When the collision energy is reso-nant with a quasi-bound state supported by the interaction potential, the incident particles can2emporarily form a long-lived complex. At these energies, a resonant partial wave ℓ res maydominate over all other partial waves, and there will be a strong enhancement of the scatteringcross section. For atom-molecule collisions, these so-called scattering resonances may be re-garded — in a simplified picture — either as the orbiting of the atom around the molecule (ashape or orbiting resonance), or as the transient excitation of the molecule to a state of higherenergy (a Feshbach resonance). After some time, however, the complex falls apart and decaysback into a separate atom and molecule ( ).Scattering resonances are the most global and sensitive probes of molecular interaction po-tentials. They depend on both the long range attractive and the short range repulsive part ofthe potential, as well as on the van der Waals well; they are not only sensitive to the shape ofthe well — as are the spectra of molecular complexes — but also to the depth of the well rela-tive to the dissociation limit. Measurements of the resonance position and width in the integralcross section (ICS) probe the energy and lifetime of the quasi-bound state from which the res-onance originates. Such observations may thus be regarded as a type of collision spectroscopy.Still, they do not yield information on the partial wave composition of the resonance. Moreinformation on the collision dynamics is inferred from the differential cross section (DCS) atthe resonance energy. The structured DCS represents the partial wave fingerprint of the colli-sion process, and offers at the resonance the opportunity to experimentally probe the relationbetween incoming, resonant, and outgoing partial waves. This gives detailed insights into themost fundamental question in molecular collision dynamics: how does the interaction potentialtransform the reagents into the collision products?Whereas scattering resonances are well-known in electron, neutron and ultracold atom scat-tering, the observation of scattering resonances in molecular systems has been a quest fordecades. Experimentally, it has proven extremely challenging to reach the required low col-lision energies and high energy resolution to observe and characterize partial wave resonances.3n crossed beam experiments, signatures of scattering resonances have been observed in reac-tive systems with a low excitation barrier, such as the benchmark F + H and F + HD reactions,using the powerful Rydberg tagging technique to record the angular distribution of the H or Dproducts ( ). In these systems, the resonance is associated with the formation of transientlystable transition-state structures. High-resolution anion photodetachment spectroscopy in com-bination with accurate calculations has recently allowed the observation and characterizationof previously unresolved reactive scattering resonances in this system ( ). Using a mergedbeam approach, resonances have recently also been observed in ICSs for Penning ionizationprocesses at collision temperatures in the mK regime (
12, 13 ). Even more recently, resonanceswere observed in inelastic scattering processes at energies near the thermodynamic thresholdfor rotational excitation of the molecule. Using cryogenically cooled molecular species suchas CO and O , partial wave resonances were observed in the state-to-state ICSs for inelasticcollisions with He and H target beams at energies down to 4 K ( ).The measurement of DCSs at scattering resonances remains a largely unexplored frontier, inparticular for species other than H and D atoms. At low collision energies, the recoil velocitiesof the scattered molecules are extremely small, and it has proven a daunting task to obtain theangular resolution required to resolve any deflection structure. Here, we report the measurementof DCSs at partial wave resonances for inelastic collisions between fully state-selected NOradicals [ X Π / , v = 0 , j = 1 / , f ( ), referred to hereafter as (1/2 f )] and He atoms in acrossed beam experiment. We combined Stark deceleration and velocity map imaging to probescattering resonances in the state-to-state and parity-resolved DCSs at energies between 13 and19 cm − , with a spectroscopic energy resolution of 0.3 cm − . The high resolution affordedby the Stark decelerator allowed us to observe structure in the very small velocity mappedscattering images, directly reflecting the DCSs. Distinct variations in the DCSs were observedas the collision energy was tuned over the resonances. At off-resonance energies, the DCSs were4ominated by quantum diffraction oscillations, whereas additional strong forward and backwardscattering was found at the resonance energies. We developed a theoretical approach similar toFeshbach-Fano partitioning (
18, 19 ) to disentangle the resonance and background contributionsto the DCSs, and we directly revealed the incoming and outgoing waves that characterize theresonances and the background.Collisions between NO and rare gas atoms represent a seminal class of systems in rota-tional energy transfer, since scattering of open-shell radical species such as NO and OH playsan important role in gas-phase chemical kinetics, combustion, and astrochemistry. Collisionsinvolving these radicals are governed by multiple potential energy surfaces (PESs), parity se-lection and propensity rules that are foreign to molecules such as CO and O ( ). We foundexcellent agreement with the DCSs predicted by ab initio quantum mechanical close-coupling(QM CC) scattering calculations based on accurate NO-He PESs.We used a crossed molecular beam apparatus with a 45 ◦ crossing angle ( ). A packet ofvelocity-controlled NO (1/2 f ) radicals, with a velocity spread of 2.1 m/s ( ) and an angularspread of 1.5 mrad, was produced using a 2.6-m long Stark decelerator. The beam of He atomswith a mean velocity between 400 m/s and 480 m/s, a velocity spread of 4.3 m/s and an angularspread of 4.8 mrad was produced by cooling an Even-Lavie valve to cryogenic temperatures. Bytuning the NO velocity between 350 m/s and 460 m/s with the Stark decelerator, the collisionenergy was varied between 13 cm − and 19 cm − with an energy resolution of 0.3 cm − . Thescattered NO radicals were state-selectively detected using a two-color laser ionization schemeand velocity mapped on a two-dimensional detector.We studied inelastic collisions that excite the NO (1/2 f ) radicals into either the (3/2 e ) orthe (5/2 f ) state. The (5/2 f ) channel opens within the experimentally accessible energy range at13.4 cm − , and we measured the threshold behavior in the ICS for this channel to calibrate boththe collision energy and energy resolution (shown in Fig. 1A). We observed a plateau just above5hreshold and a clear peak around 18 cm − , which were well reproduced by the theoretical ICSconvoluted with the experimental resolution of 0.3 cm − ( ). These features were attributed toa narrow (labeled II) and a broader resonance (labeled III) in the theoretical ICS. The theoreticalICS for the (3/2 e ) state (shown in Fig. 1B for comparison) showed a clear resonance (labeled I)at a slightly lower energy than resonance II.For both inelastic channels, scattering images were measured at selected energies (Fig. 2).Depending on the energy and the final state probed, the diameter of the low collision energyimages ranged from only a few m/s to about 60 m/s. Note that the diameter of the (5/2 f )image approached zero as the energy approached the thermodynamic threshold of the channel,effectively imaging the center of mass velocity of the NO-He pair. Despite their small sizes,distinct structure in the images could clearly be discerned. At higher energies, as illustrated bythe additional images probed at 45.0 cm − , the DCSs of both channels feature a rich diffractionpattern that extends from forward to backward scattering. Each diffraction peak in the DCStransformed into a stripe in the image due to the detection method employed in the experiment.At low collision energies the number of diffraction peaks was reduced, and an additional patternarose in the vicinity of the resonances. As the energy was varied in small energy steps over theresonances, a strong variation in the angular distribution featuring pronounced forward andbackward scattering was observed.To compare our findings with theoretical predictions, we simulated for each energy the im-age expected from the kinematics of the experiment and the DCS predicted by high-level QMCC calculations ( ). Both the experimental and simulated images were then analyzed to yieldthe angular scattering distribution, reflecting the DCS convoluted with the experimental energyand angular resolution ( ). In general, excellent agreement was found between the experimen-tal and simulated scattering distributions, although at some energies the relative intensities ofthe observed features differed from the simulated intensities ( ).6o interpret our results, we first analyzed the partial wave composition of the scattering crosssections, as well as the scattering wavefunctions ( ). We found that for the energies probed,the entrance channel is governed by partial waves up to ℓ in =
8. At the resonance energies,however, a single resonant partial wave ℓ res becomes involved in a quasi-bound state, whichcauses the ICS to rise significantly above the background. We found that resonances I and II areassociated with a quasi-bound state of the He-NO(5/2 f ) complex at 14.7 cm − dominated by ℓ res = 5 , whereas resonance III originates from a He-NO(5/2 f ) quasi-bound state at 17.9 cm − governed by ℓ res = 6 ( ).For an inelastic process, the partial wave quantum number ℓ is not conserved throughoutthe collision. The anisotropic interaction potential couples rotational states of the NO moleculewith different quantum numbers j and waves with different values of ℓ , and determines how theincoming waves ℓ in are transformed into the outgoing waves ℓ out during the collision. At theresonance energies, the quasi-bound He-NO(5/2 f ) complexes with the well-defined values of ℓ res = 5 (resonances I and II) or 6 (resonance III) decomposed to form an NO radical in eitherthe (3/2 e ) state (resonance I) or the (5/2 f ) state (resonances II and III). When the resonant(5/2 f ) state decays into the lower (3/2 e ) state, as for resonance I, not only j but also ℓ changesand ℓ out ranges from 3 to 7 ( ). When the final NO state is the same as the resonant state, asfor resonances II and III, the dominant value of ℓ out = 5 or 6 is the same as ℓ res . At energiesjust above the threshold of the (5/2 f ) inelastic channel, where almost all kinetic energy hasbeen transferred into rotational energy of the NO molecule, the (5/2 f ) state can only decay with ℓ out = 0 ( s -wave), ℓ out = 1 ( p -wave) and ℓ out = 2 ( d -wave). The observed DCS at 13.8 cm − was dominated by p -wave scattering with small contributions from s - and d -waves ( ).At the resonance energies, the DCSs contain unique information on the partial wave dy-namics of the collision process, i.e., the relation between ℓ in , ℓ res , and ℓ out . If the scatter-ing were purely determined by a resonance without any background, the scattering matrix S
24, 25 ). However, in most cases and also in our experiments, the observed ICSs andDCSs result from an interference between resonance and background contributions. We dis-entangled these contributions for each of the resonances I, II, and III by applying a theoreticalanalysis similar to Feshbach-Fano partitioning (
18, 19 ). We wrote the energy-dependent multi-channel scattering matrix as ( ) S ( E ) = S bg ( E ) U res ( E ) (1)where the background contribution S bg ( E ) is a slowly varying function of the collision energy E and the resonance contribution is given by the Breit-Wigner formula U res ( E ) = I − i A E − E res + i Γ (2) I is the unit matrix, E res is the energy of the resonance, Γ its width (or inverse lifetime), and thecomplex-valued matrix elements A αβ = a α a ∗ β contain the partial widths a α obeying the relation P α | a α | = Γ . The idea associated with the Breit-Wigner formula is that in the complex energyplane, where the bound states correspond to poles of the S -matrix on the negative real energyaxis, resonances are represented by poles below the positive real axis at positions E res − i Γ . Byanalyzing the energy dependence of the S -matrix elements in the range of each resonance withan algorithm described in the Supplementary Material ( ), we could determine the parame-ters E res , Γ , and a α . Then, we separated the resonance contributions to the scattering matrix S ( E ) from the background and applied the usual expressions ( ) to compute the ICS and DCSfrom the S -matrix, with or without resonance contributions. Figure 3 shows the results for thefinal (5/2 f ) state of NO, where the effects are most pronounced. The upper panel in this fig-ure demonstrates that the peaks in the ICS corresponding to both resonances II and III vanishwhen we only included the background contribution. The lower part of this figure illustrates8he effect of the resonances on the DCS at energies close to these resonances. The backgroundcontributions (dashed lines) show the usual pattern of diffraction oscillations, which are mostpronounced for small scattering angles and decrease in amplitude for larger angles. The effectof the resonance contributions is substantial; they lead to additional strong scattering near theforward and backward directions. Figure 3 also shows a comparison of measured and simulatedimages at 14.8, 17.1, and 18.2 cm − . The simulated images were based on DCSs calculatedin this energy range, with or without resonance contributions, by taking into account the ex-perimental energy resolution of 0.3 cm − . Clearly, the experimental images show much betteragreement with the simulations when the full DCS is taken into account.Our theoretical analysis demonstrated that the resonances strongly affect the nature of theDCSs, and allowed us to disentangle normal diffraction oscillations from the resonance finger-prints in the DCSs. The DCSs measured for collision energies in the range of the resonancesagreed very well with the DCSs obtained from the ab initio calculations, but only when the con-tributions from the resonances were fully included. This directly confirmed that our experimentindeed images the resonance fingerprints in the DCS.Our joint experimental and theoretical study showed that scattering resonances in state-to-state cross sections can now be probed with spectroscopic resolution, even for benchmark andchemically relevant systems that involve open-shell species such as NO. DCSs measured atthe resonance energies, combined with a theoretical analysis, provided detailed information onthe multichannel scattering process and explicitly revealed the effects of the resonances. Thetheoretical method developed to separate the resonant contributions to the ICSs and DCSs fromthe background will also be applicable to other systems where scattering resonances occur.9 eferences and Notes
1. The partial wave quantum number ℓ is the quantum mechanical analogue of the classicalangular momentum L = µvb , where µ is the reduced mass, v is the relative velocity of thecolliding particles, and b is the impact parameter defined as the distance of closest approachin the absence of any interaction.2. Single partial wave collisions are only found at temperatures approaching zero kelvin,where the scattering is governed by the wave with the lowest possible value for l . Colli-sions between ultracold atoms and molecules near quantum degeneracy, for instance, areexclusively governed by partial waves with l = 0 (s-wave) or l = 1 (p-wave).3. D. W. Chandler, J. Chem. Phys. , 110901 (2010).4. R. T. Skodje, et al. , Phys. Rev. Lett. , 1206 (2000).5. W. Shiu, J. J. Lin, K. Liu, Phys. Rev. Lett. , 103201 (2004).6. M. Qiu, et al. , Science , 1440 (2006).7. Z. Ren, et al. , Proc. Natl. Acad. Sci. U.S.A. , 12662 (2008).8. W. Dong, et al. , Science , 1501 (2010).9. T. Wang, et al. , Science , 1499 (2013).10. T. Yang, et al. , Science , 60 (2015).11. J. B. Kim, et al. , Science , 510 (2015).12. A. B. Henson, S. Gersten, Y. Shagam, J. Narevicius, E. Narevicius,
Science , 234(2012). 103. E. Lavert-Ofir, et al. , Nat. Chem. , 332 (2014).14. S. Chefdeville, et al. , Phys. Rev. Lett. , 023201 (2012).15. S. Chefdeville, et al. , Science , 1094 (2013).16. A. Bergeat, J. Onvlee, C. Naulin, A. van der Avoird, M. Costes,
Nat. Chem. , 349 (2015).17. The labels X Π / , v , and j indicate the electronic state, the vibrational state and rota-tional state of the NO radical, respectively. Each rotational state of NO possesses twonear-degenerate Λ -doublet components, with symmetry labels e (lower component) and f (upper component), that refer to the total parity of the electronic wavefunction, exclusiveof rotation.18. U. Fano, Phys. Rev. , 1866 (1961).19. V. Brems, T. Beyer, B. M. Nestmann, H.-D. Meyer, L. S. Cederbaum,
J. Chem. Phys. ,10635 (2002).20. M. Kirste, et al. , Science , 1060 (2012).21. Materials and methods are available as supplementary material on
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Online.22. All spreads in this manuscript refer to widths ( σ ) of a fitted Gaussian distribution.23. The calibration of the mean experimental collision energy has an uncertainty of ± − . Small differences between experimental and simulated images are foundin particular at energies where the DCS varies rapidly as a function of collision energy. Inaddition, the experiment seemed to systematically underestimate the scattering intensity atforward scattering.24. W. Erlewein, M. von Seggern, J. P. Toennies, Z. f¨ur Physik , 35 (1968).115. S. H. Sheen, J. G. Skofronick, C. R. Mueller,
Int. J. Quantum Chem. , 817 (1975).26. J. R. Taylor, Scattering theory: the quantum theory of nonrelativistic collisions (Chapter20) (Wiley, New York, 1972).27. W. A. Lester, Jr.,
The N Coupled-Channel Problem (Plenum Press, New York, 1976), pp.1–32.28. J. Onvlee, S. N. Vogels, A. von Zastrow, D. H. Parker, S. Y. T. van de Meerakker,
Phys.Chem. Chem. Phys. , 15768 (2014).29. B. Yan, et al. , Rev. Sci. Instrum. , (2013).30. U. Even, Advances in Chemistry , 636042 (2014).31. D. Townsend, M. P. Minitti, A. G. Suits,
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Acknowledgement
This work is part of the research programme of the Foundation forFundamental Research on Matter (FOM), which is supported financially by the NetherlandsOrganization for Scientific Research (NWO). S.Y.T.v.d.M. acknowledges further support fromNWO via a VIDI and a TOP grant, and from the European Research Council via a Starting12rant. We thank Andr´e van Roij, Chris Berkhout, Niek Janssen, and Peter Claus for experttechnical support. We thank Jacek Kłos for providing us with his NO-He PESs. The authorsdeclare no competing financial interests. Supplementary Material accompanies this paper.
Supplementary Materials
Materials and MethodsFigs. S1 to S10Table S1References (28-36)Movies S1-S2 13
12 16 200126 Collision energy (cm -1 ) I C S ( Å ) j = 1/23/25/2 efefef II IIII I C S ( a . u . ) AB Figure 1:14 % 100 % -1 ) I C S ( Å ) j = 1/23/25/2 efefef
14 16 18Collision energy (cm -1 )2001248 j = 1/23/25/2 efefef AB C DE F G H a bc d e f g h A (cm -1 )E coll Exp Sim100 m/sABCDEFGH
150 m/s Exp Sim75 m/s (cm -1 )E coll abcdefgh 150 m/s I C S ( Å ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) I n t . ( a . u . ) θ (degrees)0 90 180 θ (degrees)01 I n t . ( a . u . ) ExpSim ExpSim
44 46 4644 I n t . ( a . u . ) I II
III
Figure 2:15 % 100 % 0 90 18001201012
14 16 18 20 22 24 2604812 II III D C S ( Å / s r) D C S ( Å / s r) D C S ( Å / s r) θ (degrees) I C S ( Å ) Collision energy (cm -1 ) ABC
A B C
Exp Sim75 m/s Sim* SS bg Figure 3:16igure Captions:Figure 1: Collision energy dependence of the integral cross section for rotational excitation ofNO radicals by He atoms. (A) Comparison between measured (data points with error bars) andcalculated (solid curve) state-to-state inelastic scattering cross sections for excitation into the(5/2 f ) state. Experimental data in a.u., arbitrary units. Each data point is averaged over 1000laser shots with the He and NO beams overlapping (collision signal), and 1000 laser shots withthe NO beam only (reference signal). Vertical error bars represent statistical uncertainties at95% of the confidence interval. The calculated cross section was convoluted with the experi-mental energy resolution of 0.3 cm − . (B) Calculated state-to-state integral cross sections forexcitation into the (3/2 e ) state (green curve) and (5/2 f ) state (red curve). (Inset) Schematicenergy level diagram and inelastic excitation scheme of NO.Figure 2: Experimental (Exp) and simulated (Sim) ion images at selected collision energies asindicated in the top panels. Left panels: (1/2 f ) → (3/2 e ) inelastic collisions. Right panels:(1/2 f ) → (5/2 f ) inelastic collisions. The images are presented such that the relative velocityvector is oriented horizontally, with the forward direction on the right side of the image. Smallsegments of the images around forward scattering are masked due to imperfect state selectionof the NO packet. The angular scattering distributions as derived from the experimental (bluecurves) and simulated (red curves) images are shown for each channel and collision energy.Figure 3: Effect of resonances II and III on the cross sections for inelastic (1 / f ) → (5 / f ) NO-He scattering. Integral cross sections are shown above, differential below. Solid linesrepresent the complete theoretical ICSs and DCSs, dashed lines the cross sections obtained17hen only the scattering matrix S bg in Eq. (1) is included for resonances I, II, and III. Thelower panels show the measured (Exp) and simulated images based on either the completeDCSs (Sim) or the DCSs computed with the scattering matrix S bg only (Sim*) for collisionenergies of (A) 14.8 cm − , (B) 17.1 cm − , and (C) 18.2 cm − . Fig. S10 shows resonance I inthe (1 / f ) → (3 / e ))