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Yang F, Zhong Y, Diao H, Ge X, Zheng Y, Zeng Z, Xu Z. Resonance absorption of the inner shell during high-order harmonic generation. OPTICS EXPRESS 2022; 30:6577-6583. [PMID: 35299439 DOI: 10.1364/oe.452148] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
In this work, we report the observation of resonance absorption of the inner shell during the high-order harmonic generation (HHG) from xenon (Xe) and krypton (Kr). The absorption peaks show a periodic variation with the change of carrier-envelope phase of driving laser pulses and the delay of two-color laser field, which indicates the absorption peaks come from the collective multielectron effects during the HHG. With the increase of gas pressure, the depth of absorption peak will continue to increase, while due to the phase matching effect, there will be an optimal pressure for the intensity of harmonic signal. Our experimental results pave the way to uncover the physical mechanism of the collective multielectron effects involving inner-shell electrons in the HHG process.
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Ross AD, Hait D, Scutelnic V, Haugen EA, Ridente E, Balkew MB, Neumark DM, Head-Gordon M, Leone SR. Jahn-Teller Distortion and Dissociation of CCl 4+ by Transient X-ray Spectroscopy Simultaneously at the Carbon K- and Chlorine L-Edge. Chem Sci 2022; 13:9310-9320. [PMID: 36093014 PMCID: PMC9384822 DOI: 10.1039/d2sc02402k] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2022] [Accepted: 07/18/2022] [Indexed: 11/29/2022] Open
Abstract
X-ray Transient Absorption Spectroscopy (XTAS) and theoretical calculations are used to study CCl4+ prepared by 800 nm strong-field ionization. XTAS simultaneously probes atoms at the carbon K-edge (280–300 eV) and chlorine L-edge (195–220 eV). Comparison of experiment to X-ray spectra computed by orbital-optimized density functional theory (OO-DFT) indicates that after ionization, CCl4+ undergoes symmetry breaking driven by Jahn–Teller distortion away from the initial tetrahedral structure (Td) in 6 ± 2 fs. The resultant symmetry-broken covalently bonded form subsequently separates to a noncovalently bound complex between CCl3+ and Cl over 90 ± 10 fs, which is again predicted by theory. Finally, after more than 800 fs, L-edge signals for atomic Cl are observed, indicating dissociation to free CCl3+ and Cl. The results for Jahn–Teller distortion to the symmetry-broken form of CCl4+ and formation of the Cl–CCl+3 complex characterize previously unobserved new species along the route to dissociation. Dynamics of CCl4+ prepared by 800 nm strong-field ionization, as studied with X-ray transient absorption spectroscopy (XTAS) and quantum chemical calculations.![]()
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Affiliation(s)
- Andrew D Ross
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Diptarka Hait
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Valeriu Scutelnic
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Eric A Haugen
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Enrico Ridente
- Department of Chemistry, University of California Berkeley 94720 CA USA
| | - Mikias B Balkew
- School of Physics, Georgia Institute of Technology Atlanta 30332 GA USA
- Department of Chemistry, University of California Berkeley 94720 CA USA
| | - Daniel M Neumark
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Martin Head-Gordon
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
| | - Stephen R Leone
- Department of Chemistry, University of California Berkeley 94720 CA USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley 94720 CA USA
- Department of Physics, University of California Berkeley 94720 CA USA
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3
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Pandey G, Dey D, Tiwari AK. Controlling the Ultrafast Dynamics of HD + by the Carrier-Envelope Phases of an Ultrashort Laser Pulse: A Quasi-Classical Dynamics Study. J Phys Chem A 2020; 124:9710-9720. [PMID: 33191740 DOI: 10.1021/acs.jpca.0c08979] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
A theoretical study on the coupled electron-nuclear dynamics of HD+ molecular ions under ultrashort, intense laser pulses is performed by employing a well-established quasi-classical model. The influence of the laser carrier-envelope phase on various channel (H + D+, D + H+, and H+ + D+) probabilities is investigated at different laser field intensities. The carrier-envelope phase is found to govern the dissociation (H + D+ and D + H+) and Coulomb explosion (H+ + D+) channel probabilities. The kinetic energy release distributions of the fragments are also found to be sensitive to the carrier-envelope phase of the laser pulse. Our results are in agreement with the previously reported quantum dynamics studies and experiments.
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Affiliation(s)
- Gaurav Pandey
- Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
| | - Diptesh Dey
- Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
| | - Ashwani K Tiwari
- Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
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Dey D, Tiwari AK. Controlling Chemical Reactions with Laser Pulses. ACS OMEGA 2020; 5:17857-17867. [PMID: 32743156 PMCID: PMC7391256 DOI: 10.1021/acsomega.0c02098] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Accepted: 07/01/2020] [Indexed: 05/14/2023]
Abstract
One of the holy grails of contemporary science is to understand and manipulate chemical reactions to obtain desired products preferentially. To achieve this goal, chemists traditionally choose the correct starting materials and reaction conditions, but it often lacks selectivity and efficiency. A promising alternative is to design laser control schemes and apply them to guide and control chemical reactions. This mini-review attempts to provide theoretical insight into the laser-induced control of chemical reactions by highlighting some recent achievements, discussing the present challenges, and shedding some light on future prospects.
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Zhang Q, Fukahori S, Ando T, Kanya R, Iwasaki A, Rathje T, Paulus GG, Yamanouchi K. Absolute carrier-envelope-phase dependences of single and double ionization of methanol in a near-IR few-cycle laser field. J Chem Phys 2020; 152:194304. [PMID: 33687232 DOI: 10.1063/5.0006485] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We investigate the carrier-envelope phase (CEP) dependences of the single and double ionization processes of methanol (CH3OH) in an intense near-IR few-cycle laser field (2.1 × 1014 W/cm2) by the asymmetry in the ejection direction of CH3 + for the non-hydrogen migration channels and CH2 + for the hydrogen migration channels created through the C-O bond breaking after the ionization. Based on the absolute CEP values at the laser-molecule interaction point, calibrated by the method using intense few-cycle circularly polarized laser pulses [Fukahori et al., Phys. Rev. A 95, 053410-1-053410-14 (2017)], we confirm that methanol cations are produced by tunnel ionization and methanol dications are produced by the recollisional double ionization. We obtain the phase offset for the double ionization accompanying no hydrogen migration to be 1.85π as the absolute CEP at which the extent of the asymmetry becomes maximum. We interpret the phase shift of 0.85π from the phase offset of 1.0π for the tunnel ionization, estimated by a tunnel ionization model incorporating the chemical bond asymmetry, as the corresponding time delay associated with the electron recollisional ionization. The positive phase shift of 0.13π for the single ionization in the non-hydrogen migration channel is interpreted as the additional time (165 as) with which a methanol cation can be excited electronically prior to the decomposition. The additional phase shift of 0.22π for the single ionization in the hydrogen migration channel is interpreted as the additional time (280 as) required for a methanol cation to be excited electronically leading to the hydrogen migration prior to the decomposition.
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Affiliation(s)
- Qiqi Zhang
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Shinichi Fukahori
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Toshiaki Ando
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Reika Kanya
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Atsushi Iwasaki
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Tim Rathje
- Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, Max-Wien-Platz 1, 07743 Jena, Germany
| | - Gerhard G Paulus
- Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, Max-Wien-Platz 1, 07743 Jena, Germany
| | - Kaoru Yamanouchi
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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6
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Dey D, Ray D, Tiwari AK. Controlling Electron Dynamics with Carrier-Envelope Phases of a Laser Pulse. J Phys Chem A 2019; 123:4702-4707. [PMID: 31074991 DOI: 10.1021/acs.jpca.9b02870] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
A theoretical study on the ionization dynamics of carbon atom irradiated with a few-cycle, intense laser field is performed within a quasiclassical model to get mechanistic insights into an earlier reported carrier-envelope phase dependency of ionization probabilities of an atom [ Phys. Rev. Lett. 2013, 110, 083602]. The carrier-envelope phase of the laser pulse is found to govern the overall dynamics, reflecting its importance in controlling electronic motion. To understand the origin of this effect, individual trajectories were analyzed at a particular laser intensity. We found that a variation in the carrier-envelope phase affects the angle of ejection of the electrons and subsequently the attainment of the desired final state.
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Affiliation(s)
- Diptesh Dey
- Indian Institute of Science Education and Research Kolkata , Mohanpur 741246 , India
| | - Dhiman Ray
- Indian Institute of Science Education and Research Kolkata , Mohanpur 741246 , India
| | - Ashwani K Tiwari
- Indian Institute of Science Education and Research Kolkata , Mohanpur 741246 , India
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7
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Das S, Sharma P, Vatsa RK. Gas phase ionisation of carbon disulfide clusters at terawatt laser intensity: Generation of singly and multiply charged atomic and molecular ions. J Photochem Photobiol A Chem 2019. [DOI: 10.1016/j.jphotochem.2018.11.035] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Zhang M, Xu G, Lu G, Song P. A theoretical study of vibrational effects on strong field ionization of CS2. JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2018. [DOI: 10.1142/s0219633618500529] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Tunneling ionization of vibrationally excited CS2 molecules in their ground electronic state is calculated using molecular orbital Ammosov–Delone–Krainov theory (MO-ADK) considering bond length-dependence and bond angle-dependence. The tunneling ionization rates and the corresponding electron density are calculated respectively for different initial states. A relationship between laser intensity and the molecular orientation angle is determined and compared with experimental results, showing excellent agreement. Our calculations show that the primary contribution of vibration effect to CS2 in tunneling ionization is due to the symmetric expansion mode.
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Affiliation(s)
- Meixia Zhang
- College of Physics, Liaoning University, Shenyang 110036, P. R. China
| | - Guangwen Xu
- College of Physics, Liaoning University, Shenyang 110036, P. R. China
| | - Guoying Lu
- College of Physics, Liaoning University, Shenyang 110036, P. R. China
| | - Peng Song
- College of Physics, Liaoning University, Shenyang 110036, P. R. China
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Kübel M, Burger C, Siemering R, Kling NG, Bergues B, Alnaser AS, Ben-Itzhak I, Moshammer R, de Vivie-Riedle R, Kling MF. Phase- and intensity-dependence of ultrafast dynamics in hydrocarbon molecules in few-cycle laser fields. Mol Phys 2017. [DOI: 10.1080/00268976.2017.1288935] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- M. Kübel
- Department of Physics, Ludwig-Maximilians-Universität , Garching, Germany
| | - C. Burger
- Department of Physics, Ludwig-Maximilians-Universität , Garching, Germany
- Laboratory of Attosecond Physics, Max Planck Institute of Quantum Optics , Garching, Germany
| | - R. Siemering
- Department of Chemistry and Biochemistry, Ludwig-Maximilians-Universität , Munich, Germany
| | - Nora G. Kling
- Department of Physics, Ludwig-Maximilians-Universität , Garching, Germany
| | - B. Bergues
- Department of Physics, Ludwig-Maximilians-Universität , Garching, Germany
- Laboratory of Attosecond Physics, Max Planck Institute of Quantum Optics , Garching, Germany
| | - A. S. Alnaser
- Department of Chemistry and Biochemistry, Ludwig-Maximilians-Universität , Munich, Germany
| | - I. Ben-Itzhak
- J.R. Macdonald Laboratory, Physics Department, Kansas-State University , Manhattan, KS, USA
| | - R. Moshammer
- Max Planck Institute of Nuclear Physics , Heidelberg, Germany
| | | | - M. F. Kling
- Department of Physics, Ludwig-Maximilians-Universität , Garching, Germany
- Laboratory of Attosecond Physics, Max Planck Institute of Quantum Optics , Garching, Germany
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10
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Mathur D, Dota K, Dey D, Tiwari AK, Dharmadhikari JA, Dharmadhikari AK, De S, Vasa P. Selective breaking of bonds in water with intense, 2-cycle, infrared laser pulses. J Chem Phys 2015; 143:244310. [PMID: 26723674 DOI: 10.1063/1.4938500] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
One of the holy grails of contemporary science has been to establish the possibility of preferentially breaking one of several bonds in a molecule. For instance, the two O-H bonds in water are equivalent: given sufficient energy, either one of them is equally likely to break. We report bond-selective molecular fragmentation upon application of intense, 2-cycle pulses of 800 nm laser light: we demonstrate up to three-fold enhancement for preferential bond breaking in isotopically substituted water (HOD). Our experimental observations are rationalized by means of ab initio computations of the potential energy surfaces of HOD, HOD(+), and HOD(2+) and explorations of the dissociation limits resulting from either O-H or O-D bond rupture. The observations we report present a formidable theoretical challenge that need to be taken up in order to gain insights into molecular dynamics, strong field physics, chemical physics, non-adiabatic processes, mass spectrometry, and time-dependent quantum chemistry.
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Affiliation(s)
- D Mathur
- Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India
| | - K Dota
- Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India
| | - D Dey
- Indian Institute of Science Education and Research Kolkata, Mohanpur 741 246, India
| | - A K Tiwari
- Indian Institute of Science Education and Research Kolkata, Mohanpur 741 246, India
| | - J A Dharmadhikari
- Centre for Atomic and Molecular Physics, Manipal University, Manipal 576 104, India
| | - A K Dharmadhikari
- Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India
| | - S De
- Saha Institute of Nuclear Physics, Bidhan Nagar, Kolkata 700 064, India
| | - P Vasa
- Department of Physics, Indian Institute of Technology Bombay, Mumbai 400 076, India
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11
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Xie X, Lötstedt E, Roither S, Schöffler M, Kartashov D, Midorikawa K, Baltuška A, Yamanouchi K, Kitzler M. Duration of an intense laser pulse can determine the breakage of multiple chemical bonds. Sci Rep 2015; 5:12877. [PMID: 26271602 PMCID: PMC4536518 DOI: 10.1038/srep12877] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2015] [Accepted: 07/13/2015] [Indexed: 11/29/2022] Open
Abstract
Control over the breakage of a certain chemical bond in a molecule by an ultrashort laser pulse has been considered for decades. With the availability of intense non-resonant laser fields it became possible to pre-determine femtosecond to picosecond molecular bond breakage dynamics by controlled distortions of the electronic molecular system on sub-femtosecond time scales using field-sensitive processes such as strong-field ionization or excitation. So far, all successful demonstrations in this area considered only fragmentation reactions, where only one bond is broken and the molecule is split into merely two moieties. Here, using ethylene (C2H4) as an example, we experimentally investigate whether complex fragmentation reactions that involve the breakage of more than one chemical bond can be influenced by parameters of an ultrashort intense laser pulse. We show that the dynamics of removing three electrons by strong-field ionization determines the ratio of fragmentation of the molecular trication into two respectively three moieties. We observe a relative increase of two-body fragmentations with the laser pulse duration by almost an order of magnitude. Supported by quantum chemical simulations we explain our experimental results by the interplay between the dynamics of electron removal and nuclear motion.
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Affiliation(s)
- Xinhua Xie
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
| | - Erik Lötstedt
- Laser Technology Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- RIKEN Center for Advanced Photonics, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Stefan Roither
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
| | - Markus Schöffler
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
| | - Daniil Kartashov
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
| | - Katsumi Midorikawa
- Laser Technology Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- RIKEN Center for Advanced Photonics, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Andrius Baltuška
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
| | - Kaoru Yamanouchi
- Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Markus Kitzler
- Photonics Institute, Vienna University of Technology, Gusshausstrasse 27, A-1040 Vienna, Austria, EU
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Dota K, Dharmadhikari AK, Dharmadhikari JA, Patra K, Tiwari AK, Mathur D. A search for the sulphur hexafluoride cation with intense, few cycle laser pulses. J Chem Phys 2014; 139:194302. [PMID: 24320321 DOI: 10.1063/1.4830222] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
It is well established that upon ionization of sulphur hexafluoride, the SF6(+) ion is never observed in mass spectra. Recent work with ultrashort intense laser pulses has offered indications that when strong optical field are used, the resulting "bond hardening" can induce changes in the potential energy surfaces of molecular cations such that molecular ions that are normally unstable may, indeed, become metastable enough to enable their detection by mass spectrometry. Do intense, ultrashort laser pulses permit formation of SF6(+)? We have utilized intense pulses of 5 fs, 11 fs, and 22 fs to explore this possibility. Our results are negative: no evidence is discovered for SF6(+). However, multiply charged sulphur and fluorine ions from highly charged SF6(q+) ions are observed that enable us to resolve the controversy regarding the kinetic energy release accompanying formation of F(+) fragment ions. Quantum chemical computations of field-distorted potential energy curves of SF6 and its molecular ion enable us to rationalize our non-observation of SF6(+). Our findings have implications for high harmonic generation from SF6 in the few-cycle regime.
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Affiliation(s)
- Krithika Dota
- Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400 005, India
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13
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Kling NG, Betsch KJ, Zohrabi M, Zeng S, Anis F, Ablikim U, Jochim B, Wang Z, Kübel M, Kling MF, Carnes KD, Esry BD, Ben-Itzhak I. Carrier-envelope phase control over pathway interference in strong-field dissociation of H2+. PHYSICAL REVIEW LETTERS 2013; 111:163004. [PMID: 24182264 DOI: 10.1103/physrevlett.111.163004] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2013] [Indexed: 06/02/2023]
Abstract
The dissociation of an H2+ molecular-ion beam by linearly polarized, carrier-envelope-phase-tagged 5 fs pulses at 4×10(14) W/cm2 with a central wavelength of 730 nm was studied using a coincidence 3D momentum imaging technique. Carrier-envelope-phase-dependent asymmetries in the emission direction of H+ fragments relative to the laser polarization were observed. These asymmetries are caused by interference of odd and even photon number pathways, where net zero-photon and one-photon interference predominantly contributes at H+ + H kinetic energy releases of 0.2-0.45 eV, and net two-photon and one-photon interference contributes at 1.65-1.9 eV. These measurements of the benchmark H2+ molecule offer the distinct advantage that they can be quantitatively compared with ab initio theory to confirm our understanding of strong-field coherent control via the carrier-envelope phase.
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Affiliation(s)
- Nora G Kling
- J. R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA
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