S-Matrix Analysis of Vibrational and Alignment Effects in Intense-Field Multiphoton Ionization of Molecules Arvid Requate Mar. 2007 Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften an der Fakult¨atf¨urPhysik der Universit¨atBielefeld Referenten: Prof. F. H. M. Faisal Ph.D., Fakult¨atf¨urPhysik, Universit¨atBielefeld Dr. A. Becker, Max-Planck-Institut f¨urPhysik komplexer Systeme, Dresden Vorwort Die vorliegende Dissertation wurde in der Arbeitsgruppe Theoretische Atom-, Molek¨ul- und Laserphysik unter der Leitung von Prof. F.H.M. Faisal Ph.D. an der Fakult¨at f¨urPhysik der Universit¨at Bielefeld begonnen und in der Arbeitsgruppe Nichtlin- eare Prozesse in starken Feldern unter der Leitung von A. Becker an dem Max- Planck-Institut f¨urPhysik komplexer Systeme (MPIPKS) fertig gestellt. Die quan- tenchemischen Berechnungen mit dem GAMESS Paket wurden auf den Rechnern der Fakult¨atf¨urPhysik der Universit¨at Bielefeld vorgenommen, w¨ahrenddie nu- merischen S-Matrix Rechnungen zum gr¨oßten Teil auf den Linux Rechnern des MPIPKS durchgef¨uhrtwurden. Teile der in Kapitel 4 dargestellten Ergebnisse wurden in der Publikation S-matrix theory of inelastic vibronic ionization of molecules in intense laser fields A. Requate, A. Becker und F. H. M. Faisal, Phys. Rev. A 73 (2006) 033406 dargestellt. Die Arbeit wurde in englischer Sprache verfasst, um die relative Sprach- barriere gering zu gestalten, da Englisch derzeit die Hauptsprache des Fachgebiets darstellt. Außer der in der Bibliographie angegebenen Literatur wurden folgende Programm- pakete zur Erstellung und Auswertung der numerischen Simulationen sowie der Dis- sertationsschrift verwendet: GNU Compiler Collection (GCC), GNU Scientific Li- brary (GSL), CERN Class Library for High Energy Physics (CLHEP), GAMESS, GaussSum, Python, Xmgrace sowie Pybliographic, Kile, Xfig und pdfeTeX ein- schließlich verschiedener Pakete, allen voran LATEX, KOMA-Script, amsmath, graphicx, hyperref und natbib. Der verwendete Fortran Programmcode zur Berechnung der generalisierten Bessel-Funktionen ist eine Entwicklung fr¨uherer Mitglieder der Ar- beitsgruppe ”Theoretische Atom-, Molek¨ul-und Laserphysik” an der Fakult¨atf¨ur Physik der Universit¨atBielefeld. i ii Contents Vorwort i 1. Introduction and Outline 1 2. Mechanisms of Strong Field Ionization 7 2.1. Single Ionization ............................. 7 2.1.1. Quantum Mechanical Description of the Bound System .... 8 2.1.2. Tunneling Ionization ....................... 10 2.1.3. Intense-Field Multiphoton Ionization .............. 14 2.1.4. Recollision ............................. 15 2.2. Double and Multiple Ionization ..................... 15 2.3. Molecular Ionization ........................... 19 3. Overview of Theoretical Methods 25 3.1. Time-dependent Methods ........................ 25 3.1.1. Virtual NPSF Lab ........................ 26 3.2. Floquet Methods ............................. 30 3.3. Intense-Field Many-Body S-Matrix Theory ............... 32 4. S-Matrix Theory of Inelastic Vibronic Ionization of Molecules in Intense Laser Fields 41 4.1. Quantum Mechanical Description of Molecules ............. 41 4.1.1. Born-Oppenheimer Approximation ............... 41 4.1.2. Franck-Condon Approximation ................. 43 4.2. Observation of Non-Franck-Condon Distributions in Molecular Ions Generated by Intense Laser Fields .................... 44 4.3. S-Matrix Formulation of the Transition Amplitude .......... 47 4.3.1. Transition Rate .......................... 48 4.3.2. Electronic Wavefunctions ..................... 54 4.3.3. Vibrational Wavefunctions .................... 56 4.3.4. Rate Equations and Transition Yields .............. 60 4.4. S-Matrix analysis of Non-Franck-Condon Distributions in Small Di- atomics .................................. 61 4.4.1. Comparison with Experimental Data .............. 61 4.4.2. Alignment and Polarization Effects ............... 65 4.4.3. Origin of the Shift to Lower Vibrational States ......... 69 iii Contents 4.4.4. Momentum Conservation ..................... 74 4.4.5. Application to HD and D2 .................... 77 4.5. Inelastic Vibronic Ionization of Other Molecules ............ 78 4.5.1. Other Diatomics: O2 and CO .................. 78 4.5.2. Extension to Polyatomic Molecules ............... 83 5. Nonsequential Double Ionization of Diatomic Molecules 87 5.1. Electron impact ionization in a laser field ................ 89 5.1.1. Characteristic spin correlated states ............... 89 5.1.2. Collision dynamics in the laser field ............... 93 5.1.3. Ionic Recoil Momentum in Laser Assisted Electron Impact Ion- ization ............................... 99 5.1.4. Alignment Dependence for Different Orbital Symmetries ... 102 5.1.5. Spin Effects ............................ 108 5.1.6. Relation to Experiment ...................... 109 5.2. Model Formula for Nonsequential Double Ionization of Molecules ... 111 5.2.1. Results for N2 and O2 ...................... 117 6. Conclusions and Outlook 121 A. Atomic Units 127 Acknowledgments 129 Bibliography 131 iv 1. Introduction and Outline In one of his three groundbreaking papers of 1905 Albert Einstein introduced the concept that light has a corpuscular character in the process of the Photoeffect [1]. In this process an electron bound to an atom is ionized into the continuum by absorption of a single photon from a light source. Experimentally it was found that it was the frequency and not the intensity of the light, which is the crucial parameter that decided whether the process would be possible or not. Einstein’s theory was complementary to the established way to describe light as an electromagnetic wave according to Maxwell’s laws. The success of both models in different areas of physics was the foundation of the wave-particle dualism of light as a figure of thought. In the course of her Ph.D. thesis [2, 3], which was finally published 1931, Maria Goeppert- Mayer theoretically predicted the possibility of the simultaneous absorption of two photons. The cross sections for this process are considerably smaller than the ones for single photon absorption. It was not until 1950, that the first experimental evidence for processes of this kind could be found by means of radio-frequency spectroscopy [4]. About ten years later, further experimental confirmation was obtained [5] in experiments with a maser source which was realized only a couple of years earlier [6]. With the advent of sources of laser light [7, 8] of ever increasing intensity, it became possible to study the absorption of multiple photons in elementary electronic transition process in atoms and molecules [9, 10]. The electric field strength that binds the electron in its ground state to the proton inside the Hydrogen atom is ~ e 9 V |E| = 2 = 5.14 × 10 , (1.1) 4π0a0 cm where e is the charge of an electron, a0 is the Bohr radius and 0 is the dielectric constant for vacuum. The only practicable way to achieve field strengths of this order of magnitude in a laboratory is in form of the temporally varying electric field of an electromagnetic wave. The corresponding magnetic field component is found to be less important for the dynamics for intensities that are lower than the relativistic threshold intensity of about 1019 W/cm2. At that intensity the average kinetic energy of an electron in the field starts to exceed to the electronic rest energy: 2 eE0λ > 2πmec , (1.2) with the wavelength λ = c/2πω; me denotes the mass of the electron and c the speed of light. The measure for the average kinetic energy of an elementary charge 1 Introduction and Outline in an electromagnetic wave with E-field E(t) = E0 sin(ωt + φ0) is the ponderomotive energy 2 2 e E0 Up = 2 . (1.3) 4meω 16 2 At the field intensity of Ia.u. = 3.51 × 10 W/cm the strength of the electric field of the laser is of the same size as the electric field in the field free hydrogen atom. 1 I = c E2 [in S.I.], and I = E2 [in a.u.] . (1.4) 0 2 0 0 0 0 Boosted by the development of Chirped Pulse Amplification [11–13] laser technology evolved at a fast pace up to the point that today intensities above 1021 W/cm2 can be focused on atomic and molecular systems [14, 15]. But already at intensities much below Ia.u., ionization processes occur, as the bound system is distorted nonlinearly by the field and mechanisms as tunneling and intense-field multiphoton ionization are quantum mechanically possible. The usage of pulsed laser beams also introduces another interesting advantage of the laser over static fields for the study of transition processes in atoms and molecules, that is ultrashort interaction times. Nowadays already few cycle pulses are experimentally controllable [16, 17]. Since one cycle of the laser field of 800 nm wavelength takes 2.67 fs, the interaction time with the Coulombic system is on the scale of femtoseconds (1 fs = 10−15 s). Taking into account that the peak intensity of the pulse is only achieved at the central oscillation of the pulse, effective interaction times on the attosecond time scale (1 as = 10−18 s) can be realized with lasers of higher frequency, e.g. in the XUV domain (a period of 14 nm radiation extends over T0 ≈ 46 as) [18]. Series (or trains) of XUV attosecond pulses have also been produced by emission of high-harmonic frequency radiation in electron recollision processes [19]. Today it is a matter of active research to generate isolated XUV attosecond pulses, for which theoretical concepts exist (see [16] for references). Given these ultra short interaction times, that nearly reach the domain of charac- teristic times of electronic