Electron Re-scattering from Aligned Molecules using the R-matrix Method Alex Harvey A thesis submitted to University College London for the degree of Doctor of Philosophy Department of Physics and Astronomy University College London March 8, 2010 I, Alex Harvey, con(cid:12)rm that the work presented in this thesis is my own. Where infor- mation has been derived from other sources, I con(cid:12)rm that this has been indicated in the thesis. 1 Abstract Electron re-scattering in a strong laser (cid:12)eld provides an important probe of molecular structure and processes, and can allow for time resolved study of nuclear and electronic dynamics at sub-femtosecond timescales with Angstrom spatial resolution. In such exper- iments a molecule is ionised in a strong, few cycle, laser (cid:12)eld. The changes in sign of the laser (cid:12)eld during the cycle can cause the electron to re-collide with its parent molecular ion. Under these circumstances the electron can either recombine leading to high har- monic generation, or it can be re-scattered. This scattering can be thought of as electron self-di(cid:11)raction and the process has the potential to act as a detailed probe of the target molecule. It is usual for such experiments to be performed on aligned molecules, as the dynamics of the ionisation and re-collision changes with alignment. This introduces ex- tra physics compared to the standard gas-phase, electron-molecule scattering problem. It is important for the understanding and analysis of such experiments to have a physically soundtheoreticalmodelofre-scatteringwhichiscapableoftreatingquantummechanically the complicated scattering dynamics of an electron-molecular ion collision. This thesis ex- plorestheuseofsophisticatedab initio quantummechanicaltechniquestomodelthispart of the re-scattering process. Previous theoretical models of the re-collision problem have thus greatly simpli(cid:12)ed this aspect of the problem. An introduction to attosecond physics, and a review of the relevant scattering and R-matrix theory is given. A simple preliminary model not including molecular alignment isdescribedformoleculesofexperimentalinterest. Thentheformalismforscatteringfrom aligned linear molecules is presented. For linear molecules consisting of more than two 2 atoms we use the polyatomic R-matrix codes. However the polyatomic version of the code onlyusesAbelianpointgroupswhichmeansthatcalculationsonsymmetricorasymmetric linear molecules are performed using the D or C point groups respectively. A further 2h 2v step is required, involving the reconstruction of T-matrices into the linear molecule sym- metry groups D1h or C1v. The formalism for this is also presented. Finally di(cid:11)erential and integral cross sections are presented for re-scattering for H and CO . 2 2 3 Contents 1 Introduction 14 1.1 A brief history of time resolution . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Attosecond Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.1 Molecular alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.2.2 Theoretical description of strong (cid:12)eld phenomena . . . . . . . . . . . 21 1.3 The object of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 The rest of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 The R-matrix Method: Theory and Implementation 26 2.1 An introduction to scattering theory . . . . . . . . . . . . . . . . . . . . . . 26 2.1.1 Single Channel Scattering . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.2 Multi-channel scattering . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.3 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 Fixed Nuclei R-matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Constructing the wave functions . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.1 The target wave functions . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.2 Continuum wave functions. . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.3 Outer-region wave functions . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 The UK R-matrix codes structure . . . . . . . . . . . . . . . . . . . . . . . 53 2.4.1 Diatomic version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 CONTENTS 2.4.2 Polyatomic version . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Preliminary Model: H and CO 64 2 2 3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Molecular hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Carbon dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Molecular Orientation Theory 72 4.1 Neutral Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1.1 Averaging over di(cid:11)erent alignments. . . . . . . . . . . . . . . . . . . 78 4.2 Molecular Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3 Linear molecules using D symmetry . . . . . . . . . . . . . . . . . . . . . 83 2h 4.3.1 Derivation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.2 Derivation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 ALIGN and TMATMAP 89 5.1 ALIGN:Aprogramtocalculatescatteringobservablesfromalignedmolecules 89 5.1.1 The angular functions and coupling coe(cid:14)cients . . . . . . . . . . . . 90 5.1.2 Calculating scattering observables . . . . . . . . . . . . . . . . . . . 90 5.2 TMATMAP: A program to transform from D2h to D1h . . . . . . . . . . . 92 6 Scattering from aligned molecules 94 6.1 Molecular hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Carbon dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 Rotational de-excitation in electron{HD+ collisions 107 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.2 ANR Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.3 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.1 Constructing the target . . . . . . . . . . . . . . . . . . . . . . . . . 110 5 CONTENTS 7.3.2 The inner region calculation . . . . . . . . . . . . . . . . . . . . . . . 111 7.3.3 The outer region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.3.4 The ANR approximation . . . . . . . . . . . . . . . . . . . . . . . . 111 7.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8 Conclusions 115 8.1 Summary of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.1.1 Theory and code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 8.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A Conventions used in this thesis 120 A.1 Spherical harmonic de(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.2 Rotation matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.3 Normalisation conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B Code Documentation 122 B.1 ALIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B.2 TMATMAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Bibliography 131 6 List of Figures 1.1 The Horse in Motion by Eadweard Muybridge. . . . . . . . . . . . . . . . . 15 1.2 Left(cid:12)gure: BullethittinganappleH.Edgerton, right(cid:12)gure: Milkdrophit- tingaliquidsurface. Examplesofearlypump-probeexperiments. (cid:13)c Harold & Esther Edgerton Foundation, 2002. . . . . . . . . . . . . . . . . . . . . . 15 1.3 A history of laser pulse duration [1]. . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Time scales of atomic and molecular processes. . . . . . . . . . . . . . . . . 18 1.5 The various processes in intense laser-atom/molecule experiments [2]. . . . 19 1.6 Two axis molecular alignment, showing strong revivals at coincident times forbothaxes[3]. Inducedalignmentismeasuredbytheexpectationvalueof the squared direction cosine matrix element, h(cid:8)2 i, between the lab frame Fg axis, F 2 (X;Y;Z), and molecular frame axis g 2 (a;b;c). . . . . . . . . . . 22 2.1 The R-matrix sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Flow diagram of the target state calculation: Diatomic codes . . . . . . . . 55 2.3 Flow diagram of the inner-region wave functions calculation: Diatomic codes 57 2.4 Flow diagram of the outer-region calculation: Diatomic codes . . . . . . . . 59 2.5 Flow diagram of the target state calculation: Polyatomic code . . . . . . . . 61 2.6 Flow diagram of the inner-region wave functions calculation: Polyatomic code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.7 Flow diagram of the outer-region calculation: Polyatomic code . . . . . . . 63 7 LIST OF FIGURES 3.1 Totalcrosssectionasfunctionofenergyfortheelectron{H+ collisionprob- 2 lem: left hand (cid:12)gure 1(cid:6)+ total symmetry, right hand (cid:12)gure 1(cid:6)+ symmetry. 66 g u 3.2 Di(cid:11)erential cross sections for electron { H+ collisions for three electron 2 collision energies: left hand (cid:12)gure 1(cid:6)+ symmetry, right hand (cid:12)gure 1(cid:6)+ g u symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Total cross section as function of energy for the electron { CO+ collision 2 problem: upper (cid:12)gure 1(cid:6)+ total symmetry, lower (cid:12)gure 1(cid:6)+ symmetry. . . 69 g u 6.1 Total cross section as function of energy for the parallel aligned electron { H+ collision problem (Model I): left (cid:12)gure 1(cid:6)+ total symmetry, right (cid:12)gure 2 g 1(cid:6)+ symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 u 6.2 Model I: Di(cid:11)erential cross sections for parallel aligned electron { H+ colli- 2 sionsforsevenelectroncollisionenergies(Toprow: orientationallyaveraged case): left hand (cid:12)gures 1(cid:6)+ symmetry, right hand (cid:12)gure 1(cid:6)+ symmetry. . 96 g u 6.3 Model II: Di(cid:11)erential cross sections for parallel aligned electron { H+ colli- 2 sionsforsevenelectroncollisionenergies(Toprow: orientationallyaveraged case): left hand (cid:12)gures 1(cid:6)+ symmetry, right hand (cid:12)gure 1(cid:6)+ symmetry. . 97 g u 6.4 Model III: Di(cid:11)erential cross sections for parallel aligned electron { H+ colli- 2 sions for 4 electron collision energies: left hand (cid:12)gures 1(cid:6)+ symmetry, right g hand (cid:12)gure 1(cid:6)+ symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 u 6.5 Model III: Di(cid:11)erential cross sections (normalised to pure Coulomb scatter- ing) for parallel aligned electron { H+ collisions for seven electron collision 2 energies (Top row: orientationally averaged case): left hand (cid:12)gures 1(cid:6)+ g symmetry, right hand (cid:12)gure 1(cid:6)+ symmetry. Dimensionless units. . . . . . 99 u 6.6 Total cross section as function of alignment angle (cid:12) for the aligned electron { H+ collision problem at a range of energies:top (cid:12)gures Model I, bottom 2 (cid:12)gures Model II, left (cid:12)gures 1(cid:6)+ total symmetry, right 1(cid:6)+ symmetry. . . 100 g u 6.7 Model I: Total cross section as function of energy for the orientationally av- eraged electron { CO+ collision problem comparing original D and trans- 2 2h formed using TMATMAP D1h symmetries (Model I): left (cid:12)gure 1Ag and 1(cid:6)+ total symmetry, right (cid:12)gure 1B and 1(cid:6)+ symmetry. . . . . . . . . . 101 g 1u u 8 LIST OF FIGURES 6.8 Total cross section as function of alignment angle (cid:12) for the aligned electron {CO+ collisionproblematarangeofenergies: Top(cid:12)guresModelI,Bottom 2 Model II, left (cid:12)gure 1(cid:6)+ total symmetry, right 1(cid:6)+ symmetry. . . . . . . . 102 g u 6.9 Model I & Model II: Polar plots of the di(cid:11)erential cross section taken in the z(cid:0)x plane with (cid:12) = 30o for the aligned electron { CO+ collision problem, 2 top row 3 eV, middle row 6 eV, bottom row 9 eV: left (cid:12)gures 1(cid:6)+ total g symmetry, right 1(cid:6)+ symmetry. The scale is in units of a and gives the r u 0 values for the circular grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.10 ModelIII:Normaliseddi(cid:11)erentialcrosssectiontakeninthez(cid:0)xplanewith (cid:12) = 30o for the aligned electron { CO+ collision problem, backwards scat- 2 tering angles only, (cid:18) measured from z-axis. Left (cid:12)gure 1(cid:6)+ total symmetry, g right 1(cid:6)+ symmetry. Dimensionless units. . . . . . . . . . . . . . . . . . . 105 u 7.1 Integral cross section as function of energy for the electron { HD+ colli- sion problem: Left (cid:12)gure - Single symmetries. Right (cid:12)gure - Sum over symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.2 Rotationally inelastic integral cross sections for low energy electron { HD+ collisions for (cid:1)j = (cid:6)1;(cid:6)2 and (cid:6)3: Left (cid:12)gure - Excitation. Right (cid:12)gure - De-excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.3 Relative populations as a function of storage time measured at electron densities of 0.28 (low), 1.0 (medium), and 1:45 107 cm3 (high). The lines are the results of the model calculations using the theoretical SEC rate coe(cid:14)cients given in table 7.2 and multiplying them by a constant factor (cid:20). The thick solid (thin dotted) lines correspond to including (neglecting) SEC, while the thin dashed lines are obtained by including SEC at half weight, (cid:20) = 0:5 [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9