Electron re-scattering from aligned linear molecules - tampa

142 Pages·2010·2.4 MB·English

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