DEPARTMENT OF PHYSICS AND ASTRONOMY UNIVERSITY OF HEIDELBERG · rapid change in the photoelectron angular distribution, as a function of photon energy, and the relative yields of

DEPARTMENT OF PHYSICS AND ASTRONOMY

UNIVERSITY OF HEIDELBERG

A thesis submitted in partial fulfillment of the requirementsfor the degree of

Master of Science in Physicsby

Saijoscha Heckborn in Schwetzingen (Germany)

October 27, 2017

Photoelectrons in Molecular Fields: AnInvestigation of Shape Resonances and

Electron Retroaction using Coincident 3DMomentum Imaging Technique

Author:Saijoscha Heck

Max Planck Institut für Kernphysik, Heidelberg

Primary Advisor:Herr Dr., Priv.-Doz. RobertMoshammerMax Planck Institut für Kernphysik,

Heidelberg

Secondary Advisor:Herr Prof. Dr. Reinhard

DörnerInstitut für Kernphysik,

Universität Frankfurt

https://www.mpi-hd.mpg.de/pfeifer/page.php?id=25



https://www.atom.uni-frankfurt.de/

https://www.atom.uni-frankfurt.de/

iii

AbstractPhotoelectrons in Molecular Fields: An Investigation of Shape Resonances and Electron

Retroaction using Coincident 3D Momentum Imaging Technique

by Saijoscha Heck

This thesis deals with the investigation of molecular frame photoelectron emission direc-tions in valence photoionization processes of molecular Hydrogen/Deuterium (H2/D2)and Tetrafluoromethane (CF4). Experiments were carried out at beamline 10.0.1 of theAdvanced Light Source synchrotron in Berkeley, California. Photon energies were chosento be in the range of 18-22eV to study the low energy emission kinematics and dynam-ics of valence electrons and their subsequent interaction with the molecular field duringphotoionization. Using the COLd Target Recoil Ion Momentum Spectroscopy (COLTRIMS)technique, it was possible to obtain access to the molecular frame ofH2 and the recoil frameof CF4 and post-select specific polarization orientations, allowing for a highly differentialinvestigation of electron emission directions.In H2 and D2 a symmetry breaking of the molecular dissociation direction, resulting of aretroaction of the electrons Coulomb potential with the parent ion, is observed. For CF4 arapid change in the photoelectron angular distribution, as a function of photon energy, andthe relative yields of electrons towards and away from the CF+

3 fragment is found. Theseobservations are explained by a quantum mechanical description of the electron-moleculeinteraction, involving coherent coupling of two shape resonances with different symmetry,leading to the same final state.

v

ZusammenfassungPhotoelektronen in Molekülfeldern: Eine Untersuchung von Shape-Resonanzen und

Elektronen Retroaktion mittels Koinzidenter 3D Impuls Spektroskopie

von Saijoscha Heck

Diese Arbeit beschäftigt sich mit der Untersuchung von Elektron Emissionswinkeln in Pho-toionisations Prozessen von molekularem Wasserstoff/Deuterium (H2/D2) sowie Tetraflu-ormethan (CF4). Die Experimente wurden an der Beamline 10.0.1 des Synchrotrons "Ad-vanced Light Source" in Berkeley, Kalifornien ausgeführt. Es wurden Photonenenergienin einem Bereich von 18-22eV ausgewählt um Emmissions Kinematik und Dynamik vonValenzelektronen und deren anschließende Wechselwirkung mit dem Molekülfeld währendder Photoionisation zu untersuchen. Mit Hilfe der verwendeten COLTRIMS (COLd Tar-get Recoil Ion Momentum Spectroscopy) Messmethode war es möglich die Elektronen in-nerhalb des Koordinatensystem des Moleküls auszuwerten und bestimmte Ausrichtungender Photonen Polarisation relativ zur Moelkülsachse zu selektieren. Dies ermöglichte einehöchst differentielle Untersuchung der Elektronen Emissionsrichtungen.Sowohl in H2, wie auch in D2, wurde eine Symmetriebrechung entlang der molekularenDissoziationsachse beobachtet, dies ist ein Ergebnis der Wechselwirkung zwischen demCoulomb-Feld des Elektrons mit dem Mutter-Ion. InCF4 kann ein abrupter Richtungswech-sel in der Photoelektronen Emission, entlang oder entgegengesetzt des CF+

3 Fragments,durch eine kohärente Kopplung zweier Shape-Resonanzen unterschiedlicher Symmetrie,jedoch dem selben Endzustand, erklärt werden.

vii

Contents

Abstract iii

Zusammenfassung v

1 Introduction 1

2 Theoretical Background 32.1 Structure of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . 4Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 5Intersection of Potential Energy Curves . . . . . . . . . . . . . . . . . . 7

2.1.3 Description of Electronic and Molecular States . . . . . . . . . . . . . . 72.2 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Franck-Condon Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Experimental Methods 113.1 Synchrotron Radiation Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Advanced Light Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Beamline 10.0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 COLTRIMS Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Supersonic Gas Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.4 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Micro-channel Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Delay-line Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.5 Data Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Analysis 234.1 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Presorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.2 Calculations in the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 24

viii

Time-of-flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.3 Correcting Electron Momentum Kick . . . . . . . . . . . . . . . . . . . 284.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Photon Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.3 Detector Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Tilted Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Distribution on Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Selection of Reaction Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3.2 Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Jetdot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Bunchmarker Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Hot Gas Stripe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.3 Final Calibration of Momentum . . . . . . . . . . . . . . . . . . . . . . 384.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Reaction Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 H2 - Molecular Hydrogen 455.1 Breaking the Symmetry - Past Experiments . . . . . . . . . . . . . . . . . . . . 45

With External Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Without External Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46With Coulomb Field of Photoelectron . . . . . . . . . . . . . . . . . . . 47

5.2 Idea of this Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Results of H2 and D2 Photo Dissociation . . . . . . . . . . . . . . . . . . . . . . 51

5.3.1 Energy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3.2 Retroaction of Photoelectron . . . . . . . . . . . . . . . . . . . . . . . . 55

Comparing Asymmetry in H2 and D2 . . . . . . . . . . . . . . . . . . . 62Structure in Retroaction? . . . . . . . . . . . . . . . . . . . . . . . . . . . 62L- dependency of Asymmetry? . . . . . . . . . . . . . . . . . . . . . . . 64

ix

6 CF4 - Tetrafluoromethane 656.1 Past Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

RFPADs of Valence Shell Photoionization . . . . . . . . . . . . . . . . . 66Shape Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Energy Curves and Dissociation Dynamics . . . . . . . . . . . . . . . . 67

6.2 Idea of this Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Results of CF4 Photo Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.3.1 A2T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3.2 X1T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3.3 B2E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 Conclusion 77

Acknowledgements 79

A Additional Material 81A.1 Additional Material for H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.2 Additional Material for CF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Bibliography 83

xi

List of Figures

2.1 Potential energy surface of H+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Avoided Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 H2 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Undulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 ALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 ALS beamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 U100 ALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 COLTRIMS Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Gas jet Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.7 COLTRIMS spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 MCP substructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9 Micro-channel Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.10 delay-line Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Examplary ion time-of-flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Wiggle-plot of He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 KER vs electron energy, H+, 18.5eV . . . . . . . . . . . . . . . . . . . . . . . . . 314.5 Hot gas stripe on ion detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Momentum conservation between electrons and H+

2 ions . . . . . . . . . . . . 334.7 Fish plot of Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.8 Ion time-of-flight H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.9 KER vs electron energy, H+, 19.76eV . . . . . . . . . . . . . . . . . . . . . . . . 364.10 Ion momentum of H+ for 19.76eV photons . . . . . . . . . . . . . . . . . . . . 374.11 H+

2 electron energy versus various angles . . . . . . . . . . . . . . . . . . . . . 384.12 H+ energy sum versus various angles . . . . . . . . . . . . . . . . . . . . . . . 394.13 CF4 energy sum versus various angles of electron momentum vector . . . . . 404.14 CF+

3 position on detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.15 H+

2 and electron momentum, 18.56eV . . . . . . . . . . . . . . . . . . . . . . . 424.16 Electron energy versus cos(θ) of electrons. H+

2 , 19.8eV . . . . . . . . . . . . . 44

5.1 Two step process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 Density plot of Asymmetry of D+ ion emission by Ray et al. . . . . . . . . . . 47

xii

5.3 Single photon induced symmetry breaking in H2 . . . . . . . . . . . . . . . . . 485.4 Potential energy curves of H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.5 H2 MFPADs by Waitz et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 Electron Energies H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.7 H+ energy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.8 Energy sum with 18.16eV photons . . . . . . . . . . . . . . . . . . . . . . . . . 535.9 Gaussian Peak area, degeneracy corrected . . . . . . . . . . . . . . . . . . . . . 545.10 Gaussian Peak area, cross section corrected . . . . . . . . . . . . . . . . . . . . 565.11 H+ MFPADS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.12 Asymmetry parameter as function of electron energy . . . . . . . . . . . . . . 575.13 Classical calculations of coupling time and electron distance in H+

2 . . . . . . 585.14 Calculated and measured asymmetry parameter . . . . . . . . . . . . . . . . . 605.15 Calculated and measured asymmetry parameter . . . . . . . . . . . . . . . . . 615.16 D+ MFPAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.17 Comparison asymmertry in D+ and H+ . . . . . . . . . . . . . . . . . . . . . . 635.18 Asymmetry as function of KER . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.19 Delta versus KER for different rotational states of H2 . . . . . . . . . . . . . . 64

6.1 Structure of CF4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 RFPADs CF4 from Kinugawa et al. . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 Dipole transition table Td symmetry . . . . . . . . . . . . . . . . . . . . . . . . 676.4 Calculated (line) and experimental (circles) total photoionization cross sec-

tion σ and asymmetry parameter β for valence electrons in CF4 . . . . . . . . 686.5 Potential energy curves of CF+

3 − F . . . . . . . . . . . . . . . . . . . . . . . . . 686.6 Energy maps of CF4 photodissociation . . . . . . . . . . . . . . . . . . . . . . . 706.7 RFPADs of CF+

4 (A2T2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.8 RFPADs of CF+

4 (A2T2) for parallel molecular orientation . . . . . . . . . . . . 726.9 CF4 total cross section for 4t2 outer valence shell . . . . . . . . . . . . . . . . . 736.10 RFPADs of CF+

4 (X2T1) for parallel molecular orientation . . . . . . . . . . . . 736.11 CF4 total cross section for 1t1 and 1e outer valence shell . . . . . . . . . . . . . 746.12 RFPADs of CF+

4 (B2E) for parallel molecular orientation . . . . . . . . . . . . 756.13 Calculated total photoionization cross section for t2 electrons in CF4 . . . . . 756.14 RFPADs of CF4 K-edge photoionization . . . . . . . . . . . . . . . . . . . . . . 76

A.1 Calculated coupling time in H2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.2 Total Cross Section of 3t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xiii

List of Tables

3.1 Gas target parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 E- and B-field values used in the Experiments . . . . . . . . . . . . . . . . . . . 20

A.1 Character Table of Td point group. . . . . . . . . . . . . . . . . . . . . . . . . . 81

xv

Physical Constants

Speed of Light c0 = 2.997 924 58× 108 m s−1 (exact)Electrical constant ε0 = 8.854 187 817× 10−12 F/m (exact)Boltzmann constant kB = 1.380 648 52(79)× 10−23 m2kg/s2/K

Planck constant h = 6.626 069 3(11)× 10−34 Js

Reduced Planck constant h = 1.054 571 68(18)× 10−34 Js

Elementary charge e = 1.602 176 53(14)× 10−19 C

Electron mass me = 9.109 382 6(16)× 10−31 kg

Proton mass mp = 1.672 621 71(29)× 10−27 kg

xvii

Atomic Units

In order to simplify calculations and expressions in atomic and molecular physics one typ-ically uses atomic units, where the constants in the electronic Schrödinger equation are setequal to one.

h = me = e = 4πε0 = 1a.u.

Other physical quantities can be expressed using above definition.

Dimension Symbol Value in a.u. units Value in SI units 1

length a0 1 0.5292 · 10−10m

mass me 1 9.1094 · 10−31kg

speed of light c0 = 1/α 137 2.9979 · 108m · s−1

velocity v0 = a0Ehh 1 2.1877 · 108m · s−1

mass me 1 9.1094 · 10−31kg

momentum p = mev0 1 1.9929 · 10−24kg ·m · s−1

time t = hEh

1 2.4189 · 10−17s

energy Eh = h2

a20me1 4.3597 · 10−18J

charge e 1 1.6022 · 10−19C

1Values taken from https://physics.nist.gov/cuu/Constants/index.html

1

Chapter 1

Introduction

The invention of the synchrotron principle in 1944 by Vladimir Veksler [Vek44], enabledthe construction of first large scale acceleration facilities, since bending, acceleration andbeam focusing can be done in different segments. Charged particles travel around a closedloop and gain kinetic energy in electric fields, while a magnetic field is synchronized to theincreasing kinetic energy to keep the particles on track. It did not take long to observe thespecific synchrotron radiation, which is emitted when charged particles are forced on a cir-cular motion, in 1947. Mainly due to its broad spectrum and high intensity, synchrotron ra-diation was soon used for spectroscopy experiments. Since then the interest in synchrotronradiation was ever growing, soon specialized radiation facilities were build to utilize syn-chrotron radiation for science.Today the applications of synchrotron radiation are mainly found in material and biologicalsciences, where the short wavelength radiation is used for diffraction and imaging experi-ments. But synchrotron radiation is also highly useful within the physical and chemical sci-ences, where the wide bandwidth of photon energies produced by the radiating electrons isideally suited to study electronic structure in atoms and molecules. This light can be partic-ularly useful for studying valence electronic structure and chemical bonding. One methodfor studying electronic structure is photoelectron spectroscopy, where an atomic or molec-ular target is radiated with ionizing photons of well known wavelength. The energy of thephotoelectron is then measured and it is possible to gain information about the structureof the electronic state of origin. With some more advanced detection techniques it is alsopossible to retrieve angular information about the photoelectron emission and even mea-sure the corresponding ion. One of most kinematically complete detection techniques is theCOLd Target Recoil Ion Momentum Spectroscopy (COLTRIMS). Also called Reaction Mi-croscopes, these spectrometers provide the some of the most detailed views into correlateddynamics of atoms and molecules. Reaction microscopes use a combination of electric andmagnetic fields to measure the momentum vector of charged fragments after an ionizationin 4π space. Furthermore they can detect multiple fragments in coincidence, most notewor-thy the ion and corresponding electron of an ionization event. This kinetically completedescription of the reaction allows for a highly differential investigation of photoionizationand fragmentation processes; in addition to the electronic structure of atoms and molecules,

2 Chapter 1. Introduction

COLTRIMS enables the study of electron dynamics in ionization processes.As mentioned, the dynamics of valence electrons are of special interest for scientists, be-cause these weakly bound outer shell electrons are responsible for chemical reactions andmolecular bonds. The primary focus of this work lies in the emission dynamics of valenceelectrons and their interaction with the molecular field once they are in the continuum. Tothis end photoionization experiments on molecular Hydrogen (H2) as well as its heaviercounterpart molecular Deuterium (D2) and Tetrafluoromethane (CF4) were carried out atthe Advanced Light Source synchrotron in Berkeley, California. The findings of these ex-periments are presented in chapter 5 and 6, chapter 2 gives a brief introduction into thetheoretical background, while chapter 3 and 4 explains the experimental methods used andthe procedure of data analysis respectively.

3

Chapter 2

Theoretical Background

This chapter gives a brief introduction into the theoretical background needed to under-stand the physical processes covered in this work.

2.1 Structure of Molecules

Molecules can be seen as a stable arrangement of a group of nuclei and electrons. The nu-clei and electrons are in constant motion and the energy of the molecular system is storedin those motions. Supplying energy to a molecule, for example in form of a photon, can re-sult in a change of nuclear motion (vibrational and rotational modes) or electronic motion(change in electronic state). The different forms of motion can store very different amountof energies, but all of them are intrinsically quantized in their nature.When a molecule is formed by two (or more) atoms the inner shell electrons of each atomremains tightly bound and only the outermost electrons, known as valence electrons are af-fected by the other electrons and ions. Molecular binding is achieved by a lowering of thetotal energy through modification of the valence electron wavefunctions dependent of theatomic distance in a molecule. Hence the valence electrons have a central role in molecularbinding.

Quantum mechanically a molecule is described by a wavefunction ψ which solves theSchrödinger equation

Hψ = Eψ , (2.1)

where H is an energy operator called Hamiltonian and E is the energy eigenvalue of thesystem [Dem13]. When ψ is a solution of the Schrödinger equation it is called an eigenfunc-tion of H . For a neutral two atomic molecule in a field free environment the Hamiltonianconsists of a kinetic energy part T and a potential energy part V :

H = T + V = − h2

2me

2∑j=1

∇2j︸︷︷︸

Te

− h2

2

2∑k=1

∇2k

1

Mk︸︷︷︸TN

+V (~r, ~R) (2.2)

4 Chapter 2. Theoretical Background

Te and TN correspond to the kinetic energy of the electrons and nuclei respectively withme being the electron mass and Mk the mass of the nuclei. The potential energy V (~r, ~R) isdependent on the spatial coordinates ~r1,2, ~R1,2 of electrons and nuclei. It results out of thecoulomb forces between the charged particles and can be written as:

V (~r, ~R) = VNN + Vee + VNe =e2

4πε0

(Z1Z2

|~R1 − ~R2|+

1

|~r1 − ~r2|−

2∑k=1

2∑j=1

Zk

|~rj − ~Rk|

)(2.3)

WhereZ1,2 is the charge of atom 1 and 2. The positively charged ions repel each other (VNN )as well as the negatively charged electrons (Vee). Only between electrons and ions exists anattracting force (VNe).

Equation 2.1 can not be analytically solved for any molecule and even with powerful nu-merical algorithms it is essential to make well reasoned approximations. One of the mostbasic approximations commonly used is the Born-Oppenheimer approximation, which va-lidity will be discussed in the next section.

2.1.1 Born-Oppenheimer Approximation

Considering the mass ratio between the protons in the atomic core and the electrons ofmp/me ≈ 1836 it is reasonable to assume the electrons are moving much faster than theatomic core can react. Then we can consider the nuclei as a frame around which the elec-trons are moving and instantaneously adjusting to any changes in the "nuclear frame". Thissimple assumption enables us to separate nuclear from electronic motion and we can writethe molecular wavefunction as a product of nuclear (ψ(~R)) and electronic (ϕ(~r)) wavefunc-tion:

Ψ(~R,~r) = ψ(~R)ϕ(~r) (2.4)

With this Ansatz the Schrödinger equation 2.1 simplifies into two decoupled differentialequations. One describing the electrons:

Heϕ(~r)n = Ee,nϕ(~r)n (2.5)

and another one describing the nuclei:

HKψ(~R)n,i = En,iψ(~R)n,i (2.6)

ψ(~R)n,i describes the nuclear wavefunction in the electronic state n and the nuclear state i.And the Hamiltonians are He = Te + V (~r, ~R) and HN = TN + En(~R).

Adiabatic Approximation

Instead of considering the electronic motion as instantaneously to nuclear movement, itis more accurate to treat the electrons as adiabatically following the nuclei motion. The

2.1. Structure of Molecules 5

nuclei are also sitting in a potential formed by the electrons and thereby dependent on theelectronic wavefunction. With this the Hamiltonian for the nuclei changes to ([Dem13], Ch.2.1.4):

HN = TN + En(~R) +∑k

h2

2Mk

∫ (∂ϕeln

∂ ~Rk

)2

d~r (2.7)

Those approximations are generally very useful for fast ionization processes and widelyused whenever the motion of the electrons is much faster then that of the nuclei. Thereare, however, instances where the electron can not follow the nuclear motion fast enoughand it is not possible to separate the nuclear and the electronic wavefunction as we did itin equation 2.4. This happens if either the nucleus is moving to fast for some reason orthe electron is moving too slow. If this is the case we speak of a "Breakdown" of the Born-Oppenheimer approximation, which typically occurs in chemical reactions and dissociationdynamics.

2.1.2 Potential Energy Surfaces

Once the electronic Schrödinger equation is solved it is possible to plot the energy eigen-value Ee, which is generally a function of the nuclear coordinates. In the diatomic case thenuclei can be sufficiently described by their distance. The electron energy Ee includes thekinetic energy of the electron, electron/nucleus interaction and the internuclear repulsion.For each degree of freedom within the molecular coordinate system there is one dimensionneeded for the potential energy "surface". So for a diatomic molecule, which has the dis-tance between the nuclei as its only degree of freedom an energy curve is sufficient, whilethe potential energy of bigger molecules with multiple degree of freedoms are typicallyshown with an energy surface with all degrees of freedom except two held constant. Asan example the potential energy of the diatomic molecule H+

2 as a function of the nucleardistance can be seen in figure 2.1.

If the potential energy curve of a molecular state has a local minimum, it is consideredas "bound" with the equilibrium distance R0 of the nuclei being found at the minimum.States without a minimum are "unbound" and the molecule will dissociate.The total energy of the system includes is made up by the electronic energy Ee and thenuclear energy EK . The possible nuclear motions are:

• Translation

Motion of the center of mass of the molecule, which does not influence its internalenergy.

• Vibration

The bonds between atoms in a molecule act just like a spring and the resulting vibra-tional motion can be approximated by a harmonic oscillator with a spring constant k


FIGURE 2.1: Potential energy surface of H+2 . Figure taken from [YH14]

and the reduced mass µ. The angular frequency is then:

ω =

√k

µ(2.8)

and the vibrational energy is:

Eν = hω(ν +1

2) (2.9)

ν is the vibrational quantum number and can take the values ν = 0, 1, 2.... So even inthe ground state ν = 0 there is a zero point energy, which means that the molecule isnever completely standing still.Eν has a typical order of magnitude of a few 100meV .

• Rotation

The rotation of a diatomic molecule can be described by a linear rigid rotor, where thecenter of mass lies on the axis connecting the nuclei. The moment of inertia about theaxis of rotation is with the moment of inertia I depending on the reduced mass µ andthe distance R between the two atoms

I = µR2 (2.10)

Classically the energy of the rotation of a rigid rotor is [ER85]

Er =L2

2I(2.11)

where L is the angular momentum. Quantization of the magnitude of the angular mo-mentum L gives L2 = J(J + 1)h2 with the rotational quantum number J = 0, 1, 2, ...,

2.1. Structure of Molecules 7

so that the energy becomes

Er =h2J(J + 1)

2I(2.12)

Rotational energies are considerably smaller than vibrational energies and are typi-cally around 0.1− 10meV . For more information on the derivation of the above usedformulas see ([ER85], Chapter 12).

Intersection of Potential Energy Curves

One might imagine a case in which two potential curves come very close to each other oreven cross. For this case von Neumann and Wigner quantitatively formulated the so-callednon-crossing rule in 1929 [vW29], which is given here in a shortened form:

Two distinct electronic energy curves E1(R) and E2(R) can come very close and forcertain conditions even cross. The two wavefunctions can be represented as

Ψ = c1Ψ1 + c2Ψ2 (2.13)

or as a Slater determinant (H11 − E H12

H21 H22 − E

)(c1

c2

)= 0 (2.14)

with Hij = 〈Ψi|H|Ψj〉.In order for equation 2.14 to have degenerate solutions and hence have a crossing of thetwo electronic energy curves, it is necessary that

H11 = H22 and H12 = H21 = 0 . (2.15)

H12 = H21 = 0 is only fulfilled if Ψ1 and Ψ2 have different symmetry and there exist at leasttwo independently variable nuclear coordinates.Generally speaking two electronic energy curves belonging to the same symmetry can notcross. We are then speaking of an "avoided crossing" as it is visualized in figure 2.2.If the potential energy surface of two electronic states becomes degenerate at one point, thecrossing at this point is called a conical intersection.

2.1.3 Description of Electronic and Molecular States

A single electron in a diatomic molecule can be described with the main quantum numbern, the quantum number λ and the parity gerade ("g") or ungerade ("u") [HS14].

nλg/u (2.16)


FIGURE 2.2: Exemplary sketch of an avoided crossing of two energy curves.

• n characterizes the orbital of the electron

• λ is the projection of the orbital angular momentum of the electron onto the molecularaxis. It can have the values λ = 0, 1, 2, 3, ... and the notation is σ, π, δ, φ respectively.

• The parity g/u describes the symmetry of the electronic wavefunction

φg(r) = φg(−r)

φu(r) = −φu(r)

The complete molecular state with all its electrons can be described by [HS14]

2S+1Λ(+/−)Ω (2.17)

• 2S + 1 is the multiplicity and it can have the values 1,2,3,... which is described assinglet, doublet and triplet respectively. It is calculated out of the sum of the electronspins S. The projection of ~S on the molecular axis is written as Σ = S, S − 1, ...,−S.

• Λ is the quantum number for the projection of the total orbital angular momentum ~L

of all electrons on the molecular axis. It can have the values Λ = 0, 1, 2, 3..., which isdescribed with Λ = Σ,Π,∆,Φ.

• Ω is the projection of the total angular momentum ~J , which arises out of the sum ofthe orbital angular momentum and spin of the electrons ~J = ~L + ~S. Ω = |Λ + Σ| cantakes the values Ω = J, J − 1, ..., 1

2 , 0.

• +/− denotes the parity with

Ψ+(r) = Ψ+(−r)

Ψ−(r) = −Ψ−(r)

2.2. Photoionization 9

2.2 Photoionization

If the electrons in the molecular orbitals are interacting with there are several processeswhich can happen. Depending on the photon energy Eγ = hν (and the intensity) themolecule can be excited into a higher rotational, vibrational or electronic state, it can getionized and even fragmented. For excitation processes the photon needs to have just theright energy, because the energy levels in the molecule are discreet, whereas for an ioniza-tion process the energy of the photon can have some range. This is because the electroninteracting with the photon is not "lifted" to a higher energy line, but into the continuum.The minimum photon energy for this process is the ionization energy EI . The leaving elec-tron, called photoelectron, carries away the remaining energy in the form of kinetic energy

Ekin,e = EI − hν . (2.18)

If the photon energy is higher then the dissociation energy Ediss of the molecule it canfall apart and the remaining energy will be divided between the photoelectron and themolecular fragments

Eγ − Ediss = Ekin,e +KER . (2.19)

KER is the abbreviation for Kinetic Energy Release and is the sum of kinetic energy of allmolecular fragments.

2.2.1 Fermi’s Golden Rule

The transition rate between an initial state Ψi to a final state Ψf is determined by its cou-pling strength and by the number of possibilities to reach that final state. Last is determinedby the density of final states ρf . The probability of transition from initial to final state λifcan be described by a formula called Fermi’s golden rule (compare to [AF05], Chapter 6)

λif = 2πh|Mif |2ρf (2.20)

where Mif is the matrix transition element of the form

Mif = 〈Ψf |V |Ψi〉 . (2.21)

V describes the physical interaction of the coupling, which in most cases is sufficientlycharacterized by a dipole moment ~µ, so that

λif = 2πh| 〈Ψf |~µ|Ψi〉 |2ρf . (2.22)

If the final state is a continuum of states, as it is the case in non-resonant photo ionization,the density of states is written as a function of energy ρ(E) where ρ(E)dE is the number ofcontinuum states in the range E to E + dE.


2.2.2 Franck-Condon Principle

FIGURE 2.3: Potential energycurve of H2 and H2+. Taken

out of [Wai+16]

Because the nuclei are so much heavier than the electron, anelectronic transition is happening within the stationary nu-clear frame. As a result the position of the nuclei remainsunchanged and only adjusts once the electrons have adoptedtheir final distribution. This corresponds to a vertical transi-tion in the potential energy curves. Since the initial state Ψi

has some distribution the electronic transition can only hap-pen to a final state Ψf vertically above the distribution of Ψi.Typically it is assumed that the initial state is the vibrationalground state. This region is then called the Franck-Condon re-gion.The transition is more likely to happen if the vibrational wave-function of the final state has a good overlap with the vibra-tional wavefunction of the initial state.The dipole transition moment is given by (compare to [AF11])

P = 〈Ψf |~µ|Ψi〉 (2.23)

with the dipole moment operator ~µ, which depends on theposition ~ru and charge−e of electrons and the position ~Rv andcharge q of nuclei

~µ = −e∑u

ru + q∑v

Rv = ~µe + ~µN (2.24)

11

Chapter 3

Experimental Methods

The experimental work of this thesis was conducted at beamline 10.0.1 of the AdvancedLight Source (ALS) in Berkeley. The two main components of the experiments are of coursethe photon source and the reaction chamber. In the following chapter both will be outlinedin more detail.

3.1 Synchrotron Radiation Source

A synchrotron is a special kind of particle accelerator in which charged particles are acceler-ated by increasing electric fields and held on a circular path by a simultaneously increasingmagnetic field. Caused by the circular motion any charged particle is emitting synchrotronradiation due to the laws of electro dynamics. This effect is utilized by todays synchrotronfacilities, which are accelerating electrons close to the speed of light (c0 ≈ 3 · 108m/s) andthereby generating high energetic photons up into the hard X-ray regime (105eV ) 1.The power of this synchrotron radiation is strongly dependent on the kinetic energy We

and the mass m of the accelerated particle (see [GM06]):

P =e3c0

6πε0%2

( We

mc20

)4(3.1)

which is the reason electrons are preferably used. Also important is the bending radius% ofthe circular motion, since for a decreasing bending radius the power increases quadratic.

FIGURE 3.1: Schematic sketch of an Undulator (taken from [Und])

To gain maximum efficiency so-called Undulators are used to produce the radiation.Anundulator consist of an array of periodically placed magnet which force the electrons onto

1As currently possible at the synchrotron facility SPring-8 located in Hyogo, Japan [Spr]

12 Chapter 3. Experimental Methods

a sinusoidal trajectory where on each point of reversal a cone of synchrotron radiation isemitted (see figure 3.1). Because the electrons are traveling close to speed of light thoseradiation cones are constructively superimposed onto each other and quasi-monochromaticlight with a very high brilliance 2 is produced. The light cone emitted is described by itsopening angle ϑ

sin(ϑ) =

√1− v2

c20

. (3.2)

So for higher velocities the angular divergence of the photons becomes smaller and thebrilliance higher.

FIGURE 3.2: Outline of the Advanced Light Source (ALS) in Berkeley (takenfrom [ALS17])

In order to accelerate those electrons to speeds of ∼ 99.999996% [ALS17] of light theycommonly pass through three stages:

• LINACThe electrons are produced in an electron gun, which are then accelerated in a linearaccelerator (LINAC).

• Booster ringAfter the LINAC the electrons are go into the Booster ring. Here they are being accel-erated to their final speed within only 0.45s and then injected into the storage ring.

• Storage ringIn the storage ring the electrons are moving at an almost constant speed with the solepurpose of producing synchrotron radiation. They only get accelerated to correct forthe losses due to the radiation.

2Brilliance (also called brightness) is a measure of quality of the radiation. It takes into account the number ofphotons per second, the angular divergence of the photons, the cross sectional area of the beam and the amountof photons falling within 0.1% bandwidth (BW) of the central wavelength of the photons.Brilliance = photons

second·mrad2·mm2·0.1%BW

3.1. Synchrotron Radiation Source 13

As can be seen in figure 3.2 a synchrotron facility typically has many beamlines, which areequipped with different undulators to satisfy the needs of a broad spectrum of experiments.

3.1.1 Advanced Light Source

FIGURE 3.3: Beamlines of the ALS (taken from [ALS17])

The ALS is a so-called third generation synchrotron 3 with 40 beamlines. The storagering has a circumference of 196.8m and the electrons stored in it have an energy of 1.9GeV[ALS17]. In its normal operation mode there are 320 electron bunches circulating in thesynchrotron which results in a bunch spacing of 2ns. This is to short for us, since our timeresolution is only about ∼ 0.2ns and it therefore would not be able to match each photo-electron to its time zero. Fortunately for a few weeks every year the ALS is operated intwo-bunch mode, which increases the bunch spacing to 328ns. This corresponds to a repeti-tion rate of 3MHz with a pulse length of 80ps.

Beamline 10.0.1

Beamline 10.0.1 uses an undulator called U100 as insertion device. This undulator has 43 pe-riods of dipole magnets with a period length of 100mm and can be used to generate photonswith energies from∼ 12−1500eV with a peak brightness of 1019photons/(0.1%BWmm2mrad2s)

as seen in figure 3.4. The energy selection of the photons is happening right after the undu-lator with a diffraction grating and variable entrance and exit slits. For the single ionization

3The term 3rd generation synchrotron refers to the use of insertion devices like an undulator or wiggler. Thoseinsertion device are implemented at a straight part of the synchrotron and are capable of generating synchrotronradiation with a much higher brilliance than synchrotrons from previous generations, which were exclusivelyusing the bending of the ring to generate radiation.


experiments presented here the photon energies needed were found at the lower end ofthe possible energy range. So in order to achieve sufficient brightness for the COLTRIMSexperiment, the exit slit had to be opened more than under ideal conditions. A broaderslit opening after the grating corresponds to a bigger uncertainty in energy, which for theenergies used was typically around ∼ 10meV .

FIGURE 3.4: The brightness and photon flux of undulator U100 at beamline10.0.1 at the ALS. (taken from [ALS17])

3.2 COLTRIMS Apparatus

COLTRIMS is an acronym for COLd Target Recoil Ion Momentum Spectroscopy. It is animaging technique, capable of measuring the complete fragmentation of a few body sys-tem. Therefor all charged particles from a fragmentation are projected by a combinationof electric and magnetic field onto position sensitive detectors. With the position and themeasured time of flight (TOF) one obtains the three dimensional momentum vector of allparticles.

3.2.1 Vacuum Chamber

Having a good vacuum in the COLTRIMS chamber is indispensable for most parts of theapparatus. For example the photon energies of the synchrotron light are above most atomsand molecules. So without a high vacuum in the reaction chamber there would be ion-izations happening all along the light beam and it would be impossible to separate thetrue photoelectrons from the background. In our setup are 6 turbo pumps used, which arebacked by two scroll pumps and two fore pumps. With all pumps running a vacuum of

3.2. COLTRIMS Apparatus 15

FIGURE 3.5: Outline of the COLTRIMS chamber.

2 · 10−7mbar in the chamber can be achieved. To further improve this vacuum, a cold trapextending into the chamber is used. This cold trap is constantly 4 filled with liquid nitrogen(LN2) and therefore held at a temperature of under 77K, which corresponds to the boilingpoint of nitrogen. Having such a cold surface in the chamber leads to condensation of wa-ter molecules onto it. So the cold trap is effectively "trapping" the water molecules whichhappen to bump into it and thereby enhanced the vacuum to ∼ 2 · 10−8mbar. The outlineof the COLTRIMS chamber can be seen in figure 3.5.

3.2.2 Supersonic Gas Jet

As the name COLTRIMS indicates the target is a cold gas jet. This is critical since we want tomeasure the momentum of reaction products. If the molecules had a non-zero temperaturethey also would have an initial momentum distribution which would lead to a lower limitin our momentum resolution corresponding to that initial distribution. In the simplifiedpicture of an ideal gas this momentum distribution is given by the Maxwell-Boltzmanndistribution [GM06]:

f(v) = 4π( m

2πkBT

) 32v2e− mv2

2kBT (3.3)

4During the whole beamtime the cold trap was refilled every 4 hours. Which was enough that it was alwaysfilled at least one quarter of its volume.


FIGURE 3.6: Illustration of the gas target preparation via a supersonic expan-sion. Figure adopted from [Tho03].

with the particle velocity v and temperature T . The distribution gets narrower for lowertemperature. The energy stored per degree of freedom is

Ef =1

2kBT . (3.4)

For the diatomic molecule hydrogen (5 degrees of freedom, mass=2amu) that leads to athermal momentum of 4au at room temperature. This clearly demonstrates the need ofcooling down the gas target to do high resolution momentum spectroscopy.

In order to reach a low temperature in the gas target − and thereby a better momentumresolution− a supersonic expansion into the reaction chamber is used. Here the gas is pressedthrough a nozzle with an opening of 50µm into vacuum. Because of the pressure differenceof 2− 3bar to 10−4mbar the gas will adiabatically expand to supersonic speed and most ofthe thermic energy will be transformed into a highly directional transversal speed of themolecules. The expanding gas jet has the form of a cone as can be seen in figure 3.6. In thecenter region, which is called the zone of silence, the temperature is only a few kelvin. Tomake sure that only this part makes its way into the chamber we further use two skimmersto cut off the outer parts of the cone. Those two skimmers are at the same time the onlyconnection for the gas to get into the main chamber, which allows for a differential pumpingto keep a high vacuum. After the gas jet crosses to interaction region it hits the jet dump atthe other side of the chamber. The jet dump consists of a thin tube which leads the gas intoanother differential pumping stage connected with a turbo pump as can be seen in figure3.5.The temperature in the expanded gas jet Texp can be approximated by (see [Pau12] for the


motivation of following formulas):

Texp ≈f + 2

2

T0

S2(3.5)

with the degrees of freedom f and the speed ratio S.

S =vjet√

2kBTexp/m(3.6)

which was empirically calculated for small molecules to

S = 5.4(P0d)0.32 . (3.7)

Note that the nozzle pressure P0 and the nozzle diameter d need to be inserted in Torr andcm respectively. The jet velocity vjet is calculated by

vjet =

√(f + 2)T0kB

m. (3.8)

The thermal momentum of the jet can be separated into the part parallel to the jet p|| andtransversal to the jet p⊥. The half-width of those momenta is given by

∆p|| = mvjet2√ln2

S(3.9)

and∆p⊥ = mvjet

dsk2 + dnozlsk2

(3.10)

with the diameter of the second skimmer dsk2 = 0.6mm and the distance between nozzleand second skimmer lsk2 = 28mm.The values for the conditions in our experiment are listed in table 3.1. Knowing the ap-proximate temperature of the jet, it is also possible to estimate the vibrational ground statepopulation, which is important for the Franck-Condon principle as explained in chapter

Parameter H2 D2 CF4 UnitMass m 2 4 88 amu

Backing Pressure P0 15 32 40 torrinitial Temperature T0 294 294 294 K

Speed ratio S 8 10 11Jet velocity vjet 2672 2012 515 m/s

half width parallel momentum ∆p|| 0.9 1.1 5.8 a.u.half width transverse momentum ∆p⊥ 0.1 0.15 0.9 a.u.

calculated (with formula 3.5) jet temperature Texp 16 10 20 K

TABLE 3.1: Calculated jet parameters for the gases used in this experiments.


2.2.2. In statistical mechanics the ratio of the population of two sates is given by by theBoltzmann factor and only depends on the temperature and the energy difference ∆E ofthose two states

Fstate2Fstate1

= exp(− ∆E

kBT

). (3.11)

As an example for molecular hydrogen with an energy spacing of ∆E = Eν=1 − Eν=0 ≈0.5eV [Dic+13] and an estimated temperature Texp ≈ 16K the Boltzmann factor is

Fν=1

Fν=0≈ 10−126 (3.12)

which clearly shows that all of the molecules in the gas jet are in their vibrational groundstate.

3.2.3 Spectrometer

FIGURE 3.7: Illustration of the spectrometer with the ion side on the left andthe electron side on the right. Different field regions are shaded in a differ-ent gray. Visible are the ion trajectories in black and the electron trajectoriesin blue. The electrons are steered onto gyration motion caused by the mag-netic field parallel to the spectrometer. Figure is not to scale. Adapted from

[Tho03].

The spot where the jet and photon beam are crossing, called interaction zone, is placedwithin the spectrometer. The size of the interaction zone is determined by the width ofthe gas jet as well as the width of the photon beam. The diameter of the gas jet is for oursetup approximately 1.1mm in the reaction zone. The transversal width of the photon beamdepends on slit size before and after the undulator. At beamline 10 of the ALS the width


was around 200µm. Hence the reaction zone has a cigar shaped form wherein the ionizationevents are happening. Since the momentum calculation is based upon the time-of-flight andspatial coordinates on the detector, this uncertainty of the ionization of location limits theresolution. One way to counteract this uncertainty is a special spectrometer configurationdesigned by W.C. Wiley and I. H. McLaren [WM55]. Named after its inventors, the Wiley-McLaren geometry uses a ratio of 1/2 between acceleration and drift region on the electronside to focus them in time. This gives us the best resolution in the tof-direction. Followedby the jet-direction and then the beam-direction. The spectrometer can be seen in figure3.7. The electrons first get accelerated in a 68.65mm long extraction field and then they getinto a 139.13mm long, field-free drift region. After the drift region there is a 8mm long finalacceleration region with a potential difference of 2200V . The ion side of the spectrometerconsists of a 44mm acceleration region and a 22mm booster region. The acceleration regionhas a relatively low electric field (∼ 2 − 10V/cm) in which they spread out spatially. Afterthat they pass through the booster region with a potential difference of 600V and then thereis yet another 6mm long final acceleration with 2200V . A homogeneous electric field isrealized by parallel copper plates separated by 5mm thick, 600kΩ strong resistors. Theplates have a hole in their center where the electrons and ions can propagate. In addition tothat all the different field regions are separated by a mesh, which improves the homogeneityof the fields but also reduces the amount of particles making it to the detector.The electric field in the extraction region is varied depending on the expected electron andion energies and it determines the resolution in time-of-flight direction. The exact valueswhere therefore strongly dependent on the target molecule and the photon energy and willbe given in table 3.2.With the used spectrometer geometry the slow and heavy ions will be nicely spread ontothe detector. The electrons however are much faster and the electric field itself is not enoughto steer them onto the detector. Because a higher electric field would affect the resolutionwe use a magnetic field instead to limit the electrons in their spatial distribution. To puta homogeneous magnetic field parallel to the electric field inside the spectrometer, a pairof Helmholtz coils outside the chamber is used. Those coils have a diameter of 1.4m andn = 15 windings. The resulting magnetic field depends on the current flowing through thecoils and its magnitude will be chosen to optimize the spatial spread of the electrons ontothe detector. Nevertheless it does not noticeably influence the ions. The values used foreach experiment during the two beamtimes in 2016 and 2017 are given in table 3.2.

3.2.4 Detector

To detect the electrons and ions there is a detector on each side of the spectrometer. Thedetectors consist out of two parts each. The Microchannel Plate (MCP) for amplifying theevents and the delay line Anode for detecting them.


August 2016Target H2 CF4 CF4 CF4 CF4 CF4

Photon energy [eV ] 18.56 18.3 18.6 19.0 19.4 21.4Extraction field [V/cm] 2.98 2.4 2.5 2.5 2.5 4.5Magnetic field [gauss] 2.36 2.2 2.2 2.2 2.2 3.0Gyration period [ns] 152 162.4 162.4 162.4 162.4 119

June 2017Target H2 H2 H2 H2 D2 CF4 CF4

Photon energy [eV ] 18.16 18.3 18.5 19.8 18.5 20.46 31eVExtraction field [V/cm] 2 1.86 2.5 1.8 2.7 5.0 7.0Magnetic field [gauss] 1.8 2.1 2.1 2.1 2.1 3.4 7.9Gyration period [ns] 240.9 171.5 170 171.5 170 330 46.3

TABLE 3.2: Values of the extraction field and magnetic field as used for thedifferent experiments done at the ALS in Berkeley in August 2016 and June

2017.

Micro-channel Plate

FIGURE 3.8: Due to the wayMCP’s are produced they havea substructure consisting ofhexagons. Those hexagons aretypically 1mm wide and atthe edges it can come to someamplification problems. We cansee this effect here in the positionof H+

2 ions after photoionizationwith 18.56eV .

The micro-channel plate (MCP) forms the first part of thedetector. Its use is to amplify the electrons and ions and,by doing so, measuring the time of flight (TOF) of the par-ticles. The MCP used is a glass wafer interspersed by chan-nels, with a diameter of 25µm, which are tilted by 8 to thenormal vector of the surface, covering approximately 60%of the area. If a photon or a particle hits the inner wall ofthese micro-channels an electron cloud is released which isthen accelerated by a high voltage between the two sidesof the MCP with the cathode always on the entrance side.The electrons of the electron cloud can then hit the wallagain and release another cloud and so forth until the elec-trons exit the rear of the MCP. To maximize this effect, thereare two MCP’s arranged in a way the channels of the twoplates are not parallel but zig-zag to each other, which iscalled a Chevron configuration (as shown in 3.9). This wayan amplification of 106 − 107 is achieved. The voltage be-tween the outer sides of the two MCP’s is about 2200V .When the electrons exit the MCP it comes to a short voltage drop whereby the TOF is gen-erated.


FIGURE 3.9: Principle of operation of a micro-channel plate. Illustration from[Ulr11].

Delay-line Anode

A delay-line anode serves as position sensitive detector and supplies the 2-d position of theelectron avalanche from the MCP. The conventional delay-line anode consists out of twolayers of wires, wound around a frame, ordered perpendicular to each other. The distanceof the layers in only a few millimeters. When an electron cloud coming from the MCPhits the wire, a pair of signals is generated propagating in opposite directions along thewire with constant velocity vp. The travel time of each signal is evaluated and by this theposition in the delay-line’s coordinate system is calculated via

u =vp2· (t1 − t2) . (3.13)

Obviously the sum of each time pair must be a constant. Since the electron cloud from theMCP got quite big, on its way to the delay-line anode, it hits several wires, which have aspacing of 2mm. By determining the center of the pulse, a position resolution of 0.3mm canbe reached, which is even less than the distance of the wires. To reduce noise, one uses apair wires for each layer. One functions as the "signal" wire and the other as the "reference"wire. Together they form a Lecher line, showing a defined impedance. Additionally thesignal wire is charged +50V , relating to the reference wire. By this the electron avalanchepreferable hits the signal wire, while noise is induced on both wires equally. So by calculat-ing the difference the noise can be separated from the signal.In this experimental setup, a hexanode is used on the electron side and a quad-anode on theion side of the spectrometer. A hexanode consists of three layers, instead of two layer forthe quad-anode, ordered in an angle of 60 to each other. The hexanode for the electrons hasthe advantage to reduce the dead time and increase the resolution because of the additionalinformation of one layer. The configuration of the hexanode can be seen in figure 3.10.


FIGURE 3.10: Illustration of delay-line anode using three layers. This config-uration is called hexanode [Ulr11].

3.2.5 Data Handling

The signals from the detector have to be processed electronically in a way our data acquisi-tion software COBOLD can read them. Therefor all signals from the MCP and the delay-lineanode are led out of the vacuum by short wires and are capacitively decoupled. Afterwardsthe signals get amplified and then transformed into NIM (nuclear instrument methods)pulses by a constant fraction discriminator (CFD). The CFD splits the incoming signal intwo, delays and inverts one part and recombines both afterwards. The zero-crossing of thebipolar pulse, generated this way gives the time information of the signal. Hereby the timeinformation becomes independent of the pulse height. Afterwards the pulses are convertedby the TDC (time to digital converter) and stored digitally, whereby the MCP signal is usedas a trigger for the TDC, to only save those signals arriving in a time span expected out ofthe calculated time of flight of the particles. To furthermore determine the starting point ofthe ionization and thus the exact time of flight, the bunchmarker signal, given by the ALS,is stored. This signal is generated by the electron bunches passing a certain reference pointin the ring and is then processed the same way the signals of the anode are.

23

Chapter 4

Analysis

After the data is collected in the experiments there needs to be a lot of analysis done toseparate the events of interest from all the background and false signals. How this can beachieved is explained in the following chapter.

4.1 Data Processing

The analysis of the data is done in two main steps. The initial lmf file consists mainly ofnoise and unreconstructable events. So the first step is trying to limit the file to eventswhich are of interest. This is called presorting and done with the program lmf2root. As thename implies the file gets converted into a root file during this step. The analysis makes thesecond step and is done with ROOT. ROOT is a software developed at CERN (EuropeanOrganization for Nuclear Research) for the purpose of handling big data sets. It is writtenin C++ and allows the displaying of data with histograms. In the analysis the momen-tum vectors of the electrons and ions are calculated and therefrom derivable properties likeenergy or emittance angle.

4.1.1 Presorting

In order to do a physical presorting on the data and throw away useless events it is impor-tant that first the time values from the different layers of the delay-line anode are properlycalibrated. True events must produce two time signals on each layer with a constant timesum. A first step consists of determining this offset and the width in which events are to beexpected. The time signals from the anode are transformed into a position on the detector.For this transformation there is a scaling factor needed to match the position calculationfrom each layer. Since the electron detector has a redundant third layer it is likely that thecenter of the layers does not match. In this case there needs to be a correction done bydefining an offset for the third layer.

When the position calculation is calibrated correctly the amount of data can be furtherreduced by using one of the many possible presorters. In this work the presorting wasmainly done by limiting the time-of-flight of the electrons and ions to a set window. The

24 Chapter 4. Analysis

time-of-flight is generally described by a center and a width of the distribution. The elec-trons will all have the same center depending only on the electric field in the spectrometer1. Different energies will then result in a different width of the distribution with higherenergies having a broader spread. For the ions the center of the distribution will changedepending on their mass. This enables us to select only specific ions by their time-of-flight.This way one gets rid of most of the typical background gases like water (H2O) or N2 re-maining in the chamber. In figure 4.1 a typical time-of-flight distribution of the ions isshown.

FIGURE 4.1: Ion time-of-flight of single photo ionization of CF4 done with21.4eV photons at the ALS in August 2016. There are some clearly distin-guishable peaks in addition to the expected CF+

3 , which can be assigned tomolecules in the rest gas. The sharp lines on top of the distribution have theirorigin in synchrotron photons hitting the detector and have a constant spac-ing corresponding to the bunchmarker spacing of 328ns. Note that the y-axis

shows the number of hits on a logarithmic scale.

4.1.2 Calculations in the Analysis

Time-of-flight

For the reconstruction of the momentum vector it is essential to know the time-of-flightof the particles. The electrons are moving very fast through the spectrometer. They typ-ically reach the detector after ∼ 100ns. This is shorter than the bunchspacing tbs of thesynchrotron light (328ns), which is important to measure coincidences between electrons

1This is only true if the ionization happened in the reaction region.

4.1. Data Processing 25

and ions. The electron time-of-flight tofe is the difference between the time of ionizationand the time te,MCP registered at the MCP. The time of ionization is the time signal fromthe synchrotron tbm minus a certain time offset toff . This origin in this offset is that the timesignal in the synchrotron is generated by the electrons passing a reference point somewherein the storage ring. The time that passes between this reference point and the ionization isa constant offset. We can find this offset in the analysis by doing a wiggle run as will be de-scribed in the following section. Because there can be ionization events where electrons aremissed, the tofe can be larger than the bunchmarker spacing. To avoid this, tbs is subtracteduntil tofe is smaller in value:

tofe = mod((te,MCP + toff − tbm), tbs) (4.1)

Due to their large masses the ions have much higher time-of-flights. Values of a few 1000ns

are common as can be seen in figure 4.1. The ion time-of-flights are calculated with theelectron tof as reference:

tofr = tr;MCP − te,MCP + tofe (4.2)

Momenta

Knowing the position on the detector and the time-of-flight (in addition to all set parame-ters like magnetic/electric field and spectrometer length etc.) is sufficient to calculate themomentum vector of electrons and ions. To do so, one has to solve the differential equationdescribing the motion of charged particles in static electric and magnetic fields.

~F = ~FE + ~FL (4.3)

m~r = q( ~E + ~r × ~B) (4.4)

We choose our coordinate system that way our spectrometer axis is in z-direction and thelaser beam propagation in x-direction. As a result the electric and magnetic field only havecomponents in z-direction (compare to figure ??) and one gets

~r =q

m

yB

−xBE

. (4.5)

So for solving the x- and y-component there are the two equations

vx = ωvy (4.6)

vy = −ωvx (4.7)


with ω as substitution forqB

m. Now one can differentiate equation 4.6 and plug it into

equation 4.7. Doing this with both results in the differential equations

vy = −ω2vy (4.8)

vx = −ω2vx . (4.9)

With the initial conditions ~r(t0) =

0

0

0

2 and ~v(t0) =

vx0

vy0

vz0

this can be solved to

vx = v0xcos(ωt) + v0ysin(ωt) (4.10)

vy = v0ycos(ωt)− v0xsin(ωt) (4.11)

and after integrationx =

v0x

ωsin(ωt) +

v0y

ω(1− cos(ωt)) (4.12)

y = −v0x

ω(1− cos(ωt)) +

v0y

ωsin(ωt) . (4.13)

After doing some arithmetic the initial velocities are

v0x =ω

2

(x

tan(ωt2 )− y)

(4.14)

v0y =ω

2

(x+

y

tan(ωt2 )

). (4.15)

Finally with the relation p = m · v the momentum in x- and y-direction is:

p0x =mω

2

(x

tan(ωt2 )− y)

(4.16)

p0y =mω

2

(x+

y

tan(ωt2 )

). (4.17)

Due to the high mass-to-charge ratio, for the ions this simplifies to:

p0x = mx

t(4.18)

p0y = my

t. (4.19)

The z-component of the momentum is more difficult to calculate, since three differentregions in the spectrometer need to be considered, each with a different value for the electricfield. In fact this gets so complicated that we are approximating the z-momentum with the

2Since the exact position of the ionization is not known better than within the reaction zone, we can as wellassume zero. But it is important to keep in mind that the unknown offset in position limits our resolution.

4.1. Data Processing 27

Newton-Method:

p0z = m(− a1t+

√2a1l1 +

a1

a2

√2(a1l1 + a2l2)−

a1

a3

√2(a1l1 + a2l2) +

a1

a3

√2(a1l1 + a2l2 + a3l3)

) (4.20)

with the accelerationai =

qEim

and the length of the different spectrometer regions li.

Energy

The energy can easily be calculated with the momentum:

E =|~p|2

2m(4.21)

For the ions it is often more interesting to know the kinetic energy release (KER) of the re-action. The KER is the sum of the kinetic energy of all molecular fragments after a breakup(not including the electron). The KER can be calculated with help of momentum conserva-tion, as long as there is not more than one undetected fragment. For the case of a two bodybreak-up with one neutral fragment it is:

KER = Er1 + Er2 (4.22)

and~vr1mr1 = −~vr2mr2 , (4.23)

so that~vr2 = −~vr1

mr1

mr2(4.24)

and then

Er2 =mr2v

2r2

2=m2r1v

2r1

2mr2=mr1

mr2Er1 . (4.25)

So the KER is calculated withKER = Er1 +

mr1

mr2Er1 (4.26)

where r1 is the detected ionic fragment and r2 the neutral fragment.

Angles

Knowing the momentum vectors we can define the angles θ and ϕ in the labframe as shownin figure 4.2. ϕ denotes the azimuth angle in the spatial coordinates x and y. It is definedfrom 0 to 2π. While θ is the polar angle between the z-axis and the momentum vector. θ isonly defined from 0 to π. They are both calculated via the momentum:


FIGURE 4.2: Illustration of the spherical coordinate system used in this calcu-lations. X is the direction of beam propagation, y is the jet direction and thedetectors are along the z axis. Picture taken from Wikipedia, User: Ag2gaeh.

ϕ = arctan(pypx

) (4.27)

θ = arccos(pz|~p|

) (4.28)

Other possible angles are:

ϕx = arctan(pzpy

), ϕy = arctan(pxpz

)

andθx = arccos(

px|~p|

), θy = arccos(py|~p|

)

4.1.3 Correcting Electron Momentum Kick

The momentum of the electron is small compared to the momentum of the heavy ion, butdue to momentum conservation in the ionization process the parent ion receives the samemomentum as the photoelectron in opposite direction. If the molecule is subsequently dis-sociating this leads to a difference between the ion momenta in the molecular center ofmass frame ~kc.m.ion and momentum in the laboratory frame ~klabion. With the assumption thatthe ionization can be treated as a two body break up the ion momentum in the center ofmass frame is

~kc.m.ion = ~klabion +mion

mparention

~klabe . (4.29)

Here mion is the mass of the ionic fragment after the dissociation and mparention is the mass of

the complete parent ion before the break up. This correction of the momentum kick by theelectron only has a measurable effect for small ion momentum as it is the case for the abovedissociation threshold photo ionization experiments done on H2 and D2. In that case the

4.2. Calibration 29

correction done is~kc.m.ion = ~klabion + 0.5~klabe . (4.30)

4.2 Calibration

Since the momentum calculation showed above is very sensitive to the electric and mag-netic field (much more sensitive than we can adjust them) there is need to do a more precisecalibration.

4.2.1 Magnetic Field

FIGURE 4.3: Exemplary wiggle-plot of Helium. Here the photon energy was42.6eV and the electric field was set to be 1V/cm. The distance between the

nodes corresponds to the gyration period Tg .

The magnetic field is one of the easiest parameters to determine in an experiment, dueto the gyration of the electrons. To accurately measure the gyration period of the electronwe are doing a wiggle run each time the magnetic field is set to a new value. It is calleda wiggle run because with a combination of a low electric field and high energy electronscauses the electron spread in time-of-flight to be larger then the gyration period and we aretherefore able to see two or more nodes of the gyration period Tg

3. In a plot showing thetime-of-flight versus a spatial coordinate we can therefor just see the gyration period as thedistance of the wiggles. In figure 4.3 the tof is plotted versus the radius on the detector. This

3In this wiggles the spatial spread of the electrons onto the detector is minimal and hence the spatial resolu-tion is really bad. That is why we normally try to have the electron time-of-flight spread centered within twonodes of the gyration period, when choosing the parameters for an experiment.


plot is called a wiggle-plot because the wiggles can be seen very nicely. If the tof is plottedversus x or y it is called a fish-plot. The magnetic field can then be calculated with

B =2πme

eTg. (4.31)

By extending the gyration towards the zero point of time-of-flight it is also possible to getthe time offset toff . Because if there was no time offset the first node of the gyration wouldhave to be at time zero. Any offset in timing will shift the first node to the value of toff .

4.2.2 Energy Calibration

Photon Energy

The absolute value of the photon energy given adjusted with the monochrometer is typi-cally slightly off. So for the correct interpretation of the experiments it is important to do acalibration of the photon energy in order to figure out the value as exact as possible. Sincethis energy offset is constant it is sufficient to measure it for one energy. The typical wayto measure the absolute energy is by doing single ionization with a very well known ion-ization threshold. By starting below the threshold and then slowly increasing the photonenergy it is possible to determine the energy which corresponds to the ionization energyvery precisely. The discrepancy is then the offset.We were using Helium with an ionization potential of Ip,He = 24.59eV [Lid92]. The mea-sured threshold for single ionization was however Ip,measured = 24.63eV . This results inan offset of 0.04eV . In the following all the given photon energies will already have beencorrected by this offset.

Electrons

The energy calibration of the photoelectrons can also be done with Helium. Once the pho-ton energy is determined the expected electron energy is also known. This way there isanother Helium calibration run needed fro the same field parameters and similar photo-electron energies. The electron calibration is then done with the Helium run and all theparameters obtained are also used for the actual experiment.

In the case of single ionization and dissociation of molecular Hydrogen an additionalHelium run is not necessary. This is because the dissociation threshold of H2 is also verywell known and is Ediss,H2 = 18.075eV [Sha70]. At the fragmentation of H2 all the remain-ing energy has to be divided between the kinetic energy of the fragments and the electron,since the fragments are atoms and the only way to store energy would be electronic excita-tion, which is way to high. So with energy conservation we get:

Esum = Eγ − Ediss,H2 (4.32)

4.2. Calibration 31

FIGURE 4.4: Electron energy plotted versus the KER for single ionization anddissociation of H2 with 18.5eV. This data was taken at the ALS in June 2017.

Plotting the KER versus the electron energy must therefore be a diagonal, which can beused for the energy calibration of the electrons as well as the ions. An example is shown infigure 4.4.

4.2.3 Detector Orientation

FIGURE 4.5: Position on ion detector for ionization of H2 with 18.5eV pho-tons. The bright dot is made up by theH+

2 ions, while the bigger circle comesfrom H+. Clearly visible is the hot-gas stripe lying horizontally on the detec-tor. It is there because of the photons ionizing gas in the chamber along their

direction of propagation.

During the data processing and momentum calculation we are loosing the informa-tion of the detector orientation. Considering the ions and electrons separately this is not aproblem because the momentum sphere is symmetric around the axis of light polarization(z-axis), so that the x/y-plane is rotational symmetric. But to look at coincidences betweenelectrons and ions it is critical to calculate both momentum vectors in the same coordinatesystem. To do so we need to retrieve the information of the detector orientation.This can be done in two steps. First it is possible to identify the orientation of the ion de-tector due to a distinct feature introduced by the photon beam ionizing the rest gas in thechamber (See figure 4.5). Because the photon beam propagates along the x-axis, this hot-gas


stripe also has to be orientated along the x-axis. But since the direction of beam propagationis not known, the detector might be flipped along the x-direction. As long as the electrondetector has the correct position in relation to the ion detector this is not of importance.The orientation of the electron detector can be retrieved by making use of the momentumconservation during the single ionization of molecular Hydrogen. For the reaction

H2 + γ = H+2 + e− (4.33)

the momentum conservation reads as follows

mH+2~vH+

2= −me~ve (4.34)

which means plotting the single components of the ion momentum versus the respectivecomponents of the electron momentum should results in diagonal centered at zero. In figure4.6 those plots are shown for the correct orientation.

Tilted Spectrometer

It is likely that during the set up the spectrometer was not aligned totally parallel to themagnetic field. This will result in a different position of the electrons depending on thetime-of-flight, due to the orthogonal part of electric and magnetic fields. It is easy to correctthis in the analysis with a linear offset:

xnew = xold + a · tofe (4.35)

ynew = yold + b · tofe

This is best done with the fish plots from the wiggle run, which show the position versusthe time-of-flight. Because it has several nodes visible it is easy to see the slope here. Infigure 4.7 one fish plot for the y-coordinate is shown.

Distribution on Detector

Since the calculation of the momentum is based on the position on the detector we need tomake sure that the distribution on the detector is centered. For the electrons this is straightforward and best done with the fish plots. For the ions we need to consider the velocityoffset in the direction of jet propagation. So in y-direction we actually need to center thehot-gas stripe and not the ion distribution itself4.

4This is also a way to check the calculated jet velocity in section 3.2.2. If the hot-gas stripe is centered andthe jet velocity correctly calculated the ion y-momentum should be centered around zero.

4.3. Selection of Reaction Channel 33

FIGURE 4.6: Showing different components of the electron momentum onthe x-axis versus the corresponding component of the ion momentum on they-axis in coincidence. For the correct detector orientation we should ideallysee a perfect diagonal. In (c) this is very close to it, because the resolutionin time-of-flight direction is very good. The resolution in x and y are limitedby size of the reaction zone and the detector resolution. This washes out theion momentum as seen in (a) and (b), because for same momentum the ionshave a much smaller position spread than the electrons. The resolution in (a)is worst, since x is the photon direction and the reaction zone is here limitedby the width of the jet as explained in section 3.2.3. Those plots where done

with the target gas H2 and a photon energy of 19.76eV .

4.3 Selection of Reaction Channel

Doing a photo ionization experiment with molecules there are typically more then one pro-cesses happening. In the analysis it is important separate those different processes to look atthem as differential as possible. A big part of the selection was already done with the choiceof time-of-flight in the presorting but this does not get rid of all the pollution from differentchannels. "Cleaning up" the data can happen in a lot of different ways and can vary fordifferent target gases and reaction channels. In the following I only listed the methods usedfor the experiments in this thesis.


FIGURE 4.7: Shown is the y-fish plot of Helium. The photon energy was set to42.6eV with a low electric field to make several nodes of the electron gyrationvisible. Here we can see that the spectrometer and magnetic field where veryclose to parallel and we only need a small correction. The inserted red line

indicates the slope, which needs to be corrected.

4.3.1 Energy Conservation

In all reactions happening there always has to be energy conserved. And as long as wecan determine the energy of the final state5, we can "enforce" energy conservation. In thedissociation of H2 this is done with a gate of form:

(Eγ − Ediss −∆/2) < Esum < (Eγ − Ediss + ∆/2) . (4.36)

Here ∆ corresponds to the width in energy.

4.3.2 Pollution

Jetdot

Looking at the ion time-of-flight spectrum of Hydrogen in figure 4.8 we can see a few repet-itive peaks in the spacing of the bunchmarker signal (328ns). One of this repeating peaks iscaused by the dominating H+

2 ions. In the time-of-flight calculation of the ions it is possiblethat the event gets assigned to the wrong bunchmarker. As a result there will be H+

2 peaksset apart from the real peak by n · 328ns, with n being any integer number. Since the H2+

molecules only get the momentum kick by the electron, their momentum spread is verysmall compared to H+. This makes it look like a big dot within the H+ momentum and isfor that reason denoted as jetdot.The jetdot appears as a small sphere within the H+ momentum sphere as can be see infigure 4.10. Depending on the KER of the dissociation the time-of-flight spread of the H+

ions includes one or several jetdots. In figure 4.8 a comparison between the small KER

5This can be difficult if the final state includes a molecule, because we normally do not know how muchenergy went into its internal degrees of freedom.


FIGURE 4.8: Ion time-of-flight of Hydrogen. On top the photon energy is19.76eV , on the bottom 18.16eV . The big peak on the right side is H+

2 . Thebroader peak between 2000ns and 2500ns is H+. The H+

2 peak is repeatingwith a spacing of 328ns, which corresponds to the time between the electronbunches in the synchrotron. This comes from a wrong bunchmarker assign-ment for the ions. The sharper peaks are from the synchrotron light itself,

because of photons hitting the ion detector.

(KER < 0.1eV ) of dissociation with 18.16eV and a big KER (KER < 1.7eV ) with 19.76eV .The first one only has one jetdot within its time-of-flight, while the latter one has almostfour. Because we are measuring electrons and ions in coincidence and the electrons comingfrom the H+

2 have a different energy then those originating in the H+ dissociation chan-nel we should not have a problem with the jetdot if enforce the energy conservation asexplained above. But there is a chance of measuring wrong coincidences between H+

2 ionsand any electrons in a way that the energy sum is the same as we expect from the hydrogendissociation. This can be seen in figure 4.9 where the electron energy versus the KER isplotted. In the plot on the left the diagonal line as expected of the dissociation channel can


be seen. On top of it are several vertical lines caused by the jetdot. Considering only thevertical line will get rid of most of the pollution but not of the part which is directly on topof the diagonal. So to get rid of the jetdot we need to "cut" it out in the momentum space.By doing so we are loosing all the data within a defined region in the momentum space(including the relevant events). To make sure the data loss is as small as possible whilecutting out all of the jetdot we chose a spherical gate in momentum space. The "cleaned"data can be seen in figure 4.9 and 4.10 on the right side.

FIGURE 4.9: This two plots show the energy map of the hydrogen dissoci-ation channel with 19.76eV photons. The diagonal line shows the energyconservation between the electron and the KER. The plot on the left showsthe data without any gating in the analysis, while the plot on the right hasa gating on the jetdots and the bunchmarker signal, but not on the energy

conservation.

Bunchmarker Signal

There are two additional repetitive peaks in the ion time-of-flight spectrum seen in figure4.8. They are sharper than the bin size of one nano second, which is an indication that thissignal is generated by photons. And with a repetition every 328ns − which is the timebetween the light pulses of the synchrotron − we conclude that this is caused by somephotons being non resonantly scattered off the molecules in the gas jet. This scatteringprocess has to happen on a time-scale shorter than hundreds of picoseconds to generatesuch a sharp signal. At a closer look there is also a second sharp peak only 7ns after thefirst one. This one is much smaller and could be explained by photons being reflected fromthe beamdump and then again scattered at the jet. The beamdump is approximately 1m

after the jet which fits well with 2m path length resulting from 7ns. The reflected photonsare hitting the MCP all over, which causes several lines of constant z-momentum in the leftside of figure 4.10. Since this bunchmarker signal is very narrow in time-of-flight it is easyto cut it out there. By zooming into the ion time-of-flight with a higher binning the widthof the bunchmarker signal turns out to be only 200ps, which is just the time resolution of


FIGURE 4.10: H+ ion momentum sphere for 19.76eV photons, projected ontothe three planes of the coordinate system. The left row shows the momentumbefore any gating in the analysis and has logarithmic color scale. The rightshows the final data wit gates on the energy, jetdots and bunchmarker signal.

the delay line anode [Gmb17] (the synchrotron pulse length is ∼ 100ps). So by taking slicesslightly larger then 200ps in the ion time-of-flight, we get rid of the bunchmarker signalwith a minimal loss of data. The energy and momentum plots after the mentioned gatescan be seen in figure 4.9 and 4.10 on the right side.

Hot Gas Stripe

Another source of pollution is the hot gas stripe as explained in section 4.2.3. The hot gasstripe is clearly visible in the ion momentum plots (see figure 4.10), but it vanishes with thegate on energy conservation.


4.3.3 Final Calibration of Momentum

After the detector and spectrometer are calibrated and the reaction channel is selected, westill need to do some fine tuning in the momentum. This final step of the calibration needsto be done after the selection of reaction channel because it depends on the energy andmass of electrons and ions. So do ions with a different mass spend different times in thespectrometer and electrons with various energies are propagating in different regions of thespectrometer and therefore feel a slightly distinct electric field.Independently of that the electron and ion momentum spheres always need to be symmet-ric along the axis of light polarization (z-axis). And because the molecules in the jet arecompletely randomly orientated the electrons and ions are also isotropically emitted in thex- and y-plane. Being aware of this we also know that the energy of electrons and ionsneed to be constant independently of the direction they are emitted. If this is not the casethis normally means that the momentum sphere is not centered around the zero, or is not aperfect sphere. It can then be corrected by slightly shifting or stretching the momentum.

FIGURE 4.11: Electron energy for the H+2 channel plotted versus the various

angles of the electron momentum vector in the labframe (compare to 4.1.2).This was used to calibrate the electron momentum, because the energies areso distinct. The white hole in the plots shows the edge of the MCP. The fieldswere set to focus on the low energy electrons, so that the higher energies do

not fit completely on the detector. The photon energy was 18.5eV .

In hydrogen the electrons could be easily calibrated because of the very distinct energies

4.4. Error Analysis 39

the photoelectrons obtain in the non-dissociative reaction channel. They are so distinct dueto the total energy gets shared between the vibrational energy ofH+

2 and the photoelectron.By optimizing the calibration for the lowest energy electrons from this channel get veryclose to the correct calibration for the electrons in the dissociative channel.The calibration of the H+ ions can then be done with the energy sum of the electron andthe KER. Exemplary plots used for the calibration are shown in figure 4.11 and 4.12.

FIGURE 4.12: Energy sum of the KER and electron energy in the H+ channel,versus two different angles of the ion momentum vector. On the left it iscos(θ) of the z-momentum and on the right it is ϕ in the x,y plane. This

experiment was done with a photon energy of 18.5eV .

In the case of CF4 the calibration was also done with plotting the energy sum versus thevarious angle of electron and ion momentum. In the single ionization of CF4 there are threepossible final electronic states, which vary in energy (see figure 6.1 in later chapter). Thisresults in three different lines of constant energy, which all have to be straight. But since theelectron energy differs by a few electron volts, each final state has its own optimal electroncalibration. Yet the KER is almost the same, so the the calibration should not change for theions. In figure 4.13 the angles of the electron momentum vector versus the energy sum areshown.

After the final calibration is done it is necessary to adjust all the gates, since the energyand momentum will have slightly changed.

4.4 Error Analysis

It was of great importance for us to minimize uncertainties as much as possible during theexperiment and the subsequent analysis and determine all measured variables with bestaccuracy. But nevertheless it is not possible to eliminate errors. All the more it is importantto be aware of limitations of the experimental apparatus and possible inaccuracies in thecalibration.

Detector

Because each electron cloud released by the MCP results in a voltage drop on the delay-lineanode which then is detected, there is a certain time which it takes to regain the full voltageat the anode before another electron cloud can be detected. This time is called deadtime.


FIGURE 4.13: CF4 energy sum versus the angles of the electron momentumvector in the labframe. Compare to 4.1.2 for the physical meaning of thedifferent angles. There can be seen three lines of constant energy, which cor-respond to the different final electronic states X2T1, A2T2 and B2E (top tobottom). The high energy electrons of the X2T1 state are not completely cap-tured on the detector for the chosen field values and the photon energy of

19.4eV .

The deadtime limits the detection of a second electron arriving at a short (few nanoseconds)time after the first one in a distance that the electron clouds are spatially overlapping. Sincewe are not looking into double ionization processes and hence do not expect electron ar-riving at such short time differences we do not need to worry about the deadtime of thedetector.

FIGURE 4.14: CF+3 ion distribution on the detector. Top: Data taken with

21.4eV photons in August 2016. On the left side of the distribution two blindspots can be seen. Bottom: Taken with 20.4eV photons in June 2017. There

are three small spots of inefficiency visible.


The micro channels in the MCP have a limit in how many amplifications they can dowithin their lifetime. If this number of amplifications is reached they do not amplify thesignal anymore and dark spots can appear. Depending on how many micro channels in acertain region of the MCP do not work anymore there can be inefficient spots up to almostcomplete blind spots. Doing experiments over long times with non-dissociating moleculesor atoms can cause such blind spots. Because the momentum of those molecules/atomsis very small they do barely spread on the MCP and cause the micro channels at this spotto quickly reach their limit of amplifications. Since MCP’s (especially the big ion MCP)are very expensive and therefore can not be replaced that often, we have to live with thoseblind spots and inefficiencies. In the experiment done in August 2016 there was one big andone small blind spot on the MCP, while for the experiment done in June 2017 the MCP wasflipped to probe another part of the MCP. This removed the blind spots but therefore a fewsmall inefficiencies appeared. The comparison of the CF+

3 ion distribution on the detectorcan be seen in figure 4.14. Because electrons and ions are measured in coincidence, missingions because of a blind spot also means the corresponding electrons are missing. This couldpossibly cause some asymmetries in the photoelectron angular distribution and thereforewe always tested observed effects for each half of the momentum sphere separately to seeif they were caused by the missing ions.

As shown in 4.3.2 the time resolution of the detector is better than 200ps. So the uncer-tainty in z-momentum depends on magnitude of the electric acceleration field, because thisis what determines the spread in time-of-flight.

The spatial resolution of the used MCP and delay-line anode is less then 0.1mm [Gmb17].So the resulting x-and y-momentum resolution is depending on the spatial spread on thedetector. But since the reaction zone is less defined than 0.1mm our resolution is actuallylimited by the size of the reaction zone.

Reaction Zone

The size of the reaction zone is in x-direction confined by the width of the gas jet (1.1mm)and in y-and z-direction by the focal width of the photon beam (0.2mm). That results inthe worst resolution in x-direction, while the best resolution is in z-direction due to thetime focusing spectrometer. An comparison of the resolution in the different directions ofmomentum can be seen in figure 4.15. It shows the momentum spread of H+

2 ions andthe corresponding electrons. Since all of the H+

2 momentum originates from the momen-tum kick of the electron due to conservation of momentum, the magnitude of momentumshould be the same in all directions. So a larger ion momentum indicates the momentumresolution on the ion side.


FIGURE 4.15: On the left the electron momenta are shown and on the rightthe corresponding H+

2 ion momenta. The order of momentum from top tobottom is x, y and z. The momentum should look the same, since the ionmomentum consists only of the electron momentum kick. The discrepancycomes from a bad momentum resolution of the ion detector. We can see theresolution is worst in x-direction and best in z-direction as explained in thetext. This plots are from the measurement done in August 2016 with 18.56eV

photons.

Energy Resolution

The energy resolution is limited by the uncertainty in the absolute energy of the photons,which depends on the slit opening of the monochromator and the uncertainty in determin-ing the momentum vector. The first is given by the control station of the monochromatorand was between 0.01eV and 0.013eV for the photon energies used. The latter is stronglydepending on the field values of the spectrometer and is also different for the direction ofemittance. The best resolution will again be in z-direction. For the ion side we can estimatethe energy resolution via figure 4.15 to ∼ 3meV in x-direction for the data taken in August2016 with 18.56eV photons. For the electrons in the same measurement we can estimate theresolution in x-direction to ∼ 30meV and in z-direction to ∼ 15meV 6.

6This is a very coarse estimation done by comparison of the spatial/temporal resolution of the detector andthe spatial/temporal spread of the electrons. The energy resolution of the detector is depending of the spreadof the distribution and therefore can change a lot for different field parameters.


The best energy resolution is always achieved at low energies. That is because the en-ergy is calculated out of the momentum and scales with its square and momentum resolu-tion of the detectors is constant. To make best use of this high resolution at low energy wecan use energy conservation and recalculate the ion energy. This only works for the disso-ciation of H2 and D2, because it is essential to know the expected kinetic energy, in orderto make use of energy conservation. The kinetic energy release of those dissociations wascalculated with

KER =(KER′ − EeKER′ + Ee

+ 1)Eγ − Ediss

2(4.37)

where KER′ is the kinetic energy release calculated from the center of mass frame of themolecule, Ee is the calculated electron energy and Ediss is the dissociation energy of themolecule.

Another limitation is the population of several rotational states in the molecule. Due tothe cooling of the jet via the supersonic expansion we can be sure to be in the vibrationalground state, but the energy spacing of the rotational states is much smaller and it is notonly the rotational ground state which is populated. So depending on the initial rotationalstate the energy sum of the electron and KER does vary.

Calibration

An imperfect calibration can further increase the experimental uncertainties. To minimizethis contribution the calibration was done with highest accuracy possible. But for somedatasets the calibration was more difficult than for others and resulted in a higher uncer-tainty. For example the ionization of H2 with 19.8eV photons was especially difficult, be-cause the high energetic electrons flying towards the ion detector were facing field distor-tions from the fringe fields of the booster region in the ion spectrometer. Hence calibratingthose electrons with one set of parameters is tough and compromises have to be made. Infigure 4.16 the electron energy versus the cos(θ) of the electrons is shown. This plot showsthe difference between the electrons flying towards the ion detector or away most extreme.First electrons are on the right, latter on the left side of the plot. To solve this problem wehave the option to only look at the electrons which go directly towards the electron detec-tor. Since the distribution is symmetric, we could even mirror that half of the momentumsphere. Generally speaking we can see in the various calibration plots, how good the cali-bration is and estimate from there the error in the momentum and energy.


FIGURE 4.16: Electron energy versus cos(θ) of electrons for the non-dissociative ionization of H2 with 19.8eV photons. On the left side are elec-trons which fly directly towards the electron detector, while the ones on theright fly first towards the ion detector and get distorted by the electric field

from the booster field in the ion spectrometer.

45

Chapter 5

H2 - Molecular Hydrogen

H2 is the smallest and most abundant molecule in the universe and its simplicity makes itthe easiest molecular system to study. Its ground state is perfectly symmetric with electronwave functions of either gerade or ungerade parity. If the molecule is dissociating after −for example − a photo ionization the symmetry leads to an equal likelihood to localize theremaining electron at one nuclei or the other. Quantum mechanically this is because in asymmetric molecule the wave functions have a well defined parity and since all observablesare squares of wave functions or transition matrix elements they will be symmetric too.

However there are ways to break this intrinsic symmetry of a homonuclear molecule.To do so the system must be brought into a coherent superposition of gerade and ungerademolecular states. In the last years there were several experiments done showing differentways to achieve just this in H2 and D2.

5.1 Breaking the Symmetry - Past Experiments

With External Field

Probably the easiest way to break the symmetry in a symmetric molecule is with an externalfield, which couples a gerade and an ungerade state. There have been several experimentswithin the last years taking this approach ([Kli+06], [Ray+09], [Wu+13], [Mi+17]). The un-derlying principle is mostly the same:The molecule is first ionized by the laser field and a vibrational wavepacket is launched inthe 1sσg state. Dissociation is then triggered in a second step as the nuclear wavepacketgets excited into the repulsive 2pσu state. This excitation can happen via recollision of theelectron or directly by the laser field as sketched in figure 5.1. This principle is known asthe two-step process [Kli+06] and by mixing even and odd parity of the nuclear states thesymmetry of the dissociation is broken.

In the experiment done by Ray et al. [Ray+09] they identify three possible ways toachieve a coherent superposition of even and odd parity from the states 1sσg and 2pσu in atwo-color laser field. One is by recollision of the electron driven by the laser field which ex-cites the wavepacket into the repulsive 2pσu state. During the dissociation the still presentlaser field can then transfer part of the population back into the 1sσg state. Alternatively the

46 Chapter 5. H2 - Molecular Hydrogen

FIGURE 5.1: Schematic of two step dissociation process in D2. The moleculefirst gets ionized into the 1sσg state and then in a second step excited intothe 2pσu. The excitation can happen via electron recollision (dashed line) orby the laser field (Above threshold dissociation (ATD) or bond softening(BS),thin arrows). In this experiment by Ray et al. [Ray+09] the additional sec-ond harmonic (thick blue arrow) was used to enable the coherent superposi-tion of 1sσg and 2pσu, which leads to symmetry breaking. Figure taken from

[Ray+09].

excitation into the repulsive state can happen by absorbing photons from the laser field. Be-cause of the two-color field there are several combinations of absorbing and emitting pho-tons possible leading to the same kinetic energy release but with a different parity: Absorb-ing three 800nm photons and emitting one 400nm photon transfers the nuclear wavepacketinto the 1sσg state and leads to the same KER as absorbing only one 800nm photon whichbrings the nuclear wavepacket into the 2pσu state. Or absorbing one 400nm photon andemitting one 800nm photon compared to just absorbing one 800nm photon. Those possibleprocesses leading to an asymmetric ion emission are shown in figure 5.1. By controlling therelative phase between the two short laser pulses it is even possible to steer the asymmetryof ion emission for all three processes as shown in figure 5.2 from the same author [Ray+09].

Without External Field

A possible way to achieve a coherent superposition of gerade and ungerade states withoutan external laser field was shown by Martín et al. in [Mar+07]. Here they show the single-photon induced symmetry breaking in H2 and D2 via autoionizing intermediate states. Todo so the H2/D2 is resonantly excited into the first doubly excited state with a 33.25eV

5.1. Breaking the Symmetry - Past Experiments 47

FIGURE 5.2: Density plot of Asymmetry of D+ ion emission from D2 in atwo-color laser field. The asymmetry is shown as a function of the phasedifference between the two laser pulses and the ion energy. On the left panel

a log plot of the ion yield is shown. Figure from [Ray+09].

photon. Since it is a repulsive state the nuclei start moving apart and then auto ionize byemitting an electron. In this auto ionization step the H+

2 /D+2 ion can either end up in the

1sσg or the 2pσu state. If the kinetic energy release and electron energy are the same it notpossible to distinguish between those states and they need to be described by a coherentsuperposition, which in turn leads to the symmetry breaking. As seen in figure 5.3 (2) theoverlap of 1sσg and 2pσu is biggest in a region between 7 − 10eV KER. Which is also theregion where they observe the strongest asymmetry of the photoelectron emission in themolecular frame (see figure 5.3, (3)).

With Coulomb Field of Photoelectron

Jet another way to break the symmetry in a homonuclear molecule was postulated by Serovand Kheifets in 2014 [SK14]. Generally speaking the photoelectron is moving much fasterthan the nuclei and hence the dissociation is normally considered to be independent fromthe ionization. But for very slow photoelectrons this simplification is not valid anymoreand the electron and molecular ion can interact via their coulomb fields. This is becausethe electron is moving slow enough that it is still in the vicinity 1 of the molecule while itis dissociating. The electron can then localize − via its coulomb tail − the bound electronpreferentially to the nuclei farthest away from itself. The distance of the electron (whichis inversely proportional to the the strength of its coulomb field) increases with electronenergy. Thereby the asymmetry is expected to be stronger for lower electron energy and

1For a classical treatment of the electron propagation one gets a distance of 26a.u. for a kinetic energy of0.01eV after 7fs. 7fs is approximately the time the nuclear wavepacket needs to propagate to the outer turningpoint of the 1sσg curve. For a 0.5eV photoelectron the distance is already 58a.u. and for 1eV it is 82a.u..


FIGURE 5.3: Single photon induced symmetry breaking of H2 dissociation.(1) Semiclassical pathway for dissociative ionization leading to H+

2 (1sσg) aswell as H+

2 (2pσu) (respectively for D+2 ). If they both lead to the same KER

and electron energy they are undistinguishable and interfere (Note that inthey can only lead to the same energy if treated fully quantum mechanicallyand not in this simplified semiclassical picture.). (2) CalculatedD+ kinetic en-ergy distribution for dissociative ionization by absorption of a 33.25eV pho-tons. (3) Angular distribution of the electrons as a function of KER for disso-ciative ionization of D2. The molecular orientation is orthogonal to the pho-ton polarization, with the blue circle indicating the deuteron and the greencircle the deuterium. The asymmetry is strongest in the region of 7 − 10eVKER (c and d). (4) Showing the KER distribution with the blue areas indicat-

ing the regions used for the plots in (3). Plots taken out of [Mar+07].

5.2. Idea of this Experiment 49

higher KER. The breaking of the symmetry is possible because once the nuclear wavepacketreaches a region where the 1sσg and the 2pσu curve come energetically close, the electronfield couples the bound vibrational and continuum states of 1sσg to the continuum of 2pσu.A schematic of this principle is shown in figure 5.4.

FIGURE 5.4: Left: Schematic showing the coupling of 1sσg and 2pσu states bythe field of the photoelectron. Adapted from [Wai+16]. Right: Sketch of the

electron localization by the retroacting photoelectron. Figure by T. Weber.

This symmetry breaking was experimentally confirmed in the photo dissociation of H2

by M. Waitz et al. in 2016 [Wai+16]. They observe an asymmetric molecular frame photo-electron angular distribution (MFPAD), with an increasing asymmetry for increasing KER.Those results are shown in figure 5.5.

5.2 Idea of this Experiment

The motivation for this experiment is to study further and in more depth the retroaction ofthe photo electron onto its source as predicted by Serov and Kheifets [SK14] and first shownby Waitz et al. [Wai+16].To do so we photo ionize molecular Hydrogen with several photon energies just above thedissociation threshold at 18.075eV [BSS94]. This way we can access two possible channels.A non-dissociative and a dissociative one:


FIGURE 5.5: Left: MFPADs of H2 showing the symmetry breaking due toelectron retroaction. Right: Asymmetry parameter δ versus KER for different

photon energies. Both plots are the results from Waitz et al. in [Wai+16].

H2 + γ = H+2 + e− non− dissociative

H2 + γ = H +H+ + e− dissociative

Those two channels can easily be distinguished in the recoil time-of flight − because ofthe difference in mass of H+

2 and H+ − as well as in electron energy. In figure 5.6 the elec-tron energy for both channels is shown. The electrons originating from the non-dissociativechannels have a very distinct structure, while the electrons from the dissociation are con-tinuous. The reason for that can be found in the energy diagram of H+

2 (Figure 5.4). Thebound state below the dissociation threshold can only have distinct vibrational energies,while above the threshold the energy spectrum is continuous.

In contrast to the experiment done so far we use lower photon energies and longer dataacquisition resulting in much more slow photo electrons in the dissociation channel. Ourhope is that this enables us to observe a much stronger asymmetry than in 5.5 and to resolvethe relation between KER and strength of asymmetry in more detail, since we suspect morestructure in it than by the existing theory predicted.Moreover we were able to show the effect of the retroacting photo electron for the firsttime 2 on molecular deuterium. The difference of the effect between H2 and D2 helps us tounderstand the dynamics of the retroaction and is in good agreement with the expectationsby the theoretical description from Serov and Kheifets.

2To the best of our knowledge.

5.3. Results of H2 and D2 Photo Dissociation 51

FIGURE 5.6: Electron energies for photo ionization of H2 with 18.5eV . In redare the electrons from the dissociative channel, in black the electrons fromthe non-dissociative channel. The two distributions are normalized and dueto the very low cross section of the dissociation. Those electron counts are

multiplied by a factor of 10 for better comparison.

5.3 Results of H2 and D2 Photo Dissociation

5.3.1 Energy maps

In figure 5.7 the energy maps for the different photon energies are shown. As expected thedistribution lies on a diagonal with the energy sum:

Esum = hν − 18.075eV . (5.1)

We can also note a change in distribution for increasing photon energies. For the lowerphoton energies the distribution seems almost continuous along the diagonal, while for thehigh photon energy case there are almost no events with high KER and low electron ener-gies. This shift of distribution reflects the cross section for photo ionization into the 1sσg

state, which is quickly decreasing above the dissociation threshold. Another interestingfeature in the energy maps are the several diagonals we can observe for low photon ener-gies. This feature is most pronounced in the lowest photon energy case (18.16eV), wherewe can clearly identify several peaks in the energy sum shown in figure 5.8. Fitting the


FIGURE 5.7: Energy maps of H2 photo dissociation. Shown is the photo elec-tron energy versus kinetic energy release of the dissociation. Since the sumof both must be constant the distribution lies on a diagonal. As the photonenergy is increasing (marked in each upper right corner) the diagonal movesoutward and the distribution along the diagonal changes, as a sign of a de-creasing cross section to dissociate through the 1sσg state for higher KER.Also note that the distribution actually consists of more than one diagonal,indicating the source of different energetic states (more detailed explanation

in text).

distributions with four Gaussian peaks of the form:

y = y0 +A

w√π/2

exp(− 2

(x− xc)2

w2

)(5.2)

results in the best agreement.Since the synchrotron radiation is very well defined in energy, the reason for those dif-

ferent energy sums has to lie in the initial state. In a mononuclear diatomic the possibleenergetic states are limited to (compare to chapter 2.1):

• electronic excitations

• vibrational excitations


FIGURE 5.8: Left: Energy sum of fragments after photo ionization of H2

with 18.16eV photons. There can clearly be seen four peaks, which can beattributed to the initial rotational states of H2 with the rotational quantumnumber J. The fits are Gaussian peaks. Right: Sketch of two nuclear spin iso-mers of Hydrogen. Parahydrogen has nuclear spins oriented anti-parallel,which results in the spin singlet state. Orthohydrogen has parallel nuclear

spins, which results in the spin triplet state.

• rotational excitations

Electronic excitations need way too much energy. Same for vibrational excitation, wherethe first vibrational excitation takes about 0.1eV in H2. So it must be the first four rotationallevels we can resolve here. Estimating the expected rotational energies in the hydrogenmolecule with the rigid rotor:

EJ =h2

2µR2J(J + 1) (5.3)

gives values very close to the ones observed:

J 0 1 2 3

EJ calc. [eV] 0 0.0175 0.0526 0.105EJ meas. [eV] 0.03 0.047 0.075 0.127

EJ meas. -E0 [eV] 0 0.017 0.045 0.097

This good match of the peak energies leads to the conclusion that we are able to energet-ically resolve the rotational levels of hydrogen. Because the possible rotational quantumnumber of H2 is restricted by the symmetry of the nuclear spin function it follows that we


can differentiate between the spin isomers of molecular hydrogen 3. Orthohydrogen (paral-lel spin) can only exist with odd J, while Parahydrogen (anti-parallel spin) only exists witheven J (compare to figure 5.8).

Since the population of those rotational states follows the Boltzmann distribution, thisshould offer us a possibility to estimate the rotational temperature in the gas jet. To do thisestimation we first have to account for the degeneracy of the spin singlet/triplet states aswell as for the degeneracy of the rotational levels:

J 0 1 2 3

rotational degeneracy 2J + 1 1 3 5 7spin degeneracy 1 3 1 3

total degeneracy (product of both) 1 9 5 21

Plotting the peak area of the Gaussian fits in figure 5.8 divided by the factor of degeneracydoes not yet result in a Boltzmann distribution as can be seen in 5.9.

FIGURE 5.9: Area under the Gaussian peaks of fit in figure 5.8 divided by thetotal degeneracy of each rotational states plotted versus the rotational energyof each state. On the bottom axis the measured energy sum of the dissociation

is shown. The top axis shows the energy stored in rotational states of H2.

We suspect the reason for this to lie in the different transition probability into the disso-ciative state depending on the energy range accessible. E.g. because the initial state ΨJ=2

3Because the total wavefunction of H2 is antisymmetric with respect to permutation of its nuclei, the per-mutational symmetry of the rotational wavefunction is limited by the symmetry of the nuclear spin function.Parahydrogen with an antisymmetric nuclear spin wavefunction must have a symmetric rotational wavefunc-tion, while Orthohydrogen has a symmetric nuclear spin wavefunction and can only exist with a antisymmetricrotational wavefunction.


is energetically higher than ΨJ=0 there exists a broader range of final states Ψf , since thephoton energy is constant. The transition rate into a continuum can be described as follows(compare to [AF05]. Chapter 6)

P (t) =

∫range

Pf (t)ρ(E)dE (5.4)

where "range" means the integration over all final states accessible and ρ(E)dE is the num-ber of continuum states in the range E + dE. If we assume the density of states ρ(E) to beconstant in the small energy range relevant, we see that the transition probability is linearlydependent on the accessible energy range. To correct for the difference in dissociation prob-ability we further divide the area of the Gaussian peaks by their energy. The resulting plotlooks much more like the expected Boltzmann distribution. It can be seen in figure 5.10. Afit of the form

y = y0 +Ae−(x−x0)/kBT (5.5)

confirms its Boltzmann like distribution, but delivers an unrealistic temperature of T =

340K ± 30K. This value is above room temperature and hence can not be correct, as wewould expect a very cool jet due to our supersonic expansion described in chapter 3.2.2.Possible reasons we do not get a realistic value could be:

1. The correction for the range of final states is too simple.

2. Due to gating in the analysis the number of events for high and low energies gotdistorted.

3. The supersonic expansion is not as efficient in cooling down rotational motion.

4. There is no thermal equilibrium and it is not possible to assume the rotational statesare populated according to a Boltzmann distribution.

5. There could be a chance of recapturing of the very slow photoelectrons, which wouldreduce the count rate of low energy electrons more than for higher energy.

5.3.2 Retroaction of Photoelectron

The asymmetry, as predicted by Serov and Kheifets, is only visible in the photoelectrondistribution in the molecular frame.In figure 5.11 the MFPADs for dissociation of H2 are shown. Since the KER is fixed inall of those examples to be less than 0.1eV , the electron energy is decreasing with smallerphoton energies. As a result the asymmetry gets more pronounced as predicted in [SK14].In order to look at the asymmetry observed in a more quantitatively manner, we define theasymmetry parameter

δ =np − nHnp + nH

(5.6)


FIGURE 5.10: Same as in figure 5.9 but additionally divided by the energysum. The red line shows a Boltzmann fit of the data points.

FIGURE 5.11: Angular distribution of the ejected photoelectron in the molec-ular frame for all molecular orientations in regard to the light polarization.Shown is the angle between the momentum vector of the electron and theion for the photon energies 19.76eV, 18.5eV, 18.3eV and 18.16eV (top left tobottom right). Each histogram is mirrored on its horizontal axis for better vi-sual inspection. The red line shows a circle to make the asymmetry easier tosee. In all histograms the KER is restricted to be below 0.1eV. Note that the

statistical error bars are smaller than the point size.


as done in [Wai+16]. Here np denotes the count rate of the bond breaking with the ion alongthe direction of the photoelectron and nH with the bond breaking in opposite direction.

We further look at the photoelectron angular distribution for different conditions on theKER range and calculate δ for each slice of KER. To make use of the better statistics at lowKER, we chose a dynamically increasing size of the energy range. With this we are nowable to look at the energy dependence of the asymmetry for each photon energy. In figure5.12 the asymmetry parameter δ is shown as a function of electron energy.

FIGURE 5.12: Asymmetry parameter delta in H+2 dissociation as function of

electron energy for different photon energies. The error bars show the statis-tical error. Photon energies: 18.16eV (red circle), 18.3eV (magenta downward

triangle), 18.5eV (blue upward triangle) and 19.76eV (black squares).

We can observe a strong asymmetry for low electron energies, which seems to be similar,independently on the photon energy. This means the KER does not seem to play a role forlow electron energies. This changes for higher electron energies, where the KER approacheszero for the different photon energies. Here we can clearly see a fast drop in delta for KERclose to zero. In the theoretical description by Kheifets and Serov a hyperbolic decrease ispredicted, which does not seem to match our results. So what could be the reason for thisdecreasing asymmetry at low KER? The asymmetry is based on a coupling of the 1sσg and2pσU by the coulomb field of the photoelectron. This coupling is strongest when the twostates become energetically close, which is happening at a distance of about 5.5a.u.. To getthere the nuclei need some time, which is depending on there kinetic energy. If the kinetic


energy is almost zero that time should become very large and the electron therefore too faraway to have an effect. This behavior of the nuclei could explain that drop in δ we observe.

To get an idea how the time is dependent on the kinetic energy of the nuclei, we ap-proximate their motion classically and calculated the time it takes in the dissociation H+

2 toreach a nuclear distance of 5.5a.u..

tcoupling(ER) = 5.5a.u.

√mH

2ER= 5.5a.u.

√mH

KER(5.7)

This function is shown in figure 5.13a, we can see a steep increase in time for KER less than∼ 0.2eV . Of course this is only a very simple approximation and the region close to zeroKER seems very unphysical. But it matches the result of 700a.u. = 17fs for an ion energyof ER = 0.4eV (KER = 0.8eV ) as calculated in [SK14] (calculation shown in figure ??), soit does not seem to be completely off, with the exception of KER ≈ 0.

(A) Time it takes for the nuclei to reach a dis-tance of 5.5a.u. as a function of KER.

(B) Distance of the photoelectron as function of elec-tron energy. Shown for the photon energies 18.16eV

(black), 18.5eV (red) and 19.76ev (blue).

FIGURE 5.13: Classical calculations of (A) coupling time and (B) electron dis-tance in H+

2 .

We can extend this classical calculation to the motion of the photoelectron and assumethe distance of the electron from the molecule is

re(Ee,KER) =

√2Eeme· t(KER) . (5.8)

This function is now dependent on KER and electron energy, which means it looks differentdepending on the photon energy. The function is show for three exemplary photon energiesin figure 5.13b.

Now let us estimate the asymmetry to expect as a function of KER and electron energy.


We assume the asymmetry is dependent on the strength of the coulomb field from the pho-toelectron. Since the coulomb potential scales with 1/re we try to estimate the asymmetrywith

β1 = c · 1

re+ a , (5.9)

where β1 is the asymmetry parameter and c, a are constants. With those completely classicalapproximations we now try to fit the experimental data as is shown in figure 5.14.

For the simple approximations we did it actually seems surprising that the fit resem-bles the data at all. The biggest limitations are, for one assuming a slow photoelectron ismoving classically with a constant velocity without any effect by the coulomb field of theion. And two assuming a constant velocity for the nuclei at small KER, this also leads to theunphysical "explosion" of the calculated time. So especially for the regions Ee << KER

and KER << Ee the fitted curves are without much physical meaning. But it seems thedropping asymmetry for small KER are connected to a longer dissociation time.

To describe the motion of the nuclei in a better way we approximated the motion of thenuclei as particles ’rolling down4’ the potential energy curve of the 1sσg state. The time ittakes for the nuclei to reach a internuclear distance si can then be calculated by

t =

√m

2·∑i

si(KER)

(√√√√ i∑j

Ej

)−1

. (5.10)

With the sum over the distance ranging from s0 to scoupling. s0 corresponds to the internu-clear distance leading to the resulting kinetic energy release. So s0 varies, depending onthe kinetic energy of the nuclei. So far we have assumed the potential energy curves ofgerade and ungerade parity only start coupling when they reach an internuclear distanceof 5.5a.u.. This approach is problematic because the coupling of two energy curves is nothappening at one specific distance, but rather over all distances with increasing strengthas the energy difference between the states becomes smaller. Furthermore the energy dif-ference at 5.5a.u. is ∆E = U2pσu(R = 5.5a.u.) − U1sσg(R = 5.5a.u.) ≈ 1.1eV . This is arather big energy gap for, since the perturbation responsible for the coupling is caused bythe coulomb tail of a far away electron 5. A more realistic approach would be to assume thecoupling only starting to happen at a bigger internuclear distance, e.g. about R = 8.5a.u.,where the energy difference is ∆E < 0.1eV . To do this calculation the potential energycurves from figure 5.4 were divided into 10.000 segments along the x-axis, which resultsin a step size of si+1 − si = 0.000845a.u.. We then used formula 5.10 to calculate the timeusing the kinetic energy specific starting point s0(KER). With this new time we can then

4Naturally the nuclei is not literally ’rolling’ in the assumption, since there is no angular momentum in-volved. Therefore ’sliding down’ would be a more accurate description (without any friction).

5In the classical model calculated so far the value of the distance does not matter, because it is only a constantfactor in the calculation of the time. In this new approach this is not the case anymore and the value of theinternuclear distance is important.


(A) Asymmetry parameter as a function of electron energy.

(B) Asymmetry parameter as a function of kinetic energy release.

FIGURE 5.14: Classical calculations of asymmetry parameter β1 (solid lines)and measured asymmetry (scatter) for different photon energies. 18.16eV(red), 18.3eV (green), 18.5eV (blue) and 19.76eV (black). The calculated asym-metry is: β1 = 11 · 1/re + a, with a18.3eV = 0.08, a18.5eV = 0.06 and

a19.76eV = 0.03.


FIGURE 5.15: Classical calculations of asymmetry parameter β2 (solid lines)and β3 (dashed lines) together with the measured asymmetry (scatter) fordifferent photon energies. 18.16eV (red), 18.3eV (green), 18.5eV (blue) and19.76eV (black). The calculated asymmetries are: β2 = 20 · 1/re,2 + ahν , witha18.3eV = 0.04, a18.5eV = 0.03, a19.76eV = 0.02. And β2 = 10 · 1/re,3 + ahν ,

with a18.3eV = 0.08, a18.5eV = 0.04, a19.76eV = 0.02.

calculate again the distance of the photoelectron in the same way as before. And the newasymmetry parameter β2 is again β2 = c · 1

re+ a.

In a further modification of that classical model we do not assume this internucleardistance to be constant anymore, but rather to be a function of kinetic energy of the nucleiso that

U2pσu(scoupling) = KER (5.11)

This is the approach taken in [SK14] to describe the internuclear distance at which the cou-pling of states is happening. The asymmetry parameter β3 calculated this way, is shown incomparison with β2 and the measured asymmetry δ in figure 5.14. The corresponding timesthe nuclei need to reach the specific internuclear distance can be found in A.1 of AppendixA. Comparing the two differently calculated parameters β2 and β3 we can see that they areboth very similar, although β3 seems to fit better in the region with low KER, but for highKER it is worse than β2. This seems to indicate that the dynamic coupling region as a func-tion of KER, is not a good assumption for high KER. This makes sense, because the weakperturbation by the electrons coulomb field, which leads to the coupling of states 1sσg and2pσu, is presumably very weak for energy differences of ∆E > 0.1eV . We can also observethat the classical calculations are missing some important dynamics for low electron energy.This is because the assumption of a constant electron velocity is only valid for the condition


Ee >> ER. Since the retroaction of the photoelectron is a low energy effect, it is surprisingthat those calculations match the measured asymmetry at all. But especially at low electronenergy the calculations are unsatisfactory, which becomes noticeable at the increasing offsetahν needed for low photon energies. The lowest photon energy of 18.16eV , can even not befitted at all.

Comparing Asymmetry in H2 and D2

FIGURE 5.16: Same as in 5.11, but for D2 dissociation. In the left histogramMFPAD of D2 is shown with a photon energy of 18.5eV and a restricted KERof less than 0.1eV. On the right is the MFPAD forH2 with the same conditions.

Figure 5.16 shows the MFPAD of D2 in comparison with the MFPAD of H2 with thesame photon energy. It is noticeable that for the same KER and electron energy the asym-metry in the D2 dissociation is weaker than in its lighter counterpart. This behavior can bewell explained with a slower dissociation of D+

2 , due to its heavier nuclei. In our classicalcalculations the time is proportional to

√m. So for a mass which is twice as large the time

tcoupling as well as the distance of the photoelectron re will increase by a factor of√

2, whichin turn leads to an asymmetry which is weaker by the factor 1/

√2:

βD =1√2βH . (5.12)

Structure in Retroaction?

Taking a closer look into the asymmetry parameter δ versus the kinetic energy release infigure 5.18, one will notice the strong dip in the δ for 18.16eV photon energy. This dip is notexplained by any theory so far, but this might not be surprising considering that all of thedata points of this low photon energy have very small electron energies and KER (<60meV)and are therefore in a strictly non classical regime6. Unfortunately the error bars are ratherbig and this dip is only observed in a single photon energy. To get more insight into the

6One of the conditions for the theoretical model by Serov and Kheifets is Ee >> EK , because the propaga-tion of the photoelectron is assumed to be classical.


FIGURE 5.17: Asymmetry parameter δ for H+2 (blue) and D+

2 (red) versusKER. The photon energy is in both cases 18.5eV. The solid lines show the

calculated asymmetry βH,18.5eV1 (blue) and 1/√

2 · βH,18.5eV1 (red).

non-classical dynamics at low electron energies a full quantum mechanical treatment isneeded.

One possible reason for the initially decreasing and then increasing asymmetry mightbe the following. For slow nuclei (low KER), the coupling between 1sσg and 2pσu is alsopossible without the electrons coulomb field [ES99]. This is a very weak effect but it isstrongest at low KER, while the retroaction by the electron, as we observe it, is strongestfor high KER. So there might be interference between those two effects, which are bothintroducing a symmetry breaking. This could also be an explanation why the asymmetryat the 18.16eV photon energy does not drop to zero, but instead increases for KER→ 0.

FIGURE 5.18: Asymmetry parameter δ as a function of KER for several pho-ton energies in H2 and D2. A B-Spline connects the data points to guide the

eye.


L- dependency of Asymmetry?

According to [SK14] there is a weak dependency on the angular momentum L in their cal-culations. Usually it is not possible to separate between the different rotational states andmost of the molecules will be in the rotational ground state anyway, since a cold gas jet isused. But as seen in section 5.3.1, our energy resolution in the data with the lowest photonenergy is good enough to actually resolve several rotational states. In figure 5.19 we sepa-rated three different rotational states ofH2 and plot the asymmetry for each one in the usualplot versus the KER. Comparing the data points it seems like for increasing L, the ’dip’, itwas talked about in previous section, is becoming more pronounced 7. But because of lim-ited statistics and the resulting big error bars, it is not possible to make out a big difference.Within the range of error, the overall strength of the asymmetry seems to be similar.

FIGURE 5.19: Delta versus KER for different rotational states of H2 with aphoton energy of 18.16eV. The distinct initial rotational states could easily be

separated in their energy sum, as shown in figure 5.8.

7Note that in all previous plots the photon energy 18.16eV was always limited to J=1.

65

Chapter 6

CF4 - Tetrafluoromethane

Tetrafluoromethane (CF4) is an interesting polyatomic system because of its high symmetryand chemical stability, as well as its unusual dissociative behavior of its cation states. CF4

is of tetrahedral symmetry and belongs to the Td symmetry group1. The character table ofthe Td point group can be seen in the Appendix B A.1. Its high chemical stability is a resultof the four C−F bonds, which are the strongest single bonds in organic chemistry [O’H08].

FIGURE 6.1: Structure of Tetrafluoromethane.

Due to its interesting properties and role as a potent greenhouse gas in the atmosphere,CF4 has been studied extensively in past years and its electronic ground state configurationis well known to be [SDD86]:

(1t621a212a2

1)(3a212t62)(4a2

13t621e44t621t61)

At low ionization energies the first three appearing cation states are X2T1, A2T2 andB2E, which result out of an ejection of a non-bonding p-electron from the outermost or-bitals. Considering the almost spherical symmetry of CF4, one would expect the photo-electron ejection to be nearly independent of the molecular alignment relative to the po-larization of the light. An asymmetry relative to the molecular dissociation axis is to beexpected, because of the tetrahedral structure of CF4, but since the existence of four equalbond axes it should be a small effect.

1For more information on symmetry groups and the properties of the Td group look at [Cot63].

66 Chapter 6. CF4 - Tetrafluoromethane

6.1 Past Experiments

RFPADs of Valence Shell Photoionization

A past experiment on dissociative photoionization ofCF4, has actually shown this asymme-try of the photoelectron emission direction to be much more extreme than it was expected[Kin+02]. In this experiment, done by Kinugawa et al. with a velocity imaging photoion-ization coincidence (VIPCO) method, they are able to separate the A2T2 and the X1T1 stateand produce the RFPADs for those states for a specific photon energy. Those results can beseen in figure 6.2.

FIGURE 6.2: Experimental data from Kinugawa et al. showing the RFPADsfor the first two cation states of CF4. In the top half the X2T1 state is shownwith a photon energy of 16.85eV and in the lower half it is theA2T2 state witha photon energy of 21.23eV. On the left side the polarization is parallel to the

molecular orientation, while on the right side it is orthogonal.

To explain the strong orientation of photoelectrons along the molecular axis, Kinugawaet al. argue that there needs to be some interaction between the leaving electron and the ion,which leads to a symmetry breaking and the favoring of a particular molecular orientation.To explain the interaction they offer two (closely related) possibilities:

1. If the time scale of photoelectron ejection is on the same time scale as the nuclearmotion, the coulomb field of the electron might be enough to relieve the fourfolddegeneracy of the dissociation axes.

2. The distortion also might take place before the electron departs completely, within theionic core of a shape resonance 2. Since the existence of several shape resonances was

2A shape resonance is a quasi-bound state, in which an electron is temporarily trapped by an angular mo-mentum barrier due to the ’shape’ of the molecular potential. The electron will eventually tunnel through thebarrier and escape the molecule. [Deh84]

6.1. Past Experiments 67

shown within the energies of interest [SDD86], this explanations seems reasonable.

Shape Resonances

In [SDD86], Stephens, Dill and Dehmer have calculated photoionization cross sections andphotoelectron angular distributions for all occupied orbitals using the multiple-scatteringmodel. They predict two intense shape resonances below 3eV in the continuum of a1 andt2 final state symmetry. Since shape resonance reflect the final state electron-molecule dy-namics, those two shape resonances exist for all electron orbitals, as long as the the a1 andt2 state are allowed final states with a dipole transition3. All allowed dipole transitionsbetween orbitals of Td symmetry are shown in figure 6.3.

FIGURE 6.3: Dipole allowed transitions between orbitals of Td symmetry.Taken out of [SDD86].

Shape resonances can also have a strong effect on the angular correlations between thephotoelectron and the ion [Deh84]. In [SDD86], Stephens, Dill and Dehmer also calculatethe β-Parameter which is used to describe the symmetry of the photoelectron distributionsvia:

I(θ) =σ

4π(1 + βP2(cosθ)) (6.1)

Where σ is the cross section, P2 the second order Legendre polynomial and θ the angle ofthe ejected electron relative to the light polarization. In the calculated β parameter theyobserve a sudden change in the symmetry parameter at the same energies as the shaperesonances. The calculated cross section and symmetry parameter as well as experimentalresults by Carlson et al. [Car+84] are shown for the three outermost electron orbitals infigure 6.4.

Energy Curves and Dissociation Dynamics

More recently Tang et al. [Tan+13] were investigating the dissociation dynamics of CF4

using threshold photoelectron-photoion coincidence velocity imaging. By measuring theβ parameter for the CF+

3 ion distribution and additionally calculated CF+3 − F potential

3An exception are the outer valence electrons with t1 symmetry. While it is not uncommon to observe themissing of shape resonances in valence orbitals, when they would be expected due to dipole selection rules, themechanism of behind it is not yet well understood.


(A) 1e (B) 4t2 (C) 1t1

FIGURE 6.4: Calculated total photoionization cross section σ and asymmetryparameter β for the three outermost valence orbitals. Note: The low energypart of the calculated cross section is multiplied by factor 0.1 (inlet). Experi-mental results are by Carlson et al. [Car+84]. Figure is taken out of [SDD86].

energy curves they propose different dissociation mechanism for the first three ionic statesX1T1, A2T2 and B2E 4.

FIGURE 6.5: Potential energy curves of CF+3 − F as a function of the C − F

distance. Calculation by Tang et al. in [Tan+13].

In the dissociation process the symmetry of the ionic molecule changes to C3V due tothe umbrella motion of the three remaining Fluorines, while the fourth Fluorine recoils fastfrom the center of mass of CF+

3 . The time scale of the dissociation depends on the ionicstate. Tang et al. [Tan+13] measured the β-parameters for the three states X1T1, A2T2 and

4The different ionic states result out of removing an electron out of the different orbitals 1t1, 4t2 and 1erespectively.

6.2. Idea of this Experiment 69

B2E to be βX = 0.7, βA = 0.95 and βB = 0.51 respectively. With that they postulate thefastest dissociation happening inA2T2 with only a few tens of femtoseconds. The lifetime ofthe ionic ground state X1T1 is considerably longer due to its smaller β-parameter. Becauseof its bound character the dissociation of the B2E state is very different. Since there couldnot be any CF+

4 observed and the kinetic energy release was similar to that of the A2T2

state, it is assumed the dissociation is happening either by internal conversion (IC) or byradiative decay through the A2T2 state. But measuring a β-parameter of β = 0.51 indicatesa lifetime within the picosecond regime, which is shorter than expected from a fluorescentdecay. So internal decay seems to be more realistic.

6.2 Idea of this Experiment

As can be seen in the calculations shown in figure 6.4, there are shape resonances for all ofthe three outermost electronic states at low photoelectron energies. At the same energies asthose shape resonances the calculations predicts a change in asymmetry parameter β. Tothe best of our knowledge, the only experimental data measuring the photoelectron angulardistribution in the molecular recoil frame after valence shell photoionization was publishedby Kinugawa et al. [Kin+02] and is shown in figure 6.2. Those RFPADs are surprisinglystructured and a more thoroughly investigation of the photoelectron emission dynamicsseems to be appropriate.To do so, valence shell photoionization of CF4 at several photon energies ranging from18.3eV to 21.4eV was performed. In this range, photoelectron energies between 0eV and5eV are obtained, which is just in the region of the calculated shape resonances in [SDD86].

6.3 Results of CF4 Photo Dissociation

Looking at the energy maps of the CF4 photodissociation in figure 6.6, it is easy to identifythe three possible ionic states of CF+

4 , all dissociating into CF+3 + F . As expected the three

island are shifting to higher electron energies for increasing photon energies. Those islandsrepresent the three ionic states and because of their difference in electron energy it is veryeasy to separate them and analyze one at a time.

6.3.1 A2T2

If one of the 4t2 electrons is removed due to the absorption of a XUV photon this will resultin the CF+

4 state A2T2, which then dissociates to CF+3 and F . When first taking a look at

the electron angular emission direction in the molecular recoil frame for different photonenergies in this state (6.7), it is obvious that there is some change in the electron emission.At the RFPAD for 19eV photons the main peak of electron emission is towards the CF+

3


(A) hν = 18.3eV (B) hν = 19eV

(C) hν = 19.4eV (D) hν = 21.4eV

FIGURE 6.6: Electron energy versus KER for the photodissociation with dif-ferent photon energies. Note the changing scale of the electron energy.

ion, while for 21.4eV photon energy this changes to a peak in direction of the neutral F. Sofar this is integrated over all molecular orientations relative to the light polarization.

If the molecular orientation is limited to be parallel 5 to the polarization, much morestructured RFPADs emerge. In figure 6.8 the RFPADs for four different photon energies areshown. While the one with a photon energy of 21.4eV displays the same structure as in thepast experiment by Kinugawa et al. (figure 6.2), the low energy ones show a completelyflipped peak in the angular distribution. More interestingly this extreme change in electronemission direction seems to happen over a range of less than 2eV.

5In practice the orientation can not be limited to be perfectly parallel, but rather to a range of orientationswhich is close to parallel. How small this range can be depends on the available statistics. In this case themolecular axis was set to be ±15 in relation to the light polarization.

6.3. Results of CF4 Photo Dissociation 71

(A) hν = 19eV (B) hν = 21.4eV

FIGURE 6.7: Recoil Frame Photoelectron Angular Distribution (RFPAD) ofCF+

4 (A2T2) for 19eV (A) and 21.4eV (B) photon energies. Note that the sta-tistical error bars are smaller than the bin size.

Theoretical calculations, done by C. Trevisan, T. Resigno, R. Lucchese and C. W. Mc-Curdy with a Complex Kohn Variational Method, show a good agreement with the exper-imental RFPADs and in particular it exhibits the same flip of electron emission direction.Further calculations of the total cross section (shown in figure 6.9) of that particular statefeatures the two shape resonances of different symmetry, which were already proposed in[SDD86]. Those two resonances are exactly in the energy range of interest and in orderto achieve matching RFPADs, as in figure 6.8, it is necessary to coherently couple the twoelectronic wavefunctions underlying the shape resonances. So responsible for the abruptchange of the RFPADs is an interference between the continuum electron of symmetry t2and a1:

(CF+3 e−) final state

(4t52 kt12) 1T2

(4t52 ka11) 1T2

6.3.2 X1T1

The total cross section for the state X1T1 as calculated by C. Trevisan, shows also the sharpfeature of a shape resonance for low electron energy. This state is achieved by removing of a1t1 electron in the photoionization process. Even though continuum electrons of symmetryt2, t1, e and a2 are dipole allowed transitions (compare to 6.3), there is only one observedshape resonance of t2 symmetry. Nevertheless there is a major change in the electron emis-sion direction in the molecular recoil frame for low photoelectron energy in comparisonwith higher energy. In figure 6.10 the RFPADs for 18.6eV and 31eV photon energy areshown. This corresponds to an electron energy of 2.3eV and 14.7eV respectively.


(A) hν = 18.3eV (B) hν = 19eV

(C) hν = 19.4eV (D) hν = 21.4eV

FIGURE 6.8: RFPADs CF+4 (A2T2) for parallel (±15) molecular orientation

relative to the light polarization for several photon energies. Black dots showexperimental data, the red line shows the calculated RFPADs for electron en-ergies of 2.2eV, 3eV, 3.4eV and 5eV - suggesting an offset of 1.5eV electronenergy in the calculations. Note that the error bars only account for the sta-

tistical error (This will also be the case in the following RFPADs).

The angular distribution of the photoelectrons shows again a flip, but this time the the-ory does not fit as good as in the 4t2 state. This is likely to be caused by the experimentallimitation of the molecular orientation relative to the polarization. In the theory a perfectlyparallel aligned molecule is assumed and therefore the features in the RFPADs are verysharp. In the experimental analysis there has to be an angular range of alignment, whichwas in this case set to±8. This will inevitably wash out the features. Another contributionto washing out the feature is the chosen bin size in the experimental data. Considering this

6.3. Results of CF4 Photo Dissociation 73

FIGURE 6.9: CF4 total cross section for 4t2 outer valence shell electrons. Theleft peak is caused by a shape resonance with a1 symmetry, the right peak bya shape resonance with t2 symmetry. Calculations done by Cynthia Trevisan.

(A) hν = 18.6eV (B) hν = 31eV

FIGURE 6.10: RFPADs CF+4 (X2T1) for parallel (±8) molecular orientation

relative to the light polarization for 18.6eV (A) and 31eV (B) photon energy.Black dots show experimental data, the red line shows the calculated RF-

PADs.

the theory fits all the features of the experimental RFPADs and provides good agreement.

The cause for this dramatic change in the photoelectron angular distribution is likelydue to the existence of the single shape resonance at low energy. When the electron isincreasing in energy and crossing the resonance, there is rapid increase in the asymptoticphase shift of the electron wavefunction induced [Deh84], which leads to changes in theelectron emission direction.


(A) 1t1 (B) 1e

FIGURE 6.11: CF4 total cross section for 1t1 (A) and 1e (B) outer valenceshell electrons. Both cross sections show a t2 shape resonance for low energy

electrons. Calculations done by Cynthia Trevisan.

6.3.3 B2E

The total cross section of the state B2E 6.11, which results of removing a 1e electron in thephotoionization, shows a shape resonance of t2 symmetry for low energies as well. Forthe same reasoning as above there is a change in electron emission for low photoelectronenergies compared to higher energies. In figure 6.12 the experimental and theoretical RF-PADs are shown for photon energies of 20.5eV and 31eV. The overlap between theory andexperiment is the worst for this state. The reason for this is likely to be found in the - yetunresolved - dissociation dynamics of theB2E ionic state. The state is known to be a boundstate but it must auto dissociate quickly because there have not been measure anyCF+

4 ionsso far and the RFPADs are surprisingly structured. But nevertheless the unknown lifetimeof the B2E state is likely to wash out the RFPADs resulting in a bad agreement with thetheory.

6.4 Outlook

Understanding the cause of the dramatic asymmetry change of the RFPAD seen in the 4t2

electrons, the question arises if this is an isolated phenomenon of the valence shell electronsin CF4, or if the coupling of two shape resonances, resulting in interesting dynamics of theelectron angular emission direction, can also be observed in other molecules 6. Since shaperesonances reflect the final state electron-molecule dynamics [SDD86], the same shape res-onances observed in the outer valence shell electrons of 4t2 can be expected in lower lyingelectron orbitals with t2 symmetry. And indeed Stephens, Dill and Dehmer calculate almostidentical shape resonances in the CF4 total photoionization cross sections of 1t2, 2t2 and 3t2

6To the best of our knowledge this effect has not yet been observed in other molecules.

6.4. Outlook 75

(A) hν = 20.5eV (B) hν = 31eV

FIGURE 6.12: RFPADs CF+4 (B2E) for parallel (±8) molecular orientation

relative to the light polarization for 20.5eV (A) and 31eV (B) photon en-ergy. Black dots show experimental data, the red line shows the calcu-lated RFPADs. The experimental data corresponds to electrons of energies2eV (±0.15eV ) and 12eV (±0.5eV ), while the theory has energies of 2.5eV and

13eV

(figure 6.13), which is also partially confirmed in a recent calculation by Robert Lucchese ofthe 3t2 total cross section shown in figure A.2 in the Appendix.

(A) 1t2 (B) 2t2 (C) 3t2

FIGURE 6.13: Calculated total photoionization cross section σ for electrons oft2 symmetry in different orbitals of CF4. Note: The low energy part of thecalculated cross section in (c) is multiplied by factor 0.1 (inlet). Figures taken

from [SDD86].

Considering the flip of photoelectron emission direction is caused by the coupling oftwo continuum electrons with different symmetry, it stands to reason to predict similareffects in the RFPADs for core and inner valence shell photoionization of CF4.

As it happens an investigation of the RFPADs in the K-shell photoionization of CF4 hasalready been conducted by C.W. McCurdy et al. [McC+17]. Although the focus of that re-cent publication lies on the observation of core-hole localization in the F-atoms, there can


also be a a flip in the RFPAD observed 7 (figure 6.14). Interestingly the used photon energiesof 698eV, 700eV and 707eV, over which the change in the electron emission direction is seen,fall well into the region where the shape resonances of a1 and t2 symmetry are expected (fig-ure 6.13a). At the time of writing calculations were still carried out by Cynthia Trevisan inorder to check if this change of the electron emission direction in the photoionization nearthe fluorine K-edge is caused by the same mechanism as in the valence photoionizationdiscussed in this work. There are also calculations being currently performed by C. Tre-visan for the RFPADs of the 3t2 state in the inner valence shell. It would not be surprisingto observe the same effect in this state, which could then easily be confirmed by anotherCOLTRIMS experiment at any synchrotron source capable of producing 20-30eV photonenergies.

FIGURE 6.14: RFPADs for CF4 K-edge photoionization for photoelectron en-ergies of (left to right) 3eV, 5eV and 12 eV and polarization parallel to theC-F recoil axis. The respective photon energies are 698eV, 700eV and 707eV.

Figure taken out of [McC+17].

7The appearance of a flip in the electron emission direction is not discussed in mentioned publication.

77

Chapter 7

Conclusion

This thesis presented the results of several valence photoionization experiments on themolecular targets of H2, D2 and CF4. Here, it was possible to observe two distinct effectsinvolving interaction between the photoelectron and the molecular field during ionization.In H2 and D2, a symmetry breaking of the Molecular Frame Photoelectron Angular Distri-bution (MFPAD) could be explained by the retroaction of the photoelectron onto its source.

During dissociation, the long-ranging coulomb-tail of the photoelectron can localize theremaining electron on the proton furthest from the continuum electron and therefore causethe dissociation to be preferentially aligned with the proton in the same direction as thephotoelectron. Quantum mechanically, this symmetry breaking is caused by a coupling ofthe states 1sσg and 2pσu with different symmetry. This coupling is strongest when these twostates are energetically degenerate at a large internuclear distance. Since D2 takes longerto dissociate due to its heavier nuclei, the coupling and the observed asymmetry is weakerby a factor of ∼ 1√

2. The dependence of a defined asymmetry parameter on the nuclear

momentum shows an interesting possibility to measure nuclear dynamics. The strength ofthis symmetry breaking is strongly dependent on the electron energy and a classical cal-culation gives decent agreement for the higher electron energies. However, for the verylow electron energies we observed some interesting structure in the asymmetry as a func-tion of energy, which is so far not covered by any existing calculations and requires a fullyquantum-mechanical treatment.InCF4, results of the valence photoionization of the three outermost orbitals were presentedtogether with Komplex-Kohn variational method calculations1 for same states. For the 4t2

electrons, it could be shown that a rapid change in the photoelectron angular distribution,as a function of photon energy, is caused by a coherent coupling of two continuum elec-trons with different symmetries a1 and t2, each passing through a distinct shape resonancebut leading to the same final state. Based on a similar observation in the K-shell photoion-ization of CF4 [McC+17], where the same shape resonances and electron symmetries areinvolved, it is speculated that this rapid change in photoelectron angular distribution takesplace in other electron orbitals possessing t2 symmetry as well.If a 1t1 or 1e electron is removed from the outer shell, a change in the photoelectron angular

1Calculations were done by Cynthia Trevisan, Tom Rescigno, Robert Lucchese and Bill McCurdy.

78 Chapter 7. Conclusion

distribution, as a function of photon energy, is also observed, but not as rapid as in the 4t2

electrons. In these cases the change could be contributed to the asymptotic phase shift inthe electron wavefunction, when the increasing energy of the photoelectron passes throughthe shape resonance.In all three outer valence electrons of CF4, a strongly asymmetric photoelectron emissionwas observed, which is surprising considering the nearly spherical symmetry of the parentmolecule. The reason for this is likely due to the interaction of the electron with the parention via a shape resonance. This small perturbation by the photoelectron may cause a col-lapse from Td to C3V symmetry during the dissociation, which breaks the symmetry of thenormally four equivalent axes.These results demonstrate numerous significant effects that can arise from the interactionof photoelectrons with their parent ion in various photoionization processes.

79

AcknowledgementsAfter filling many pages in pursuit of the Master of Science it is now the time to thank thepeople who supported me along the way.Insbesondere möchte ich mich bei Thorsten Weber bedanken, der mich für ein ganzes Jahrso herzlich in seiner Arbeitsgruppe im wunderschönen Berkeley aufgenommen hat undimmer ein offenes Ohr für meine Fragen hatte. Genau so will ich mich bei Reinhard Dörnerund Robert Moshammer bedanken, die beide so bereitwillig die Betreuung meiner Mas-terarbeit übernommen haben und mir damit überhaupt erst diesen Aufenthalt ermöglichthaben.In Berkeley I had a lot of great people helping me with my work. First of all I want to thankAve for explaining me lmf2root and the procedures of doing a proper analysis. You spenda lot of time solving the problems I couldn’t solve on my own!A big ’thank you’ goes to Kirk, who helped me a lot to figure out the science and did nothesitate to correct me if I got something wrong. Although we never got behind the mysteryof the Sacramento point source... Also thanks for everything else you did to help me. I’mgoing to miss those coffee breaks on the balcony!Another big help in "figuring out the science" was Elio. Thank you! And I want to thankeveryone else in of the AMOS group for including me and helping me to understand thephysics. Especially thank you Dan for giving me advice and borrowing me your car justlike that to go on an amazing road trip.Then I want to thank Cynthia Trevisan, Bill McCurdy, Robert Lucchese and Tom Rescignofor the strong theoretical support and for the amazingly quick calculations done on CF4.Letztendlich der größte Dank geht an meine Eltern. Ihr seid die Personen, ohne derenständige und bedingungslose Unterstützung ich es niemals bis hierhin geschafft hätte.

81

Appendix A

Additional Material

A.1 Additional Material for H2

(A) (B)

FIGURE A.1: Classical calculations of the time it takes for the nuclei ofH+

2 to reach an internuclear distance of (A) 8.5a.u. and (B) R(KER), withU2pσu

(R) = KER.

A.2 Additional Material for CF4

Td E 8C3 3C2 6S4 6σdA1 1 1 1 1 1 x2 + y2 + z2

A2 1 1 1 -1 -1E 2 -1 2 0 0 (2z2 − x2 − y2, x2 − y2)T1 3 0 -1 1 -1 (Rx, Ry, Rz)T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)

TABLE A.1: Character Table of Td point group. Taken out of [Cot63].

82 Appendix A. Additional Material

FIGURE A.2: Calculated total and partial cross sections of possible electronicwavefunctions of a 3t2 hole state. Calculation done by Robert Lucchese.

83

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Erklärung

Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als dieangegebenen Quellen und Hilfsmittel benutzt habe.

Berkeley (Kalifornien), den 27. Oktober 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . .

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