ELECTRON COINCIDENCE STUDIES OF MOLECULES

Transcription

1 ELECTRON COINCIDENCE STUDIES OF MOLECULES A thesis submitted in fulfilment of the requirements of the degree of Doctor of Philosophy By Danielle S. ATKINS née MILNE-BROWNLIE B.Sc. (Hons.) Griffith University Griffith University School of science January 2007

2 This work has not previously been submitted for a degree or diploma in any university. To the best of my knowledge and belief, the thesis contains no material previously published or written by another person except where due reference is made in the thesis itself. Danielle S. ATKINS née MILNE-BROWNLIE ii

3 Abstract The electron-electron coincidence (e,2e) technique yields complete kinematical information about the electron impact ionization process. The (e,2e) technique has been widely used to study dynamical effects in ionizing collisions with atomic targets, however studies of molecular ionization using this technique have been very limited. Recently further experimental studies of small molecules have been proposed, as the cross sections of small molecules are now computable using sophisticated theoretical approaches [77, 24]. This thesis presents dynamical investigations for the electron impact ionization of the molecular targets H 2 OandH 2, employing the (e,2e) technique to experimentally measure the triple differential cross section (TDCS). The TDCS is defined as the probability that a bound electron will be ejected from the target atom or molecule (into a particular direction with a defined energy) and the initial electron will be scattered into a particular direction with a particular energy. All TDCSs presented within this thesis were performed using an electron coincidence spectrometer in the coplanar asymmetric geometry at intermediate incident electron energies. This thesis presents the electron impact ionization TDCSs of H 2 O. A series of measurements were performed using H 2 O in the vapour form. Measurements of the TDCS are presented for the 2a 1 atomic-like orbital and the 1b 2,3a 1 and 1b 1 molecular orbitals at a common incident electron energy of 250eV, ejected electron energy of 10eV and scattering angle of -15 o. The experimental TDCSs are compared with theoretical cross sections that were calculated by Champion et al [25, 26] using a distorted wave Born approach (DWBA). TDCS measurements for the single ionization of the hydrogen molecule, H 2 were performed as in recent years there has been evidence that indicates the ejected electron angular distribution is perturbed due to Young-type interference effects. The oscillatory structure which is predicted in the cross section is due to the two-centred nature of the molecule [27, 29]. This thesis presents experimental TDCSs for the ionization of H 2 which are compared to TDCSs of helium. A series of measurements for the TDCSs of H 2 andhearepresentedatacommonincidentelectronenergyof 250eV and scattering angle of -15 o, for a range of ejected electron energies between 10eV and 100eV. The experimental TDCSs are compared with two types of theoretical calculations. iii

4 Acknowledgments There has been a large number of people who have assisted me during the course of my PhD. Without the help and encouragement of these people, the work contained within this thesis would not have been completed. I would like to take this opportunity to personally thank several people for their assistance, and support over the past five years. First, a special thanks goes to my supervisor, Prof. Birgit Lohmann. Thank you for allowing me the opportunity to complete my PhD under your supervision and for your encouragement and assistance which has helped me during the course of my PhD project. Your guidance and influence has helped me develop not only my experimental skills, but also my writing and public speaking abilities. Thanks go also to Dr. Robert Sang, my associate supervisor and Prof. William MacGillivray for their support and encouragement and for always taking an interest in my project. To my fellow PhD students and research fellows in the laser lab, Matthew Haynes, Michael Went, Jonathan Ashmore (now Doctors), Mark Baker, Adam Palmer, Kristen Matherson, Bill Guinea, Peter Johnson and Dr Mark Stevenson. Thank you for your friendship, encouragement, inspiration and entertainment. Of the above mentioned people I would like to say a special thanks to Matthew Haynes for his assistance in learning the experimental apparatus during the first year of my PhD, to Michael Went whose assistance with general topics was greatly appreciated and to Dr. Mark Stevenson for his assistance with experimental problems and guidance during the writing of this thesis. I would also like to thank all the members of the Centre for Quantum Dynamics who I have not mentioned personally for their friendship and support. My appreciation goes to the members of the mechanical and electrical workshops, Bruce Stevens, Kevin Selway, Kevin Hoatson, David Palmer, Dennis Smith, Ian Ashworth, Russell Saunders, Mal Kelson, Steven Beams, Steve Barrett, Ken Parkyn and Luke Rigo for the time, effort and advice they have contributed towards my project. Of these people I would like to say a special thanks to David Palmer, Kevin Hoatson and Ken Parkyn who have contributed most towards my project. iv

5 Finally, I would also like to say a special thanks to those I hold closest to my heart, my family and friends. To my loving husband Danny, thank you for always being there when I need someone to lean on. Thanks goes to my mum and dad along with my brother Aaron and sister Cherie for supporting and encouraging me not only over the past five years but through my entire life. To Cath, John, Trish, Libby, Louie and Jessie, thank you for accepting me so warmly into your family and for the support you have given me during my studies. To Kristen and Alisa thanks for being there to just chat when I needed a break. Danny, Mum, Dad, Aaron, Cherie, Cath, John, Trish, Libby, Louie, Jessie, Kristen and Alisa thank-you for your love and support and for always being there when I needed it most. v

6 Contents Abstract Acknowledgments List of Tables List of Figures iii iv viii ix 1 Introduction 1 2 Theory of Electron Impact Ionization Introduction Ionization DifferentialCrossSections SingleDifferentialCrossSection DoubleDifferentialCrossSection TripleDifferentialCrossSection FormalScatteringTheory BornApproximations OtherFormsofApproximations Experimental Apparatus Introduction ExperimentalApparatus VacuumChamber GasSource ElectronGun Analysers FastTimingElectronics ComputerControlandMonitoring vi

7 3.3 CoincidenceTechnique CoincidenceTimingandDataCollection CoincidenceEnergyResolution AlignmentofSpectrometer ExperimentalCalibration EnergyCalibration AngularCalibration SpectrometerConsistency Coplanar Asymmetric (e,2e) Measurements on H 2 O Introduction MolecularStructure DetailsofTheoryusedforComparison ExperimentalConsiderations ResultsandDiscussion MolecularOrbitals AtomicOrbitals Coplanar Asymmetric (e,2e) Measurements on H 2 and He Introduction DetailsofTheoryusedforComparison InterferenceFactor InclusionofInterferencewithinTDCS ExperimentalConsiderations ResultsandDiscussion H 2 and He TDCS Measurements with a Comparison to the Theoretical Model ofstia[29] H 2 TDCS Measurements with a Comparison to the Theoretical Model of Foster,GaoandMadison[91] Conclusion SummaryofResults FutureDirections A H 2 O Experimental Data 116 B He and H 2 Experimental Data 119 Bibliography 123 vii

8 List of Tables 3.1 ElectronGunVoltages ArgonAugerLineEnergies Ionization Energies of H 2 O Kinematic Conditions of H 2 O Kinematic Conditions of H 2 andhe A.1 H 2 O Experimental Data for the Summed 1b 1 and 3a 1 MolecularOrbitals A.2 H 2 O Experimental Data for the 1b 1 and 3a 1 MolecularOrbitals A.3 H 2 O Experimental Data for the 1b 2 MolecularOrbital A.4 H 2 O Experimental Data for the 2a 1 AtomicOrbital B.1 H 2 and He Experimental Data for E b =10eV B.2 H 2 and He Experimental Data for E b =20eV B.3 H 2 and He Experimental Data for E b =50eV B.4 H 2 and He Experimental Data for E b = 100eV viii

9 List of Figures 2.1 ASingleDifferentialCrossSection TheMomentumTransferVector KinematicGeometries ExampleTDCSsforAsymmetricKinematics ElectronRecoilMomentumCrossSections Photographof(e,2e)Spectrometer PhotographoftheVacuumChamber PhotographoftheElectronicRacks SchematicDiagramoftheVacuumChamber ThreeElementCylinderLensSystem FiveElementCylinderLensSystem SchematicoftheCliftronInc.5-CElectronGun SchematicofLocallyConstructedElectronGun SchematicoftheHemisphericalElectronEnergyAnalyser ElectronTrajectoriesThroughaHemisphericalAnalyser ChannelElectronMultiplier Multi-ChannelElectronPlate CoincidenceTimingElectronics TimingSpectrum SchematicDiagramoftheScatteringVolume ArgonAugerLinesMeasuredbytheEjectedAnalyser BindingEnergySpectrum Argon60eVElasticDifferentialCrossSection Argon 100eV Elastic Differential Cross Section ScatteredAnalyserDoubleDifferentialCrossSection Helium Double Differential Cross Section for E b =10eV Helium Double Differential Cross Section for E b =50eV HeliumTripleDifferentialCrossSection ix

10 4.1 Theoretical TDCS of H 2 O Flow Down a Tube for λ r Flow Down a Tube for λ r H 2 OVapourOscillations H 2 OBindingEnergySpectrum TDCSofH 2 O for the Summed 1b 1 and 3a 1 MolecularOrbitals TDCSofH 2 O for the 1b 1 MolecularOrbital TDCSofH 2 O for the 3a 1 MolecularOrbital TDCSofH 2 O for the 1b 2 MolecularOrbital TDCS of H 2 O for the 2a 1 AtomicOrbital Recent H 2 OTheoreticalCalculations Recent H 2 OTheoreticalCalculations InterferenceFactorasaFunctionoftheEjectedElectronAngle Binding Energy Spectra for H 2 andhe TDCSofH 2 and He for E b =10eV TDCSofH 2 and He for E b =20eV TDCSofH 2 and He for E b =50eV TDCSofH 2 and He for E b = 100eV TDCSofH 2 and He for with a Comparison to the Theoretical Calculations of Foster, GaoandMadison[91] TDCS of H 2 with a Comparison to the Theoretical Calculations of Foster, Gao and Madison[91] x

11 Chapter 1 Introduction The electron impact ionization process is one of the most fundamental collision processes in atomic physics. Electron collision experiments can probe the kinematics of the particles which emerge from an ionizing collision event. Complete knowledge of the ionization process occurs when the energy and momentum of all collision products are determined. Cross section determinations from electron collisions are used in the modeling of fusion plasmas, the modeling of radiation effects for both materials and medical research, and astronomy. A technique that measures the properties of all collision particles either directly, or via conservation rules, is the electron-electron coincidence (e,2e) technique. (e,2e) experiments refer specifically to electron impact ionization experiments in which a projectile electron (the incident electron, e o ) ionizes a target atom or molecule thus ejecting a bound electron from an orbital; the two outgoing electrons are then detected. Quantum mechanically the two outgoing electrons are indistinguishable, however the faster electron is generally referred to as the scattered electron (e a ) while the slower electron is referred to as the ejected electron (e b ). The most information about the electron impact ionization process is gained by using the (e,2e) coincidence technique to measure the triple differential cross section (TDCS). The TDCS is the probability that after an ionizing collision event the two outgoing electrons (with defined energies and momenta) will be scattered in particular directions. Atomic collision experiments can be a powerful tool for investigating the structure of a target particle (atom or molecule) and/or the dynamics of a collision event. Spectroscopy experiments in which the absorption or emission of a photon was observed were the first atomic physics experiments performed. While this type of experiment is valuable, it is passive in nature. A more intrusive approach 1

12 2 CHAPTER 1. INTRODUCTION to atomic physics started with the development of scattering experiments. The first scattering experiment was performed by Rutherford in 1911, who scattered α and β particles through a thin plate of matter. Rutherford realized that by measuring the angular distribution of the scattered particles, information relating to the internal structure of the target could be determined [1]. In 1921 Ramsauer performed the first electron scattering experiment. Ramsauer used a single electron beam technique to measure a total electron-atom collision cross section [2]. Pioneering work was carried out in 1922 by Townsend and Bailey who investigated the motion of electrons through Ar. They showed that the effect of an electron collision with a target particle is dependent on the velocity of the electron [3]. In the late 1920 s the first experimental electron impact ionization experiments were performed. In 1928 Langmuir and Jones measured the energy lost by the incident electron (with velocities between 30eV and 250eV) during a collision with gas molecules and presented the resonance energies and ionization potentials they measured [4]. Rudberg performed energy loss experiments for N 2 in 1930, in an attempt to gain a higher accuracy for the resonance and ionization energies [5]. The importance of these experiments was described in 1930 by Bethe who used the First Born Approximation (FBA) to calculate theoretical cross sections for elastic and inelastic collisions and for ionization processes [6]. Most of the earlier experiments mentioned above focused on measuring cross sections which were only differential in energy. More information regarding the ionization process can be obtained by measuring the double differential cross section (DDCS), which is a measure of the angular or energy distribution for either of the outgoing electrons. In 1932 Hughes and McMillen measured the DDCS of Ar for a range of electron energies between 50eV and 550eV [7]. In 1938 Hughes and Mann developed a new method for investigating atomic velocities where energy analysis techniques were used to measure electron Compton Profiles [8]. While important information can be gained by measuring the DDCS, some of the parameters in the collision process are undetermined so the information gained is limited. By measuring the TDCS using (e,2e) coincidence techniques, the ionization process is fully determined with respect to all kinematics of the scattered and ejected electrons. The first coincidence (e,2e) studies were proposed by Smirnov and co-workers [9, 10]. In 1969 the first experimental (e,2e) results were published simultaneously by Amaldi et al [11] and Ehrhardt et al [12] under different kinematic conditions and with different aims. The (e,2e) experiments performed by Amaldi et al were at an incident energy of 15keV for a thin carbon film target. Amaldi et al used a coplanar symmetric geometry where the two outgoing

13 3 electrons were detected at an equal energy and angle; the incident electron energy was varied [11]. This type of measurement where only the incident electron energy is varied, is now known as a binding energy spectrum. Ehrhardt et al performed (e,2e) coincidence experiments that were differential in angle with respect to the ejected electron; the scattered electron angle was fixed. The TDCSs measured by Ehrhardt et al were performed at incident electron energies of 114eV and 50eV with ejected electron energies of 15eV and 10.5eV respectively, for a He target [12]. The development of the (e,2e) coincidence technique has led to a great deal of experimental and theoretical work over the last 36 years. As a result the (e,2e) experimental technique has become a powerful tool for investigating the dynamics of the ionization process. The experimental methods used to study electron impact ionization have now developed far enough to allow the investigation of a range of atomic and molecular ionization processes. The main aims of electron collision experiments are to determine the validity of the theoretical calculations and to describe the physical properties of the process under investigation. Currently satisfactory agreement between theory and experiment is obtainable for simple atomic systems such as H and He. The theoretical TDCS calculations at high incident electron energies show reasonable agreement with the experimental observations for heavier atoms. However, understanding the complex interactions when polyatomic systems are ionized using low to intermediate energy electrons (dynamical investigations) remains one of the greatest challenges facing atomic physics today. Literature dedicated to the dynamical (e,2e) study of molecules is limited. Numerous studies of the ionization of single atoms have been performed, but few measurements of the TDCS for molecules exist. TDCSs of molecular targets are important as they display the molecular effects that can perturb the cross section. A quantity which may influence the TDCS is the non-spherical shape of the molecular target. Another factor which may contribute to the details of the cross section is that the molecular ion may be left in one of several rotational-vibrational electronic ground states after a direct single ionizing collision [13]. Experimental difficulties in measuring the TDCS of molecules arise as it can be difficult to resolve different molecular electronic states due to their close spacing and contributions from the vibrational and rotational states. Experimental results are available for the TDCS of H 2 [13, 14], N 2 [13, 15, 16, 17, 18], O 2 [19], CO [17], CO 2 [20] N 2 O [21] and C 2 H 2 [22] with comparisons to theoretical calculations where available. The number of theoretical molecular TDCS calculations is also limited; this is due to the complexity of the theoretical model needed to describe the cross

14 4 CHAPTER 1. INTRODUCTION section. The theories available for comparison with experimental data include the first Born approximation (FBA), the FBA with orthogonalized Coulomb waves (FBA-OCW), the plane wave impulse approximation (PWIA) and the distorted wave impulse approximation (DWIA). Jung et al [13] performed a series of coplanar asymmetric measurements (see chapter two for a definition of all kinematic regimes) on H 2 at incident electron energies of 100eV and 250eV and ejected electron energies of 4.5eV and 9eV for a range of scattering angles between 4 o and 25 o.observations were made in order to measure the effect of the recoiling molecular ion when it was left in one of numerous possible rotational-vibrational states of its electronic ground state. Results showed the measured recoil peaks were generally weak which is contrary to the observations of atomic targets under similar kinematic regimes. Experimental H 2 measurements by Chérid et al [14] were performed at the much higher incident electron energy of 4087eV in the coplanar asymmetric geometry with ejected electron energies of 20eV and 100eV. A comparison between the experimental data and theoretical calculations using the FBA and the PWIA was made. The FBA calculation showed good agreement when compared to the experimental results while the PWIA was completely wrong in predicting the shape and magnitude of the TDCS. It was expected that the PWIA describing the 20eV ejected electron TDCS would have difficulty predicting the cross section as the impulsive conditions were not met. At an ejected electron energy of 100eV the conditions correspond to the bound Bethe ridge condition (the ejected electron energy is equal to half the square of the momentum transferred, ie. the ejected electron momentum is equal to the momentum transfer). Previous PWIA calculations performed for He under bound Bethe ridge conditions showed total agreement between experiment and theory. Discrepancies between the theoretical calculation and the experimental data may be due to the crudeness of the wavefunction used to describe the initial state of H 2 thus further theoretical calculations are required. Similar to the measurements they performed on H 2, Jung et al [13] also performed a series of coplanar asymmetric measurements on N 2 at an incident energy of 100eV and ejected energies of 3eV and 4eV for three scattering angles of 8 o,15 o and 25 o to observe the effect of the recoiling molecular ion. The observations resemble the H 2 experimental data and show that the recoil peak is weak and in some cases there is almost a total absence of the recoil peak. TodeterminethemagnitudeofaTDCStheexperimentalobservationsneedtobeplacedonan absolute scale. Absolute N 2 TDCSs utilizing coplanar asymmetric kinematics were performed by Avaldi et al [15] at an incident energy of approximately 300eV for an ejected energy of 10eV at scattering angles of 3 o and 3.5 o and an ejected energy of 18.4eV at scattering angles of 7.2 o and

15 5 8 o. The TDCS experimental results were unable to be compared to a theoretical model, however a comparison was made to a He TDCS under the same kinematic conditions. The measurements of Avaldi et al were able to provide evidence that the shape resonances of the 3σ g orbital (known to produce significant deviations to the vibrational branching ratios) perturb the TDCS. Doering and Yang [16] performed a series of coplanar asymmetric TDCS measurements for the ionization of the 3σ g and 1π u orbitals of N 2. Observationsweremadeatanincidentelectronenergy of 100eV for a range of ejected electron energies between 3eV and 13eV at scattering angles less than 5 o. The N 2 cross sections showed a shift in the position of the binary and recoil peaks (the peaks were not centered about the momentum transfer direction as anticipated). Also despite only a 1.1eV difference in ionization potentials there are distinctive differences in the TDCS of the two orbitals. This would indicate that the TDCS is sensitive to the initial and final states of the molecular target under this kinematic condition. Measurements of the N 2 TDCS performed by Rioual et al [17] were carried out using coplanar symmetric kinematics at incident electron energies between 90eV and 400eV. Due to the high resolution of the apparatus used during the measurements, satellite structures arising from several different states of the molecular target overlapping, were observed. The experimental TDCSs were also compared to PWIA calculations, however the theoretical calculations did not accurately describe the experimental cross sections. Coplanar symmetric measurements of N 2 at incident electron energies of 25.6eV and 76.7eV for the 3σ g and 1π u orbitals were performed by Hussey and Murray [18]. Their results showed that the cross section is sensitive to contributions from shape resonances, as the results of Avaldi et al [15] did for asymmetric kinematics. Gao et al have presented a DWIA calculation for the electron impact ionization of the 3σ g state of N 2 for coplanar symmetric kinematics at incident electron energies between 35.6eV and 400eV [23]. The DWIA used a molecular wavefunction which was averaged over all orientations (DWIAOA, where OA stands for orientation average). The theoretical DWIAOA calculations have been compared to the experimental data of Rioual et al [17] and Hussey and Murray [18]. The DWIAOA showed reasonable agreement with the experimental results at intermediate and high incident electron energies, however the agreement deteriorated for low incident energies. In a comparison between the DWIAOA and a PWIA, the DWIAOA showed better agreement with the experimental data, particularly at intermediate incident electron energies. The PWIA was unable to predict the secondary peak, however while the DWIAOA was able to predict the secondary peak the calculation

16 6 CHAPTER 1. INTRODUCTION only gave qualitative agreement with the experimental data. Gao et al suggest that the distorted wave Born approximation (DWBA) would be better in achieving a more detailed understanding of the secondary peak observed at low incident electron energies. Yang and Doering [19] performed a series of asymmetric (e,2e) measurements for the ionization of the 1π g,1π u and 3σ g orbitals of O 2 at a common incident electron energy of 100eV and scattering angle of 4 o, with ejected electron energies of 3.5eV, 6eV and 11eV. The observations of Yang and Doering are compared to N 2 observations, as both molecular targets have the same type of orbitals (1π u and 3σ g ), however the orbital energies are reversed. In the binary region the O 2 TDCSs are quite similar to the corresponding N 2 orbital TDCSs. Observations show that the binary lobe maxima are shifted away from the momentum transfer direction to a larger angle. This suggests that the shape of the molecular orbital is more important than the energy scheme of the orbital. A comparison between O 2 and N 2 in the recoil region shows the TDCS to be completely different and currently a physical explanation for the recoil differences is unknown. TDCS measurements for the 5σ, 1π, 4σ and 3σ orbitals of CO have been performed by Rioual et al [17] using the coplanar symmetric kinematic geometry. A series of results were obtained for incident electron energies between 90eV and 400eV. Similar to the measurements they performed on N 2, several satellite structures due to overlapping states of the molecule were seen in the measured cross sections. In a comparison of the CO experimental data to a PWIA calculation using equivalent energy conditions, the theoretical cross section was not able to reproduce the experimental TDCS. Hussey and Murray [20] performed a series of coplanar symmetric measurements on the 1π g and 4σ g orbitals of CO 2 at incident electron energies between 25eV and 100eV. As no theoretical calculations are available, they compared the results to previous N 2 observations [18] as the characteristics of the molecular orbitals are similar (π and σ). Observations showed a clear difference between the TDCSs of CO 2 and N 2. Hussey and Murray attribute these differences to the configuration of each molecular target. Several N 2 O TDCSs were obtained by Cavanagh and Lohmann [21] using the coplanar asymmetric kinematic geometry at a common incident electron energy of 900eV and ejected electron energy of 25eV. Measurements were performed for the valence 2π orbital at scattering angles of 2 o,5 o and 10 o, and for the inner-shell 4σ orbital at a scattering angle of 5 o. For the 2π orbital of N 2 Othe most pronounced feature of the TDCS was seen at a scattering angle of 10 o which corresponds to the Bethe ridge condition; under this condition a double lobe structure is seen in the binary region. At a scattering angle of 5 o the minimum of the binary double lobe is scarcely pronounced. For the

17 7 scattering angle of 2 o the magnitude of the recoil peak was larger than expected and the maxima of the binary and recoil lobes were shifted; this observation is similar to the observations of Doering and Yang [16] for N 2.TheN 2 O separation spectra indicated that there are many satellite structures between 20eV and 42eV. Thus ionization of the broad 4σ orbital of N 2 O, which lies within this region, could not be completely resolved. Observations for the 4σ orbital showed a smaller than expected binary to recoil ratio which may have occurred due to a localized interaction of the ejected electron with the molecular target. Ionization of the innermost Cσ1s orbital of C 2 H 2 was investigated by Avaldi et al [22] using the (e,2e) technique under coplanar asymmetric conditions. Measurements were performed for a common scattering electron energy of 1500eV with ejected electron energies of 9.6eV and 41eV at small scattering angles. Observations show that the TDCS is not symmetric about the momentum transfer direction with both the binary and recoil peaks having a larger than expected shift in the positions of the maxima. The TDCSs also showed that the magnitude of the recoil peak was larger then previously measured atomic targets under similar kinematic conditions. When compared to a FBA calculation, the theoretical model didn t fully account for the post-collision interaction and higher order scattering effects. The magnitude of the recoil peaks was underestimated and the calculation did not account for the severe shift in the position of the binary and recoil maxima. The above mentioned molecular experimental TDCSs performed at low and intermediate electron energies have shown that the TDCS is very sensitive to the initial and final electronic states of the target molecule. Theoretical comparisons with experimental data using FBA methods have been able to predict cross sections at high incident electron energies, but are unable to reliably predict the TDCS at lower electron energies. Comparisons between experimental data and theoretical calculations using a PWIA calculation showed that the PWIA is unable to correctly predict the shape and magnitude of the TDCS of molecules. While a DWIAOA calculation is an improvement over a PWIA for intermediate electron energies in the coplanar symmetric regime, a DWBA would be better for achieving a more detailed understanding of the TDCS. In the last few years there has been some theoretical activity in modeling the (e,2e) process for small molecules such as H 2, D 2, T 2 and H 2 O using a distorted wave Born approach (DWBA) [24, 25, 26]. DWBA theoretical models should improve agreement between experiment and theory for low and intermediate electron energies as the model uses a distorting potential which can include the interactions between the target particle and the incoming and outgoing electrons. Currently there is a limited amount of experimental data available for comparison to test the new theoretical models.

18 8 CHAPTER 1. INTRODUCTION Furthermore, there has also been considerable interest in diatomic molecules which is stimulated by the possibility of interference effects. The observation of interference effects via the electron impact ionization of a diatomic molecule may be expected due to the two molecular centres playing the role of slits. Theoretical studies of the TDCS for the ionization of H 2 by photon impact have predicted an oscillatory structure due to the dual centre nature of the molecule [27]. The interference can be considered as arising from the quantum mechanical interference of two amplitudes, for ionization from each nucleus. Recent experimental studies of the DDCS of D 2 show that interference effects due to electron impact ionization are observable at low ejected electron energies [28]. Theoretical studies by Stia et al [29] have predicted that these interference effects should also be observable in the TDCS for electron impact ionization of H 2. No evidence has previously been found for interference effects in the TDCS for electron impact ionization of H 2. This thesis presents a range of experimental TDCSs for the molecular targets H 2 OandH 2 at an intermediate incident electron energy. All TDCSs presented were measured using an electron coincidence spectrometer in coplanar asymmetric geometry. Where available, a comparison has been made between the experimental data and a theoretical calculation. The majority of electron impact ionization processes are direct single ionization of a target particle; chapter two will provide an overview of the coincidence and non-coincidence techniques used to probe this type of ionization process. Four types of cross sections will be defined with emphasis being placed on the TDCS. When measuring the TDCS, five different kinematic geometries can be utilized (non-coplanar symmetric, non-coplanar asymmetric, coplanar symmetric, coplanar asymmetric and coplanar constant mutual angle). An overview of each kinematic geometry will be presented with a thorough description of the coplanar asymmetric geometry. The formal scattering theories for electron impact ionization will also be discussed in chapter two. These theories include the FBA, PWBA, DWBA, PWIA and DWIA. The third chapter will discuss the experimental and technical characteristics of the electron coincidence spectrometer utilized during the measurement of the TDCSs presented within this thesis. A detailed description of the individual components of the spectrometer including the vacuum chamber, gas source, electron gun, hemispherical analysers, fast timing electronics and the computer control and monitoring is provided. Chapter three also includes an overview of the coincidence technique focussing on the coincidence energy resolution and coincidence timing and data collection. Finally, several steps have been undertaken to ensure the reliable operation of the experimental apparatus. These procedures, the alignment of the spectrometer, energy and angular calibration and spectrometer consistency checks will be discussed and outcomes presented.

19 9 Chapter four will present the first dynamical (e,2e) results for the electron impact ionization of the water molecule (H 2 O). The TDCSs were measured for a common incident electron energy of 250eV, ejected electron energy of 10eV and a fixed scattering angle of -15 o. Experimental data is presented for the ionization of the 1b 1,3a 1 and 1b 2 molecular orbitals and the 2a 1 atomic orbital. The results presented are on a relative scale and are compared with calculations by Champion et al which are based on the distorted-wave Born approximation [25, 26]. Chapter five presents experimental results for the electron impact ionization of the hydrogen molecule (H 2 ) with a comparison to helium (He) under the same kinematic conditions. Several (e,2e) experiments on H 2 and He were conducted in coplanar asymmetric geometry at a common intermediate incident energy of 250eV and a fixed scattering angle of -15 o for a range of ejected electron energies from 10eV to 100eV. The results presented are on a relative scale and are compared with theoretical calculations which include quantum mechanical interference effects. The final chapter, chapter six will summarize the work presented within the thesis. outline the future directions to be undertaken both experimentally and theoretically. It will also Appendix A contains the experimental data for the H 2 O TDCSs and appendix B contains the experimental data for the H 2 and He TDCSs. Throughout the thesis atomic units ( h =m e = e = 1) have been used except where stated otherwise. In keeping with the convention of experimental atomic physics, all electron energies are stated in electron volts (ev).

20 Chapter 2 Theory of Electron Impact Ionization 2.1 Introduction An electron collision experiment involves a projectile electron, the incident electron which collides with a target atom or molecule. During the collision event, several collision processes may occur. These collision processes including superelastic scattering, elastic scattering and inelastic scattering. For superelastic scattering to occur, a laser is used to excite the target; the target is then de-excited by the projectile electron which is superelastically scattered (the electron gains energy during the collision process). From an atomic point of view, in an elastic collision there is no change in the internal structure of the two colliding particles. During an inelastic collision the target particle undergoes a change in its internal structure. Two collision processes which change the internal structure of the target particle are ionization and excitation events. This thesis is concerned with the electron impact ionization process. Electron impact ionization is the removal of one or more electron(s) from a target particle due to a collision between an electron and the target particle. There are many different types of ionization processes such as direct and resonant, single and multiple, inner and outer shell ionization of atoms and molecules. Direct ionization is the removal of an electron(s) from the target particle with little interaction with the target core. For resonant ionization, the ionization process occurs via some resonance process after a collision has taken place. An example of a resonance effect is autoionization, in which the projectile electron excites two of the outer-shell electrons in the target species; the target then decays to a lower energy state by the emission of an electron, ionizing the target. Single ionization is when the ion is left with only one unit of positive charge; the ion can also 10

21 2.2. IONIZATION 11 have multiple positive charges due to multiple ionization or other effects such as the Auger effect. The majority of ionization processes are direct single ionization of the target particle. 2.2 Ionization The direct single ionization of a target particle A (assumed to be in the ground state) by electron impact can be described by e + A A + + e + e (2.1) where A + is the ion produced by the collision. The most probable process is an electron being ejected from the valence shell with the ion remaining in the ground state. The motion of the remaining ion can be neglected as the mass of the ion is very large in comparison to the mass of an electron. If themotionoftheremainingionisneglectedequation2.1canberewrittenas: e (E o, k o )+A A + + e (E a, k a )+e (E b, k b ) (2.2) where E o,e a,e b and k o, k a, k b are the kinetic energies and momenta of the incident, scattered and ejected electrons respectively. During the collision, energy and momentum must be conserved. For the ionization process energy conservation demands that, E o = ε i + E a + E b (2.3) where ε i is the ionization potential of an electronic orbital in the target species. The recoil energy of the ion is small compared to the energy of the other particles and is neglected. Using the conservation of momentum, the momentum imparted to the ion is q = k o k a k b. (2.4) The existence of three free moving particles allows four types of cross sections to be defined: the total cross section and the single, double and triple differential cross sections. 2.3 Differential Cross Sections The results obtained from (e,2e) collision experiments are represented as a cross section. A cross section measures the probability that a given type of collision process will occur and is defined as the ratio of scattered particles per unit time, per unit of target number density, per unit of incident flux [30]. The four types of cross sections mentioned above are differential in energy and/or the

22 12 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION angle for the incoming and/or outgoing electron(s). The cross section which yields the least information regarding the ionization process is the total cross section (TCS) represented by σ(e o ). The TCS is a function of only the incident electron energy E o and is a measure of the total number of ions produced due to the collision process independent of the parameters E a,e b, k a and k b. The TCS can be obtained by summing and integrating over all differential cross sections [31]. Differential cross sections yield more information than the TCS as they are dependent on the energies of the collision particles and/or the direction in which the particles are scattered/ejected Single Differential Cross Section The single differential cross section (SDCS) represented by dσ de, (2.5) is a measure of the energy distribution of the two outgoing electrons. Quantum mechanically the two outgoing electrons are indistinguishable; however for asymmetric kinematics the slower electron is referred to as the ejected electron and the fast electron is referred to as the scattered electron. The scattered electron has an electron energy close to the excess energy (defined as E o - ε i ), while the ejected electron has a low energy close to zero. An example of a SDCS is given in figure 2.1. From figure 2.1 it is seen that a minimum in the cross section occurs at (E o - ε i )/2; this indicates that equal energy sharing between the scattered and ejected electrons is the least probable process. The SDCS can not be measured directly. The cross section is obtained by numerical integration of the double differential cross section with respect to all angles of emission of the outgoing electrons [32]. There is a second form of the SDCS which is obtained by the numerical integration of the double differential cross section with respect to all energies. This form of the SDCS has not been measured as it has little physical importance [31] Double Differential Cross Section The double differential cross section (DDCS) represented by d 2 σ dωde, (2.6)

23 2.3. DIFFERENTIAL CROSS SECTIONS 13 d /de Slow Ejected Ionization E a + E b 2 Elastic Scattering (E o ) Excitation Fast Scattered Super Elastic Scattering ( >E o ) E a + E b Electron Energy (ev) E o - i E o Figure 2.1: This diagram represents a single differential cross section. E o is the incident electron energy, E a and E b are the scattered and ejected electron energies respectively and ε i is the ionization potential of an orbital in the target particle [33].

24 14 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION is a measure of the energy and angular distribution of the scattered or ejected electron after an electron impact ionization event. The DDCS can be measured using a cross beam experiment where a beam of energy selected electrons crosses a beam of target particles at 90 o. A single outgoing ejected/scattered electron is then detected according to its energy and/or angle. At high and intermediate incident electron energies, the high energy scattered electron is preferentially emitted in the forward direction, ie., these electrons are scattered in a narrow angular range about the unscattered incident beam of electrons. The low energy ejected electrons are mostly ejected isotropically in all directions. Ejected electrons with higher energies produce some structure in the cross section due to a binary collision between the incident electron and an electron from the target species [32]. The problem with DDCS measurements, is the uncertainty in the processes observed and the state which is ionized for a given excess energy. The theories used to describe the DDCS must include all possible processes and states that could contribute to the cross section. To overcome this problem the energy and momentum of the scattered and ejected electrons must be determined so that the particular ionization process is specific. The (e,2e) technique provides such information via measurement of the triple differential cross section (TDCS) Triple Differential Cross Section The triple differential cross section (TDCS) represented by d 5 σ dω a dω b de a, (2.7) is a measure of the probability that after ionization of a target particle by a projectile with energy E o and momentum k o, two electrons will be produced with energies E a and E b and momenta k a and k b into the solid angles dω a and dω b. Ionizing collision experiments in which all the kinematics of the particles are determined assist in the understanding of the ionization mechanism and of the momentum distribution of the ionized target electron [34]. The electron-electron (e,2e) coincidence technique is used to determine all the kinematics (excluding spin) of the electrons via the measurement of the TDCS. In the (e,2e) technique the scattered and ejected electrons are detected in coincidence. This allows them to be connected to the same ionization event. As the ionization process is fully determined (excluding spin) with respect to all kinematics of the scattered and ejected electrons, theories describing the process are no longer summed over unobservable parameters; therefore TDCS measurements are of considerable importance for comparison with theory.

25 2.3. DIFFERENTIAL CROSS SECTIONS 15 The two most important kinematic parameters required to understand the ionizing collision process are the momentum imparted to the ion and the momentum transfer vector. The momentum imparted to the ion, q is obtained via the conservation of momentum equation (equation 2.4). The momentum transfer vector κ, givenby κ = k o k a (2.8) is defined as the momentum change between the incident electron and the outgoing scattered electron [32]. Figure 2.2 is a vector diagram illustrating the transfer of momentum during an ionizing collision event. k b a k a q k o Figure 2.2: Vector diagram illustrating the transfer of momentum during an ionizing collision event. θ a is the angle the scattered electron has relative to the incident electron, θ κ is the angle at which the momentum is transferred and θ α = θ κ - θ b where θ b is the angle the ejected electron has relative to the incident electron. The amount of momentum transferred during a collision process is important. A TDCS with a high incident electron energy and a small momentum transfer magnitude (κ <1 a.u.) is proportional to the optical oscillator strength. The cross section is therefore dependent on the configuration of the ejected electron within the target particle before it has been ionized [32]. Under this kinematic condition, the collision process is equivalent to photoionization experiment with a similar level of accuracy. This type of electron collision experiment has been extensively used to determine optical oscillator strength measurements, for example see Wiel and Brion [35]. TDCSs under these kinematic conditions also yield additional information relating to the structure of the target particle. Structural studies yield information about the orbital ionization potentials, orbital momentum distributions and the target and ion correlations of the target particle. Structural cross section measurements also measure orbital-specific momentum densities. The measurement of these momentum densities allows a clear picture of an electron s momentum distribution around the target particle nucleus to be determined [36]. When determining structural information, the ionization process is only used as a tool and little or no information about the dynamics of the collision process is obtained [37, 32]. Information relating to the dynamics of the collision event is provided by electron collision experiments with a medium to large momentum transfer magnitude (κ >1 a.u.). Cross

26 16 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION section measurements from dynamical collision events differ from structural measurements as they provide detailed information which relates to processes (such as distortion effects and the interactions between particles) that occur during an ionizing collision event. As there are a large number of kinematic parameters (including k o,k a,k b, θ a and θ b )thatcan be varied when obtaining a TDCS; different kinematic variations of the TDCS allow information concerning different physical properties (relating to the electron impact ionization process) to be obtained. The five kinematic geometries employed to measure the TDCS are: coplanar symmetric, non-coplanar symmetric, coplanar asymmetric, non-coplanar asymmetric and coplanar mutual angle. These five kinematic geometries are shown in figure 2.3. Coplanar depicts a kinematic geometry in which the incident electron, scattered electron and the ejected electron are detected in a single plane, known as the scattering plane. Geometries in which one of the electrons is at an angle φ to the scattering plane are termed non-coplanar. Information obtained from non-coplanar TDCS measurements differ from coplanar measurements as the recoil momentum (q) lies in a plane perpendicular to k o. A symmetric geometry refers to the kinematic arrangement where the scattered and ejected electrons are equal in energy and angle (E a =E b, θ a = θ b ). The kinematic arrangement termed asymmetric is the geometry in which the scattered electron is detected at a fixed forward angle θ a (normally between 5 0 and 20 0 ) while the ejected electron angle θ b is varied within the scattering plane. In the mutual constant angle geometry, the angle between the two outgoing electrons θ ab remains constant. Symmetric Kinematics In the symmetric kinematic regime there is equal energy sharing between the two outgoing electrons (E a =E b ). As seen from the SDCS (figure 2.1), equal energy sharing is the least probable process, thus for symmetric kinematics the cross section is difficult to measure. However these kinematics provide unique information about the dynamics of the collision process. As E a =E b in the symmetric kinematic regime, almost half of the incident electron s momentum is transferred to the atomic electron thus κ is always large. The amount of momentum transferred to the ejected electron is not constant because both θ a and θ b are varied through the scattering plane. From figure 2.2, for a scattering angle of θ a the momentum transfer is defined as κ 2 = k o 2 + k a 2 2k o k a cos(θ a ). (2.9) The structure seen in the TDCS for symmetric kinematics is caused by three main factors, (i) the

27 2.3. DIFFERENTIAL CROSS SECTIONS 17 (a) E o, k o E b, k b E a, k a a b b is varied 0 (d) E o, k o o E b, k b b is varied (b) a is varied E o, k o E b, k b E a, k a a b a is fixed E o, k o E b, k b b is varied 0 E a, k a a b a is varied (c) b is varied 0 (e) E o, k o b E b, k b b E a, k a a a is varied 0 b is varied E a, k a ab 0 a is varied Figure 2.3: Schematic diagram of the five kinematic geometries. (a) Coplanar symmetric geometry where Ea = Eb and θa = θb; θa and θb are varied within the scattering plane, (b) Coplanar asymmetric geometry where Ea Eb, θa is fixed and θb is varied within the scattering plane, (c) Coplanar mutual angle geometry where Ea = Eb and the angle between the outgoing electrons θab is kept constant, (d) Non-coplanar symmetric geometry where Ea = Eb, θa = θb and φo makes an angle to the scattering plane; φo is fixed while θa and θb are varied within the scattering plane and (e) Non-coplanar symmetric geometry where Ea = Eb, θa = θb and φb is out of the scattering plane; φb is varied with respect to the scattering plane.

28 18 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION Coulomb density-of-state factors, (ii) binary collisions, and (iii) quantum interference between different contributions to the total transition amplitude [38]. High energy (e,2e) experiments in the non-coplanar symmetric kinematic regime are generally referred to as Electron Momentum Spectroscopy (EMS) experiments. EMS experiments are normally in a regime where the plane wave impulse approximation is valid (incident electron energies of the order of 1000eV) and are measured for each ion state over a range of ion recoil momentum q from about 0 to 2.5 a.u.. EMS experiments will not be discussed further in this thesis; for a detailed review of EMS see McCarthy and Weigold (1976) and McCarthy (1998) [39, 40, 42]. Asymmetric Kinematics TDCSs measured using asymmetric kinematics differ from non-coplanar symmetric TDCSs as they can obtain either dynamical or structural information depending on the kinematics. The physical properties of the collision process are described by the characteristics of the forward and backward lobes in the TDCS. The typical shape of the TDCS, for the ionization of s-type and p-type electrons in the three energy regions is shown in figure 2.4. The results in figure 2.4 are only relevant if the kinematics are near to the Bethe ridge conditions. Under Bethe ridge conditions all the momentum is transfered to the ejected electron such that κ = k b and the collision can be considered a binary collision. The forward lobe is due to a binary collision that involves only a bound electron; the backward lobe, commonly known as the recoil peak, is due to a binary collision followed by a collision with the target nucleus. Measurement of the TDCS using an asymmetric geometry can be grouped into three energy regions; the low, the intermediate and the high incident electron energy regions. The low energy region refers to the region where 1.5ε i E o 5ε i. In this region exchange processes play an important role as the particles have time to interact with other particles. The expression intermediate energy region is where 5ε i E o 20ε i. In this region the ionization mechanisms are similar to those in the high energy region (exchange processes are negligible), however the higher order terms in the interaction potential are important. The high energy region refers to the energy region where E o 20ε i ;ina high energy collision the particles have very little time to interact [32]. At low and intermediate electron impact ionization energies κ is medium to large. In these interaction energy regimes information relating to the dynamics of the collision process is obtained. At high incident energies there is little to no interaction between the target particle and the incident electron and κ is small. In this regime the incident electron is equivalent to a photon (charge is irrelevant as the particles have very little time to interact) [32] and measurements of the TDCS

29 2.3. DIFFERENTIAL CROSS SECTIONS 19 Figure 2.4: The conjectured shape of TDCSs measured using asymmetric kinematics for s type electrons and p type electrons ordered with respect to the amount of momentum transfer [32]. The conjectured cross sections are only relevant for medium and high electron impact ionization energies. (a) (b) Figure 2.5: (a) A helium differential cross section, differential with respect to p, for the ionization of an s-type electron and (b) An argon differential cross section, differential with respect to p, for the ionization of a p-type electron [39].

30 20 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION yield information which relate to the structure of the target particle. The advantage of using the electron impact ionization technique to obtain the ionization potentials for a target particle is that the incident electron energy can be varied easily in comparison to photoionization techniques [39]. This type of high incident electron energy (e,2e) experiment is generally referred to as Binary (e,2e). The shape of the TDCS for an asymmetric kinematic is determined by the structural or dynamical conditions of the collision process. An example of the TDCS for each energy regime for a s type electron and a p type electron is given in figure 2.4. In TDCSs obtained for s type or p type electrons at high incident electron energies (low κ), a symmetric peak about κ is produced in the binary region and a second peak is produced in the recoil region about -κ. For the intermediate electron energy regime the s type electron produces a single peak in the binary region and a second peak in the recoil region which has a smaller magnitude. The p-type electron produces a double peak in the binary region and a single peak in the recoil region (with a smaller magnitude). How pronounced the double peak structure is, depends on how close the kinematics are to the Bethe ridge condition. In the intermediate energy range there is a shift in the position of the binary and recoil peaks, they are no longer symmetric about κ. For large momentum transfer magnitudes (low incident electron energies) a single binary peak about κ is produced in the binary region for an s type electron; for a p type electron, the TDCS has a double peak, which is symmetric about κ in the binary region [32]. The shape of the TDCS is determined by the momentum distribution of the bound electron before it is ejected. For a p-type orbital the probability of finding an electron when the electron recoil momentum, p is zero, goes to zero. Thus a minimum in the binary region of the TDCS is formed in the direction of the momentum transfer vector. For an s type orbital the probability of finding an electron when p is zero, is at a maximum. Thus a maximum in the binary region of the TDCS is formed in the direction of the momentum transfer vector [31]. The probability of finding a s-type or p-type electron for a range of electron recoil momenta is illustrated in figure 2.5. Mutual Constant Angle Kinematics In the mutual constant angle geometry there is equal energy sharing between the two outgoing electrons (E a =E b ) and the angle between the two outgoing electrons θ ab remains constant. Due to equal energy sharing the physical properties measured in this geometry are similar to symmetric kinematic properties. However when the angle θ ab is chosen so k o k b = 0 the contributions of the single and triplet two electron states to the TDCS are enhanced. Thus under this condition the Coulomb density of state factors are not dependent on the angle of the outgoing electrons (θ ab )and the single or double collision effects are highlighted by this geometry [38].

31 2.4. FORMAL SCATTERING THEORY Formal Scattering Theory The aim of scattering theory is to model the development of the collision system in the interaction region, for comparison with observations measured by experiments. The ultimate goal is to establish uniquely the relationship between the wavefunctions which describe the initial and final states of the system. Due to the current level of computational power a complete theoretical description for all kinematic parameters is not available. However, simplified theoretical models can provide an understanding of the ionization process. The dynamics of a collision system are governed by the principles of quantum mechanics. The Hamiltonian, H is the observable of the system which corresponds to the total energy for a conservative system and is defined as H = K + V (2.10) where K is the total kinetic energy and V is the total potential energy. For every free moving particle in a conservative field there exists an associated wavefunction ψ where its eigenstates Ψ are the quantum states of the system. The eigenvalue equation for the Hamiltonian is the Schrödinger equation, (E H) Ψ =0, (2.11) while the asympototic wavefunction Φ, which is an eigenstate of K can be obtained from, (E K) Φ =0. (2.12) To obtain a cross section for a collision system, the time-independent Schrödinger equation can be solved. The time-independent Schrödinger equation assumes the whole system has reached a stationary state. In a scattering experiment, the incident beam of electrons can be considered as a continuous beam as the electron guns used to produce the electron beam run for long periods of time. A solution of the time-independent Schrödinger equation is given by, ( h2 2m 2 + V (r))ψ(r) =Eψ(r) (2.13) wherev(r)isthescatteringpotentialandtheenergyeoftheelectronis, E = ρ2 2m = h2 k 2 2m = 1 2 mv2. (2.14)

32 22 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION By assuming that the potential, V(r) tends to zero faster than r 1 as r, a particular solution for equation 2.13 is the stationary scattering wave function ψ(r) r A(e iki r + f(k, θ, φ) eikr ), (2.15) r which satisfies the asymptotic boundary conditions. A is a normalization constant which is independent of r and the angles θ and φ and f is the scattering amplitude. The cross section for the collision system can than be related to the asymptotic behavior of this wave function [30]. For a collision event, theoretically the differential cross section is proportional to the scattering amplitude (f ), dσ dω f(k, Ω) 2. (2.16) For an electron collision event, the scattering amplitude f has an explicit dependence on the scattering angles θ and φ and the energy (E) of the particle. f relates the asymptotic behavior of the incident wavefunction to the magnitude of scattering in a particular direction. Using Dirac notation thescatteringamplitudeisgivenby f = 2π 2 Φ kf V Ψ ki. (2.17) Ψ ki describes the incident wavefunction, Φ kf describes the final wavefunction and V is the scattering potential which describes the interaction between the incident electron and the target particle. Equation 2.17 may also be written in terms of the collision matrix. The collision matrix relates the wavefunction that describes the system in the past to the wavefunction that describes the system in the future [30]. There are several forms of the collision matrix including the S-matrix, T-matrix, R-matrix and K-matrix. This thesis is only concerned with the T-matrix, for further information relating to other collision matrices refer to Joachain (1983) and McCarthy and Weigold (1995) [30, 31]. The T-matrix or transition matrix (T if ) element represents the transition between the incident wavefunction Φ ki and the final wavefunction Φ kf. The T-matrix is defined as T Φ ki = V Ψ ki, (2.18) thus equation 2.17 can be re-written in terms of the T-matrix as f = 2π 2 Φ kf T Φ ki. (2.19) This general approach can be applied in the specific case of electron impact ionization to yield the

33 2.4. FORMAL SCATTERING THEORY 23 TDCS in atomic units, d 5 σ dω a dω b de a =(2π) 4 k ak b k o Σ av. T if 2, (2.20) where the summation represents the average of all spin orientations and magnetic sublevels for the final state of the target particle. A complete theoretical description of the ionization process is not available so approximation methods are necessary to evaluate the T-matrix. There are several valid approximations, however before discussing approximation methods it is important to define the Lippmann-Schwinger equation. The Lippmann-Schwinger Equation It is possible to obtain a solution for the wavefunction Ψ using the Lippmann-Schwinger equation which takes the boundary conditions of the scattering problem into account. We begin by rewriting the time-dependent Schrödinger equation as [ 2 + k 2 ]Ψ(r) =V (r)ψ(r), (2.21) where the general solution of this equation may be written as Ψ(r) =Φ(r)+ G O (r, r )V (r )Ψ(r )dr. (2.22) Φ(r) is a solution to the Homogeneous equation [ 2 + k 2 ]Φ(r) = 0 (2.23) and G O (r,r ) is the Green s function for the incoming (+) and outgoing (-) waves defined using Dirac notation as G (±) O (r, r )= r G (±) O r. (2.24) From this the operator called the resolvent or Green s function operator is defined as G O (E (±) ) 1 E (±) K. (2.25) Utilizing the Green s function operator, the wavefunction can be rewritten as Ψ(r) =Φ(r)+G (±) O V Ψ(±), (2.26) the Lippmann-Schwinger equation. The Schrödinger equation can be substituted by the Lippmann- Schwinger equation when discussing various approximation methods used to obtain a theoretical TDCS.

34 24 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION Born Approximations To obtain the general solution of the Schrödinger equation (2.22) we start with a zero-order approximation and increase the number of orders in the approximation to produce a sequence of functions. Ψ 0 (r) = Φ(r) Ψ 1 (r) = Φ(r)+. Ψ n (r) = Φ(r)+ G O (r, r )V (r )Ψ 0 (r )dr G O (r, r )V (r )Ψ n 1 (r )dr. (2.27) If we then assume that this sequence will converge to an exact solution we can obtain the Born series, Ψ(r) =Φ(r)+ G O (r, r )V (r )Ψ 0 (r )dr + G O (r, r )V (r )Ψ 1 (r )dr +... (2.28) where the Born series is defined as a perturbation type expansion of the wavefunction or scattering amplitude in powers of the interaction potential. By substituting the Lippmann-Schwinger equation (2.26) into the Born series we are then able to use the Born approximation to find an expression for the scattering amplitude, f = 2π 2 Φ kf V + VG (±) O V + VG(±) O VG(±) O V +... Φ k i. (2.29) First Born Approximation The first Born approximation (FBA) is used to obtain an approximate quantum mechanical prediction for the differential cross sections where a potential, V(r) scatters a particle in three dimensions. A FBA cross section is obtained using the first term of the Born series, f FBA = 2π 2 Φ kf V Φ ki.. (2.30) For example, for the elastic scattering process, the FBA is described using plane waves where the incident wave (Φ ki ) and the scattered wave (Φ kf ) are defined as Φ ki (r) =e ik i r, (2.31) Φ kf (r) =e ik f r. (2.32)

35 2.4. FORMAL SCATTERING THEORY 25 The scattering amplitude may then be written as f FBA = 2π 2 = 2π 2 e i(k i k f ) r V (r)dr e iκ r V (r)dr (2.33) where κ is the momentum transferred between the incoming and outgoing particles (equation 2.8). Equation 2.33 shows that the f FBA is proportional to the Fourier transform of the potential. From equation 2.16 we also see that f FBA 2 is proportional to the differential cross section. Thus the differential cross section remains unchanged when the sign of the potential is changed [41]. As the interaction potential, V(r) of the function doesn t include all of the interactions within the collision event, the FBA is only able to reproduce experimental cross sections at high incident electron energies. As discussed in chapter 1 the above statement agrees with the observations seen when theoretical calculations using the FBA were compared to experimental results. Higher Order Born Approximations It could be expected that the deficiencies in the FBA could be reduced or eliminated by calculating higher order terms. The inclusion of higher order Born approximation terms in the scattering amplitude (equation 2.29) account for the incident electron interacting multiple times with the potential and then propagating freely between the interactions. The Green s function, G (±) o is the propagator which carries the wave from the scattering point to the observation point. However using the Born series with higher order terms to calculate a cross section is difficult as a large amount of computational power is required. Another difficulty in using higher order Born Approximations is that for low and intermediate electron energies the Born Series will diverge as the potentials are strong enough to bind the particles [30]. Furthermore the FBA and Born Approximations with higher order terms fail to adequately represent a final three body wavefunction. Plane Wave Born Approximation The application of the FBA to the (e,2e) process is via the plane wave Born approximation (PWBA). As the electron impact ionization of a target particle is a three body problem the Hamiltonian in equation 2.10 is re-defined for a three body system. For a three body problem K is given by, K =(K 1 + U 1 )+(K 2 + v 2 ) (2.34) where K 1 is the kinetic energy of the incident/scattered electron, K 2 is the kinetic energy of the bound/ejected electron, U 1 is the distorting potential of the target particle and v 2 is the interaction

36 26 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION between the incident/scattered electron and the target particle. V is given by, V = v 1 + v 3 U 1 (2.35) where v 1 is the interaction between the bound/ejected electron and the target particle and v 3 is the interaction between the incident/scattered electron and the bound/ejected electron. Thus the Hamiltonian (equation 2.10) for a three body problem is defined as, H = K 1 + K 2 + v 1 + v 2 + v 3. (2.36) In the PWBA the incident wavefunction Φ i which describes the incident electron and the target particle is defined as, Φ i = ψ i k o (2.37) where ψ i is the wavefunction for the target particle when it is in its electronic ground state and k o is the incident electron which is represented by a plane wave. The final wavefunction is defined as, Φ f = ψ f k a k b (2.38) where ψ f is the wavefunction of the ion which represents the final target particle state and k a and k b represent the plane waves for the scattered and ejected electrons respectfully. Substituting equations 2.37 and 2.38 into the FBA (equation 2.30) we obtain the scattering amplitude for the PWBA, f PWBA = 2π 2 ψ f k a k b V ψ i k o. (2.39) For convenience it can be assumed that the ion is in the ground electronic state, so the wavefunction of the ion ψ f can be omitted. The TDCS in the PWBA is written as, d 5 σ dω a dω b de a =(2π) 4 k ak b k o Σ av. k a k b V ψ i k o 2. (2.40) A TDCS calculated using the PWBA only yields a binary peak as higher order terms of the Born series are omitted. As discussed earlier, higher order terms account for multiple scattering processes. As the recoil peak results from an interaction between the incident electron and the bound electron followed by an interaction of the bound/ejected electron with the nucleus, multiple scattering processes need to be accounted for.

37 2.4. FORMAL SCATTERING THEORY 27 Distorted Wave Born Approximation As an alternative to including higher order terms in the interaction potential it is simpler to describe the initial and final wavefunctions using distorted waves. When discussing collision theory for a problem with charged particles, the distorted wave transformation involves choosing a local central potential, U called the distorting potential. The scattering amplitude (equation 2.17) is then reformulated in terms of the distorting wave eigenstate of U rather then in terms of plane waves. This is done by partitioning the Hamiltonian (equation 2.10) as follows, H =(K + U)+(V U). (2.41) By substituting equation 2.41 into the Schrödinger equation (2.11) we obtain an inhomogeneous equation of the form (E (±) K U) Ψ (±) =(V U) Ψ (±) (2.42) where K = K o +H t. K o is the kinetic energy operator of the incident electron and H t is the Hamiltonian of the target particle. The homogenous equation with a distorted wave is then written as, (E (±) K o U H t ) jχ (±) j = 0 (2.43) where jχ (±) j is the distorted wave channel state. This equation is separable into projectile and target operators: (E (±) K o U) χ (±) j = 0 (2.44) (ε i H t ) j = 0. (2.45) The j represents a particular channel. j = o, a, b corresponds to the incident electron channel, the scattered electron channel and the ejected electron channel respectively. χ (±) j are the distorted waves for the incident or outgoing waves which replace the plane waves k j in equations 2.37 and 2.38 [31]. For the electron impact ionization of a target particle, the one electron distorted waves are defined as (E o K 1 U 1 ) χ (+) (k o ) = 0 (2.46) (E a K 1 U 1 ) χ ( ) (k a ) = 0 (2.47) (E b K 2 v 2 ) χ ( ) (k b ) = 0 (2.48) (ε i K 2 v 2 ) α = 0 (2.49) where α is the state of the bound electron, thus it is the target state.

38 28 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION Substituting the plane waves in equations 2.37 and 2.38 for distorted waves, the T matrix element is, k a k b T if ψ i k o = χ (+) (k a )χ ( ) (k b ) U 1 αχ (+) (k o ). (2.50) The optimum choice for the distorting potential U 1 is the ground state average of the projectile-target potential, U 1 = α v 1 + v 3 α. (2.51) The central potential v 1 -U 1, does not contribute to the states α and χ ( ) (k b ) of the bound/ejected electron as they are orthogonal [31]. Thus this approximation uses a first order two-body Coulomb potential, 1 v 3 = (2.52) r a r b where r a and r b are the position vectors of the scattered and ejected electrons respectively. The calculation for the distorted waves χ (+) (k o ) and χ ( ) (k a ) are done in the potential U 1 and the calculation of χ ( ) (k b ) is done in the potential v 2. TheTDCSintheDWBAiswrittenas, d 5 σ dω a dω b de a =(2π) 4 k ak b k o Σ av. χ (+) (k o )χ ( ) (k b ) v 3 αχ (+) (k o ) 2. (2.53) Modifications to Born Approximations As Born approximations are not an exact solution to the Schrödinger equation they can not completely describe all processes/interactions that can occur during an ionizing collision. There are several modifications that can be employed to improve TDCS calculations. The three major inclusions are post collisional interaction (PCI), polarization effects and electron exchange. The inclusion of PCI effects accounts for the Coulomb repulsion seen between the two out going electrons. The simplest approach for treating PCI is by multiplying the TDCS by the Gamow factor which adjusts the magnitude of the TDCS [43, 44]. While this method is able to provide accurate TDCSs for He and H at high incident electron energies, it is unable to accurately predict low and intermediate energy ionization cross sections. At low/intermediate energies the outgoing electrons interact for a longer period of time, thus Coulomb repulsion effects have a greater possibility of perturbing the trajectories of the outgoing electrons. Brauner, Briggs and Klar (BBK) [45] calculated the TDCS for incident electron energies of 150eV and 250eV using plane waves for the

39 2.4. FORMAL SCATTERING THEORY 29 incoming electron, Coulomb waves for the outgoing electrons and a three Coulomb (3C) wavefunction for the final state; a phase factor was also included in the calculation. Jones et al [46] then modified the BBK model by using distorted waves for both the incoming and outgoing electrons and an electron-electron correlation factor. This method has been successful in describing the TDCS in the intermediate electron energy range, however it still fails to accurately predict the position of the binary peak for low incident electron energies. To reduce the discrepancies seen when theoretical calculations are compared to experimental data, the correlation factor has been modified further by Jones and Madison [47] who included short range static electron target interactions for the initial and final state wavefunctions. These inclusions have shown better agreement with experimental data at low/intermediate electron energies. The inclusion of polarization effects when describing electron collision events is important, particularly at low electron energies as charge cloud polarization of the target orbitals can produce strong static fields. Charge cloud polarization can be induced by the incident electron and can alter the TDCS. To account for this effect several different forms of polarization potentials (V pot ) have been developed [48]. The inclusion of polarization potentials is accomplished by adding V pot to the static exchange potential used to calculate the incident wave. The simplest form of V pot is given by Padial and Norcross [49] and has a 1/r 4 dependence, where r is the atomic radius of the target particle. From this it can be seen that for inner shell ionization inclusion of the static potential can not be neglected as the ionization process takes place particularly close to the nucleus and therefore in a region where the static potential is strongest. Currently use of the polarization potential for small targets such as H and He has described the TDCS well but fails for heavier targets. As discussed previously, for the electron impact ionization of a target particle, the faster outgoing electron is referred to as the scattered electron and the slower outgoing electron is referred to as the ejected electron, however this is not always true. Electron exchange effects occur when the two outgoing electrons interchange. If the spin of the electrons is considered, then as a direct consequence of the Pauli exclusion principle the total wavefunction must be antisymmetric. For an unpolarized mixture of initial spin states and for all final spin states, 1/4 of the cross section must be calculated with a symmetric wavefunction. The remaining 3/4 is calculated using an antisymmetric wavefunction [30]. As the exchange potential (V exchange ) is spin independent, difficulties arise due to the choice between singlet or triplet states [30]. These difficulties can be avoided by using the Hartree-Fock (HF) approximation of Froese-Fischer [50].

40 30 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION Other Forms of Approximations Plane and Distorted Wave Impulse Approximations Consider a T-matrix which describes the scattering of an incident electron by two particles (the nucleus and the bound electron) as if the particles were free. The relevant T-matrix element is given by the Faddeev-Watson multiple scattering expansion [30], T = t 2 + t 3 + t 2 G O t 3 + t 3 G O t 2 + t 1 G O t 3 + t 2 G O t 1 G O t (2.54) where t i = v i + v i G O t i for i =1, 2, 3. (2.55) i = 1 is the interaction between the bound/ejected electron and the nucleus, i = 2 is the interaction between the incident electron and the nucleus and i = 3 is the interaction between the incident electron and the bound/ejected electron [30]. It is important to note that interactions between the bound/ejected electron and the nucleus only occur in the higher order terms of the Faddeev-Watson scattering series. The Impulse Approximation only takes the first order terms of the Faddeev-Watson multiple scattering expansion, thus the T-matrix is given by, T IA = t 2 + t 3. (2.56) If the only interaction is impulsive, ie. one where there is a clean knock-out of the bound/ejected electron by the incident electron, then the binary encounter approximation can be applied and the T-matrix becomes T IA = t 3. (2.57) By substituting the T-matrix used in the PWBA (equation 2.39) for this T-matrix (2.57), the scattering amplitude for the Plane Wave Impulse Approximation (PWIA) is k a k b T IA αk o = k a k b t 3 αk o. (2.58) Assuming that the Binary Encounter Approximation (equation 2.57) commutes with the ion coordinates, a complete set of plane waves q for the ejected electron can be introduced, k a k b f T IA αk o = d 3 q k a k b t 3 qk o qf α. (2.59) f is the state of the ion and q is the momentum imparted to the ion (equation 2.4). The quantity qf α is the momentum space orbital of the ejected electron, qf α = ϕ α (q) (2.60)

41 2.4. FORMAL SCATTERING THEORY 31 where ϕ α is the Fourier transform of ψ α (r) from position space to momentum space. InthePWIApartofequation2.59isrewrittenas where k a k b T IA qk o = k t 3 k δ(q k a k b + k o ) (2.61) k = 1/2(k o + q) (2.62) k = 1/2(k a k b ) (2.63) and the delta function δ(q-k a - k b +k o ) comes from the translation invariance [51]. The TDCS in the PWIA is then given by d 5 σ =(2π) 4 k ak b f ee Σ av. ϕ α ( q) 2 (2.64) dω a dω b de a k o where the summation (Σ av ) is an average over the initial degeneracies and a sum over the final ones. f ee is the spin averaged two-electron collision factor defined as f ee =Σ av. k t 3 k 2. (2.65) An analytic form of this factor was defined by Ford [52] as f ee = 1 2πη (2π 2 ) 2 e 2πη [ 1 k o k a k o k b k o k a 2 k o k b 2 cos(η ln k o k b 2 )] (2.66) k o k a 2 where η = 1 k. (2.67) The TDCS in the PWIA is thus directly proportional to the momentum space distribution of the ejected electron (equation 2.64). As no information relating to the interaction of the bound/ejected electron with the nucleus is obtained only a binary collision is seen, thus the PWIA only applies to high energy collisions. To apply the Impulse Approximation to lower energies the plane waves can be replaced by distorted waves. The distorted waves are equivalent to the distorted waves used in the Born approximation. By substituting distorted waves into equation 2.59 the scattering amplitude in the distorted wave impulse approximation(dwia) is k a k b f T IA αk o = d 3 q χ (+) (k a )χ ( ) (k b ) t 3 αχ (+) (k o ). (2.68) TheTDCSintheDWIAisthendefinedas d 5 σ =(2π) 4 k ak b f ee Σ av. χ (+) (k a )χ ( ) (k b ) t 3 αχ (+) (k o ) 2. (2.69) dω a dω b de a k o The DWIA is easier to compute then the DWBA as the factor f ee is analytic, however the DWIA can only accurately predict the TDCS at high electron energies and under Bethe ridge conditions.

42 32 CHAPTER 2. THEORY OF ELECTRON IMPACT IONIZATION Convergent Close Coupling The Convergent Close Coupling approximation (CCC) solves the Schrödinger equation by expanding the total wavefunction by diagonalizing the target Hamiltonian in an orthogonal Laguerre basis (of size N) until convergence is obtained. It is assumed that the N states are closely coupled together. That is all the target states lie close to the initial and final states in energy, otherwise the expansion is slow to converge. Thus CCC can effectively predict cross sections for collisions of low energy electrons with small target particles, however it is unsuitable for heavier atoms or molecules [33]. As this thesis is only concerned with molecules this theoretical method is not appropriate; for a complete review of this approximation see Bray and Fursa [53].

43 Chapter 3 Experimental Apparatus 3.1 Introduction An electron impact ionization triple differential cross section (TDCS), is the probability that a bound electron will be ejected from the target atom or molecule into a particular direction with a particular energy. To gain complete knowledge of the ionization process, the ejected electron must be correlated with the final energy and scattering direction of the electron which initiated the collision event. This is achieved by employing the (e,2e) coincidence technique, where measurements are performed by experimentally selecting characteristics of the incoming and outgoing electrons, and then observing the behavior of the collision system. In this thesis, this is achieved using an electron coincidence spectrometer in coplanar asymmetric geometry, where the scattered analyser is fixed at a forward angle θ a while the ejected angle θ b is varied within the scattering plane. This chapter will discuss the components of the electron coincidence spectrometer, the coincidence technique, the alignment of the spectrometer and the experimental calibration of the spectrometer. 3.2 Experimental Apparatus The experimental TDCS measurements for H 2 O, H 2 andhewereperformedwithinavacuumchamber containing the electron coincidence spectrometer. This section will describe the individual components of the apparatus from the vacuum chamber to the fast timing electronics. Photographs of the apparatus are shown in figures 3.1, 3.2 and 3.3. Figure 3.1 is a photograph of the electron coincidence spectrometer which is contained within the scattering chamber, figure 3.2 is a photograph of the scattering chamber and figure 3.3 is a photograph of the electronics used to control and monitor the (e,2e) spectrometer. 33

44 34 CHAPTER 3. EXPERIMENTAL APPARATUS Figure 3.1: A photograph of the (e,2e) spectrometer showing the two electrostatic hemispherical energy analysers mounted on the separate turntables, the electron gun, the Faraday cup and the Faraday cage.

45 3.2. EXPERIMENTAL APPARATUS 35 Figure 3.2: A photograph of the vacuum chamber which contains the (e,2e) spectrometer.

46 36 CHAPTER 3. EXPERIMENTAL APPARATUS Figure 3.3: A photograph of the electronic racks and computers used to control and monitor the proposed experimental measurements.