^{1}, R. Panajotović

^{1}, S. Bellm

^{1}and E. Weigold

^{1}

### Abstract

A new spectrometer is described for measuring the momentum distributions of scatteredelectrons arising from electron-atom and electron-molecule ionization experiments. It incorporates and builds on elements from a number of previous designs, namely, a source of polarized electrons and two high-efficiency electrostaticelectron energy analyzers. The analyzers each comprise a seven-element retarding-electrostatic lens system, four toroidal-sector electrodes, and a fast position-and-time-sensitive two-dimensional delay-line detector. Results are presented for the electron-impact-induced ionization of helium and the elastic scattering of electrons from argon and helium which demonstrate that high levels of momentum resolution and data-collection efficiency are achieved. Problematic aspects regarding variations in collection efficiency over the accepted momentum phase space are addressed and a methodology for their correction presented. Principles behind the present design and previous designs for electrostaticanalyzers based around electrodes of toroidal-sector geometry are discussed and a framework is provided for optimizing future devices.

We thank J. Williams and P. Hayes for interesting discussions in the early stages of the analyzer design. We also thank A. Dorn, H. Schmidt-Böcking, M. Smolarski, C. Höhr, A. Knapp, M. Hattas, and K. Ullmann-Pfleger for stimulating discussions and their contributions to this work. We also acknowledge the technical expertise of Mr. Kevin Roberts who constructed large portions of the spectrometer and provided valuable design input. The assistance of the Australian-German Research Cooperation Scheme and the Australian Research Council through Grant No. DP0452553 and a 1998 ARC RIEF grant is gratefully acknowledged.

I. INTRODUCTION

II. BACKGROUND

III. TOROIDAL ANALYZERS

A. Toroidal geometry

B. Focusing properties of the toroidal field

C. Focusing in the radial planes

D. Focusing in the axial planes

E. Double-toroidal analyzers

1. Background

2. Present design

3. Optimization of design parameters

4. Angular resolution considerations

5. Focusing aberrations in the radial planes

6. Timing resolution

F. Entrance lens design

1. Background

2. Present design

3. Operating principles

4. Axial focusing

G. Analyzer construction

IV. THE ANU SPECTROMETER

V. ELECTRONDETECTORS AND PROCESSING ELECTRONICS

VI. ENERGY RESOLUTION

VII. AZIMUTHAL-PLANE ANGULAR RESOLUTION

VIII. EFFICIENCY CORRECTIONS FOR DATA

IX. RESULTS

X. CONCLUSION

### Key Topics

- Electrodes
- 83.0
- Electron scattering
- 49.0
- Electron spectrometers
- 39.0
- Electron beams
- 35.0
- Spatial resolution
- 34.0

## Figures

The geometry of a toroidal surface is characterized by the “cylindrical radius” and “spherical radius” . The limiting case of the “cylindrical-to-spherical-radius ratio” corresponds to the family of spherical surfaces, while describes cylindrical surfaces. represents the axis of cylindrical symmetry. is the azimuthal angular displacement measured clockwise around the axis and with respect to the positive axis. Polar angles in the radial planes are measured clockwise about the spherical-radius origin located on the positive axis and with respect to the positive axis.

The geometry of a toroidal surface is characterized by the “cylindrical radius” and “spherical radius” . The limiting case of the “cylindrical-to-spherical-radius ratio” corresponds to the family of spherical surfaces, while describes cylindrical surfaces. represents the axis of cylindrical symmetry. is the azimuthal angular displacement measured clockwise around the axis and with respect to the positive axis. Polar angles in the radial planes are measured clockwise about the spherical-radius origin located on the positive axis and with respect to the positive axis.

A toroidal-sector analyzer comprises a pair of nested toroidal-sector electrodes spanning truncated azimuthal and polar ranges and , respectively, where and . The two electrodes are characterized by an identical cylindrical radius , but different spherical radii and (held at potentials and , respectively). To first order and neglect fringing fields, electrons emanating with parallel trajectories from an extended source in the axial plane centered on the cylindrical-symmetry axis come to focus at a unique polar deflection angle ( in this example).

A toroidal-sector analyzer comprises a pair of nested toroidal-sector electrodes spanning truncated azimuthal and polar ranges and , respectively, where and . The two electrodes are characterized by an identical cylindrical radius , but different spherical radii and (held at potentials and , respectively). To first order and neglect fringing fields, electrons emanating with parallel trajectories from an extended source in the axial plane centered on the cylindrical-symmetry axis come to focus at a unique polar deflection angle ( in this example).

A radial-plane cross section through two pairs of concentric toroidal electrodes. For each pair, the inner and outer electrodes are maintained at different potentials. The figure shows the dependence of the shape of the interelectrode equipotential surfaces on the value of the “cylindrical-to-spherical-radius ratio” . In case (i) (small value) significant deviations from circular symmetry are observed in the range . In case (ii) (large value) the equipotentials are well approximated by circles.

A radial-plane cross section through two pairs of concentric toroidal electrodes. For each pair, the inner and outer electrodes are maintained at different potentials. The figure shows the dependence of the shape of the interelectrode equipotential surfaces on the value of the “cylindrical-to-spherical-radius ratio” . In case (i) (small value) significant deviations from circular symmetry are observed in the range . In case (ii) (large value) the equipotentials are well approximated by circles.

Radial-plane cross section through concentric toroidal-sector electrodes showing simulated electron trajectories for the five entrance energies (95, 97.5, 100, 102.5, and ) and for the entrance angles of 0°, , and . The potentials and applied to the inner- and outer-toroidal-sector electrodes are and , respectively. Electrons of the same entrance energy, but different entrance angles, come to focus (to first order) after a deflection angle . The effects of fringing fields have been neglected in this simulation.

Radial-plane cross section through concentric toroidal-sector electrodes showing simulated electron trajectories for the five entrance energies (95, 97.5, 100, 102.5, and ) and for the entrance angles of 0°, , and . The potentials and applied to the inner- and outer-toroidal-sector electrodes are and , respectively. Electrons of the same entrance energy, but different entrance angles, come to focus (to first order) after a deflection angle . The effects of fringing fields have been neglected in this simulation.

Graph showing the functional dependencies of the deflection angle for “point-to-point” focusing in the radial plane (solid line) and the deflection angle for “parallel-to-point” focusing angle in the axial plane (dashed line) as functions of the parameter value. Simultaneous focusing in both planes occurs at the deflection angle for . Also shown is the function (dash-dot curve), pertinent to the discussion of the double-toroidal analyzer, which crosses the function at the coordinates (, ).

Graph showing the functional dependencies of the deflection angle for “point-to-point” focusing in the radial plane (solid line) and the deflection angle for “parallel-to-point” focusing angle in the axial plane (dashed line) as functions of the parameter value. Simultaneous focusing in both planes occurs at the deflection angle for . Also shown is the function (dash-dot curve), pertinent to the discussion of the double-toroidal analyzer, which crosses the function at the coordinates (, ).

Simulated electron trajectories through the present double-toroidal analyzer. Electrons emitted in the scattering plane are deflected through 270° and focused on to a position-sensitive delay-line detector from which their emission energies , azimuthal emission angles , and arrival-time coordinates are deduced. Trajectories corresponding to five different scattered electron energies (95, 97.5, 100, 102.5, and ) and five different polar entrance angles are shown for the pass energy . The inner- and outer-toroidal-sector electrodes for the 180° pair are held at the respective potentials and and for the 90° pair at the respective potentials and . For this simulation, the deceleration ratio was used (see text for details).

Simulated electron trajectories through the present double-toroidal analyzer. Electrons emitted in the scattering plane are deflected through 270° and focused on to a position-sensitive delay-line detector from which their emission energies , azimuthal emission angles , and arrival-time coordinates are deduced. Trajectories corresponding to five different scattered electron energies (95, 97.5, 100, 102.5, and ) and five different polar entrance angles are shown for the pass energy . The inner- and outer-toroidal-sector electrodes for the 180° pair are held at the respective potentials and and for the 90° pair at the respective potentials and . For this simulation, the deceleration ratio was used (see text for details).

Schematic representation of alternative analyzer designs for different values of the angular variable (see text for details). Each design comprises two toroidal-sector electrode pairs of respective cylindrical radii and and respective polar sector angles and . Continuity between inner- and outer-electrode surfaces is maintained by the collinearity of the spherical radius origins ( and ) and , the point where the central electron path intersects the interface between the two electrode pairs. Only for is .

Schematic representation of alternative analyzer designs for different values of the angular variable (see text for details). Each design comprises two toroidal-sector electrode pairs of respective cylindrical radii and and respective polar sector angles and . Continuity between inner- and outer-electrode surfaces is maintained by the collinearity of the spherical radius origins ( and ) and , the point where the central electron path intersects the interface between the two electrode pairs. Only for is .

Radial-plane view of the interface between the two toroidal-sector electrode pairs comprising the present double-toroidal-sector analyzer. Equipotential lines are shown for (a) the case of equal pass energies and (b) the case where the pass energy of the second toroidal-sector electrode pair is twice that of the first. For case (b), the applied potentials correspond to the simulation of Fig. 6. See text for details.

Radial-plane view of the interface between the two toroidal-sector electrode pairs comprising the present double-toroidal-sector analyzer. Equipotential lines are shown for (a) the case of equal pass energies and (b) the case where the pass energy of the second toroidal-sector electrode pair is twice that of the first. For case (b), the applied potentials correspond to the simulation of Fig. 6. See text for details.

The seven-element angle-resolving electrostatic lens system used to angularly select and focus electrons emitted from the interaction region at angles . It consists of a grounded first lens , two defining slits and , and five focusing lenses of independently adjustable potentials. The final lens is fixed to a potential value determined by the analyzer pass energy . Each lens comprises an identical pair of annular electrodes. Indicated dimensions are in millimeters. See text for details.

The seven-element angle-resolving electrostatic lens system used to angularly select and focus electrons emitted from the interaction region at angles . It consists of a grounded first lens , two defining slits and , and five focusing lenses of independently adjustable potentials. The final lens is fixed to a potential value determined by the analyzer pass energy . Each lens comprises an identical pair of annular electrodes. Indicated dimensions are in millimeters. See text for details.

Divergence of initially parallel electron trajectories as they pass between two juxtaposed pairs of annular-lens electrodes operated at different potentials (axial-plane view). The dimensions, electron energies, and potentials were chosen for illustrative purposes only. Parallel off-axial rays are deflected in the axial plane as they traverse the lens-pair interface. The degree of axial-plane deflection increases monotonically with increasing displacement of the trajectory from that of an electron emanating from the axis of cylindrical symmetry into the same azimuthal scattering angle . Note that deviations shown are due to curvature of lens surfaces, not due to the apparent “discrete” nature of the simulation grid.

Divergence of initially parallel electron trajectories as they pass between two juxtaposed pairs of annular-lens electrodes operated at different potentials (axial-plane view). The dimensions, electron energies, and potentials were chosen for illustrative purposes only. Parallel off-axial rays are deflected in the axial plane as they traverse the lens-pair interface. The degree of axial-plane deflection increases monotonically with increasing displacement of the trajectory from that of an electron emanating from the axis of cylindrical symmetry into the same azimuthal scattering angle . Note that deviations shown are due to curvature of lens surfaces, not due to the apparent “discrete” nature of the simulation grid.

Double-toroidal-sector electron analyzer (lens system removed). For clarity, field correcting side plates which assist in correcting for fringe fields at the edges of both the lens system and toroidal-sector analyzer have also been removed. Supporting bar, is for structural stability only, and is removed when side plates are in place. See text for details.

Double-toroidal-sector electron analyzer (lens system removed). For clarity, field correcting side plates which assist in correcting for fringe fields at the edges of both the lens system and toroidal-sector analyzer have also been removed. Supporting bar, is for structural stability only, and is removed when side plates are in place. See text for details.

Schematic representation of the polarized-electron spectrometer comprising polarized-electron source, two double-toroidal-sector electron analyzers, and a Mott polarimeter (not shown). See text for details.

Schematic representation of the polarized-electron spectrometer comprising polarized-electron source, two double-toroidal-sector electron analyzers, and a Mott polarimeter (not shown). See text for details.

Schematic representation of the electron detection and instrumental control electronics. It comprises microchannel-plate electron multipliers (MCP), two delay-line position-and-time-sensitive detectors, amplifiers, and discriminators, fast coincidence unit, CAMAC-based 16-channel time-to-digital converter, spin-tag electronics, liquid crystal retarders and their control electronics, and computer with data-acquisition card and running COBOLD analysis-and-display software. See text for details.

Schematic representation of the electron detection and instrumental control electronics. It comprises microchannel-plate electron multipliers (MCP), two delay-line position-and-time-sensitive detectors, amplifiers, and discriminators, fast coincidence unit, CAMAC-based 16-channel time-to-digital converter, spin-tag electronics, liquid crystal retarders and their control electronics, and computer with data-acquisition card and running COBOLD analysis-and-display software. See text for details.

Coincidence-timing spectrum measured for analyzer pass energies of and deceleration ratios . coincidence counts are displayed as a function of the time difference , where and represent the respective arrival times for the two electrons comprising each recorded electron pair. See text for details.

Coincidence-timing spectrum measured for analyzer pass energies of and deceleration ratios . coincidence counts are displayed as a function of the time difference , where and represent the respective arrival times for the two electrons comprising each recorded electron pair. See text for details.

Detector image resulting from the measurement of electrons scattered elastically from helium. The electron energy was stepped from at intervals of . Electrons of a given energy trace out an arc on the detector. The radius of the arc varies monotonically with electron energy while the angular position along each varies linearly with the azimuthal electron-emission angle . See text for details.

Detector image resulting from the measurement of electrons scattered elastically from helium. The electron energy was stepped from at intervals of . Electrons of a given energy trace out an arc on the detector. The radius of the arc varies monotonically with electron energy while the angular position along each varies linearly with the azimuthal electron-emission angle . See text for details.

Differential cross section for electrons elastically scattered from argon at an impact energy of . Open circles: present results. Solid circles: experimental results of Panajotovic *et al.* (Ref. 58). Solid line: relativistic polarized-orbital calculation by McEachran (Ref. 59) convoluted with a 2° FWHM Gaussian. The present results have been normalized to the data of Panajotovic *et al.* at a scattering angle of 50°.

Differential cross section for electrons elastically scattered from argon at an impact energy of . Open circles: present results. Solid circles: experimental results of Panajotovic *et al.* (Ref. 58). Solid line: relativistic polarized-orbital calculation by McEachran (Ref. 59) convoluted with a 2° FWHM Gaussian. The present results have been normalized to the data of Panajotovic *et al.* at a scattering angle of 50°.

Measured angular distribution for elastic scattering from helium at . Solid circles: present results. Open circles connected by straight lines: experimental results of Register *et al.* (Ref. 60). Large scatter in present data points primarily attributable to spatial gain variations in microchannel-plate detection efficiency. See text for details.

Measured angular distribution for elastic scattering from helium at . Solid circles: present results. Open circles connected by straight lines: experimental results of Register *et al.* (Ref. 60). Large scatter in present data points primarily attributable to spatial gain variations in microchannel-plate detection efficiency. See text for details.

Elastic scattering measurement from background helium gas. Experimental conditions as for Fig. 17, except that the target gas is introduced from a position well removed from the axis.

Elastic scattering measurement from background helium gas. Experimental conditions as for Fig. 17, except that the target gas is introduced from a position well removed from the axis.

Schematic representation of scattering geometry and coordinate system defining the triple-differential ionization cross section. An incident electron of energy ionizes a target atom, resulting in the ejection of two electrons of respective energies and into the polar and azimuthal emission angles and , and and , respectively. For the present spectrometer design, only those electron pairs emitted into a common plane containing the incident electron trajectory (the so-called scattering plane) are measured.

Schematic representation of scattering geometry and coordinate system defining the triple-differential ionization cross section. An incident electron of energy ionizes a target atom, resulting in the ejection of two electrons of respective energies and into the polar and azimuthal emission angles and , and and , respectively. For the present spectrometer design, only those electron pairs emitted into a common plane containing the incident electron trajectory (the so-called scattering plane) are measured.

Relative cross section for the ionization of ground-state helium where the residual ion is excited to the manifold. The incident beam energy was , the fast and slow scattered electrons of 200 and respective mean energies. For these kinematics, ionization leading to this excited state is around 0.4% of the intensity of transitions leaving the residual ion in the ground state. Data-collection time was around . See text for details.

Relative cross section for the ionization of ground-state helium where the residual ion is excited to the manifold. The incident beam energy was , the fast and slow scattered electrons of 200 and respective mean energies. For these kinematics, ionization leading to this excited state is around 0.4% of the intensity of transitions leaving the residual ion in the ground state. Data-collection time was around . See text for details.

Doubly symmetric measurement on helium where the residual ion is left in the ground electronic state. Incident beam energy , electron scattering angles . Solid circles: present data, open squares: Murray and Read (Ref. 63). Present data have been normalized to Murray and Read data for best visual fit. Error bars on Murray and Read data not shown as they are smaller than symbols. See text for details.

Doubly symmetric measurement on helium where the residual ion is left in the ground electronic state. Incident beam energy , electron scattering angles . Solid circles: present data, open squares: Murray and Read (Ref. 63). Present data have been normalized to Murray and Read data for best visual fit. Error bars on Murray and Read data not shown as they are smaller than symbols. See text for details.

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