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This page was updated on 20th January 1999.
Look at the (e,2e) Computer Controlled Spectrometer Hardware
Look at the Symmetric (e,2e) Data collected by this spectrometer
Look at the 64.6eV Data where the detected electrons have unequal energies
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Link to the Manchester Physics & Astronomy Department Home Page
1. Introduction.
Electron-electron angular correlation experiments, or so-called (e,2e) coincidence experiments, provide the most detailed test of theories which attempt to describe the ionisation of a target atom by an incoming electron.
The experimental and theoretical aspects of this field have been extensively reviewed by :
to name but a few. These experiments measure the angular correlation between a projectile electron which is scattered from a target and a resulting electron which is ionised from the target.
This process, for direct ionisation from the ground state of the neutral target without excitation of the ion may be written:
e+ A Æ A+ + ea+ eb
where (ea, eb) signify the two electrons resulting from the collision, which respectively carry
The parameters (Ea,qa,fa,Eb,qb,fb) are freely selectable experimentally within the constraints that energy and momentum must be conserved during the collision. For an unpolarised target and unpolarised incident electron beam the only azimuthal angle of significance is given by f = fa - fb.
Figure 1 shows a schematic of the general (e,2e) process. When the electrons are detected in the perpendicular plane, the two angles qa and qb are both equal to 90°. The (e,2e) Differential Cross Section (DCS) in this geometry is then
where Wa, Wb
are the solid angles of detection of analysers that select electrons of
energy Ea, Eb at polar
co-ordinates
(90°,fa),
(90°,fb)
respectively, with f = fa-
fb. The perpendicular plane
is accessed mainly through double collision processes (Hawley-Jones et
al (1992), which represent only part of the full dynamical picture.
Figure 1. The Interaction region and Detection plane as depicted in conventional Ehrhardt Geometry. The Perpendicular Plane is defined when qa =qb = 90°
In the experiments described here the DCS has been measured in the perpendicular plane using a helium target with incident electron energies ranging from 10eV to 80eV above the ionisation threshold.
Eight different incident electron energies from 34.6eV to 104.6eV have been used for the symmetric energy case, where the selected electrons following the collision have equal energies (i.e. Ea = Eb), and five different incident energies from 34.6eV to 74.6eV for the non-symmetric case (i.e. Ea Eb).
The Electron Coincidence Spectrometer.
The electron coincidence spectrometer was designed to measure angular and energy correlations between low energy electrons emerging from near-threshold electron impact ionisation of helium and is capable of accessing all geometries from the perpendicular plane to coplanar geometry.
The initial experiments as detailed in
were confined to the perpendicular plane, and were designed to test the theories of the Wannier model for near threshold ionisation processes. For a review of the Wannier model, see
A number of modifications to the original spectrometer as described in Hawley-Jones et al (1992) have been implemented and are detailed elsewhere.
The Computer Optimisation and Control (Brief Review).
Unique to this (e,2e) electron spectrometer is the computer interface which controls and optimises the spectrometer during normal operation.
The computer controls all aspects of the spectrometer from
Figure 2 is a block diagram of the hardware interface between the computer and the spectrometer.
Figure 2. The Hardware which Interfaces the Computer to the Spectrometer (Block Diagram)
At the heart of the operation is an IBM 80286 PC which controls the spectrometer and receives information about the system status from monitors around the spectrometer.
The electrostatic lens and deflector voltages in the system are supplied by separate optically isolated active supplies controlled by 12 bit digital to analogue converter (DAC) cards, while a digital monitor measures the voltages and currents on the various lens and deflector elements around the system.
The analyser positions are measured by potentiometers located in the drive shafts external to the spectrometer, and are monitored by a datalogger located on the main PC bus, which also monitors the EHT supplies and the vacuum pressure in the system.
Count rates from the channel electron multipliers and photomultiplier tube are monitored by a 32 bit counter board, and the TAC output is sent directly to an MCA card installed in the main PC.
The software controlling the electron coincidence spectrometer is written to address specific tasks unique to the coincidence experiment.
The main PC controls the spectrometer lens and deflector tuning during operation, optimising these voltages using a Simplex method.
Figure 3 illustrates the software control.
Figure 3. The Software which controls the Computer running the Spectrometer (Block Diagram)
Following input of relevant control information, the system is switched to computer control. The computer loads all the voltage supplies from either a manually selected set of voltages or from the results of a previous optimisation run. The procedure which establishes a coincidence signal is then implemented :
A correlation function between the scattered and ejected electrons is therefore accumulated over many sweeps of the detection plane until the results are statistically significant. Data accumulation is therefore carefully controlled since :
During operation the running conditions are monitored every 50 seconds by the data logger, allowing any anomalies in the data to be accounted for when final results are considered.
Experimental Perpendicular Plane Results with Symmetric Energy Sharing.
Results from experiment for the symmetric case, where the outgoing electrons following the collision have equal energy, are shown in the figures 4a - 4h.
Figure 4. Symmetric Energy (Ea = Eb) Experimental results for the DCS placed on an absolute scale by comparison with the measurements of Gélébart and Tweed (1990). The estimated uncertainty in the normalisation to this absolute scale is ±44%.
In these experiments the incident electron energy varies from 10eV above the ionisation threshold of helium (24.6eV), to 80eV above this threshold, in steps of 10eV increments.
The absolute values of the cross sections have been estimated by normalising the relative results to a common point and relating this to the absolute cross sections determined at 100eV incident energy in the coplanar symmetric experiments of Gélébart and Tweed (1990).
A full discussion of this procedure is given below.
In the perpendicular plane, the symmetry about the incident electron beam direction requires that the DCS must be symmetric about f = 180° and this is evident in the data.
The evolution of the peaks observed in the angular correlation as the energy changes is clearly seen :
Inspection of the 94.6eV data (35eV scattered and ejected electron energies) indicates that this result does not follow the trends of the other data in the set. This is considered to be due to the contribution to the cross section of a number of resonances in helium for scattered and ejected electron energies around 35eV.
The resolution of the spectrometer (~ 1eV) does not allow these contributions to be excluded, and so the 94.6eV data set cannot strictly be considered as part of the overall set of data which is presented here.
Experimental Perpendicular Plane Results with Non-Symmetric Energy Sharing.
Experiments have also been carried out where one of the detected electrons has an energy of 5eV and the other has an energy that varies from 5 to 45eV, in incremental steps of 10eV.
The first result in this data set therefore is the same as the first result in the previous set, allowing a convenient point for common normalisation of the data sets.
Figures 5a - 5e indicate the results so obtained.
Figure 5. Non-Symmetric Energy (Ea Eb) Experimental results for the DCS placed on an absolute scale by comparison with the measurements of Gélébart and Tweed (1990). The estimated uncertainty in the normalisation to this absolute scale is ±44%.
At the two lower energies the results are similar to the symmetric case.
Above 54.6eV incident energy the results differ markedly, the angular correlation evolving into a single broad peak instead of the three distinct peaks observed in the symmetric case.
This is most clearly evident at 74.6eV incident energy, where the symmetric case shows three distinct peaks of approximately equal intensity, whereas the non-symmetric case shows only a broad featureless structure centred about 180°.
Relative Normalisation Procedure for the Data at Different Incident Energies.
The sets of data shown in figures 4 and 5 were each normalised relative to the result obtained at 34.6eV incident energy, where both the electrons emerge from the interaction region with an energy of 5eV.
The relative angular data presented at any particular energy is normalised by consideration of the electron singles counting rate, which depends on the Double Differential Cross Section at 90° but is independent of the azimuthal angle f.
To normalise the data for different energies requires knowledge about
The procedure adopted here allows an upper and lower bound to be estimated, as no facility exists to allow these overlap volumes to be accurately determined experimentally as is implemented by other electron scattering groups. For details of other techniques adopted, see
To evaluate the analyser efficiency and solid angle of detection, the Double Differential Cross Section (DDCS) data of
is considered.
The DDCS gives the probability of obtaining an electron from the scattering event at an angle q1 in a solid angle dW1 at an energy E1 ± dE1. In the present experiment,
Thus letting eF(EA) be the efficiency of analyser A and dWA(EA,qA,fA) be the solid angle of detection, the singles count NA(Einc.,E1) obtained at the channeltron output for an incident electron energy Einc. and a detected energy E1 as a function of the DDCS may be written
where
Assuming that the profiles of the gas and electron beam are constant over the interaction volume, this integral reduces to the double integral :
where
The DDCS varies only slowly over the energy range dE accepted by the analyser (Jones et al 1991) as does the efficiency and solid angle of detection of the analyser as determined from electrostatic lens studies.
The energy resolution DEA is independent of E1 because the pass energy of the hemispherical deflector is kept constant. For an incident energy Einc. and constant target gas density and interaction current the ratio of the counts obtained at different analyser energies is therefore given by :
This expression can be rearranged to yield the ratios of analyser efficiency and solid angle in terms of the DDCS, singles count rates and overlap volumes at the two energies E1 and E2 for an incident energy Einc..
Thus:
where
The ratios on the left-hand side of this equation were obtained from the DDCS data of Müller-Fiedler et al (1986) at an incident energy of 200eV for each analyser at the selected energies of interest.
In a similar way the coincidence count rate may be evaluated in terms of the DCS.
In this case the volume integral required is the four way overlap integral between
Hence if C (Einc.,E1) is the coincidence count rate and I.P. is the energy required for ionisation, we obtain
The first three terms in this expression are evaluated directly from the experiment, while the fourth and fifth terms are taken from the data of Müller-Fiedler et al (1986).
This leaves the final three terms in the expression.
Figure 6 indicates the experimental geometry as viewed by the analysers in the perpendicular plane, where the atomic beam direction is at 45° to the incident electron beam trajectory.
Figure 6. The estimated interaction volume. The diameter of the image as viewed by the analysers varied from 4.8mm to 3.9mm over the range of detected energies.
Letting GV be the product of the final three volumetric terms in the above expression,
For the worst case, the interaction volume as seen by the analyser for a 1mm diameter electron beam is given by :
V(Ei) ~ p x (0.5)^2 x D(Ei)
where
Thus :
VO/lap(Einc,Ei) = min (VA(Ei),VB(Einc-IP-Ei))
The data is normalised to the TDCS at 34.6eV where each analyser selects 5eV electrons.
For the symmetric data the worst case ratios GV are therefore given by :
Table 1 indicates the calculated relationship GV as determined from the SIMION ray tracing model, and these values have been used to calculate the normalisations as given in figure 4. For the non-symmetric data no compensation is required for volume effects since GV = 1.0 at all values of Ei.
|
Energy (eV) |
34.6eV |
44.6eV |
54.6eV |
64.6eV |
74.6eV |
84.6eV |
94.6eV |
104.6eV |
|
Factor Gv |
1.0 |
0.91 |
0.83 |
0.82 |
0.82 |
0.82 |
0.87 |
0.87 |
Table 1. The Volumetric Factors GV as a function of Incident Electron Energy.
The errors associated with the normalisation are determined using standard error analysis applied to the above equations :
Calculation of the Absolute Differential Cross Sections.
No experimental facilities exist to allow determination of absolute differential cross sections, however it is possible to obtain these values relative to the work of Gélébart and Tweed (1990), who measured absolute cross sections for 100eV electron impact ionisation of helium in the symmetric energy-sharing geometry.
This is possible since the q = 90° coplanar symmetric energy-sharing result is identical to the perpendicular plane f = 180° energy-sharing result.
Since data was not obtained at an incident energy of 100eV in these experiments, an interpolation between the energy normalised symmetric results yields a relative value.
It should be noted that the 94.6eV data was not used in this interpolation due to the presence of the strong (2p)D and (2s2p)P resonances at this energy as detailed in
The much weaker (2p2)1S and (2s3s)1S resonances at 100eV and (2p4p)1D, (sp24+)1P, (2s5s)1S and (sp25+)3P resonances at 104.6eV incident energy have been ignored, since their contribution to the differential cross sections at these energies is expected to be negligible.
The absolute differential cross sections as presented in figures 4 and 5 therefore have uncertainties of approximately ±44%.
Discussion of the Results.
Theoretical investigations of the DCS in the perpendicular plane have been instigated for the Symmetric energy configuration by
These latter calculations, normalised to one energy, produced results that were close to those of the experiment.
A simple descriptive model for the symmetric case illuminates a possible scattering processes involved.
On the other hand the peak that is observed at 180° can be explained in terms of either single or multiple scattering.
This simple intuitive model cannot be readily applied to the non-symmetric case, since the experimental results indicate that as the energy increases the peaks observed at 180° ± 90° are not enhanced, but tend to merge into the central peak at 180° until at 74.6eV incident energy there is only a single broad peak observed.
The mass equivalence of the scattered and ejected electrons requires that a quasi-free collision results in electrons emerging at approximately 90° to each other irrespective of their resulting energies, and so application of this model would once again yield three peaks, although the relative peak intensities might be expected to change.
It is therefore necessary to consider a different scattering mechanism, and this has been done by
These results predict a single peak at 180° due to a mechanism following elastic scattering of the incident electron from the nucleus that is more delicate and complex than the quasi-free collision, however the wings at the lower energies in the experiment are not predicted by this model.
This may not be entirely unexpected as the DWBA model gives better results at higher energies.
The central peak predicted by the theory also differs in width and height from the results presented here, when normalised to the 34.6eV data.
Look at the (e,2e) Computer Controlled Spectrometer Hardware
Look at the Symmetric (e,2e) Data collected by this spectrometer
Look at the 64.6eV Data where the detected electrons have unequal energies
Link to the Atomic Molecular & Laser Manipulation Group Home Page
Link to the Manchester Physics & Astronomy Department Home Page