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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 parameters (Ea,qa,fa,Eb,qb,fb) are freely selectable experimentally within the constraints that energy and momentum must be conserved during the collision. 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.
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. A number of modifications to the original spectrometer as described in Hawley-Jones et al (1992) have been implemented and are detailed elsewhere.
Unique to this (e,2e) electron spectrometer is the computer interface which controls and optimises the spectrometer during normal operation. Figure 2 is a block diagram of the hardware interface between the computer and the spectrometer. 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 12A 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.
Following input of relevant control information, the system is switched to computer control. The electron gun is then optimised to the counts detected by the photomultiplier tube, thus ensuring that the electron current is focussed to a beam of diameter approximately 1mm at the gas jet.
The analyser input lenses and deflectors are then adjusted by the computer to optimise the non-coincidence counting rate of each analyser. A coincidence correlation spectrum is then acquired for a predetermined time, after which the analysers are moved to a new position and the optimisation procedure repeated in the inner loop.
Once the analysers have moved around the detection plane, the whole system is reoptimised and the process of data collection repeated. 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. 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.
Results from experiment for the symmetric case, where the outgoing electrons following the collision have equal energy, are shown in the figures 4a - 4h. In the perpendicular plane, the symmetry about the incident electron beam direction requires that the DCS must be symmetric about f = 180A° and this is evident in the data. 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. 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.
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. 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 180A°. 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 90A° but is independent of the azimuthal angle f. 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.
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 A± dE1.

The solid angle of detection depends upon the geometry and operating potentials of the electrostatic lenses, and is therefore also a function of the detected electron energy. VA(E1) is given by the three way overlap of the gas beam, electron beam and volume as seen by the analyser.
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. 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..
The ratios on the left-hand side of this equation were obtained from the DDCS data of MA?ller-Fiedler et al (1986) at an incident energy of 200eV for each analyser at the selected energies of interest. 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 MA?ller-Fiedler et al (1986). As no experimental facilities existed to measure these parameters, a SIMION ray-tracing model was set-up to allow a computational determination of these volumetric terms.
The analyser input electrostatic lenses were also modelled to determine the volume viewed at the interaction point for the varying energies of electrons selected.
Figure 6 indicates the experimental geometry as viewed by the analysers in the perpendicular plane, where the atomic beam direction is at 45A° to the incident electron beam trajectory.
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.
The dominant error in the expression is given by the double differential cross sections as measured by MA?ller-Fiedler et al (1986), since these terms arise four times in the expression, and typically have measurement errors of A± 15%. This is possible since the q = 90A° coplanar symmetric energy-sharing result is identical to the perpendicular plane f = 180A° 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.
The uncertainty in the normalisations of the relative energy is approximately A±30%, as discussed in the previous section. The absolute differential cross sections as presented in figures 4 and 5 therefore have uncertainties of approximately A±44%. 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 180A° can be explained in terms of either single or multiple scattering.
In terms of single scattering it is due to the incident electron selecting a bound electron whose momentum is equal in magnitude but opposite in direction to its own momentum. In terms of multiple scattering, particularly at the lower incident energies, the peak at 180A° has additional contributions that may be described by outgoing electron correlations as in the Wannier model. The mass equivalence of the scattered and ejected electrons requires that a quasi-free collision results in electrons emerging at approximately 90A° 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. These results predict a single peak at 180A° 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.
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For an unpolarised target and unpolarised incident electron beam the only azimuthal angle of significance is given by f = fa - fb.
When the electrons are detected in the perpendicular plane, the two angles qa and qb are both equal to 90A°.

The perpendicular plane is accessed mainly through double collision processes (Hawley-Jones et al (1992), which represent only part of the full dynamical picture. The computer loads all the voltage supplies from either a manually selected set of voltages or from the results of a previous optimisation run. This decouples the tuning of the electron gun from any tuning requirements of the electron analysers. This should be compared with the data obtained near threshold by Hawley-Jones et al (1992), which also shows a single peak with an approximately Gaussian structure.
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 model indicated that the electron gun operating at the voltages determined by the optimisation routines produced an electron beam of diameter approximately 1.0mm at the interaction region, consistent with the tuning of the electron beam onto counts from the photomultiplier which views a 1mm3 volume. For the non-symmetric data no compensation is required for volume effects since GV = 1.0 at all values of Ei. The electrons are then able to emerge at approximately 90A° with respect to each other in the perpendicular plane, as is observed. As the momentum distribution of the bound electron peaks at zero for helium, the peak at 180A° diminishes with increased incident electron energy as the probability of finding a bound electron with the required momentum decreases. It means that you can place the tool on a removable device (like a USB flash drive) and directly run its executable file on any computer.
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