Page constructed by Andrew Murray

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**

**Link to the Atomic Molecular & Laser Manipulation Group
Home Page**

**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 :

- Ehrhardt H, Jung K, Knoth G and Schlemmer P 1986 Z Phys D
**1**3 - Byron F W and Joachain C J 1989 Physics Reports
**179**211-272 - Lahmam-Bennani A 1991 J Phys B
**24**2401

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

- energy (Ea,Eb) and
- momentum
**k**a(qa,fa) and**k**b(qb,fb) away from the reaction zone.

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

- Hawley-Jones T J, Read F H, Cvejanovic S, Hammond P and King G 1992
J Phys B
**25**2393

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

- Read F H 1985
*Electron Impact Ionisation*(Eds G Dunn and T Märk, Springer, Berlin) 42-88

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

- the tuning of the electron gun and
- the tuning of the analysers to
- the setting of the analyser positions and
- subsequent data collection.

**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 :

- The
**electron gun**is optimised initially to the current detected in the**Faraday cup** - 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. This decouples the tuning of the electron gun from any tuning requirements of the electron analysers. Further it allows the electron gun to be aligned and focussed accurately onto the interaction region whose centre is located at the intersection of the axes of rotation of the analysers and the electron gun. This is essential for experiments where all of these may be varied over 3-Dimensional space. - The
**analysers**are then moved to their starting positions - 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. Data accumulation is
therefore
carefully controlled since :

- The
**optimisation**of the analysers in the inner loop allows for any variation of the analyser positions and image as the analysers sweep around the detection plane, whereas - the
**optimisation**in the outer loop allows for any long term variation in the running conditions of the system due to either changes in the gas density or variations in the electron gun conditions.

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 :

- At the
**lowest**incident energy which is measured there is a single peak at 180°. 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. - At
**higher incident energies**peaks appear in the region of f = 180° ± 90°. - There is a cross over point around 74.6eV where the peak at 180° is of equal intensity to the peaks at 180° ± 90°, &
- as the
**incident energy****increases**the 180° peak**diminishes**with respect to the peaks in the side lobes.

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 overlap volume between the gas jet, the electron beam and the volume viewed by each analyser, as well as
- the efficiency of detection and the solid angle viewed by each analyser.

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

- Rösel T, Dupre C, Röder J, Duguet A, Jung K, Lahmam-Bennani
A and Ehrhardt H 1991 J Phys B
**24**3059

To evaluate the analyser efficiency and solid angle of detection, the
Double Differential Cross Section (**DDCS**) data of

- Müller-Fiedler R, Jung K and Ehrhardt H 1986 J Phys B
**19**1211

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,

- each analyser observes the interaction region with an efficiency given by the transmission efficiency of the electron optics and the detection efficiency of the channeltron.
- 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.

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

- J(V) is the electron beam current density and
- r(V) is the target gas density in the interaction volume.
- VA(E1) is given by the three way overlap of the gas beam, electron beam and volume as seen by the analyser.

Assuming that the profiles of the gas and electron beam are constant over the interaction volume, this integral reduces to the double integral :

where

- pr2 is the area swept out by the electron beam traversing the interaction volume as defined by axial symmetry, and
- IF is the current as detected in the Faraday cup.

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

- DDCS (Einc,Ei) represents the double differential cross section at qi = 90°.

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

- the gas beam,
- the electron beam and
- the volumes viewed by both analysers, VO/lap.

Hence if C (Einc.,E1) is the coincidence count rate and I.P. is the energy required for ionisation, we obtain

- Taking ratios of the coincidence counts for different incident energies,
- noting the above relationship for the ratios of efficiencies and solid angles of the detectors and
- rearranging this expression therefore yields :

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.

- 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 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.
- The analyser input electrostatic lenses were also modelled to determine the volume viewed at the interaction point for the varying energies of electrons selected. These calculations indicated that the diameter of the viewed volume varied from 4.8mm to 3.9mm over the selected energy range 5 to 40eV.

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,

- the
**worst case**occurs when the volume common to the gas jet and electron beam completely overlaps the region viewed by the analysers and the volumetric profiles of the gas jet and electron beam are uniform rather than centrally peaked as might normally be expected, while - the
**best case**(GV = 1) occurs when this common volume is always smaller than the region viewed by the analysers.

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

- D(Ei) is the diameter of the region viewed by the analyser at the energy Ei, while
- the overlap volume VO/lap(Einc,Ei) will be given by the smallest volume as observed by the two analysers.

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 :

- The count rates as measured from the experiment are considered to be Poissonian, whereas
- the error in the coincidence signal is determined using standard techniques
- (e.g. Eminyan M, MacAdam K B, Slevin J and Kleinpoppen H 1974 J Phys
B
**7**1519). - The dominant error in the expression is given by the double
differential
cross sections as measured by Müller-Fiedler
*et al*(1986), since these terms arise four times in the expression, and typically have measurement errors of ± 15%. - This leads to a statistical uncertainty in the normalisation of approximately ± 30%.

**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

- Hicks P J and Comer J 1975 J Phys B
**8**1866.

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 DCS at q = 90° for 100eV impact energy as measured by Gélébart and Tweed (1990) is s(90°) = 3.09 ± 0.79 (x 10E-4) a.u., whereas
- the interpolation to an energy of 100eV in this experiment has an uncertainty estimated to be ±20%.
- The uncertainty in the normalisations of the relative energy is approximately ±30%, as discussed in the previous section.

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

- Mota-Furtado F and O'Mahony P F 1989 J Phys B
**22**3925 using a second Born approximation and by - Zhang X, Whelan C T and Walters H R J 1990 J Phys B
**23**L173 using a DWBA calculation.

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.

- The peaks in the region of f = 180° ± 90° are thought to be due to an initial elastic scattering of the incident electron by the atomic system through an angle of approximately 90° followed by a quasi-free collision between the elastically scattered electron and a bound electron. The electrons are then able to emerge at approximately 90° with respect to each other in the perpendicular plane, as is observed. This type of double collision process must therefore be modelled using higher order scattering terms as is done using the DWBA.

On the other hand the peak that is observed at 180° 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. As the momentum distribution of the bound electron peaks at zero for helium, the peak at 180° diminishes with increased incident electron energy as the probability of finding a bound electron with the required momentum decreases.
- In terms of multiple scattering, particularly at the lower incident energies, the peak at 180° has additional contributions that may be described by outgoing electron correlations as in the Wannier model.

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

- Zhang X, Whelan C T and Walters H R J 1990 J Phys B
**23**L173.

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**