The (e,2e) Parameterisation using a set of Irreducible Angular Tensorial Functions

page prepared by Andrew Murray (20th January, 1999)

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A large body of experimental data now exists for (e,2e) differential cross section (DCS) ionisation studies in which the scattered and ejected electrons are detected with the same energy and at the 'same' asymptotic scattering angles with respect to the incident electron direction. These experiments have recently been carried out principally by two groups at Kaiserslautern group and at Manchester (see references for examples). It is the results of the Manchester group that are considered here.

Figure 1 shows the scattering geometry for these experiments.

In all experiments carried out so far the condition xa = xb = x has been chosen, yielding a common normalisation point at xa = xb = p/2 for all gun angles y.

General geometry

Figure 1. The (e,2e) Scattering geometry. See text for details.

The energy region from around 3eV to 100eV excess energy is defined here as the 'intermediate energy' region.

Studies at lower energies are considered to lie in the threshold region where correlation effects between outgoing electrons dominate the reaction.

Differential cross section studies at energies in excess of 100eV are predominantly governed by single binary collisions between the incident and ejected electrons, the core playing a decreasing role in the reaction as this energy increases. This region is now successfully modelled to a large degree using the Born approximation and its associated derivatives.

Ionisation in the Threshold Region.

The threshold region has been studied using different models ranging from:

Common to all these models is the importance of correlations between the outgoing electrons brought about through electrostatic coupling as they emerge from the reaction zone (see figure 2).

As the excess energy above ionisation decreases, the outgoing electrons have more time to mutually interact and so the probability that they will emerge asymptotically at a mutual angle of p radians increases, whereas their 'memory' of the incident electron direction correspondingly decreases.

The symmetric geometry chosen with xa = xb = x thus increasingly constrains the differential cross section to a narrowing maximum at xa = xb = p/2 as the excess energy decreases toward zero.

threshold correlations

Ionisation in the Intermediate Energy Region

In the intermediate energy region all the complexities of

All can play a significant role in the reaction process (see figure 3). Although theoretical interest in this region is increasing, no theory adequately explains the experimental data accumulated so far.

intermediate correlations

In this page a model-independent parameterisation of the ionisation differential cross section is discussed.

The parameterisation is in terms of a complete orthogonal set of irreducible tensorial angular functions which define correlation between any three vectors in space.

In the present geometry as shown in figure 1, it is sensible to choose these vectors to be

This parameterisation was first proposed for the (e,2e) process by Klar and Fehr (1992), and has a long and distinguished history in the field of nuclear particle interactions proceeding through sharply defined states, where the analysis gives information about the angular momenta carried into and out of the reaction.

By contrast, in the (e,2e) reaction the intermediate state is not normally sharply defined, and so the parameters do not directly reveal the angular momenta associated with the reaction, but rather indicate the degree of correlation between ingoing and outgoing electrons.


Parameterisation of experimental data in the threshold region using the Wannier model.

Here the energy independent fitting parameters suggested by Fournier-Lagarde et al (1984) in the Wannier model and derived from experimental results in the paper by Hawley-Jones et al (1989) provide interpolated data between :

The (e,2e) differential cross section is parameterised as a coherent superposition of partial waves up to L = 2 defining scattering in the reaction zone which is modulated by a Gaussian function modelling correlations between the outgoing electrons.

The differential cross section s is given by:


where in the geometry chosen here g = p - qab and:

Eexc is the excess energy shared between the outgoing electrons.

For the symmetric geometry all triplet amplitudes f1 are zero since they depend upon the difference between the angles qa and qb

This leaves only singlet partial wave contributions to the differential cross section. The DCS is normalised to the S-wave amplitude, yielding singlet partial waves up to L = 2 given by:



and where alpha = 0.1865.

The energy independent parameters in the above expressions calculated using the results of the Manchester and Paris groups yield:

The differential cross section is finally obtained in the (y,x) geometry by applying the transformations:

1eV data & fit graph

Figure 4 shows:

These results have been re-normalised to unity at the common point xa = xb = p/2.

There is no evidence of the complex structure discussed above for the intermediate energy regime. The strong outgoing electron correlations brought about through electrostatic repulsion only allow the single structure at x = p/2 to be observed.


Experimental data in the Intermediate Energy Region.

Comprehensive experimental data exists for ionisation of helium in the intermediate energy region over a very wide range of scattering geometries.

These experiments have been carried out from the coplanar geometry (y = 0°) to the perpendicular plane geometry (y = p/2) for 7 excess energies :

The experimental results can be accessed in detail, a summary being shown in figure 5.

In these results the DCS is assumed to be zero when x = 0 and x = p since the probability of detection of two electrons of equal energy emerging in the same direction is very small.

The regions between x = 0° and 35° and between x = 125° and 180° are experimentally inaccessible due to the physical size of the detectors and electron gun, and in these regions the cross section is assumed to be positive and monotonically decreasing towards zero.

Parameterising the Experimental Data.

The differential cross section has symmetries which are defined by the experimental constraints:

These symmetries allow additional data to be inferred in the region p/2 >y > 2p . In the experimentally inaccessible regions centred at x = 0° and x = p the DCS is constrained to be positive and to have no points of inflection, and at the angles x = 0° and p themselves the DCS is put equal to zero.

A. General Model of the Parameterisation.

When electron spin is not considered the (e,2e) differential cross section is a function only of the momenta of the ingoing and outgoing electrons. Hence :


A complete orthogonal basis upon which to project the angular dependence of the cross section is given by the functions :


where Ylm(k) is a spherical harmonic. Expanding the differential cross section in terms of this basis then gives


where the parameters Blalbl0 define all non-angular dependent terms characterising the reaction.

The choice of angular functions Ilalbl0 required to characterise the (e,2e) DCS is restricted by the symmetries inherent in the ionisation process.

It follows that for the symmetric case as considered here :


The summation in equation (6) can therefore be restricted to lb >=la.

Defining the z-axis along k0 and letting the x-axis lie in the detection plane spanned by ka and kb the polar angles of ka and kb are (qa, fa) and (qb, fb) and so the angular functions have the form:


In general the complete set of angular functions is required to fully characterise the differential cross section, however with the present geometry the relationships

Thus for certain values of la, lb and l0 :


Only a subset of the full range of angular functions is required. The choice of la, lb and l0 used to define the subset is arbitrary within the constraint that the angular functions must be unrelated.

As an example, the functions I134 and I235 can be written as follows :


It is possible to build 3-D images of the angular functions used in the parameterisation. These images are established by plotting the magnitude of the angular function against the angles theta and phi that are defined in the function.

As an example, consider the angular function where l0 = 0 and la and lb are equal to 1. This I110 function when in symmetric geometry reduces to the form:

where theta and phi take on the usual spherical polar co-ordinates.

It is clear that plotting the absolute magnitude of the I110 function as a radial vector when scanning over all possible values of theta and phi produces a 3-D surface which represents the function.

Since the magnitude can be positive as well as negative, it is important to represent this facet when describing the function in this way. It is convenient to use two colours for this representation, as shown in figure below.

The complete set of 3-D images of the 44 Angular Functions Ila,lb,l0 required to parameterise the symmetric (e,2e) Differential Cross Section over the energy range from 1eV to 50eV above the Helium ionisation threshold have been generated.

These images, together with their appropriate functional form and a downloadable postscript file of each figure can be accessed by linking to the appropriate page for each of the seven groups l0 = 0 to l0 = 6:



B. Fitting the Parameterisation to the Experimental Data

The parameterisation was fitted to the data using a least squares fitting procedure.

The quantity to be minimised is the chi-squared function :


The summation is over the experimental data points i each with an associated weight wi evaluated from the uncertainty in the data.

Two fitting methods were adopted:

simplex figure

The maximum significant values of la, lb and l0 were established using a statistical F-test. Consideration of the data sets at all excess energies established that 44 angular functions up to la, lb = 7 and l0 = 6 gave the best overall fit to the data.

Figure 8 shows the 44 fitted Blalbl0 parameters from the Simplex fitting routine for the eight data sets all normalised to unity at x = p/2. The fitting parameters can be downloaded as a TAB deliminated text file by linking to the file Blalbl0.dat.

There is a clear difference between the parameters in the threshold region compared to those in the higher energy region :


Figure 8. The fitted parameters Blalbl0 for the 8 data sets. Higher resolution images together with a downloadable EPS file for each of the data sets can be obtained by linking to either of the above low resolution images.

Figure 9 shows 3-D representations of the (e,2e) differential cross sections at each energy from 1eV to 50eV excess energy as obtained from fitting the data to the parameterisation. Further information can be obtained by linking from the figures to the appropriate WWW page.

The results shown in figure 9 show 3 dimensional representations of the (e,2e) differential cross sections derived from the parameterisation at the energies where experiments were conducted.

It is useful to establish a technique which allows interpolation between these results at the eight discrete excess energies, to enable the differential cross section to be estimated as a function of the three angular and one energy variables over the complete range of energies from threshold to 50eV excess energy.

These images can then be projected as a moving film, where the time from the start to the end of the film depicts the energy change from 1eV to 50eV excess energy.

This program is becoming the industry standard and so is chosen here. The 3D Quicktime movie can be downloaded at the end of this page by linking to the appropriate file.

Parameterising the DCS in terms of Excess Energy.

Since the DCS are normalised to unity at the common point x = p/2, only the relative changes as a function of energy can be determined.

As no physical model yet exists for the variation of the Blalbl0 with energy, the following procedure is adopted:

This method was used as there is no simple general analytical form for a cubic spline. The polynomial fit then gives a compact set of 396 parameters blalbl0j :


Although the b-parameters do not have a direct physical meaning, they allow the normalised DCS surfaces to be generated at any energy from 1eV to 50eV as has been discussed.

More information on the (e,2e) experiment can be found in the following pages:

Application of the Parameterisation to the deconvolution of the effects of experimental solid angles on the DCS

Go to the (e,2e) home page

Exploring the (e,2e) Hardware

The (e,2e) symmetric data

Go to the Atomic Molecular & Laser Manipulation group Home Page