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

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

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**Look
at the Symmetric (e,2e) Data collected by this
spectrometer**

**Look
at the (e,2e) Computer Controlled Spectrometer
Hardware**

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

**Link
to the Manchester Physics & Astronomy Department Home
Page**

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.

- A detection plane spanned by the scattered and ejected electron momenta
**ka**and**kb**is initially defined. - The incident electron momentum
**k0**makes an angle y, the 'gun angle' with respect to this plane, and - the projection of this vector onto the detection plane defines the scattering angles xa and xb of the two outgoing electrons.

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.

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.

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

- the classical Wannier picture and its semi-classical and quantum mechanical derivatives, through
- classical trajectory studies to
- purely quantum mechanical treatments of ionisation.

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

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.

Figure 2. **Ionisation in the threshold region**. The
incident electron ionises the target, releasing a valence electron. This
electron, and the incident electron that has lost almost all of its energy
in the reaction have similar energies and therefore as they emerge from
the reaction zone feel the Coulombic influence of each other and the ion
core following ionisation. The low relative velocity of the electrons
allows
them to mutually interact for sufficient time to tightly correlate their
respective asymptotic directions.

In the intermediate energy region all the complexities of

- exchange
- capture
- incoming channel distortions
- outgoing channel distortions and
- short and long range correlations between
*all*electrons and the core must be included

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.

Figure 3. Ionisation in the intermediate energy region.
The incident electron again ionises the target, releasing a valence
electron.
The scattering is more complex than in the threshold region, since the
incident electron can penetrate deeper into the neutral target electron
cloud, and therefore there is more complex interactions between the
electrons
and the core. There is now a finite probability of exchange, capture,
ingoing
and outgoing correlations between all charged particles in the reaction.
This leads to both forward and backward scattering possibilities, depending
upon the complexity of these interactions.

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

- the ingoing electron momentum
**k0**and - the outgoing electron momenta
**ka**and**kb**.

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.

Rösel, T, Dupré C, Röder J, Duguet A, Jung K,
Lahmam-Bennani
A. and Ehrhardt H. (1991) *Coplanar symmetric (e,2e) cross sections on
helium and neon*, J. Phys.B At. Mol. Opt. Phys. **24**:3059.

Rösel T, Röder J, Frost L, Jung K, Ehrhardt H, Jones S. and
Madison D.H. (1992), *Absolute triple differential cross section for
ionisation of helium near threshold*, Phys Rev A **46**:2539.

Murray A.J. and Read F.H. (1992), *Novel exploration of the helium
(e,2e) Ionisation process*, Phys Rev Letters **69**:2912

Murray A.J. and Read F.H. (1993), *Evolution from the coplanar to
the perpendicular plane geometry of helium (e,2e) differential cross
sections
symmetric in scattering angle and energy*, Phys Rev A **47**:3724

Murray A.J, Woolf M.B.J. and Read F.H. (1992), *Results from symmetric
and non-symmetric energy sharing (e,2e) experiments in the perpendicular
plane*, J.Phys.B At. Mol. Opt. Phys. **25**:3021

Wannier G.H. (1953), *The threshold law for single ionisation of atoms
or ions by electrons*, Phys. Rev. **90**:817

Peterkop R. (1971), *WKB approximation and threshold law for electron
atom ionisation*, J. Phys. B. At. Mol. Phys. **4**:513

Rau A.R.P. (1971), *Two electrons in a coulomb potential. Double
continuum
wave functions and threshold law for electron atom ionisation*, Physics
Review A. **4**:207

Crothers D.S.F. (1986), *Quantal threshold ionisation*, J. Phys.
B. At. Mol. Phys. **19**:463

Altick P.L. (1985), *Use of a long range correlation factor in
describing
triply differential electron impact ionisation cross sections*, J. Phys.
B. At. Mol. Phys. **18**:1841

Klar H. and Fehr M. (1992), *Parameterisation of multiply differential
cross sections*, Z. Phys. D. **23**:295

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
**coplanar**results of Selles*et al*(1987a, 1987b) and - the
**perpendicular**plane measurements of Hawley-Jones*et al*(1989).

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:

.........................(1)

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:

where:

...............(2)

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:

Figure 4. The Paris coplanar and Manchester Perpendicular plane results at an excess energy of 1eV ionising Helium as the target. The Wannier model parameterisation transformed to the (y,x) geometry is shown as a fit to the data. Results from this parameterisation are additionally shown at intermediate incident electron beam angles of 30°, 60° & 80°.

**Figure 4 **shows:

- the experimental results and
- the calculated DCS in the (y,x) symmetric geometry at an excess energy E = 1eV for incident electron angles y = 0°, 30°, 60°, 80° & 90°.

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

- The perpendicular plane results have an approximately Gaussian angular distribution, in contrast to
- The coplanar results which have a slight asymmetry in the backscatter direction (x > 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.

Fournier-Lagarde P, Mazeau J. and Huetz A. (1984), *Electron impact
ionisation of helium - a measurement of (e,2e) differential cross sections
close to threshold*, J. Phys. B. At. Mol. Phys. **17**:L591

Selles P., Mazeau J. and Huetz A. (1987), *Wannier theory for P and
D states of two electrons*, J. Phys. B. At. Mol. Phys.
**20**:5183

Selles P, Huetz A. and Mazeau J. (1987), *Analysis of e-e angular
correlations in near threshold electron impact ionisation of helium*,
J. Phys. B. At. Mol. Phys. **20**:5195

Hawley Jones T.J, Read F.H, Cvejanovic S., Hammond P. and King G.C.
(1992), *Measurements in the perpendicular plane of angular correlations
in near threshold electron impact ionisation of helium*, J. Phys.B At.
Mol. Opt. Phys. **25**:2393

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 :

- 3eV above threshold
- 5eV above threshold
- 10eV above threshold
- 20eV above threshold
- 30eV above threshold
- 40eV above threshold
- 50eV above threshold.

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

Figure 5. The experimental results from 1eV to 50eV above the ionisation threshold of helium. More detail is available by linking to the appropriate page.

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**.

- At 1eV excess energy
**a single structure is observed**due to strong outgoing electron correlation, as previously noted. - At 3eV and 5eV above ionisation both
**a forward and a backward structure start evolving**from the central peak at low incident electron angles y, whereas at higher angles y the DCS is still approximately Gaussian.

**Backward scattering**, probably dominated by elastic scattering of the incoming electron from the atom into the backward direction in the detection plane followed by a binary collision with a valence electron,**is the dominant contribution to the overall DCS structure**.

- At around
**25eV****forward and backward peaks become approximately equal in magnitude**.

- At the higher excess energy (30 - 50eV)
**forward scattering starts to dominate**, reflecting**the importance of single binary collisions**. - This effect increasingly dominates as the energy progresses into the high energy regime.

- In the
**perpendicular plane**(y = p/2) the single peak at low energies evolves into a three peak structure as the energy increases. - The relative magnitude of the central peak at x = p/2 compared to the peaks at low and high values of x decreases with increasing energy
- at 50eV excess energy the three peaks are of approximately the same magnitude.

- Only the central peak is attainable through a single collision process and there will additionally be significant contributions to this peak through outgoing electron correlations at the lower energies.
- The non-central peaks can only arise through higher order collision processes.

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

**Indistinguishability of the outgoing electrons**requires the DCS at (y,x) to be equated to that at (y, 2mp-x) where m is an integer.**Reflection symmetry**is necessary in the detection plane since no preferred direction is defined with respect to this plane, and hence s(y,x) = s(-y,x).**As the gun angle reaches p/2**the projection of**k0**onto the detection plane becomes undefined and so only the angle (xa + xb) is relevant, as a consequence of which s(p - y, x) = s(y, p - x).

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.

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 :

..................................(4)

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

.............(5)

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

.......................(6)

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 :

................(7)

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:

......(8)

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

- qb = qa and
- fb = p - fa linearly relate many of the angular functions.

Thus for certain values of *la*, *lb*
and *l0* :

.........(9)

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.

- Here the set for which
*la*=*lb*when*l0*is even and (*la*,*l0*) <*lb*when*l0*is odd is chosen. - All other angular functions can be related to this set using the above relationship.

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

..........(10)

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.

Figure 6. The I110 function plotted as a surface in 3-D
space. The function can either be positive or negative depending upon the
value of theta and phi, and the sign is therefore represented by two
colours,
**PURPLE **indicating a positive magnitude
and **RED **representing a negative
magnitude.

- It is necessary to consider the magnitude and sign of the functions
as the (e,2e) Differential Cross Section is parameterised as a
**linear sum**of the angular functions (see equation 6), and so for various values of theta and phi the functions can cancel each other to allow zeroes in the cross section.

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:

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

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

..........(11)

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

- In the first procedure a
**linear least-squares fit**was used to obtain the unknown Blalbl0. Extra pseudo-points were added to the data in the inaccessible regions produced from a spline fit through the data which was constrained to pass through zero at x = -p/2, 0°, +p/2 and +p. - Experimental data in the regions -p/2 < x < 0° and +p/2 < x < +p were generated using the experimental symmetries. These artificially generated points were given low weighting to minimise the constraints placed upon the fit.
- This linear fitting was fast and reliable, yielding excellent starting parameters for the second non-linear fitting method.

- In the second procedure no artificial pseudo-points were added.
- A
**Simplex method**was used to minimise the c2 function by adjusting the Blalbl0 - Any trial for which the fitting function did not obey the assumptions was rejected and the Simplex routine was reset to explore alternative regions of the c2 surface.
- This method converges far more slowly than the linear fitting method but requires no inferred data.

Figure 7. The Simplex fitting method. The example here
shows the method for finding the minimum of a 3-D Gaussian function of
the form z = - exp(-x^2 + y^2). The Simplex, in this case a 2-D triangle
whose vertices are defined by points on the surface, proceeds by
reflection,
contraction, expansion or shrinking along the surface as shown. Testing
for the minimum vertex with each iteration allows the Simplex to proceed
down the well until the minimum is located.

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 :**

- At 1eV excess energy only
*l0*= 0 to 4 contribute, and the*la*and*lb*terms are all of the same sign, reinforcing each other in each*l0*manifold. - The limitation on
*l0max*reflects the bilinear fit to the L=0, 1 & 2 partial wave coherent summation in the Wannier model, whereas - the equal signs of the
*la*and*lb*terms reflect the simplicity of the lobe structure. - The relative magnitudes of the terms in each manifold combine to produce the lobe structure at the mutual angle of p radians.

- As the energy increases this regular pattern quickly disappears as the forward and backward lobes evolve from the central 'Gaussian' profile.
- Higher order
*l0*correlation terms become significant as the intermediate energy region is entered, indicating an increasing contribution of higher order partial waves to the DCS. - Additionally the
*la*and*lb*terms start to compete in magnitude and sign to produce the more complex structures observed. - At excess energies above about 30eV the parameters tend to stabilise in sign and vary smoothly with energy.
- The maximum significant number of
*la*and*lb*terms in each*l0*manifold also decreases as the energy increases, indicating that the number of partial waves required in the outgoing channel decreases with increasing energy.

**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 incident electron beam direction is shown in each figure.
- The 3-D surface is generated from a vector whose
**length**is given by the**magnitude**of the differential cross section as the surface is generated throughout space for all angles (q,f). - The backward scattering region is towards the viewer, whereas the forward scattering region is away from the viewer.

- At 1eV only 2 lobes are seen, these being opposite each other.
- At 3eV and 5eV the lobes start to evolve into a 4 lobe structure.
- At 10eV and 20eV there is a greater probability of backward scattering, whereas
- at 30eV to 50eV forward scattering starts to dominate the reaction process.

**Figure 9**. 3-D representations of the DCS calculated
from the fitted parameters Blalbl0 for the 8 data sets from 1eV to 50eV.
Further details on each of the data sets can be obtained by accessing the
links to each figure.

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.

- The (e,2e) differential cross section s
is then projected onto a
**four dimensional space**, whose axes are defined by the independant variables (s, q,f & Eexc).

- As this cannot be visualised as a single projection on these pages,
**it is instructive to map one of these variables onto the time co-ordinate**.

In this case, we choose the **excess energy to be varied in time**,
whereas the other 3 variables are mapped onto conventional 3-dimensional
co-ordinate space, as has been done in the previous figures.

Hence, **once the DCS is parameterised as a function of energy as well
as a function of the scattering angle**, this parameterisation can be
used to derive 3-D images at discrete energy intervals.

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.

- Such a movie has been constructed using the
**Apple Quicktime compression and display program**that is available for a number of different computers, including

- MacIntosh and
- IBM PC compatible computers.

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.

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:

- A cubic spline is interpolated through the parameters as a function of the ratio of excess energy to the ionisation energy of 24.6eV.
- 1000 equally spaced points are then derived from the cubic spline
- These are fitted using a
**ninth order polynomial fit**which closely emulates the cubic spline through the data.

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 **:

..............(12)

- Details of these
**cubic spline fits to the Blalbl0 parameters**can be found by accessing the appropriate interlink page**splinefit.htm**. - The b-parameters which define the normalised
DCS as a function of energy and angle can be downloaded here as a
**text file**.

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.

- 100 of these images have been generated at 0.5eV intervals, and compressed into a Quicktime movie.

- This movie can be downloaded (and played using some WWW viewers such as Netscape Gold) by clicking either on the file e2emovie.mov
- Or for a
**dark background**the file DarkDCS.mov can be accessed and viewed.

- Note that
**these files are around 3MB in size**, so may take some time to download should internet access be busy.

- Additionally
**the evolution of the 44.6eV DCS as successive angular functions are added together can be seen**! - This movie can be downloaded by clicking on the Quicktime file 44evolution.mov.

*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 Atomic Molecular & Laser Manipulation group Home Page