Jan 14, 2001 (last modified by
A.K.)
COMPLETE
EXPERIMENTS FOR THE DOUBLE IONISATION OF He:
(e,3e) CROSS
SECTIONS AT 1keV IMPACT ENERGY
AND SMALL
MOMENTUM TRANSFER
A. Lahmam-Bennani(a,b), A. Duguet(a), M. N. Gaboriaud(a), I. Taouil (a), M. Lecas,(a)
A. Kheifets(c), J. Berakdar(d) and C. Dal Cappello(e)
(a)Laboratoire des Collisions Atomiques et Moléculaires
(UMR 8625)
Bât. 351, Université de Paris-Sud XI, F-91405 Orsay Cedex, France
(c)Research School of Physical Sciences, The Australian
National University,
Canberra ACT 0200, Australia
(d)Max-Planck Institut für Mikrostruktur Physik, Weinberg
2, D-06120 Halle, Germany
(e) Institut de Physique, LPMC, 1 Boulevard Arago,
Technopôle 2000, F-57078 Metz, France
(b) Author to whom all correspondance should be addressed
e-mail : azzedine@lcam.u-psud.fr
Physics Abstracts Classification :
3450H, 3480D
Short Title : (e,3e) cross sections
for He at 1 keV
submitted to J.Phys.B
: At.Mol.Opt.Phys., January xxx, 2001
Abstract
The relative, coplanar angular distributions of electrons, produced in
an electron-impact double ionization of helium (e,3e reaction), have been
measured at 1.1 keV impact energy. The momentum transfer was 0.45 a.u. and the
two "ejected"
electrons were detected with the same
energy of 10 eV. The general features of the angular distributions are
discussed. The data are analysed in different angular modes which allows a
detailed comparison with state-of the art calculations. For high incident
energy and small momentum transfer, as in the present case, the (e,3e) cross
section can be related to the single -photon double ionisation (PDI). We
exploit this fact and compare the present findings with the PDI and identify
the contribution of non-dipole effects.
1. Introduction
The study of atomic electron-impact double ionisation,
also called the (e,3e) reaction, provides a powerful and straightforward tool
to investigate the dynamics of interacting, highly excited few-electron systems (three highly excited
electrons in the field of a residual ion). A rather complete picture of the
(e,3e) process is obtained by
performing an experiment in which the momentum vectors of all the
particles are determined (a complete experiment would require resolving the
spins as well). In the present experiment the reaction cross section is
measured while the solid emission angles of the three electrons and the
energies of two of them are determined, i.e. a five-fold differential cross
sections (5DCS) is registered. Other kinematical variables of the involved
particles are obtained from the energy and (linear) momentum conservation laws.
Recently, we conducted a systematic study of the
(e,3e) reaction employing various targets (Kr [1], Ar [2], Ne [3] and He
[4-6]). Basically, all these measurements haven been carried out at a
relatively high impact energy ( ~5.5 keV) and a small momentum transfer to the
target. Therefore, the corresponding theoretical models [5-10] have been
designed in the spirit of the first Born approximation (FBA) for the
projectile-target interaction. Most of
the theoretical results have been obtained using He as a target since the
residual ion (the alpha particle) has no relevant internal structure leading
thus to a simplified theoretical treatment. The comparison of these theories
with the experiments have lead to the following observation. The absolute
magnitude of the calculated cross sections was largely different from one
calculation to another and from the experimental absolute data. In general, however, the qualitative
features of the measured angular correlation patterns at various fixed ejection
angles were reproduced by the theories. In several cases the calculations deviated significantly from the
experiments with regards to the shape of the angular correlation patterns. This
disagreement was attributed, at least partly, to the non-first Born effects
which were not included in most of the theoretical models (note that the C4FS
model described in [5] went somehow beyond the FBA). Similar effects have also
been observed in Ar [2] and, to some extent, in Ne [3]. In addition, it was
found that the dipole limit was approached differently at various electrons
ejection angles despite the fact that the amount of momentum transferred from
the projectile to the target remained constant. This means that in the case of
double ionisation the optical limit depends not only on the incident energy and
the amount of momentum transfer, as is the case for single ionisation, but it
also depends on the emission direction of the slow ejected electrons, see [11].
These deviations are seen to be energy dependent and are expected to become
larger at lower incident energy in which case the optical limit is increasingly
violated. We have therefore performed new low impact-energy experiments ( ~1
keV) in a similar scattering geometry to the work reported in [5] at 5.5 keV.
The goal is to address the origin of the deviations between the experiments and
FBA type calculations. To this end, our data will be compared to results from
three standard theoretical models, which have been shown to be successful for
the description of single ionisation processes or photo double ionisation
processes. Two of these models utilise the FBA in a sense described above. This
approximation reduces the four-body problem (three electrons in the field of
residual ion) to a three-body one (the
two slow electrons in the residual-ion field). The latter problem is then
approached within the framework of the three Coulomb wave method (3C) [12] or
using the convergent close coupling (CCC) formalism [13]. Alternatively, the
third model employes a correlated four-body final state (C4FS) as described in
[5] and references therein and goes beyond the FBA via the introduction of
effective charges. However, as noted in [6], the variation of the effective
charges is small, and the non-first Born effects due to this model should be
weak.
Throughout this paper, the same notations as in [5]
are used. In particular, positive scattering and ejection angles are counted
counter-clockwise, starting at the incident beam direction.
2. Experimental
2.1 Experimental procedure
The experimental set-up and procedure have been
described extensively elsewhere [2, 6, 14]. Briefly, a coplanar arrangement is
used where the incident and the three outgoing electrons lie in the same plane.
The impact energy is E0 = 1099 eV. The scattered electron is observed at a fixed angle, qa= +1.10°, with an energy Ea = 1000 eV (corresponding to a fixed momentum
transfer, K = 0.45 au, in the direction qK = - 21.6°). The value of the scattering angle is
measured with an accuracy of ± 0.02°, whereas the spectrometer acceptance angle
is Dqa = ± 0.10°, which corresponds to a high momentum
transfer resolution DK < ± 0.006 au, and a small uncertainty in the
momentum transfer direction, < ± 0.9°. The two ejected electrons have
identical energies, Eb = Ec = 10 eV.
They are selected in two opposite half planes with respect to the electron beam
in a double toroidal electrostatic analyser. The angular information contained
in the collision plane (k0, ka), i.e. the ejection angles qb and qc, is preserved upon arrival on the position sensitive
detectors. Therefore, multi-angle collection of the ejected electrons is
realized over the useful angular ranges 20° < qb <160° and 200° < qc < 340°. The energy and angle resolutions for the
ejected electrons are fixed in the off-line analysis [14] to DEb = DEc = ± 2 eV, and Dqb = Dqc = ± 8°. As far
as the present paper is concerned, the data have been sorted into three modes:
(i) the so-called 'fixed ejected angle mode' where the escape direction qb or qc of one electron
is fixed and the other one is mapped on to the opposite half-plane, (ii) the
'fixed mutual angle mode’ at fixed qbc = (kb, kc) but varying qb or qc, and (iii) 'the
symmetric geometry mode' where both electrons emerge with equal but opposite
angle with respect to the incident beam, qb = - qc. A fourth mode,
the so-called 'summed mutual angle mode' with varying qbc but summing over all individual directions qb or qc which lead to
the same qbc has been discussed in [11] and will not be repeated
here.
The
long accumulation time needed to achieve a reasonable statistical error (~ 32
days of continuous, non-stop acquisition for all the data presented in this
paper) resulted in a fatigue effect on the detectors, which was corrected as
explained in [2, 3, 6, 14] by daily recording 'reference' (e,2e) distributions.
These distributions were also used as an angular calibration of the toroidal
analysers by comparing the measured (e,2e) spectra with well-established
theoretical ones such as the orthogonalized Coulomb wave (OCW) calculations or
the convergent close coupling (CCC) calculations. After these corrections
(which amount to less than 10 - 15%), all the angular distributions presented
in this paper are obtained on the same relative scale. In this work, main
emphasis is put on the accurate determination of the shape of the distributions
to be compared with the different theoretical models. Therefore, no attempt was
made to determine the absolute scale for the measured cross sections.
Finally, it should be emphasised
that, as discussed in [6], the finite angular and energy resolutions used in
this work is not expected to have a severe effect on the measured (e,2e) or
(e,3e) distributions since it is unlikely that these distributions will have to have to exhibit any sharp structures.
2.2 Results
The calculated
and measured 5DCS are shown in figure 1 (a) to (v) and in figure 2 (a) to (f)
according to the 'fixed ejected angle mode', that is as angular distributions
of one electron for fixed emission direction of the second one. In these plots
and in the following discussion, we denote by qfix and qvar the fixed
and the variable electron angles, respectively. Figure 3 (a) to (c) corresponds
to the so-called 'fixed mutual angle mode', that is with variable qb and qc angles while
keeping the mutual angle qcb fixed.
Finally, figure 4 presents the so-called symmetric geometry where both
electrons emerge at equal angles on both sides of the incident beam, qb = - qc. The fast
electron is observed at an angle qa = +1.10°, not
shown in the figures, hence the +K direction at qK = -21.6° as indicated in Fig. 1(a).
Since the data are only relative, we arbitrarily
choose to plot all the results by renormalising them to the absolute scale
given by the CCC results. In doing so, and for the sake of clarity of the
figures, we have used different scaling
factors between experiment and CCC. Similarly, though the 3C and the C4FS
results are of course absolute, we have chosen to plot them after a rough
renormalisation to the CCC ones, as indicated in the figures, in order to put
the emphasis on the shape comparison. (This choice is arbitrary and is not
meant to favour one model over the others). However, we recall that the
experimental data are obtained on the same relative scale. Therefore,
the internal change in the scaling factor between the experiment and each
theoretical model is a measure of the ability (or unability) of this model to
reproduce the relative scale of the data, whereas its variation from one model
to another is a measure of the consistency of the theoretical models to
reproduce the absolute scale.
2.2 1. Fixed ejected angle mode
As a first observation from figure 1, one may compare
between the three theoretical results. They generally all yield a two lobes
structure for the 5DCS angular distributions. The two lobes are separated on
the one hand by a strict zero intensity for the parallel emission (see however the discussion
below of figure 3 (c)), due to the Coulomb repulsion between two electrons with
equal energies emerging in the same direction, and on the other hand by a deep
minimum of intensity for the anti-parallel or back-to-back emission. This
minimum is reminiscent of the node observed in photo double ionisation (PDI)
where the back-to-back emission is forbidden, due to the electron pair final
state symmetry [15]. The origin of the dips and maxima in the (e,3e) cross
section and the connection to their PDI counterparts have been explored in Ref.
[5].
Noticeable
exceptions to the two lobes structure are:
(a) on the one hand, cases (a), (t),
(u) and (v) corresponding to the fixed electron being emitted forward with
respect to the moentum transfer vector, K. Here, a third intensity
maximum appears roughly in the direction opposite to the fixed electron
direction. This maximum appears as a distinct, small lobe in the CCC results
and a much wider structure in the 3C and C4FS results. As the back-to-back
emission of the two photoelectrons is forbidden in PDI process, the finite
(e,3e) intensity in this back-to-back configuration must be attributed to non
dipolar contributions. As discussed in [20], within the FBA, only even-parity
multipoles of the Born operator contribute to the back-to-back emission. To see
this directly we write the (e,3e) transition amplitude in the form:
where and are the wave functions of the two slow
electrons in the final and initial state, respectively. In general, the final-state wave function
does not possess a defined parity. However, in the case kc=-kb
(i.e., in the back-to-back configuation) it has an even parity. This follows
directly and in an exact manner from the structure of the Schrödinger equation
that dictates: . Therefore,
only the cosine term in the expression for T(e,3e) (1) contributes in the back-to-back emission case.
(b) Cases (j), (k), (l) and to some
extent case (i) corresponding to the fixed electron being emitted backward with
respect to the K vector. Here, a strong filling of the minimum
corresponding to the back-to-back emission is observed, though less strong in
the CCC than in the C4FS results. This is again a clear evidence of non-dipolar
contributions which even dominate the
dipole term. Or more precisely, as discussed above, the filling of the back-to
back emission node in PDI is due to first term in Eq.(1), i.e. to the odd part
of the transition operator. The three theoretical models shown here do not
predict the same relative importance for these non-dipole terms. Such effects
are appreciably larger than those reported at 5.5 keV impact energy in [6] and
[11], as it is expected due to the
lower incident energy and to the larger momentum transfer employed in the
present work.
(g) apart from these cases where the
fixed electron is observed backward or forward in the vicinity of the incident
beam direction, the three theories predict a small probability for the
back-to-back emission in agreeemnt with the PDI expectation, meaning that the
optical limit is here closely approached. All observation (a)-(g) confirm the
conclusion made in [5] and [11] that the dipolar limit is reached differently
depending on the directions of the momentum vectors kb and kc, even though the momentum transfer, the incident energy and the
outgoing electrons’ energies are kept fixed.
Now let us have a closer look at the shape of the
theoretical distributions in Fig. 1. The intensity ratio of the two lobes, when
present, is quite different from one model to another. For instance, CCC
calculations predict one large and one small lobe in case (b) where qfix = 22°, while both C4FS and 3C calculations yield two
almost identical lobes. The situation is practically reversed in case (o) where
qfix = 242°, with two identical lobes for CCC and very
asymmetric lobes for C4FS and 3C. Generally speaking, the distributions
obtained with the C4FS and 3C models are very close to each other as to the
shape, and are significantly different from the CCC ones. This can be explained
by the way the final state wave function is calculated in different models. The
C4FS and 3C final state wave functions are most accurate at the asymptotic
region of large distances from the nucleus.
On the contrary, the CCC final state wave function is accurate at a
close and intermediate range from the nucleus and looses its accuracy in the asymptotic
region. Nevertheless, the CCC calculations with the same final state wave
function provide reliable and accurate
absolute cross-sections for the related PDI process. In addition, recent (e,3e)
measuremtnts of Dorn et al [22] confirmed prediction of the CCC model
for a similar kinematics of a large incident energy and a small momentum
transfer.
Another observation concerns the relative magnitude of
the cross sections given by the three models. With the normalisation procedure
explained above, one sees that there is a factor of 10 difference in magnitude
between CCC and 3C results in cases (g) to (p), 3C yielding larger cross
sections. This factor reduces to 4.5 in the other cases. We note that cases (g)
to (p) correspond to the fixed electron being observed in the 'backward half
plane' (with respect to the incident direction), that is 90° < qfix < 270°,
while the other cases correspond to the fixed electron being observed in the
'forward half plane' , that is -90°
< qfix < 90°.
Similarly, the C4FS results are a factor of 10 to 50 larger than the CCC ones
depending on the qfix value, with more or less the same remark applying as
to the backward and forward half planes. On the other hand, the ratio of 3C to
C4FS results is almost constant, C4FS yielding about ~ 5 times larger
cross sections at practically all qfix values.
Therefore, as noted above for what concerns the shapes, the C4FS and 3C models
yield comparable results but both
deviate significantly from the CCC
calculations. Our tentative explanation of these differences in magnitude
relies on the following observation: the C4FS results presented here make use
of a three-term Hylleraas type ground state description of the He atom.
Calculations within the same model have also been performed using a poorer
description of the initial state, namely a Slater type wavefunction. The
results (not presented here) are very similar in shape to the Hylleraas ones,
but differ in magnitude being a factor ~6 larger, the use of a yet another
choice of initial-state wave function leads basically to the same the shape of
the angular patterns but alters significantly the absolute value of the cross
sections. This may be taken as a good illustration of the sensitivity of the
results to the initial state description. The 3C calculations utilise a
Silvermann ground state wavefunction that partly includes both radial and
angular correlations. The CCC calculations are performed with the more
elaborate 20-term Hylleraas wavefunction (CCC calculations in the veloctiy
gauge of the Born operator, not shown here, have also been performed with an
18-term MCHF expansion, yielding very similar results). Therefore, it is likely
that most of the difference in absolute scale between the three theories has
its origin in the quality of the ground state wavefunction. The CCC claculation
is the most stable as it does not show any sensitivity neither to the ground
state wave function nor to the gauge of the Born operator (length or velocity).
In general, all three models do grossly reproduce the shape
of the experimental 5DCS distributions of Fig. (1) within the uncertainly given
by the error bars of the experiments and by the fact that the data are only
recorded in one half plane. Nonetheless, there are obvious substantial
deviations between theories and experiment:
(i) First, none of the considered models does
correctly reproduce the relative scale of the experimental data. If cases (b)
and (c) are taken as a reference where the
normalisation constant between experiment and each theory is arbitrarily given
a value equal to 1, then it can easily be inferred from Fig. 1 that it would be
necessary to divide the experiment by a variable factor which takes values
between 0.35 and 2.8 in order to bring the measured data in cases (d) - (u) in
reasonable agreement with the calculated results. This cannot be attributed to an
experimental fault since, as noted above, there is also no internal consistency
in the relative scale between the different theoretical results[1].
(ii) In a number of cases, e.g. (d), (h) and (o), the
angular position of the calculated lobes is in a nice agreement with the
measured ones, whereas for several other cases, e.g. (b), (g), (n) and (q)
large shifts are observed, which may amount up to some 45°. In this respect,
the C4FS and 3C models yield identical positions of the lobes, whereas the main
lobe in the CCC model generally appears to be rotated backwards by ~ 10°
with respect to the other models.
(iii) cases (u) and (k) are of particular interest
since the fixed electron is observed along +K or (approximately) along -K,
respectively. Under these conditions, it was shown [16] that any first order
model must yield a symmetrical distribution about ±K. This is the case
for all three models, including the C4FS which goes beyond FBA through the
introduction of effective charges. This either means that the non-first order
effects are intrinsically small under these geometries, or, as previously noted
in [6] and [8] that their contribution to the C4FS cross sections is small.
Unfortunately, the limited range of the experimental data does not allow
concluding about a possible breaking of symmetry. It is also interesting to
note that the three models predict a strong (CCC) or very strong (C4FS and 3C)
relative contribution of the back-to-back emission under this geometry (this
contribution is in fact maximum there, see figure 3 (a)). This appears to be in
conflict with the experimental observation in case (u) where the measured
back-to-back intensity approaches zero.
(iv) As noted in [6] and [11], the PDI cross sections
distributions obtained for two electrons fixed angles which are symmetrical
with respect to the electric field direction e must be mirror images of each other with respect to this e direction. In the electron impact case, K plays the role of
e direction in the dipole limit. Therefore, we exploit
this mirror symmetry in figure 2 in which some of the data of figure 1 are
accordingly superimposed on each other. Comparison is made with the CCC results
obtained under the same transformation. We see clearly from figure 2 that this
symmetry transformation does not produce much change in the CCC results which
are thus quite close to the dipole limit. On the other hand, even considering
the large error bars, the experimental data are not invariant under such
transformation. This might show how important are the deviations from the
optical limit, but it might also indicate the large role played by non-first Born
processes, even though the incident energy is large (1.1 keV) and the momentum
transfer is small (0.45 au). A similar observation was made in [6], but in
general the deviations from 'perfect' symmetry were smaller, as one would
expect from the higher incident energy (5.5 keV) and smaller momentum transfer
(0.24 au) used in [6].
2.2.2. Fixed mutual angle mode
Figure 3 presents angular distributions in the
so-called fixed mutual angle mode, that is with variable qb and qc angles while keeping the mutual angle qbc fixed. The experiments are compared to the CCC and
the 3C results (C4FS results are almost identical to the 3C ones except for the
magnitude). In Fig. 3 (a) and (b) the two ejected electrons emerge at large
angle from each other, whereas in Fig. 3 (c) they asymptotically fly out close
to each other which enhances the final state Coulomb repulsion between them. In
all cases the two models do not produce the same magnitude of the cross
sections as discussed above, hence different normalisation factors are used (as
indicated on the figures) in order to put the emphasis on the shape comparison.
- Case (a) with a mutual angle qbc = p is a
very interesting one since it corresponds to the forbidden back-to-back
emission in PDI (i.e., in PDI this graph would be a strict zero everywhere). In
other words, this graph represents a direct measure of the non dipolar
effects involved in the present (e,3e) process, in the sense discussed
following Eq.(1). As noted above, theories predict these non dipolar effects to
be maximum when the pair of electrons is ejected along ±K, whereas the
maximum probability in the experiments occurs when the electron-pair axis is
rotated by ~60° from this direction. Moreover, the two first order theories
show the expected symmetry about ±K direction (at 338° and 158°,
respectively) whereas the experiment does not. Both these angular shift and
breaking of symmetry may be directly attributed to second or higher order
effects in the projectile-target interaction process.
- In Fig. 3 (b) the data are too scarce to come to a
definite conclusion. However, the data seem to indicate a minimum where the
calculations show a maximum.
- In fig. 3 (c), though no data could be measured, it
is interesting to compare the result of the two calculations. At 40° mutual
angle both models yield almost undistinguishable angular distributions as to
the shape. However, in contrast with all the above observations, the CCC cross
sections are now a factor of 4 larger than the 3C ones. This might find
its explanation in the 0° behaviour of the CCC. Indeed, at 0° mutual angle the 3C model (and the
C4FS model) yield a vanishing cross section as expected from the Coulomb
repulsion which forbids the two ejected electrons to emerge in the same
direction with the same velocity. In contrast, CCC model fails to strictly
fulfill this selection rule as it predicts a non-zero intensity. However, the
corresponding cross sections are small, about two orders of magnitude smaller
than a typical CCC lobe intensity in figure 1, and roughly one order of
magnitude smaller than the "DPI-forbidden" intensities involved in
Fig. 3 (a).
2.2.3. Symmetric geometry
In figure 4, the data have been sorted for the two
ejected electrons to emerge at equal but opposite angles, qb = - qc. In (e,2e)
processes, such symmetric geometry is well known to be very sensitive at large
angles to second order effects [17], and has allowed to identify the observed
(e,2e) large angle peak [18, 19] as being due to a double mechanism involving a
backward elastic scattering of the projectile followed by a binary e -e collision. In this
representation, the present theoretical and experimental results show two
peaks, one forward at about 60° and one backward at ~ 130° in the
experiment and ~110° in the theories. The experiment shows a backward to
forward peak ratio of ~ 0.5. The CCC ratio of ~ 0.8 is close to the
measurements, whereas the other two theories show ratios larger than 1, roughly
1.5. In the lower panel of the figure is plotted the corresponding magnitude of
the ion recoil momentum, kion. Comparing the upper and the lower panels of the figure, a striking
similarity is observed as to the position of the minimum (especially with the
theoretical minimum) at qb ~ 80°. When kion is minimum, the Bethe sphere is closely approached
(the Bethe sphere is reached when the momentum transfer K is fully
absorbed by the pair of ejected electrons, the ion remaining spectator, i.e. kion = 0). It has been argued [20] that under these
kinematical conditions the cross
section should be maximal, while it is observed here to be minimal. This
apparent contradiction was also noted in [5] and was attributed to the fact that
under the present near-dipole conditions, the optical transition is forbidden
for two 'free' electrons which is the condition for the Bethe sphere. The fact
that the experimental minimum occurs at a larger angle than in the calculations
might be due to the non-dipole contributions which are not fully taken into
account in the models as suggested above, e.g. in Fig. 3 (a).
3. Conclusion
We presented a large body of new
experimental data on the (e,3e) fully resolved cross sections for the double
ionization of helium, in the coplanar geometry and under equal energy sharing
10 + 10 eV. The data were compared in different angular modes with the best
available first-order theories, namely the 3C, C4FS and CCC models.
The three theories do not predict the same
absolute scale for the 5DCS. This could be mostly due to the different
descriptions of the initial state used, from poor to more elaborate. But, when
compared to the experiments, they do not predict the correct relative scale for
different angular distributions neither.
The experiments as well as all theories display a
strong filling of the node corresponding to the forbidden back-to-back emission
in PDI, or even display an additional lobe at this position. This behaviour
obviously reflects a strong manifestation of non-dipolar effects in the
projectile-target interaction which become prominent when the fixed electron is
emitted either forward or backward around the momentum transfer direction .
However, the three theoretical models do not predict the same relative
importance for these effects. Such
effects are appreciably larger than observed at 5.5 keV, as it is expected from
the lower impact energy and larger momentum transfer of this work. Moreover,
these effects do appreciably vary with the emission directions, which confirms our earlier conclusion [5,11] that
the dipolar limit is reached differently depending on the orientation of the kb and kc vectors, even though the momentum transfer, the
incident energy and the outgoing energies are kept fixed. A similar experiment
at much higher incident energy, 10 keV or more, is highly desirable in order to
probe when and how the optical limit is reached.
On the other hand, while the CCC theory almost fulfils
the 'mirror symmetry' when superimposing angular distributions taken at
symmetrical qfix angles with respect to K, the experiments do not. This might indicate
the large role played by non-first Born processes. Another indication for the
importance of these effects is found when the pair of electrons is detected
back-to-back while rotating the e - e axis (figure 3) : the three first order
theories show the expected symmetry about ±K direction, whereas
this symmetry is broken in the experimental data. A similar experiment at lower
incident energy is also desirable in order to enhance the observed effects.
Such experiment has been recently performed in our group and is currently under
analysis.
say something about symmgeo???
Acknowledgments
The authors are grateful to Dr A. Huetz for helpful
discussions. They also acknowledge the technical assistance and skills of Mr A.
Abadia. One of the authors (AK) wishes to thank the Université de Paris-Sud XI
for a Professeur invité position.
References
Dal Cappello still needs to complete ref. # 7, 8
Jamal needs to complete ref. # 10, 11
Azzedine needs to complete ref. # 9, 17 to 19
[1] Lahmam-Bennani
A, Duguet A, Grisogono A M and Lecas M 1992 J. Phys. B : At. Mol. Phys. 25
2873
[2] El Marji B, Schröter C, Duguet A,
Lahmam-Bennani A, Lecas M and Spielberger L 1997 J. Phys. B : At. Mol. Opt.
Phys. 30 3677
[3] Schröter C, El Marji B, Lahmam-Bennani
A, Duguet A, Lecas M and Spielberger L 1998 J. Phys. B : At. Mol. Opt. Phys.
31 131
[4] Taouil I, Lahmam-Bennani A, Duguet A
and Avaldi L 1998Phys Rev Lett 81
4600
[5] Lahmam-Bennani A, Taouil I, Duguet A,
Lecas M, Avaldi L and Berakdar J 1999Phys Rev A 59 3548
[6] Kheifets A, Bray I, Lahmam-Bennani A,
Duguet A and Taouil I 1998 J. Phys. B : At. Mol. Opt. Phys. 32
5047
[7] elmkhanter dalcap
[8] article grin dalcap
[9] nebdi louvain thèse
[10] Popov Yu V, Dal Cappello C and Ancarani
L U. 2000 in Many Paritcle Spectroscopy of Atoms and Molecules, Clusters and
Surfaces ed. by J. Berakdar and J. Kirschner, Plenum/Kluwer Press (2001).
[11] Lahmam-Bennani A, Duguet A, Taouil I and
Gaboriaud M. N. in Many Paritcle
Spectroscopy of Atoms and Molecules, Clusters and Surfaces ed. by
J.Berakdar and J. Kirschner, Plenum/Kluwer Press (2001)
[12] Brauner M, Briggs J and Klar H 1989 J.
Phys. B : At. Mol. Opt. Phys. 22 2265
[13] Kheifets A S and Bray I 1998Phys Rev A
58 4501
[14] Duguet A, Lahmam-Bennani A, Lecas M and
El Marji B 1998 Rev Sci Instrum 69 3524
[15] Huetz A, Selles P,
Waymel D, and Mazeau J 1991 J. Phys. B : At. Mol. Opt. Phys. 24,
1917
[16] Dal
Cappello C and Le Rouzo H 1991 Phys. Rev. A 43 1395
[17] Joachain on H
[18] Pochat symm geometry on He
[19] our symmetric geometry on Ne
[20] Berakdar J and Klar H 1993 J. Phys.
B : At. Mol. Opt. Phys. 26 4219
[21] A. Dorn, A. S. Kheifets, C.D. Schröter,
B. Najjari, C.Höhr, R. Moshammer and J. Ullrich, 2001 Phys Rev Lett, submitted
Caption to figures
Figure
1. (e,3e) fivefold differential
cross sections (5DCS) for coplanar double ionization of helium at an impact
energy E0 = 1099 eV, momentum transfer K = 0.45 au
(qa = 1.1°), and
equal ejected energies Eb = Ec = 10
eV. The momentum transfer direction is indicated by the bold arrow in panel
(a). One electron is detected at a fixed angle, qfix,
shown by the thin arrow and the
labeling, while the second electron is mapped in the plane. Experiments are
shown as full circles, and the error bars represent one standard deviation
statistical error. Full line : CCC results. Dashed line : 3C results. Dotted
line : C4FS results with a Hylleraas initial state wavefunction, except in (k)
and (v) where a Slater wavefunction is used. The experimental data are obtained
on the same relative scale, but for the sake of clarity they are here
normalized to the CCC results using different scaling factors, as indicated on
each diagram. The absolute scale shown is the CCC one, and is given in 10-4 atomic
units. Similarly, the 3C and C4FS results are renormalized to the CCC ones,
also using different scaling factors, as indicated on each diagram.
Figure
2. As for figure 1, (e,3e) 5DCS at
Eb = Ec = 10 eV. The experimental data and the CCC results
obtained at two qfix angles
which are symmetrical with respect to K direction are here superimposed
(qfix<180° : full
circles and full curve. qfix>180° : open circles and dashed
curve.) As in figure 1, the CCC
absolute scale is used. For the scaling, the experimental data have been
divided by the following factors, respectively from (a) to (f) : 1; 1; 2; 1.6;
2.5; 2.
Figure
3. (e,3e) cross sections in the
fixed mutual angle mode, qbc (see
text). Full circles : experiments. Full line : CCC results. Dashed line : 3C
results divided by 25 in (a) and (b) and multiplied by 4 in (c). (a) : qbc =
180°, and the dotted line is a polynomial fit to the experimental data used to
guide the eyes; (b) : qbc =140°;
(c) : qbc = 40°,
and in addition the dotted line represents CCC results at qbc = 0° magnified by 10. Experimental
results are normalized to CCC by multiplying them by 1 in (a) and by 7 in
(b). Kinematical parameters as in
figure 1.
Figure 4 . (a)
: (e,3e) cross sections in the symmetric geometry mode, qb = - qc (see
text). Full circles : experiments. Full
line : CCC results. Dashed line : 3C results divided by 4.5. Dotted line : C4FS
results divided by 25. Kinematical parameters as in figure 1. (b) The
corresponding ion recoil momentum as a function of qb.
12+10 graphs??? A. Lahmam-Bennani et al
Figure 1 (a) - (h)
A. Lahmam-Bennani
et al
Figure 1 (i) - (p)
A. Lahmam-Bennani
et al
Figure 1 (q) - (v)
A. Lahmam-Bennani
et al
Figure 2 (a) - (f)
A. Lahmam-Bennani
et al
Figure 3
A. Lahmam-Bennani
et al
Figure 4
FAX To Prof. Claude
DalCappello fax #
Dr Jamal Berakdar fax #
Dr Anatoli Kheifets fax #
including
this one
from
A. Lahmam-Bennani 00 331 6915 7671
Orsay
, january 5th, 2001
Dear
Claude, Jamal, and Anatoli
Let
me first wish you all a very happy new year, full of success and hapiness.
Here
is my new year's present : the manuscript of the e3e paper for our 1 keV data,
the one I was promising myself to write and submit since already ... (I am
ashamed to say it) more than a year!!! But I have many excuses. I have used all
your calculations you sent me a while ago. I would like you to please take the
time to carefully read it, and make all the suggestions, amendments, etc ... that
you fell needed.
As
you will see, the manuscript only contains the first order models (3C, C4FS and
CCC). After the discusion in november-december between jamal and anatoli, and a
little bit myself, the "beyond first Born project" is postponed until
Jamal comes in (next spring? Incha'Allah) with his Green results. It does not
matter, I think that the present manuscript contains enough information as
such.
I
will also e-mail you the whole stuff. Hopefully it will work (I am not sure
that all of you can read my Kaleidagraph files, this is the reason for this
fax).
Enjoy your reading, and sharpen your criticisms.
Looking forward to hearing from you,
all the best to all of you
Azzedine