Influence of Pauli-exchange on net cross sections
In this section we investigate how different
ansätze for the effective target potential influence net cross sections.
The effective potential is just the screening
potential of the undisturbed target atom - a ground state potential
that can be calculated with specific approximations of ground state density functional theory (DFT).
One can split it into a Hartree and an exchange-correlation part. This
xc part reflects the quantum character of the system because for a
classical charge-distribution only the first term exists. The question
we are interested in then is: Are inelastic transitions in inter-atomic collisions
finally sensitive to this nonclassical contribution to the
potential? In particular are they sensitive to exchange effects?
In order to answer this question we have applied three approximations to this
potential, where the xc part is treated differently.
First, we have used a HFS potential, second a LDA potential. Both
potentials rely on the density dependence of the homogeneous electron gas,
so the exchange potentials have the same functional form but they differ
in strength. The HFS is stronger by a factor of 1.5 . Additionally the
LDA potential contains a correlation contribution which of course is
lacking in the HFS approximation. Both potentials are modified a posteriori
to ensure the correct asymptotic behavior of the x potential. So, a
1/r-tail is added. This is very important for the discussion of
scattering problems because the asymptotic form determines the binding of
the outer electrons - the ionization potential and this is an important
parameter if one is interested in ionization.
Third, we have used an OPM potential [1] where
the x part of the potential is treated exactly. In this sense the OPM
potential is superior to LDA and HFS potentials and the question then is
if this more accurate description of electronic exchange effects is visible
in the inelastic transitions in the ion-atom collisions we investigate.
In Figure 1 we compare the results obtained from the different atomic potentials
for the net capture. With the LDA potential the experimental results are
considerably overestimated over the entire energy range; at 5 keV it is
factor of five. The reason for this failure is that the binding energy
of the outer
2p electrons is close to the hydrogen ground state energy - so the
LDA exchange, which is too weak, causes an artificial quasi-resonant
charge transfer process.
The HFS potential also yields a coupling to the hydrogen
ground state, which is too strong,
although the effect is much smaller and the results are in
good agreement except at the lowest impact energies.
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Figure 1. p+Ne: net electron capture. Theory: present calculations for different Ne potentials. Experiment: full circles: capture [2].
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Figure 2. p+Ne: net electron ionization. Theory: as in figure 1. Experiment: open circles: ionization [3]
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Figure 2 shows the same analysis for the net ionization:
Again the LDA potential gives too large cross sections for all energies.
The binding of the neon 2p electrons is too weak and thus, the ionization
yields are too large. Remarkably, the HFS potential gives results which are
in closer agreement with the experiment than the OPM potential.
If one disregards the OPM results for a moment one is led to the conclusion
that response effects are not important for the net ionization, at all.
The cross sections are calculated accurately with a ground state HFS
potential.
On the other hand, it is very clear that the atomic ground state is described
more accurately by the OPM potential; thus, it is only meaningful to
evaluate the qualitiy of the
frozen-potential approximation if one uses the OPM rather than
the HFS potential. The HFS potential mimics binding properties
which lead to agreement with experiment but favours a wrong interpretation.
From the results with the OPM we know that response effects are important
at the intermediate energies and will reduce the cross sections there
whereas the HFS potential gives accidental agreement.
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Figure 3. p+Ar: net electron loss. Theory: present calculation as in figure 1. Experiment: full circles: loss [3], open circles: ionization [2]
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The problems with the HFS potential become more obvious if one considers
p + Ar collisions. Figure 3 shows results for net electron loss as
obtained from LDA, HFS and OPM potentials. The open circles denote experimental
results for net ionization which we have not extracted in this case from the
projection of the solution onto the capture channels. Again, in case of the OPM
potential the
agreement with experiment is good for low and for high energies. At intermediate
energies the disagreement is a bit enlarged (compared to neon). This is due
to the fact, that the binding of the outer electrons is weaker in the case
of argon, and thus the contribution of multiple processes is larger.
We overestimate the experiment here because of the neglect of response effects
in the effective potential.
The HFS potential yields cross sections which are larger than the experimental values for
almost all impact energies - in particular at high energies where the
net electron loss can be identified with net ionization.
Thus, the HFS potential behaves not systematically if the
target system is changed. This is not the case if OPM or LDA potentials are
used. The LDA overestimates the net electron loss for neon as well as for
argon. The agreement of the calculations with the OPM potential can be
understood for both targets from the range of applicability of the frozen
potential approximation.
The important conclusion that can be drawn from these figures is that
the correct description of electronic exchange effects is important if
one wants to obtain accurate results for net electron capture and ionization.
This is remarkable, because these observables are rather global. One can
state that Pauli-exchange - which is a quantum property - is mirrored in
these results.
References:
- E. Engel and S.H. Vosko,
Phys. Rev. A47, 2800 (1993);
- M.E. Rudd, Y.-K. Kim, D.H. Madison and J.W. Gallagher,
Rev. Mod. Phys. 57, 965 (1985);
- M.E. Rudd, R.D. DuBois, L.H. Toburen, C.A. Ratcliffe and T.V. Goffe,
Phys. Rev. A28, 3244 (1983);
created: 28.4.99
most recent update: 28.4.99
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