D>
D. M. Mitnik, D. C. Griffin,
Department of Physics, Rollins College, Winter Park, Florida 32789, USA
and M. S. Pindzola,
Department of Physics, Auburn University, Auburn, Alabama 36849, USA
March 3, 2002
PACS number(s) 32.80.Dz, 31.70.Hq
Revolutionary advances in experimental techniques and theoretical methods along with spectacular increases in computer power now provide opportunities for the development of a much more profound understanding of the atomic few-body problem. In recent years, a number of quantal non-perturbative methods have been developed which provide benchmark accuracy for electron-impact ionization of simple atoms and their ions [1,2,3,4]. Of particular interest are cases in which the incident electron has an energy equal to an autoionization resonance. It is then impossible to distinguish between electrons that have been ejected directly from the atom from those that are first excited to an autoionizing level and subsequently ejected. As a result, these two processes will interfere and interference can be sensitive to physical effects not present in the (e,2e) process far from resonance [5,6]. This problem has been largely investigated by using perturbative methods (see, for example [7] and references therein). However, due to the delicate interplay between three-body effects in the incident channel and postcollisional Coulombic interactions between the three free particles in the final state, a fully quantum treatment of this problem is needed. Examples of the difficulties arising in a non-perturbative calculation of breakup processes through resonant states have been pointed out in the model calculations (involving only short-range interactions) in Ref. [8].
Among the fully quantum nonperturbative theories, the time-dependent close-coupling (TDCC) method has been successfully employed for calculations of electron-impact ionization [9,10,11]. However, before undertaking a TDCC calculation of the resonant contributions to ionization, further work is needed in order to learn how to treat resonance problems within this theoretical framework. It is in this spirit that we have undertaken the study of dielectronic capture into doubly excited resonances, that after autoionization, contribute to elastic scattering. This requires the development of methods for generating accurate wavefunctions for doubly-excited autoionizing states and time-dependent close-coupling calculations of the capture and the subsequent decay of autoionizing states.
In this work, we study dielectronic capture and autoionization
in two-electron systems by the direct solution of the time-dependent
close-coupling equations.
In this letter we will only present results for capture into the
autoionizing state of neutral He.
For an electron colliding with a He+
ion, the target electron is
initially described by the radial wavefunction
obtained from the diagonalization of the one-electron Hamiltonian
on the radial grid.
The projectile electron at an initial time
, with an energy Ei
and a momentum
, is represented in coordinate space by a
Gaussian wavepacket of width w, centered at a position s
sufficiently
far from the target electron,
The time-dependent wavefunction for a given LS
symmetry is expanded in
coupled spherical harmonics:
For the
symmetry, we start with an initial radial wavefunction
of the form
The probability amplitude for dielectronic-capture into a doubly excited
state at time
is determined from the total wavefunction using the
projection:
With no constraints, this imaginary time propagation will relax to the solution with the smallest eigenvalue of H; thus, after many iterations (renormalizing the wavefunction continuously), only the ground-level eigenfunction will survive. Higher energy eigenfunctions can be determined by imposing constraints during the iteration that require the desired states to be orthogonal to all lower states. Although this procedure has been used for calculation of the the low-lying singly excited states (e.g., ), it does not work for the calculation of the doubly-excited states. However, the doubly excited, autoionization states may be determined by imposing an additional constraint on the relaxation that projects out the one-electron components of the lower-energy wavefunctions.
For a numerical grid having a mesh spacing
a.u., the energies
of the doubly-excited
states obtained using this modified relaxation
technique are within a few percents of the accepted values.
These results can be improved significantly by using a finer
radial mesh.
However, our purpose here is to study the dynamics
of capture into a doubly excited state, rather than obtain the most accurate
values for the energies of these states.
Examples of the wavefunctions obtained with this damped relaxation
method are given in Figure
1.
The figure shows the total probability
, for the
,
,
, and
wavefunctions
(for brevity, we will drop the
specification in every state).
The results of our TDCC calculations of the probability amplitude for
dielectronic capture (Eq.
5) into at the resonance energy
(E
eV) are
presented in Fig.
2.
The different curves are obtained with different initial wavepackets,
having a spatial widths w of 5 a.u., 10 a.u., 15 a.u., and 30 a.u.
(
= 8.2 eV, 4.2 eV, 2.8 eV, and 1.4 eV), respectively.
In this figure,
corresponds to the time
when the center of the incoming wavepackets arrives at the origin.
We see that the probability amplitude rises rapidly as the
incoming wavepacket begins to overlap with the target,
with a slope that increases inversely with the width w
of the incoming packet.
After
reaches a maximum, it begins a gradual decline
with a slope that is a direct measure of the rate for autoionization from
state to the
ground state of He
.
The dynamical behavior of autoionization in two-electron systems has been
studied previously by monitoring the decay of the autoionizing
state in time[13].
This requires the computation of the autocorrelation function defined by
By extending the autocorrelation method to include the and
channels,
we can compare the decay of the
state obtained from the
autocorrelation method with the decay of the
state following
dielectronic capture.
The TDCC result for
is also shown in Fig.
2 by the thin dashed line
along the
curve with a width of 30 a.u..
In order to compare these two curves, we multiplied
with
at
= 0.
As is shown in the figure, the agreement between the two calculations
is excellent.
An exponential fit of the form exp(
to the data
yields a value of
a.u., compared to the
width determined by Bhatia and Temkin [12] of
a.u.. Our result can be improved by using a finer radial mesh.
To confirm this, we repeated both the
and
calculations with
= 30 a.u., using a mesh separation of
a.u.
An autoionization width of
a.u. was obtained from both
curves.
A careful examination of the curves with different widths
indicates that there are oscillations
with small amplitudes in the decay of the autoionizing state.
This is shown more clearly in the inset in Fig.
2.
This effect is most pronounced for the curve corresponding to an initial
wavepacket with
= 5 a.u. (
eV).
In energy space, this packet is sufficiently wide to coherently populate
both the
and the
states (separated by 4.56 eV);
these two states both autoionize to a single continuum state and this
produces the observed beating.
The period of these oscillations is about
=37.3 a.u.,
corresponding to
= 4.6 eV, confirming the
origin of these oscillations.
A similar effect has been observed in the calculation of intense field
photoionization of He, in which two autoionizing states are populated by
a broad laser pulse [16].
As expected, the oscillations decrease in amplitude as the energy width of
the packets decreases and are completely absent for
= 30 a.u..
In order to obtain a quantitative measure in the variation of the capture
probability with energy, we calculated the dielectronic capture for
incoming wavepackets with = 30 a.u. at different incident energies,
and some examples are presented in Fig.
3 (solid lines).
We notice that the projection of the total wavefunction on the
state gives an unusually high overlap at energies far from resonance.
This overlap rises and then falls off rapidly as the wavepacket moves out
beyond those radii where the 2s
has any appreciable amplitude.
The capture amplitude then falls off exponentially at the characteristic
rate determined by the autoionizing width.
In order to gain an understanding of the shape of these curves, we employed
a simple analytic one-dimensional model based on the perturbative approach
for a discrete state interacting with a single continuum state
,
as developed by Fano[17].
The eigenfunction of the Hamiltonian matrix for eigenvalue
has the form
The curves of at different energies allow us to perform fits
at times where the decay is clearly exponential.
Using these exponential fits, the curves were extrapolated back to
,
and the resulting probability amplitudes were employed to calculate maximum
probabilities for capture into the
state.
The curve of the maximum probability as a function of the incident energy
is in agreement with the Fourier transform of a Gaussian wavepacket with
= 30 a.u. and E
eV.
Thus, with a sufficiently narrow wavepacket in energy
space, one should be able to use the TDCC technique to map out the shape of
a resonance.
One of the nice features of the time-dependent method, is that it allows one
to study the formation and decay of autoionizing states as a function
of time.
In Fig.
4, we show ``snapshots" of the probability density
near the origin, at different times during the collision.
In this figure only,
is the time at the beginning of the iterations
( i.e.,
).
In the part (a) of the figure, the incoming electron has an energy
E
= 25.0 eV, far below any resonance.
As time evolves, the peak of the probability density centered originally
at 110 a.u. moves toward the
origin. The center of the packet arrives to the origin at
a.u.,
and then bounces back.
At
= 140 a.u., the density is contained in the outgoing
wavepacket along the coordinate axes (elastic scattering).
At this time, there is no trace of any
component, as expected from
the very small dielectronic capture amplitude at this energy.
In the part (b) of the figure, the incoming electron has an energy
E = 31.6 eV, close to the energy of the
resonance.
In this case, the packet arrives at the origin at
a.u..
A significant fraction of the total wavefunction is concentrated
around the origin, and this is the part that is captured into the
state.
As is seen in the subsequent figures (b), the general shape
of the total wavefunction near the origin is not changing with time,
only decreasing in magnitude, as the doubly-excited state
decays.
A completely different picture is shown in Figs.
4(c),
where the incoming electron has an energy close to the resonance
(E
= 36.1 eV). The shape of the wavefunction is clearly
changing in time, oscillating between the
and the
functions. An incident wavepacket with an energy
width of
E = 1.4 eV can not resolve the
and the
states (separated by 0.4 eV), and large-period
oscillations
are found in the dielectronic capture probabilities.
In conclusion, we have been able to carry out the first
fully quantal time-dependent study of the dielectronic capture
of an incident electron into a doubly excited state followed by autoionization.
With the procedure we have outlined, the effects of numerical approximations
can, in principle, be made arbitrarily small.
The theoretical framework demonstrated here provides a basis for
developing practical methods for treating resonances in the problem.
We would like to thank Dr. Francis Robicheaux for a number of helpful suggestions. This work was supported in part by the US Department of Energy, and a subcontract from Los Alamos National Laboratory. Computational work was carried out at the National Energy Research Supercomputer Center in Oakland, CA.