Double ionization processes in the R-matrix Floquet approach
H W van der Hart, L Feng
Department of Applied Mathematics and Theoretical Physics, Queen’s University Belfast, Belfast BT7 1NN
Main contact email address h.vanderhart@am.qub.ac.uk
Introduction
The theoretical description of atoms and ions in strong laser
fields is of essential importance to understand the atomic
response to the laser field in detail. One of the topics that has
received great attention over the last decade is the relative
importance of double and single ionization. This ratio is of
interest in two different types of laser fields: low frequency and
high intensity, as well as high frequency and ‘low’ intensity.
Many theoretical methods have focused on double versus single
ionization for atoms subjected to a high-intensity low-frequency
laser field. Walker et al1) demonstrated that double ionization
was significantly more pronounced in the atomic response than
could be expected from a single-active-electron picture. These
experiments provided the impetus for many theoretical
approaches to describe the non-perturbative behaviour of twoelectron systems in intense laser fields2,3,4,5).
Recent developments in the generation of high-intensity X-ray
laser sources6) have recently stimulated interest in the
theoretical description of double ionization processes at high
frequencies. At these frequencies only a few photons are
required to eject two electrons from the atom. However, such
photons may not only ionize the atom; they may also be
sufficiently energetic to ionize singly charged ions. At these
photon energies, the competition between sequential and nonsequential double ionization probes the influence of electronic
interactions on the atomic behaviour.
Several different types of methods exist for the description of
atoms in strong laser fields. In a time-dependent approach, one
solves the Schrödinger equation for an atom subjected to a short
intense laser pulse directly. In a time-independent approach, one
uses the Floquet-Fourier Ansatz to transform the timedependent Schrödinger equation into a time-independent one.
The time-independent R-matrix Floquet approach5) is best
suited when the number of photons absorbed is relatively small.
As a consequence of the Fourier-Floquet Ansatz, the atomic
wavefunction must be determined for each net number of
photons absorbed or emitted. In order to keep the calculations
feasible, it is necessary to restrict the total net number of
photons absorbed or emitted.
The R-matrix Floquet approach has presently only been applied
to investigate single ionization processes. In order to describe
double ionization processes within the R-matrix Floquet
approach, we have combined R-matrix Floquet theory with Bspline basis sets for the description of the two-electron
continuum. B-spline basis sets were introduced for atomic
physics calculations about 15 years ago7). Since B-spline basis
sets allow continuum effects to be included in a straightforward
manner, they are now widely used in theoretical atomic
physics8).
We have chosen to study He for our initial calculations on
double photoionization within the R-matrix Floquet approach,
since He is the simplest two-electron atom. A schematic
diagram of the He energy levels is given in figure 1, where we
have highlighted the energy levels that we are particularly
interested in: the 1s2 ground state, and the lowest doubly excited
state 2s2. These states are chosen because of the large amount of
experimental and theoretical data available for the 1s2 state, and
because the competition between autoionization and
photoionization for doubly-excited states is important for the
dynamics of atoms in strong laser fields.
Figure 1: Schematic energy diagram of He. Our focus is on
double and single photoionization of He initially in the 1s2 and
2s2 states. Two distinct photoabsorption pathways for the 2s2
state are indicated.
Double photoionization using B-spline basis sets
We have first established that B-spline basis sets are well suited
for the description of the double continuum by examining the
relation between double and single photoionization of He in the
1s2 ground state within perturbation theory. This process has
been studied extensively over the last decade, since this process
measures the importance of electron interactions within atoms.
The photon is absorbed by a single electron; a many-electron
response thus indicates energy transfer from one electron to the
other.
In order to describe the two-electron continuum, we describe it
in terms of products of B-spline basis functions. B-splines are
piecewise polynomials with maximum smoothness. They are
highly suited to describe smooth functions, such as atomic wave
functions. Although B-splines are non-orthogonal functions,
their limited extent means that each B-spline has a non-zero
overlap with a small number of other spline functions. This
means that the matrix calculations generally involve banded
matrices.
Results for the ratio between single and double ionization of
ground-state He9) are shown in figure 2, and are compared to
other theoretical and experimental data10,11). The agreement
with the converged close-coupling calculations10) is highly
satisfactory. The differences close to threshold are due to the
present method having difficulty in describing outgoing
electrons with identical energy. These differences can in
principle be reduced by enlarging the box size. The agreement
with experiment11) is also quite good with a difference of less
than 5% above a photon energy of 120 eV.
Combining B-spline basis sets and R-matrix Floquet theory
After demonstrating the appropriateness of the B-spline basis
set for describing the two-electron continuum, we have
integrated the B-spline basis sets into R-matrix Floquet theory.
We thus describe the full motion of the two electrons in each
Floquet block. The main changes involve the determination of
the Floquet R-matrix and the data input routines
between double and single ionization, involving the absorption
of a single photon, decreases rapidly at a final-state energy of
0.6 a.u. This decrease is absent in the He results since the
ground state of He cannot autoionize.
Figure 2: Ratio between double and single photoionization for
ground-state He. Results using the B-spline basis (solid black
line) are compared to experimental results11) (purple circles) and
other theoretical results10) (dashed blue line).
Firstly, we have investigated single photoionization from the
ground state of He within this approach. The results obtained
using the new code are in excellent agreement with the previous
results. It should be emphasized that the R-matrix Floquet
approach employs a more sophisticated approach to describing
the asymptotic wave functions than the previous calculations,
and that minor differences between the two sets of results would
not be unexpected. However, the agreement gives us confidence
in the accuracy of the approach.
One of the problems of the R-matrix Floquet approach for
double ionization is the computational requirement. A proper
description of the double continuum requires an extensive basis
set. In the R-matrix Floquet approach, we need to multiply this
basis set by the number of Floquet blocks included in the
calculation. In order to keep the calculation manageable on a
2Gb PC, we have only included the minimum number of
Floquet blocks: 2.
Despite the limitation to 2 Floquet blocks, there are still new
physical processes that can be investigated using this approach:
the competition between double and single ionization when the
initial state is a doubly excited state. In this case, as shown in
figure 1, several pathways involving the absorption of a photon
lead to ionization. Firstly, direct photoabsorption can lead to
single as well as double photoionization of 2s2. Secondly, the
2s2 state may autoionize to the 1s state of He+ with the emission
of an electron. Following autoionization, the 1s state of He+
may absorb a photon, possibly exciting the 2p state of He+.
Figure 3 shows the ratio for double to single photoionization for
the 2s2 state of He embedded in a laser field with an intensity of
5*1011 W/cm2 as obtained by the R-matrix Floquet approach
with B-spline basis sets12). The results are compared to
experimental results for double photoionization of Be13), and
He11). Since the two valence electrons of the Be ground state are
in 2s2, the theoretical results and the experimental results for Be
are expected to be similar. The similarity with the He results is
more surprising, although electron-electron effects are expected
to be strong for both types of states.
Figure 3 shows that the ratio between double and single
photoionization is very similar for the He 2s2 state and the Be
1s22s2 state with a relative difference of about 20%. The main
difference with the He results can be seen around a final-state
energy of 0.72 a.u. At this particular final-state energy, the 1s –
2p transition in He+ becomes resonant, so that the dominant
photoabsorption process is autoionization of 2s2 followed by
excitation of the residual He+ ion to the 2p state. Since reaching
this final state requires absorption of one photon, the ratio
Figure 3: Ratio between double and single ionization involving
photoabsorption. Present results for the 2s2 state of He (solid
line, filled circles) are compared to experimental results for Be
(filled triangles) and ground-state He (open circles).
Conclusions
We have extended the R-matrix Floquet approach to enable the
study of double ionization of atoms embedded in strong laser
fields. This has been achieved by combining R-matrix-Floquet
theory and B-spline basis sets. The accuracy of the approach is
verified by examining the ratio between single and double
photoionization of ground-state He. We have determined the
branching ratios for several photoabsorption processes that can
lead to either single or double ionization. We have
demonstrated that the ratio between double and single
photoionization may be crucially dependent on the presence of
resonances for the singly ionized target.
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