Electron, Positron and Photon Collisions with Atoms and Molecules
Bspline methods for electron scattering and photoionization
Since the
start of his Ph.D. research, Dr.
van der Hart has been interested in the application of
Bspline basis sets in computational atomic physics. Bsplines
are piecewise polynomials, defined using a knot set [1]. An
example of Bsplines defined using a linear knot set is shown
in the diagram above. The socalled knot set can be tailored
to the problem under investigation, which allows for great
flexibility in the Bspline basis set. Within atomic physics,
these basis sets can be used when the wave functions of interest
can be represented well using piecewise polynomials (and this
is generally the case).
One of the main techniques in computational atomic physics is Rmatrix theory.
Bspline basis sets are particularly useful within this theory. Bsplines are defined
on a closed interval and the inner region in Rmatrix theory is an example of such
an interval. The use of Bspline basis sets within Rmatrix theory thus seems logical.
Continuum wave functions are oscillating functions, so proper results can be expected
as long as the density of points in the knot set reflects the frequency of these
oscillations.
The flexibility of Bspline basis sets was first demonstrated in electron scattering
on H, for which excellent agreement with other leading approaches was obtained [2].
More recently, it was demonstrated in photoionization calculations for He [3]. By
describing the residual states of He^{+} in terms of Bsplines, we were able
to obtain photo ionization spectra of He with excitation of the residual He^{+}
state up to n=8. At these nvalues, irregularities appear in the total and partial photo
ionization spectra due to the increased overlap of Rydberg series converging to different
thresholds.
Bspline basis sets allow highprecision calculations since eigenfunctions obtained
using these basis sets form a discretised continuum. This discretised continuum can also
be exploited for the investigation of multipleionization processes. For example,
Bspline basis sets were used to investigate, for example, electronimpact ionization
of He^{+} (a process of importance in the understanding of recollision
processes in strong laser fields) [4] and double photo ionization of He and excited
states of He [5].
Bspline basis set techniques developed for electronscattering processes and photo
ionization processes have subsequently been combined with
RmatrixFloquet theory. This
combination has proven to give an accurate description of a wide variety of
multiphoton processes, such as multiphoton ionization of He irradiated by 390nm
laser light [6] and twophoton double ionization of He [7].
Recently, the Rmatrix II codes have been modified so that all orbitals,
including both boundstate and continuum orbitals, are described in terms of
Bsplines [8]. This modification was necessary for the development of the
timedependent Rmatrix theory
and associated codes, since the entire continuum needs to be described. Using
these basis sets, no Buttle correction is required within the calculations to
compensate for highenergy continuum orbital neglected in the standard approach.
In addition, the accuracy of all integrations is significantly enhanced, so that
cross sections in electronscattering or photo ionization calculations converge
far more rapidly in the Bspline approach than in the standard approach.
 C. de Boor, A Practical guide to Splines (Spinger: New York, 1978)
 H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 30, 453 (1997)
 H.W. van der Hart and C.H. Greene, Phys.Rev. A66, 022710 (2002)
 H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 34, L147 (2001)
 H.W. van der Hart and L. Feng, J.Phys.B: At.Mol.Opt.Phys. 34,L601 (2001)
 H.W. van der Hart, B.J.S. Doherty, J.S. Parker and K.T. Taylor, J.Phys.B: At.Mol.Opt.Phys. 38,L207 (2005)
 L. Feng and H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 36,L1 (2003)
 H.W. van der Hart, M.A. Lysaght and P.G. Burke, Phys.Rev. A76, 043405 (2007)
