Electron, Positron and Photon Collisions with Atoms and Molecules
B-spline methods for electron scattering and photoionization
start of his Ph.D. research, Dr.
van der Hart has been interested in the application of
B-spline basis sets in computational atomic physics. B-splines
are piecewise polynomials, defined using a knot set . An
example of B-splines defined using a linear knot set is shown
in the diagram above. The so-called knot set can be tailored
to the problem under investigation, which allows for great
flexibility in the B-spline 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 R-matrix theory.
B-spline basis sets are particularly useful within this theory. B-splines are defined
on a closed interval and the inner region in R-matrix theory is an example of such
an interval. The use of B-spline basis sets within R-matrix 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
The flexibility of B-spline basis sets was first demonstrated in electron scattering
on H, for which excellent agreement with other leading approaches was obtained .
More recently, it was demonstrated in photoionization calculations for He . By
describing the residual states of He+ in terms of B-splines, we were able
to obtain photo ionization spectra of He with excitation of the residual He+
state up to n=8. At these n-values, irregularities appear in the total and partial photo
ionization spectra due to the increased overlap of Rydberg series converging to different
B-spline basis sets allow high-precision calculations since eigenfunctions obtained
using these basis sets form a discretised continuum. This discretised continuum can also
be exploited for the investigation of multiple-ionization processes. For example,
B-spline basis sets were used to investigate, for example, electron-impact ionization
of He+ (a process of importance in the understanding of recollision
processes in strong laser fields)  and double photo ionization of He and excited
states of He .
B-spline basis set techniques developed for electron-scattering processes and photo
ionization processes have subsequently been combined with
R-matrix-Floquet 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 390-nm
laser light  and two-photon double ionization of He .
Recently, the R-matrix II codes have been modified so that all orbitals,
including both bound-state and continuum orbitals, are described in terms of
B-splines . This modification was necessary for the development of the
time-dependent R-matrix 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 high-energy continuum orbital neglected in the standard approach.
In addition, the accuracy of all integrations is significantly enhanced, so that
cross sections in electron-scattering or photo ionization calculations converge
far more rapidly in the B-spline 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)