# Electron, Positron and Photon Collisions with Atoms and Molecules

## B-spline 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 B-spline basis sets in computational atomic physics. B-splines are piecewise polynomials, defined using a knot set [1]. 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 oscillations.

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 [2]. More recently, it was demonstrated in photoionization calculations for He [3]. 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 thresholds.

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) [4] and double photo ionization of He and excited states of He [5].

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 [6] and two-photon double ionization of He [7].

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 [8]. 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.

1. C. de Boor, A Practical guide to Splines (Spinger: New York, 1978)
2. H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 30, 453 (1997)
3. H.W. van der Hart and C.H. Greene, Phys.Rev. A66, 022710 (2002)
4. H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 34, L147 (2001)
5. H.W. van der Hart and L. Feng, J.Phys.B: At.Mol.Opt.Phys. 34,L601 (2001)
6. 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)
7. L. Feng and H.W. van der Hart, J.Phys.B: At.Mol.Opt.Phys. 36,L1 (2003)
8. H.W. van der Hart, M.A. Lysaght and P.G. Burke, Phys.Rev. A76, 043405 (2007)