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We present an efficient generalization of the exterior complex scaling (ECS) method to extract discrete inelastic and ionization amplitudes for electron-impact scattering of atomic hydrogen. This fully quantal method is demonstrated over a range of energies for the collinear and Temkin-Poet models and near-threshold ionization is examined in detail for singlet and triplet scattering. Our numerical calculations for total ionization cross sections near threshold strongly support the classical threshold law of Wannier [Phys. Rev. 90, 817 (1953)] (σ∝E1.128±0.004) for the L=0 singlet collinear model and the semiclassical threshold law of Peterkop [J. Phys. B 16, L587 (1983)] (σ∝E3.37±0.02) for the L=0 triplet collinear model, and are consistent with the semiclassical threshold law of Macek and Ihra [Phys. Rev. A 55, 2024 (1997)] (σ∝exp[(-6.87±0.01)E-1/6]) for the singlet Temkin-Poet model.

We present accurate ab initio numerical solutions of the full Schrödinger equation for the electron-impact ionization of hydrogen near threshold using the propagating exterior complex scaling method. They provide strong support for the Wannier threshold law [ Phys. Rev. 90 817 (1953)], giving σ∝E1.122±0.015, and also give the energy dependence of the electrons' angular distribution as (π-θ12)FWHM≈3.0E1/4, in general agreement with classical and semiclassical predictions.

Electron-hydrogen fully differential ionisation cross sections are studied using the convergent close-coupling (CCC) method that utilises box-based eigenstates. Convergence as a function of the box size parameter R0 is presented for the coplanar case of 15.6 eV incident electrons with two 1 eV outgoing electrons.

A newly-derived iterative coupling procedure for the propagating exterior complex scaling (PECS) method is used to efficiently calculate the electron-impact wavefunctions for atomic hydrogen. An overview of this method is given along with methods for extracting scattering cross sections. Differential scattering cross sections at 30 eV are presented for the electron-impact excitation to the n = 1, 2, 3 and 4 final states, for both PECS and convergent close coupling (CCC), which are in excellent agreement with each other and with experiment. PECS results are presented at 27.2 eV and 30 eV for symmetric and asymmetric energy-sharing triple differential cross sections, which are in excellent agreement with CCC and exterior complex scaling calculations, and with experimental data. At these intermediate energies, the efficiency of the PECS method with iterative coupling has allowed highly accurate partial-wave solutions of the full Schrödinger equation, for L ≤ 50 and a large number of coupled angular momentum states, to be obtained with minimal computing resources.

Recent progress in the calculation of electron-atom scattering, with particular reference to electron-hydrogen ionization is presented. There have been substantial developments and improvements computationally, and also perhaps unexpectedly, in the formal theory. We conclude that within the frame of non-relativistic scattering theory electron-hydrogen scattering is practically a solved problem.

The single-centred convergent close-coupling method has been applied to positron-helium scattering at energies from the first ionization threshold to 1 keV for calculating one-electron excitation and ionization processes. Total scattering, positronium formation plus single ionization (unresolved), and target excitation cross sections have been calculated. It is found that the often-used frozen-core approximation for the target structure leads to a small but systematic overestimation of the cross sections. Generally, the agreement between the present and most recent calculations, as well as experiment, is satisfactory.

We use three different implementations of the non-relativistic close-coupling method to investigate the convergence of theoretical predictions for electron scattering from magnesium at impact energies below 1 eV. It is found that the convergence of the close-coupling expansion is extremely slow, particularly for the L = 1 partial wave that is dominated by a shape-resonance common to many similar quasi-two-electron systems. An accurate prediction of the scattering length (we obtain −2.4 ± 0.1 a0) is also a nontrivial problem. Although the shape resonance in the L = 1 channel can be predicted in agreement with experiment already in small close-coupling expansions with three or nine coupled states (see Bartschat and Sadeghpour 2003 J. Phys. B: At. Mol. Opt. Phys. 36 L9 for details), this fortuitous agreement is due to simultaneously inaccurate descriptions of both the N-electron target and the (N + 1)-electron collision problems. We demonstrate how a more systematic though slow convergence pattern of the close-coupling expansion can be achieved by concentrating on a highly accurate target description before attempting to solve the collision problem.

The Laguerre-based convergent close coupling method is used to calculate the electron-impact excitation cross sections of the auto-ionizing states 2s2s1S, 2s3s1S and 2s3s3S, various ionization, as well as other cross sections for scattering from the 11S, 2 1S and 23S initial states of helium. The calculations are done within the S-wave model. Unlike the case of bound-state excitation, convergence studies need to take into account occasional mixing between the core manifolds which can severely affect the structure of the auto-ionizing states and those corresponding to ionization with excitation. A method to take this into account is given and it is shown that convergent results may be obtained, Comparison with the 'Z = 2' time-dependent calculations of Pindzola et al (1999 Phys. Rev. A 59 4390-8) is satisfactory.

The electron-impact ionization cross sections for all neutral ground-state elements H-Xe (Z = 1 to Z = 54) have been calculated using the plane wave Born approximation. The atomic orbital wavefunctions have been approximated by Roothaan-Hartree-Fock Slater functions, and the scattered and ejected electrons have been approximated by a plane wave and Coulomb wave, respectively. Full orthogonalization of the Coulomb wave with the occupied atomic orbitals has been performed, which improves the correspondence with experimental data at low- and mid-energies. Results are presented for the total ionization cross section and the partial ionization cross sections of the largest contributing orbitals for each atom.

The differential ionization cross sections for charged hydrogenic targets with low Z at low to moderate energies were calculated using propagating exterior complex scaling (PECS) method. The PECS method with iterative coupling proved to be highly efficient, providing an estimated 100-fold reduction in total computation time compared with the PECS method using direct coupling. Scattering wave functions were calculated for the electron impact of hydrogenic targets with nuclear charge Z≤4. The results show that the angular distributions of the differential cross sections change systematically with increasing nuclear charge for energies above the peak total ionization cross section.

The existing formulations of electron-impact ionization of a hydrogenic target suffer from a number of formal problems including an ambiguous and phase-divergent definition of the ionization amplitude. An alternative formulation of the theory is given. An integral representation for the ionization amplitude which is free of ambiguity and divergence problems is derived and is shown to have four alternative, but equivalent, forms well suited for practical calculations. The extension to amplitudes of all possible scattering processes taking place in an arbitrary three-body system follows. A well-defined conventional post form of the breakup amplitude valid for arbitrary potentials including the long-range Coulomb interaction is given. Practical approaches are based on partial-wave expansions, so the formulation is also recast in terms of partial waves and partial-wave expansions of the asymptotic wave functions are presented. In particular, expansions of the asymptotic forms of the total scattering wave function, developed from both the initial and the final state, for electron-impact ionization of hydrogen are given. Finally, the utility of the present formulation is demonstrated on some well-known model problems.

The Convergent Close Coupling (CCC) method is applied, for the first time, to the scattering of positrons on helium. The helium target wave functions are obtained within various configuration interaction (CI) expansions. In the full CI expansion the two electrons are treated equally and thus all electron-electron correlations are taken into account. In the frozen-core (FC) approximation the CI expansion fixes one of the electrons to be described by a pure 1s orbital of He+, while maintaining the required singlet and triplet symmetries. Lastly, the multi-configuration (MC) approximation relaxes the FC approximation to allow the description of the inner electron to include several low-lying orbitals and is therefore more accurate than the FC approximation. The accuracy of the target wave functions is tested by comparing the calculated energy levels with the experimental data. Based on positron-hydrogen scattering, comprehensive close-coupling formulas for positron-helium scattering are developed. The reduced two-centre V-matrix elements are derived in momentum space for various channels. These include direct, excitation and rearrangement channels, i.e. positronium formation. We first consider low energy positron-helium elastic scattering for energies below the positronium formation threshold of 17.8 eV. Utilizing a single-centre expansion the elastic cross section and phase shifts have been calculated as a function of the positron incident energy. The calculations agree very well with the experimental data and the variational calculations, but not previous single- or double-centre close-coupling calculations. We then consider energies above the first ionization threshold (24.6 eV) and calculate helium elastic, excitation, fragmentation and total cross sections within the single-centre expansion approach. Good agreement with the available experimental and other theoretical results has been obtained. The studies have proved that a single-centre expansion, with accurate

An important model in the study of electron-atom collisions was development by Temkin and Poet. Although the model has been used to test many theories and approximation methods, the novel method of solution used by Poet has not been developed further to any great extent. The Temkin-Poet model of electron scattering simplifies the three-body scattering problem by suppressing all angular dependence of the wavefunction. In this thesis we return to Poet's method and apply it to a range of calculations for the hydrogen-isoelectronic series. Firstly it is demonstrated that the method provides high-precision solutions for elastic and inelastic scattering. these solutions will then be used to investigate various aspects of electron scattering including the ionisation of hydrogen in the near-threshold region, resonant states in helium, and the scaling of elastic cross sections for a number of hydrogen-like ions.

The convergent close-coupling (CCC) method is applied to the calculation of electron–helium ionization doubly differential cross sections (DDCSs) at low to intermediate incident energies. We re-examine the CCC calculations and measurements of Röder et al (1997 J. Phys. B: At. Mol. Opt. Phys. 30 1309–22) by making allowance for the step-function behaviour of the underlying CCC-calculated amplitudes. As done previously, the experimental DDCS were normalized at energies below 100 eV using the 100 eV CCC calculation to determine analyser properties at several secondary energies. In addition, substantially larger calculations are presented, to check the convergence. The agreement between the experiment and the calculations as a whole is much improved on the situation reported earlier.

Born total ionization cross section was calculated by an algebraic computing program using Maple. The atomic orbitals were approximated by Hartree-Fock Slater functions, and the ejected electron was modeled by a hydrogenic Coulomb wave. The numerical integrations were performed using FORTRAN by automatically converting the analytic matrix element for each orbital into a FORTRAN subroutine.

The convergent close coupling method has been applied to positron-helium scattering for energies below the positronium formation threshold (17.8 eV). Convergence is studied as a function of the Laguerre basis size. Rapid convergence to the accurate phase shifts given by Van Reeth and Humberston (1999 J. Phys. B: At. Mol. Opt. Phys. 32 3651-67) is demonstrated. The resultant cross sections are in excellent agreement with the experimental data of Stein et al (1978 Phys. Rev. A17 1600-8) and Mizogawa et al (1985 Phys. Rev. A 31 2171-9).

We apply a new implementation of the convergent close-coupling (CCC) method to electron–helium scattering. The target states are obtained from one-electron He+ box-based eigenstates rather than the usual Laguerre-based orbitals. The utility of the new method is demonstrated for 50 eV electron-impact ionization of helium with three different energy sharings between the two outgoing electrons. Excellent agreement is found between previous and new CCC predictions, and also with experimental data.

A relationship between the total wave function describing electron-impact ionization of hydrogen and the one representing scattering of two electrons and a proton in the continuum is revealed. On the basis of this relationship, forms of the scattered wave for the ionization process valid in all asymptotic domains are obtained. When all interparticle distances become large, the new wave functions reduce to the well-known Peterkop asymptotic wave function obtained in the hyperspherical approach. In particular, the Peterkop wave function is obtained by direct application of the present approach. This allows one to resolve the long-standing amplitude-phase ambiguity problem, which is an artifact of the hyperspherical approach to the ionization process. The Peterkop wave function is invalid when the two electrons are close to each other. This causes problems in practical calculations even in the domain where all particles are far apart. Our formulation provides a solution to this problem.

We present an implementation of the convergent close-coupling (CCC) method, in which the Laguerre-based target states are replaced with box-based eigenstates. Discrete energies are obtained by taking the states that vanish at the box radius R0, with convergence being established by simply varying R0. Excellent agreement between box- and Laguerre-based CCC, exterior complex scaling, and renormalized experiment is obtained for one of the most difficult three-body Coulomb breakup problems, near-threshold electron-impact ionization of atomic hydrogen. The simplicity of the box-based CCC method is expected to lead to a greater understanding of the close-coupling formalism generally.

According to quantum collision theory, scattering amplitudes are complex numbers, which are completely defined by their magnitude and phase. Although the phase information is generally not determined entirely in collision experiments, the phases are well defined and can be used to check computational models. We use four state-of-the-art approaches to calculate the magnitude and phase of the electron-hydrogen ionization amplitude in the Temkin-Poet S-wave model. We demonstrate that the correct phase can be extracted for each method by using the appropriate final-state continuum functions.

Asymptotic forms of the scattered wave for a system of three arbitrary charged particles valid in all domains relevant to breakup are derived. The derivations are based on the relationship between the total wave function of a breakup process in a three-body system with Coulomb interactions and the wave function of the process of scattering of all three particles of the system in the continuum. The results are free from amplitude-phase ambiguity problems associated with previously known forms. A similar technique is used to obtain asymptotic forms of the three-body Coulomb Green’s function.

The problems related to the formal theory of electron-atom ionization were studied. An integral representation for the ionization amplitude, which was free of ambiguity and divergence problems, was presented. Electron impact ionization of atomic hydrogen was considered, and it was assumed that the proton was infinitely heavy compared to the electrons and remained at rest. The results show that the proposed representation is readily applicable to extract the exact breakup amplitudes in direct calculations of general few-body systems.

We outline progress in a broad range of research projects that are currently being undertaken in our group. These include the formulation of ionisation amplitudes in Coulomb few‐body problems, a new convergent close‐coupling implementation for electron‐helium collisions, elastic electron‐magnesium scattering at very low energies, and the calculation of electron‐cesium spin asymmetries and differential cross sections.