Pseudostate Method Research Articles

Comparative Study of Three-Body Models for Continuum States

We discuss properties of three-body continuum states in the hyperspherical formalism. The associated coupled-channel system is solved on a Lagrange basis, and is complemented by the R-matrix theory for the treatment of scattering states. Three-body α + n + n phase shifts are determined under various conditions. Two approximations of the continuum are considered: the complex scaling and the pseudostate methods. Both are briefly outlined, and compared with the R-matrix approach for the dipole-strength distribution of 6 He. A fair agreement is obtained between the three methods. We also discuss the influence of the α + n and n + n interactions, as well as the effect of the Pauli principle. Some differences are obtained, but all calculations predict a broad 1 − resonance near 1 MeV.

Electron-impact ionization of C+excited states

Nonperturbative close-coupling and perturbative distorted-wave methods are used to calculate electron-impact ionization cross sections for excited states of C${}^{+}$. Both the time-dependent close-coupling and the time-independent distorted-wave methods are used to calculate direct ionization cross sections for the $1{s}^{2}2{s}^{2}3l$ configurations. An $R$ matrix with pseudostates method is used to calculate total ionization cross sections for the $1{s}^{2}2{s}^{2}3l$ configurations, including both direct and indirect excitation-autoionization contributions. Comparison of the results shows both a substantial reduction of the direct ionization cross sections due to nonperturbative effects and a large contribution to the total ionization cross sections due to indirect contributions coming from $2s\ensuremath{\rightarrow}2p$ excitations. The ionization cross-section results will permit the generation of more accurate generalized collisional-radiative ionization coefficients needed for the modeling of moderately dense carbon plasmas.

Positron collisions with acetylene calculated using the R-matrix with pseudo-states method

Eigenphase sums, total cross sections and differential cross sections are calculated for low-energy collisions of positrons with C2H2. The calculations demonstrate that the use of appropriate pseudo-state expansions very significantly improves the representation of this process giving both realistic eigenphases and cross sections. Differential cross sections are strongly forward peaked in agreement with the measurements. These calculations are computationally very demanding; even with improved procedures for matrix diagonalization, fully converged calculations are too expensive with current computer resources. Nonetheless, the calculations show clear evidence for the formation of a virtual state but no indication that acetylene actually binds a positron at its equilibrium geometry.

R-matrix with Pseudo-States (RMPS) method: application to CH+ resonances curves

In a series of calculations on both electron and positron collisions with small molecules the R-Matrix with Pseudo-States (RMPS) method has been found to recover polarisation effects neglected in other close-coupling methods including the standard R-matrix procedure. The molecular R-Matrix and RMPS methods is being applied to determine low-lying resonance states of CH+ as a function of internuclear separation. Initial results are presented for both a standard R-matrix close-coupling model and for an RMPS calculation. Eigenphase sums and resonances below the 3Π threshold are presented for 2Π total symmetry. These resonances are classified by their quantum defects and compared to previous results. Prospects for these and other calculations using the RMPS method are discussed.

Tests of the discretized-continuum method in three-body dipole strengths

We investigate the 6He dipole distribution in a three-body α+n+n model. Two approaches are used to describe the three-body 1− continuum: the discretized-continuum method, where the scattering wave functions are approximated by square-integrable functions, and the R-matrix formalism, where their asymptotic behaviour is taken into account. We show that some ambiguity exists in the pseudostate method, owing to the smoothing technique, necessary to derive continuous distributions. We show evidence for the important role of the halo structure in the E1 dipole strength. We also address the treatment of Pauli forbidden states in the three-body wave functions.

The influence of electron collisions on non-LTE Li line formation in stellar atmospheres

Chemical abundances as measured from stellar spectral lines are often subject to uncertainties due to lack of accurate data for inelastic collisions, which is needed for non-local thermodynamic equilibrium (non-LTE) modelling. For cool stars, understanding of collision processes with electrons and hydrogen atoms is required to achieve high precision measurements. In this thesis, I have investigated the role of these collisions on the non-LTE formation of Li and Mg spectral lines in late-type stars.In the case of Li, electron impact excitation processes were calculated using the R-matrix with pseudo states method and the results found to agree well with recent calculations using the convergent close-coupling technique. These modern data were employed in non-LTE calculations by updating an existing model atom, which already included modern data for hydrogen collision processes. Our results were compared with calculations using older semi-empirical approximation calculations and only small differences were found: about 0.01 dex (~ 2%) or less in the abundance corrections. We therefore conclude that the influence of uncertainties in the electron collision data on non-LTE calculations is negligible. Indeed, together with the collision data for the charge transfer process Li + H ↔ Li+ + H- now available, and barring the existence of an unknown important collisional process, the collisional data in general is not a source of significant uncertainty in non-LTE Li line formation calculations.In the case of Mg, electron impact excitation processes were again calculated with the Rmatrix with pseudo states method, and used together with recent hydrogen collision calculations to build and test a model atom, without free parameters, for non-LTE modelling. Both electron and hydrogen collision processes, including charge transfer and excitation, are found to be important thermalising agents in various cases. The modelled spectra agree well with observed spectra from benchmark stars in the optical and infrared. The modelling predicts non-LTE abundance corrections ∆A(Mg)NLTE–LTE in dwarfs, both solar metallicity and metal-poor, to be very small (of order 0.01 dex), even smaller than found in previous studies. In giants, corrections vary greatly between lines, but can be as large as 0.4 dex. Results of calculations in a large grid of 1D model atmospheres are presented, and the implications for studies of Mg discussed. The propagation of uncertainties in the inelastic collision data to those in stellar abundances is investigated, and found to lead to small uncertainties, once again typically less than 0.01 dex (2%), although for few stellar models in specific lines (e.g., metal-poor suns, in the 7691 A line) uncertainties can be as large as 0.03 dex (7%).

Positron collisions with molecular hydrogen: cross sections and annihilation parameters calculated using the R-matrix with pseudo-states method

The molecular R-matrix with pseudo-states (MRMPS) method is employed to study positron collisions with H2. The calculations employ pseudo-continuum orbital sets containing up to h (l = 5) functions. Use of these high l functions is found to give converged eigenphase sums. Below the positronium formation threshold, the calculated cross sections agree with other high-accuracy theories and generally with the measurements. Calculation of the positron annihilation parameter Zeff with the MRMPS wavefunctions gives values significantly higher than other R-matrix wavefunctions but still do not completely converge with h functions. Extrapolation to higher l-values leads to a predicted value of Zeff for H2 of about 10.4. The MRMPS method is both completely general and ab initio; it can therefore be applied to positron collisions with other molecular targets.

Pseudostate methods and differential cross sections for antiproton ionization of atomic hydrogen and helium

A relaxed form of a recent impact parameter coupled pseudostate approximation of McGovern et al. [Phys. Rev. A 79, 042707 (2009)] for calculating differential ionization cross sections is proposed. This greatly eases the computational burden in cases where a range of ejected electron energies has to be considered. The relaxed approximation is tested against exact first Born calculations for antiproton impact on H and nonperturbatively for the highly nonperturbative system of Au${}^{53+}$ incident upon He. The approximation performs well in these tests. It is shown how, with a little further approximation, the relaxed theory leads to a widely used prescription for the total ionization cross section. Results for differential ionization of H and He by antiprotons are presented. These reveal the growing dominance of the interaction between the antiproton and the target nucleus at low impact energies and show the changing importance of the role of the postcollisional interaction between the antiproton and the ejected electron.

Two-photon 1s-3d and 1s-4d transitions in the He+ ion embedded in strongly-coupled plasmas with ion-sphere model

Here we present an investigation of 1s-3d and 1s-4d two-photon transitions in the He+ ion embedded in strongly-coupled plasmas. Using the ion-sphere (IS) model to represent the plasma effects, and with B-spline basis for the wave functions, we have calculated absorption coefficients for 1s-3d and 1s-4d two-photon transitions with different sizes of the ion-sphere by the pseudo-state method.

Nonperturbative calculations of double photoionization of Ne: AnRmatrix with pseudostate-convergence study

The $R$ matrix with pseudostate (RMPS) method has been employed to perform nonperturbative calculations of the total cross sections for single-photon double photoionization of ground-state Ne. A series of calculations was performed in which the number of pseudostates included in the close-coupling expansion was varied. Within the constraints of currently available massively parallel computers, we were able to achieve reasonably good convergence of the double photoionization cross section for Ne. The final RMPS results in both the length and the velocity gauges are compared to double photoionization cross sections determined from many-body perturbation-theory calculations and synchrotron measurements.

Three-photon transitions in the hydrogen atom immersed in Debye plasmas

Plasma-screening effects are investigated on three-photon bound-bound transitions in hydrogen atoms embedded in weakly coupled plasmas. In the present context, we discuss a set of basis functions for the pseudostate method. The basis functions are obtained from the modified wave functions for screening Coulomb potential (Debye model) using the Ritz variation method. The calculated energy eigenvalues for various Debye lengths agree well with the other theoretical results. Our calculated transition amplitudes for free-atom case agree excellently with the existing results in the literature. We have calculated the three-photon transition matrix elements for various Debye lengths, showing the strong plasma effects on the three-photon bound-bound transition amplitudes for the plasmas-embedded hydrogen atom.

Polarization effects in electron collisions with Li2: application of the molecular R-matrix method with pseudostates

Li2 has a huge polarizability and a huge elastic cross section for collisions with low-energy electrons. The recently developed molecular R-matrix with pseudostates (MRMPS) method is applied to electron collisions with Li2 at energies below 5 eV. The calculations, which are shown to be stable with respect to the choice of pseudostate basis, demonstrate the power of the MRMPS method for representing target polarizabilities and polarization effects in general. A previously identified low-lying 2Πu shape resonance is found at about 0.05 eV and several other low-lying resonances and resonance-like features are identified. Cross sections for elastic and electronically inelastic electron collisions are calculated and compared with previous studies.

Dirac R-matrix with pseudo-states

We describe the development of the Dirac R-matrix with pseudo-states (DRMPS) method for electron and photon collisions with arbitrary atoms and ions. An N-electron atom is represented by an anti-symmetrized product of single-particle spinors comprising the usual four-component Dirac spinors as well as paired two-component Laguerre spinors (L-spinors). The convergent L-spinor basis forms a discretization of the electron and positron continua. This representation has been implemented quite generally within the general relativistic atomic structure package (grasp) specifically grasp0. An (N + 1)th ‘scattering’ electron is represented by the exact same L-spinor basis plus the usual R-matrix box-state spinors which are chosen so as to form a combined complete finite linearly independent orthogonal basis. The (non-diagonal) Buttle correction is determined consistently. This representation has been implemented quite generally within the Dirac atomic R-matrix code (DARC). Results of some simple model problems are presented which demonstrate the correctness of the implementation.

Low and intermediate energy electron collisions with the C−2 molecular anion

Calculations are presented which use the molecular R-matrix with pseudo-states (MRMPS) method to treat electron impact electron detachment and electronic excitation of the carbon dimer anion. Resonances are found above the ionization threshold of C−2 with 1Σ+g, 1Πg and 3Πg symmetry. These are shape resonances trapped by the effect of an attractive polarization potential competing with a repulsive Coulomb interaction. The Πg resonances are found to give structure in the detachment cross section similar to that observed experimentally. Both excitation and detachment cross sections are found to be dominated by large impact parameter collisions whose contribution is modelled using the Born approximation.

Two-photon transitions in hydrogen atoms embedded in weakly coupled plasmas

The pseudostate method has been applied to calculate energy eigenvalues and corresponding eigenfunctions of the hydrogen atom in Debye plasma environments. Resonant two-photon transition rates from the ground state of atomic hydrogen to 2s and 3s excited states have been computed as a function of photon frequency in the length and velocity gauges for different Debye lengths. A two-photon transparency is found in correspondence to each resonance for 1s–3s. The transparency frequency and resonance enhancement frequency vary significantly with the Debye length.

Polarizabilities of the Ps negative ion

We have calculated polarizabilities (${\ensuremath{\alpha}}_{1}$, ${\ensuremath{\beta}}_{1}$, ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\alpha}}_{2}$, ${\ensuremath{\beta}}_{2}$, and ${\ensuremath{\gamma}}_{2}$) of ${\mathrm{Ps}}^{\ensuremath{-}}$ by the pseudostate method. These parameters can be used to calculate Rydberg states of ${\mathrm{Ps}}^{\ensuremath{-}}$ in the presence of an external electron with high quantum numbers $N$ and $L$. They are also of importance in a system containing ${\mathrm{Ps}}^{\ensuremath{-}}$ bound to a proton [PsH] and also Rydberg states of ${\mathrm{Ps}}_{2}$.

Electron-impact excitation of neutral boron using the R-matrix with the pseudostates method

We have carried out a large R-matrix with pseudostates (RMPS) calculation of the electron-impact excitation of neutral boron. The RMPS method has been employed for the excitation/ionization of many light fusion related species, but primarily for one or two active electrons. The present 640-term close-coupling calculation included three distinct pseudostate expansions. This enabled us to both converge the N-electron structure for three active open-shell electrons and accurately represent the high Rydberg and target continuum states. The derived Maxwell-averaged effective collision strengths will be employed to model the spectral emission from boron, which is of importance to both current (TEXTOR) and future (ITER) magnetic fusion reactors. The full set of effective collision strengths is available at the Oak Ridge National Laboratory Controlled Fusion Atomic Data Center Web site.

Doubly excited resonances in the photoionization spectrum of Li+

Absolute cross-section measurements for resonant double photoexcitation of Li+ions followed by autoionization have been performed at high resolution in the photon energy range from 148 eV, just below the (2s2p, 2(0,1)+2) resonance to 198 eV (the region of the double ionization threshold). The measurements have been made using the photon-ion merged-beam endstation at the Advanced Light Source, Lawrence Berkeley National Laboratory, USA. The absolute cross-section measurements show excellent agreement with theoretical results from the R-matrix plus pseudo-state (RMPS) method.

Doubly excited resonances in the photoionization spectrum of Li+: experiment and theory

Absolute cross-section measurements for resonant double photoexcitation of Li+ ions followed by autoionization have been performed at high resolution in the photon energy range from 148 eV, just below the (2s2p, 2(0, 1)n+) resonance, to 198 eV (the region of the double ionization threshold). The measurements have been made using the photon–ion merged-beam endstation at the Advanced Light Source, Lawrence Berkeley National Laboratory, USA. The absolute cross-section measurements show excellent agreement with theoretical results from the R-matrix plus pseudo-state (RMPS) method. Comparisons between theory and experiment for the Auger resonance energies, autoionization linewidth (Γ) and the Fano line profile index q for several members of the principal (2snp, 2(0, 1)n+) and (3snp, 3(1, 1)n+) Rydberg series found in the photoionization spectra for the 1Po symmetry show satisfactory accord.

Collisional-radiative calculations of He line emission in low-temperature plasmas

We present spectral modeling results for neutral helium. Our underlying atomic data contains radiative transition rates that are generated from atomic structure calculations and electron-impact excitation rates, that are determined from both the standard $R$-matrix method and the $R$-matrix with pseudostates (RMPS) method. In this paper, we focus on transitions of particular importance to diagnostic line ratios. For example, our calculated rate coefficient for the electron-impact transition $1s3s\phantom{\rule{0.2em}{0ex}}^{1}S\ensuremath{\rightarrow}1s3p\phantom{\rule{0.2em}{0ex}}^{1}P$, which has a pronounced effect on the $728.1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ diagnostic spectral line, is found to be in good agreement with previous experimental measurements. We also consider transitions from the $1{s}^{2}\phantom{\rule{0.2em}{0ex}}^{1}S$ ground and $1s2s\phantom{\rule{0.2em}{0ex}}^{3}S$ terms to terms of the $n=4$ shell. They are found to be affected significantly by coupling of the bound states to the target continuum (continuum coupling), which is included in our RMPS calculation, but not in our standard $R$-matrix calculation. We perform collisional-radiative calculations to determine spectral line intensity ratios for three ratios of particular interest, namely the $504.8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}∕471.3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, $492.2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}∕471.3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, and $492.2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}∕504.8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ line ratios. Comparing our results determined from the RMPS excitation rates with those from the standard $R$-matrix excitation rates, we find that continuum coupling affects the rate coefficients significantly, leading to different values for all three line ratios. We also compare our modeling results with spectral measurements taken recently on the Auburn Helicon plasma device, finding that the ground and metastable populations are not in equilibrium, and that the experimental measurements are more consistent with the $1s2s\phantom{\rule{0.2em}{0ex}}^{3}S$ metastable term populations being short lived in the plasma.