R-matrix Approach Research Articles

Description of two-proton radioactivity on the basis of methods of the quantum theory of ternary nuclear fission

The two-proton decay of spherical nuclei is investigated on the basis of the formalism developed in constructing the quantum-mechanical theory of ternary fission. The proposed method for determining the amplitudes of partial widths with respect to two-proton decay and the asymptotic behavior of the wave function for a decaying nucleus makes it possible to solve the problem of describing two-proton radioactivity without recourse to the traditionally used (in R-matrix approaches) cumbersome procedure of matching the internal and the external wave function for the decaying nucleus within the three-body formulation. In the diagonal approximation and with allowance for the properties of the potential describing the interaction of the products of two-proton decay, the structure of the wave function for the Cooper pair of two protons bound in the parent nucleus is analyzed, along with the behavior of the wave function describing the potential scattering of the products of binary decay, the coupling of decay channels being taken into account in this analysis.

R-matrix calculations of the integral electron-impact excitation cross sections of the ground state of the barium atom

The R-matrix approach has been used to calculate the integral cross sections of excitation of the barium atom by low-energy electrons. The collision strengths have been calculated in the 38-state close coupling approximation, including 16 physical states and 22 pseudostates of the target. The target wave functions have been obtained in terms of the multiconfiguration Hartree-Fock method in the LS coupling scheme. The cross sections calculated exhibit a pronounced resonance structure, which contributes significantly to the total excitation cross sections. The results obtained have been compared to available theoretical and experimental data.

Photoionization of atomic phosphorus from ground and excited states

Resonance structures in the photoionization of atomic phosphorus for the ejection of 3p or 3s electron from the ground 3s23p34So and excited metastable 2Do and 2Po states have been studied in the R-matrix approach. Photoionization cross sections and photoelectron angular distribution asymmetry parameters are dominated by Rydberg series of autoionization resonances converging to several ionic thresholds. There are window resonances due to the 3s3p3(5So)np, 3s3p3(3Do)np and 3s3p3(3Po)np Rydberg series in the inner-shell photoionization from the ground state. Autoionizing states are identified and analysed to obtain resonance positions Er, effective quantum numbers n* and widths Γr using a procedure of eigenphase gradients. Our results for the 3s3p3(5So)np window resonances are in excellent agreement with the experimental values. Total and partial cross sections together with asymmetry parameter for leaving the ion in the ground 3s23p23P state have been calculated in both length and velocity formulations that exhibit very good agreement with each other.

Effective Collision Strengths for Fine‐Structure Transitions in S i

Electron collisional excitation strengths for transitions between the 3s23p4 3P2,1,0, 1D2, and 1S0 levels and from these levels to the 58 fine-structure levels of the excited configurations of neutral sulfur are calculated using the B-spline R-matrix approach. The nonorthogonal orbitals are used for an accurate representation of both the target states and the R-matrix basis functions. The close-coupling expansion consists of the 27 spectroscopic bound and autoionizing target states that give rise to 63 fine-structure levels. We included the lowest 22 LS states of the 3s23p4, 3s23p34s, 3s23p34p, 3s23p33d, 3s23p34d, 3s23p34f, 3s23p35s, 3s23p35p, 3s23p36s, and 3s3p5 configurations, and five higher lying 3s23p3(2Po)4s 3Po, 1Po, 3s23p3(2Do)3d 3Po, 1Po, and 3s3p5 1Po states that have strong dipole connection with the ground terms. The collision strengths for transitions between fine-structure levels are calculated by transforming the LS-coupled K-matrices to K-matrices in an intermediate coupling scheme. The effective collision strengths are determined by integrating collision strengths over a Maxwellian distribution of electron energies. The effective collision strengths are tabulated for log T from 3.3 to 5.6 K suitable for applications in astrophysical plasma modeling.

Electron impact excitation of fine-structure levels in Cl II

The B-spline basis has been used in the R-matrix approach to calculate electron impact excitation collision strengths for transitions between the 3s 2 3p 4 3 P 0,1,2 , 1 D 2 and 1 S 0 levels and from these levels to the fine-structure levels of the excited 3s3p 5 , 3s 2 3p 3 4s, 3s 2 3p 3 3d, 3s 2 3p 3 4p and 3s 2 3p 3 4d configurations of Cl II. We considered 23 LS states 3s 2 3p 4 3 P, 1 D, 1 S, 3s3p 5 3 P°, 3 s 23p 3 ( 4 S°)4s 5.3 S°, 3s 2 3p 3 ( 4 S°)3d 5,3 D°, 3s 2 3p 3 ( 2 D°)3d 1 P°, 1 S°, 3 F°, 3 D°, 3 P°, 3s 2 3p 3 ( 2 D°)4s 3,1 D°, 3s 2 3p 3 (2P°)4s 3,1 P°, 3s 2 3p 3 ( 4 S°)4p 5,3 P, 3s 2 3p 3 ( 2 P°)3d 3,1 P°, 3 D° and 3s 2 3p 3 ( 2 D°)4d 3 P° in the close-coupling expansion that give rise to 51 fine-structure levels. The non-orthogonal orbitals are used for an accurate representation of both the target wavefunctions and the R-matrix basis functions. The collision strengths for transitions between fine-structure levels are calculated by transforming the LS-coupled K-matrices to K-matrices in an intermediate coupling scheme. The Rydberg series of resonances converging to excited state thresholds make substantial contributions to collision strengths. The thermally averaged collision strengths have been obtained by integrating collision strengths over a Maxwellian distribution of electron energies and these are listed in Table 2 for log T from 3.3 to 5.6 K.

Self-consistent R-matrix approach to photoionization and unified electron–ion recombination

A unified scheme using the R-matrix method has been developed for electron–ion recombination subsuming heretofore separate treatments of radiative and dielectronic recombination (RR and DR). The ab initio approach within the coupled channel approximation has several inherent advantages in addition to the natural unification of resonant and non-resonant phenomena. It enables a general and self-consistent treatment of photoionization and electron–ion recombination employing identical wavefunction expansion. Detailed balance takes account of interference effects due to resonances in cross sections, calculated explicitly for a large number of recombined (e+ion) bound levels over extended energy regions. The theory of DR by Bell and Seaton is adapted for high- n resonances in the region below series limits. The R-matrix method is employed for (A) partial and total photoionization and photorecombination cross sections of (e+ion) bound levels, and (B) DR and (e+ion) scattering cross sections. Relativistic effects and fine structure are considered in the Breit–Pauli approximation. Effects such as radiation damping may be taken into account where necessary. Unified recombination cross sections are in excellent agreement with measurements on ion storage rings to about 10–20%. In addition to high accuracy, the strengths of the method are: (I) both total and level-specific cross sections and rate coefficients are obtained, and (II) a single (e+ion) recombination rate coefficient for any given atom or ion is obtained over the entire temperature range of practical importance in laboratory and astrophysical plasmas, (III) self-consistent results are obtained for the inverse processes of photoionization and recombination; comprehensive datasets have been computed for over 50 atoms and ions. Selected data are presented for iron ions.

Electronic transport in nanosystems

We demonstrate the application of the Landauer–Büttiker formalism (LBF) to MIS (metal–insulator–semiconductor)-type quantum nanostructures and analyze their static and dynamical response properties. For the time-dependent problem, we employ the random phase approximation (RPA) and calculate the irreducible polarization from the one-particle scattering wave functions in the Hartree approximation. For their calculation, we employ the R-matrix approach which we have developed over the past years. We find in the low frequency limit a reasonable agreement with the static Hartree calculations and experimental capacitance data.

Electron Collisional Excitation Rates for O i Using the B ‐Spline R ‐Matrix Approach

The B-spline R-matrix approach has been used to calculate electron collisional excitation strengths and rates for transitions between the 3P, 1D, and 1S states of ground configuration and from these states to the states of the excited 2s(sup 2)2p(sup 3)ns (n = 3-5), 2s(sup 2)2p(sup 3)np (n = 3-4), 2s(sup 2)2p(sup 3)nd (n = 3-4), 2s(sup 2)2p(sup 3)4f, and 2s2p(sup 5) configurations. The nonorthogonal orbitals are used for an accurate description of both the target wave functions and the R-matrix basis functions. The thermally averaged collision strengths are obtained from the collision strengths by integrating over a Maxwellian velocity distribution of electron energies, and these are tabulated over a temperature range from 1000 to 60,000 K. The parametric functions of scaled energy have also been obtained to represent collision strengths over a wide energy range or thermally averaged collision strengths at any desired temperature.

Effective collision strengths for electron impact excitation of Si VIII

Effective collision strengths for electron-impact excitation of the nitrogen-like ion Si VIII are presented over the wide range of electron temperatures log T(K)=4.0–6.5. All 231 fine-structure transitions among the 22 fine-structure levels arising from the lowest 11 LS target states (2s 22p 3, 2s2p 4, 2p 5, and 2s 22p 23s) are considered in the tabulation. The collision strengths are evaluated in a multi-channel R-matrix approach, and the corresponding effective collision strengths are obtained by averaging these over a Maxwellian distribution of electron velocities. Comparisons are made with recent distorted-wave results at high incident electron energies. Differences of up to 20% are found, particularly for some allowed transitions.

Asymptotic normalization coefficients from proton transfer reactions and astrophysical S factors for the CNO 13C(p, γ) 14N radiative capture process

The 13C(p, γ) 14N radiative capture reaction is analyzed within the R-matrix approach with the aim to determine the astrophysical S factor. The low-energy astrophysical S factor has important contributions from both resonant and nonresonant captures. The contribution of the nonresonant component of the transition to a particular 14N bound state is expressed in terms of the asymptotic normalization coefficient (ANC). The experimental ANCs induced from the 13C( 14N, 13C) 14N and 13C( 3He, d) 14N reactions are used in the analysis. The fits of the calculated S factors to the experimental data are sensitive to the ANC values and are used to test the extracted ANCs. It is found that the ANCs determined from the transfer reactions provide the best fit for transitions to all the states in 14N, except for the third excited state. The obtained astrophysical factor, S(0)=7.6±1.1 keV b, is in excellent agreement with published values.

Relativistic theory of photoionization of one-electron systems: an eigenchannel R-matrix approach

A theoretical background of an eigenchannel R-matrix approach to photoionization of relativistic one-electron systems is outlined.

Determination of the astrophysicalSfactor for11C(p,γ)12Nfrom the12N→11C+pasymptotic normalization coefficient

The evolution of very low-metallicity, massive stars depends critically on the amount of CNO nuclei that they produce. Alternative paths from the slow $3\ensuremath{\alpha}$ process to produce CNO seed nuclei could change their fate. The ${}^{11}\mathrm{C}(p,\ensuremath{\gamma}{)}^{12}\mathrm{N}$ reaction is an important branch point in one such alternative path. At energies appropriate to stellar evolution of very low-metallicity, massive stars, nonresonant capture dominates the reaction rate. We have determined the astrophysical S factor for $\mathrm{the}{ }^{11}\mathrm{C}(p,\ensuremath{\gamma}{)}^{12}\mathrm{N}$ reaction using the asymptotic normalization coefficient for ${}^{12}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{N}}}^{11}\mathrm{C}+p$ to fix the nonresonant capture rate. In our experiment, a 110 MeV ${}^{11}\mathrm{C}$ radioactive beam was used to study ${\mathrm{the}}^{14}\mathrm{N}{(}^{11}\mathrm{C}{,}^{12}\mathrm{N}{)}^{13}\mathrm{C}$ peripheral transfer reaction and the asymptotic normalization coefficient, ${(C}_{{p}_{\mathrm{eff}}}^{{}^{12}N}{)}^{2}{=(C}_{{p}_{1/2}}^{{}^{12}N}{)}^{2}{+(C}_{{p}_{3/2}}^{{}^{12}N}{)}^{2}=1.73\ifmmode\pm\else\textpm\fi{}0.25{\mathrm{fm}}^{\ensuremath{-}1},$ was extracted from the measured cross section. The contributions from the second resonance and interference effects were estimated using an R-matrix approach with the measured asymptotic normalization coefficient and the latest value for ${\ensuremath{\Gamma}}_{\ensuremath{\gamma}}.$ We find the S factor for ${}^{11}\mathrm{C}(p,\ensuremath{\gamma}{)}^{12}\mathrm{N}$ is significantly larger than previous estimates. As a result, the required density for it to contribute is reduced, and more CNO material may be produced.

Photoionization of C2+ ions

We have investigated valence-shell photoionization (PI) of 1S ground state and 3Po metastable states of C2+ ions. Absolute PI cross-sections were measured in the range 40.7–56.4 eV with an energy spread of 30 meV using a photon–ion merged beams arrangement at the Advanced Light Source (ALS). Detailed calculations based on the semi-relativistic Breit–Pauli R-matrix approach suggest a fraction of 40% of metastable ions in the primary beam of the experiment. Additional high resolution measurements have been carried out with an energy spread as low as 7.5 meV. The new data permit a sensitive test of theory with respect to the individual PI cross-section contributions originating from the different 3P0,1,2 fine-structure components of the initial triplet states.

AstrophysicalSfactorfor 13C(p,γ)14Nand asymptotic normalization coefficients

We reanalyze the ${}^{13}\mathrm{C}(p,\ensuremath{\gamma}{)}^{14}\mathrm{N}$ radiative capture reaction within the R-matrix approach. The low-energy astrophysical S factor has important contributions from both resonant and onresonant captures. The normalization of the nonresonant component of the transition to a particular ${}^{14}\mathrm{N}$ bound state is expressed in terms of the asymptotic normalization coefficient (ANC). In the analysis we use the experimental ANC's inferred from $\mathrm{the}{}^{13}\mathrm{C}{(}^{14}\mathrm{N}{,}^{13}\mathrm{C}{)}^{14}\mathrm{N}$ and ${}^{13}\mathrm{C}{(}^{3}\mathrm{He}{,d)}^{14}\mathrm{N}$ reactions. The fits of the calculated S factors to the experimental data are sensitive to the ANC values and are used to test the extracted ANC's. We find that for transitions to all the states $\mathrm{in}{}^{14}\mathrm{N},$ except the third excited state, the ANC's determined from the transfer reactions provide the best fit. The astrophysical factor we obtain, $S(0)=7.7\ifmmode\pm\else\textpm\fi{}1.1 \mathrm{keV}\mathrm{}\mathrm{b},$ is in excellent agreement with previous results.

Electron impact collisionstrengths and rates for neutral sulphur using the B-splineR-matrix approach

Our previous calculations based on the use of the B-splinebasis in the R-matrix approach for low-energy electroncollisions with atomic sulphur (Zatsarinny O and Tayal S S 2001J. Phys. B: At. Mol. Opt. Phys. 34 3383) have beenextended to determine electron collisional excitation strengthsand rates for transitions between the 3s23p4 3P,1D and 1S LS states and from these states to the 24states of the excited configurations. The close-couplingexpansion consists of the 27 spectroscopic bound andautoionizing target states. The collision strengths arerepresented by simple parametric functions of scaled energy.Their values in a very wide energy range are easily obtainedusing the tabulated parameters. The same parameters can be usedto determine effective collision strengths over a wide electrontemperature range. Branching ratios are presented that can beused to estimate cascade contributions and to determine emissioncross sections for transitions between low-lying states.

Permanent and transition dipole moments in CaF and CaCl

Permanent and transition dipole moments have been calculated for various states of CaF and CaCl by using an effective one-electron variational eigenchannel R-matrix approach combined with generalized quantum defect theory. The ion core dipole moment has been evaluated ab initio. The calculations reproduce the existing measurements involving the lowest states of these compounds to within about 5%. The calculations yield for the first time electronic one-photon transition moments for excitation of the higher Rydberg series members, from the A 2Π, C 2Π or D 2Σ+ intermediate states. Permanent dipole moments for the higher Rydberg states have also been evaluated. It is found that the dipolar ion core causes an “internal” Stark effect to appear, such that certain series are characterized by strong positive dipole moments and other series are characterized by strong negative dipole moments.

Theoretical study on dielectronic recombination ofO6+ions in metastable states

A computational scheme, based on the theory of the continuum-bound transitions of Bell and Seaton [J. Phys. B 18, 1589 (1985)] and the close-coupling R-matrix approach, has been developed to treat dielectronic recombination (DR) in high-lying resonance-energy regions. This scheme and our presented numerical method to compute DR in low-lying resonance-energy regions [Phys. Rev. A 62, 022706 (2000)] have been applied together to elucidate the experimental spectra of the DR of ${\mathrm{O}}^{6+}$ ions in the metastable $1s2s{}^{3}S$ and $1s2s{}^{1}S$ states. For comparison, a perturbative theoretical calculation of DR for ${\mathrm{O}}^{6+}$ has also been accompanied. The reasonable representation of the general dielectronic spectral shape is yielded by both our close-coupling and perturbative calculations. However, both the methods do not reproduce the experimental double-peak structure at $\ensuremath{\sim}6$--8 eV. This shows that the further investigation on DR of this kind of ions is required both experimentally and theoretically.

Vibrational excitation of H2O by electron impact

We present results for the vibrational excitation of H2O bylow-energy electrons (<10 eV). We calculate differentialcross sections for the first excited states for all three modesusing an R-matrix approach, at the static-exchange level, and theChase approximation. Correction to the scattering amplitude forall partial waves (l >3) is made using a closureapproximation. The results are consistent with other theoreticaltreatments. For the bending mode there is agreement with twoolder experiments but disagreement with a more recent experimentat 7.5 eV. However, the present results for the stretch modes areless than the experimental values.

Polarization ofBalmer-α radiation following electron impact on atomichydrogen

The polarization of Balmer-α radiation excited incollisions of electrons with atomic hydrogen is presented for anelectron energy range from threshold to 1000 eV. Measurementsare in good agreement with calculations carried out usingeither convergent-close-coupling or R-matrix withpseudo-states approaches. Cascade is demonstrated tohave a significant effect. Balmer-α excitation functiondata are also presented. A previous measurement of thepolarization of Balmer-α following dissociativeexcitation of H2 by electrons isconfirmed and extended.

A radiation-damped R-matrixapproach to the electron-impact excitation of helium-like ions fordiagnostic application to fusion and astrophysical plasmas

Electron-impact excitation collision strengths for transitions between allsingly excited levels up to the n = 4 shell of helium-like argon and then = 4 and 5 shells of helium-like iron have been calculated using aradiation-damped R-matrix approach. The theoretical collision strengthshave been examined and associated with their infinite-energy limit valuesto allow the preparation of Maxwell-averaged effective collisionstrengths. These are conservatively considered to be accurate to within20% at all temperatures, 3×105-3×108 K forAr16+ and 106-109 K for Fe24+. Theyhave been compared with the results of previous studies, where possible,and we find a broad accord.The corresponding rate coefficients are required for use in the calculationof derived, collisional-radiative, effective emission coefficients forhelium-like lines for diagnostic application to fusion and astrophysicalplasmas. The uncertainties in the fundamental collision data have beenused to provide a critical assessment of the expected resultantuncertainties in such derived data, including redistributive and cascadecollisional-radiative effects. The consequential uncertainties in theparts of the effective emission coefficients driven by excitation from theground levels for the key w, x, y and z lines vary between 5% and 10%.Our results remove an uncertainty in the reaction rates of a key class ofatomic processes governing the spectral emission of helium-like ions inplasmas.