Frequency-independent Breit Interaction Research Articles

Kr][formula omitted], The final layzer quenched configuration for highly charged ions

We discuss the Layzer quenched ground state configuration of [Kr]4d104f13 along the Praseodymium (Pr) isoelectronic sequence between 80≤Z≤100. The 2F fine structure energy is examined from the point of view of the various contirbutions: correlation, frequency-independent and dependent Breit interaction and the two QED terms, self energy and vacuum polarisation. We find the fine structure energy to be dominated by the frequency-independent Breit interaction, correlation is relatively small, which is to be expected for this Layzer-quenched term. We also calculate the self energy using three different formulations, all included in the new version of the GRASP2K codes.

Conforming the measured lifetimes of the5d2D3/2,5/2states in Cs with theory

We find very good agreement between our theoretically evaluated lifetimes of the $5d \ ^2D_{3/2}$ and $5d \ ^2D_{5/2}$ states of Cs with the experimental values reported in [Phys. Rev. A {\bf 57}, 4204 (1998)], which were earlier evinced to be disagreeing with an earlier rigorous theoretical study [Phys. Rev. A {\bf 69}, 040501(R) (2004)] and with another precise measurement [Opt. Lett. {\bf 21}, 74 (1996)]. In this work, we have carried out calculations of the radiative transition matrix elements using many variants of relativistic many-body methods, mainly in the coupled-cluster theory framework, and analyze propagation of the electron correlation effects to elucidate their roles for accurate evaluations of the matrix elements. We also demonstrate contributions explicitly from the Dirac-Coulomb interactions, frequency independent Breit interaction and lower order quantum electrodynamics (QED) effects. Uncertainties to these matrix elements due to different possible sources of errors are estimated. By combining our calculated radiative matrix elements with the experimental values of the transition wavelengths, we obtain the transition probabilities due to both the allowed and lower order forbidden channels. Adding these quantities together, the lifetimes of the above two states are determined precisely and plausible reasons for the reported inconsistencies between the earlier theoretical calculations and the experimental results have been pointed out.

Spectral properties of a few F-like ions

Aims. We intend to provide accurate data for the oscillator strengths, transition probabilities, lifetimes, and hyperfine shifts of the atomic energy levels in the fluorine (F)- like Ti, V, Cr, Mn, Co, Ni, Cu, Zn, and Mo ions for the diagnostic of the astrophysical plasma. Furthermore, we propose these ions for probing possible variation in the fine structure constant (αe) by observing their transition lines from the distant astronomical objects. Methods. We have employed an all-order perturbative method in the relativistic coupled-cluster framework using the Dirac-Coulomb Hamiltonian for calculating the atomic wave functions in the considered F-like ions. We adopted the Fock-space formalism to account for the correlation effects among the occupied electrons by allowing all possible singly and doubly excited configurations in two steps: first considering the correlations between the electrons in a closed-shell configuration, and second constructing the openshell configurations of the first three low-lying states in the considered atomic systems by removing the respective electron from the core orbitals. This procedure enormously simplifies the computational complexity and overcomes the need to select important configuration state functions in determining the atomic state functions in contrast to a truncated configuration interaction method. Moreover, corrections from the frequency-independent Breit interaction and the lowest order quantum electrodynamic effects are incorporated self-consistently. Results. We present the calculated ionization potential energies of the electrons that are detached to form the above considered F-like ions. The correctness of our estimated results are ensured by comparing them with the available experimental values and other calculations. From these calculations, we evaluate the relativistic sensitivity coefficients that we propose here to be used for probing possible temporal variation of the fine structure constant (αe) through the precise astrophysical observation of the spectral lines of the transitions involving the above low-lying states. The accuracies of these sensitivity coefficients are appraised from the comparison of our calculated energies with the experimental values. We also present the line strengths by calculating the reduced matrix elements of the allowed transitions and of the M1 and E2 forbidden transitions. By combining our calculated line strengths with the observed wavelengths, we determine the oscillator strengths, transition probabilities, and lifetimes of the first two excited states of the above ions and compare them with the other theoretical calculations. The hyperfine splittings of the states are also presented by calculating their magnetic dipole and electric quadrupole hyperfine structure constants.

Self-consistent treatment ofthe frequency-independent Breit interaction in Dirac-Fockcalculations of atomic structures. II. He- and Be-like ions

We systematically study the effect of the inclusion of thefrequency-independent Breit interaction - treated perturbativelyas well as variationally - on Dirac-Fock total energies and⟨rk⟩ expectation values for He- and Be-likeions using an algorithm recently described by Reiher andHinze. Fully numerical, highly accuratesolution methods are employed throughout for solving theDirac-Fock-Coulomb-Breit (DFCB) equations. These methods allow us to investigate the change of the wavefunction upon inclusion ofthe Breit interaction in the self-consistent field procedureof an atomic structure program.The dependence of expectation values on different finite-nucleusmodel (FNM) potentials for the electron-nucleus attraction is alsostudied. It is shown that, in general, the choice of theFNM for the electron-nucleus potential hardlyaffects the difference between Dirac-Fock-Coulomb (DFC) andDFCB expectation values. In the case ofpointlike nuclei one should treat the Breit interaction variationally instead of perturbatively.For energy eigenvalues, we find that the difference betweenthe Breit interaction treated self-consistently or perturbativelyis negligibly small - even for highly charged ions - if anextended nucleus model is used. The data given may serve as areference for more approximate treatments involving, for example,basis-set approaches with limited basis-set size. In thisextensive study of the SCF effect on the Breit interaction,highly accurate, variationally obtained DFCB total energies and⟨rk⟩ expectation values are given for differentmodels of the electron-nucleus interaction.

Self-consistent treatment of the frequency-independent Breit interaction in Dirac-Fock and MCSCF calculations of atomic structures: I. Theoretical considerations

The self-consistent treatment of the Breit interaction in fully numerical atomic structure calculations is cumbersome due to the computationally demanding evaluation of two-electron integrals as they occur in the original formulation (Grant I P and Pyper N C 1976 J. Phys. B: At. Mol. Phys. 9 761). We present a reformulation of the frequency-independent Breit interaction operator in spherical coordinates and derive the corresponding matrix elements over spinors. With this formulation it becomes possible to compute the matrix elements of the Breit interaction efficiently and analogously to those of the Coulomb interaction: i.e., by determining the corresponding interaction potential functions using Poisson equations. The derived formulae will equally simplify computations using either basis sets or a numerical representation of the orbitals such that the Breit interaction can be included effectively in CI and SCF calculations for atoms and molecules. Of course, the computation of the Breit contribution to the total electronic energy as a first-order perturbation correction is also simplified. Furthermore, the frequency-dependent Breit interaction could be treated analogously.

Dirac-Fock-Breit self-consistent-field method: Gaussian basis-set calculations on many-electron atoms.

The self-consistent-field treatment of the frequency-independent Breit interaction is reviewed with applications to many-electron atoms. The implementation of the matrix Dirac-Fock-Breit self-consistent-field procedure is presented for Gaussian-type basis sets that show no near-linear dependency problem. The matrix Dirac-Fock-Breit procedure has the advantage over the finite-difference approach that it does not complicate the self-consistent-field procedure in basis-set expansion calculations. Basis sets of even- and well-tempered Gaussian functions were used to expand the large and small components of Dirac four-spinors. Expressions are derived for evaluating the matrix elements of the Dirac-Fock-Breit equations. Calculations done on rare-gas atoms He, Ne, Ar, Kr, and Xe and alkaline-earth metals Be, Mg, Ca, and Sr are presented.

Atomic Dirac-Fock-Breit self-consistent field calculations

The frequency-independent Breit interaction, which gives the leading correction to the instantaneous Coulomb interaction, is treated self-consistently in matrix Dirac–Fock–Breit self-consistent field (SCF) calculations. Calculations were done on He, He-like ions, Be, Be-like ions, Ne, and Ar. Basis sets of even- and well-tempered Gaussian functions were used to expand the large and small components of Dirac 4-spinors. The results of the variational calculations are compared with Desclaux's benchmark numerical perturbative calculations.

Dirac-Fock Gaussian basis calculations: Inclusion of the Breit interaction in the self-consistent field procedure

The frequency-independent Breit interaction is treated self-consistently in Dirac-Fock Gaussian basis set calculations on the Be atom and Ne 6+, Ar 14+ and Sn 46+ ions. The results of the calculations on Be are in excellent agreement with both Desclaux's benchmark numerical perturbative calculations and the Slater basis set calculations of Quiney, Grant and Wilson. The Gaussian basis set calculations yield accurate results for the highly ionized systems, and show none of the signs of near-linear dependence difficulties reported with Slater basis sets.

The Dirac equation in the algebraic approximation. V. Self-consistent field studies including the Breit interaction

It is argued that the Breit interaction can be included in the self-consistent-field procedure of Dirac-Hartree-Fock calculations, and need not necessarily be treated as a first-order perturbation, as is widely believed. The matrix Dirac-Hartree-Fock equations including the frequency-independent Breit interaction are presented and discussed. Prototype calculations for helium-like and beryllium-like ions are presented.