Breit Operator Research Articles

Relativistic corrections in a series of helium excited states

Relativistic corrections to order (υ/ c) 2 are applied to the helium excited states 2 1,3P, 3 1,3D,…, 8 1,3K. Simple correlated open-shell wavefunctions are employed and the Breit operators H 1 through H 5 treated as perturbations. Account is also taken of mass polarization and lowest-order one electron Lamb shift. The energies thus calculated agree with experiment to within 2.2 cm −1 or better.

An expansion of Hartree-Fock and correlation energies, relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈HB>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s2 2sn1 2pn2 (2 ≧n1≧ 0, 6≧ n2 ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).

Wavelengths and transition probabilities for the nitrogen isoelectronic series

The energies of transitions of the type 1 s2 2 s2 2 p3-1 s2 2 s 2 p4-1 s2 2 p5 were computed for isoelectronic sequence of nitrogen. The account was taken of the Hartree-Fock and correlation energies in the first two-three orders of the perturbation theory and the relativistic corrections in terms of Breit operator. The computed values were compared with experimental results for the isoelectronic oxygen series. Sufficiently good agreement of the computed and experimental results for all isoelectronic sequence manifests rather fast convergence in the series of powers of 1/Z. The perturbation theory was employed in computing the transition probabilities of 1 s2 2 s2 2 p3-1 s2 2 s 2 p4-1 s2 2 p5. The data computed may be used in the identification of the coronal solar spectra and the spectra of high-temperature plasma obtained under the laboratory conditions.

Second-Order Corrections to the Fine Structure of Helium. III. Improved Calculation of SomeP3Contributions

Spin-dependent perturbations from intermediate $^{3}P$ states are calculated by expanding the perturbations to the wave function in terms with one negative power of one of the electron or interelectron coordinates, in addition to standard Hylleraas terms. The nonstandard terms reflect singularities of the Breit operators. A method for solving linear equations with "almost-singular" coefficient matrices is devised. Significant improvement in accuracy is achieved for five contributions, but the procedure does not work well in all cases where the coefficient matrix is unsymmetrical.

Second-Order Corrections to the Fine Structure of Helium. II. Contributions fromP1andD3Intermediate States

The contributions from the three spin-dependent Breit operators in second-order perturbation theory are calculated when the symmetries of the intermediate states are odd $^{1}P$ or $^{3}D$. Standard Hylleraas expansions with up to 165 terms are used for the perturbations of the wave functions. The large interval of the fine structure of the $2^{3}P$ level in helium is increased by 2.17(2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ by $^{1}P$, and by 0.0089(6)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ by $^{3}D$. Two methods for handling angular integration over $D$ tensors are described.

Second-Order Corrections to the Fine Structure of Helium

The fine-structure constant can be determined to high accuracy from precise measurements of the fine structure of the $2^{3}P$ level in helium. One of the necessary calculations is to compute the contributions from the six Breit operators and the mass-polarization operator in second-order perturbation theory. The eighteen spin-dependent perturbations from intermediate $^{3}P$ states are calculated by solving an inhomogeneous Schr\"odinger equation for the perturbation of the wave function by the variational method. The second-order contributions are then given by a single integral. These corrections are calculated using standard Hylleraas expansions with up to 165 terms for the perturbed wave functions, resulting in contributions to the two fine-structure intervals of the order of ${10}^{\ensuremath{-}4}$ ${\mathrm{cm}}^{\ensuremath{-}1}$, but only four of the results are sufficiently accurate.

Relativistic and Magnetic Spin Interactions in Helium-Like Atoms

Relativistic and magnetic spin corrections to the ionization energy of He and ${\mathrm{O}}^{6+}$ are given. The magnetic spin interaction given by Sessler and Foley has been derived by a careful evaluation of the Breit operator. The relativistic corrections obtained by reduction of the Dirac equation differ from some previous expressions. With all known corrections taken into account, there remains a discrepancy with experiment of about 30 ${\mathrm{cm}}^{\ensuremath{-}1}$ in the case of He. Within the rather large experimental and theoretical uncertainties, there appears to be no residual discrepancy in the case of ${\mathrm{O}}^{6+}$.