Quantum Electrodynamic Corrections Research Articles

Precision Measurement of the1s2pP32−P13Fine Structure Interval in Heliumlike Fluorine

Using Doppler-tuned fast-beam laser spectroscopy with co- and counterpropagating beams we have measured the three hyperfine components of the $1s2p^{3}P_{2}\ensuremath{-}^{3}P_{1}$ fine structure interval in ${}^{19}{\mathrm{F}}^{7+}$. Our result for the centroid is $957.6679(10){\mathrm{cm}}^{\ensuremath{-}1}$. Allowing for the hyperfine interaction the ``pure'' fine structure interval is determined to be $957.8730(12){\mathrm{cm}}^{\ensuremath{-}1}$. This result tests $O({\ensuremath{\alpha}}^{7}{m}_{e}{c}^{2})$ quantum electrodynamic corrections to high precision calculations which will be used to obtain a new value for the fine structure constant from the fine structure of helium.

Precision prediction for the big-bang abundance of primordial4He

Within the standard models of particle physics and cosmology we have calculated the big-bang prediction for the primordial abundance of ${}^{4}\mathrm{He}$ to a theoretical uncertainty of less than $0.1% (\ensuremath{\delta}{Y}_{P}l\ifmmode\pm\else\textpm\fi{}0.0002),$ improving the current theoretical precision by a factor of 10. At this accuracy the uncertainty in the abundance is dominated by the experimental uncertainty in the neutron mean lifetime, ${\ensuremath{\tau}}_{n}=885.4\ifmmode\pm\else\textpm\fi{}2.0 \mathrm{sec}.$ The following physical effects were included in the calculation: the zero and finite-temperature radiative, Coulomb and finite-nucleon-mass corrections to the weak rates; order-$\ensuremath{\alpha}$ quantum-electrodynamic correction to the plasma density, electron mass, and neutrino temperature; and incomplete neutrino decoupling. New results for the finite-temperature radiative correction and the QED plasma correction were used. In addition, we wrote a new and independent nucleosynthesis code designed to control numerical errors to be less than 0.1% . Our predictions for the ${}^{4}\mathrm{He}$ abundance are presented in the form of an accurate fitting formula. Summarizing our work in one number, ${Y}_{P}(\ensuremath{\eta}=5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10})=0.2462\ifmmode\pm\else\textpm\fi{}0.0004 (\mathrm{expt})\ifmmode\pm\else\textpm\fi{}l0.0002 (\mathrm{theory}).$ Further, the baryon density inferred from the Burles-Tytler determination of the primordial D abundance, ${\ensuremath{\Omega}}_{B}{h}^{2}=0.019\ifmmode\pm\else\textpm\fi{}0.001,$ leads to the prediction ${Y}_{P}=0.2464\ifmmode\pm\else\textpm\fi{}0.0005 (\mathrm{D}/\mathrm{H})\ifmmode\pm\else\textpm\fi{}l0.0002 (\mathrm{theory})\ifmmode\pm\else\textpm\fi{}0.0005 (\mathrm{expt}).$ This ``prediction'' and an accurate measurement of the primeval ${}^{4}\mathrm{He}$ abundance will allow an important consistency test of primordial nucleosynthesis.

Transition probability between the hyperfine structure components of hydrogenlike ions and bound-electron g-factor

The transition probability between the hyperfine splittingcomponents of a hydrogenlike ion is expressed in terms of thebound-electron g-factor. It allows a simple calculationof the QED (quantum electrodynamic) and nuclear-size corrections to the transition probability by using thecorresponding corrections to the bound-electron g-factor.In experimentally interesting cases of Ho, Re, Pb, and Bithe QED corrections increase the transition probability by about 0.3%.Using a recent experimental value for the transition probabilitybetween the hyperfine structure components in 207Pb81+it is foundthat the bound-electron g-factor in lead amounts to 1.78(12).The corresponding theoretical value is 1.7383.\\\\PACS Nos.:12.20Ds, 31.30Gs, and 31.30Jv

Influence of changes in the valence electronic configuration on theKβ-to-Kαx-ray intensity ratios of the3dtransition metals

Very extensive multiconfiguration Dirac-Fock calculations including the transverse (Breit) interaction and quantum electrodynamics corrections have been carried out for all $3d$ transition metals to explain reliably the dependence of $K\ensuremath{\beta}$-to-$K\ensuremath{\alpha}$ x-ray intensity ratios on the changes in configurations of the valence electrons. For all considered atoms the greatest values of the $K\ensuremath{\beta}$-to-$K\ensuremath{\alpha}$ intensity ratios are for ${3d}^{m\ensuremath{-}2}{4s}^{2}$ configuration type, then for ${3d}^{m\ensuremath{-}1}{4s}^{1},$ and the smallest values are for ${3d}^{m}$ type. It has been found that for each type of electronic configuration the $K\ensuremath{\beta}$-to-$K\ensuremath{\alpha}$ intensity ratios increase evidently with the atomic number and for a particular atom they are quite sensitive to the changes of the valence electronic configuration. The greatest relative increase of the $K\ensuremath{\beta}$-to-$K\ensuremath{\alpha}$ intensity ratios (4%) as a result of transition from electronic configuration of the ${3d}^{m\ensuremath{-}1}{4s}^{1}$ type to the ${3d}^{m\ensuremath{-}2}{4s}^{2}$ type takes place for Sc and the smallest relative increase (2%) takes place for Cu. The presented results make it possible to carry out reliable interpretation of various experimental $K\ensuremath{\beta}$-to-$K\ensuremath{\alpha}$ x-ray intensity ratios for $3d$ transition metals in their compounds and alloys and can also provide quantitative information about the changes of the valence electronic configurations of these metals in considered systems.

Influence of nuclear size on QED corrections in hydrogenlike heavy ions

The influence of the nuclear size and shape on quantum electrodynamic corrections to electron binding energies in heavy hydrogenlike ions is evaluated. For the self-energy, numerical values for the energy shift that take into account the size of the nucleus are given for ${1S}_{1/2}$-, ${2S}_{1/2}$-, and ${2P}_{1/2}$-states in the $Z$ range from 26 to 110. Simple parametrizations are derived for the dependence of the self-energy and the vacuum polarization on the nuclear radius, which allow corrections to be made for small deviations from the currently known nuclear radii. The influence of nuclear size on second-order quantum electrodynamic corrections is examined for the two-loop vacuum-polarization contributions. The contribution is found to be experimentally indistinguishable from other contributions to the energy shift which are of nonquantum electrodynamic origin.

Application of B-splines finite basis sets to relativistic two-photon decay rates of $\mathsf{2s}$ level in hydrogenic ions

A theoretical study of the one- and two-photon spontaneous emission rates from the 2s1/2 state of one-electron ions is presented. High-precision values of the relativistic emission rates for ions with nuclear charge Z up to 100 are obtained through the use of finite basis sets for the Dirac equation constructed from B-splines. Furthermore, we analyze the influence of the inclusion of quantum electrodynamics corrections in the initial and final state energies.

The spectrum and term system of C IV

This paper reports an investigation of the spectrum and term system of Li-like carbon, C IV. Between 200 and 7750 Å a total of 105 C IV lines have been identified by combining previously known data with new spectra recorded by the beam-foil technique and from observations at the JET tokamak. A comprehensive term system, with accurate energies for all levels up to n = 12, is constructed from the identified transitions. Semiempirical series formulae are also given, allowing reliable extrapolations to still higher n values. The ionization energy is determined to (520 175.3 ± 1.5) cm−1, which confirms recent theoretical predictions that include quantum electrodynamical (QED) corrections.

Large-scale relativistic configuration-interaction calculation of the3s21S0-3s3p1,3P1transition energies in magnesiumlike ions

We have calculated the ${3\mathrm{s}}^{21}$${\mathrm{S}}_{0}$-${3\mathrm{s}3\mathrm{p}}^{1,3}$${\mathrm{P}}_{1}$ transition energies for neutral magnesium and Mg-like ${\mathrm{Ar}}^{6+}$, ${\mathrm{Cu}}^{17+}$, ${\mathrm{Kr}}^{24+}$, and ${\mathrm{Mo}}^{30+}$ using the relativistic configuration-interaction (CI) method. These calculations are based on the relativistic no-pair Hamiltonian which includes Coulomb and retarded Breit interactions and employ finite B-spline basis functions. Quantum electrodynamic and mass polarization corrections are also calculated. For Mg-like ions studied here, intravalence correlations are treated exactly by saturating the basis with all configurations that arise from valence-valence excitations. Contributions from core polarization are also calculated with large-scale CI expansions by including dominant configurations from core-valence excitations. Agreement between theory and experiment are good for all the Mg-like ions studied here.

K-SHELL AUGER AND RADIATIVE TRANSITIONS IN THE CARBON ISOELECTRONIC SEQUENCE, 6 ≤Z≤ 54

K-shell energies and transition rates for the Auger and radiative decays in carbon-like ions with 1s2sm2pn(m + n= 5) configurations have been calculated using the multiconfiguration Dirac–Fock method including Breit interaction and quantum electrodynamic corrections. Seventeen ions with atomic numbers 6 ≤Z≤ 54 were considered. Transition rates were evaluated in intermediate coupling with configuration interaction within the same complex. Line fluorescence yields were also computed.

Strengths and weaknesses of relativistic atomic structure calculations

Among the relativistic atomic structure theories developed in the last two decades, two methods - multiconfiguration Dirac-Fock (MCDF) and relativistic many-body perturbation theory (RMBPT) - have extensively been applied to calculate spectroscopic properties of neutral atoms as well as highly charged ions. The strengths and weaknesses of these methods are discussed with some examples. The MCDF method is based on the shell structure familiar to atomic physicists, and hence intuitively easier to understand. On the other hand, the RMBPT method is more systematic in introducing correlation and relativistic corrections and logically simpler to include quantum electrodynamic (QED) corrections and higher order relativistic effects than the MCDF method. A recent development in extensive relativistic configuration interaction is also described.

Energy levels of the ground state and the 2s2pn(J=1) excited states of berylliumlike ions: A large-scale, relativistic configuration-interaction calculation

We have calculated energy levels of the ${2\mathrm{s}}^{2}$${\mathrm{}}^{1}$${\mathrm{S}}_{0}$ ground state and the 2s2p${\mathrm{}}^{1,3}$${\mathrm{P}}_{1}$ excited states of berylliumlike ions with Z=10--92 using the relativistic configuration-interaction method. These calculations are based on the relativistic no-pair Hamiltonian which includes Coulomb and retarded Breit interactions and employ finite B-spline basis functions. Quantum electrodynamic and mass polarization corrections are also calculated. Good agreement between theory and experiment for the ${2\mathrm{s}}^{2}$${\mathrm{}}^{1}$${\mathrm{S}}_{0}$-2s2p${\mathrm{}}^{1,3}$${\mathrm{P}}_{1}$ transition energies is found throughout the isoelectronic sequence.

Electronic Configurations of Superheavy Elements

We study electronic configurations of superheavy elements around one of islands of stabilities, from element 121 to 131, using the relativistic density functional theory with quantum electrodynamical corrections (Breit interaction) by MacDonald and Vosko. It is found that these corrections are important for studying electronic configurations of superheavy elements, and that g electrons in the atom first appear from element 126. We draw a table for the electronic configuration of elements including these superheavy elements.

Hyperfine-Induced 1s2s 1S0-1s2p 3P0 Transition and Fine-Structure Measurement in Heliumlike Nitrogen.

We have observed the 1{ital s}2{ital s}{sup 1}{ital S}{sub 0}-1{ital s}2{ital p}{sup 3}{ital P}{sub 0} transition in {sup 14}N{sup 5+} using Doppler-tuned fast-beam laser spectroscopy. This transition, absolutely forbidden for atoms with zero nuclear spin, occurs due to mixing of states of different {ital J} by the hyperfine interaction. By measuring the difference between the 2{sup 1}{ital S}{sub 0}-2{sup 3}{ital P}{sub 1,{ital F}} and 2{sup 1}{ital S}{sub 0}-2{sup 3}{ital P}{sub 0} transition energies, we have obtained a value for the 1{ital s}2{ital p}{sup 3}{ital P}{sub 1}-2{sup 3}{ital P}{sub 0} fine-structure splitting in N{sup 5+}. Our result will test quantum-electrodynamic corrections to recent high precision calculations [Z.-C. Yan and G.W.F. Drake, Phys. Rev. Lett. {bold 74}, 4791 (1995)]. {copyright} {ital 1996 The American Physical Society.}

Relativistic configuration-interaction calculations for the 2s-2p3/2 transition energies of uranium ions.

The 2s-2${\mathit{p}}_{3/2}$ transition energies in Li-like through F-like uraniums are calculated using the relativistic configuration-interaction method with B-spline basis functions. These calculations are based on the relativistic no-pair Hamiltonian, which includes both the Coulomb and retarded Breit interactions. Davidson's method [E. R. Davidson, J. Comput. Phys. 17, 87 (1975)] is employed to solve the eigenvalue problem for the lowest few eigenenergies and eigenfunctions. Quantum electrodynamic and mass polarization corrections are also calculated. Good agreement with experiment for all uranium ions is found. \textcopyright{} 1996 The American Physical Society.

Relativistic configuration-interaction calculations for the n=2 states of lithiumlike ions.

Term energies are determined for the {ital n}=2 states of lithiumlike ions with nuclear charges in the range 10{le}{ital Z}{le}92. These calculations are based on the relativistic no-pair Hamiltonian, which includes both the Coulomb and the retarded Breit interactions. Single-particle wave functions are expanded in a {ital B}-spline basis constructed from Dirac-Slater orbitals restricted to a finite cavity. The Hamiltonian matrix is evaluated and Davidson`s method is used to determine the lowest few eigenenergies and eigenfunctions for each angular symmetry. Quantum electrodynamic and mass polarization corrections are also calculated. Precise agreement between theory and experiment for the 2{ital s}-2{ital p} transition energies is found throughout the Periodic Table.

Full QED calculations of two-photon exchange for heliumlike systems: Analysis in the Coulomb and Feynman gauges.

A complete, numerical calculation of the effect of the exchange of two virtual photons between the electrons in the ground states of heliumlike systems is presented. Feynman diagrams with uncrossed and crossed photons are evaluated, using the Furry interaction picture [Phys. Rev. 81, 115 (1951)], neglecting nuclear recoil. The calculations are carried out in the Feynman gauge as well as in the Coulomb gauge, and the gauge invariance of this set of diagrams is verified with high numerical accuracy. The numerical technique employed is similar to that recently developed for our self-energy calculations. [H. Persson, I. Lindgren, and S. Salomonson, Phys. Scrip. T 46, 125 (1993); I. Lindgren, H. Persson, S. Salomonson, and A. Ynnerman, Phys. Rev. A 47, R4555 (1993)]. An explicit summation is performed over a complete set of intermediate states with positive and negative energy, generated with the technique of discretization. The photon propagators are expanded in spherical waves, and the radial integrations are performed numerically using analytical Bessel functions. Moreover, the integration over the photon energy is performed numerically. The calculations are performed for different values of the nuclear charge in the range Z=2--92. Corresponding calculations are also performed without retardation and neglecting the effect of negative-energy states (virtual electron-positron pairs), thus simulating relativsitic many-body calculations in the no-virtual-pair approximation. The difference, which in this way is obtained with high numerical accuracy, represents the ``quantum electrodynamics (QED) correction,'' which should be added to the many-body result in a combined QED--many-body procedure. The various contributions to the two-photon exchange have been analyzed in detail and compared with the analytical results to order (Z\ensuremath{\alpha}${)}^{3}$ of Sucher [Phys. Rev. 109, 1010 (1958)]. From our analysis, general conclusions can also be drawn concerning the accuracy of various relativistic many-body approaches.

Many-body calculation of the 2p1/2,3/2-2s1/2 transition energies in Li-like 238U.

We present a relativistic many-body calculation of the 2[ital s][sub 1/2] and 2[ital p][sub 1/2,3/2] ionization energies in Li-like [sup 238]U, based on the coupled-cluster singles and doubles approximation with the inclusion of the unretarded Breit interaction, using Dirac-Fock-Breit orbitals. A detailed comparison is made with a similar calculation using ordinary Dirac-Fock orbitals. The calculation yields the transition energies, and by comparison with the recently published experimental values we find the value for the remaining quantum-electrodynamical correction, which is compared with recent Lamb-shift calculations. The importance of an accurate nuclear model for evaluating the nuclear-size effect is also discussed.

Relativistic configuration-interaction calculations for the ground state and n=2 singlet states of heliumlike ions.

Term energies are determined for the ground state and n=2 singlet states of heliumlike ions with nuclear charges in the range 4\ensuremath{\le}Z\ensuremath{\le}92. These calculations are based on the relativistic no-pair Hamiltonian which includes both the Coulomb interaction and the retarded Breit interaction. Single-particle wave functions are expanded in a B-spline basis constructed from Dirac-Coulomb orbitals restricted to a finite cavity. The Hamiltonian matrix is evaluated and Davidson's method is used to determine the lowest few eigenenergies and eigenfunctions for each angular symmetry. Quantum electrodynamic corrections are also calculated. We find good, but not precise, agreement between theory and experiment for the K\ensuremath{\alpha} x-ray energies. It is argued that high-precision x-ray measurements for highly charged ions are needed to further test the theory.

Relativistic all-order many-body calculations of the n=1 and n=2 states of heliumlike ions.

An all-order procedure is used to determine the energies of the n=1 and n=2 states of heliumlike ions with nuclear charges in the range Z=3--100. The Coulomb energies and wave functions are obtained from a no-pair Hamiltonian by an iterative scheme that sums all orders of many-body perturbation theory. The Breit energy is determined by taking the expectation value of the instantaneous Breit Hamiltonian between the Coulomb wave functions. Corrections for the frequency dependence of the Breit interaction and for two Breit interactions are included. Quantum-electrodynamic and mass-polarization corrections are taken from Drake's unified method [Can. J. Phys. 66, 586 (1988)]. The resulting energies are complete through order (Z\ensuremath{\alpha}${)}^{3}$ a.u. and include new terms of order (Z\ensuremath{\alpha}${)}^{4}$ a.u. Comparisons are made with experiment and with recent configuration-interaction calculations.

Absolute term energy of core-excited doublet states of lithium.

Thirteen core-excited lithium doublet states of the $^{2}$P, $^{2}$${\mathit{P}}^{\mathit{o}}$, $^{2}$D, and $^{2}$${\mathit{D}}^{\mathit{o}}$ symmetries are calculated with a multiconfiguration-interaction wave function. A variation method is used for the bound states whereas the saddle-point and saddle-point complex-rotation methods are used for the autoionizing states. A restricted variation method is used for the saturated wave functions. Fine structures, relativistic corrections, and mass polarization corrections are calculated. The quantum electrodynamic corrections are also estimated. The absolute term energy is predicted with an uncertainty which is much smaller than that in the literature. Among these core-excited states, ten doublet-doublet transitions and one doublet-quartet transition have been measured. The largest error bar for the transition energy has been 6.2 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ whereas others range from 0.5 to 3.3 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. Our predicted transition wavelengths agree with the experiment for 10 of the 11 transitions. The only discrepancy is about 2 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. In the case where there is a disagreement between different experiments, our results show a clear preference in the comparison. The transition rates calculated in this work suggest that other spectral lines may also be observed in future experiments.