Quantum Electrodynamic Corrections Research Articles

Ionization potential for excitedSstates of the lithium atom

Nonrelativistic, relativistic, quantum electrodynamic, and finite nuclear mass corrections to the energy levels are obtained for the ${\mathit{nS}}_{1/2},n=3,\dots{},9$ states of the lithium atom. Computational approach is based on the explicitly correlated Hylleraas functions with the analytic integration and recursion relations. Theoretical predictions for the ionization potential of ${\mathit{nS}}_{1/2}$ states and transition energies ${\mathit{nS}}_{1/2}\ensuremath{\rightarrow}2{S}_{1/2}$ are compared to known experimental values for $^{6,7}\mathrm{Li}$ isotopes.

Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms

We present quantum electrodynamic (QED) calculations within the picture of bound-state QED for the frequency-dependent Breit interaction between electrons, the vacuum polarization, and the electron self-energy correction starting from the Dirac-Coulomb Hamiltonian for the ionization potentials of the group 1, 2, 11, 12, 13, and 18 elements of the periodic table, and down to the superheavy elements up to nuclear charge Z=120. The results for the s-block elements are in very good agreement with earlier studies by Labzowsky et al. [Phys. Rev. A 59, 2707 (1999)]. We discuss the influence of the variational versus perturbative treatment of the Breit interaction for valence-space ionization potentials. We argue that the lowest-order QED contributions become as important as the Breit interaction for ionization potentials out of the valence s shell.

Finite basis set approach to the two-centre Dirac problem in Cassini coordinates

We lay out a new approach to solving the two-centre Dirac eigenvalue problem. It is based on the application of B-spline basis sets constructed in Cassini coordinates. The approach provides a very promising way for the theoretical description of various atomic processes such as charge transfer, excitation, ionization and electron–positron pair production that accompany slow collisions of heavy ions. Moreover, yielding a (quasi-)complete set of eigensolutions, the finite basis set method allows a systematic analysis of the quantum electrodynamics (QED) corrections to the energy levels of heavy quasi-molecules formed in these collisions.

Proton radius, Darwin-Foldy term and radiative corrections

We discuss the role of the so-called Darwin-Foldy term in the evaluation of the proton and deuteron charge radii from atomic hydrogen spectroscopy and nuclear scattering data. The question of whether this term should be included or excluded from the nuclear radius has been controversially discussed in the literature. We attempt to clarify which literature values correspond to which conventions. A detailed discussion of the conventions appears useful because a recent experiment [R. Pohl et al., Nature 466, 213 (2010)] has indicated that there is a discrepancy between the proton charge radii inferred from ordinary (“electronic”) atomic hydrogen and muonic hydrogen. We also investigate the role of quantum electrodynamic radiative corrections in the determination of nuclear radii from scattering data, and propose a definition of the nuclear self energy which is compatible with the subtraction of the radiative corrections in scattering experiments.

Two-electron one-photon transitions in atoms with 12 ⩽ Z ⩽ 80

Relativistic configuration interaction results are presented for the Kαα x-rays arising from the decay of doubly ionized K shell of atoms with 12 ⩽ Z ⩽ 80. The x-ray energies and dipole rates are calculated in the active space approximation using multi-configuration Dirac–Fock wavefunctions with the inclusion of the Breit interaction and quantum electrodynamics corrections. The calculations include all single and double excitations from core-valence interactions. The resulting length and velocity gauge transition rates are well converged and are in good agreement with the allowed transition, while the intercombination line rate is seen to be gauge dependent. Conclusions concerning the relative importance of correlation and the Breit interaction on the intensity ratios are obtained in some detail. The sensitivity of the relative intensities of the fine structure lines to the coupling variations in both length and velocity gauges is analysed. The calculated results are compared with previous experimental and theoretical data.

Isotope shifts of the1s22s2(1S0)→1s22p2(1S0) transition in the doubly ionized carbon ionC2+

Highly accurate quantum mechanical calculations are performed for the $1{s}^{2}2{s}^{2}$ (${}^{1}{S}_{0}$) $\ensuremath{\rightarrow}$ $1{s}^{2}2{p}^{2}$ (${}^{1}{S}_{0}$) transition energy in the isotopomers of ${\mathrm{C}}^{2+}$ ion to determine the isotope shifts. Explicitly correlated Gaussian functions and a variational approach that explicitly includes the nuclear motion are employed in the calculations. The leading relativistic and quantum electrodynamics corrections to the transition energy are also calculated using the perturbation theory with the nonrelativistic wave function as the zero-order approximation. It is determined that the ${}^{12}{\mathrm{C}}^{2+}$ transitions energy, which is obtained from the calculations to be 182 519.031 cm${}^{\ensuremath{-}1}$ (vs the experimental value of 182 519.88 cm${}^{\ensuremath{-}1}$, an excellent sub-wave-number agreement) up-shifts by 1.755 cm${}^{\ensuremath{-}1}$ for ${}^{13}{\mathrm{C}}^{2+}$ and by additional 1.498 cm${}^{\ensuremath{-}1}$ for ${}^{14}{\mathrm{C}}^{2+}$. Those shifts are sufficiently large to be measured experimentally.

Theoretical study on mechanisms of anomalous fine structure in the magnesium isoelectronic sequence

Using the multi-configuration Dirac–Fock self-consistent field method and the relativistic configuration interaction method, the calculations are carried out for the fine structure (FS) energy levels of Mg I, Al II, Si III, S V excited states (3s3d 1,3D, 3s4d 1,3D, 3s4s 1,3S, 3s3p 1,3P and 3s4p 1,3P). Breit interactions and quantum electrodynamics corrections are added as corrections. For Mg and Mg-like ions studied here, electron correlations are treated adequately, intravalence electron correlations in particular. The calculation results are found to be in good agreement with the experimental data and compare favourably with other theoretical results. By means of precise computational methods, we elucidate four competitive mechanisms which affect the interesting FS splitting orderings in the magnesium isoelectronic sequence.

Isotope shifts of the three lowest S1 states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections

We present very accurate quantum mechanical calculations of the three lowest S-states [1s(2)2s(2)((1)S(0)), 1s(2)2p(2)((1)S(0)), and 1s(2)2s3s((1)S(0))] of the two stable isotopes of the boron ion, (10)B(+) and (11)B(+). At the nonrelativistic level the calculations have been performed with the Hamiltonian that explicitly includes the finite mass of the nucleus as it was obtained by a rigorous separation of the center-of-mass motion from the laboratory frame Hamiltonian. The spatial part of the nonrelativistic wave function for each state was expanded in terms of 10,000 all-electron explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians were variationally optimized using a procedure involving the analytical energy gradient determined with respect to the nonlinear parameters. The nonrelativistic wave functions of the three states were subsequently used to calculate the leading alpha(2) relativistic corrections (alpha is the fine structure constant; alpha=1/c, where c is the speed of light) and the alpha(3) quantum electrodynamics (QED) correction. We also estimated the alpha(4) QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement. The discrepancy is smaller than 0.4 cm(-1).

Theoretical Study of Interesting Fine-Structure Splittings Based on a Scenario for Precise Calculations

Based on the multi-configuration Dirac–Fock self-consistent field method, a scenario has been presented to calculate the fine-structure energy levels of C2+ and Si2+ excited states (31 D2 and 33 D1,2,3). The Breit interactions and quantum electrodynamics corrections are added as perturbations. The present calculation results are found to be in excellent agreement with the experimental data. By means of the precise calculation procedure, we elucidate that four competitive mechanisms influence the interesting fine-structure splittings in C2+ and Si2+, such as spin-orbit interactions, relativistic corrections of exchange interactions, the Breit interactions and electron correlation effects. Furthermore, the mechanism of relativistic correction of exchange interactions has been studied clearly. We elucidate that the inner shell 2p1/2,3/2 orbitals are essential to relativistic corrections of exchange interactions which are crucial for the final anomalous fine-structure splittings.

Correlation, relativistic, and quantum electrodynamics effects on the atomic structure of eka-thorium

Large-scale multiconfiguration Dirac-Fock calculations have been performed for the superheavy element eka-thorium, $Z=122$. The resulting atomic structure is compared with that obtained by various computational approaches involving different degrees of approximation in order to elucidate the role that correlation, relativistic, Breit, and quantum electrodynamics corrections play in determining the low-energy atomic spectrum. The accuracy of the calculations is assessed by comparing theoretical results obtained for thorium with available experimental data.

Isotope shift in the electron affinity of lithium

Very accurate electron affinity (EA) calculations of (6)Li and (7)Li (and (infinity)Li) have been performed using explicitly correlated Gaussian functions and a variational approach that explicitly includes the nuclear motion in the calculations (i.e., the approach that does not assume the Born-Oppenheimer approximation). The leading relativistic and quantum electrodynamics corrections to the electron affinities were also calculated. The results are the most accurate theoretical values obtained for the studied systems to date. Our best estimates of the (7)Li and (6)Li EAs are 4984.9842(30) and 4984.9015(30) cm(-1), respectively, and of the (7)Li/(6)Li EA isotope shift is 0.0827 cm(-1).

Theoretical Determination of the Dissociation Energy of Molecular Hydrogen.

The dissociation energy of molecular hydrogen is determined theoretically with a careful estimation of error bars by including nonadiabatic, relativistic, and quantum electrodynamics (QED) corrections. The relativistic and QED corrections were obtained at the adiabatic level of theory by including all contributions of the order α(2) and α(3) as well as the major (one-loop) α(4) term, where α is the fine-structure constant. The computed α(0), α(2), α(3), and α(4) components of the dissociation energy of the H2 isotopomer are 36 118.7978(2), -0.5319(3), -0.1948(2), and -0.0016(8) cm(-1), respectively, while their sum amounts to 36 118.0695(10) cm(-1), where the total uncertainty includes the estimated size (±0.0004 cm(-1)) of the neglected relativistic nonadiabatic/recoil corrections. The obtained theoretical value of the dissociation energy is in excellent agreement with the most recent experimental determination 36 118.0696(4) cm(-1) [J. Liu et al. J. Chem. Phys. 2009, 130, 174 306]. This agreement would have been impossible without inclusion of several subtle QED contributions which have not been considered, thus far, for molecules. A similarly good agreement is observed for the leading vibrational and rotational energy differences. For the D2 molecule we observe, however, a small disagreement between our value 36 748.3633(9) cm(-1) and the experimental result 36 748.343(10) cm(-1) obtained in a somewhat older and less precise experiment [Y. P. Zhang et al. Phys. Rev. Lett. 2004, 92, 203003]. The reason of this discrepancy is not known.

Five lowestS1states of the Be atom calculated with a finite-nuclear-mass approach and with relativistic and QED corrections

We have performed very accurate quantum mechanical calculations of the five lowest $S$ states of the beryllium atom. In the nonrelativistic part of the calculations we used the variational method and we explicitly included the nuclear motion in the Schr\odinger equation. The nonrelativistic wave functions of the five states were expanded in terms of explicitly correlated Gaussian functions. These wave functions were used to calculate the leading ${\ensuremath{\alpha}}^{2}$ relativistic correction ($\ensuremath{\alpha}$ is the fine structure constant) and the ${\ensuremath{\alpha}}^{3}$ quantum electrodynamics (QED) correction. We also estimated the ${\ensuremath{\alpha}}^{4}$ QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement.

Relativistic(Zα)2corrections and leading quantum electrodynamic corrections to the two-photon decay rate of ionic states

We calculate the relativistic corrections of relative order (Z alpha)^2$ to the two-photon decay rate of higher excited S and D states in ionic atomic systems, and we also evaluate the leading radiative corrections of relative order alpha (Z alpha)^2 ln[(Z alpha)^(-2)]. We thus complete the theory of the two-photon decay rates up to relative order alpha^3 ln(alpha). An approach inspired by nonrelativistic quantum electrodynamics is used. We find that the corrections of relative order (Z alpha)^2 to the two-photon decay are given by the zitterbewegung, the spin-orbit coupling and by relativistic corrections to the electron mass, and by quadrupole interactions. We show that all corrections are separately gauge-invariant with respect to a hybrid transformation from velocity to length gauge, where the gauge transformation of the wave function is neglected. The corrections are evaluated for the two-photon decay from 2S, 3S, 3D, and 4S states in one-electron (hydrogenlike) systems, with 1S and 2S final states.

Characteristic trends in the x-ray rates from the2s3pconfiguration of He-like ions

Relativistic configuration interaction results are presented for the $K\ensuremath{\beta}$ x rays from $2s3p\text{\ensuremath{-}}1s2s$ transitions in He-like ions. The x-ray energies and rates are calculated for 23 ions in the range $Z=14$ to 54 using multiconfiguration Dirac-Fock wave functions in the active space approximation. The calculations include finite nuclear size effect, Breit interaction, and quantum electrodynamic corrections. The influence of configuration mixing on the converged x-ray rates is analyzed. Our calculations show that strong configuration mixing reduces substantially the single configuration electric dipole allowed as well the ${^{3}\text{P}}_{1}\text{\ensuremath{-}}{^{1}\text{S}}_{0}$ spin-forbidden transition rates and its contribution decreases with increasing $Z$. While configuration mixing consistently reduces the monoconfiguration rate of ${^{1}\text{P}}_{1}\text{\ensuremath{-}}{^{3}\text{S}}_{1}$ transition in ions with $Z<22$ by as much as 94%, it enhances the spin-forbidden rate of ions with $Z\ensuremath{\ge}23$. The contributions from higher order relativistic corrections are seen to be negligible for ${^{1}\text{P}}_{1}\text{\ensuremath{-}}{^{1}\text{S}}_{0}$ and ${^{1}\text{P}}_{1}\text{\ensuremath{-}}{^{3}\text{S}}_{1}$ transitions whereas the effect of Breit interaction is appreciable for the other transitions to ${^{3}\text{S}}_{1}$ state, especially for the high-$Z$ ions. The results are compared with other available values.

Classical and quantum radiation reaction in conformally flat spacetime

We investigate the physics of a charged scalar particle moving in conformally flat spacetime with the conformal factor depending only on time in the framework of quantum electrodynamics (QED). In particular, we show that the radiation-reaction force derived from QED agrees with the classical counterpart in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ using the fact that to lowest order in $\ensuremath{\hbar}$ the charged scalar field theory with mass $m$ in conformally flat spacetime with conformal factor $\ensuremath{\Omega}(t)$, which we call Model B, is equivalent to that in flat spacetime with a time-dependent mass $m\ensuremath{\Omega}(t)$, which we call Model A, at tree level in this limit. We also consider the one-loop QED corrections to these two models in the semiclassical approximation. We find nonzero one-loop corrections to the mass and Maxwell's equations in Model A at order ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. This does not mean, however, that the corresponding one-loop corrections in Model B are nonzero because the equivalence of these models through a conformal transformation breaks down at one loop. We find that the one-loop corrections vanish in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ in Model B.

J Dependence of Isotope Shifts at High-Lying Levels of Gd I

In the past decades, gadolinium has been the subject of much interest to spectroscopists as it possesses rich optical transitions with various atomic configurations associated with 4f, 5d, 6s, and 6p open electrons. The higher-order effect of the isotope shift (IS), i.e., the crossed-second-order (CSO) effect, has been an interesting subject, which results in the J and term dependences of ISs. From the point view of nuclear laser spectroscopy, understanding of such a CSO effect is indispensable in deriving nuclear information from measured ISs. On the other hand, recent theoretical calculations of ISs using the configuration interaction method and many-body perturbation theory have reached high precision even for heavy elements with a few valence electrons such as Mg, Na, K, and Ca. High-order effects such as the quantum electrodynamic correction and the nuclear polarizability have also been calculated very recently for Li and He. The CSO effects in the ground configurations of Gd I 4f5d6s and Gd II 4f5d6s were studied early and those in the 4f5d6s and 4f6s6p configurations of Gd I with lower excited energies were reported later. The J dependences of ISs in the D and F terms of 4f5d6s6p at energies of about 18000 cm 1 were also measured. These experiments were performed in the visible and nearinfared regions. Recently, IS measurements at wavelengths of about 405 nm have been reported. However, the J dependences of ISs at high-lying levels such as at energies of about 26000 cm 1 have not yet been reported. Such J dependences, particularly for high-lying levels with a possibly strong configuration mixing, will provide a challenge for atomic theoretical calculation. In this paper, we report the high-resolution atomic-beam laser spectroscopy of Gd I in the ultraviolet (UV) region. ISs are measured for seven UV transitions including the previously reported three transitions. The J dependences of ISs in the three atomic configurations, related to high-lying levels, are obtained and discussed. The present experiment was performed using a UV laser beam and an atomic beam. Using a cw frequency doubler (Spectra-Physics WAVETRAIN), a UV laser beam with a wavelength of about 394 nm was obtained by the frequency doubling of a diode-laser beam produced from a commercial tunable diode laser (Newport 2010M). An atomic beam was produced by heating a molybdenum oven using an electronbombardment method and was made to intersect with a laser beam perpendicularly. Fluorescence from the atomic beam was detected with a cooled photon-counting photomultiplier (Hamamatsu R2257P). A confocal Fabry–Perot interferometer (FPI) with a free spectral range of 300MHz was used for relative frequency calibration. The experimental setup is essentially identical to that used in our previous work. Seven transitions in Gd I were studied in this experiment. Figure 1 shows the wavelengths of studied transitions together with atomic configurations, terms and total electronic angular momentums J of their lower and upper levels. These transitions are all from the ground D term of the 4f5d6s configuration, and their upper levels, with energies of about 26000 cm , relate to three configurations of 4f5d6s6p, 4f5d6p and 4f5d6s with four different terms of D, G, G, and G. Figure 2 shows the observed fluorescence spectrum of the 4f 5d6s D3–4f 5d6p G4 transition at 394.263 nm. It can be seen from Fig. 2 that all peaks are clearly observed for even-mass isotopes, including the lowest abundance (0.20%) isotope Gd. For the 395.868, 394.554, 395.337, and 394.180 nm transitions, no peaks of Gd could be observed owing to their weak transition intensities. The full width at half maximum (FWHM) of the peaks is about 23MHz, which is mainly due to the natural width of the upper level of transition and the residual Doppler broadening of the atomic beam. For measured spectra, peak centers were determined from a least-squares fit with a Lorentz function and calibrated with the FPI spectrum. For each transition, measurement was performed about 20 times. Thus, the ISs between evenmass isotopes were obtained and are presented in Table I for the seven transitions studied; the ISs of three transitions were reported in our previous paper. The uncertainties of the measured ISs, 1–6MHz, include the error of peak-center determination, the error of the free spectral range of the FPI (0.046MHz), and the error of linearity correction for frequency scanning. Hyperfine structures of the odd-mass isotopes Gd and Gd are discussed elsewhere. The IS difference between different transitions with identical lower levels yields the IS difference between different upper levels, i.e., the residual IS Tres. For example, the IS difference between the 394.324 and 394.557 nm transitions yields the IS difference between the upper D1 D2 D3 D5 D4 D6 D3 D2 D1 G4 G6 G4 G6

Quantum-electrodynamic effects for atoms in a laser field

Quantum-electrodynamic radiation corrections to atomic energy levels in the presence of monochromatic laser field are considered as radiation shifts of quasienergies. General expressions are obtained for self-energy part of radiation shifts for an arbitrary multilevel atomic system. As an appendix, the radiation shifts of spectral lines of atomic resonance transitions are investigated.

Effect of Electron Correlations and Breit Interactions on Ground-State Fine-Structures along the Nitrogen-Like Isoelectronic Sequence

The accurate atomic data of nitrogen and nitrogen-like ions have an importance role in fusion plasma studies and astrophysics studies. The precise calculation of fine-structures is required to obtain such atomic data. Along the whole nitrogen isoelectronic sequence, the contributions of the electron correlations, the Breit interactions and the quantum electrodynamics corrections on the ground-state fine-structures are elucidated. When Z is low, the electron correlations are important, and the Breit interactions, which cannot be neglected cause interesting anomalous fine-structure splittings. When Z is high, the electron correlations are less important, and the Breit interactions are important in addition to spin-orbit interactions for precise calculations.

High-Precision Calculations of the Hyperfine Constants for the Low-Lying Excited 2S States of Be+

High-precision Hylleraas-type calculations of the hyperfine constants for the four lowest excited 3 (2)S, 4 (2)S, 5 (2)S, and 6 (2)S states of the (9)Be(+) ion are reported. Small adjustments to the hyperfine constants arising from effects that include the finite nuclear mass, the magnetization density distribution over the nucleus, the Breit-Rosenthal correction for finite nuclear size, and lowest-order relativistic and quantum electrodynamic corrections are considered. The final values obtained for the hyperfine constants for the n (2)S states were: -158.78, -62.43, -30.66, and -17.29 MHz for n = 3, 4, 5, and 6, respectively.