# Publikationen von

Dr. Andrey Volotka

Alle Publikationen des HI Jena

## 2022

**QED approach to valence-hole excitation in closed-shell systems**

**106**, 012802 (2022)

**Abstract:** An ab initio QED approach to treat a valence-hole excitation in closed-shell systems is developed in the framework of the two-time Greens-function method. The derivation considers a redefinition of the vacuum state and its excitation as a valence-hole pair. The proper two-time Greens function, whose spectral representation confirms the poles at valence-hole excitation energies, is proposed. An contour integral formula which connects the energy corrections and the Greens function is also presented. First-order corrections to the valence-hole excitation energy involving self-energy, vacuum polarization, and one-photon-exchange terms are explicitly derived in the redefined vacuum picture. Reduction to the usual vacuum electron propagators is shown, which agrees in the Breit approximation with the many-body perturbation theory expressions for the valence-hole excitation energy.

**Measurement of the bound-electron g-factor difference in coupled ions**

**606**, 479 (2022)

**Abstract:** Quantum electrodynamics (QED) is one of the most fundamental theories of physics and has been shown to be in excellent agreement with experimental results(1-5). In particular, measurements of the electron\textquotesingle s magnetic moment (orgfactor) of highly charged ions in Penning traps provide a stringent probe for QED, which allows testing of the standard model in the strongest electromagnetic fields(6). When studying the differences between isotopes, many common QED contributions cancel owing to the identical electron configuration, making it possible to resolve the intricate effects stemming from the nuclear differences. Experimentally, however, this quickly becomes limited, particularly by the precision of the ion masses or the magnetic field stability(7). Here we report on a measurement technique that overcomes these limitations by co-trapping two highly charged ions and measuring the difference in their g factors directly. We apply a dual Ramsey-type measurement scheme with the ions locked on a common magnetron orbit(8), separated by only a few hundred micrometres, to coherently extract the spin precession frequency difference. We have measured the isotopic shift of the bound-electrongfactor of the isotopes Ne-20(9+) and Ne-22(9+) to 0.56-parts-per-trillion (5.6 x 10(-13)) precision relative to their g factors, an improvement of about two orders of magnitude compared with state-of-the-art techniques(7). This resolves the QED contribution to the nuclear recoil, accurately validates the corresponding theory and offers an alternative approach to set constraints on new physics.

**Polarization of Photons Scattered by Ultra-Relativistic Ion Beams**

**534**, 2100252 (2022)

**Abstract:** A theoretical investigation of the elastic resonant scattering of photons by ultra-relativistic and partially stripped ions is presented. Particular attention in the study is given to the angular distribution and polarization of scattered photons as \textasciigrave \textasciigrave seen'' in both the ion-rest and laboratory reference frames. In order to evaluate these angular and polarization properties, the irreducible polarization tensor approach is combined with the density matrix theory. If, furthermore, the ion-photon coupling is treated within the electric dipole approximation, this framework enables one to obtain simple analytical expressions for both the emission pattern and the polarization Stokes parameters of the outgoing radiation. These (analytical) expressions for the nS0 -> n \textasciigrave P1 -> nS0\textbackslash \textdollar n S\_0 \textbackslash rightarrow n<\^>\textbackslash \textbackslash prime \textbackslash P\_1 \textbackslash rightarrow n S\_0\textbackslash \textdollar , nS1/2 -> n \textasciigrave P1/2 -> nS1/2\textbackslash \textdollar n S\_\textbackslash 1/2\textbackslash \textbackslash rightarrow n<\^>\textbackslash \textbackslash prime \textbackslash P\_\textbackslash 1/2\textbackslash \textbackslash rightarrow n S\_\textbackslash 1/2\textbackslash \textbackslash \textdollar , and nS1/2 -> n \textasciigrave P3/2 -> nS1/2\textbackslash \textdollar n S\_\textbackslash 1/2\textbackslash \textbackslash rightarrow n<\^>\textbackslash \textbackslash prime \textbackslash P\_\textbackslash 3/2\textbackslash \textbackslash rightarrow n S\_\textbackslash 1/2\textbackslash \textbackslash \textdollar transitions are displayed and analyzed , that are of interest for the Gamma Factory project and whose realization is currently under discussion at CERN. Based on the performed analysis, it is demonstrated that the resonantly scattered photons can be strongly (linearly or circularly) polarized, and that this polarization can be well controlled by adjusting either the emission angle and/or the polarization state of the incident radiation. Moreover, the potential of the photon scattering for measuring the spin-polarization of ion beams is also discussed in detail.

**g Factor of Lithiumlike Silicon and Calcium: Resolving the Disagreement between Theory and Experiment**

**128**, 103001 (2022)

**Abstract:** The bound-electron g factor is a stringent tool for tests of the standard model and the search for new physics. The comparison between an experiment on the g factor of lithiumlike silicon and the two recent theoretical values revealed the discrepancies of 1.7 sigma [Glazov et al. Phys. Rev. Lett. 123, 173001 (2019)] and 5.2 sigma [Yerokhin et al. Phys. Rev. A 102, 022815 (2020)]. To identify the reason for this disagreement, we accomplish large-scale high-precision computation of the interelectronic-interaction and many-electron QED corrections. The calculations are performed within the extended Furry picture of QED, and the dependence of the final values on the choice of the binding potential is carefully analyzed. As a result, we significantly improve the agreement between the theory and experiment for the g factor of lithiumlike silicon. We also report the most accurate theoretical prediction to date for lithiumlike calcium, which perfectly agrees with the experimental value.

**Relativistic calculations of two-color two-photon K-shell ionization**

**76**, 18 (2022)

**Abstract:** We investigate the two-color two-photon K-shell ionization of neutral atoms based on the relativistic second-order perturbation theory and independent particle approximation. Analytical expressions for the relativistic and nonrelativistic total cross sections are derived in terms of radial transition amplitudes and Stokes parameters. Particular attention is paid especially to how the two-photon ionization total cross section depends on the energy sharing and polarization of the two incident photons. We construct the nonrelativistic expressions of cross section ratios for different polarization combinations of the two incident photons. The numerical results of total cross section and cross section ratios show that the energy sharing of the two incident photons plays an essential role in two-photon K-shell ionization. Particularly, if the energies of the two incident photons are identical, the total cross section and cross section ratios will reach the minimum or maximum value. Moreover, due to the strong screening effects, we find strong deviations of the cross section ratios near the two-photon ionization threshold of the Ne atom.

## 2021

**Possible Polarization Measurements in Elastic Scattering at the Gamma Factory Utilizing a 2D Sensitive Strip Detector as Dedicated Compton Polarimeter**

**2100285**, 2100285 (2021)

**Steering of circular dichroism in biharmonic ionization of atoms**

**104**, L031103 (2021)

**Elliptical dichroism in biharmonic ionization of atoms**

**104**, 013102 (2021)

**Abstract:** In multiphoton ionization of atoms, elliptical dichroism may arise in the photoelectron angular distributions due to the interference of the possible ionization pathways. We here consider the interaction of atoms with an elliptically polarized biharmonic $(\omega + 2\omega)$ field which simultaneously allows one- and two-photon ionization of the atoms. The interference between these two ionization pathways introduces contributions to the elliptical dichroism in addition to the dichroism that arises from the two-photon ionization alone. We show that these additional dichroism contributions can lead to a stronger dichroism in comparison to the one arising from two-photon ionization only. We present a relativistic analysis of the corresponding photoelectron angular distributions and discuss individual contributions to the dichroic phenomena. Detailed computations have been performed for biharmonic ionization of neutral helium atoms.

**Many-Electron QED with Redefined Vacuum Approach**

**13**, 1014 (2021)

**Abstract:** The redefined vacuum approach, which is frequently employed in the many-body perturbation theory, proved to be a powerful tool for formula derivation. Here, we elaborate this approach within the bound-state QED perturbation theory. In addition to general formulation, we consider the particular example of a single particle (electron or vacancy) excitation with respect to the redefined vacuum. Starting with simple one-electron QED diagrams, we deduce first- and second-order many-electron contributions: screened self-energy, screened vacuum polarization, one-photon exchange, and two-photon exchange. The redefined vacuum approach provides a straightforward and streamlined derivation and facilitates its application to any electronic configuration. Moreover, based on the gauge invariance of the one-electron diagrams, we can identify various gauge-invariant subsets within derived many-electron QED contributions.

**Redefined vacuum approach and gauge-invariant subsets in two-photon-exchange diagrams for a closed-shell system with a valence electron**

**103**, 042818 (2021)

**Abstract:** CThe two-photon-exchange diagrams for atoms with single valence electrons are investigated. Calculation formulas are derived for an arbitrary state within the rigorous bound-state QED framework utilizing the redefined vacuum formalism. In contrast to other methods, the redefined vacuum approach enables the identification of eight gauge-invariant subsets and, thus, efficiently checks the consistency of the obtained results. The gauge invariance of found subsets is demonstrated both analytically (for an arbitrary state) as well as numerically for 2s, 2p(1/2), and 2p(3/2) valence electrons in Li-like ions. Identifying gauge-invariant subsets in the framework of the proposed approach opens a way to tackle more complex diagrams, e.g., three-photon exchange, where the fragmentation on simpler subsets is crucial for its successful calculation.

**Polarization studies on Rayleigh scattering of hard x rays by closed-shell atoms**

**103**, 012801 (2021)

**Abstract:** We present a theoretical study on the elastic Rayleigh scattering of x-ray photons by closed-shell atoms. Special attention is paid to the transfer of linear polarization from the incident to the outgoing photons. To study this process, we apply the density-matrix formalism combined with the relativistic perturbation theory. This formalism enables us to find general relations between the Stokes parameters of the incident and scattered photons. By using these expressions, we revisit the recent proposal to use Rayleigh scattering for the analysis of the polarization purity of synchrotron radiation. We show that this analysis can be performed without any need for the theoretically calculated scattering amplitudes, if the linear polarization of the scattered light is measured simultaneously at the azimuthal angles 0 degrees and 45 degrees with respect to the plane of the synchrotron. To illustrate our approach, we present detailed calculations for scattering of 145 keV photons by lead atoms.

## 2020

**Elastic photon scattering on hydrogenic atoms near resonances**

**8**, 12 (2020)

**Abstract:** Scattering of light on relativistic heavy ion beams is widely used for characterizing and tuning the properties of both the light and the ion beam. Its elastic component-Rayleigh scattering-is investigated in this work for photon energies close to certain electronic transitions because of its potential usage in the Gamma Factory initiative at CERN. The angle-differential cross-section, as well as the degree of polarization of the scattered light are investigated for the cases of 1s - 2p1/2 and 1s - 2p3/2 resonance transitions in H-like lead ions. In order to gauge the validity and uncertainty of frequently used approximations, we compare different methods. In particular, rigorous quantum electrodynamics calculations are compared with the resonant electric-dipole approximation evaluated within the relativistic and nonrelativistic formalisms. For better understanding of the origin of the approximation, the commonly used theoretical approach is explained here in detail. We find that in most cases, the nonrelativistic resonant electric-dipole approximation fails to describe the properties of the scattered light. At the same time, its relativistic variant agrees with the rigorous treatment within a level of 10% to 20%. These findings are essential for the design of an experimental setup exploiting the scattering process, as well as for the determination of the scattered light properties.

**Nonlinear Cooper minimum as a precise tool for understanding multiphoton photoionization**

**1412**, 152017 (2020)

**Abstract:** A new approach to accurately assess multiphoton ionization is suggested. Vanishing of the dominant ionization channel in nonresonant (direct) multiphoton ionization is predicted for a specific incident photon energy. The exact energy position of such nonlinear Cooper minimum can be accurately measured and requires calculations of the complete electronic spectrum. Measurements of various observables at these photon energies are desirable for further evaluation of theoretical calculations at hitherto unreachable accuracy.

**Direct measurements for the fine-structure splitting of S VIII and Cl IX**

**102**, 042817 (2020)

**Enhanced polarization transfer to the characteristic L alpha x-ray lines near the nonlinear Cooper minimum of two-photon ionization**

**102**, 042807 (2020)

**Rayleigh scattering of linearly polarized light: Scenario of the complete experiment**

**102**, 042814 (2020)

**Vacuum polarization and finite-nuclear-size effects in the two-photon decay of hydrogenlike ions**

**102**, 042811 (2020)

**Photoelectron Angular Distributions of Nonresonant Two-Photon Atomic Ionization Near Nonlinear Cooper Minima**

**8**, 54 (2020)

**Abstract:** Photoelectron angular distributions of the two-photon ionization of neutral atoms are theoretically investigated. Numerical calculations of two-photon ionization cross sections and asymmetry parameters are carried out within the independent-particle approximation and relativistic second-order perturbation theory. The dependence of the asymmetry parameters on the polarization and energy of the incident light as well as on the angular momentum properties of the ionized electron are investigated. While dynamic variations of the angular distributions at photon energies near intermediate level resonances are expected, we demonstrate that equally strong variations occur near the nonlinear Cooper minimum. The described phenomena is demonstrated on the example of two-photon ionization of magnesium atom.

**Measurement of the P1/2 2 - P3/2 2 fine-structure splitting in fluorinelike Kr, W, Re, Os, and Ir**

**102**, 032803 (2020)

**Abstract:** Quantum electrodynamics (QED) is currently considered to be one of the most accurate theories of fundamental interactions. As its extraordinary precision offers unique scientific opportunities, e.g., search for new physics, stringent experimental tests of QED continue to be of high importance. To this end, highly charged ions represent an exceptional test-bed due to enhanced QED effects. Recently, forbidden transitions in F-like ions have been analyzed to few ppm precision, resolving previous discrepancies between theory and experiment. Here we further test the accuracy of QED calculations with three new (Re, Os, Ir), and two improved (Kr, W) measurements of the P1/22-P3/22 transition energy in F-like ions using the NIST electron-beam ion trap and extreme-ultraviolet and x-ray spectrometers. Good agreement between theoretical and experimental energies is found for all considered elements.

**Many-electron effects in the hyperfine splitting of lithiumlike ions**

**2**, 013364 (2020)

**Abstract:** The rigorous QED evaluation of the one- and two-photon exchange corrections to the ground-state hyperfine splitting in Li-like ions is presented for the wide range of nuclear charge number Z = 7-82. The calculations are carried out in the framework of the extended Furry picture, i.e., with the inclusion of the effective local screening potential in the zeroth-order approximation. The interelectronic-interaction contributions of the third and higher orders are taken into account in the framework of the Breit approximation employing the recursive perturbation theory. In comparison to the previous theoretical calculations, the accuracy of the interelectronic-interaction contributions to the ground-state hyperfine splitting in Li-like ions is substantially improved.

**Breakdown of the electric dipole approximation at Cooper minima in direct two-photon ionisation**

**10**, 3617 (2020)

**Abstract:** We predict breakdown of the electric dipole approximation at nonlinear Cooper minimum in direct two-photon K–shell atomic ionisation by circularly polarised light. According to predictions based on the electric dipole approximation, we expect that tuning the incident photon energy to the Cooper minimum in two-photon ionisation results in pure depletion of one spin projection of the initially bound 1s electrons, and hence, leaves the ionised atom in a fully oriented state. We show that by inclusion of electric quadrupole interaction, dramatic drop of orientation purity is obtained. The low degree of the remaining ion orientation provides a direct access to contributions of the electron-photon interaction beyond the electric dipole approximation in the two-photon ionisation of atoms and molecules. The orientation of the photoions can be experimentally detected either directly by a Stern-Gerlach analyzer, or by means of subsequent Kα fluorescence emission, which has the information about the ion orientation imprinted in the polarisation of the emitted photons.

**g factor of the [(1s)2(2s)22p]2P3/2 state of middle-Z boronlike ions**

**49**, 143 (2020)

**Abstract:** Theoretical g-factor calculations for the first excited 2P3/2 state of boronlike ions in the range Z=10–20 are presented and compared with the previously published values. The first-order interelectronic-interaction contribution is evaluated within the rigorous quantum electrodynamics (QED) approach in the effective screening potential. The second-order contribution is considered within the Breit approximation. The QED and nuclear recoil corrections are also taken into account.

**Lifetime measurements of ultrashort-lived excited states in Be-like ions**

**49**, 165 (2020)

**Abstract:** We propose to measure the lifetime of short-lived excited states in highly charged ions by pump-probe experiments. Utilizing two synchronized and delayed Femtosecond pulses allows accessing these lifetimes with Femtosecond precision. Such measurements could provide sensitive tests of state-of-the art atomic structure calculations beyond the capabilities of established methods.

## 2019

**g Factor of Lithiumlike Silicon: New Challenge to Bound-State QED**

**123**, 173001 (2019)

**Abstract:** The recently established agreement between experiment and theory for the g factors of lithiumlike silicon and calcium ions manifests the most stringent test of the many-electron bound-state quantum electrodynamics (QED) effects in the presence of a magnetic field. In this Letter, we present a significant simultaneous improvement of both theoretical gth=2.000 889 894 4 (34) and experimental gexp=2.000 889 888 45 (14) values of the g factor of lithiumlike silicon 28Si11+. The theoretical precision now is limited by the many-electron two-loop contributions of the bound-state QED. The experimental value is accurate enough to test these contributions on a few percent level.

**Ab initio QED Treatment of the Two-Photon Annihilation of Positrons with Bound Electrons**

**123**, 093401 (2019)

**Abstract:** The process of a positron—bound-electron annihilation with simultaneous emission of two photons is investigated theoretically. A fully relativistic formalism based on an ab initio QED description of the process is worked out. The developed approach is applied to evaluate the annihilation of a positron with K-shell electrons of a silver atom, for which a strong contradiction between theory and experiment was previously stated. The results obtained here resolve this longstanding disagreement and, moreover, demonstrate a sizable difference with approaches so far used for calculations of the positron—bound-electron annihilation process, namely, Lee’s and the impulse approximations.

**Fluorescence polarization as a precise tool for understanding nonsequential many-photon ionization**

**100**, 011401 (2019)

**Abstract:** Nonsequential two-photon ionization of inner-shell np subshell of neutral atoms by circularly polarized light is investigated. Detection of subsequent fluorescence as a signature of the process is proposed and the dependence of fluorescence degree of polarization on incident photon beam energy is studied. It is generally expected that the degree of polarization remains approximately constant, except when the beam energy is tuned to an intermediate n′ resonance. However, strong unexpected change in the polarization degree is discovered for nonsequential two-photon ionization at specific incident beam energy due to a zero contribution of the otherwise dominant ionization channel. Polarization degree of the fluorescence depends less on the beam parameters, and its measurements at this specific beam energy, whose position is very sensitive to the details of the employed theory, are highly desirable for evaluation of theoretical calculations of nonlinear ionization at hitherto unreachable accuracy.

**QED radiative corrections to the ²P₁/₂-²P₃/₂ fine structure in fluorinelike ions**

**100**, 010502 (2019)

**Abstract:** calculations of QED radiative corrections to the 2P1/2 - 2P3/2 fine-structure transition energy are performed for selected F-like ions. These calculations are nonperturbative in αZ and include all first-order and many-electron second-order effects in α. When compared to approximate QED computations, a notable discrepancy is found especially for F-like uranium for which the predicted self-energy contributions even differ in sign. Moreover, all deviations between theory and experiment for the 2P1/2 - 2P3/2 fine-structure energies of F-like ions, reported recently by Li et al., Phys. Rev. A 98, 020502(R) (2018), are resolved if their highly accurate, non-QED fine-structure values are combined with the QED corrections ab initially evaluated here.

**Ground-state hyperfine splitting of B-like ions in the high-Z region**

**99**, 062503 (2019)

**Abstract:** The hyperfine splitting of the ground state of selected B-like ions within the range of nuclear charge numbers Z=49–83 is investigated in detail. The rigorous QED approach together with the large-scale configuration-interaction Dirac-Fock-Sturm method are employed for the evaluation of the interelectronic-interaction contributions of first and higher orders in 1/Z. The screened QED corrections are evaluated to all orders in αZ by using an effective potential. The influence of nuclear magnetization distribution is taken into account within the single-particle nuclear model.

**High-precision calculations of the 1s²2s2p ¹P₁->1s²2s² ¹S₀ spin-allowed E1 transition in C iii**

**99**, 062511 (2019)

**Abstract:** Large-scale relativistic calculations are performed for the transition energy and line strength of the 1s22s2p 1P1− 1s22s2 1S0 transition in Be-like carbon. Based on the multiconfiguration Dirac-Hartree-Fock~(MCDHF) approach, different correlation models are developed to account for all major electron-electron correlation contributions. These correlation models are tested with various sets of the initial and the final state wave functions. The uncertainty of the predicted line strength due to missing correlation effects is estimated from the differences between the results obtained with those models. The finite nuclear mass effect is accurately calculated taking into account the energy, wave functions as well as operator contributions. As a result, a reliable theoretical benchmark of the E1 line strength is provided to support high precision lifetime measurement of the 1s22s2p 1P1 state in Be-like carbon.

**g Factor of Boronlike Argon ⁴⁰Ar¹³⁺**

**122**, 253001 (2019)

**Abstract:** We have measured the ground-state g factor of boronlike argon 40Ar13+ with a fractional uncertainty of 1.4×10−9 with a single ion in the newly developed Alphatrap double Penning-trap setup. The value of g=0.663 648 455 32(93) obtained here is in agreement with our theoretical prediction of 0.663 648 12(58). The latter is obtained accounting for quantum electrodynamics, electron correlation, and nuclear effects within the state-of-the-art theoretical methods. Our experimental result distinguishes between existing predictions that are in disagreement, and lays the foundations for an independent determination of the fine-structure constant.

**The hyperfine puzzle of strong-field bound-state QED**

**240**, 51 (2019)

**Abstract:** The hyperfine splitting in heavy highly charged ions provide the means to test QED in extremely strong magnetic fields. In order to provide a meaningful test, the splitting has to be measured in H-like and Li-like ions to remove uncertainties from nuclear structure. This has been achieved at the experimental storage ring ESR but a discrepancy to the theoretical prediction of more than 7s was observed. We report on these measurements as well as on NMR measurements that were performed to solve this issue.

**Excitation of the ²²⁹Th nucleus via a two-photon electronic transition**

**99**, 042517 (2019)

**Abstract:** We investigate the process of nuclear excitation via a two-photon electron transition (NETP) for the case of the doubly charged thorium. The theory of the NETP process was originally devised for heavy-helium-like ions. In this work, we study this process in the nuclear clock isotope 229Th in the 2+ charge state. For this purpose we employ a combination of configuration interaction and many-body perturbation theory to calculate the probability of NETP in resonance approximation. The experimental scenario we propose for the excitation of the low-lying isomeric state in 229Th is a circular process starting with a two-step pumping stage followed by NETP. The ideal intermediate steps in this process depends on the supposed energy ℏωN of the nuclear isomeric state. For each of these energies, the best initial state for NETP is calculated. Special focus is put on the most recent experimental results for ℏωN.

## 2018

**Ground-state g factor of middle-Z boronlike ions**

**1138**, 012003 (2018)

**Abstract:** Theoretical calculations of the interelectronic-interaction and QED corrections to the g factor of the ground state of boronlike ions are presented. The first-order interelectronic-interaction and the self-energy corrections are evaluated within the rigorous QED approach in the effective screening potential. The second-order interelectronic interaction is considered within the Breit approximation. The nuclear recoil effect is also taken into account. The results for the ground-state g factor of boronlike ions in the range Z = 10-20 are presented and compared to the previous calculations.

**Diagnostics of polarization purity of x rays by means of Rayleigh scattering**

**98**, 053403 (2018)

**Abstract:** Synchrotron radiation is commonly known to be completely linearly polarized when observed in the orbital plane of the synchrotron motion. Under actual experimental conditions, however, the degree of polarization of the synchrotron radiation may be lower than the ideal 100%. We demonstrate that even tiny impurities of polarization of the incident radiation can drastically affect the polarization of the elastically scattered light. We propose to use this effect as a precision tool for the diagnostics of the polarization purity of the synchrotron radiation. Two variants of the diagnostics method are proposed. The first one is based on the polarization measurements of the scattered radiation and relies on theoretical calculations of the transition amplitudes. The second one involves simultaneous measurements of the polarization and the cross sections of the scattered radiation and is independent of theoretical amplitudes.

**Testing atomic wave functions in the nuclear vicinity: The hyperfine structure with empirically deduced nuclear and quantum electrodynamic effects**

**98**, 032504 (2018)

**Abstract:** Calculations of the magnetic hyperfine structure rely on the input of nuclear properties—nuclear magnetic moments and nuclear magnetization distributions—as well as quantum electrodynamic radiative corrections for high-accuracy evaluation in heavy atoms. The uncertainties associated with assumed values of these properties limit the accuracy of hyperfine calculations. For example, for the heavy alkali-metal atoms Cs and Fr, these uncertainties may amount collectively to almost 1% or 2%, respectively. In this paper, we propose a method for removing the dependence of hyperfine structure calculations on assumed values of nuclear magnetic moments and nuclear magnetization distributions by determining these effects empirically from measurements of the hyperfine structure for high states. The method is valid for s, p1/2, and p3/2 states of alkali-metal atoms and alkali-metal-like ions. We have shown that for s states, the dependence on QED effects may also be removed to high accuracy. The ability to probe the electronic wave functions, through hyperfine comparisons, with significantly increased accuracy is important for the analysis of atomic parity violation measurements, and it may enable the accuracy of atomic parity violation calculations to be improved. More broadly, it paves the way for further development of high-precision atomic many-body methods.

**Hyperfine interaction with the ²²⁹Th nucleus and its low-lying isomeric state**

**98**, 020503 (2018)

**Abstract:** The thorium nucleus with a mass number A=229 has attracted much interest because its extremely low-lying first excited isomeric state at about 8 eV opens the possibility for the development of a nuclear clock. Both the energy of this state as well as the nuclear magnetic dipole and electric quadrupole moment of the 229mTh isomer are subjects of intense research. The latter can be determined by investigating the hyperfine structure of thorium atoms or ions. Due to its electronic structure and the long lifetime of the nuclear isomeric state, Th2+ is especially suitable for such kinds of studies. In this Rapid Communication, we present a combined experimental and theoretical investigation of the hyperfine structure of the 229Th^(2+) ion in the nuclear ground state, where a good agreement between theory and experiment is found. For the nuclear excited state we use our calculations in combination with recent measurements to obtain the nuclear dipole moment of the isomeric state μ_iso=−0.35 μN, which is in contradiction to the theoretically predicted value of μ_iso=−0.076 μN.

**Maximum Elliptical Dichroism in Atomic Two-Photon Ionization**

**121**, 053401 (2018)

**Abstract:** Elliptical dichroism is known in atomic photoionization as the difference in the photoelectron angular distributions produced in nonlinear ionization of atoms by left- and right-handed elliptically polarized light. We theoretically demonstrate that the maximum dichroism |ΔED|=1 always appears in two-photon ionization of any atom if the photon energy is tuned in so that the electron emission is dominantly determined by two intermediate resonances. We propose the two-photon ionization of atomic helium in order to demonstrate this remarkable phenomenon. The maximum elliptical dichroism could be used as a sensitive tool for analyzing the polarization state of photon beams produced by free-electron lasers.

**Dispersive refraction of different light to heavy materials at MeV γ-ray energies**

**97**, 063843 (2018)

**Abstract:** The dispersive behavior of materials with atomic charge numbers varying from Z=4 (beryllium, Be) to Z=82 (lead, Pb) was investigated experimentally and theoretically at gamma-ray energies up to 2 MeV. The experiment was performed at the double-crystal gamma spectrometer GAMS6 of the Institut Laue-Langevin in Grenoble. The experimental results were compared with theoretical calculations which account for all major elastic processes involved. Overall, we found a good agreement between theory and experiment. We find that, for the development of refractive optics at $\gamma$-ray energies beyond those currently in use, high-Z materials become increasingly attractive compared to the beryllium lens-stacks used at x-ray energies.

**Influence of a stray magnetic field on the measurement of long-range spin-spin interaction**

**2**, 055025 (2018)

**Abstract:** We study the influence of an additional uncontrolled (stray) magnetic field upon the measurement of long-range spin-spin interaction strength of two spin-1/2 valence electrons bound in two separate ions at well-defined distances from each other. This stray field, which is neither perpendicular nor parallel to the line connecting two ions, could appear due to the Earth magnetic field, or, due to the slight angular misalignment between the applied magnetic field and the line connecting two ions. It is found that the presence of the stray magnetic field plays an important role in the dynamics of the spin-states of two electrons. If neglected in the analysis, moreover, such a stray field may affect the measurement of the spin-spin interaction strength, especially at smaller inter-spin distances.

**Line strengths of QED-sensitive forbidden transitions in B-, Al-, F- and Cl-like ions**

**97**, 052506 (2018)

**Abstract:** The magnetic dipole (M1) line strength between the fine-structure levels of the ground configurations in B-, F-, Al-, and Cl-like ions are calculated for the four elements argon, iron, molybdenum, and tungsten. Systematically enlarged multiconfiguration Dirac-Hartree-Fock (MCDHF) wave functions are employed to account for the interelectronic interaction with the Breit interaction included in first-order perturbation theory. The QED corrections are evaluated to all orders in αZ utilizing an effective potential approach. The calculated line strengths are compared with the results of other theories. The M1 transition rates are reported using accurate energies from the literature. Moreover, the lifetimes in the range of millisecond to picosecond are predicted including the contributions from the transition rate due to the E2 transition channel. The discrepancies of the predicted rates from those available from the literature are discussed and a benchmark data set of theoretical lifetimes is provided to support future experiments.

**Interelectronic-interaction contribution to the nonlinear Zeeman effect in boronlike ions**

**97**, 043402 (2018)

**Abstract:** Relativistic calculations of the second- and third-order contributions in magnetic field to the Zeeman splitting in boronlike ions are presented for the wide range of nuclear charge numbers Z=6−92. The interelectronic-interaction correction of the first order in 1/Z is evaluated to all orders in αZ. The higher order corrections in 1/Z are taken into account approximately by means of effective screening potentials. The obtained results are important for interpretation of experimental data on the Zeeman splitting in boronlike ions, in particular, for the ARTEMIS experiment presently implemented at GSI.

**Rayleigh scattering of twisted light by hydrogenlike ions**

**97**, 023802 (2018)

**Abstract:** The elastic Rayleigh scattering of twisted light and, in particular, the polarization (transfer) of the scattered photons have been analyzed within the framework of second-order perturbation theory and Dirac's relativistic equation. Special attention was paid hereby to the scattering on three different atomic targets: single atoms, a mesoscopic (small) target, and a macroscopic (large) target, which are all centered with regard to the beam axis. Detailed calculations of the polarization Stokes parameters were performed for C5+ ions and for twisted Bessel beams. It is shown that the polarization of scattered photons is sensitive to the size of an atomic target and to the helicity, the opening angle, and the projection of the total angular momentum of the incident Bessel beam. These computations indicate more that the Stokes parameters of the (Rayleigh) scattered twisted light may significantly differ from their behavior for an incident plane-wave radiation.

**New Nuclear Magnetic Moment of ²⁰⁹Bi: Resolving the Bismuth Hyperfine Puzzle**

**120**, 093001 (2018)

**Abstract:** A recent measurement of the hyperfine splitting in the ground state of Li-like 208Bi80+ has established a "hyperfine puzzle" - the experimental result exhibits a 7σ deviation from the theoretical prediction. We provide evidence that the discrepancy is caused by an inaccurate value of the tabulated nuclear magnetic moment (μI) of 209Bi. We perform relativistic density functional theory and relativistic coupled cluster calculations of the shielding constant that should be used to extract the value of μI(209Bi) and combine it with nuclear magnetic resonance measurements of Bi(NO3)3 in nitric acid solutions and of the hexafluoridobismuthate(V) BiF−6 ion in acetonitrile. The result clearly reveals that μI(209Bi) is much smaller than the tabulated value used previously. Applying the new magnetic moment shifts the theoretical prediction into agreement with experiment and resolves the hyperfine puzzle.

**The nuclear magnetic moment of 208Bi and its relevance for a test of bound-state strong-field QED**

**779**, 324 (2018)

**Abstract:** The hyperfine structure splitting in the 6p3S3/24→6p27sP1/24 transition at 307 nm in atomic 208Bi was measured with collinear laser spectroscopy at ISOLDE, CERN. The hyperfine A and B factors of both states were determined with an order of magnitude improved accuracy. Based on these measurements, theoretical input for the hyperfine structure anomaly, and results from hyperfine measurements on hydrogen-like and lithium-like 209Bi80+,82+, the nuclear magnetic moment of 208Bi has been determined to μ(Bi208)=+4.570(10) μN. Using this value, the transition energy of the ground-state hyperfine splitting in hydrogen-like and lithium-like 208Bi80+,82+ and their specific difference of −67.491(5)(148) meV are predicted. This provides a means for an experimental confirmation of the cancellation of nuclear structure effects in the specific difference in order to exclude such contributions as the cause of the hyperfine puzzle, the recently reported 7-σ discrepancy between experiment and bound-state strong-field QED calculations of the specific difference in the hyperfine structure splitting of 209Bi80+,82+.

## 2017

**Ground-state hyperfine splitting for Rb, Cs, Fr, Ba⁺, and Ra⁺**

**96**, 062502 (2017)

**Abstract:** We have systematically investigated the ground-state hyperfine structure for alkali-metal atoms 87Rb, 33Cs, and 211Fr and alkali-metal-like ions 135Ba^+ and 225Ra^+, which are of particular interest for parity violation studies. The quantum electrodynamic one-loop radiative corrections have been rigorously evaluated within an extended Furry picture employing core-Hartree and Kohn-Sham atomic potentials. Moreover, the effect of the nuclear magnetization distribution on the hyperfine structure intervals has been studied in detail and its uncertainty has been estimated. Finally, the theoretical description of the hyperfine structure has been completed with full many-body calculations performed in the all-orders correlation potential method.

**Higher-order perturbative relativistic calculations for few-electron atoms and ions**

**408**, 46 (2017)

**Abstract:** An effective computational method is developed for electronic-structure calculations in few-electron atoms and ions on the basis of the Dirac-Coulomb-Breit Hamiltonian. The recursive formulation of the perturbation theory provides an efficient access to the higher-order contributions of the interelectronic interaction. Application of the presented approach to the binding energies of lithiumlike and boronlike systems is demonstrated. The results obtained are in agreement with the large-scale configuration interaction Dirac-Fock-Sturm method and other all-order calculations.

**Quadratic Zeeman effect in boronlike argon**

**408**, 70 (2017)

**Abstract:** Abstract A theoretical investigation of the second-order Zeeman effect in boronlike ions is presented. Rigorous calculations of the one-photon-exchange and one-loop QED corrections allow for predictions of the corresponding theoretical values for boronlike argon with an accuracy of about 2%. The obtained results are important in view of the forthcoming measurements of the Zeeman splitting in 40Ar13+ at GSI (ARTEMIS experiment).

**Third-order Zeeman effect in highly charged ions**

**408**, 80 (2017)

**Abstract:** The contribution of the third order in magnetic field to the Zeeman splitting of the ground state of hydrogenlike, lithiumlike, and boronlike ions in the range Z=6-82 is investigated within the relativistic approach. Both perturbative and non-perturbative methods of calculation are employed and found to be in agreement. For lithiumlike and boronlike ions the interelectronic-interaction effects are taken into account within the approximation of the local screening potential. The contribution of the third-order effect in low- and medium-Z boronlike ions is found to be important for anticipated high-precision measurements.

**Theoretical analysis of the electron bridge process in 229Th3+**

**408**, 84 (2017)

**Abstract:** We investigate the deexcitation of the 229Th nucleus via the excitation of an electron. Detailed calculations are performed for the enhancement of the nuclear decay width due to the so called electron bridge (EB) compared to the direct photoemission from the nucleus. The results are obtained for triply ionized thorium by using a B-spline pseudo basis approach to solve the Dirac equation for a local xα potential. This approach allows for an approximation of the full electron propagator including the positive and negative continuum. We show that the contribution of continua slightly increases the enhancement compared to a propagator calculated by a direct summation over bound states. Moreover we put special emphasis on the interference between the direct and exchange Feynman diagrams that can have a strong influence on the enhancement.

**Nuclear magnetic shielding in boronlike ions**

**408**, 89 (2017)

**Abstract:** The relativistic treatment of the nuclear magnetic shielding effect in boronlike ions is presented. The leading-order contribution of the magnetic-dipole hyperfine interaction is calculated. Along with the standard second-order perturbation theory expression, the solutions of the Dirac equation in the presence of magnetic field are employed. All methods are found to be in agreement with each other and with the previous calculations for hydrogenlike and lithiumlike ions. The effective screening potential is used to account approximately for the interelectronic interaction.

**Binding energies of the 1s2 2s2 2pj states in boronlike argon**

**408**, 103 (2017)

**Abstract:** The binding energies of the ground 1s2 2s2 2p1/2 and first excited 1s2 2s2 2p3/2 states in boronlike argon are rigorously evaluated. The calculations are performed by the QED perturbation theory in the framework of the extended Furry picture taking into account all the relevant first- and second-order radiative and correlation corrections. The third- and higher-order interelectronic-interaction effects are considered within the Breit approximation. The relativistic nuclear recoil effect is taken into account. In comparison with the previous calculations of the binding energies in boronlike argon the accuracy of the theoretical predictions has been significantly improved.

**Relativistic effects in the non-resonant two-photon K-shell ionization of neutral atoms**

**408**, 125 (2017)

**Abstract:** Relativistic effects in the non-resonant two-photon K-shell ionization of neutral atoms are studied theoretically within the framework of second-order perturbation theory. The non-relativistic results are compared with the relativistic calculations in the dipole and no-pair approximations as well as with the complete relativistic approach. The calculations are performed in both velocity and length gauges. Our results show a significant decrease of the total cross section for heavy atoms as compared to the non-relativistic treatment, which is mainly due to the relativistic wavefunction contraction. The effects of higher multipoles and negative continuum energy states counteract the relativistic contraction contribution, but are generally much weaker. While the effects beyond the dipole approximation are equally important in both gauges, the inclusion of negative continuum energy states visibly contributes to the total cross section only in the velocity gauge.

**Dielectronic recombination of highly charged ions with spin-polarized electrons**

**408**, 130 (2017)

**Abstract:** Angular distribution and linear polarization of photon emission following dielectronic recombination of initially lithium-like ions with spin-polarized electrons are studied. In particular, a general expression is derived for the alignment parameter of the doubly excited states produced via the resonant capture of spin-polarized electrons. By means of the alignment parameter, moreover, the angular distribution and linear polarization of the subsequently emitted photons are further obtained. Detailed computations are performed for the 1s2 2s J0=1/2+εe-→1s2s2 2p1/2 J=1→1s2 2s2 Jf=0+γ resonant electron capture and subsequent radiative decay of iodine ions. It is found that the spin polarization of the incident electrons changes only the q=±1 components of the alignment parameter A2q. As a consequence, the electron spin polarization contributes weakly to the γ photon angular distribution and linear polarization that are dominantly determined by the A20 parameter.

**Ab initio calculations of energy levels, transition rates and lifetimes in Ni xii**

**469**, 4620 (2017)

**Abstract:** We report large-scale multi-configuration Dirac–Hartree–Fock calculations and relativistic configuration interaction calculations for allowed E1 and forbidden transitions (M1, E2, M2) among the fine structure levels of the 3s^2 3p^5, 3s 3p^6 and 3s^2 3p^4 3d configurations for Ni xii. In our systematically enlarged wave functions, we incorporated the effects of relativity, all important electron correlations and rearrangement of the bound electron density within two different computational models. We compare our calculated energies for the fine structure levels with previous calculations and experiments. We validate all the tentative experimental lines recently identified by Del Zanna & Badnell with one exception. We discuss the consistency of our transition rates in comparison to semi-empirical predictions. We present ab initio lifetime values by taking into account all allowed E1 and forbidden transitions (M1, E2, M2) rates among lowest 31 levels. Our results for lifetime values are better than previously reported ab initio and semi-empirical values as compared to available experiments, thus, providing reliable predictions in the prospects of future experiments.

**Ground-state ionization energies of boronlike ions**

**96**, 022512 (2017)

**Abstract:** High-precision QED calculations of ground-state ionization energies are performed for all boronlike ions with nuclear charge numbers in the range 16≤Z≤96. Rigorous QED calculations are performed within the extended Furry picture and include all many-electron QED effects up to the second order of the perturbation theory. The contributions of third- and higher-order electron-correlation effects are accounted for within the Breit approximation. Nuclear recoil and nuclear polarization effects are taken into account as well. In comparison with previous evaluations of the ground-state ionization energies of boronlike ions the accuracy of the theoretical predictions is improved significantly.

**Angle-resolved x-ray spectroscopic scheme to determine overlapping hyperfine splittings in highly charged heliumlike ions**

**96**, 012503 (2017)

**Abstract:** An angle-resolved x-ray spectroscopic scheme is presented for determining the hyperfine splitting of highly charged ions. For heliumlike ions, in particular, we propose to measure either the angular distribution or polarization of the 1s2p ^3P_{1,F} -> 1s^2 ^1S_{0,F_f} emission following the stimulated decay of the initial 1s 2s ^1S_{0,F_i} level. It is found that both the angular and polarization characteristics of the emitted x-ray photons strongly depend on the (relative) splitting of the partially overlapping hyperfine 1s 2p ^3P_{1,F} resonances and may thus help resolve their hyperfine structure. The proposed scheme is feasible with present-day photon detectors and allows a measurement of the hyperfine splitting of heliumlike ions with a relative accuracy of about 10^{-4}.

**Photoelectron distribution of nonresonant two-photon ionization of neutral atoms**

**96**, 013409 (2017)

**Abstract:** Photoelectron angular distributions following the nonresonant two-photon K-shell ionization of neutral atoms are studied theoretically. Using the independent particle approximation and relativistic second-order perturbation theory, the contributions of screening and relativistic effects to the photoelectron angular distribution are evaluated. A simple nonrelativistic expression is presented for the angle-differential cross section in dipole approximation for two-photon ionization by elliptically polarized photons, and its limitations are analyzed numerically. Moreover, we show that screening effects of the inactive electrons can significantly affect the photoelectron distributions and can also lead to a strong elliptical dichroism. Numerical results are presented for the case of two-photon K-shell ionization of neutral Ne, Ge, Xe, and U atoms.

## 2016

**Relativistic calculations of the nonresonant two-photon ionization of neutral atoms**

**94**, 063412 (2016)

**Abstract:** The nonresonant, two-photon, one-electron ionization of neutral atoms is studied theoretically in the framework of relativistic second-order perturbation theory and independent particle approximation. In particular, the importance of relativistic and screening effects in the total two-photon ionization cross section is investigated. Detailed computations have been carried out for the K-shell ionization of neutral Ne, Ge, Xe, and U atoms. The relativistic effects significantly decrease the total cross section; for the case of U, for example, they reduce the total cross section by a factor of two. Moreover, we have found that the account for the screening effects of the remaining electrons leads to occurrence of an unexpected minimum in the total cross section at the total photon energies equal to the ionization threshold; for the case of Ne, for example, the cross section drops there by a factor of three.

**Nuclear Excitation by Two-Photon Electron Transition**

**117**, 243001 (2016)

**Abstract:** A new mechanism of nuclear excitation via two-photon electron transitions (NETP) is proposed and studied theoretically. As a generic example, detailed calculations are performed for the E1E1 1s2sS01→1s2S01 two-photon decay of a He-like Ac87+225 ion with a resonant excitation of the 3/2+ nuclear state with an energy of 40.09(5) keV. The probability for such a two-photon decay via the nuclear excitation is found to be PNETP=3.5×10−9 and, thus, is comparable with other mechanisms, such as nuclear excitation by electron transition and by electron capture. The possibility for the experimental observation of the proposed mechanism is thoroughly discussed.

**Level sequence and splitting identification of closely spaced energy levels by angle-resolved analysis of fluorescence light**

**93**, 063413 (2016)

**Abstract:** The angular distribution and linear polarization of the fluorescence light following the resonant photoexcitation is investigated within the framework of density matrix and second-order perturbation theory. Emphasis has been placed on “signatures” for determining the level sequence and splitting of intermediate (partially) overlapping resonances, if analyzed as a function of photon energy of incident light. Detailed computations within the multiconfiguration Dirac–Fock method have been performed, especially for the 1s^2 2s^2 2p^6 3s,Ji=1/2+γ1 → (1s^2 2s 2p^6 3s)_1 3p3/2,J=1/2,3/2 → 1s^2 2s^2 2p^6 3s,Jf=1/2+γ2 photoexcitation and subsequent fluorescence emission of atomic sodium. A remarkably strong dependence of the angular distribution and linear polarization of the γ2 fluorescence emission is found upon the level sequence and splitting of the intermediate (1s^2 2s 2p^6 3s)_1 3p3/2,J=1/2,3/2 overlapping resonances owing to their finite lifetime (linewidth). We therefore suggest that accurate measurements of the angular distribution and linear polarization might help identify the sequence and small splittings of closely spaced energy levels, even if they cannot be spectroscopically resolved.

**Many-electron effects on x-ray Rayleigh scattering by highly charged He-like ions**

**93**, 023418 (2016)

**Abstract:** The Rayleigh scattering of x rays by many-electron highly charged ions is studied theoretically. The many-electron perturbation theory, based on a rigorous quantum electrodynamics approach, is developed and implemented for the case of the elastic scattering of (high-energetic) photons by heliumlike ions. Using this elaborate approach, we here investigate the many-electron effects beyond the independent-particle approximation (IPA) as conventionally employed for describing the Rayleigh scattering. The total and angle-differential cross sections are evaluated for the x-ray scattering by heliumlike Ni^26+, Xe^52+, and Au^77+ ions in their ground state. The obtained results show that, for high-energetic photons, the effects beyond the IPA do not exceed 2% for the scattering by a closed K shell.

## 2015

**Relativistically prolonged lifetime of the 2s2p 3P0 level of zero nuclear-spin beryllium-like ions**

**17**, 103009 (2015)

**Abstract:** The E1M1 transition rate of the 2s2p{\ ³P₀\to 2s²\ ¹S₀} line in beryllium-like ions has been calculated within the framework of relativistic second-order perturbation theory. Both multiconfiguration and quantum-electrodynamical computations have been carried out independently to better understand and test for all major electron–electron correlation contributions in the representation of the initial, intermediate and final states. By comparing the results from these methods, which agree well for all ions along the beryllium isoelectronic sequence, the lifetime of the metastable 2s2p 3P0 level is found to be longer by about 2–3 orders of magnitude for all medium and heavy elements than was estimated previously. This makes the 3P0 level of beryllium-like ions to one of the longest living (low-lying) electronic excitations of a tightly bound system with potential applications for atomic clocks and in astro physics and plasma physics.