# Publications by

Randolf Beerwerth

## 2019

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

**1**, 1 (2019)

**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.

**Electron Correlation in Relativistic Multiconfiguration Calculations of Isotope Shift Parameters, Hyperfine Coupling Constants and Atomic Processes**

Friedrich-Schiller-Universität Jena, Physikalisch-Astronomische Fakultät (2019)

**Abstract:** Electron correlation denotes the corrections to central field approximations applied in Hartree--Fock methods that arise from the electron-electron interaction. As a consequence, wave functions for atomic states are represented as a mixture of different electronic configurations. Corresponding highly correlated multiconfiguration wave functions allow precise computations of atomic parameters such as energy levels, transition rates, isotope shift parameters and hyperfine coupling constants.

In this work, multiconfiguration Dirac–Hartree–Fock computations are utilized to compute precise isotope shift parameters and hyperfine coupling constants for actinium, nobelium and iron. As a prerequisite, extensive computations of the atomic level structure for actinium were performed to assign the computed energies to measured transitions, and as a consequence several unknown levels are predicted. In order to estimate uncertainties of the computed results, systematically enlarged configuration spaces are utilized and the results of several model computations that probe different correlation effects are compared.

Furthermore, electron correlation is crucial to describe higher order processes such as shake transitions that accompany photoionization or Auger processes. These processes are in addition caused by the non-orthogonality of the single electron orbitals obtained in Hartree–Fock computations. The latter can be transformed into electronic correlation by a biorthonormal transformation and we evaluate its application to the efficient computation of Auger transition rates. With this approach, large scale calculations for complex atoms with multiple open shells can be extended to include shake transitions. These transition rates are utilized in Auger cascade models that describe the ionization or excitation of core electrons from atoms or ions into highly excited states and the subsequent decay of these inner-shell holes by the emission of a cascade of Auger electrons.

## 2018

**First Ionization Potentials of Fm, Md, No, and Lr: Verification of Filling-Up of 5f Electrons and Confirmation of the Actinide Series**

**140**, 14609 (2018)

**Abstract:** We report the first ionization potentials (IP1) of the heavy actinides, fermium (Fm, atomic number Z = 100), mendelevium (Md, Z = 101), nobelium (No, Z = 102), and lawrencium (Lr, Z = 103), determined using a method based on a surface ionization process coupled to an online mass separation technique in an atom-at-a-time regime. The measured IP1 values agree well with those predicted by state-of-the-art relativistic calculations performed alongside the present measurements. Similar to the well-established behavior for the lanthanides, the IP1 values of the heavy actinides up to No increase with filling up the 5f orbital, while that of Lr is the lowest among the actinides. These results clearly demonstrate that the 5f orbital is fully filled at No with the [Rn]5f147s2 configuration and that Lr has a weakly bound electron outside the No core. In analogy to the lanthanide series, the present results unequivocally verify that the actinide series ends with Lr.

**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.

**Probing Sizes and Shapes of Nobelium Isotopes by Laser Spectroscopy**

**120**, 232503 (2018)

**Abstract:** Until recently, ground-state nuclear moments of the heaviest nuclei could only be inferred from nuclear spectroscopy, where model assumptions are required. Laser spectroscopy in combination with modern atomic structure calculations is now able to probe these moments directly, in a comprehensive and nuclear-model-independent way. Here we report on unique access to the differential mean-square charge radii of 252,253,254No, and therefore to changes in nuclear size and shape. State-of-the-art nuclear density functional calculations describe well the changes in nuclear charge radii in the region of the heavy actinides, indicating an appreciable central depression in the deformed proton density distribution in 252,254No isotopes. Finally, the hyperfine splitting of 253No was evaluated, enabling a complementary measure of its (quadrupole) deformation, as well as an insight into the neutron single-particle wave function via the nuclear spin and magnetic moment.

**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.

**Isotope shifts from collinear laser spectroscopy of doubly charged yttrium isotopes**

**97**, 042504 (2018)

**Abstract:** Collinear laser spectroscopy has been performed on doubly charged ions of radioactive yttrium in order to study the isotope shifts of the 294.6-nm 5s2S1/2→5p2P1/2 line. The potential of such an alkali-metal-like transition to improve the reliability of atomic-field-shift and mass-shift factor calculations, and hence the extraction of nuclear mean-square radii, is discussed. Production of yttrium ion beams for such studies is available at the IGISOL IV Accelerator Laboratory, Jyväskylä, Finland. This newly recommissioned facility is described here in relation to the on-line study of accelerator-produced short-lived isotopes using collinear laser spectroscopy and application of the technique to doubly charged ions.

**Neon in ultrashort and intense x-rays from free electron lasers**

**51**, 055602 (2018)

**Abstract:** We theoretically examine neon atoms in ultrashort and intense x-rays from free electron lasers and compare our results with data from experiments conducted at the Linac Coherent Light Source. For this purpose, we treat in detail the electronic structure in all possible nonrelativistic cationic configurations using a relativistic multiconfiguration approach. The interaction with the x-rays is described in rate-equation approximation. To understand the mechanisms of the interaction, a path analysis is devised which allows us to investigate what sequences of photoionization and decay processes lead to a specific configuration and with what probability. Thereby, we uncover a connection to the mathematics of graph theory and formal languages. In detail, we study the ion yields and find that plain rate equations do not provide a satisfactory description. We need to extend the rate equations for neon to incorporate double Auger decay of a K-shell vacancy and photoionization shake off for neutral neon. Shake off is included for valence and core ionization; the former has hitherto been overlooked but has important consequences for the ion yields from an x-ray energy below the core ionization threshold. Furthermore, we predict the photon yields from xuv and x-ray fluorescence; these allow one insights into the configurations populated by the interaction with the x-rays. Finally, we discover that inaccuracies in those Auger decay widths employed in previous studies have only a minor influence on ion and photon yields.

**Ultrafast quantum control of ionization dynamics in krypton**

**9**, 719 (2018)

**Abstract:** Ultrafast spectroscopy with attosecond resolution has enabled the real time observation of ultrafast electron dynamics in atoms, molecules and solids. These experiments employ attosecond pulses or pulse trains and explore dynamical processes in a pump–probe scheme that is selectively sensitive to electronic state of matter via photoelectron or XUV absorption spectroscopy or that includes changes of the ionic state detected via photo-ion mass spectrometry. Here, we demonstrate how the implementation of combined photo-ion and absorption spectroscopy with attosecond resolution enables tracking the complex multidimensional excitation and decay cascade of an Auger auto-ionization process of a few femtoseconds in highly excited krypton. In tandem with theory, our study reveals the role of intermediate electronic states in the formation of multiply charged ions. Amplitude tuning of a dressing laser field addresses different groups of decay channels and allows exerting temporal and quantitative control over the ionization dynamics in rare gas atoms.

**LCLS in-photon out: fluorescence measurement of neon using soft x-rays**

**51**, 034003 (2018)

**Abstract:** We measured the fluorescence photon yield of neon upon soft x-ray ionization (∼1200 eV) from the x-ray free-electron laser at Linac Coherent Light Source, and demonstrated the usage of a grazing incidence spectrometer with a variable line spacing grating to perform x-ray fluorescence spectroscopy on a gas phase system. Our measurements also allowed us to estimate the focal size of the beam from the theoretical description developed, in terms of the rate equation approximation accounting for photoionization shake off of neutral neon and double auger decay of single core holes.

## 2017

**First determination of ground state electromagnetic moments of ⁵³Fe**

**96**, 054314 (2017)

**Abstract:** The hyperfine coupling constants of neutron deficient 53Fe were deduced from the atomic hyperfine spectrum of the 3d^6 4s^2 ^5D_4 <-> 3d^6 4s 4p ^5F_5 transition, measured using the bunched-beam collinear laser spectroscopy technique. The low-energy 53Fe beam was produced by projectile-fragmentation reactions followed by gas stopping, and used for the first time for laser spectroscopy. Ground state magnetic-dipole and electric-quadrupole moments were determined as μ = -0.65(1)μ_N and Q = +35(15)e^2 fm^2, respectively. The multiconfiguration Dirac-Fock method was used to calculate the electric field gradient to deduce Q from the quadrupole hyperfine coupling constant, since the quadrupole coupling constant has not been determined for any Fe isotopes. Both experimental values agree well with nuclear shell model calculations using the GXPF1A effective interaction performed in a full fp shell model space, which support the soft nature of the 56Ni nucleus.

**In-gas laser ionization and spectroscopy of actinium isotopes near the N=126 closed shell**

**96**, 054331 (2017)

**Abstract:** The in-gas laser ionization and spectroscopy (IGLIS) technique was applied on the 212-215Ac isotopes, produced at the Leuven Isotope Separator On-Line (LISOL) facility by using the in-gas-cell and the in-gas-jet methods. The first application under on-line conditions of the in-gas-jet laser spectroscopy method showed a superior performance in terms of selectivity, spectral resolution, and efficiency in comparison with the in-gas-cell method. Following the analysis of both experiments, the magnetic-dipole moments for the 212-215Ac isotopes, electric-quadrupole moments and nuclear spins for the 214,215Ac isotopes are presented and discussed. A good agreement is obtained with large-scale nuclear shell-model calculations by using a 208Pb core.

**Near L-edge Single and Multiple Photoionization of Singly Charged Iron Ions**

**849**, 5 (2017)

**Abstract:** Absolute cross-sections for m -fold photoionization (m=1, ... , 6 ) of Fe+ by a single photon were measured employing the photon–ion merged-beams setup PIPE at the PETRA III synchrotron light source, operated by DESY in Hamburg, Germany. Photon energies were in the range 680–920 eV, which covers the photoionization resonances associated with 2p and 2s excitation to higher atomic shells as well as the thresholds for 2p and 2s ionization. The corresponding resonance positions were measured with an uncertainty of ±0.2 eV. The cross-section for Fe+ photoabsorption is derived as the sum of the individually measured cross-sections for m -fold ionization. Calculations of the Fe+ absorption cross-sections were carried out using two different theoretical approaches, Hartree–Fock including relativistic extensions and fully relativistic multiconfiguration Dirac–Fock. Apart from overall energy shifts of up to about 3 eV, the theoretical cross-sections are in good agreement with each other and with the experimental results. In addition, the complex de-excitation cascades after the creation of inner-shell holes in the Fe+ ion were tracked on the atomic fine-structure level. The corresponding theoretical results for the product charge-state distributions are in much better agreement with the experimental data than previously published configuration-average results. The present experimental and theoretical results are valuable for opacity calculations and are expected to pave the way to a more accurate determination of the iron abundance in the interstellar medium.

**MCDF calculations of Auger cascade processes**

**71**, 253 (2017)

**Abstract:** We model the multiple ionization of near-neutral core-excited atoms where a cascade of Auger processes leads to the emission of several electrons. We utilize the multiconfiguration Dirac-Fock (MCDF) method to generate approximate wave functions for all fine-structure levels and to account for all decays between them. This approach allows to compute electron spectra, the population of final-states and ion yields, that are accessible in many experiments. Furthermore, our approach is based on the configuration interaction method. A careful treatment of correlation between electronic configurations enables one to model three-electron processes such as an Auger decay that is accompanied by an additional shake-up transition. Here, this model is applied to the triple ionization of atomic cadmium, where we show that the decay of inner-shell 4p holes to triply-charged final states is purely due to the shake-up transition of valence 5s electrons.

**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.

**Auger decay of 4d inner-shell holes in atomic Hg leading to triple ionization**

**96**, 012505 (2017)

**Abstract:** Formation of triply ionized states upon the creation of 4d inner-shell holes in atomic Hg is investigated by using synchrotron radiation of 730 eV photon energy and a versatile multielectron coincidence detection technique in combination with multiconfiguration Dirac-Fock calculations. By carefully selecting Coster-Kronig electrons detected only in coincidence with a 4d photoelectron, the Coster-Kronig spectrum has been extracted and the corresponding branching ratios of the 4d hole have been determined. The results are found to differ from previously established experimental ratios based on electron impact ionization but to agree now better with theory. We also present an Auger cascade analysis of pathways leading to triply ionized states of atomic Hg upon removal of a 4d inner-shell electron.

**Auger cascades in resonantly excited neon**

**95**, 053407 (2017)

**Abstract:** The Auger cascades following the resonant 1s → np (n = 3, 4) excitation of neutral neon are studied theoretically. In contrast to previous investigations, we here model the complete cascade from the initially core-excited 1s⁻¹3p ¹P₁ and 1s⁻¹4p ¹P₁ levels of Ne up to the doubly-ionized Ne²⁺ ions. Extensive multiconfiguration Dirac-Fock (MCDF) calculations are carried out, combined with a proper cascades model, to incorporate as many decay branches as possible, including all major single-electron shake-up and shake-down processes. We simulate the electron spectra and predict shake probabilities, ion yields, as well as the relative population of the intermediate and final states. Experimentally known level energies for neutral, singly- and doubly-ionized neon are utilized whenever possible in order to improve the predictions. Most features from experiment can be reproduced with quite good agreement if a sufficiently large basis is taken into account. These simulations therefore demonstrate not only the required computational effort, but also that it is nowadays possible to predict whole Auger spectra of decay cascades, a central feature for further exploring electron coincidence maps as obtained at synchrotrons and free-electron lasers.

**Towards high-resolution laser ionization spectroscopy of the heaviest elements in supersonic gas jet expansion**

**8**, 14520 (2017)

**Abstract:** Resonant laser ionization and spectroscopy are widely used techniques at radioactive ion beam facilities to produce pure beams of exotic nuclei and measure the shape, size, spin and electromagnetic multipole moments of these nuclei. However, in such measurements it is difficult to combine a high efficiency with a high spectral resolution. Here we demonstrate the on-line application of atomic laser ionization spectroscopy in a supersonic gas jet, a technique suited for high-precision studies of the ground- and isomeric-state properties of nuclei located at the extremes of stability. The technique is characterized in a measurement on actinium isotopes around the N=126 neutron shell closure. A significant improvement in the spectral resolution by more than one order of magnitude is achieved in these experiments without loss in efficiency.

## 2016

**Multiconfiguration calculations of electronic isotope shift factors in Al I**

**94**, 062508 (2016)

**Abstract:** The present work reports results from systematic multiconfiguration Dirac–Hartree–Fock calculations of electronic isotope shift factors for a set of transitions between low-lying levels of neutral aluminium. These electronic quantities together with observed isotope shifts between different pairs of isotopes provide the changes in mean-square charge radii of the atomic nuclei. Two computational approaches are adopted for the estimation of the mass- and field-shift factors. Within these approaches, different models for electron correlation are explored in a systematic way to determine a reliable computational strategy and to estimate theoretical error bars of the isotope shift factors.

**Charge Radii of Neutron Deficient ⁵²,⁵³Fe Produced by Projectile Fragmentation**

**117**, 252501 (2016)

**Abstract:** Bunched-beam collinear laser spectroscopy is performed on neutron deficient Fe52,53 prepared through in-flight separation followed by a gas stopping. This novel scheme is a major step to reach nuclides far from the stability line in laser spectroscopy. Differential mean-square charge radii δ⟨r2⟩ of Fe52,53 are determined relative to stable Fe56 as δ⟨r2⟩56,52=−0.034(13) fm2 and δ⟨r2⟩56,53=−0.218(13) fm2, respectively, from the isotope shift of atomic hyperfine structures. The multiconfiguration Dirac-Fock method is used to calculate atomic factors to deduce δ⟨r2⟩. The values of δ⟨r2⟩ exhibit a minimum at the N=28 neutron shell closure. The nuclear density functional theory with Fayans and Skyrme energy density functionals is used to interpret the data. The trend of δ⟨r2⟩ along the Fe isotopic chain results from an interplay between single-particle shell structure, pairing, and polarization effects and provides important data for understanding the intricate trend in the δ⟨r2⟩ of closed-shell Ca isotopes.

**Changes in nuclear structure along the Mn isotopic chain studied via charge radii**

**94**, 054321 (2016)

**Abstract:** The hyperfine spectra of Mn51,53−64 were measured in two experimental runs using collinear laser spectroscopy at ISOLDE, CERN. Laser spectroscopy was performed on the atomic 3d54s2S5/26→3d54s4pP3/26 and ionic 3d54s5S2→3d54p5P3 transitions, yielding two sets of isotope shifts. The mass and field shift factors for both transitions have been calculated in the multiconfiguration Dirac-Fock framework and were combined with a King plot analysis in order to obtain a consistent set of mean-square charge radii which, together with earlier work on neutron-deficient Mn, allow the study of nuclear structure changes from N=25 across N=28 up to N=39. A clear development of deformation is observed towards N=40, confirming the conclusions of the nuclear moments studies. From a Monte Carlo shell-model study of the shape in the Mn isotopic chain, it is suggested that the observed development of deformation is not only due to an increase in static prolate deformation but also due to shape fluctuations and triaxiality. The changes in mean-square charge radii are well reproduced using the Duflo-Zuker formula except in the case of large deformation.

**Photoionization of neutral atoms by X waves carrying orbital angular momentum**

**94**, 041402 (2016)

**Abstract:** In contrast to plane waves, twisted or vortex beams have a complex spatial structure. Both their intensity and energy flow vary within the wave front. Beyond that, polychromatic vortex beams, such as X waves, have a spatially dependent energy distribution. We propose a method to measure this (local) energy spectrum. The method is based on the measurement of the energy distribution of photoelectrons from alkali-metal atoms. On the basis of our fully relativistic calculations, we argue that even ensembles of atoms can be used to probe the local energy spectrum of short twisted pulses.

**Prominent role of multielectron processes in K-shell double and triple photodetachment of oxygen anions**

**94**, 041401 (2016)

**Abstract:** The photon-ion merged-beam technique was used at a synchrotron light source for measuring the absolute cross sections of the double and triple photodetachment of O− ions. The experimental photon energy range of 524–543 eV comprised the threshold for K-shell ionization. Using resolving powers of up to 13 000, the position, strength, and width of the below-threshold 1s2s22p6 S2 resonance as well as the positions of the 1s2s22p5 P3 and 1s2s22p5 P1 thresholds for K-shell ionization were determined with high precision. In addition, systematically enlarged multiconfiguration Dirac-Fock calculations have been performed for the resonant detachment cross sections. Results from these ab initio computations agree very well with the measurements for the widths and branching fractions for double and triple detachment, if double shakeup (and shakedown) of the valence electrons and the rearrangement of the electron density is taken into account. For the absolute cross sections, however, a previously found discrepancy between measurements and theory is confirmed.

## 2015

**Triple ionization of atomic Cd involving 4p⁻1 and 4s⁻1 inner-shell holes**

**92**, 023414 (2015)

**Abstract:** The triple ionization spectrum of atomic Cd formed upon the removal of a 4p or a 4s inner-shell electron and subsequent Auger decays has been obtained at 200 eV photon energy. By using a versatile multielectron coincidence detection technique based on a magnetic bottle spectrometer in combination with multiconfiguration Dirac-Fock calculations, Auger cascades leading to tricationic final states have been analyzed and final-state configurations have been identified. The most prominent Auger cascades leading to the ground state of Cd³⁺ have been identified in good agreement with theory.