Atomic Structure Calculations Research Articles

Determination of nuclear charge radius by extreme-ultraviolet spectroscopy of Na-like ions

We report on a method for determining the absolute nuclear charge radius of high-Z elements using extreme-ultraviolet spectroscopy of highly charged Na-like ions in tandem with highly accurate atomic structure calculations of transition energy differences. The application of this method has reduced the nuclear charge radius uncertainty of Ir191 by a factor of 8 from the currently accepted literature value, with a recently reported charge radius of 5.442(12) fm. The result reduces the charge radius uncertainty along the full Ir isotopic chain when combined with prior optical isotope shift measurements. The technique utilizes only a few million ions stored in an ion trap, which should apply to measurements with small quantities of radioactive nuclei. Published by the American Physical Society 2025

Theoretical analysis and predictions for the two-neutrino double electron capture of 124Xe

Abstract We provide a complete theoretical description of the two-neutrino electron capture in 124Xe, improving both the nuclear and the atomic structure calculations. We improve the general formalism through the use of the Taylor expansion method, leading to higher-order terms in the decay rate of the process. The nuclear part is treated with pn-QRPA and interacting shell model (ISM) methods. The nuclear matrix elements (NMEs) are calculated with the pn-QRPA method with isospin restoration by fixing the input parameters so that the experimental decay rate is reproduced, resulting in values significantly lower than in previous calculations. The validity of the pn-QRPA NMEs is tested by showing their values to be comparable with the ones for double-beta decay with the emission of two antineutrinos of 128,130Te, which have similar pairing features. Within the ISM, we reproduce the total experimental half-life within a factor of two and predict the capture fraction to the KK channel of about 74%. We also predict the capture fractions to other decay channels and show that for the cumulative decay to the KL1–KO1 channels, a capture fraction of about 24% could be observed experimentally. On the atomic side, calculations are improved by accounting for the Pauli blocking of the decay of innermost nucleon states and by considering all s-wave electrons available for capture, expanding beyond the K and L1 orbitals considered in previous studies. We also provide improved atomic relaxation energies of the final atomic states of 124Te, which may be used as input for background modeling in liquid Xenon experiments.

Machine Learning-Assisted Hartree-Fock Approach for Energy Level Calculations in the Neutral Ytterbium Atom.

Data-driven machine learning approaches with precise predictive capabilities are proposed to address the long-standing challenges in the calculation of complex many-electron atomic systems, including high computational costs and limited accuracy. In this work, we develop a general workflow for machine learning-assisted atomic structure calculations based on the Cowan code's Hartree-Fock with relativistic corrections (HFR) theory. The workflow incorporates enhanced ElasticNet and XGBoost algorithms, refined using entropy weight methodology to optimize performance. This semi-empirical framework is applied to calculate and analyze the excited state energy levels of the 4f closed-shell Yb I atom, providing insights into the applicability of different algorithms under various conditions. The reliability and advantages of this innovative approach are demonstrated through comprehensive comparisons with ab initio calculations, experimental data, and other theoretical results.

Spectrum and energy levels of the high-lying singly excited configurations of Nd III. New Nd III experimental energy levels and wavelengths, with transition probability and ionisation energy calculations

We aim to accurately determine bound-to-bound transition wavelengths and energy levels of the high-lying open-shell configurations 4f$^3$7s, 4f$^3$6d, and 4f$^3$5f of doubly ionised neodymium (Nd III, $Z=60$) through high-resolution spectroscopy and semi-empirical calculations. This study is motivated by lanthanide atomic data requirements in astronomy, such those involved in investigations of kilonova spectra. Fourier transform spectra of Nd Penning and hollow cathode discharge lamps were recorded within the region 32\,500--54\,000 and grating spectra of Nd vacuum sliding sparks were recorded within the regions 820--1159 and 1600--3250 . New energy levels were found using the observed wavelengths measured accurate to a few parts in $10^8$ in Fourier transform spectra and to a few parts in $10^7$ in grating spectra. Atomic structure and transition probability calculations of Nd III were carried out using the Cowan codes, where energy parameters were adjusted to fit all known Nd III levels. Finally, Nd-rich stellar spectra were also used to aid in the analysis. In total, 355 transitions were classified from observed spectra involving 116 previously experimentally unknown energy levels of the 4f$^3$7s, 4f$^3$6d, and 4f$^3$5f configurations of Nd III, all reported here for the first time. Three newly identified levels of the 4f$^3$5d configuration and one newly identified 4f$^4$ level are also reported. Typical level energy uncertainties are 0.01 cm$^ $ for the 4f$^3$7s and 4f$^3$6d levels and 0.3 cm$^ $ for the 4f$^3$5f levels. In addition, calculated energy levels up to 130\,936 $ are presented, including eigenvector composition and calculated level lifetimes. Calculated transition probabilities and wavelengths between 1900--50\,000 are also given. Using newly established levels of the 4f$^3$7s configuration and the recently established levels of the 4f$^3$6s configuration, the ionisation energy of Nd III has been estimated at $178\,090 $. This result offers up to twice the accuracy of the most recently published value.

Spectral scheme for atomic structure calculations in density functional theory

We present a spectral scheme for atomic structure calculations in pseudopotential Kohn-Sham density functional theory. In particular, after applying an exponential transformation of the radial coordinates, we employ global polynomial interpolation on a Chebyshev grid, with derivative operators approximated using the Chebyshev differentiation matrix, and integrations using Clenshaw-Curtis quadrature. We demonstrate the accuracy and efficiency of the scheme through spin-polarized and unpolarized calculations for representative atoms, while considering local, semilocal, and hybrid exchange-correlation functionals. In particular, we find that O(200) grid points are sufficient to achieve an accuracy of 1 microhartree in the eigenvalues for optimized norm conserving Vanderbilt pseudopotentials spanning the periodic table from atomic number Z=1 to 83.

Theoretical investigation of iodine plasma through detailed electron impact excitation cross-section calculations and collisional-radiative model analysis

Abstract In the present study, we consider the electron impact excitation of neutral iodine and its application in developing a collisional-radiative model for plasma diagnostics of iodine plasma. Electron impact excitation cross- sections are calculated using fully relativistic distorted wave theory. The required atomic structure calculations for iodine are carried out using the relativistic multi-configuration Dirac-Fock method. The excitation energies obtained are reported and compared with values from the NIST database. The electron impact excitation cross-sections are calculated from the ground (5p5 2P3/2) and first- excited state (5p5 2P1/2) to the several upper fine structure levels. Our cross- section results align well with previous studies, reported by Ambalampitiya et al. [Atoms, 9, 103, 2021] using the Breit-Pauli B-spline R-matrix method. Further calculated cross-section results are incorporated into the collisional-radiative model. Our model also includes electron impact ionization, de-excitation, three- body recombination, and radiative decay. We validate our model using emission spectra from inductively coupled iodine plasma. For this purpose, we used the recorded intensities for the emission lines, 486.23 nm, 511.92 nm, 523.45 nm, 589.40 nm, and 608.24 nm are compared with those obtained from our model to calculate electron temperature and electron density of the iodine plasma.

Electron-impact excitation data for W2+ in support of tungsten spectroscopy and re-deposition measurements for magnetically-confined plasmas

Abstract To better understand plasma wall interactions involving tungsten, accurate atomic structure and electron-impact driven collisional processes for near-neutral ion stages of tungsten are required. Complementing existing work on neutral and singly ionised tungsten, atomic structure and collisional calculations for W2+ electron-impact excitation have been completed. These excitation calculations are an important component of S/XB coefficients for near-neutral charge states, which may be used to spectroscopically infer re-deposition of tungsten at the plasma-solid boundary of fusion relevant devices. With W2+ in particular having emission lines that can be observed at ultraviolet (UV) wavelengths, while higher charge states of tungsten are unlikely to have lines possible to observe outside of the vacuum UV range. The atomic structure was generated using the General-purpose Relativistic Atomic Structure Package (GRASP0), implementing the Multi-configuration Dirac Fock approach. This structure was the basis for a subsequent Dirac R-matrix electron-impact excitation calculation to provide Maxwellian averaged rate coefficients. A synthetic spectrum was generated from this data using a collisional-radiative model to predict the strongest W III spectral lines and these lines were compared to emission from the Compact Toroidal Hybrid (CTH) plasma device. Several of the strongest W III lines are observed in CTH and agree well with the modelled line wavelengths and intensities, a table of these lines is provided that could be observed in other devices.

Hanle Effect for Lifetime Determinations in the Soft X-Ray Regime.

By exciting a series of 1s^{2} ^{1}S_{0}→1snp^{1}P_{1} transitions in heliumlike nitrogen ions with linearly polarized monochromatic soft x rays at the Elettra facility, we found a change in the angular distribution of the fluorescence sensitive to the principal quantum number n. In particular it is observed that the ratio of emission in directions parallel and perpendicular to the polarization of incident radiation increases with higher n. We find this n dependence to be a manifestation of the Hanle effect, which served as a practical tool for lifetime determinations of optical transitions since its discovery in 1924. In contrast to traditional Hanle effect experiments, in which one varies the magnetic field and considers a particular excited state, we demonstrate a "soft x-ray Hanle effect" which arises in a static magnetic field but for a series of excited states. By comparing experimental data with theoretical predictions, we were able to determine lifetimes ranging from hundreds of femtoseconds to tens of picoseconds of the 1snp^{1}P_{1} levels, which find excellent agreement with atomic-structure calculations. We argue that dedicated soft x-ray measurements could yield lifetime data that are beyond current experimental reach and cannot yet be predicted with sufficient accuracy.

Distribution of angular momenta ML and MS in non-relativistic configurations: statistical analysis using cumulants and Gram-Charlier series

Abstract The distributions P(ML,MS) of the total magnetic quantum numbers ML and MS for N electrons of angular momentum l, as well as the enumeration of LS spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for P(ML,MS) is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the P(ML,MS) distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of P(ML,MS). It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent.

Absolute nuclear charge radius by Na-like spectral line separation in high-Z elements

Abstract We describe a novel technique to determine absolute nuclear radii of high-Z nuclides. Utilizing accurate theoretical atomic structure calculations together with precise measurements of extreme ultraviolet transitions in highly charged ions this method allows for precise determinations of absolute nuclear charge radii based upon the well-known nuclear radii of their neighboring elements. This method can work for elements without stable isotopes, and its accuracy may be competitive with current methods (electron scattering and muonic x-ray spectroscopy).

Matrix methods for opacity calculations

Atomic structure calculations are the first step in an opacity calculation. They provide the self-consistent electronic potential and electron wavefunctions that are needed to evaluate the Slater integrals and dipole matrix elements. In this paper, we show that simple matrix methods can be used to perform Hartree–Fock–Slater or LDA calculations on standard atomic structure grids, including relativistic corrections, yielding bound and continuum wavefunctions that can be used, in conjunction with statistical broadening techniques, to obtain realistic average-atom opacities.

Identification of new gold lines in the 350–1000 nm spectral region using laser produced plasmas

We present results from a pilot study, using a laser-produced plasma, to identify new lines in the 350–1000 nm spectral region for the r-process element gold (Au), of relevance to studies of neutron star mergers. This was achieved via optical-IR spectroscopy of a laser-produced Au plasma, with an Au target of high purity (99.95%) and a low vacuum pressure to remove any air contamination from the experimental spectra. Our data were recorded with a spectrometer of 750 mm focal length and 1200 lines mm−1 grating, yielding a resolution of 0.04 nm. We find 54 lines not previously identified and which are not due to the impurities (principally copper (Cu) and silver (Ag)) in our Au sample. Of these 54 lines, we provisionally match 21 strong transitions to theoretical results from collisional-radiative models that include energy levels derived from atomic structure calculations up to the 6s level. Some of the remaining 33 unidentified lines in our spectra are also strong and may be due to transitions involving energy levels which are higher-lying than those in our plasma models. Nevertheless, our experiments demonstrate that laser-produced plasmas are well suited to the identification of transitions in r-process elements, with the method applicable to spectra ranging from UV to IR wavelengths.

Radiative and opacity data obtained from large-scale atomic structure calculations and from statistical simulations for the spectral analysis of kilonovae in their photospheric and nebular phases: the sample case of Er III

Radiative and opacity data obtained from large-scale atomic structure calculations and from statistical simulations for the spectral analysis of kilonovae in their photospheric and nebular phases: the sample case of Er III

Theoretical exploration of atomic structure calculation of sodium atom confined in the fullerene cage: a primer investigation for EIT applications

Alterations in the electronic structural properties of a sodium atom confined inside a fullerene cage are investigated in this manuscript; the engineered atom is explored to suit for the quantum technique applications. A Gaussian annular square well (GASW) model has been used to mimic the fullerene environment. Electronic properties such as energy levels, probabilities, the lifetime of the levels, etc, have been studied for various confinement depths of GASW using the multi-configuration Dirac Hartree–Fock technique. The wavelengths of the D lines of the sodium atom have been shifted from the typical yellow to the blue and then to the ultraviolet region upon confinement. Using the range-shifted blue D lines of the engineered atom, a typical Λ-type electromagnetically induced transparency configuration is proposed. Furthermore, we have shown the explicit dependency of relevant atomic parameters on the confinement depth, which showcases the tunability to suit the engineered atoms for various applications; for instance, increasing the confinement depth increases the lifetime of 3D5/2 and 3D3/2 states significantly from the typical lifetime value, which has potential applications in quantum technologies.

Comparative study of kilonova opacities for three elements of the sixth period (hafnium, osmium, and gold) from new atomic structure calculations in Hf I–IV, Os I–IV, and Au I–IV

Aims. It is now well established that a large amount of heavy (trans-iron) elements are produced during neutron star (NS) mergers. These elements can be detected in the spectra of the kilonova emitted from the post-merger ejected materials. Due to the high level densities that characterize the complex configurations belonging to heavy elements, thus giving rise to millions of absorption lines, the kilonova ejecta opacity is of significant importance. The elements that contribute the most to the latter are those with an unfilled nd subshell belonging to the fifth and the sixth rows of the periodic table, and those with an unfilled nf subshell belonging to the lanthanide and actinide groups. The aim of the present work is to make a new contribution to this field by performing large-scale atomic structure calculations in three specific sixth-row 5d elements, namely hafnium, osmium, and gold, in the first four charge stages (I–IV), and by computing the corresponding opacities, while focusing on the importance of the atomic models used. Methods. The pseudo-relativistic Hartree–Fock (HFR) method, including extended sets of interacting configurations, was used for the atomic structure and radiative parameter calculations, while the expansion formalism was used to estimate the opacities. Results. Theoretical energy levels, wavelengths, and oscillator strengths were computed for millions of spectral lines in Hf I–IV, Os I–IV, and Au I–IV ions, the reliability of these parameters being assessed through detailed comparisons with previously published experimental and theoretical results. The newly obtained atomic data were then used to calculate expansion opacities for typical kilonova conditions expected one day after the NS merger; these are a density of ρ = 10−13 g cm−3 and temperatures ranging from T = 5000 K to T = 15 000 K. Some agreements and differences were found when comparing our results with available data, highlighting the importance of using sufficiently complete atomic models for the determination of opacities.

Performance assessment of the effective core potentials under the fermionic neural network: First and second row elements.

The rapid development of deep learning techniques has driven the emergence of a neural network-based variational Monte Carlo (VMC) method (referred to as FermiNet), which has manifested high accuracy and strong predictive power in the electronic structure calculations of atoms, molecules, and some periodic systems. Recently, the implementation of the effective core potential (ECP) scheme has further facilitated more efficient calculations in practice. However, there is still a lack of comprehensive assessments of the ECP's performance under the FermiNet. In this work, we set sail to fill this gap by conducting extensive tests on the first two row elements regarding their atomic, spectral, and molecular properties. Our major finding is that, in general, the qualities of ECPs have been correctly reflected under FermiNet. Two recently built ECP tables, namely, correlation consistent ECP (ccECP) and energy consistent correlated electron pseudopotential (eCEPP), seem to prevail in terms of overall performance. In particular, ccECP performs slightly better on spectral precision and covers more elements, while eCEPP is more systematically built from both shape and energy consistency and better treats the core polarization. On the other hand, the high accuracy of the all-electron calculations is hindered by the absence of relativistic effects as well as the numerical instabilities in some heavier elements. Finally, with further in-depth discussions, we generate possible directions for developing and improving FermiNet in the near future.

Relativistic atomic structure calculations of Li-like ions used for plasma diagnostic studies

We have carried out atomic structure calculations using systematically enlarged multiconfiguration Dirac-Fock wavefunctions of Li-like ions of the most prominent plasma impurities (Ar, Ti, Fe, Ni, Kr and W) found in presently working tokamaks. Relativistic Breit interaction and quantum electrodynamic (QED) corrections such as vacuum polarization and self-energy corrections are also included in the calculations prior to the evaluation of low lying energy levels, transition probabilities, oscillator strengths and line strengths. Selective radiative data for electric dipole and magnetic quadrupole transitions are also reported. Special emphasis is given in the computations of fundamental quantities such as oscillator strengths as they are widely used in atomic data and analysis structure (ADAS) databases to evaluate quantities such as effective collision strengths. Present computed values are compared with existing available results on NIST database and few similar earlier computations and a good agreement has been found. We believe that the detailed atomic data with the relativistic and QED corrections will assist in spectroscopic studies such as accurate line identification and plasma modelling work in tokamak plasma, laser induced breakdown spectroscopy (LIBS), highly charged ions clocks and astrophysical observations.

Investigation of optical spectroscopy in W13+ ions

This study presents the investigation of magnetic dipole transition lines of W13+ ions in the wavelength range of 200–700 nm at a low-energy electron beam ion trap. Seventeen lines emitted from W13+ ions have been observed with three being reported experimentally for the first time. The atomic structure calculation and spectra simulation of W13+ ions have been carried out using the multiconfiguration Dirac–Hartree–Fock method combined with the relativistic configuration interaction approach and the collisional–radiative model, respectively. The theoretical calculations are in reasonable agreement with the experimental observations, and most of the observed lines have been identified.

Targeted optimization in small-scale atomic structure calculations: application to Au I

The lack of reliable atomic data can be a severe limitation in astrophysical modelling, in particular of events such as kilonovae that require information on all neutron-capture elements across a wide range of ionization stages. Notably, the presence of non-orthonormalities between electron orbitals representing configurations that are close in energy can introduce significant inaccuracies in computed energies and transition probabilities. Here, we propose an explicit targeted optimization (TO) method that can effectively circumvent this concern while retaining an orthonormal orbital basis set. We illustrate this method within the framework of small-scale atomic structure models of Au I, using the Grasp2018 multiconfigurational Dirac–Hartree–Fock atomic structure code. By comparing to conventional optimization schemes we show how a TO approach improves the energy level positioning and ordering. TO also leads to better agreement with experimental data for the strongest E1 transitions. This illustrates how small-scale models can be significantly improved with minor computational costs if orbital non-orthonormalities are considered carefully. These results should prove useful to multi-element atomic structure calculations in, for example, astrophysical opacity applications involving neutron-capture elements.

Accurate and experimentally validated transition data for Si I and Si II

Aims. The aim of this study is to provide radiative data for neutral and singly ionised silicon, in particular for the first experimental oscillator strengths for near-infrared Si I lines. In addition, we aim to perform atomic structure calculations both for neutral and singly ionised silicon while including lines from highly excited levels. Methods. We performed large-scale atomic structure calculations with the relativistic multiconfiguration Dirac-Hartree-Fock method using the GRASP2K package to determine log(𝑔ƒ) values of Si I and Si II lines, taking into account valence-valence and core-valence electron correlation. In addition, we derived oscillator strengths of near-infrared Si I lines by combining the experimental branching fractions with radiative lifetimes from our calculations. The silicon plasma was obtained from a hollow cathode discharge lamp, and the intensity-calibrated high-resolution spectra between 1037 and 2655 nm were recorded by a Fourier transform spectrometer. Results. We provide an extensive set of accurate experimental and theoretical log(𝑔ƒ) values. For the first time, we derived 17 log(𝑔ƒ) values of Si I lines in the infrared from experimental measurements. We report data for 1500 Si I lines and 500 Si II lines. The experimental uncertainties of our ƒ-values vary between 5% for the strong lines and 25% for the weak lines. The theoretical log(𝑔ƒ) values for Si I lines in the range 161 nm to 6340 nm agree very well with the experimental values of this study and complete the missing transitions involving levels up to 3s23p7s (61 970 cm−1). In addition, we provide accurate calculated log(𝑔ƒ) values of Si II lines from the levels up to 3s27f (122 483 cm−1) in the range 81 nm to 7324 nm.