Multiconfiguration Dirac-Fock Method Research Articles

Semi-Empirical and Theoretical Calculation of L1, L2, and L3 Subshell Fluorescence Yields

In this study, semi-empirical Li subshells fluorescence yields (ωLi) were calculated from the available experimental data covering the period from 1955 to 2022 for elements with 28 ≤ Z ≤ 96 for L1 subshell, and 23 ≤ Z ≤ 96 for L2 and L3 subshells. First, we critically examine these data and obtain weighted average values (ωL1W,ωL2W,andωL3W) using a formula based on experimental (ωEXP) values and measurement errors. New recommended values ωWR are obtained by calculating the ratio S = ωEXP/ωW and removing out-of-range values (less than 0.8 or greater than 1.2). Semi-empirical values were derived for the three subshells using two interpolations: the analytical function [ωWR/(1 − ωWR)]1/4 and the fitting ratio S = ωEXP/ωWR, both as function of the atomic number Z. Furthermore, new theoretical calculations based on the Multiconfiguration Dirac-Fock Method have been performed for some elements and are presented in this work. Finally, our semi-empirically and theorical calculated Li subshell fluorescence yields were compared with other theoretical, experimental, and empirical values from the literature.

Theoretical investigation of magnetic-field and hyperfine-induced transition parameters for magnesium-like and chlorine-like ions

In this paper, we report an extensive theoretical study of the magnetic-field and hyperfine-induced transition parameters for magnesium-like ions from Cr XIII to Zn XIX and chlorine-like ions from Mn IX to Co XI. These ions have a high abundance in the astrophysical plasma. Zeeman splitting level energies, hyperfine- and/or magnetic-field dependent wavelengths, and transition rates of the 3s3p 3P2,1,0→3s21S0 (for magnesium-like ions) and the 3p43d 4D7/2→ 3p52P3/2 line (for chlorine-like ions), are computed using the multiconfiguration Dirac–Fock method. Some higher-order effects, such as the Breit interaction and quantum electrodynamics corrections, are included in the relativistic configuration interaction method, as this makes the results more accurate. A comparison between the our predictions and other theoretical and experimental values is made. A satisfactory agreement is achieved. Besides providing valuable information for spectral line analysis, the present work will have a variety of implications for fusion and astrophysical plasma sources, for example, in the determination of magnetic fields in solar coronas and other stars.

A Study of the Atomic Processes of Highly Charged Ions Embedded in Dense Plasma

The study of atomic spectroscopy and collision processes in a dense plasma environment has gained a considerable interest in the past few years due to its several applications in various branches of physics. The multiconfiguration Dirac-Fock (MCDF) method and relativistic configuration interaction (RCI) technique incorporating the uniform electron gas model (UEGM) and analytical plasma screening (APS) potentials have been employed for characterizing the interactions among the charged particles in plasma. The bound and continuum state wavefunctions are determined using the aforementioned potentials within a relativistic Dirac-Coulomb atomic structure framework. The present approach is applied for the calculation of electronic structures, radiative properties, electron impact excitation cross sections and photoionization cross sections of many electron systems confined in a plasma environment. The present study not only extends our knowledge of the plasma-screening effect but also opens the door for the modelling and diagnostics of astrophysical and laboratory plasmas.

Hyperfine-induced effects on linear polarization following electron-impact excitation of heliumlike Tl79+ ions with nuclear spin

A most general density-matrix formalism is presented to investigate linear polarization of characteristic lines following electron-impact excitation of atoms or ions with arbitrary nuclear spin, which can account for depolarization of energy levels and multipole mixing of radiation fields. It is then applied to the linear polarization of the line radiated from heliumlike ions with nuclear spin . As an example, detailed calculations are performed for Tl79+ ions using the multi-configurational Dirac-Fock method and relativistic distorted-wave theory. It is found that the effect of the hyperfine interaction on the linear polarization depends dominantly on impact electron energy. For low impact energies close to the excitation threshold, the hyperfine interaction results in an enhancement of the linear polarization, especially for those photons emitted perpendicularly to the impact electron beam. In contrast, such a hyperfine-induced effect diminishes quickly with increasing impact energy and vanishes at medium and high energies, which is very different from the results for the case of radiative electron capture (Surzhykov et al 2013 Phys. Rev. A 87 052507). The present study is experimentally accessible at both electron-beam ion traps and ion storage rings and, thus, accurate polarization measurements at low energies can be utilized to probe the hyperfine interaction in highly charged few-electron ions.

Multipole mixing effects on the angular distribution and polarization of dielectronic hypersatellite lines from highly charged helium-like ions

The angular distribution and the degree of linear polarization of the dielectronic hypersatellite lines of highly charged helium-like ions are studied by using the multiconfiguration Dirac-Fock method and density matrix theory with a focus on the multipole mixing effects. Detailed calculations have been carried out for the emission of characteristic X-rays from the (2s2p)P1,23 and P11 resonant states decay to the (1s2s)S13 state following the KLL dielectronic recombination of initial hydrogen-like ions with atomic nuclear charge from Z=22 to Z=100. It is shown that the mixing of electric dipole (E1) and magnetic quadrupole (M2) radiations significantly alters the angular distribution and the degree of linear polarization, in which the shifts are dependent on the resonant capture states and the atomic nuclear charges. The E1+M2 mixing has opposite effects for the (2s2p)P23 and (2s2p)P11 resonant states, except for the (2s2p)P11 resonance of the heavy ions with Z≳90, where the profile of angular distribution is reversed by strong mixing effects due to the contribution of the M2 term and its interference with the E1 radiative channel. The results should be of particular interest to the diagnosis of anisotropic plasmas and the test of fundamental physics in few-electron heavy ions.

Theoretical investigation on extreme ultraviolet radiative opacity and emissivity of Sn plasmas at local-thermodynamic equilibrium

Sn is the material for an extreme ultraviolet (EUV) light source working at 13.5 nm, therefore the radiative properties of Sn plasma are of great importance in designing light source. The radiative opacity and emissivity of Sn plasma at local thermodynamic equilibrium are investigated by using a detailed-level-accounting model. In order to obtain precise atomic data, a multi-configuration Dirac-Fock method is used to calculate energy levels and oscillator strengths of <inline-formula><tex-math id="M3">\begin{document}${\rm{Sn}}^{6+}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M3.png"/></alternatives></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}${\rm{Sn}}^{14+}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M4.png"/></alternatives></inline-formula>. The electronic correlation effects of <inline-formula><tex-math id="M5">\begin{document}$4{\rm d}^m\text{-}4{\rm f}^m$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M5.png"/></alternatives></inline-formula>(<inline-formula><tex-math id="M6">\begin{document}$m=1, 2, 3, 4$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M6.png"/></alternatives></inline-formula>) and <inline-formula><tex-math id="M7">\begin{document}$ 4\mathrm{p}^n\text{-}4\mathrm{d}^n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M7.png"/></alternatives></inline-formula>(<inline-formula><tex-math id="M8">\begin{document}$n=1, 2, 3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20230455_M8.png"/></alternatives></inline-formula>) are mainly considered, which dominate the radiation near 13.5 nm. The number of fine-structure levels reaches about 200000 for each ionization stage in the present large-scale configuration interaction calculations. For the large oscillator strengths (> 0.01), the length form is in accord with the velocity form and their relative difference is about 20%–30%. The calculated transmission spectra of Sn plasma at 30 eV and 0.01 g/cm<sup>3</sup> are compared with the experimental result, respectively, showing that they have both good consistency. The radiative opacity and emissivity of Sn plasma at the temperature in a range of 16–30 eV and density in a scope of of 0.0001–0.1 g/cm<sup>3</sup> are investigated systematically. The effects of the plasma temperature and plasma density on radiation characteristics are studied. The results show that the radiative properties near 13.5 nm are broadened with the increase of density at a specific temperature, while it is narrowed with the increase of temperature for a specific density. The present investigation should be helpful in designing and studying EUV light source in the future.

Single and double electron excitation transition rates and oscillator strengths calculations in the study of the x-ray absorption spectrum of Neon

We report a new study of the x-ray absorption spectrum of Neon near the K-edge and near the KL-edge. The study is performed by calculating the transition probabilities and oscillator strengths for excitations from the ground state to singly and doubly excited states, the calculations were performed using multi-configurational Dirac-Fock (MCDF) method. Breit interaction between electrons and finite nuclear size effects were taken into account during the calculations, the transition probabilities were convoluted into Lorentzian line shape profiles. The results of the calculations were then compared to previous measurements of the x-ray absorption spectrum of Neon, good agreement was observed and many features in the spectrum were explained.

Updated database, semi-empirical and theoretical calculation of K[formula omitted]/K[formula omitted] intensity ratios for elements ranging from [formula omitted]Na to [formula omitted]Cm

A collection of 1118 experimental Kβ/Kα X-ray intensity ratios induced by photon impact was published recently by Daoudi et al. [1]. There were errors in some tabulated values. These errors are corrected here, as well as a substantial update including all experimental Kβ/Kα X-ray intensity ratios obtained from various sources for elements ranging from 11Na to 96Cm, when bombarded with photons, protons, deuterons, alpha particles, and electrons. More than 2100 experimental Kβ/Kα X-ray intensity ratios published between 1969 and 2020 (196 papers) are summarized in a table. A thorough analysis of these data is provided, as well as a table with weighted average intensity ratio values Kβ/KαW. Subsequently new semi-empirical average Kβ/Kα X-ray intensity ratios for elements in the range 11 ≤ Z ≤ 96 have been determined using an interpolation procedure which involves the analytical function Kβ/KαW/1−Kβ/KαW1/4 as a function of the atomic number Z and then fitting the experimental to weighted average ratios S=Kβ/KαEXP/Kβ/KαW. Furthermore, new theoretical calculations based on the Multiconfiguration Dirac–Fock Method have been performed for some elements and are presented in this work. The compilation provides a snapshot of the current status of Kβ/Kα X-ray atomic data. The need for additional quality measurements with detailed uncertainty assessments is clear. Considerable expert judgment will be needed in the future to shift through the database to create a statistically robust evaluation.

Diagnosing the electron density of channel plasma in artificially triggered lightning

Transition rates, collision excitation cross sections, and collision excitation rate coefficients of neutral N I atom and N II ions were calculated by Flexible Atomic Code that is based on the multiconfiguration Dirac-Fock method. The line intensity of some strong spectral lines were calculated. Then the dependent relationship among the ratio of line intensity I′I, the electron density ne and the electron temperature T were obtained. The calculation shows that the electron density has some dependence on the temperature and the ratio of line intensity. Furthermore, through applying the theoretical calculations to the channel plasma spectra of artificially triggered lightning, the electron densities were diagnosed at different channel locations and times. The accuracy of the electron density was estimated using the degree of accordance, which can pick out the diagnostic data that can best reflect the real state of plasma from a large number of statistical data.

Experimental and theoretical approaches for determining the K-shell fluorescence yield of carbon

The knowledge of atomic fundamental parameters, such as the fluorescence yields with low uncertainties, is of decisive importance in elemental quantification involving X-ray fluorescence analysis techniques. However, especially for the low-Z elements, the available data in the literature is either of poor quality, of unknown or very large uncertainty, or both. For this reason, the K-shell fluorescence yield of carbon was determined in the PTB laboratory at the synchrotron radiation facility BESSY II. In addition, theoretical calculations of the same parameter were performed using the multiconfiguration Dirac–Fock method, including relativistic and quantum electrodynamics (QED) corrections. Both values obtained in this work are compared to the corresponding available published data.

Energy Levels and Radiative Lifetimes for Four Times Ionized Xenon (Xe V)

By employing the general-purpose relativistic atomic structure package (GRASP) based on the fully relativistic multiconfiguration Dirac-Fock (MCDF) method, energies and radiative lifetimes of levels have been reported for four times ionized xenon (Xe V). In calculations, the Breit interactions for relativistic effects, quantum electrodynamical (QED) contributions have been included as a perturbation, as well as electron correlation effects (VV, CV, and CC). We have compared the results obtained from this work with other available experimental and theoretical values in available literature for Xe V. We have obtained good agreement with previous works.

Photoelectron spectra of shake-up processes from the 2p subshell of sodium atoms in the excited states 2p 63p

The 2p photoionization of sodium atoms from the initial states 2p 63p 1/2,3/2 have been investigated, with special emphasis placed on the relaxation and correlation effects. To understand the details of the spectra, a series of ab initio calculations based on the multi-configuration Dirac–Fock method have been performed. Theoretical predictions have been compared with the high-resolution spectra measured using a synchrotron radiation source, and suitable agreement between experiment and theory has enabled the identification of fine-structure spectra. The prominent influence of the relaxation and correlation effects on the photoelectron spectra has been analysed according to the detailed calculations.

Z -dependent crossing of excited-state energy levels in highly charged galliumlike lanthanide atomic ions

Extreme ultraviolet (EUV) optical transitions of lanthanide highly charged ions have been studied. The $[\mathrm{Ni}]4{s}^{2}4p--[\mathrm{Ni}]4{s}^{2}4d$ and $[\mathrm{Ni}]4{s}^{2}4p--[\mathrm{Ni}]4s4{p}^{2}$ transition lines of galliumlike lanthanide elements have been spectroscopically measured in large helical device (LHD) plasma and electron-beam ion trap (EBIT) plasma. The wavelengths for Eu ions agreed within $0.03%$ between the LHD and EBIT measurements. Due to the enhancement of spin-orbit interactions along with the increase of atomic number $Z$ in $4p$ atomic orbitals, energy splitting between $4{p}_{1/2}$ and $4{p}_{3/2}$ orbitals increases significantly from lanthanum to lutetium, causing the crossing of $Z$-dependent wavelength curves. The spectral line positions and strengths are theoretically calculated by means of the multiconfiguration Dirac-Fock method. The mixing of the $[\mathrm{Ni}]4{s}^{2}4d$ and $[\mathrm{Ni}]4s4{p}^{2}$ configurations leads to an avoided crossing in the apparent wavelength curves. The configuration mixing makes both the $[\mathrm{Ni}]4{s}^{2}4p--[\mathrm{Ni}]4{s}^{2}4d$ and $[\mathrm{Ni}]4{s}^{2}4p--[\mathrm{Ni}]4s4{p}^{2}$ lines well visible in the EUV spectra near the level crossing point. The levels are found to cross between $Z=62$ and 63.

Electron Impact Excitation of Ge-like Te20+–Cd16+ Ions

We study electron impact excitation of dipole allowed transitions in the extreme ultraviolet range—8–55 nm—for the germanium isoelectronic sequence Te20+–Cd16+. The fine structure transitions between the ground state having configuration 4s24p2 and the excited states with configurations 4s4p3 and 4s24p4d are considered for these ions. We employ the relativistic distorted wave method to calculate the excitation cross sections in the incident electron energy range from the excitation threshold to 5000 eV. To obtain the required ionic bound state wavefunctions we have used the multi-configuration Dirac-Fock method with correlations within the n = 5 complexes as well as performed relativistic configuration interaction calculations to include the quantum electrodynamic effects. The accuracy of these wavefunctions is established by comparing our calculated wavelengths and oscillator strengths of the considered transitions with the previously reported measurements and other available theoretical results. We also provide the fitting parameters of the calculated cross sections and the excitation rate coefficients for their direct applications in plasma modeling.

Non-dipole effects on angular distribution of photoelectrons in sequential two-photon double ionization of Ar atom and K<sup>+</sup> ion

Owing to the development of XUV and X ray of the free-electron lasers, the photoelectron angular distribution in the sequential two-photon double ionization has received increasing attention of theorists and experimentalists, because it provides the valuable information about the electronic structure of atom or molecule systems and allows the obtaining of additional information about mechanisms and pathways of the two-photon double ionization. In this paper, the expression of the sequential two-photon double ionization process of the photoelectron angular distributions, including the non-dipole effects, is obtained based on the multi-configuration Dirac-Fock method and the density matrix theory, and the corresponding calculation code is also developed. Based on the code, the sequential two-photon double ionization process of the 3p and 2p shells of Ar atom and K<sup>+</sup> ion are studied, in which, the dipole and the non-dipole parameters of photoelectron angular distribution are investigated systematically. It is found that the angular distributions of the first- and second-step electrons in sequential two-photon double ionization are similar and the two photoionization processes affect each other. Near the ionization threshold, the photoionization cross-sections and anisotropy parameters for the 3p shell and the 2p shell show a large difference. While away from the threshold, the cross-section and angular anisotropy parameters of the 3p and 2p shells show similar behaviors. At the position of Cooper minimum of the photoionization cross section, the contribution of the electric dipole is suppressed, and the non-dipole effect is obvious. The non-dipole effect leads to a forward-backward asymmetric distribution of photoelectrons relative to the direction of incident light. The results of this paper will be helpful in studying the nonlinear processes of photon and matter interaction in the XUV range.

Plasma environment effects on K lines of astrophysical interest

Aims. We calculate the plasma environment effects on the ionization potentials (IPs) and K-thresholds used in the modeling of K lines for all the ions belonging to the isonuclear sequences of abundant elements apart from oxygen and iron, namely: carbon, silicon, calcium, chromium, and nickel. These calculations are used to extend the data points for the fits of the universal formulae, first proposed in our fourth paper of this series, to predict the IP and K-threshold lowerings in any elemental ion. Methods. We used the fully relativistic multi-configuration Dirac–Fock method and approximated the plasma electron-nucleus and electron-electron screenings with a time-averaged Debye–Hückel potential. Results. We report the modified ionization potentials and K-threshold energies for plasmas characterized by electron temperatures and densities in the ranges of 105−107 K and 1018−1022 cm−3. In addition, the improved universal fitting formulae are obtained. Conclusions. We conclude that since explicit calculations of the atomic structures for each ion of each element under different plasma conditions is impractical, the use of these universal formulae for predicting the IP and K-threshold lowerings in plasma modeling codes is still recommended. However, their comparatively moderate to low accuracies may affect the predicted opacities with regard to certain cases under extreme plasma conditions that are characterized by a plasma screening parameter of μ > 0.2 a.u., especially for the K-thresholds.

Electron Impact Excitation of Extreme Ultra-Violet Transitions in Xe7–Xe10 Ions

In the present work, a detailed study on the electron impact excitation of Xe7+, Xe8+, Xe9+ and Xe10+ ions for the dipole allowed (E1) transitions in the EUV range of 8–19 nm is presented. The multi-configuration Dirac–Fock method is used for the atomic structure calculation including the Breit and QED corrections along with the relativistic configuration interaction approach. We have compared our calculated energy levels, wavelengths and transition rates with other reported experimental and theoretical results. Further, the relativistic distorted wave method is used to calculate the cross sections from the excitation threshold to 3000 eV electron energy. For plasma physics applications, we have reported the fitting parameters of these cross sections using two different formulae for low and high energy ranges. The rate coefficients are also obtained using our calculated cross sections and considering the Maxwellian electron energy distribution function in the electron temperature range from 5 eV to 100 eV.

Fine structure calculations of excitation energies, lifetimes and radiative properties of S-like Kr XXI

Extensive and systematic calculations for energy levels and lifetimes are carried out for the lowest 145 levels arising from the 3s23p4, 3s3p5, 3s23p33d, 3p6, 3s3p43d and 3s23p23 d2 configurations in S-like Kr XXI ion using the multiconfiguration Dirac-Fock (MCDF) method. Additionally, radiative data such as transition wavelengths, absorption oscillator strengths, line strengths and radiative decay rates have also been computed for electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1) and magnetic quadrupole (M2) transitions from the ground state. The importance of Breit interaction (BI) and quantum electrodynamics (QED) corrections on the energy levels has also been estimated. To assess the credibility and integrity of our MCDF energies, independent calculations using the relativistic configuration interaction (RCI) approach have also been performed. Detailed comparisons are made between our two sets of results, as well as with other existing experimental and theoretical values, and a good agreement is achieved. The present results will be helpful for the modeling and diagnostics of fusion plasmas.

Forbidden transitions between the 2p3/2 and 2p1/2 states of ground B-like ions with 14 ≤ Z ≤ 92

The magnetic-dipole M1 and electric quadrupole E2 transition energies and rates between the two fine structure states of 1s22s22p configuration for B-like ions with 14 92 have been calculated using the multi-configuration Dirac-Fock (MCDF) method. The interactions with atomic core have been considered and the core-valence correlation contributions to the level energies are evaluated in the active space approximation. The finite nuclear size, the frequency dependent and independent Breit interaction and quantum electrodynamics (QED) corrections have been included in the calculations. To improve the accuracy of the present data on the forbidden line energies and rates, the QED corrections to the transition energies have been adjusted using QEDMOD code with MCDF wavefunctions. The sensitivity of the transition parameters on the nature of the spin orbitals has been probed by considering two sets of calculations, one with fully relaxed and the other with a common set of radial functions. The present line energies and lifetime of the 2s22p 2 p 3/2 state are found to be in excellent agreement with available experimental and theoretical data.

Sensitivity of correlation effects and Shannon entropy in Be@C60 to the nature of confinement potentials

The multi-configuration Dirac-Fock (MCDF) method is used to study the correlation effects in Be@C60 by investigating the localization properties of the electronic probability density distribution. The confinement environment is simulated by two types of model potentials: (A) the commonly employed compact annular square well (ASW) and (B) the diffusive Gaussian annular square well (GASW). The present work aims to investigate the sensitivity of correlation effects and localization properties to the contrasting nature of the confinement potentials. To accomplish this, the variation of the correlation energy and Shannon’s correlation information entropy with the depth of both confinement potentials are presented to give an idea of the sensitivity of these parameters to the nature of the confining potential. The present work concludes that the correlation energy and the associated electronic structure attributes are extremely susceptible to the nature of the confining potential.