Dirac Fock Research Articles

Calculations of energy levels, radiative transition parameters, hyperfine structure constants [formula omitted] - [formula omitted], Landé [formula omitted] factors and isotope shifts for Sc XX using the MCDF-RCI method

Large scale calculations for the energy levels, transition rates, oscillator strengths, lifetimes, hyperfine interaction constants, Landé gJ factors, and isotope shift factors have been performed for 1s2 and 1snl(n=2−8andl≤n−1) levels of He-like Sc XX ion. The general-purpose relativistic atomic structure package (GRASP2018) based on the fully relativistic multiconfiguration Dirac–Fock (MCDF) method is used to carry out the calculations. The leading quantum electrodynamic corrections, Breit interaction and nuclear recoil effects are also included in the succeeding relativistic configuration interaction (RCI) calculations. The relativistic isotope shift (RIS4) programme is used to determine the mass and field shifts factors. Furthermore, the percentage uncertainty in the transition parameters and lifetimes is estimated. A detailed comparison of the present results with the corresponding values from the NIST database and other theoretical and experimental works, wherever available, has been done, and an excellent agreement is achieved. A large section of the results is reported for the first time in the present work.

Relativistic calculations of molecular electric dipole moments of singly charged aluminium monohalides

In this work, we have studied the permanent electric dipole moments of singly charged aluminum monohalides in their electronic ground state, , using Kramers-restricted relativistic configuration interaction method. We report our results from this method in the singles and doubles approximation with those of Dirac–Fock calculations. For our finite field computations, quadruple zeta basis sets were employed. We discuss the electron correlation trends that we find in our calculated properties and have compared our results with those from literature, wherever available.

Detailed Analysis of Spectra from Ga-like Ions of Heavy Elements Observed in High-Temperature Plasmas

This study has systematically investigated the atomic number (Z) dependence of spectra from gallium-like (Ga-like) ions of heavy elements. We have mainly analyzed the experimental spectra recorded in high-temperature plasmas produced in the Large Helical Device (LHD) for various elements with atomic numbers from 57 onward. The measured wavelengths are compared with theoretical values calculated with a multi-configuration Dirac Fock code. As a result, we have successfully obtained Z-dependent wavelengths of several prominent transitions of Ga-like ions, including a magnetic dipole (M1) transition. Many of them have been experimentally identified for the first time in this study. The present results manifest the significant effects of configuration interaction and spin–orbit interaction for highly charged heavy ions.

Douglas-Kroll and infinite order two-component transformations of Dirac-Fock operator.

We extended the conventional Douglas-Kroll (DK) and infinite order two-component (IOTC) methods to a technique applicable to Fock matrices, called extended DK (EDK) and extended IOTC (EIOTC), respectively. First, we defined a strategy to divide the Dirac-Fock operator into zero- and first-order terms. We then demonstrated that the first-order extended DK transformation, which is the Foldy-Wouthuysen transformation for the zero-order term, as well as the second- and third-order EDK and EIOTC, could be well defined. The EDK- and EIOTC-transformed Fock matrix, kinetic energy operator, nuclear attraction operator, and density matrix were derived. These equations were numerically evaluated, and it was found that these methods were accurate. In particular, EIOTC was consistent with the four-component approach. Four-component and extended two-component calculations are more expensive than non-relativistic calculations due to small-component-type two-electron integrals. We developed a new approximation formula, RIS-V, for small-component-type two-electron integrals, including the spin-orbit interaction between electrons. These results suggest that the RIS-V formula effectively accelerates the four-component and extended two-component methods.

Detection of cesium in the atmosphere of the hot He-rich white dwarf HD 149499B

ABSTRACT We report the first detection of cesium (Z = 55) in the atmosphere of a white dwarf. Around a dozen absorption lines of Cs iv, Cs v, and Cs vi have been identified in the Far Ultraviolet Spectroscopic Explorer spectrum of the He-rich white dwarf HD 149499B (Teff = 49 500 K, log g = 7.97). The lines have equivalent widths ranging from 2.3 to 26.9 mÅ. We performed a spectral synthesis analysis to determine the cesium content in the atmosphere. Non-local thermodynamic equilibrium atmosphere models were computed by considering cesium explicitly in the calculations. For this purpose we calculated oscillator strengths for the bound–bound transitions of Cs iv–Cs vi with both autostructure (multiconfiguration Breit–Pauli) and grasp2k (multiconfiguration Dirac–Fock) atomic structure codes as neither measured nor theoretical values are reported in the literature. We determined a cesium abundance of log N(Cs)/N(He) = −5.45 ± 0.35, which can also be expressed in terms of the mass fraction log XCs = −3.95 ± 0.35.

Transition energies and transition probabilities for the core-excited state [formula omitted] of Li-like ions with [formula omitted] 6–92

The transition energies and radiative transition probabilities for the transition from the (1s2s2p)3/2 group to the ground state 1s22s involving the electric dipole (E1) and magnetic quadrupole (M2) transitions have been systematically calculated for ions along the Li isoelectronic sequence from Z=6 to Z=92. The calculations are performed by using the full relativistic multi-configuration Dirac–Fock (MCDF) method, in which the leading quantum electrodynamics (QED) shifts and the frequency dependence of the Breit interaction are also considered. The present calculation results are in good agreement with available previous results. It is found that the transition probabilities generally increase along with nuclear number Z; however, the abnormal behavior with the minima transition probabilities at Z=28 and Z=25 emerges in the [(1s2s)02p3/2]3/2→1s22s via E1 transition and the [(1s2s)12p3/2]3/2→1s22s via M2 transition, respectively; the M2 transition probabilities become comparable with that of E1 in mid-Z ions, and particularly for the q line of Z=28 ion, the M2 and E1 transition probabilities would be of the same order of magnitude. In addition, the present results show that the electron correlations impact on the energy levels and transition probabilities mainly work on the low-Z ions, while, the QED and Breit effects mainly work on the high-Z ions. This work should be of particular interest to the analysis of the spectrum and the effects of multipole mixing on the X-ray properties.

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.

K- and L-shell theoretical fluorescence yields for the Fe isonuclear sequence

In this work, we present K- and L- shell fluorescence yield values of the full isonuclear sequence of Fe ions, using a state-of-the-art multiconfiguration Dirac–Fock approach. These results may be of importance for spectral fitting and plasma modeling, both in laboratory and astrophysical studies, where Fe is an important benchmark element. The K-shell fluorescence yields were found to be very similar up to the removal of 14 electrons.

Combined effect of plasma screening and electric field on the He atom

The combined effect of plasma screening and electric field on the structural properties of atoms/ions is investigated with the help of the relativistic computational method. Two concepts, the ion-sphere model potential and the electric field perturbation operator, which are crucial to perform the current investigation, are introduced to solve the Dirac–Fock equations based on the self-consistent field theory. The developed algorithm is applied to calculation of the plasma-embedded He atom in the case of the field strength 0 ≤F≤ 0.1 a.u. The influences of plasma screening and applied electric field strength, separately and combined, on the energies and radiative transition properties are analyzed in detail. Our results reveal that adding shielding effect destabilizes the atomic system (reduces the energy eigenvalues), and the applied electric field counteracts this effect, thereby increasing the stability of the atomic system. On the whole, the effect of electric field on the electronic structures and spectral properties of the guest atom is a deflection shielding effect, implying that the latter effect is stronger. The present results are in good agreement with the values from the previous literature.

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.

Electron impact excitation cross section calculations of the fine structure transitions of Mo and their applications in the diagnostics of the laser induced Mo plasma

The current work reports a detailed calculation of electron impact excitation cross sections for the fine structure transitions of Mo from the manifolds 4d 55s, 4d 45s 2, 4d 6 and 4d 55p to the manifolds 4d 55s, 4d 45s 2, 4d 6, 4d 55p, 4d 45s5p, 4d 55d, 4d 56s, 4d 45s6s, and 4d 57s, using the relativistic distorted wave approximation for the applications in plasma modelling. Multi-configurational Dirac–Fock wave functions are used in the calculations. The oscillator strength and cross section results are compared with the previous calculations and measurements. A comprehensive collisional radiative (CR) model is developed and used to characterize laser induced molybdenum plasma to ensure that the calculated cross sections can be used for various plasma modelling applications. The current CR model has taken into account the electron impact excitation and de-excitation processes using the calculated consistent cross sections. The electron induced processes are dominant kinetic processes in the laser induced plasma. Furthermore, the diagnostics of the laser induced Mo plasma is done by coupling the current CR model with the experimental laser induced breakdown spectroscopic measurements of Mal et al (2021 Appl. Phys. B 127 52). The plasma parameter, i.e., electron temperature has been calculated using nine measured intensities of the emission lines of Mo, with wavelengths 406.9, 423.3, 438.2, 453.7, 476.0, 550.7, 553.3, 557.0 and 592.9 nm. The results are also compared with the values reported in the Boltzmann plot at various delay times ranging from 0.5 to 5.0 μs.

Contribution of near-edge processes to the L X-ray emission lines for various lanthanum (III) compounds

The present measurement illustrates the study of near-edge processes and the intensity ratios [ILl/ILα, ILβ1/ILα and ILl/ILβ1] for Mn Kα1,2 X-rays across the L2 edge of lanthanum in its various chemical forms viz., lanthanum oxide (La2O3), lanthanum oxalate (C6La2O12) and lanthanum acetate (C6H9LaO6) at four different scattering angles i.e. 90°, 105°, 115° and 125° with a lab-scale X-ray fluorescence (XRF) spectrometer. The current observation reveals that the Lβ1 X-ray emission lines exhibit a variance of ∼ (21–40) % between measured and calculated fluorescence cross-sections. Apart from this, the measured values of intensity ratios have been compared with the three data sets of calculated values using L shell emission rates based on the Dirac-Fock (DF) model, total X-ray emission rates and photoionization cross-section values based on the relativistic Hartree-Slater (RHS) model and three sets of Coster-Kronig (CK) and fluorescence yields. The correlation between crystal structure, bond distance and intensity ratio are also discussed in this work.

Relativistic Two-Photon Matrix Elements for Attosecond Delays

The theory of one-photon ionization and two-photon above-threshold ionization is formulated for applications to heavy atoms in attosecond science by using Dirac–Fock formalism. A direct comparison of Wigner–Smith–Eisenbud delays for photoionization is made with delays from the Reconstruction of Attosecond Beating By Interference of Two-photon Transitions (RABBIT) method. Photoionization by an attosecond pulse train, consisting of monochromatic fields in the extreme ultraviolet range, is computed with many-body effects at the level of the relativistic random phase approximation (RRPA). Subsequent absorption and emission processes of infrared laser photons in RABBIT are evaluated by using static ionic potentials as well as asymptotic properties of relativistic Coulomb functions. As expected, light elements, such as argon, show negligible relativistic effects, whereas heavier elements, such a krypton and xenon, exhibit delays that depend on the fine-structure of the ionic target. The relativistic effects are notably close to ionization thresholds and Cooper minima with differences in fine-structure delays predicted to be as large as tens of attoseconds. The separability of relativistic RABBIT delays into a Wigner–Smith–Eisenbud delay and a universal continuum–continuum delay is studied with reasonable separability found for photoelectrons emitted along the laser polarization axis in agreement with prior non-relativistic results.

Study of electron impact elastic scattering from Kr@C60 and Xe@C60 using a fully relativistic approach

In the present work, a detailed study has been reported on electron impact elastic scattering from krypton (Kr) and xenon (Xe) atoms when confined in two different types of C60 potentials viz (a) hard annular square well (ASW) and (b) diffused Gaussian annular square well (GASW). The Dirac equations are solved using these potentials for encaged Kr and Xe in C60. First, bound state Dirac–Fock wave functions of these encaged Kr and Xe atoms are found by utilizing modified general relativistic atomic structure package and thereafter, the charge densities and static potentials of the endohedral Kr@C60 and Xe@C60 are obtained. Further, using these, the Dirac equations are solved by the relativistic partial wave phase shift analysis method and the scattering amplitudes in terms of phase shifts are obtained. Thereafter, the electron elastic differential and integrated cross sections of Kr@C60 and Xe@C60 along with the C60 are calculated in the range of 0.1–15 eV incident electron energies. Presently, no experimental and theoretical results are available to compare our electron scattering cross section results from Kr@C60 and Xe@C60; thus, we have shown the cross section results obtained from ASW and GASW potential and compared them.

Unusual behavior of Cooper minima of ns subshells in high- Z atoms

A study of Cooper minima (CM) arising from the photoionization of $6s$, 5s, and $4s$ subshells of high-$Z$ atoms has been performed using Dirac-Fock (DF), two-channel relativistic-random-phase approximation (RRPA), and fully coupled RRPA. The results show huge splittings between $ns\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{3/2}$ and $\mathrm{ns}\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{1/2}$ CM which increase with $Z$ owing primarily to the relativistic interactions (spin orbit) that are attractive for the $\ensuremath{\varepsilon}{p}_{1/2}$ final state but repulsive for the corresponding $\ensuremath{\varepsilon}{p}_{3/2}$. In addition, it was found that correlation in the form of interchannel coupling (essentially configuration interaction in the final continuum states) plays a huge role in determining the location of the CM. For $6s$ photoionization, the $6s\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{3/2}$ and $6s\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{1/2}$ CM behave completely different as a function of $Z$. It was also found that for $5s$ and $4s$ photoionization, the CM move below the threshold, with increasing $Z$, and, at high enough $Z$, the $5s\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{3/2}$ and $4s\ensuremath{\rightarrow}\ensuremath{\varepsilon}{p}_{3/2}$ CM re-emerge into the continuum. The calculations have been carried out for the $ns$ subshells of Hg ($Z=80$), Rn ($Z=86$), Ra ($Z=88$), No ($Z=102$), Cn ($Z=112$), and Og ($Z=118$).

Energy levels and magnetic dipole transition parameters for the nitrogen isoelectronic sequence

Theoretical calculations of the energy levels and magnetic dipole transition parameters for the 1s22s22p3 and 1s22p5 configurations of nitrogen isoelectronic sequence with Z = 21–30 are performed using multi-congfiguration Dirac–Fock (MCDF) method. Based on the relativistic computational code GRASP2k compiled within the framework of MCDF method, the electron correlations, Breit interaction and QED effects are well treated in detail. The energy levels, line strengths and transition rates of magnetic dipole transition are obtained and compared with the experimental data available. For most cases, good agreements are achieved and the relative differences of them are less than 0.114%, 8.43% and 9.80%, respectively. The scaling laws of the fine structure splitting and transition rate are obtained on the isoelectronic sequence and the corresponding physical mechanisms are discussed. The data sets for tables are openly available at https://www.doi.org/10.57760/sciencedb.j00113.00022.

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.

Measurements of L-shell X-ray production cross sections for Sn and Sb using 6–14 keV synchrotron radiation

The synchrotron radiation induced Lkk=l,η,α,β1,3,4,6,β2,15,9,10,γ1,5,γ2,3,4 X-ray production cross-sections (XRPCS) have been measured for Sn and Sb at energies ranging 6–14 keV. These present measured Lk XRPCS have been compared with different sets of theoretical cross sections calculated using the Dirac-Fock (DF) model based X-ray emission rates, the non-relativistic Hartree-Fock-Slater (HFS) model based photoionization cross-sections and three sets of fluorescence and Coster-Kronig (CK) yields based on the Dirac-Hartree-Slater (DHS) model, the semi-empirical values tabulated by Krause, and the recently reported experimental values. The present measured Lk XRP cross sections for Sn and Sb are found to differ by up to 25% from different sets of presently calculated values. Further, the average L-shell fluorescence yields (ω¯L) have been deduced from the present measured total L XRPCS using the theoretical photoionization cross sections and are compared with the values available in literature.

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.

Development of various methods to the investigation of the spectral properties and collision dynamics of H‐like ions taking place in dense and hot plasma environments

AbstractThe correct understanding of the behavior of highly stripped ions immersed in a dense plasma environment is a hitherto challenge for theory and it is of importance in various applications. Here, we developed four kinds of novel theoretical models to the determination of the spectral properties and collision dynamics of H‐like ions in dense and hot plasmas by employing the analytical potential [Li et al., Phys. Plasmas 2019, 26, 033301] to describe the interactions among the charged particles: (i) The electron photon processes in plasma model within the relativistic scheme; (ii) The multiconfiguration Dirac–Fock model within the framework of relativistic self‐consistent iteration approximation; (iii) The generalized pseudospectral method within the relativistic framework; and (iv) An analytic formula based on the Hartree–Fock method and the irreducible tensor theory within the non‐relativistic regime, derived from a solution of Schrödinger equation. As a typical example, energy eigenvalues, radiative transition properties, spectral line shifts, and electron impact excitation and ionization cross sections of the selected N6+ and Ne9+ ions are determined for several temperature‐density cases, characteristic of inertial confinement fusion plasmas. A comparison of these results with each other and the results from earlier calculations as well as with the available experimental data is provided. Systematic trend is determined for all the properties under study concerning increased screening. The present study not only extends our understanding of the dense plasma shielding effects, but also opens the way for future investigations allowing accurate predictions of spectral properties of ions in dense and hot plasmas aimed at providing precision data for various practical applications.