Dirac Fock Research Articles

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.

Breit interaction in dielectronic recombination of hydrogenlike xenon ions: storage-ring experiment and theory

Abstract Electron-ion collision spectroscopy of the KLL dielectronic recombination (DR) resonances of hydrogenlike xenon ions was performed at a heavy-ion storage ring with a resolving power that is competitive with x-ray spectroscopy of inner-shell transitions in highly charged ions. The $$KL_{1/2}L_{1/2}$$ K L 1 / 2 L 1 / 2 , $$KL_{1/2}L_{3/2}$$ K L 1 / 2 L 3 / 2 , and $$KL_{3/2}L_{3/2}$$ K L 3 / 2 L 3 / 2 resonance groups and even parts of their fine structure are individually resolved. The resonance strengths were measured on an absolute scale and compared with results from multi-configuration Dirac–Fock (MCDF) calculations. These are in excellent agreement with the experimental findings when QED effects on the resonance energies and the Breit interaction are considered. As already found for DR of hydrogenlike uranium (Bernhardt et al. in Phys Rev A 83:020701(R), 2011), this interaction is particularly strong for the $$KL_{1/2}L_{1/2}$$ K L 1 / 2 L 1 / 2 resonance group. For U$$^{91+}$$ 91 + , it increases the $$KL_{1/2}L_{1/2}$$ K L 1 / 2 L 1 / 2 DR resonance strength by 40%. For Xe$$^{53+}$$ 53 + , the increase is found to amount to 25%, confirming the prediction that the influence of the Breit interaction grows with increasing nuclear charge. A comprehensive appendix treats the derivation of experimental and theoretical merged-beams recombination rate coefficients for interacting beams of relativistic electrons and ions. Graphical abstract

Investigating empirical and theoretical calculations for intensity ratios of L-Shell X-ray transitions in atoms with 39 ≤ Z ≤ 94

The significance of theoretical, experimental, and analytical methods in calculating the intensity ratios of L-shell transitions for diverse elements lies in their widespread applications across various domains, including physical chemistry and medical research. In the present paper, empirical values for intensity ratios were computed through polynomial interpolations using experimental databases within the scope of atomic number 39 ≤ Z ≤ 92 for ILβILα and ILγILα, and extending to the range of 39 ≤ Z ≤ 94 for ILlILα. Additionally, new theoretical calculations were conducted using the Multiconfiguration Dirac–Fock Method for specific elements. The obtained results were compared with standard theoretical, experimental, and empirical values, showing a reasonable agreement with them.

Relativistic calculations for total energies, ionization energies, and one-electron binding energies for Neodymium ions Nd I to Nd[formula omitted]

We present the results of calculations for the total energies, ionization energies, and one-electron binding energies for ground-state configurations of neodymium ions Nd I to Nd59+. These calculations are based on the Dirac–Fock approximation taking the Breit and quantum electrodynamics corrections into account. The configuration interaction method, taking into account all relativistic configurations corresponding to the non-relativistic one, is used to obtain total energies and wave functions in the intermediate coupling scheme. Comparison is given with other available data.

Existence of Minimizers for the Dirac–Fock Model of Crystals

Existence of Minimizers for the Dirac–Fock Model of Crystals

Accurate ab initio calculation and neural network prediction of the atomic properties of Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI

In this paper, we present an ab initio theoretical calculation of the atomic parameters, including energy levels, wavelengths, lifetimes, and transition parameters, associated with the 1s22snl and 1s22pnl (n= 2–4, l≤n−1) configurations of Be-like sequence: Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI. Our analysis is based on the relativistic wavefunctions derived from the multi-configuration Dirac–Fock (MCDF) method, which are implemented in the relativistic atomic structure package GRASP2018. The correlations within the (principal quantum number) n= 10 complex are accounted for. The Breit interaction and quantum electrodynamical effects are added in the relativistic configuration interaction (RCI) calculation. Our results are compared with the existing data. The uncertainties of the dipole transition line strengths are assessed. In addition, we propose a feed forward neural network to predict the energy levels of Au LXXVI. This is a powerful machine learning tool capable of learning from existing data and predicting unknown data. The network is trained using theoretical energy levels of Os LXXIII, Ir LXXIV, and Pt LXXV obtained from the MCDF method. Our neural network exhibits average relative deviations of approximately 0.01% and 0.02% for trained and predicted energy levels, respectively, when compared to the MCDF results. This level of deviation suggests that our results are reasonably accurate. The data set presented in this study is useful for modeling fusion plasmas.

Large-scale relativistic calculations of spectral data in Zr XXVIII, Nb XXIX and Tc XXXI of tokamak interest

Large-scale relativistic multiconfiguration Dirac–Fock calculations of energy levels and lifetimes of the 206 states arising from the 3s23p, 3p3, 3s3p3d, 3p3d2, 3s24p, 3s24f, 3s3p2, 3s23d, 3s3d2, 3s24d, 3s24s, 3s3p4s, 3s3p4d, 3p23d, 3d3, and 3s3p4p configurations of Zr XXVIII, Nb XXIX, and Tc XXXI of tokamak interest are carried out. The active space approximation is employed for the calculations. To obtain accurate and convergent results, the calculations take into account the valence and core-valence correlations within the n= 9 complex, as well as the Breit interaction, self-energy corrections, and vacuum polarization corrections. Detailed information regarding the decay properties of these ions is provided, including the transition wavelengths, line strengths, and radiative transition rates for various types (electric dipole, electric quadrupole, magnetic dipole, and magnetic quadrupole transitions) of transitions among the levels of the aforementioned configurations. The uncertainties of each dipole transition are evaluated. The present data sets are compared to previous results and a good agreement is observed, which is valuable for emission line identification and fusion modeling, especially in situations where experimental data are scarce.

Measurement of infrared magic wavelength for an all-optical trapping of 40Ca+ ion clock

For the first time, we experimentally determine the infrared magic wavelength for the 40Ca+ 4s2S1/2→3d2D5/2 electric quadrupole transition by observation of the light shift canceling in 40Ca+ optical clock. A ‘magic’ magnetic field direction is chosen to make the magic wavelength insensitive to both the linear polarization purity and the polarization direction of the laser. The determined magic wavelength for this transition is 1056.37(9) nm, which is not only in good agreement with theoretical predictions (‘Dirac–Fock plus core polarization’ method) but also more precise by a factor of about 300. Using this measured magic wavelength, we also derive the differential static polarizability to be −44.32(32) a.u., which will be an important input for the evaluation of the blackbody radiation shift at room temperatures. Our work paves a way for all-optical-trapping of 40Ca+ optical clock.

Fine structure resolved excitation cross sections of singly ionized Ga for the modeling and diagnostics of Ga plasmas

Calculation of electron impact excitation cross sections for singly charged Ga ions plays a crucial role in plasma modeling, facilitating the comprehension of plasma behavior, characteristics, and dynamics in diverse domains, such as astrophysics, fusion research, the semiconductor industry, etc. In the available literature, there is a notable scarcity of, or even a complete absence of, these cross sections. Hence, in the present work, electron impact excitation cross sections are calculated for the transitions from the fine structure resolved energy levels of the configurations 4s2 and 4s4p to the fine structure resolved energy levels of the configurations 4s4p , 4s5s , 4p2 and 4s4d of the singly charged Ga ion (Ga+) using the relativistic distorted wave approximation theory with the target states represented by multi configurational Dirac Fock wavefunctions. The cross sections are calculated for projectile electron energy varying from threshold to 500 eV. Furthermore, the electron impact excitation rate coefficients for all the transitions under investigation are also calculated for electron temperatures ranging from 0.5 to 5 eV. In addition, analytic fitting of the rate coefficients is also performed, providing a practical resource for directly utilizing in plasma modeling applications.

Lisbon Atomic Database (LISA): a compilation of calculated fundamental atomic parameters

The Lisbon Atomic Database (LISA) has a dedicated mission of compiling and providing a comprehensive collection of atomic parameters for the study of the interaction of X-rays with matter. This encompassing array of parameters spans a broad spectrum, extending from the calculation of electron impact ionization cross sections (EIICS) using the Modified Relativistic Binary Encounter Bethe model (MRBEB) , to pivotal data such as fluorescence and Coster–Kronig yields of atomic subshells, binding energies, and the full suite of radiative and non-radiative atomic transition parameters. Except for the EIICS values, all these parameters are obtained through ab initio calculations. These calculations are carried out using a self-consistent ab initio Multi-Configuration Dirac–Fock approach, supported by a specialized code developed by Desclaux, Indelicato, and others (MCDFGME).Graphical abstract

A rigorous justification of the Mittleman’s approach to the Dirac–Fock model

A rigorous justification of the Mittleman’s approach to the Dirac–Fock model

Basis Set Calculations of Heavy Atoms

Most modern calculations of many-electron atoms use basis sets of atomic orbitals. An accurate account for electronic correlations in heavy atoms is a very difficult computational problem, and an optimization of the basis sets can reduce computational costs and increase final accuracy. Here, we propose a simple differential ansatz to form virtual orbitals from the Dirac–Fock orbitals of the core and valence electrons. We use basis sets with such orbitals to calculate different properties in Cs including hyperfine structure constants and QED corrections to the valence energies and to the E1 transition amplitudes.

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.

Theoretical investigation on the spectra of M1 transition of W53+ ion

The properties and spectrum of the highly charged tungsten ions draw more and more attention, due to its potential application in the next generation fusion reactor. The transition wavelength and rate of the magnetic-dipole transition between the levels of the ground configuration of W53+ ion in the range of 10−20 nm were calculated by Dirac–Fock–Slater (DFS) method with a local central potential and Multi-Configuration Dirac-Hartree-Fock (MCDHF) method, respectively. The results are consistent with the previous EBIT experimental data and other theoretical data. A collisional-radiative model was constructed, and the spectrum of magnetic-dipole transition of W53+ ion observed on EBIT (Y. Ralchenko et al., Phys. Rev. A. 83, 032517, 2011) was reproduced theoretically. Another two new transitions near the wavelength of 16.683 nm and 18.018 nm are also assigned to the M1 transition of W53+ ion.

A New Definition of the Dirac–Fock Ground State

A New Definition of the Dirac–Fock Ground State

Relativistic and magnetic Breit effects for the reaction Sg + 6CO → Sg(CO)6 and Sg(OC)6: Prediction of the existence and atomization energy of the isomer Sg(OC)6

Our ab initio all-electron fully relativistic Dirac–Fock (DF) and Dirac–Fock–Breit–Gaunt (DFBG) and nonrelativistic (NR) Hartree–Fock (HF) calculations for seaborgium hexacarbonyl Sg(CO)6 predict atomization energies (Ae) of 68.81, 69.28, and 67.69 eV, respectively, at the corresponding optimized octahedral geometry. However, our DF, DFBG, and NR HF calculations for the isomer Sg(OC)6 yield atomization energies of 64.30, 64.77, and 62.62 eV, respectively, at the optimized geometry for this species. The optimized Sg–C and C–O bond distances (in Å) for octahedral Sg(CO)6 using our DF (NR) calculations are 2.15 (2.32) and 1.11 (1.11), respectively. However, the optimized Sg–O and O–C bond distances (in Å) for the isomer octahedral Sg(OC)6 obtained with our DF (NR) calculations are 2.80 (2.73) and 1.10 (1.11), respectively. Our prediction of the greater stability of Sg(CO)6 isomer at both the relativistic (DF and DFBG) and the NR HF levels of theory lends further support to the detection of Sg(CO)6 in the state-of-the-art gas-phase experimental studies of the carbonyl complex of seaborgium reported by Even et al. [Science 345(6203), 1491 (2014)].

Measurement and modelling of resonant inelastic X-ray scattering and many-body processes using extended-range high energy resolution fluorescence detection and multiconfigurational Dirac–Fock calculations

Measurement and modelling of resonant inelastic X-ray scattering and many-body processes using extended-range high energy resolution fluorescence detection and multiconfigurational Dirac–Fock calculations

Electron impact fine-structure cross-section calculations of Cu and their applications in the diagnostics of laser-produced Cu-plasma through the collisional radiative model

We present the calculations of electron impact excitation cross-sections for the transitions from the ground state of Cu 3d104sS1/22 to the excited fine structure energy levels of the configurations 3d94s2,3d94s4p,3d104l,(l=p,d,f),3d105l,(l=s,p,d),3d10nl,(n=6,7;l=s,p) using the fully relativistic distorted wave theory. In addition, cross-sections for the transitions from the metastable states 3d94s2D5/2,3/22 to the upper excited levels are also presented. For calculating these cross-sections, we have obtained the atomic wave functions of copper using the multi-configuration Dirac–Fock approximation theory. To ensure the accuracy of these wave functions, we compared our calculated oscillator strengths of some dipole-allowed transitions of Cu with the available earlier reported values. We have utilized all the calculated electron impact fine-structure level excitation cross-sections from the ground as well as the metastable states of Cu to develop a fine structure-resolved collisional-radiative (CR) model for the diagnostics of laser-produced Cu plasma at atmospheric pressure. The model has been applied for diagnostics by coupling it with the spatially resolved optical emission spectroscopy (OES) measurements of Singh and Sharma [Phys. Plasmas 23 122104 (2016)] in the presence and absence of a static magnetic field. For diagnostics, the intensities of three OES observed strong atomic emission lines of Cu viz, 510.5, 515.3, and 521.8 nm have been utilized in the absence and presence of a magnetic field at different axial lengths (z = 0.5–6.5 mm) from the Cu target. Also, these intensities are corrected using the internal reference self-absorption method to overcome the effect of radiation trapping. The normalized intensities of the considered lines obtained from the CR model have been compared with their corresponding OES-measured values to extract the plasma parameters, i.e., electron temperature and electron density. Further, the obtained electron temperatures from the CR model are compared with the corresponding values reported in the experimental paper of Singh and Sharma [Phys. Plasmas 23 122104 (2016)] obtained using the Boltzmann plot method.

Energy levels and radiative transition properties of the 2s2p double K-shell vacancy state in He-like ions ([formula omitted

The energy levels of 1s2, 1s2s, and 2s2p configurations for He-like ions (4≤Z≤54) are calculated using the multi-configuration Dirac–Hartree–Fock​ methods. The transition energies, transition rates, and oscillator strengths of 2s2p→1s2 and 2s2p→1s2s are also calculated. A parallel calculation based on the Dirac–Fock–Slater method with a local central potential is performed using the Flexible Atomic Code to cross-check the accuracy of the data. The Breit interaction and quantum electrodynamics effects, including self-energy and vacuum polarization, are included as perturbations in these calculations. The average relative standard deviation of the energy levels from the available NIST data is about 0.2% for both methods. The transition energies and rates are in reasonable agreement with other available theoretical and experimental data, and the consistency between the transition energies calculated by the two methods reaches 99.9%. These results are expected to help future investigations on double K-shell vacancy decays of He-like ions.

Relativistic calculation of electron impact excitation of tin: cross sections of importance in plasma modeling

The present work includes the calculation of electron impact excitation cross sections for the fine structure resolved transitions in Sn I and Sn II, using the relativistic distorted wave approximation with the aid of Dirac-Fock multiconfigurational wavefunctions. For Sn I, the cross sections are calculated for the excitations from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds for a wide range of projectile electron energy from 0 to 500 eV. Further, the cross sections are also calculated for the excitations of Sn II from the fine structure resolved energy levels of the manifolds, , , , and including the ground state, to the fine structure resolved levels of , , , , and manifolds. All the cross sections for Sn I except for the excitation from the ground state to the fine-structure resolved excited states of , and manifolds and all the cross sections for Sn II are reported for the first time. In order to ensure the consistency of the present calculations, a comparison of the oscillator strengths and transition probabilities is carried out with the available calculations and measurements. The present cross sections are also compared with previous calculations and observed to be in reasonable agreement. The calculated cross sections are of great interest in the modeling and diagnostic of various plasmas such as laser induced plasma, fusion plasma, etc. Further, to ease the use of calculated cross sections, the analytic fitting of the cross sections is also provided.