Dirac Fock Research Articles

Measuring the mean inner potential of Al2O3 sapphire using off-axis electron holography

The mean inner potential (MIP) of a single crystal α-Al2O3 sapphire was measured using off-axis electron holography. To measure the MIP, we use mechanically polished wedge specimens for transmission electron microscopy (TEM). This approach also enabled us to measure the plasmon mean free path for inelastic scattering (IMFP). The wedge specimen, chosen here at an angle of approximately 45°, allows to determine the MIP by measuring the gradient of phase variations of the reconstructed electron wave over extended regions across the sample. The angle of the wedge was measured to an accuracy of better than 1° by two methods: first, perpendicular sectioning in a focused ion beam for direct measurement by TEM and second, by a non-destructive approach of confocal optical microscopy. The validity of this methodology was examined on a single crystal Si(001) sample showing that the mechanically polished wedge approach can be applied to a wide range of materials. Our measurements concluded that the MIP of sapphire is V0 = 16.90 ± 0.36 V. Furthermore, the IMFP of sapphire was measured at 136 ± 2 nm for 197 keV electrons with a collection angle of 18mrad. The measured MIP of sapphire reflects its degree of ionicity, which lies between theoretical calculations based on electron scattering factors of charged and neutral isolated atoms obtained by Dirac–Fock calculations. Our MIP measurements tend to the expected value for this predominantly ionic material. To account for chemical bonding and the role of the crystallographic plane at the surface of the sample, we compared the experimental measurements to density-functional-theory calculations of the MIP. Calculations of α-Al2O3 slabs cut along (0001) and (1–100) planes obtained MIP values of 15.7 V and 16.7 V, respectively.

KLL Dielectronic Satellite Spectra from the Photorecombination of He-like Fe and Ni Ions

Abstract The radiative decay of a doubly excited autoionizing state is characterized by the dielectronic recombination in the photorecombination (PR) process. We calculate PR cross sections for the ground states of the Fe xxv and Ni xxvii ions by employing the Dirac R-matrix technique. The autoionizing states are treated in the framework of Davies and Seaton’s theory to account for the radiative decay. The strengths of the KLL resonances that equate directly to the dielectronic satellite line strengths are evaluated. Besides, the Multiconfiguration Dirac Fock formalism is utilized to obtain the dielectronic satellite line strengths under the isolated resonance approximation. A very good agreement is found for the wavelengths and transition parameters belonging to the KLL autoionizing states with the previous theoretical and experimental results. The dominant KLL dielectronic satellite line strengths from both theoretical models are consistent with each other and are in good agreement with the established theoretical models and measurements at the electron beam ion trap.

L X-ray fluorescence cross-sections measurements for elements Ba, La and Ce at synchrotron radiation energies 7, 8, 9 and 10 keV

Ll, Lα, Lβ and Lγ1 X-ray fluorescence cross-sections have been measured for the elements Ba, La and Ce at 7, 8, 9 and 10 keV synchrotron radiations. A Peltier cooled Vortex solid state detector (SII Nano Technology, USA) with an energy resolution of 138 eV at 5.96 keV X-rays was employed for the analysis. The experimental results were compared with the theoretical values estimated from the data of Puri et al. (1993) and Campbell (2003). Two different models Dirac-Hartree-Slater (DHS) and Dirac-Fock (DF) were used to compare the theoretical and experimental results. Most of the experimental measurements were found in good agreement with the theoretical values of Puri et al. (1993) based on DF model.

Theoretical wavelengths, energy levels, lifetimes, and transition probabilities for the silicon isoelectronic sequence Cd XXXV-Ba XLIII, W LXI

We have performed the theoretical calculations of wavelengths and excitation energies from ground state for 56 fine-structure levels as well as of oscillator strengths and radiative decay rates for all the electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), and magnetic quadrupole (M2) transitions among these levels of the terms belonging to the $$3s^{2}3p^{2}$$ , $$3s^{2}3p3d$$ , $$3s3p^{3}$$ , $$3s3p^{2}3d$$ , $$3s^{2}3d^{2}$$ , and $$3p^{4}$$ configurations for the silicon isoelectronic sequence Cd XXXV-Ba XLIII, W LXI. The calculation is based on the multiconfiguration Dirac–Fock approach, and Breit interaction and quantum electrodynamic corrections are included. Electron correlation effects are taken into account by means of the expansions in terms of a basis of configuration state functions. To achieve the accuracy of the data, independent calculation is provided with similar data obtained from the flexible atomic code, taking W LXI as an example. Our results are compared with each other, and other available theories/experimental values and the energy levels/transition rates for a majority of strong transitions are found to be accurate to be better than 0.5%/10%. The present atomic data should be useful in controlled thermonuclear fusion research and technical plasma modeling.

A study for hydrogen-like lawrencium

Studying hydrogenic ions with high Z is an occasion to understand atomic structure. It also provides a reliable test of methods used to determine atomic structures. Many fields and applications require precise atomic data. For this reason, a hydrogen-like study is performed for lawrencium (Lr102+, Z = 103). The energy levels of hydrogen-like lawrencium are calculated with both multiconfiguration Hartree–Fock (MCHF) and multiconfiguration Dirac–Fock (MCDF) methods. The calculations contain Breit–Pauli relativistic corrections in MCHF calculation and the transverse photon and quantum electrodynamics (QED) effects in MCDF calculation along with electron correlations. In addition, some transition parameters (wavelengths, λ, logarithmic weighted oscillator strengths, log(gf) value, and transition probabilities, Aki) for allowed (E1) and forbidden (E2 and M1) transitions are investigated. The results from this study are compared with only a few theoretical works, but there is no available experimental data yet for Lr102+.

Relativistic calculations on the x-ray satellites of platinum with multiply ionized L shell

The energies, electric dipole rates and oscillator strengths of Kα x-ray satellites from 1s–2p transitions in platinum with a single vacancy in the K shell and variously ionized L shell are calculated using multi-configuration Dirac Fock wavefunctions with the inclusion of Breit interaction and quantum electrodynamics effects. The computations have been performed for low ionized (Pt1+–Pt6+) and also for highly ionized (Pt69+–Pt77+) Pt. As Kα x-ray satellites with spectator vacancies only in the L shell, to the best of our knowledge, have been studied for the first time, to check the accuracy of the present work, calculations have also been performed for H-, He-, Li-, Be-, B- and F-like Pt for which theoretical data for comparison exist. The purpose of choosing a high Z element is that the detailed decay properties of heavy ions due to systematic 1s–2p transitions with spectator vacancies in the L shell using fully relativistic treatment are unexplored and the increasing need for reliable data from such ions in various applications that are under investigations in recent years. Also, for heavy ions like Pt, the contribution from the competing non-radiative transitions is less and hence the atomic properties are largely due to radiative transitions. The K X ray satellite energies, rates and resonance photo- ionization cross sections from H-, He-, Li-, Be-, B- and F-like Pt are compared with available data.

Electron impact excitation and ionization cross section of tungsten ions, W[formula omitted

Atomic structure data, electron impact excitations and ionization cross sections of tungsten ions, W44+, have been calculated using the multi-configuration Dirac–Fock and R-matrix methods for a set of configurations consists of 3d104s2, 3d104s4l, and 3d104p4l. A set of configuration state functions (CSFs) with doubly excitations up to 5l orbitals have been included within the calculations of energy levels and transition probabilities. As well as, 4-electron excitation to 4l-levels and 3-electron to 5l-levels have been involved in the computation process. R-matrix calculations of collision strength and cross section have been executed for the intercombination transitions between 3d104s2, 3d104s4p, 3d104s4d, and 3d104s4f at different angular momenta using DARC code. Resonance collision strengths have been observed at low energy of incident electron in the range of ∼ 6 – 40 Ryd, however, the strong lines are limited to 6–25 Ryd. For cross section of various transitions, resonances arising from these transitions are observed at low-electron energies in the range of 6–25 Ryd. To estimate the accuracies of calculated energy levels and transition probabilities, comparisons between experimental observations and theoretical calculations are presented. The convergence between different methods for collision strengths and cross sections are observed. Different calculations of electron impact ionization cross section of the present ion in the collision energy range of 100–1000 Ryd are evaluated.

Theoretical study of 4p5nln'l'LSJstates of Sr ion excited by electron impact on Sr atom

Calculations of energy levels and electron-impact excitation cross sections of the Auger autoionizing states 4p5nln'l'(nl= 4d, 5s, 5p; n'l'= 4d, 5s, 5p, 5d, 6s, 6p, 6d, 7s, 7p) of Sr ion were performed. A large-scale configuration interaction method on the basis of the solutions of Dirac–Fock–Slater equations was used. The cross sections of electron-impact simultaneous ionization and excitation, and Auger decay of the electron-impact excited states of Sr atom to the 4p-core excited autoionizing states of Sr ion were calculated for the first time and used to estimate the intensity of ejected electron lines. Tentative identification of the Auger electron lines of Sr ion registered in a number of experiments is presented.

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V4 with the signature (−1,−1,−1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.

Extended atomic structure calculations for energy levels and radiative data of Xe XLIII

We report an extensive and elaborate theoretical study of atomic data for Xe XLIII. We have calculated energies for the lowest 141 fin. structure levels by using Multi-Configuration Dirac Fock (MCDF) method. We have also considered Quantum Electrodynamics (QED) and Breit corrections in our calculations. We have presented the radiative data for electric dipole (E1) transitions among lowest 141 levels. We have made comparisons of our calculated excitation energies and EUV (Extreme Ultraviolet) transition wavelengths with the theoretically computed and experimentally observed energy levels and wavelengths at Electron Beam Ion Trap (EBIT) and a good agreement is achieved. We have also computed energy levels by performing similar relativistic distorted wave calculations using Flexible Atomic Code (FAC). Additionally, we have provided new atomic data for Xe XLIII which are not published elsewhere in the literature. We believe that our results may be beneficial in fusion plasma research and detection and investigation of EUV and Soft X-ray (SXR) spectral lines.

Atomic structure calculations, partition function, and thermodynamic parameters for W LXXII and Au LXXVII

In this work, we present a theoretical study of W LXXII (Li-like W71+) and Au LXXVII (Li-like Au76+) by employing a multi-configuration Dirac Fock method approach within the framework of the Dirac-Coulomb Hamiltonian including quantum electrodynamics and Breit corrections. We have presented energies of the lowest 42 fine structure energy levels and radiative data for electric dipole (E1) transitions within the lowest 42 levels. We have made comparisons with available energy levels compiled by NIST and achieved good agreement. We have also discussed discrepancies between our results and other theoretical results with possible reasons of discrepancy. Due to inadequate data in the literature for higher excited states, analogous relativistic distorted wave calculations have also been performed using the flexible atomic code to assess the reliability and accuracy of our results. Furthermore, we have calculated the partition function and thermodynamic properties by the fine structure approach. The variation of the partition function, internal energy, and specific heat with temperature is also discussed. We believe that our presented work and future comparisons with experimental observations will be beneficial in fusion plasma.

Min–max and max–min approaches to the solution of wave equation in relativistic quantum chemistry

The variation problem associated with the solution of Dirac’s relativistic electron equation is reviewed here. Derivations of the min–max and max–min theorems are discussed: A new observation is that the spurious roots of negative energy satisfy a max–min theorem. Min–max principle (MMP) for solution of Dirac equation, extendable to the Dirac–Fock case, is concisely inspected and illustrated. MMP for two-electron Dirac–Coulomb equation is also considered. The min–max relation is physically interpreted for both Dirac and Dirac–Coulomb problems. Applications to chemical systems are investigated. Mathematical advances are reviewed. Limitations of MMP are spelt out, recent mathematical developments are collated, and an outline is given of the associated theoretical and computational developments. For a complementary purpose, the case of spin-zero is briefly examined. The variation principle for Dirac fermions in nanoscience is discussed. A brief outline of future prospects is given.

Extended calculations of energy levels, radiative properties, AJ, BJ hyperfine interaction constants, and Landé gJ-factors for Nitrogen-like Se XXVIII

Employing two state-of-the-art methods, multiconfiguration Dirac–Fock and second-order many-body perturbation theory, highly accurate calculations are performed for the lowest 316 fine-structure levels arising from the 2s22p3, 2s2p4, 2p5, 2s22p23l, 2s2p33l, 2p43l, and 2s22p24l configurations in nitrogen-like Se XXVIII. Complete and consistent atomic data, including excitation energies, lifetimes, wavelengths, hyperfine structures, Landé gJ-factors, and E1, E2, M1, M2 line strengths, oscillator strengths, and transition rates among these 316 levels are provided. Comparisons are made between the present two data sets, as well as with other available experimental and theoretical values. Previous measured lines of Se XXVIII are affected by large errors for some transitions, and should be remeasured. The present calculations provide a consistent and accurate data set for line identification and modeling purposes, which can also be considered as a benchmark for other calculations. The present data are accurate enough for identification and deblending of emission lines involving the n=3,4 levels, and are also useful for modeling and diagnosing fusion plasmas.

Spectroscopic study of EUV and SXR transitions of Cs XXV

We report an extensive and elaborate theoretical study of atomic data for Cs XXV by using multi-configuration Dirac–Fock method and calculated energy levels for the lowest 110 fine structure levels. We have presented the radiative data for electric and magnetic dipole (E1, M1) and quadrupole (E2, M2) transitions among lowest 110 levels. We have made comparisons of our calculated excitation energies with theoretically calculated and experimentally observed energy levels. We have studied the effect of correlation by introducing more configurations in our calculations. We have also computed energy levels by performing similar relativistic distorted wave calculations using Flexible Atomic Code. Additionally, we have also provided new atomic data for Cs XXV and identified extreme ultraviolet and soft X-ray spectral lines with gA spectra for E1 and M2 transitions, which are not published elsewhere in the literature. We believe that our results may be beneficial in fusion plasma research and applications.

Experimental and theoretical determination of the L-fluorescence yields of bismuth

The knowledge of atomic fundamental parameters such as the mass attenuation coefficients or fluorescence yields with low uncertainties, is of decisive importance in elemental quantification involving x-ray fluorescence analysis techniques. For example, several databases giving the mass attenuation coefficients are accessible and frequently used within a large community of users. These compilations are most often in good agreement for photon energies in the hard x-ray ranges. However, they significantly differ for low photon energies and around the absorption edges of the elements. Some review articles present compilations of measured fluorescence yields values found in the literature. However reliable measurements are not so many illustrating the inherent difficulties. Mass attenuation coefficients of bismuth were determined experimentally in the photon energy range from 100 eV to 30 keV by using monochromatized synchrotron radiation at SOLEIL (France). The fluorescence yields of the three L-subshells of bismuth were also determined using a reflection geometry setup. The application of high-accuracy experimental techniques resulted in low uncertainty mass attenuation coefficients and L-subshell fluorescence yields that are directly compared to existing databases and with updated relativistic configuration mixing Dirac–Fock calculations including QED corrections.

Maxwellian rate coefficients for electron-impact ionization of W26+

Excitation-autoionization (EA) and direct ionization (DI) Maxwellian rate coefficients (MRC) are presented for the levels of the ground configuration of the W26+ ion. The study of EA and DI cross sections is performed by applying the Dirac–Fock–Slater method within the level-to-level distorted-wave approximation including radiative damping. Excitations to the high-nl shells (n ≤ 25) are taken into account for the EA process. It is shown that electron-impact ionization from the different levels of the ground configuration produces a significant variation of MRC. Excitations to configurations with energy levels that straddle the ionization threshold play the main role in enhancing MRC for the highest level of the ground configuration. The difference of 40% among the EA MRC is determined for the levels of the ground configuration.

Electric dipole transitions between low-lying levels in doubly ionized krypton, xenon, and radon

Using the general-purpose relativistic atomic structure package (GRASP) based on a fully relativistic multiconfiguration Dirac–Fock (MCDF) method, the transition parameters, such as transition rates (probabilities), oscillator strengths, and line strengths for the electric dipole transitions between low-lying levels are evaluated for doubly ionized krypton, xenon, and radon. Breit interactions for relativistic effects and quantum electrodynamical (QED) contributions besides valence and valence–core correlation effects are taken into account in calculations. We compare the results obtained with the available data in the literature and discuss them, when possible.

Core-polarization corrected relativistic energies and radiative data for one-electron spectra of Bi V through U XIV

The relativistic core-polarization corrected Dirac–Fock energy levels, oscillator strength, and lifetimes are predicted for highly ionized spectra of gold isoelectronic sequence from Bi V through U XIV. It is demonstrated that core polarization (core–valence electron correlation) strongly affects oscillator strengths and lifetimes of 6s–6p, 6p–5d, and 6d–5f transitions (the latter particularly in the vicinity of the collapse region) even for highly ionized systems of gold isoelectronic sequence.

Single and double Auger decay of 4f-ionized mercury including cascade and direct processes**Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402300), the National Natural Science Foundation of China (Grant Nos. 11774344 and 11474033), and the Joint Foundation of the National Natural Science Foundation and the China Academy of Engineering Physics (Grant No. U1330117).

Single (SA) and double (DA) Auger decay including cascade and direct processes are investigated for Hg 4f−1 with multiconfiguration Dirac–Fock method and two-step approaches, i.e., knockout and shakeoff mechanisms. Due to the computational effort, only the major transitions are considered to describe the SA and DA decays for the Hg+ ions with complex electronic configurations. In order to estimate the Auger transition energies and amplitudes, the reference configuration sets producing the configuration state functions are carefully chosen for balancing electron correlations among the successive singly, doubly and triply ionized mercury. The Auger rates and electron spectra, DA probabilities as well as the populations of the final Hg3+ states are obtained. Our results well explain the recent experimental data about the 4f hole states of Hg [Palaudoux J et al., Phys. Rev. A 91 012513 (2015)], and could provide guidance for further studies on complex atoms. Particularly for the DA decay, the contributions of the direct processes, which are neglected in their calculations, are found to be important, accounting for as high as about 38% and 34% of the total DA decays for the and , respectively.

Ionic scattering factors of atoms that compose biological molecules.

Ionic scattering factors of atoms that compose biological molecules have been computed by the multi-configuration Dirac-Fock method. These ions are chemically unstable and their scattering factors had not been reported except for O-. Yet these factors are required for the estimation of partial charges in protein molecules and nucleic acids. The electron scattering factors of these ions are particularly important as the electron scattering curves vary considerably between neutral and charged atoms in the spatial-resolution range explored in structural biology. The calculated X-ray and electron scattering factors have then been parameterized for the major scattering curve models used in X-ray and electron protein crystallography and single-particle cryo-EM. The X-ray and electron scattering factors and the fitting parameters are presented for future reference.