Dirac-Fock Calculations Research Articles

Relative K X-ray intensity ratios of the first and second transition elements in the magnetic field

In this research, EDXRF (Energy Dispersive X-Ray Fluorescence) technique was used for obtaining Kβ-to-Kα X-ray intensity ratios (IKβ/IKα) of 3d and 4d transition metal elements prepared at 200 bar pressure and high temperatures under the magnetic field and non-magnetic field, firstly. 3d and 4s4p electron numbers for 3d transition elements were calculated by comparing of (IKβ/IKα) obtained with (MCDF) multi-configurations Dirac-Fock calculations values, secondly. The results measured without magnetic field were compared with the theoretical predictions of some studies and were found approximately compatible with the early studies. Finally, it was seen that the magnetic field affects IKβ/IKα values and valence electron numbers of the transition metal elements.

Relativistic photoionization cross section calculations and resonance parameters for Mg-like Se XXIII

Photoionization cross sections of the ground state 2p63s2 (1S0) and the first three excited states 2p63s3p (3Po0,1,2) of Mg-like Se XXIII have been reported by employing the fully relativistic Dirac atomic R-matrix code (DARC). For construction of target wavefunctions, multiconfiguration Dirac-Fock calculations are performed for the lowest 21 fine structure energy levels belonging to 2p6nl (3 ≤ n ≤ 5) configurations. Our calculated target state energies of the core ion Se XXIV show close agreement with the energies tabulated in the NIST database. In order to assess the accuracy of our DARC results, photoionization cross section calculations of the ground state configuration have also been performed using the close-coupling Breit-Pauli R-matrix (BPRM) method. Our two independent calculations show a good agreement both in magnitude and resonant photoionization cross sections. The resonance energies (Er) and widths (Γ) of the 3pns (3P1,1P1), 3pnd (3D1, 3P1, 1P1), 3dnp (3D1, 3P1, 1P1) and 3dnf (3D1, 3P1, 1P1) series due to ejection of a 3s electron from the ground state 2p63s2 (1S0) have also been predicted by adopting Quigley and Berrington (QB) method. We believe that our results might be beneficial for the modelling and diagnostics of fusion and laser generated plasmas.

Multiconfiguration Dirac–Fock calculations of Zn K‐shell radiative and nonradiative transitions

Zinc K‐shell radiative and radiationless transition rates are calculated using the multiconfiguration Dirac–Fock method. Correlation up to the 4p orbital is included in almost all transition rate calculations. Calculated radiative transition rates and transition probabilities are compared with Scofield's Dirac–Hartree–Slater and Dirac–Hartree–Fock calculations, presenting good agreement with the later. Radiative transition intensity ratios involving the strongest lines are compared with theoretical, experimental, and empirical‐fit values. Most ratios are in close agreement with the empirical‐fit values from NIST's Fundamental Parameters database. Calculated radiationless transition rates and ratios are compared with Chen et al.'s Dirac–Fock values and Safronova et al.'s Dirac–Fock values. The K‐LL transition rates are overall lower than Chen et al.'s values, whereas the K‐LX and K‐XY transition rates are overall higher. Calculated K‐LX/K‐LL and K‐XY/K‐LL ratios are relatively close to the experimental values compared. Some calculated intensities relative to K‐L are in good agreement with the experimental values, whereas others present worse agreement. The calculated fluorescence yield is higher than all theoretical, experimental, and empirical‐fitted values compared, probably because the total radiationless transition rate value calculated in the present work is relatively low.

Fock-space multireference coupled cluster calculations of Auger energies of noble gas elements using relativistic spinors.

We report the Auger and Coster-Kronig transition energies (related to double ionization potentials) of noble gas elements obtained using the Fock-space multireference coupled cluster (FSMRCC) method with relativistic spinors. The resulting Auger and Coster-Kronig lines are found to be in agreement with the experimental data and with other reference theoretical estimates. To the best of our knowledge, no prior report of relativistic calculations is available for Auger transition energies at the FSMRCC level of theory. The ionization potentials resulted from this method with no extra cost are also found to be in agreement with experiment, particularly the outer-valence ones. Interestingly, the FSMRCC and the multiconfiguration Dirac-Fock calculations exhibit an inversion in the 3P energy levels of the xenon atom for N4,5-O23O23 Auger transitions, where the 3Pj state energies appear in the order J = 1, 0, 2, a feature which can be verified experimentally.

Collinear laser spectroscopy at ion-trap accuracy: Transition frequencies and isotope shifts in the 6s2S1/2→6p2P1/2,3/2 transitions in Ba+

The rest-frame transition frequencies of the $6s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}\ensuremath{\rightarrow}6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}$ (${D}_{1}$) and $6s{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2}\ensuremath{\rightarrow}6p{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2}$ (${D}_{2}$) lines in the stable isotopes of ${\mathrm{Ba}}^{+}$ were measured with an accuracy better than $200\phantom{\rule{4pt}{0ex}}\mathrm{kHz}$ through (quasi)simultaneous collinear and anticollinear laser spectroscopy, leading to an improvement in the accuracy of the isotope shifts by more than an order of magnitude compared to previous data. The ratio of the field-shift constants in the ${D}_{1}$, ${D}_{2}$ fine-structure doublet has been determined to $f=1.0186(9)$ through a King plot analysis. These collinear laser spectroscopy measurements have reached an accuracy comparable to ion-trap measurements on allowed dipole transitions. Our result provides an accurate benchmark for atomic many-body calculations and is compared to pure Dirac-Hartree-Fock calculations and to elaborate multiconfigurational Dirac-Fock calculations. The influence of correlation effects and the Breit interaction on the isotope shift in ${\mathrm{Ba}}^{+}$ are discussed.

Investigating XRF parameters and valance electronic structure of the Co, Ni, and Cu spinel ferrites

In the present study, K X-ray emission spectra of Cu, Ni, Co spinel ferrite (MFe2O4) alloys obtained with a HPGe detector by inducing 59.54 keV and 22.1 keV γ-rays emitted from 3 Ci 241Am annular source and 40 mCi 109Cd point source. The data obtained were evaluated in terms of the Kβ/Kα X-ray intensity ratios, the full width at half maximum (FWHM) values, asymmetry index (As), peak energies and chemical shifts (ΔE) depending on the alloying effect. In addition, calculated Kβ/Kα intensity ratios, which is used to determine the electron number changes in the orbitals for Cu, Ni, Co, Fe elements in the alloys, were compared with the results of Multiconfiguration Dirac-Fock (MCDF) calculations. It has been observed that chemical shifts in the spectra have been occurred upon alloying due to crystal structure, valence electron structure, bond type and length. It has been also concluded that the X-ray intensity ratios and the 3d electron densities of the pure ferrite have influenced by the alloying transition elements. It has been observed that the charge transfer mechanism predominates mainly due to the different electronegativity of the elements in the alloys.

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.

Precise measurement of αK and αT for the 39.8-keV E3 transition in Rh103 : Test of internal-conversion theory

Neutron-activated sources of $^{103}$Ru and $^{103}$Pd both share the isomeric first-excited state in $^{103}$Rh as a daughter product. From independent measurements of both decays, we have measured the $K$-shell and total internal conversion coefficients, $\alpha_K$ and $\alpha_T$, for the 39.8-keV $E$3 transition, which de-excites that state in $^{103}$Rh, to be 141.1(23) and 1428(13), respectively. When compared with Dirac-Fock calculations, our new results disagree with the version of the theory that ignores the $K$-shell atomic vacancy, which is consistent with our conclusion drawn from a series of measurements on high multipolarity transitions in nuclei with higher $Z$. Calculations that include the atomic vacancy indicate that the transition actually has a small $M$4 component with mixing ratio $\delta$ = 0.023(5).

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.

High-resolution tungsten spectroscopy relevant to the diagnostic of high-temperature tokamak plasmas

The x-ray transitions in Cu- and Ni-like tungsten ions in the 5.19--5.26 \AA{} wavelength range that are relevant as a high-temperature tokamak diagnostic, in particular for JET in the ITER-like wall configuration, have been studied. Tungsten spectra were measured at the upgraded Shanghai- Electron Beam Ion Trap operated with electron-beam energies from 3.16 to 4.55 keV. High-resolution measurements were performed by means of a flat Si 111 crystal spectrometer equipped by a CCD camera. The experimental wavelengths were determined with an accuracy of 0.3--0.4 m\AA{}. The wavelength of the ground-state transition in Cu-like tungsten from the $3{p}^{5}3{d}^{10}4s4d\phantom{\rule{4pt}{0ex}}[{(3/2,{(1/2,5/2)}_{2}]}_{1/2}$ level was measured. All measured wavelengths were compared with those measured from JET ITER-like wall plasmas and with other experiments and various theoretical predictions including cowan, relac, multiconfigurational Dirac-Fock (MCDF), and fac calculations. To obtain a higher accuracy from theoretical predictions, the MCDF calculations were extended by taking into account correlation effects (configuration-interaction approach). It was found that such an extension brings the calculations closer to the experimental values in comparison with other calculations.

High-precision measurements of n=2→n=1 transition energies and level widths in He- and Be-like argon ions

We performed a reference-free measurement of the transition energies of the $1s 2p\,^1P\_1\to 1s^2 \,^1S\_0$ line in He-like argon, and of the $1s 2s^2 2p\,^1P\_1\to 1s^2 2s^2\,^1S\_0$ line in Be-like argon ions. The highly-charged ions were produced in the plasma of an Electron-Cyclotron Resonance Ion Source. Both energy measurements were performed with an accuracy better than 3 parts in $10^6$, using a double flat-crystal spectrometer, without reference to any theoretical or experimental energy. The $1s 2s^2 2p\,^1P\_1\to 1s^2 2s^2\,^1S\_0$ transition measurement is the first reference-free measurement for this core-excited transition. The $1s 2p\,^1P\_1\to 1s^2 \,^1S\_0$ transition measurement confirms recent measurement performed at the Heidelberg Electron-Beam Ion Trap (EBIT). The width measurement in the He-like transition provides test of a purely radiative decay calculations. In the case of the Be-like argon transition, the width results from the sum of a radiative channel and three main Auger channels. We also performed Multiconfiguration Dirac-Fock (MCDF) calculations of transition energies and rates and have done an extensive comparison with theory and other experimental data. For both measurements reported here, we find agreement with the most recent theoretical calculations within the combined theoretical and experimental uncertainties.

Multielectron spectroscopy: energy levels of Kn+ and Rbn+ ions (n = 2, 3, 4)

A magnetic bottle time-of-flight spectrometer has been used to perform spectroscopy of Kn+ and Rbn+ states with ionization degrees n of 2, 3 and 4. Energy levels are directly measured by detecting in coincidence the n electrons that are emitted as a result of single photon absorption. Experimental results are compared with the energies from the NIST atomic database and ab initio multiconfiguration Dirac–Fock calculations. Previously unidentified 3p4(3P)3d1 4D energy levels of K2+ are assigned.

Auger decay of 4d inner-shell holes in atomic Hg leading to triple ionization

Formation of triply ionized states upon the creation of $4d$ inner-shell holes in atomic Hg is investigated by using synchrotron radiation of 730 eV photon energy and a versatile multielectron coincidence detection technique in combination with multiconfiguration Dirac-Fock calculations. By carefully selecting Coster-Kronig electrons detected only in coincidence with a $4d$ photoelectron, the Coster-Kronig spectrum has been extracted and the corresponding branching ratios of the $4d$ hole have been determined. The results are found to differ from previously established experimental ratios based on electron impact ionization but to agree now better with theory. We also present an Auger cascade analysis of pathways leading to triply ionized states of atomic Hg upon removal of a $4d$ inner-shell electron.

Unraveling the origin of the complex structure of the thorium Lγ x-ray lines in high-resolution spectra induced by heavy projectiles

The origin of the complex structure of $L{\ensuremath{\gamma}}_{1}$ $({L}_{2}{N}_{4})$, $L{\ensuremath{\gamma}}_{2}$ $({L}_{1}{N}_{2})$, $L{\ensuremath{\gamma}}_{3}$ $({L}_{1}{N}_{3})$, and $L{\ensuremath{\gamma}}_{6}$ $({L}_{2}{O}_{4})$ x-ray lines of thorium in high-resolution spectra induced by oxygen projectiles has so far resisted reliable quantitative interpretation. Therefore, the detailed structure predictions for satellite (additional holes in $M,N,O$, and/or $P$ shells) and hypersatellite (additional hole in $L$ shell) $L\ensuremath{\gamma}$ x-ray lines of thorium being the results of extensive multiconfiguration Dirac-Fock calculations have been performed. Our predictions reproduce with the excellent precision the positions and shapes of characteristic parts of above high-resolution $L\ensuremath{\gamma}$ x-ray spectrum of thorium what indicates that our study has been carried out with very high accuracy. The results of this study are the precursor for proper quantitative interpretation of the complex origin of $L{\ensuremath{\gamma}}_{1},L{\ensuremath{\gamma}}_{2},L{\ensuremath{\gamma}}_{3}$, and $L{\ensuremath{\gamma}}_{6}$ lines in high-resolution x-ray spectra of thorium induced by different light and heavy projectiles.

Precise measurement of αK and αT for the 109.3-keV M4 transition in Te125 : Test of internal-conversion theory

We have measured the K-shell and total internal conversion coefficients (ICCs), alpha_K and alpha_T, for the 109.3-keV M4 transition in 125Te to be 185.0(40) and 350.0(38), respectively. Previously this transition's ICCs were considered anomalous, with alpha values lying below calculated values. When compared with Dirac-Fock calculations, our new results show good agreement. The alpha_K result agrees well with the version of the theory that takes account of the K-shell atomic vacancy and disagrees with the one that does not. This is consistent with our conclusion drawn from a series of high multipolarity transitions.

Precise test of internal-conversion theory: αK measurements for transitions in nine nuclei spanning 45 ≤ Z ≤ 78

We have been testing the theory used to calculate internal-conversion coefficients (ICCs) by making a series of measurements of αK values with precision better than ±2%. So far we have measured E3 transitions in three nuclei, 103Rh, 111Cd and 134Cs; and M4 transitions in six nuclei, 119Sn, 125Te, 127Te, 137Ba, 193Ir and 197Pt. Together, these span a wide range of A and Z values. In all cases, the results strongly favor Dirac-Fock calculations in which the final-state electron wave function has been computed in an atomic field that includes the vacancy created by the internal-conversion process.

A project based on multi-configuration Dirac–Fock calculations for plasma spectroscopy

We present a project dedicated to hot plasma spectroscopy based on a Multi-Configuration Dirac–Fock (MCDF) code, initially developed by J. Bruneau. The code is briefly described and the use of the transition state method for plasma spectroscopy is detailed. Then an opacity code for local-thermodynamic-equilibrium plasmas using MCDF data, named OPAMCDF, is presented. Transition arrays for which the number of lines is too large to be handled in a Detailed Line Accounting (DLA) calculation can be modeled within the Partially Resolved Transition Array method or using the Unresolved Transition Arrays formalism in jj-coupling. An improvement of the original Partially Resolved Transition Array method is presented which gives a better agreement with DLA computations. Comparisons with some absorption and emission experimental spectra are shown. Finally, the capability of the MCDF code to compute atomic data required for collisional-radiative modeling of plasma at non local thermodynamic equilibrium is illustrated. In addition to photoexcitation, this code can be used to calculate photoionization, electron impact excitation and ionization cross-sections as well as autoionization rates in the Distorted-Wave or Close Coupling approximations. Comparisons with cross-sections and rates available in the literature are discussed.

Auger cascades in resonantly excited neon

The Auger cascades following the resonant $1s\ensuremath{\rightarrow}np$ ($n=3,4$) excitation of neutral neon are studied theoretically. In contrast to previous investigations, we here model the complete cascade from the initially core-excited $1{s}^{\ensuremath{-}1}3p\phantom{\rule{0.16em}{0ex}}^{1}P_{1}$ and $1{s}^{\ensuremath{-}1}4p\phantom{\rule{0.16em}{0ex}}^{1}P_{1}$ levels of Ne up to the doubly ionized ${\mathrm{Ne}}^{2+}$ ions. Extensive multiconfiguration Dirac-Fock calculations are carried out, combined with a proper cascades model, to incorporate as many decay branches as possible, including all major single-electron shake-up and shake-down processes. We simulate the electron spectra and predict shake probabilities, ion yields, as well as the relative population of the intermediate and final states. Experimentally known level energies for neutral, singly, and doubly ionized neon are utilized whenever possible in order to improve the predictions. Most features from experiment can be reproduced with quite good agreement if a sufficiently large basis is taken into account. These simulations therefore demonstrate not only the required computational effort, but also that it is nowadays possible to predict whole Auger spectra of decay cascades, a central feature for further exploring electron coincidence maps as obtained at synchrotrons and free-electron lasers.

Theoretical study of the dielectronic recombination process of Li-like Xe51+ ions

The dielectronic recombination of Li-like Xe51+ (2s) ions was studied using the flexible atomic code based on the relativistic configuration interaction method. The resonance energies, radiative and autoionization rates, and resonance strengths were calculated systematically for the doubly excited states (2p 1/2 n l j ) J (n = 18−32) and (2p 3/2 n′l j ) J (n′ = 9−27) of Be-like Xe50+ ions. For the higher Rydberg resonance states with n ≥ 33 and n′ ≥ 28, the resonance energies and strengths were obtained by extrapolation based on quantum defect theory. The theoretical rate coefficients, covering the center-of-mass energy range 0–505 eV, are in a better agreement with the experimental results measured at the heavy-ion storage ring ESR than the Multi-Configuration Dirac-Fock calculations, especially at the resonance energy range close to the series limits.

Precise measurement of αK for the 88.2-keV M4 transition in Te127 : Test of internal-conversion theory

We have measured the $K$-shell internal conversion coefficient, ${\ensuremath{\alpha}}_{K}$, for the 88.2-keV $M4$ transition from the 106-day isomer to the ground state in $^{127}\mathrm{Te}$ to be 484(6). When compared with Dirac-Fock calculations of ${\ensuremath{\alpha}}_{K}$, this result agrees well with the version of the theory that incorporates the effect of the $K$-shell atomic vacancy and disagrees with the one that does not. As a byproduct of this measurement, we have determined the $\ensuremath{\beta}$ branching from the isomer to be 2.14(3)%.