Hartree-Fock-Slater Research Articles

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.

Experimental validation of theoretically predicted cut-off of L1–L3M4,5 transitions at Z = 50 through measurements of fluorescence and Coster–Kronig yields for Sn and Sb

In the present work, we have deduced the fluorescence (ω 1, ω 2, ω 3) and Coster–Kronig (CK)(f 12, f 13, f 23) yields for Sn (Z = 50) and Sb (Z = 51) from the L i (i = 1–3) sub-shell x-ray intensities measured using the energy tunable synchrotron radiation employing the selective photoionization method. For both the elements, yields have been obtained using two sets of theoretical photoionization cross sections based on the non-relativistic Hartree–Fock–Slater (HFS) model and the self-consistent Dirac–Hartree–Fock (DHF) model. In case of Sb, we have obtained a third set of measured yields also by using the experimental photoionization cross sections evaluated from independent measurements of the mass-attenuation coefficients. The experimental yields for Sb are reported for the first time by us. We have compared the present deduced fluorescence and CK yields with the Dirac–Hartree–Slater model based values, the semi-empirical values tabulated by Krause and the earlier reported values. In case of Sn, using the DHF and the HFS model based photoionization cross sections, two sets of present measured L1 sub-shell fluorescence yields (ω 1) are found to be 0.039 ± 0.007 and 0.036 ± 0.003, and the CK yields (f 13) are found to be 0.428 ± 0.107 and 0.405 ± 0.028, respectively. In case of Sb, using three sets of the photoionization cross sections (DHF, HFS and recent experimental values), the ω 1 values are measured to be 0.042 ± 0.007, 0.040 ± 0.004 and 0.047 ± 0.005, and the CK yields are measured to be 0.343 ± 0.085, 0.297 ± 0.021 and 0.247 ± 0.022, respectively. The comparison of these present measured yields with the theoretical values provided a reliable experimental evidence indicating cut-off of the intense L1–L3M4,5 CK transitions at Z = 50.

Cascade Mi (i = 1–5) subshell X‐ray emission at incident photon energies across the Lj (j = 1–3) subshell absorption edges of 66Dy

The total M shell and the Mk (k = ξ, αβ, γ, m) X‐ray production cross sections for 66Dy have been measured at incident photon energies across its Lj (j = 1–3) subshell absorption edge energies, ranging 7.8–9.2 keV. This study aims to investigate the evolution of the probability for cascade decay of Lj subshell vacancies as the tunable incident energy ionizes progressively different 66Dy Lj subshells. The experimental X‐ray production cross sections have been compared with theoretical ones calculated using the nonrelativistic Hartree–Fock–Slater (HFS) model‐based photoionization cross sections; three sets of the X‐ray emission rates, fluorescence and Coster–Kronig yield based on the nonrelativistic Hartree–Slater (NRHS) model, Dirac–Hartree–Slater (DHS) model and Dirac–Fock (DF) model; the Lj (j = 1–3) subshell to the Mi (i = 1–5) subshell vacancy transfer probabilities evaluated in the present work. Presently measured total M shell and the Mαβ X‐ray production cross sections are found to be significantly lower than the theoretical ones evaluated using physical parameters based on the relativistic Dirac–Fock/Dirac–Hartree–Slater model calculations, whereas a much better agreement is observed with respect to the NRHS model‐based calculations; however, the measured X‐ray production cross sections are still systematically lower than the NRHS values.

Measurements of mass attenuation coefficients and determination of photoionization cross sections at energies across the Li (i=1–3) edges of 66Dy

The absolute values of the mass attenuation coefficients have been measured at sixty two photon energies across the Li (i=1–3) sub-shell absorption edges of 66Dy covering the region 7.6–14.0keV in order to investigate the influence of near-edge processes on the attenuation coefficients. The present measured attenuation coefficients are found to be higher by up to 10% than the theoretical values evaluated from the computer code XCOM (Berger et al., 2010) and the self-consistent Dirac-Hartree-Slater (DHS) model based values tabulated by Chantler (1995) over the energy region 7.6–14.0keV, except at energies in vicinity (few eV) of the Li (i=1–3) sub-shell absorption edge energies where the measured values are significantly higher (up to 37%) than both the sets of theoretical values. Further, the Li (i=1–3) sub-shell photoionization cross sections, (σLiP)exp, deduced from the present measured mass attenuation coefficients are compared with the non-relativistic Hartree-Fock-Slater (HFS) model based values tabulated by Scofield (1973) and those evaluated from the theoretical total photoionization attenuation coefficients tabulated by Chantler (1995). The deduced (σLiP)exp(i=1–3) values are found to be in better agreement with those evaluated from the tabulations given by Chantler (1995) than the values given by Scofield (1973) over the energy region 7.8 – 14.0keV included in this study. However, at photon energies up to few eV above the Li edges, the deduced (σLiP)exp(i=1–3) values are found to be significantly higher (up to 32%) than both the sets of theoretical values.

Emission properties of non-equilibrium krypton plasma in the water-window region

The line emission properties of non-equilibrium krypton plasma are examined and the optimal emission temperature conditions for soft x-ray emission output in the water-window region are explored. The kinetic parameters for non-equilibrium plasma including major inelastic ion interaction processes, radiation and emission data are obtained with an approach based on the Hartree–Fock–Slater (HFS) quantum-statistical model and a distorted wave approximation. A nonmaxwellian electron distribution is used as well for calculating collisional rates. At a temperature of 70 eV the emission spectral efficiency for Kr equilibrium plasma is about 10%, and it jumps to a value greater than 70% at 100 eV. A similar spectral efficiency is achieved at a lower temperature e.g. 80 eV in non-equilibrium plasma with 7.5 keV fast electron average energy.

Measurements of X-ray production cross sections at photon energies across the Li (i = 1–3) sub-shell absorption edges of 74W and 76Os using synchrotron radiation

The cross sections for production of the Lk (k=l, η, α, β1,3,4,6, β2,15,5,7, γ1,5, γ2,3) X-rays have been measured at different incident photon energies ranging 10.2–13.1keV across the Li (i=1–3) sub-shell absorption edge energies of 74W and 76Os. These measured XRP cross sections, σLk(exp), exhibited significant differences with two sets of the theoretical values calculated using the Dirac–Fock (DF) model based X-ray emission rates provided by Campbell and Wang (1989), Dirac–Hartree–Slater (DHS) model based fluorescence (ωi) and Coster–Kronig (fij) yields given by Puri et al. (1993) and two sets of the Li (i=1–3) sub-shell photoionization cross sections based on the non-relativistic Hartree–Fock–Slater (HFS) model tabulated by Scofield (1973) and those deduced from the self-consistent Dirac–Hartree–Fock (DHF) model based theoretical total photoionization attenuation coefficients tabulated by Chantler (1995). However, the branching ratios (ILk/ILα)exp (k=l, β2,15) and(ILk/ILβ1)exp (k=γ1,5, η) are found to agree with the theoretical values.

Ytterbium to thorium semi-empirical average M-shell fluorescence yields

Abstract By using the weighted and unweighted mean values of the experimental average M-shell fluorescence yield reported in the literature covering the period from 1955 to 2005 and the theoretical values based on the non-relativistic Hartree–Fock–Slater (HFS) calculations of McGuire (1972) , we produce good estimations for the semi-empirical values of elements in the range of 70≤ Z ≤90. These values are obtained by the interpolation of the normalized weighted and unweighted mean values to their corresponding theoretical ones. The results have been compared to other theoretical, empirical and experimental data reported in the literature. Reasonable agreement has been obtained between our result and other works.

An accurate determination of the K‐shell X‐ray fluorescence yield of silicon

A measurement of the K‐shell fluorescence yield of silicon is undertaken in which identified sources of systematic errors in previous measurements are reduced or eliminated. This enables a stringent test of the only two sets of theoretical predictions available for atomic numbers less than 18. Our result ωK = 0.0495 ± 0.0015 is very slightly lower than the non‐relativistic Hartree‐Fock‐Slater (HFS) prediction of 0.0514. This stringent test of the HFS predictions helps to refine the fundamental parameter database of the X‐ray fluorescence analysis technique, whose importance for light elements is increasing. Our work indicates the need for new theoretical calculations of K‐shell fluorescence yields for these elements. Copyright © 2012 John Wiley & Sons, Ltd.

Modeling of ionization composition in argon plasma with fast electrons

Discharge and laser produced plasmas often have non-equilibrium electron distributions containing fast electrons that may have a profound effect on the ionization balance. The influence of high energy electrons on the ionization balance in the collisional-radiative equilibrium model for optically thin plasma is considered forthwith. A nonmaxwellian electron distribution with fast electrons at energies and concentrations is used for calculating collisional rates; this according to the Hartree-Fock-Slater (HFS) quantum-statistical model in the average atom approximation. The calculative approach is based on the distorted wave approximation along with numerical and semiclassical wave functions calculated in the self-consistent HFS potential. The results from ionization equilibrium modeling in argon plasma at temperatures T ∼ 5–1000 eV with fast electron energies ranging from 1 to 10 keV and their different relative concentrations (from 0.1–10%) are analyzed. It is shown that fast electrons increase the impact ionization rates and significantly change the ionization stage as compared with equilibrium plasma.

Generalized scattered-wave approach to molecular-orbital theory

The scattered-wave approach to molecular-orbital theory is generalized to the case where the one-electron potential can be represented by a model nonlocal operator which is dependent on the orbital energy and angular-momentum components. In its most elementary form, this operator reduces to the previously applied example of nonover-lapping, spherically averaged Hartree–Fock–Slater (HFS) or Slater Xα potentials. More general forms of the potential operator allow one to simulate the effects of truly overlapping potentials, without significantly sacrificing the computational advantages of the scattered-wave technique.

2p3/2 −13x−1−3x−13d−1 X-ray satellites spectra in the Lα1 region of 4d transition elements

The X-ray satellite spectra arising due to 2p3/2−13x−1−3x−13d−1 (x ≡ s, p, d) transition array, in elements with Z = 40 to 48, have been calculated, using available Hartree-Fock-Slater (HFS) data on 1s−1−2p−1 3x− and 2p3/2−1−3x−1,3x−1 Auger transition energies. The relative intensities of all the possible transitions have been estimated by considering cross — sections for the Auger transitions simultaneous to a hole creation and then distributing statistically the total cross sections for initial two hole states 2p3/2−1−3x−1 amongst various allowed transitions from these initial states to 3x−1 3d−1 final states by Coster-Kronig (CK) and shake off processes. In both these processes initial single hole creation is the prime phenomenon. Each transition has been assumed to give rise to a Gaussian line and the overall spectrum has been computed as the sum of these Gaussian curves. The calculated spectra have been compared with the measured satellite energies in Lα1 spectra. Their intense peaks have been identified as the observed satellite lines. The peaks in the theoretical satellite spectra were identified as the experimentally reported satellites α3, α4 and α5, which lie on the high-energy side of the Lα1 dipole line.

L i ( i = 1–3) subshell X-ray production cross sections and fluorescence yields for some elements with 56 ⩽ Z ⩽ 68 at 22.6 keV

The L k ( k = l, α, β 1,4, β 3,6, β 2,15,9,10,7, γ 1,5 and γ 2,3,4) X-ray production (XRP) cross sections have been measured for six elements with 56 ⩽ Z ⩽ 68 at 22.6 keV incident photon energy using the EDXRF spectrometer. The incident photon intensity, detector efficiency and geometrical factors have been determined from the K X-ray yields emitted from elemental targets with 22 ⩽ Z ⩽ 42 in the same geometrical setup and from knowledge of the K XRP cross sections. The L 1 and L 2 subshell fluorescence yields have been deduced from the present measured L k XRP cross sections using the relativistic Hartree-Fock-Slater (HFS) model based photoionization cross sections. The present deduced ω 1 (exp) values have been found to be, on an average, higher by 15% and 20% than those based on the Dirac–Hartree–Slater (DHS) model and the semi-empirical values compiled by Krause, respectively, for elements with 60 ⩽ Z ⩽ 68.

On the optimal value of α for the Hartree–Fock–Slater method

We have examined the performance of the analytic Hartree–Fock–Slater (HFS) method for various α values and empirically determined the optimal α value by minimizing the mean absolute error (MAE) in atomization energies of the G2 set of molecules. At the optimal α the HFS method’s performance is far superior with the MAE of 14 kcal/mol than that of the local density approximation (MAE ∼ 36 kcal/mol) and the Hartree–Fock theory (MAE ∼ 78 kcal/mol). The HFS exchange functional with α = 0.7091 performs significantly better than the Kohn–Sham exchange functional for equally weighted atoms H–Kr.

A Tale of Two Compilations: Quadrupole Deformations and Internal Conversion Coefficients

(i) The deformation of the nuclear ground state is a fundamental property that can be deduced in even-even nuclei from the B(E2) value for the first 2+ (21+) state. In 1987, we published a compilation of such B(E2: 01+→ 21+) values for ~ 280 even-even nuclei. Very recently, we have updated the 1987 compilation and have compared the adopted B(E2: 01+ → 21+) values (for ~ 330 even-even nuclei) with those given by various theoretical models (Finite-Range Droplet Model, Woods Saxon Model, Relativistic Mean-Field Calculations, Extended Thomas-Fermi Strutinski Integral Method, Hartree-Fock+BCS Calculations, and Dynamical Microscopic Model). Agreement between theory and experiment varies from “reasonable” to “excellent”.(ii) Currently, the most widely used Internal Conversion Coefficient (ICC) compilations are those by Hager and Seltzer, Rösel et al., and Band and Trzhaskovskaya. In all three cases, the calculation of electron wave functions is carried out in the framework of the Hartree-Fock-Slater (HFS) method. It has been suspected for some time that there is a “2–3% problem” — that is, the HFS models give results that are systematically larger than the experimental data by 2–3%. ICCs are now available from relativistic self-consistent Dirac-Fock (DF) calculations in which the exchange terms of DF equations are included exactly. The DF model gives results that are very close to the experimental data. The long-standing “2–3% problem” has apparently now been solved.

Repulsive interatomic potentials calculated using Hartree-Fock and density-functional theory methods

The repulsive part of the interatomic potential affects the outcome of computer simulations of many irradiation processes of practical interest, like sputtering and ion irradiation range distributions. The accuracy of repulsive potentials is studied by comparing potentials calculated using commonly available density-functional theory (DFT) and Hartree-Fock (HF) methods to highly accurate fully numerical HF and Hartree-Fock-Slater (HFS) calculations. We find that DFT calculations utilizing numerical basis sets and HF calculations using decontracted standard basis sets provide repulsive potentials which are significantly improved compared to the standard universal ZBL potential. The accuracy of the calculated potentials is almost totally governed by the quality of the one-particle basis set. The use of reliable repulsive potentials open up new avenues for analysis of ion irradiation experiments.

A Fitting Formula for Screening Constants of Partially Ionized Atoms in Plasma

This paper presents a simple fitting formula for the numerical screening constants which were obtained for each of 33 materials recently by the authors [Jpn. J. Appl. Phys. 32 (1993) 5681]. Using the screening constants, the atomic energy levels are obtained for the finite-temperature materials based on the screened-hydrogen model. The formula is convenient for users of the screening constants and reproduces well the Hartree-Fock-Slater (HFS) energy levels: the deviations from the HFS levels are 5.99% for n=1 and 55.3% for n=2 by averaging results for the 33 materials. Here n is the principal quantum number.

Improved Screening Constants for Plasma

We present the new screening constants, with which the atomic energy levels for the finite temperature materials are obtained by the screened-Hydrogen model. The new screening constants are obtained for each of 33 materials. Each set of screening constants has a temperature dependence in order to realize a better reproducibility of the Hartree-Fock-Slater (HFS) energy levels. The deviations from the HFS calculation are 5.15% for n=1 and 54.1% for n=2, based on the average of 33 materials. Here n is the principal quantum number.

Comparison of the Hartree-Fock, M�ller-Plesset, and Hartree-Fock-Slater method with respect to electrostatic properties of small molecules

The results of various quantum chemical calculations, the Hartree-Fock (HF) method, the Moller-Plesset perturbation theory (MP2), and the Hartree-Fock-Slater (HFS) method are compared. Atomic charges, dipole moments, topological properties of the electron density distribution and polarizabilities, and first hyperpolarizabilities are calculated. Atomic charges obtained with the HFS method are found to be very close to those calculated with the MP2 method, from which we conclude that the HFS method describes to some extent electron correlation effects. Performing an MP2 calculation after an HF calculation improves the molecular dipole moments considerably, yielding values close to the experimental ones. HFS calculations are computationally less demanding than MP2 and yield comparable results for the electron density distributions, dipole moments and polarizabilities.

Analytical representation of atomic shellwise electron densities and applications

Analytical expressions of atomic shellwise electron densities have been constructed. The authors' approach was based on the superposition of Yukawa potentials for the atomic electrostatic potential. The electron density distribution was then established by the Poisson equation. Parameters characterizing the shellwise electrostatic potential and the corresponding electron density distribution were obtained by a fitting procedure utilizing the Hartree-Fock-Slater (HFS) data. These parameters were determined by the atomic number and electron occupation numbers in various states. The accuracy of constructed electron density distributions was tested by comparing their various moments with the corresponding results obtained from other methods. Applications of these atomic shellwise electron densities were made for calculations of the elastic scattering cross section and the mean excitation energy.

Density of states and energy levels of BaF 2:Sm 2+ using nonrelativistic and relativistic cluster models

Density of states and energy levels for BaF 2:Sm 2+ are calculated self-consistently, using both one-electron local density discrete variational non-relativistic Hartree-Fock-Slater (HFS) and relativistic Dirac-Slater (DS) cluster models. The finite clusters calculated include SmF 8 and SmF 8Ba 12 for BaF 2:Sm 2+, which are embedded in a crystal environment. The spin-orbit coupling parameter ζ 4f derived from the DS model is equal to 1247 cm −1. The energy gap derived from the HFS model with SmF 8Ba 12 clusters embedded in the crystal is 9.56 eV, which is near the experimental results.