Relativistic Dirac-Fock Research Articles

Spin polarisation of electrons elastically scattered from mercury

Spin polarisation parameters have been calculated for the elastic scattering of electrons from mercury atoms at incident energies up to 180 eV. These results were obtained by solving the Dirac equations, which included relativistic Dirac-Fock static and non-local exchange potentials plus a non-relativistic polarisation potential. A comparison with recent experiments of the results for the spin polarisation parameters S, T and U as well as for the difference in phase between the direct and exchange scattering amplitudes yields very good agreement at energies above 15 eV. The authors also present results for elastic differential, total and momentum transfer cross sections.

A Monte Carlo Simulation of Auger Cascades

The energy imparted to biological tissue after the decay of incorporated Auger emitters stems from two sources: (a) energy deposition by the Auger and Coster-Kronig electrons and (b) the charge potential which remains on the multiple ionized atom after the end of the cascade. For the numerical assessment of both the kinetic energy of the released electrons and the charge potential, a new and--for purposes of microdosimetry--precise method is presented. Based on relativistic Dirac-Fock calculations and a rigorous bookkeeping, this method provides a perfect energy balance of the considered atomic system when applied to Monte Carlo simulations of Auger cascades. By comparing the results for charge distribution for krypton and iodine with experimental data and the electron spectrum of 125I with theoretical data, it can be shown that the approach followed in this work is reasonable and appropriate for the determination of the energy deposited by incorporated Auger emitters in small volumes of condensed matter. The total energy deposited by 125I in a volume of 20-nm diameter is 2.03 keV which is made up by multiple ionization (1.07 keV) and energy deposition by the emitted Auger electrons (0.96 keV).

An intermediate neglect of differential overlap (INDO) technique for lanthanide complexes: studies on lanthanide halides

An intermediate neglect of differential overlap method of use for examining the electronic structure of lanthanide complexes is developed. It is characterized by a basis set obtained from relativistic Dirac-Fock atomic calculations, the inclusion of all one-center two-electron integrals, and a parameter set based on molecular geometry. Lanthanide halides MX2, MX3 and MX4 are studied here, as well as initial results for the twelve coordinate Ce(NO3)6 −2 ion. Geometries obtained are in excellent agreement with experimental values when available. Many MX3 complexes are found to be pyramidal, and EuCl2 and YbCl2 are calculated to be bent even at the SCF level. Models invoking London type forces are therefore not required. Ionization potentials are calculated for the trihalides (δSCF) and are in reasonable agreement with experiment. Contrary to conclusion of others, f-orbital participation, although small, is required — at least in this model — to obtain the spread of metal to halide bond distance observed in these complexes. However f-orbital participation does not seem to be significant even in the twelve coordinate Ce(NO3)6 −2 complex: rather the large coordination number seems to be a consequence of the relatively large size of the lanthanide ion.

The electron momentum spectroscopy of lead

Results of electron momentum spectroscopy measurements on lead vapour at a total energy of 1000 eV are presented in conjunction with a theoretical analysis using relativistic structure calculations. The 6p-1 transition is seen to be split, resulting in transitions to the 6p1 and 6p3/2 ion states. The measured branching ratio for these transitions is reproduced when a multiconfiguration Dirac-Fock wavefunction is used for the target ground state. The transition involving ejection of an electron from the 6s subshell is also seen to undergo significant splitting. The splitting in this case is mainly due to final-state configuration interaction, although a multiconfiguration target state must again be used to obtain detailed agreement with experiment. The measured momentum probability distributions for the 6p and 6s electrons agree with the orbital momentum distributions given by relativistic Dirac-Fock wavefunctions, although distortion effects are evident at high momenta for the 6s-1 transitions. The corresponding non-relativistic Hartree-Fock momentum distributions are in poorer agreement with the data. An inner valence 5d-1 transition is also seen, but the cross section is too small for detailed comparison with theory.

Energy shifts of M x rays.

The recent experiment by Springer, Bennett, and Baisden on M-x-ray energy shifts is reanalyzed, taking advantage of existing extensive work on K x-rays. The shifts, calculated by use of relativistic Dirac-Fock wave functions, are about a factor of 3 smaller than the experimental values. It is pointed out that x-ray energy shifts can provide information on the healing process in metals.

Accurate wavenumbers of the first resonance doublet of francium atoms and relativistic contribution to isotope shifts in the resonance doublets of francium and caesium atoms

The results of a new experiment using high-resolution Doppler-free laser spectroscopy are reported. Firstly, the wavenumbers of the D1 and D2 lines of Fr atoms have been accurately measured: sigma (D1)-12237.409 (2) cm-1 and sigma (D2)=13923.998 (2) cm-1. Secondly, the isotope shifts of the D1 and D2 lines have been compared for Fr and Cs atoms. The contribution of the p1/2 electron to the volume shift has been measured relative to that of the s1/2 electron as 4.75(25)*10-2 for Fr and 1.4(3)*10-2 for Cs. They are discussed in terms of semi-empirical calculations as well as relativistic Dirac-Fock ones.

Isotope shift and hyperfine structure in low-lying levels of Pb II

The Fabry-Perot technique in the far UV range was used to investigate isotope shifts and the hyperfine structure of resonance transitions in singly ionised lead. The experimental ground-configuration data were compared with those computed with Desclaux's (1975) program which uses the relativistic Dirac-Fock method. A good agreement is observed between experimental results and theoretical evaluations, for both the isotope shift and the hyperfine structure.

X-ray attenuation cross section measurements in the K edge region of krypton

The absolute X-ray attenuation cross section of krypton in the vicinity of the K edge was measured. The data are analysed within the framework of the screened hydrogenic approximation using various sets of screening constants. Those obtained from relativistic Dirac-Fock calculations give the best agreement with the data, although significant deviations occur from the threshold up to about 100 eV above it. No deviations from monotonicity are observed, and the data seem to support recent theoretical predictions on the decisive role of relaxation and overlap effects in determining the shape of the cross section immediately above threshold. Jump ratios obtained from the data and from theoretical calculations are also presented and discussed.

Absolute total static relaxation and extra-atomic relaxation terms for volatile phosphorus compounds obtained from the Auger parameter and relativistic

Reassessment of previous experimental results using the Auger parameter of Lang and Williams, plus a relativistic Dirac—Fock atomic calculation for the phosporus atom, yields absolute relaxation and extra-atomic relaxation contributions for phosporus compounds in a straightforward way; all terms, except the extra-atomic relaxation contribution arising from the ligands, are defined relative to the central probe phosphorus atom. The extra-atomic relaxation contributions so obtained are found to be non-additive but averaged. The relaxation contributions deduced from the data obtained for the phosphorus compounds are not identical to those deduced for sulfur compounds for comparable ligands, but the relative trends are parallel. The absolute relaxation values are compared with various empirically derived relaxation terms. Initial-state effects are also deduced from relaxation corrected experimental binding-energy shifts.

Relativistic Gaussian basis set calculations on one-electron ions with a nucleus of finite extent

The disadvantages of using Gaussian functions (GTOs) in SCF electronic structure calculations are well known. The difficulties associated with Gaussians are likely to be more severe in relativistic Dirac-Fock calculations. Relativistic effects are important for core electrons, whereas GTOs give a poor description near the nucleus. By changing the boundary condition on the wavefunction at the origin, it may be possible to remove some of the difficulty. In calculations on O 7+ and Hg 79+ we explore the effect of replacing the point model of the atomic nucleus by a nucleus of finite size. We find that smaller basis expansions are required for the heavier nucleus in the finite-nucleus model.

Theoretical investigation of hyperfine interactions and isotope shifts in gold

Use of relativistic many-body perturbation theory is made to calculate the hyperfine splitting of the ground 62S1/2 state in 197Au (I=3/2). The calculated splitting of 6068 MHz is in good agreement with the experimental value of 6099.320184(13) MHz determined by an atomic beam magnetic resonance technique. Electron charge densities at the nuclear surface corresponding to various configurations of 197Au are calculated in an ab initio way by numerically solving the relativistic Dirac-Fock equations incorporating the nuclear charge distribution. The electronic factor relevant to the volume isotope shift for the resonance transition 6s2S1/2-6p2P1/2 in gold is also calculated and compared with other available results.

Relativistic Dirac—Fock—Slater potential-energy curve for N 2

A fully relativistic SCF Dirac—Fock—Slater program to calculate diatomic molecules has been developed. Numerical wavefunctions of atomic DFS calculations are used as basis functions. All integrations as well as the solution of the Poisson equation are performed fully numerically. As a first test, the relativistic potential-energy curve of N 2 is presented and compared with non-relativistic Hartree—Fock—Slater calculations.

Volume isotope shifts in low lying transitions of Au I

Electronic factors in the volume isotope shift have been calculated in anab initio way with the relativistic Dirac-Fock method for a number of different optical single/and two-photon transitions in Au I. The agreement with a semi-empirical method is within 10% for the resonance transition. For this one and a few other transitions the effect of core excitation has been analyzed with the Multi-configuration Dirac-Fock method as well, and it was found to reduce the electronic factor in the order of 5%.

N4,5N6,7X Coster-Kronig spectra of free Hg atoms

The N4,5N6,7X (X=N6,7, O2,3, O4,5) Coster-Kronig-type Auger spectra of free mercury atoms have been measured and analysed by comparing the experimental results with the calculated profiles. The spectra have been measured with the use of electron beam excitation. The kinetic energies of the Auger lines have been calculated by the Delta SCF method using a relativistic Dirac-Fock program. The relative energies of the line components have also been computed non-relativistically and a comparison has been made between the non-relativistic and the relativistic results. The intensities of the lines have been calculated in the mixed coupling scheme applying jj coupling to the initial states and intermediate coupling to the final states. The calculated profile reproduces the fine structure in the experimental spectrum rather well. The experimental intensity ratios between different line groups are between the experimental results of Matthew et al. (1980) and Larkins et al. (1977) for solid Au, and in obvious discrepancy with the calculated group intensity ratios based on McGuire's results (1974).

Study of M 4,5N 4,5N 4,5 auger energy shifts of Cs and I in free CsI molecules

High-resolution electron beam excited M 4,5N 4,5N 4,5 Auger electron spectra of Cs and I have been measured from CsI vapour. The Auger energies of both Cs and I observed from gaseous CsI are higher than the corresponding free-atom energies due to extra-atomic relaxation. The molecular Auger results have been compared with corresponding photoelectron measurements and free-atom data. Estimates for extra-atomic relaxation energies have been extracted from the changes of the Auger parameter between molecular and atomic species and from the difference between experimental energies and energies calculated with a relativistic Dirac-Fock program, applying the point-charge model for the CsI molecule.

Effective core potentials for fully relativistic Dirac–Fock calculations

A fully relativistic effective core potential method has been developed for use in the valence-only Dirac–Fock self-consistent-field calculations for atoms and molecules. The effective potentials are constructed so that the atomic valence-only Dirac–Fock calculations reproduce the energies and the outer region of the corresponding valence spinors obtained from the full Dirac–Fock SCF calculations.

Photoionization of the4d10subshell of cadmium: Photoelectron angular distributions and polarization of fluorescence radiation

Relativistic Dirac-Fock calculations are presented for the photoelectron angular distribution asymmetry parameter $\ensuremath{\beta}$ corresponding to $4d$-subshell photoionization in Cd and for the linear polarization ${P}_{L}$ of the subsequent fluorescence radiation, for photon energies less than 185 eV. Near threshold, our results for $\ensuremath{\beta}$ lie lower than previous relativistic Dirac-Slater and nonrelativistic many-body-perturbation-theory (MBPT) calculations. The agreement with very recent measurements by Heinzmann and Sch\onhense is very good. Our calculations show that ${P}_{L}(^{2}D_{\frac{5}{2}}\ensuremath{\rightarrow}^{2}P_{\frac{3}{2}})=\ensuremath{-}0.059 \mathrm{and} \ensuremath{-}0.058$ at incident photon energies $h\ensuremath{\nu}=40.2 \mathrm{and} 45.6$ eV, respectively. These results lie above the nonrelativistic independent-electron theory absolute maximum of -0.061 for ${P}_{L}$, but the agreement with recent measurements lying above this maximum is still not very good. Except for this peaking of ${P}_{L}(^{2}D_{\frac{5}{2}}\ensuremath{\rightarrow}^{2}P_{\frac{3}{2}})$ above the nonrelativistic theory maximum for $40.2\ensuremath{\le}h\ensuremath{\nu}\ensuremath{\le}45.6$ eV, our results at other energies are in general agreement with the nonrelativistic MBPT calculations of Carter and Kelly. We also find that ${P}_{L}(^{2}D_{\frac{3}{2}}\ensuremath{\rightarrow}^{2}P_{\frac{3}{2}})=+0.134$ at $h\ensuremath{\nu}=21.2$ eV, in excellent agreement with the experimental value of + 0.12\ifmmode\pm\else\textpm\fi{}0.04 of Caldwell and Zare.

On the core binding energies of ions: The 3d levels of I− and Cs+

The 3d5/2 binding energies of cesium and iodine in gaseous CsI have been measured by x-ray photoelectron spectroscopy. The values obtained are 732.3±0.1 eV for Cs and 623.6±0.2 eV for I. A simple electrostatic model is used to compute ionic and atomic 3d5/2 binding energies from the CsI measurements. We have also obtained these energies by relativistic Dirac–Fock calculations. Comparison is made with analogous nonrelativistic results and with experiment.

X-ray emission from highly stripped atomic ions

Abstract Relativistic Dirac-Fock (DF) and central-field Dirac-Fock-Slater (DFS) calculations of X-ray emission from highly stripped atomic lead ions have been performed. The X-ray line shifts provide a nonmonotone characteristic of outer-shell ionization. The intensities of the most intense X-ray lines are largely unaffected by the removal of outer-shell electrons.

Self-consistent relativistic density-functional theory: Application to neutral uranium atom and some ions of lithium isoelectronic sequence

In a relativistic density-functional theory the importance of the transverse photon contribution to the exchange potential in the high-density regime is pointed out. A neutral atom $^{92}\mathrm{U}$ is studied and its orbital energies calculated in the present scheme are compared with previous calculations based on relativistic Hartree, Dirac-Slater, and Dirac-Fock schemes and with experimental values from (electron spectroscopy for chemical analysis) ESCA studies. The present scheme is also used to calculate the orbital and total energies of some highly stripped ions of the Li isoelectronic sequence which occur as impurities in controlled thermonuclear plasma. These energies are found to be in close agreement with more involved Dirac-Fock results. Also, the relativistic effects significantly increase in going from ${\mathrm{C}}^{3+}$ to ${\mathrm{W}}^{71+}$ because the electronic density reaches values for which relativistic effects become large.