Hartree Wave Functions Research Articles

Charge-Exchange Calculations with Hartree Wavefunctions

Symmetric resonance charge-exchange cross sections are calculated, in the quasiclassical approximation. Previous calculations of the same type, due to Rapp and Francis, give systematically lower cross sections than the experimental ones. It is suggested that this is a result of the rough form of the atomic wavefunctions used by Rapp and Francis. Here, Hartree–Fock–Slater wavefunctions are used. The necessary integrations are done numerically. On the average, much better agreement with experimental results is achieved. The calculations were performed for He, K, Cs, and Hg.

Quadrupole Polarizabilities of Various Ions and the Alkali Atoms

The quadrupole polarizability ${\ensuremath{\alpha}}_{q}$ which measures the induced quadrupole moment produced by an external field gradient has been calculated for three ions: ${\mathrm{Rb}}^{+}$, ${\mathrm{Cs}}^{+}$, and ${\mathrm{Bi}}^{5+}$, and for the neutral alkali atoms. The present calculations for ${\mathrm{Rb}}^{+}$ and ${\mathrm{Cs}}^{+}$, using the appropriate Hartree-Fock wave functions, represent a considerable improvement over our previous calculations, in which Hartree wave functions (without exchange) were employed. The values obtained for ${\ensuremath{\alpha}}_{q}$ of the alkali atoms are very large, ranging from 60.0 ${\mathrm{\AA{}}}^{5}$ for Li to 440.8 ${\mathrm{\AA{}}}^{5}$ for Cs. In Sec. III, we have collected the values of ${\ensuremath{\alpha}}_{q}$ for 11 ions, as obtained using the author's method of direct solution of the inhomogeneous wave equation. Wherever possible, these values have been compared to those obtained by other methods. The agreement is generally very good.

Electronic Polarizabilities of the Alkali Atoms. II

The electronic dipole polarizabilities ${\ensuremath{\alpha}}_{d}$ of the alkali atoms have been recalculated with the inclusion of the shielding effect of the ion core in the perturbation equations. The resulting values of ${\ensuremath{\alpha}}_{d}$ are close to those obtained in a previous calculation, and are also in very good agreement with the recent experimental values of Hall and Zorn. The dipole polarizabilities ${\ensuremath{\alpha}}_{d}$ of the ${\mathrm{Rb}}^{+}$ and ${\mathrm{Cs}}^{+}$ ions have been recalculated, using the appropriate Hartree-Fock wave functions (including exchange), which have recently become available. The resulting values of ${\ensuremath{\alpha}}_{d}({\mathrm{Rb}}^{+})$ and ${\ensuremath{\alpha}}_{d}({\mathrm{Cs}}^{+})$ are in much better agreement with experiment than those obtained in a previous calculation using Hartree wave functions (without exchange).

Lamb shift in the metastable states of the helium atom

The radiative corrections of order α 3 rydbergs are evaluated for the ionization energy of the metastable states 2 1 , 3 S, of the helium atom. In the calculation of the average excitation energy k 0 , the main contribution comes from the transition to and ( ms, np ) and ( ms, ∊p ) states. The oscillator strengths for transitions to (1 s, ∊p ), (2 s, ∊p ) and (3 s, ∊p ) states are evaluated by using six-parameter wavefunction for the metastable states and a product of a hydrogenic wavefunction with Z = 2 for the s electron and a wavefunction analogous to the Hartree wavefunction for the excited p electron. Making use of these oscillator strengths and a method used by Pekeris, the values of the average excitation energies for the singlet and triplet states are found to be 77.09 ± 1.6 and 79.84 ± 1.0 rydbergs respectively. With these values of the average excitation energies, the Lamb shift corrections, including the estimate of a α 4 Ry order corrections, to the ionization energies of the singlet and triplet states become – 0.106 ± 0.018 cm -1 and –0.129 ± 0.013 cm –1 respectively. When they are added to the theoretical values of the ionization energies obtained by Pekeris, the values of the ionization energies become 32033.212 ± 0.018 an d 38454.698 ± 0.013 cm -1 compared with Herzberg’s experimental values of 32033.24 ± 0.05 an d 38454.73 ± 0.05 cm -1 for the singlet and triplet states respectively.

A new calculation of electron scattering cross sections and a theoretical discussion of image contrast in the electron microscope

The current basis for the theoretical discussion of the contrast in an image in the electron microscope is the 1954 Lenz theory. The basis of this theory is reviewed and the reasons are given for the two interpretations of Lenz's theory in its application to all the elements which have arisen in the literature; sample calculations based on these interpretations and theoretical reasons for favouring one of them are given. Since 1954, theoretical descriptions of the atom have improved and the Hartree-Fock wave functions have replaced the Hartree wave functions and the Thomas-Fermi-Dirac potentials have superseded the Thomas-Fermi potentials. In this article the results from the newer atomic models are incorporated into the theory of single-electron scattering as applied to electron microscopy. The new method of calculation avoids the rigid expressions used by Lenz for the potential distribution within an atom and the atomic radius by using directly the values of the atomic scattering factors for X-rays and for electrons evaluated from the latest atomic models. A feature of the new expressions for the elastic and inelastic atomic scattering cross sections is their flexibility; the results of future developments of the atomic models can be readily incorporated into the calculations. The variations of the elastic, inelastic and total mass-scattering cross sections and differential atomic scattering cross sections with the accelerating voltage of the electron beam, the angular aperture of the objective lens and the atomic number of the amorphous specimen giving rise to the scattering, are discussed in detail for both the newly developed theory and the Lenz theory in its two interpretations. Most of the results are given for accelerating voltages in the range 20-100 kV but a few results for the new theory are given for voltages of 500 kV and 1000 kV. The differences and similarities between the results of the two theories are discussed and a section is included on the agreement between theoretical and experimental parameters describing the image contrast. The theoretical values of the total mass-scattering cross sections are appreciably higher than the experimental values, but the theoretical predictions about the variations of the cross sections with the atomic number of the scattering element and the instrumental parameters are in general agreement with the experimental results. Methods are suggested by which the basis of the new calculations of the cross sections may be improved.

The Hartree Perturbation Method for Helium

The Hartree perturbation method is used to calculate the dipole polarizability, the electric field at the nucleus, the quadrupole polarizability and the nuclear shielding factor of helium using a twelve-term representation of the unperturbed Hartree wave functions. It is pointed out that the method yields lower bounds for the dipole and quadrupole polarizabilities. The absolute error in the predicted values of any of the quantities is less than 4.3%.

Single Electron Shakeoff Probability Following the Beta Decay of Krypton

In view of the experimental work of A. H. Snell and F. Pleasonton, which indicates that the product atom following the beta decay of krypton is frequently left in high states of ionization, it is of interest to attempt a calculation of shakeoff probabilities for all the outermost occupied configurations. The method of calculation used here essentially follows the formulations of A. Migdal, E. L. Feinberg, J. S. Levinger, H. M. Schwartz, and earlier workers. The details of the calculations are governed to a considerable extent by the availability of Rb/ sup +/ Hartree wave functions. The innovations which are attempted here involve the use of newly determined analytical approximations to Hartree functions, the use of a simple scaling technique, and the extension of the work to the outermost occupied levels. Comparisons of the calculated shakeoff probabilities with those inferred from experiment by Snell and Pleasonton are made. The agreement is fairly satisfactory. It is believed that some of the innovations used here might be useful for other types of atomic calculations. (auth)

Coulomb Scattering of Electrons at Very Small Angles

Electrostatic scattering of highly relativistic electrons at very small angles is discussed. The Hartree wave function is used to calculate the screening effects for the smallest angles. The calculations are facilitated by a theorem which states that, in the Born approximation for elastic scattering by any physical central potential, the total cross section in the relativistic limit is independent of energy.Both the "elastic" (nuclear) and "inelastic" (electronic) contributions to the scattering are investigated. It is found that at very small angles the inelastic differential cross section exceeds by many times the elastic differential cross section. Numerical calculations were carried out for scattering by nitrogen atoms, where the total cross section of the scattering by atomic electrons is 3.6 times the total cross section due to screened nuclear scattering.

Computed Transition Probabilities for X-Ray Continua of Potassium

The rate of change of the oscillator strength at the $K$-edge of ionized potassium has been computed in a number of ways and a few computations have been made on the $L$-edge. The work has been carried through by using both hydrogenic and Hartree wave functions at the level of the single-electron approximation and Hartree and Hartree-Fock wave functions when the single-electron approximation was not used. Calcium wave functions both with with and without exchange have also been employed. The moment, momentum, and acceleration matrix elements have been computed. The momentum matrix element gives values most closely in agreement with experiment and is most stable with respect to changes in the wave functions. It is followed by the acceleration and the moment in that order. The single electron approximation gives good results. If the single-electron approximation is not used, it is essential that exchange effects be included by employing the determinant form of the wave functions, and the Hartree-Fock one-electron wave functions then give considerably better results than the ordinary Hartree functions. It also proves to be possible to substitute the wave functions of the unexcited ion of next higher atomic number, calcium, for the wave functions of potassium excited in an x-ray level without noticeable deterioration in the results.

Improved Theory of the Zeeman Effect of Atomic Oxygen

The previous calculations of Abragam and Van Vleck on the $g$ factors of the $2{p}^{4}^{3}P$ terms of atomic oxygen are refined by including exchange, using better wave functions, and not assuming that the charge distribution in which a $2p$ electron moves is spherically symmetric. In Sec. 2, a general technique is developed for calculating the relativistic and diamagnetic corrections to the $g$ factors of complex atoms (Breit-Margenau and Lamb terms, etc.) by an expansion in spherical harmonics which is an extension of the well-known Slater procedure for computing the angular dependence of electrostatic energy. The integrals encountered are usually either of the Condon and Shortley $F$, $G$ type, or variants thereof. The most important members are computed by means of the wave functions of Hartree, Hartree, and Swirles, based on a Fock self-consistent field, but some of the minor contributions are estimated with Slater wave functions improved by introducing slightly different screening constants for the $2s$ and $2p$ terms. When all corrections are comprised, including also the Schwinger electrodynamic shift, motion of nucleus, and deviations from Russell-Saunders coupling, the calculated values of the $g$ factors for the $^{3}P_{1}$ and $^{3}P_{2}$ terms are, respectively, 1.500974, 1.500913, in excellent agreement with Rawson and Beringer's experimental determinations 1.500971, 1.500905.

The Continuous Radiative Absorption Cross Section of FeXIV and the Coronal Temperature

The continuous radiative absorption cross section of FeXIV was calculated, using a Hartree wave function to evaluate the matrix element for recombination to the ground state. This matrix element was considerably smaller than the value obtained by Hill (1950, 1951) using the hydrogen-like approximation, but the total cross section for recombination was not greatly different.

The Effects of Configuration Interaction on the Atomic Hyperfine Structure of Gallium

The effects on the hyperfine structure of the mixing of higher configurations with the ground configuration ${(4s)}^{2}4p$ of gallium is studied. It is found that the only configuration which need be considered is the $4s4p5s$. A Hartree wave function for the excited $5s$-state of gallium is calculated, and the extent of the mixing of this excited configuration with the ground configuration is determined. The coupling constants for magnetic and electric quadrupole hyperfine structure are computed with the use of the wave function for the ground state of gallium including the excited configuration. The determination of quadrupole moments from atomic hyperfine structure measurements is discussed and various methods of making this calculation compared. Beside the determination of the quadrupole moments of gallium, the quadrupole moments of aluminum, indium, and chlorine are estimated.

Patterns of term structure in simple spectra.

view Abstract Citations (3) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Patterns of term structure in simple spectra. Layzer, David Abstract Theoretically, a pure configuration of electrons outside closed subshells has a definite, calculable pattern of term structure. For example, the two intervals between the three terms of (closed sub- shells) (p1) have the theoretical ratio 1.5. The observed patterns, however, often vary widely from atom to atom. For example, the above-mentioned ratio for (Is ) ( p1) is about -5 in Bei and 8 in BIl. Discrepancies of this kind are generally, and in my opinion correctly, ascribed to the mixing of configurations. Two obstacles have stood in the way of a successful quantitative theoretical description of configuration mixing: the lack of reliable criteria for recognizing the important components in a configuration mixture; and the lack of workable mathematical machinery for calculating the degree of mixing of given configurations. In this paper I try to overcome both these obstacles. My first point is that the data can be grouped so as to exhibit useful regularities in the structure of configuration-sequences. A configuration-sequence is defined as a set of identical configurations in an iso-electronic sequence, e.g., (Is2) ( p2) in Bei, BIl, CIII If 1(Z) is the interval between two terms of a configuration in such a sequence, Z denoting the atomic number, then the interval-difference I (Z I) - 1(Z) becomes sensibly constant for moderately large Z. The set of all independent interval-differences for a given sequence characterizes the sequence as a whole. A simple theoretical argument now shows that the structure of this set is influenced only by a very special kind of configuration mixing, which I call stable mixing. Only a small number of configurations can participate in a stable mixture, and these may be readily enumerated in every instance. I have found a method of predicting the structure of the set of interval-differences characterizing a configuration-sequence, and have successfully applied it to the sequence (is ) ( p2). This method, moreover, may be used to predict patterns of term structure in individual atoms. The crux of the method lies in the construction of analytic one-electron wave functions of a very general form. The 15, 25, and 2p wave functions are as accurate as the corresponding numerical wave functions of Hartree and Fock. The variational wave functions are used as the basis for a perturbation treatment, which can be carried out to any desired degree of accuracy. The Observatory, University of Michigan, Ann Arbor, Mich. Publication: The Astronomical Journal Pub Date: April 1951 DOI: 10.1086/106496 Bibcode: 1951AJ.....56...43L full text sources ADS |

A Quantum Mechanical Treatment of the Lithium Fluoride Crystal

A simplified quantum-mechanical model for the LiF crystal is described. The lattice parameter, cohesive energy and compressibility are calculated using three different choices of wave function for the fluorine ion in the crystal; these are (i) the Hartree wave functions for the free ion F-, (ii) wave functions orthogonalized to the 1s-orbitals of the six nearest neighbour lithium ions, (iii) functions similar to (ii) but based on wave functions for the free ion which have been contracted to give agreement with the diamagnetism of F-. The results obtained in (i) are in fair agreement with the empirical values Large discrepancies are observed when the more complete wave functions (ii) are employed. This is taken as indicating that the Hartree wave functions for the free ion are too diffuse and the results of the third calculation, which agree well with experiment, substantiate this interpretation.

Prediction of Interatomic Distances and Force Constants in Diatomic Hydrides

The interatomic distance in hydrides is roughly the radius in the united atom at which Zeff=1.0. The force constant is roughly 4πρe2 where ρ is the electron density at this radius. These results are justified by a perturbation treatment. The best fit to experimental values is obtained using Hartree wave functions. Slater functions account for the general trends and breaks in the curves of distance and force constant vs. atomic number, and for the anomalous values noted by the Sheline in the transition elements.

Interference in the Zeeman Effect of Forbidden Lines

The Zeeman effect of forbidden lines involving simultaneous electric quadrupole and magnetic dipole radiation will exhibit interference between these two different modes of radiation. The predicted intensity of any Zeeman component includes, in general, a term dependent on the fact that both modes of radiation are possible, as well as the usual transition probabilities for independent electric quadrupole and magnetic dipole radiation. Such additional interference terms appear only in the Zeeman effect and not in total line intensities. Proofs of these assertions and formulas for the Zeeman intensities are developed. The latter are compared with observations of the Zeeman effect of the forbidden lines of the $6{p}^{2}$ configuration of Pb I, by Jenkins and Mrozowski. Good agreement with experiment is obtained only if interference is taken into account. Their observations on the Zeeman effect correspond to a somewhat smaller value of the quadrupole moment of the transition electron in Pb I than is computed from comparison of total line intensities. Both these values of the quadrupole moment of the transition electron are 100 percent or more smaller than that given by a rough estimate from screening constant data using hydrogenic wave functions and the known positions of the levels to evaluate the effective nuclear charge. In the absence of Hartree wave functions for Pb I, no better estimate seems possible. A brief discussion of the hyperfine structure of forbidden lines is included, in which it is shown that the rules for determining the relative intensities of electric dipole hyperfine multiplets also give the intensity ratios in the hyperfine structure of magnetic dipole lines.

The Variation of theKResonating Strength with Atomic Number

The resonating strength of the $K$-shell or the effective number ${N}_{K}$ of $K$ electrons is less than 2 and varies with atomic number. It is determined in the present paper by an approximate theory using Hartree wave functions. The results are compared with ${N}_{k}$ values obtained by anomalous dispersion, by computation from the most reliable absorption experiments, and by an approximate absorption theory of H\onl and Williams. Below atomic number 40 the present theory gives good results and allows of the determination of the ${N}_{k}$ values which are needed in finding absolute x-ray wave-lengths from dispersion measurements.

Isotope Shift in Neon

The theory of isotopic displacement due to the motion of the nucleus has been extended to atoms with any number of electrons (assuming Russell-Saunders coupling) and applied to the transitions $2{p}^{5}3s\ensuremath{-}2{p}^{5}3p$ in neon. Approximate Hartree wave functions have been obtained and orthogonalized. The theory gives a shift in singlet states of 0.0195 ${\mathrm{cm}}^{\ensuremath{-}1}$ more than that in triplet states, as against approximately 0.017 ${\mathrm{cm}}^{\ensuremath{-}1}$ measured by Nagaoka and Mishima. However, the calculated shift for the singlet transitions is 0.0038 ${\mathrm{cm}}^{\ensuremath{-}1}$, as against the experimental value of 0.0332 ${\mathrm{cm}}^{\ensuremath{-}1}$. This discrepancy may disappear if better wave functions are found.