Properties of the quasiparticle excitations in 207Pb and 209Pb from an extrapolation of the optical-model potential

A previously developed dispersion relation approach is used to calculate the shell-model potential in the case of neutrons in ²⁰⁸Pb, in the energy domain (-50 MeV, 0). This potential contains a dispersive contribution besides a Hartree-Fock type component, and thereby includes correlation and polarization effects. The shell-model and the Hartree-Fock type potentials are assumed to have Woods-Saxon shapes with diffuseness a_v = 0.70 fm; the energy dependence of their depths and radii is calculated. The energy dependence of the shell-model potential is characterized by the effective mass, whose dependence upon radial distance and neutron energy is determined. The effective mass is a sensitive function of energy, in contrast to its Hartree-Fock type component which is nearly independent of energy. Attention is drawn to the fact that the effective mass in nuclear matter cannot be straightforwardly identified with the effective mass at the nuclear centre. The effective mass presents a sharp peak at the nuclear surface near the Fermi energy and a dip at the surface for energies 10 to 20 MeV away from the Fermi energy. The spectroscopic factors of single-particle excitations in ²⁰⁷Pb and ²⁰⁹Pb are calculated from the difference between the effective mass and its Hartree-Fock type component. The predicted values of the valence single-particle wave functions at large radial distances are in fair agreement with experimental values deduced from analyses of sub-Coulomb pickup reactions. It is shown that the dispersive contribution increases the level density parameter by about 25%, in agreement with previous microscopic or semi-phenomenological models; the calculated level density parameter is in good agreement with the empirical value.     

Single-particle potential in a relativistic Hartree-Fock mean field approximation

A relativistic Hartree-Fock mean field approximation is investigated in a model in which the nucléon field interacts with scalar and vector meson fields. The Hartree-Fock potential felt by individual nucléons enters in a relativistic Dirac single-particle equation. It is shown that in the case of symmetric nuclear matter one can always find a potential which is fully equivalent to the most general mean field and which is only the sum of a Lorentz scalar, of one component of a Lorentz tensor and of the fourth component of a Lorentz vector. A non-relativistic potential is derived which yields exactly the same single-particle energies and elastic scattering phase shifts as the relativistic Hartree-Fock potential. Analytical results are presented in the case of nuclear matter. A local density approximation is constructed which enables one to consider finite nuclei. The input parameters of the model can be chosen in such a way that the empirical saturation properties of nuclear matter are well reproduced. Good agreement is obtained between the calculated non-relativistic potential and the empirical value of the real part of the optical-model potential at low and at intermediate energy. At intermediate energy, the wine-bottle bottom shape which had previously been found for the potential in the framework of the relativistic Hartree approximation is maintained when the Fock contribution is included.     

A treatment of renormalization,potential energy of intermediate states and re-arrangement effects in brueckner calculations of closed shell nuclei

In the Brueckner-Hartree-Fock theory of finite nuclei, the problems of renormalization, potential energy of intermediate states and re-arrangement effects are examined. For practical calculations it is shown that satisfactory solutions to them can be obtained if the parametrization chosen for the single-particle potential in finite nuclei is linked closely to nuclear matter results. We make the link with similar problems in the density dependent Hartree-Fock theory and emphasize the possibility of such a parametrization by recalling the existence of a formal solution for the Hartree-Fock single-particle potential if the effective interaction is of the δ-function type. A method of solution of the Bethe-Goldstone equation is then presented which separates the intermediate states into those of an “open shell” and of a “continuum”. Finally results of model calculations of ¹⁶O and ⁴⁰Ca with harmonic oscillator functions are presented in which the parametrization chosen for the BHF single-particle potential is taken from the Skyrme-Vautherin δ-function force. A self-consistent determination of certain parameters of this form of force leads to values in close agreement with the empirical estimate made by Vautherin and Brink in ¹⁶O, with the exception of the spin-orbit splittings. Limitations and possible improvements of this type of approach are discussed for ⁴⁰Ca.     

Nuclear compression moduli and density-dependent forces

Density-dependent zero-range forces of the form of the modified delta interaction (MDI) are generalized (MDI3, MDI4) in order to yield reasonable values of the compression modulus in nuclear matter (K_N = 200 MeV). This low value can be fitted by introducing two terms with different density dependence in the force. The four free parameters of MDI3 are adjusted to reproduce the nuclear matter values of the binding energy, density and compression modulus, and to fulfil the condition that the total energy of ¹⁶O in harmonic oscillator wave functions has a minimum at the oscillator length b = 1.75 fm, corresponding to the correct rms radius. MDI4 contains in addition a two-body spin-orbit interaction. The five parameters of MDI4 are fitted to the above three nuclear matter data and by requiring that Hartree-Fock (HF) calculations in ²⁰⁸Pb yield the experimental charge rms radius and reasonable values of certain single-particle spin-orbit splittings. The quality of MDI4 is checked by comparing calculated rms radii, binding energies, and elastic electron scattering cross sections with available experimental data for doubly closed shell nuclei. As a test the energy levels and the nuclear monopole polarization of muonic ²⁰⁸Pb are calculated self-consistently yielding impressive agreement with experiment.     

Do electron scattering and muonic x-ray data really require a central depression in the charge distribution of208Pb?

The charge distribution of²⁰⁸pb calculated in the Hartree-Fock (HF) approach using the density dependent nucleon-nucleon interaction of Ehlers and Moszkowski is tested by simultaneous comparison with experimental data from 502 MeV elastic electron scattering and muonic atoms. In both cases the agreement is very good and nearly as good as the best fits with a phenomenological charge distribution of Fermi type, if the effect of the polarization of the nucleus due to the presence of the muon is properly taken into account. In contradiction to the Fermi fits the HF distribution shows a hump at the center of the nucleus.     

Electronic structure,field-induced magnetization density and neutron magnetic form factor of palladium

The magnetic field induced magnetization density and neutron magnetic form factor of Pd metal is obtained from an abinitio APW energy band study of its electronic structure and properties. The magnetization consists of the spin density calculated for states on the Fermi surface and a much smaller orbital contribution. The solid state wavefunctions are found to yield a spatial localization of the spin density which is greater than that of the very contracted Hartree-Fock density of the free Pd²⁺ ion. The theretical magnetic form factor, which is dominated by the contribution of the fifth band, is found to be in excellent agreement with the measurements of Cable, Wollan, Felcher, Brun and Hornfeldt.     

The electronic states of silicon monofluoride: Low-lying valence states

Hartree-Fock and configuration interaction calculations are reported for the lowest valence state of the SiF molecule. Deduced spectroscopic constants are in good agreement with the experimental values. The dipole moment calculated for SiF is consistent with a charge distribution corresponding to a Si⁺F^? configuration.     

The shell correction and self-consistent deformation energies

The Hartree-Fock deformation energy of the nucleus is represented as the sum of two terms one of which (E^σ) is due to the re-distribution of the nuclear density and depends on the microscopically non-self-consistent parameters σ of the nuclear shape. The other component (E^π) is related to the coherent distortion of the quasiparticle wave functions in the occupied states and is the same as the deformation energy considered in theories of microscopic vibrations for a fixed quasiparticle distribution. Quantities averaged over the particle-hole distribution are introduced which satisfy the condition of statistical self-consistency. It is shown that the shell correction energy represents the averaged effect of the re-distribution of the single-particle states. Finally, corrections are formulated for the shell-model potential which does not satisfy the condition of statistical self-consistency.     

Neutron density distributions of neutron-rich nuclei studied with the isobaric yield ratio difference

The isobaric yield ratio difference (IBD) between two reactions of similar experimental setups is found to be sensitive to nuclear density differences between projectiles. In this article, the IBD probe is used to study the density variation in neutron-rich ⁴⁸Ca . By adjusting diffuseness in the neutron density distribution, three different neutron density distributions of ⁴⁸Ca are obtained. The yields of fragments in the 80A MeV ^{40, 48}Ca + ¹²C reactions are calculated by using a modified statistical abrasion-ablation model. It is found that the IBD results obtained from the prefragments are sensitive to the density distribution of the projectile, while the IBD results from the final fragments are less sensitive to the density distribution of the projectile.     

Hartree-Fock distortion energy of the nucleus

The Hartree-Fock energy of a nucleus in an external field is expressed in terms of two auxiliary density matrices. One of them is a smoothed distribution $\text{?}$ which generates the non-selfconsistent deformed shell-model potential and is directly related to the nuclear shape. The other represents the volume polarization effect by the external field. The final expression suggests a possibility of a two-mode model for the nuclear vibrations. Renormalisation of smoothed quantities by means of phenomenological models is indicated.     

A microscopic approach to the multipole resonances in 20Ne

A fully microscopic study of the proton radiative capture on ¹⁹F with excitation of the multipole resonances in ²⁰Ne is presented. The calculation makes use of a version of the Feshbach formalism of nuclear reactions. The bound states are described as linear combinations of angular momentum projected one particle — one hole configurations in a deformed Hartree-Fock basis. The agreement for the 90° yields with the experimental data is excellent. Gross features as well as intermediate structures are nicely reproduced.     

Determination of the proton and neutron densities at the nuclear periphery with antiprotonic X-rays and \overline p -nucleus reactions

The nucleon density at the nuclear periphery has been investigated with two different methods. The first one is the measurement of the energy shift and the width of antiprotonic X-ray lines. Besides various other nuclides the X-rays from the isotopes ¹⁷²Yb and ¹⁷⁶Yb have been measured very extensively. From these data the absorption widths of six antiprotonic levels and the shifts of the lowest visible transitions could be determined. The comparison of the experimental intensities of the transitions with results from calculations of the antiprotonic cascade showed a good agreement. The second method is the determination of the yield of the annihilation products from the reaction of the antiprotons with nuclei. From this the neutron-to-proton density ratio at a distance of about 3 fm outside the half-density radius can be determined. This method was extended to short-lived residual nuclei with a half-life down to 5 s. The experiment confirmed the previously found negative correlation between the size of the neutron skin and the binding energy of the last neutron. The results have been compared with calculations. For many nuclei a good agreement was found between experiment and calculation, however for some nuclei measured and calculated values are at variance. This revised version was published online in August 2006 with corrections to the Cover Date.     

Measurement of the fine-structure splitting of the 42 D state of lithium using level-crossing spectroscopy

The fine-structure splitting of the 4² D state in lithium was measured using stepwise laser excitation combined with level-crossing spectroscopy. The splitting was determined to be 456.2(0.8) MHz, a considerably higher value than the one obtained in a recent level-anticrossing measurement. The new value is in good agreement with the result of a Hartree-Fock calculation, 460.2 MHz.     

Modification of Dispersion Relations and Generalization of the Hartree-Fock Method for Hard-Core Interactions in Nuclear Matter

The influence of a hard-core part in the interaction on dispersion relations for the generalized optical potential (mass operator) and the T-matrix of nuclear matter is investigated in the frame-work of the A⁰⁰-approximation. The model is based on the two-nucleon scattering problem in vacuo, for which a hard-core generalization of the Low equation is derived. As a consequence, T-matrix and mass operator are shown to split into a polynomial of the first order in the energy variable and a dispersion integral generalized by a limiting process, so that dispersion relations of the twice subtracted type result. Restriction to a self-consistent calculation of the non-dispersive term of the mass operator leads to a close analogue of the Hartree-Fock equations for non-singular interactions. This simple approximation which avoids the full-nucleon problem is shown to yield a qualitatively correct density dependence of the ground-state energy possibly to be improved by more realistic interactions. A formulation as an eigenvalue problem for finite nuclei is also given.     

Numerical solutions of the orbital equations for diatomic molecules

A basis of Hermite splines is used in conjunction with the collocation method to solve the orbital equations for diatomic molecules. Accurate solutions of the Hartree-Fock equations are obtained using iterative methods over most regions of space, while solving the equations by Gaussian elimination near the nuclear centres. In order to improve the speed and accuracy of our iterative scheme, a new self-adjoint form of the Hartree-Fock equation is derived. Using this new equation, our iterative subroutines solve the Hartree-Fock equations to one part in 10⁶. The Gaussian elimination routines are accurate to better than one part in 10⁸.     

Magnetic and Mössbauer spectral studies of nano crystalline cobalt substituted magnesium ferrites (MgxCo1 − xFe2O4)

The electronic structure of the heme unit of deoxyhemoglobin including the proximal imidazole has been studied using the first-principles Hartree-Fock procedure. Our results for the ^57mFe isomer shift and asymmetry parameter are in very good agreement with the values obtained from Mössbauer spectroscopy measurements. The ^57mFe nuclear quadrupole coupling constant is smaller than the experimental result and possible ways to improve the agreement in the future are discussed. Improved analysis of the Mössbauer data, removing some approximations made for deriving the magnetic hyperfine tensor for the ^57mFe nucleus, is suggested to allow quantitative comparison with our results in the future.     

Nuclear charge distribution of mass-separated isobars from thermal-neutron-induced fission of 235U

The nuclear charge distribution of fission products with mass numbers A = 90, 91, 94, 99, 100, 101 and 104 provided by the mass separator “Lohengrin” was measured. Adjacent elements in the group of the light fission products could be separated by their different energy loss in a carbon absorber. The Z-yields were found to be strongly dependent on the kinetic energy of the fission products. The widths of the nuclear charge distributions are very small, in general, and strongly dependent on A as well as on the kinetic energy. The influence of the neutron evaporation and odd-even effects are clearly detected. An asymmetric nuclear charge distribution was found for A = 104 indicating the suppression of fission fragments with Z = 43. The average nuclear charges of the fission products at their average kinetic energy are in good agreement with the results from measurements of the number of β-decays and K X-ray measurements. The average nuclear charge of the isobar A = 132 was measured at its average kinetic energy with a calibrated secondary electron detector to be Z = 51.14 ± 0.15 which is in very good agreement with the radiochemical results. Thus previous physical measurements indicating a large independent yield for the doubly magic nucleus ¹³²Sn could not be confirmed.     

Measurement of interaction cross sections using isotope beams of Be and B and isospin dependence of the nuclear radii

Interaction radii for ^11,12,14Be and ^{8,12,13,14,15}B have been determined by the measurement of interaction cross sections at 790 MeV/nucleon. Effective root-mean-square (RMS) radii of nucleon distribution of these nuclei have been deduced using a Glauber-model calculation. Isospin dependence of nuclear radii are presented for isobars of mass numbers from 6 to 12, and compared with theoretical predictions by the droplet model and by a Hartree-Fock (HF) model. The HF model with the Skyrme-III potential, which includes a strong density-dependent force, reproduce the observed isospin dependence.     

Study of the local spin-density functional method for calculating the contact term in atoms

Using various models for the exchange-correlation potential, the spin density of electrons at the nuclei of light atoms (up to Ni) is calculated. The magnitude of the Fermi-contact term is then determined and it is compared with the unrestricted Hartree-Fock calculations and with the available experimental results. It appears that for the alkali metal ions most of the local spin density models give results which are in good agreement with experiment. On the other hand, for atoms, where the existence of contact term is connected with the polarization of the core s-electrons, none of the models yield satisfactory results.The author is indebted to Dr. L.Farkas for valuable discussion concerning the self-consistent-field atomic program.