Coulomb energies of spherical nuclei

Abstract

A phenomenological Coulomb energy equation has been derived for spherical nuclei with diffuse surfaces. Contributions from the direct and exchange Coulomb energy and from the electromagnetic spin-orbit interaction are included explicitly. The experimental Coulomb displacement energies of 42 essentially spherical nuclei with 2 2 28 have been subjected to a least-squares analysis with the shape parameters of the ground state charge distributions required to agree with those obtained from electron scattering and muon capture X-ray experiments. Agreement between the experimental and calculated energies with a standard deviation of about 20 keV is obtained if (i) the ratio between the rms radii of the neutron excess and the proton core is taken between about 1.10 and 1.15, (ii) a smoothly A-dependent correction term which varies between about +450 keV in the light nuclei and +350 keV in the heavy nuclei is introduced, and (iii) an averaged contribution from the electromagnetic spin-orbit interaction is used rather than the expected shell-mode1 contribution. Justification for this averaging may be provided by amounts of not less than 15 % of excitations into higher shellmodel orbits for all nuclei. The Coulomb displacement energies of vibrational and rotational nuclei are observed to be smaller by up to 200 keV than corresponding spherical nuclei. Neutron halos have been calculated and are found to be in very good agreement with the results of Hartree-Fock calculations, the droplet mode1 of Myers and Swiatecki and the calculations by Nolen and Schiffer using wave functions generated in a Woods-Saxon potential well. The calculated neutron halo for zo8Pb is 0.25kO.05 fm.

Oklo reactors and implications for nuclear science

“…In fact, the results of Ref. [95] imply that, for the isotopes considered in Ref. [72] (and subsequent analyses of Oklo data), all the abovementioned effects induce a decrease of less than 3% in a ground state Coulomb energy.…”

“…The importance of such corrections can most simply be gauged with the analytic expressions for Coulomb energies of section 9 in Ref. [95]. These formulae have been inferred from a phenomenological study of Coulomb displacement energies for spherical nuclei with Z ≥ 28 and contain multiplicative corrections accommodating the diffuseness of the charge distribution's surface.…”

Analytical solution of the Coulomb potential for spherical nuclei

“…Substituting Eqs. (7), (20) and (21) into (14), we obtain the following analytical expression for the Coulomb potential: (24) and, the values of coefficients as follows:…”

Table of calculated nuclear properties

“…The expression used for the liquid-drop energy includes volume and surface terms, a combined volume and surface charge asymmetry term in the denominator form, 16 a curvature term, a Coulomb energy term including effect of surface diffuseness on the effective nuclear radius, 17 In Eq. (1), the coefficients a, g, £, y, and <|> are adjustable, as are the Coulomb energy parameter r and the parameters of the shell correction.…”

“…The surface, Coulomb, and curvature shape dependences (B , B , and B. , respectively) of the liquid drop are calculated as integrals 19 over the shapes and they are normalized to unity for spherical shapes. We include a Coulomb energy given by Janecke 17 in which the direct, exchange, and spin-orbit interaction contributions to tup energy have been expanded in powers of the ratio of su/'face diffuseness to radius, a/R, up to the fourth order. The equivalent radius R is defined as the radius of a uniform spherical charge distribution with the same root mean square radius as the Fermi distribution.…”