Coulomb energies and nuclear radii in the energy density formalism

The relation between Coulomb displacement energies,ΔE _c , andΔr=r _n -r _p , the difference between the rms radii of neutrons and protons in nuclei, is investigated within the energy density formalism (EDF). The variational equation, obtained by minimizing the Coulomb plus symmetry energies, is solved assuming the symmetry interaction is a simple functional of the local nuclear matter density. Varying parameters of the model, rather unique relation betweenΔE _c andΔr is obtained (within ±50 keV).ΔE _c isindependent ofr _ex, the rms radius of the excess neutrons distribution. Using the experimental values ofr _p and adjusting the model to reproduce the recent data onΔr (Δr∽~0.05 fm for⁴⁸Ca and²⁰⁸Pb), which are significantly smaller than those obtained from current Hartree-Fock calculations, the calculatedΔE _c agree with the experimental results. Using the value ofΔr～0.05 fm and the experimental values ofr _ex, a small compression (<0.02 fm) of the proton core in the analogue state relative to its parent state emerges, thus contributing an additional electrostatic term to the Coulomb displacement energy. The size of this relative core-compression effect depends on the values assumed forΔr andr _ex, it increases with the decreasing ofΔr and the increasing ofr _ex. IfΔr～0.05 fm the effect is large enough to remove the long standing Coulomb energy anomaly. The main result of the present work is the correlation betweenΔE _c andΔr, suggesting that the difficulties of current Hartree-Fock calculations in reproducing isotope shifts ofr _p , the small value ofr _n ?r _p and the values ofΔE _c may all be different manifestations of some missing residualp n effective interaction.     

Matrices and eigenfunctions induced by Markov maps

The matrix M induced by a Markov map has 1 as the eigenvalue of maximum modulus. If M is also irreducible, then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1. Let g=(g₁,g₂…g_n) be an n-row-vector such that g₁=c> and g_ic, i?2. Then every such g is the unique (upto constant multiples) left eigenvector associated with the eigenvalue 1 of a matrix M induced by a Markov map.     

Continual measurements in the spacetime formation of nonrelativistic quantum mechanics

We study the nature and effects of some continual measurements in nonrelativistic quantum mechanics, a concept introduced by R. P. Feynman in his path integral formulation of quantum mechanics. We prove the existence in various senses of mathematically rigorous objects formally equivalent to the path integrals associated with such measurements, by means of both a limiting definition and more general techniques; and we derive some quantitative results concerning the effects on the wave-function and the numerical value of certain associated probabilities.     

Models having the thermodynamic properties of aqueous alkaline earth halides and NaCl-MgCl2 mixtures

A model studied earlier in relation to aqueous 1–1 electrolytes has been applied to the interpretation of the excess free energies of aqueous solutions of the alkaline earth halides up to 1 M ionic strength. Several variations on the model could be fitted to the data. In the one which seems most consistent with the other observations, the cospheres (hydration layers) on the M⁺⁺ ions were two water molecules thick rather than one as for the alkali metal, tetraalkylammonium, and halide ions. The overlap of cospheres of an M⁺⁺, M⁺⁺ pair was found to make a small attractive contribution to the interionic force, while the overlap of cospheres of an M⁺⁺, X^– pair makes a small positive contribution which is almost the same for X^–=Cl^–, Br^–, or I^–. Thus, ionic hydration shells which are not penetrated by other ions are not required to account for the observed excess free energies. The model is also fitted to NaCl-MgCl₂ mixtures to see whether the new cosphereoverlap parameter which must be determined is consistent with the others. The calculation also shows how the thermodynamic behavior of the mixtures is consistent with the mixture limiting law for unsymmetrical mixtures. The singularities in the mixing coefficients g₀ and w₀ at infinite dilution dominate their concentration dependence only up to ionic strengths of about 0.01 M in models which fit the data.Most of this report is abstracted from a thesis presented by A. Smitherman to the Graduate School of the State University of New York at Stony Brook in partial fulfillment of the requirements for the M.S. degree, August 1972.     

A-dependence of high-energy inelastic electron-nucleus scattering

Cross sections for inclusive inelastic electron scattering from H, D, Be, Al, Cu and Au have been measured at 6° for incident energies up to 19.5 GeV. Within the stated errors, the cross sections were found to be directly proportional to the sum of the constituent nucleon cross sections.