The coordination geometry of water in some salt hydrates

The orientations of water molecules next to metal ions in salt hydrates are useful indications of the orientations which may be expected for water molecules which are next to metal ions in aqueous solution. A survey of the data for the salt hydrates shows that a water molecule may often be tetrahedral (i.e., coordinated to the metal ion, accepting a hydrogen bond from another water molecule, and donating two hydrogen bonds) if it is next to a +1 ion, a+2 ion, or a+3 ion of the rare earth series, although apparently not for smaller +3 ions.     

The maximum entropy determination of nuclear densities of calcium isotopes from elastic scattering of alpha particles

Radial moments of the real part of the optical potential for elastic scattering of 104 MeV α particles are used as constraints, in determining the nuclear density of maximal entropy. The potential is related to the density by the folding model.     

The influence of an electric field on a hydrogen atom confined to boxes of different shapes

The ground state energy of a hydrogen atom confined to a box, in the presence of an electric field is calculated. Boxes of various shapes are applied in order to demonstrate the shape's influence on that energy level. The procedure consists of numerically solving the two-dimensional Schrödinger equation using a finite element package.     

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.