Coulomb Energy Of Nucleus

Based on the Skyrme energy density functional and the extended Thomas-Fermi approach(ETF),we calculate the nucleus ground state density distribution by using the restricted density variational method.Then by solving the Poisson equation to obtain the Coulomb potential,we calculate the corresponding Coulomb energy with a numerical integration over the Coulomb potential.For any form of proton density dis-tribution,one can calculate the Coulomb energy through integrals over the Coulomb potential,which is much time-saving than the 6-dimensional integrals from the def-inition. For a spherical nucleus with uniform charge distribution and a system with gaussian charge distribution,the corresponding Coulomb energy can be accurately ob-tained by analytical formulas.We first check the results of the numerical integration for these two cases.It shows that the results of these two methods are very close to each other.The relative deviations are about 10-4-10-5.With the same numerical approach,we calculate the Coulomb energies of a num-ber of spherical nuclei from A=16 to 300 with Woods-Saxon charge distribution. The nuclear surface diffuseness a varies from 0.2 to 1.2fm.By fitting the calculated Coulomb energies of these nuclei,we obtain an analytical formula for the Coulomb en-ergy of spherical nucleus with surface diffuseness.The Coulomb energy of a spherical nucleus can be reliably obtained with the formula. For almost all calculated nuclei, the relative deviation is smaller than 0.05%.Based on the results of spherical nucleus,using the same method, we also calcu-late the Coulomb energies of a number of axially deformed nuclei from A=16 to 300 with Woods-Saxon charge distribution.The nuclear surface diffuseness a varies from 0.2 to 0.6fm and the deformation parameterβ2 varies from-0.5 to 0.5.By fitting the calculated Coulomb energies of these nuclei,we obtain an analytical formula for the Coulomb energy of axially deformed nucleus in which the nuclear surface diffuseness and deformation parameter is taken into account.For almost all calculated nuclei,the relative deviation is smaller than 0.1%.For intermediate and heavy nuclei(a/R 0.12), the relative deviation is smaller than 0.05%.