Comparison of Hartree-Fock and energy -density formalism calculations for some spherical nuclei

The purpose of this paper is twofold. We calculate the parameters of the nuclear droplet model introduced by Myers and Swiatecki (an extension of the conventional liquid drop model) to higher powers of $A^{-\frac{1}{3}}$. We do Hartree-Fock (HF) calculations for a large number of spherical nuclei, using the modified $\delta$ interaction version of the Skyrme interaction and average over shell structure by making a least squares fit to the Hartree-Fock energies. We also compare the shell-averaged HF results with those obtained by using a simple version of the energy density formalism developed by Brueckner, Lombard, and others. We assume the simple form of the energy density first introduced by Skyrme for $N=Z$ nuclei, and minimize the total energy with respect to the actual density and diffuseness, subject to the constraint of fixed number of particles. Detailed comparison with the HF results was made for two $A=100\mathrm{}$ nuclei with $Z=50\mathrm{} \mathrm{and} 40$. We find that the droplet model energies calculated either exactly (numerically) or analytically by expansion of the radial integrals in powers of ($\frac{a}{R}$) ($a=\mathrm{diffuseness}$, $R=\mathrm{half}\mathrm{density}\mathrm{radius}\mathrm{for}\mathrm{a}\mathrm{Fermi}\mathrm{density}$) are in good agreement with the shell averaged HF results although there is some uncertainty in the latter. However, the central density is too small by about 10% for the cases considered. Yet, for the case $Z=40\mathrm{}$, $A=100\mathrm{}$, the ratio of neutron and proton densities is close to the Hartree-Fock value. Thus, we find the energy density formalism can reproduce quite closely the results of more complicated Hartree-Fock calculations.