Thomas-Fermi Theory of Nuclei

A Thomas-Fermi theory of large, finite nuclei is developed. Realistic nuclear forces with repulsive core are assumed, and maximum use is made of the theory of nuclear matter. Simplifications are introduced wherever permissible. The local-density approximation with a certain correction is found to be valid. Tensor forces are replaced by a density-dependent, effective central force, the repulsive core by a density-dependent $\delta$-function interaction. The Thomas-Fermi expression for kinetic energy is shown to be good whenever the density is at least 17% of nuclear-matter density; under the same conditions, the Slater approximation to the mixed density $\rho (r_1, r_2)$ is valid. From the total energy of the nucleus, an integral equation is derived for the density $\rho \mathrm{}\left(r\right)\mathrm{}$. This is approximated by a differential equation which is solved analytically. The resulting density distribution has both similarities with and differences from the conventional, Fermi-type distribution. Our density agrees as well with electron-scattering experiments as the Fermi type does. The thickness of the nuclear surface comes out about 10% too large from our theory; the surface energy is in good agreement with the semiempirical value. So far, the number of neutrons and protons has been assumed equal, and the Coulomb force has been neglected.