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Single Particle Energies in Skyrme Hartree Fock and Woods Saxon Potentials Brian D Newman Cyclotron Institute Texas A M University Mentor Dr Shalom Shlomo Introduction Atomic nuclei exhibit the interesting phenomenon of single particle motion that can be described within the mean field approximation for the many body system We have carried out Hartree Fock calculations for a wide range of nuclei using the Skyrme type interactions We have examined the resulting mean field potentials UHF by fitting r2UHF to r2UWS where UWS is the commonly used Woods Saxon potential We consider in particular the asymmetry x N Z A dependence in UWS and the spinorbit splitting in the spectra of 17F8 and the recently measured spectra of 23F14 Using UWS we obtained good agreement with experimental data Mean Field Approximation Many body problem for nuclear wave function generally cannot be solved analytically H E In Mean Field Approximation each nucleon interacts independently with a potential formed by other nucleons Mean Field Approximation R 0 0 10 Single Particle Schr dinger Equation 20 30 A Nucleon Wave Function 40 50 Vo 60 A Anti Symmetrization operator for fermions 2 Ui r 4 6 8 10 12 Mean Field cont The anti symmetric ground state wave function of a nucleus can be written as a Slater determinant of a matrix whose elements are single particle wave functions Single particle wave functions i are determined by the independent single particle potentials Due to spherical symmetry the solution is separable into radial component angular component spherical harmonics and the isospin function Hartree Fock Method The Hamiltonian operator is sum of kinetic and potential energy operators where The ground state wave function should give the lowest expectation value for the Hamiltonian Hartree Fock Method cont We want to obtain minimum of E with the constraint that the sum of the single particle wave function integrals over all space is A to conserve the number of nucleons We obtain the Hartree

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