Reaction matrix theory for a finite nucleus by the separation method

Abstract

Binding energy, single particle energies and radius of oxygen-16 were calculated by the reaction matrix method. The single particle wave-functions were taken to be those of a hormonic oscillator. For the nucleon-nucleon potential we took a hard core repulsion of radius 0.4 fm and an exponential shaped attraction with intrinsic range 2.5 fm and infinite scattering length. We assumed this spin-independent potential to act only in s-states.The reaction matrix was calculated by the separation method.We obtained saturation at an r.m.s. radius of 2.74 fm and a binding energy of 3.4 MeV per nucleon. The experimental values are 2.58 fm and 8 MeV per nucleon. The above potential will give too little binding also for infinite nuclear matter, at somewhat too small density.The characteristic features of the reaction matrix is discussed and an approximate but explicit expression given.It is brought out that the correlation between two nucleons depends mainly on the states or potential energies of the two nucleons and is larger for two nucleons deep down in the Fermi sea than at the top. We conclude this would lead to peculiar properties of nuclear matter.With the total energy given in the reaction matrix approximation the variational method gives a shell model potential consisting of essentially three parts. One corresponds to the ordinary Hartree-Fock potential. Another has closely the same form as the Hartree-Fock potential and decreases the strength of this by about 15% for infinite nuclear matter of normal density. The weakest part is proportional to the square of the local density and is at mal density ≈ −1 MeV. The rearrangement energy for a nucleon at the top of the Fermi sea is thus obtained to be ≈ 8 MeV.The single particle energies of oxygen-16 come out in fair agreement with experiments.The rearrangement energy, usually neglected when calculating the reaction matrix, gives a correction of ≈ −1 MeV per nucleon in infinite nuclear matter and ≈ −0.1 MeV per nucleon for oxygen-16.