The L O C V finite n u c l e o n i c matter three-body cluster energy in the periodic boxes

Highlights: • The $PBLOCV$ $nucleonic$ matter three-body cluster energies become roughly constant. • The $\left(PB\right)LOCV$ $nucleonic$ matter energies at the three-body cluster level agree with those of $\left(PB\right)FHNC$. • The $PBLOCV$ approximation is valid for the Bethe homework problem. The lowest order constrained variational ($LOCV$) method is used for the finite $nucleonic$ matter in the periodic boxes ($PBLOCV$). Using the cluster expansion of the energy, the $PBLOCV$ symmetric nuclear (pure neutron) matter energies for the different number of nucleons are found at the three-body cluster level. In these computations, the Bethe homework, as well as the central part of the $A{V}^{\prime }8$ (${\left(A{V}^{\prime }8\right)}_c$) potentials, are employed. Considering the three-body cluster energy, the $\left(PB\right)LOCV$ energies become consistent with those of $\left(PB\right)$ Fermi $hypernetted$ chain ($\left(PB\right)FHNC$). By increasing the number of nucleons, the $PBLOCV$ symmetric nuclear (pure neutron) matter three-body cluster energy becomes almost constant (changes at most 15%). It is demonstrated that including the Bethe homework interaction, the ratio of the three-body cluster energy to that of two-body ($R$) is less than 0.17. The corresponding $PBLOCV$ $SNM$ $R$ for the ${\left(A{V}^{\prime }8\right)}_c$ potential is about 1 (1.9), at low (high) densities. As a result, the approximation of the $PBLOCV$ formalism is valid for the pure neutron matter with the Bethe homework potential. Employing the ${\left(A{V}^{\prime }8\right)}_c$ interaction in the $PBLOCV$ $SNM$ calculations, the $LOCV$ normalization constraint probably needs to be extended to the three-body cluster term to improve the $PBLOCV$ approximation.