Assessment of effect of local approximation on single folding potential at low and intermediate incident energies

To the single folding potentials (SFPs) for the nucleon-nucleus ($N$-$A$) elastic scatterings, local approximations (LAs) have customarily been applied. The LA discussed by Brieva and Rook has been well-known, which only needs the density profile as the structure information of the target nucleus. By applying the M3Y-P6 interaction both to the target wave functions and the real part of SFP, supplemented with the Koning-Delaroche phenomenological imaginary potential, the precision of the Brieva-Rook LA on the SFP is investigated for the proton-nucleus elastic scatterings at $\epsilon_p=16\,-\,80\,\mathrm{MeV}$ incident energies. The analyzing powers as well as the differential cross sections are in reasonable agreement with the available data. The precision of the LA for the central and LS channels is distinctly examined. Although the LA works well at small angles ($\theta_\mathrm{c.m.}\lesssim 30^\circ$), it gives rise to sizable deviation from the results of the non-local SFP (\textit{i.e.}, without the LA) at larger angles. The results of the non-local SFP are always in better agreement with the data. The LA for the LS channel influences the differential cross-sections, and the LA for the central channel does the spin observables. It is found that the precision of the LA well correlates to the momentum transfer $q$, and the discrepancy becomes sizable at $q\gtrsim 1.5\,\mathrm{fm}^{-1}$. The LA is also examined for a halo nucleus, by taking $^{86}$Ni as an example. The precision is slightly worse than in stable nuclei. Difference from the prediction of the empirical potential in the observables of the $p$-$^{86}$Ni scattering is discussed.