Theoretical interpretation of the systematics of effective single-particle level densities from (n,p) reactions at 14.8 MeV energy

Abstract

The universal experimental data for the energy-integrated angular distribution and the angle-integrated energy spectra for protons emitted from the ($n$,$p$) reactions at 14.8 MeV incident energy were analyzed earlier using the computer code preco-d2, developed on the basis of Kalbach's semiempirical model for the preequilibrium reactions. The results of the analysis provided the semiempirical systematics of the single-particle level densities ${g}_{R}(\mathrm{exp})$ (effective in the residual nuclei) and ${g}_{c}(\mathrm{exp})$ (effective in the composite system). In order to interpret these results, we have carried out the theoretical calculations with Shlomo's theory developed on the basis of the Green's function approach. The theoretical values based on Shlomo's theory for ${g}_{Rn}^{Th}({\ensuremath{\varepsilon}}_{Rn})$ and ${g}_{Rp}^{Th}({\ensuremath{\varepsilon}}_{Rp})$ for the residual nucleus and ${g}_{cn}^{Th}({\ensuremath{\varepsilon}}_{cn})$ and ${g}_{cp}^{Th}({\ensuremath{\varepsilon}}_{cp})$ for the composite systems, respectively, were calculated at various excitation energies by using a reasonable single-particle nuclear potential strength ${V}_{0}$, available from systematics in literature. Here, ${\ensuremath{\varepsilon}}_{Rn}$ and ${\ensuremath{\varepsilon}}_{Rp}$ are the single-particle excitation energies for single-particle level densities for the residual nucleus, and ${\ensuremath{\varepsilon}}_{cn}$ and ${\ensuremath{\varepsilon}}_{cp}$ are the single-particle excitation energies of effective single-particle level densities for the composite system for neutrons and protons, respectively. The Coulomb interaction potential ${V}_{c}$ was included for protons over and above the nuclear potential ${V}_{0}$. The total theoretical values were taken as ${g}_{R}^{TTh}({\ensuremath{\varepsilon}}_{R})={g}_{Rn}^{Th}({\ensuremath{\varepsilon}}_{Rn})+{g}_{Rp}^{Th}({\ensuremath{\varepsilon}}_{Rp})$ for residual nuclei and ${g}_{c}^{TTh}({\ensuremath{\varepsilon}}_{c})={g}_{cn}^{Th}({\ensuremath{\varepsilon}}_{cn})+{g}_{cp}^{TTh}({\ensuremath{\varepsilon}}_{cp})$ for the composite system. ${\ensuremath{\varepsilon}}_{R}$ is the Fermi energy of the total effective single-particle level density for residual nuclei, and ${\ensuremath{\varepsilon}}_{c}$ is the excitation energy of the effective single-particle level density of the composite system. Careful comparison of theoretical and experimental results shows that ${g}_{R}(\mathrm{exp})$ matches with ${g}_{R}^{TTh}({\ensuremath{\varepsilon}}_{f})$ for ${V}_{0}^{i}=45\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}\phantom{\rule{4pt}{0ex}}\mathrm{where}\phantom{\rule{4pt}{0ex}}({\ensuremath{\varepsilon}}_{f})\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{Fermi}\phantom{\rule{4.pt}{0ex}}\text{energy}.$ The theoretical values ${g}_{R}^{TTh}({\ensuremath{\varepsilon}}_{f})$ also nearly agree with the previously reported values by various groups, obtained through different approaches for a large number of cases. The values of ${g}_{c}(\mathrm{exp})$ always are found to be greater than ${g}_{R}(\mathrm{exp})$ and match with ${g}_{c}^{TTh}({\ensuremath{\varepsilon}}_{c})$ at effective excitation energies ${\ensuremath{\varepsilon}}_{c}$, which are found to be invariably much higher than the respective ${\ensuremath{\varepsilon}}_{f}$ and are positive and follow the nuclear shell model structure when plotted against $A$. This supports the concept that the values of effective ${g}_{c}(\mathrm{exp})$ are decided mainly by dominant transition, which occurs during initial stages of the multistep statistical direct preequilibrium process that involves unbound states. The values of ${g}_{R}(\mathrm{exp})$, on the other hand, are found to correspond to the effective ${\ensuremath{\varepsilon}}_{R}$ around the Fermi energies of the bound states of residual nuclei involved in the decay processes. The ratios of ${g}_{c}(\mathrm{exp})/{g}_{R}(\mathrm{exp})$ are found to follow the ${(\frac{\ensuremath{\langle}{\ensuremath{\varepsilon}}_{c}\ensuremath{\rangle}}{\ensuremath{\langle}{\ensuremath{\varepsilon}}_{f}\ensuremath{\rangle}})}^{1/\phantom{\rule{0.0pt}{0ex}}2}f$ description and to match Shlomo's theory.