Location of Single-Particle Levels in Medium Mass Nuclei

The levels of ${\mathrm{Ni}}^{59}$, ${\mathrm{Ni}}^{61}$, ${\mathrm{Fe}}^{57}$, and ${\mathrm{Ti}}^{49}$ are studied with ($d, p$) reactions; angular distributions and absolute cross sections are analyzed with the aid of distorted-wave Born approximation calculations. Essentially all levels in the Ni isotopes up to about 1-Mev binding energy are assigned to a shell-model state and their reduced widths are measured. From this, the "center of gravity" of each shell-model state is determined. The results are very much different from those of Schiffer, Lee, and Zeidman, and much closer to theoretical expectations. For example, there is a large energy gap between major shells, the energy of the $3s_{\frac{1}{2}}$ state agrees with other evidence, the semiclosed-shell behavior of 40 and 56 neutrons is explained, etc. The widths of the energy distributions of levels belonging to single shell-model states agree well with the predictions of the giant resonance theory of Lane, Thomas, and Wigner.