Shell-cluster transition in $^{48}$Ti

Background: Whether or not the $\alpha$ ($^4$He nucleus) clustering exists in the medium-mass region of nuclear systems is a fundamental and intriguing question. However, the recent analysis of the $\alpha$ knockout reaction on $^{48}$Ti [Phys. Rev. C 103, L031305 (2021)] poses a puzzle: The microscopic wave function gives an $\alpha$ knockout cross section that is two orders of magnitude smaller than the experiment, while basic nuclear properties such as the charge radius and the electromagnetic transition probabilities are well explained. Purpose: The ground-state structure of $^{48}$Ti is investigated by using proton- and $\alpha$-nucleus elastic scattering at a few to several hundred MeV, which offers different sensitivity to the region of the nuclear density profiles. Method: Four types of density distributions, the $jj$-coupling shell model and three cluster model configurations, are generated in a single scheme by the antisymmetrized quasi-cluster model (AQCM). The angular distribution of the proton- and $\alpha$-$^{48}$Ti elastic scattering cross sections are obtained with a reliable high-energy reaction theory, the Glauber model. Results: The $jj$-coupling shell model configuration is found to best reproduce the proton-nucleus elastic scattering cross section. On the other hand, the trace of the $\alpha$ cluster structure in the tail region of the wave function is embedded in the $\alpha$-nucleus elastic scattering cross section. Conclusion: Our results suggest that the structure of the nucleus changes as a function of distance from the center, from the $jj$-coupling shell model structure in the surface region to the $\alpha$+$^{44}$Ca cluster structure in the tail region. This picture is consistent with the finding of the $\alpha$ knockout reaction on $^{48}$Ti.