Existence of higher nodal band states with $\alpha$+$^{48}$Ca cluster structure in $^{52}$Ti

It is shown that recently observed $\alpha$ cluster states a few MeV above the $\alpha$ threshold energy in $^{52}$Ti correspond to the higher nodal band states with the $\alpha$+$^{48}$Ca cluster structure, i.e. a vibrational mode in which the intercluster relative motion is excited. The existence of the higher nodal states in the $^{48}$Ca core region in addition to the well-known higher nodal states in $^{20}$Ne and $^{44}$Ti reinforces the importance of the concept of vibrational motion due to clustering even in medium-weight nuclei with a $jj$-shell closed core. The higher nodal band and the shell-like ground band in $^{52}$Ti are described in a unified way by a Luneburg lens-like deep local potential due to the Pauli principle, which explains the emergence of backward angle anomaly (anomalous large angle scattering (ALAS)) at low energies, prerainbows at intermediate energies and nuclear rainbows at high energies in $\alpha$+$^{48}$Ca scattering. The existence of a $K=0^-$ $\alpha$ cluster band analog to $^{44}$Ti midway between the ground band and the higher nodal band is inevitably predicted.