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

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

Figure 1

The angular distributions of cross sections (ratio to Rutherford scattering) in $\alpha$-particle scattering from ${}^{48}\mathrm{Ca}$ calculated with the optical model potentials in Table 1 (solid lines) are compared with the experimental data (points) taken from Refs. [38, 39, 40]. $A1$ indicates the Airy minimum.

Figure 2

The calculated angular distributions of cross sections (ratio to Rutherford scattering) in $\alpha$-particle scattering from ${}^{48}\mathrm{Ca}$ (solid lines) at (a) $E_L=18$ and (b) 29 MeV in Fig. 1 are decomposed into the internal-wave (dashed lines) and the barrier-wave (dashed-dotted lines) contributions. The points are the experimental data from Ref. [38]. In the insets, the reflection coefficient ($|S_L|$) is decomposed into the internal-wave ($|S_L^{\left(I\right)}|$) and the barrier-wave ($|S_L^{\left(B\right)}|$). The lines are to guide the eye.

Figure 3

Calculated energy levels of ${}^{52}\mathrm{Ti}$ are compared with the experimental ground band [45] and the newly observed three excited $\alpha$ cluster states (red thick solid lines) [33].

Figure 4

The $\alpha -{}^{48}\mathrm{Ca}$ potential at $E_L=18\mathrm{MeV}$ (solid line) is compared with the Luneburg lens potential with $R_0=5.75\mathrm{fm}$ and $V_0=180\mathrm{MeV}$ (points). The energy-independent equivalent local potential (for $L=0$) in the GCM calculation from Ref. [52] is shown by the dashed line.