Evidence of a higher nodal band $\alpha +{}^{44}\mathrm{Ca}$ cluster state in fusion reactions and $\alpha$ clustering in ${}^{48}\mathrm{Ti}$

In the nucleus ${}^{48}\mathrm{Ti}$, whose structure is essential in evaluating the half-life of the neutrinoless double-$\beta$ decay ($0\nu \beta \beta$) of ${}^{48}\mathrm{Ca}$, the existence of the $\alpha$ cluster structure is shown for the first time. A unified description of scattering and structure is performed for the $\alpha +{}^{44}\mathrm{Ca}$ system. By using a global potential, which reproduces experimental $\alpha +{}^{44}\mathrm{Ca}$ scattering over a wide range of incident energies, $E_{\alpha }=18\text{–}100\mathrm{MeV}$, it is shown that the observed ($\alpha +{}^{44}\mathrm{Ca}$)-fusion excitation function at $E_{\alpha }=9\text{–}18\mathrm{MeV}$ is described well. The bump at $E_{\alpha }=10.2\mathrm{MeV}$ is found to be due to a resonance which is a $7^{-}$ state of the higher nodal band with the ($\alpha +{}^{44}\mathrm{Ca}$)-cluster structure in ${}^{48}\mathrm{Ti}$. The local potential ($\alpha +{}^{44}\mathrm{Ca}$)-cluster model locates the ground band of ${}^{48}\mathrm{Ti}$ in agreement with experiment and reproduces the enhanced $B\left(E2\right)$ values in the ground band well. This shows that collectivity due to $\alpha$ clustering in ${}^{48}\mathrm{Ti}$ should be taken into account in the evaluation of the nuclear matrix element in the $0\nu \beta \beta$ double-$\beta$ decay of ${}^{48}\mathrm{Ca}$.

Figure 1

Angular distributions in $\alpha +{}^{44}\mathrm{Ca}$ scattering at $E_{\alpha }=18$ and 24.1 MeV calculated with the optical potentials in Table 1 (solid lines) are compared with the experimental data (circles) [49]. The calculated angular distributions are decomposed into the barrier-wave (dotted lines) and the internal-wave (dashed lines) contributions. At 18 MeV the solid line overlaps with the dotted line, and the dashed line is negligibly small and is not drawn.

Figure 2

Calculated excitation function of the ($\alpha +{}^{44}\mathrm{Ca}$)-fusion cross sections (solid lines) is compared with the experimental data (filled circles) [54].

Figure 3

Calculated excitation functions of the ($\alpha +{}^{44}\mathrm{Ca}$)-fusion cross sections (solid lines) are decomposed into the partial-wave contributions; even $L$ (dotted lines) and odd $L$ (dashed lines). The inset is for $E_{\alpha }=8.4\text{–}9.0\mathrm{MeV}$.

Figure 4

The phase shifts for ($\alpha +{}^{44}\mathrm{Ca}$)-scattering calculated with the D181 real potential. The vertical axis is ${\delta }_L\text{−}{m}_L\pi$ where $m_L$ is the number of the Pauli-forbidden states. See the text. The resonances for odd $L$ and even $L$ correspond to $N=15$ and $N=16$, respectively.

Figure 5

Energy levels of ${}^{48}\mathrm{Ti}$ with respect to the $\alpha$ threshold calculated with the potentials (a) $V=181\mathrm{MeV}$ and (b) $V=171\mathrm{MeV}$ are compared with (c) the experimental ground-state band [66]. The energy levels are calculated in the bound-state approximation except the $N=15$ band resonances. The horizontal dotted lines indicate the incident energy $E_{\alpha }=8.4\text{–}18\mathrm{MeV}$ of the fusion excitation functions in Fig. 3.

Figure 6

The D181 $\alpha +{}^{44}\mathrm{Ca}$ potential at $E_{\alpha }=18\mathrm{MeV}$ (solid line) is compared with the Luneburg lens potential [44, 45, 46] with $R_0=5.68\mathrm{fm}$ and $V_0=173\mathrm{MeV}$ (points) and the energy-independent equivalent local potential for $L=0$ in the GCM calculation [41] (medium dashed line). For comparison the $\alpha +{}^{40}\mathrm{Ca}$ potential used in the $\alpha$-cluster calculation of ${}^{44}\mathrm{Ti}$ [6] (long dashed line) and the $\alpha +{}^{48}\mathrm{Ca}$ potential at $E_{\alpha }=18\mathrm{MeV}$ [15] (dashed-dot line) are also displayed.