$\alpha +{}^{92}\mathrm{Zr}$ cluster structure in ${}^{96}\mathrm{Mo}$

In the evaluation of the half-life of the neutrinoless double-$\beta$ decay ($0\nu \beta \beta$) of a doubly closed-subshell nucleus ${}^{96}\mathrm{Zr}$, the structure of the nucleus ${}^{96}\mathrm{Mo}$ is essentially important. The $\alpha$-clustering aspects of ${}^{96}\mathrm{Mo}$ are investigated for the first time. By studying the nuclear rainbows in $\alpha$ scattering from ${}^{92}\mathrm{Zr}$ at high energies and the characteristic structure of the excitation functions at the extreme backward angle at the low-energy region, the interaction potential between the $\alpha$ particle and the ${}^{92}\mathrm{Zr}$ nucleus is determined well in the double folding model. The validity of the double folding model was reinforced by studying $\alpha$ scattering from neighboring nuclei ${}^{90}\mathrm{Zr}, {}^{91}\mathrm{Zr}$, and ${}^{94}\mathrm{Zr}$. The double-folding-model calculations reproduced well all the observed angular distributions over a wide range of incident energies and the characteristic excitation functions. By using the obtained potential the $\alpha +{}^{92}\mathrm{Zr}$ cluster structure of ${}^{96}\mathrm{Mo}$ is investigated in the spirit of a unified description of scattering and structure. The existence of the second-higher nodal band states with the $\alpha +{}^{92}\mathrm{Zr}$ cluster structure, in which two more nodes are excited in the relative motion compared with the ground band, is demonstrated. The calculation reproduces well the ground-band states of ${}^{96}\mathrm{Mo}$ in agreement with experiment. The experimental $B\left(E2\right)$ value of the transition in the ground band is also reproduced well. The effect of $\alpha$ clustering in ${}^{96}\mathrm{Mo}$ on the the half-life of the $0\nu \beta \beta$ double-$\beta$ decay of ${}^{96}\mathrm{Zr}$ is discussed.

Figure 1

The angular distributions in $\alpha +{}^{92}\mathrm{Zr}$ scattering at $E_{\alpha }=35.4$, 40, 65, 90, and 120 MeV calculated with the optical potential model with the double folding potential (solid lines) are compared with the experimental data (filled circles) [59, 60]. The calculated farside (dotted lines) and nearside (dashed lines) contributions are also indicated.

Figure 2

The calculated angular distributions (solid lines) in (a) $\alpha +{}^{90}\mathrm{Zr}$ scattering at $E_{\alpha }=21$, 23.4 and 25 MeV and (b) $\alpha +{}^{91}\mathrm{Zr}$ scattering at $E_{\alpha }=21$, 23, and 25 MeV are compared with the experimental data (filled circles) [66].

Figure 3

The calculated excitation functions in $\alpha$ scattering from ${}^{90}\mathrm{Zr}, {}^{91}\mathrm{Zr}, {}^{92}\mathrm{Zr}$, and ${}^{94}\mathrm{Zr}$ at the extreme backward angle $\theta =176.2^{\circ }$ (${\theta }_{\mathrm{Lab}}={176}^{\circ }$) (solid lines) are compared with the experimental data (filled circles) [66].

Figure 4

(a)The excitation functions at the extreme backward angle $\theta =176.2^{\circ }$ in $\alpha +{}^{92}\mathrm{Zr}$ scattering calculated with the reduced strengths of the imaginary potential $W=0$ (dotted lines), $W_0$/8 (dashed lines), $W_0$/4 (long dashed lines), $W_0$/2 (dash-dotted lines), and $3W_0$/4 (dashed-and-double-dotted lines) are compared with the original one with $W=W_0=10.7$ MeV (solid lines). (b) The calculated partial wave cross sections of elastic scattering under $W=0$ for even $L$ and for (c) odd $L$.

Figure 5

Phase shifts in $\alpha +{}^{92}\mathrm{Zr}$ scattering calculated with the double folding potential with $N_R=1.22$, (a) even $L$ partial waves and (b) odd $L$ partial waves, are displayed for $L\le 15$. The lower abscissa shows center-of-mass energy $E$ and the upper abscissa shows laboratory incident energy $E_{\alpha }$.

Figure 6

(a) Energy levels of ${}^{96}\mathrm{Mo}$ calculated in the $\alpha +{}^{92}\mathrm{Zr}$ cluster model with the double folding potential with $N_R=1.22$. The calculated excitation energy of the $N=16$ ground-band states, $0^{+}, 2^{+}, 4^{+}, 6^{+}$, and $8^{+}$, which look compressed, increases as the spin increases. (b) The $N=16$ band energy levels calculated using the double folding model with $L$ dependence. (c) Experimental energy levels of the ground band. The horizontal dashed lines (blue) correspond to $E_{\alpha }=18$ MeV (center-of-mass energy $E=17.3$) and $E_{\alpha }=26$ MeV ($E=24.9$ MeV), between which the characteristic dip in the excitation function appears.

Figure 7

The calculated $u\left(R\right)$ of the relative wave function $u\left(R\right)/R$ of the ${10}^{+}$ and ${12}^{+}$ states of the $N=20$ band with the $\alpha +{}^{92}\mathrm{Zr}$ cluster structure in ${}^{96}\mathrm{Mo}$. The wave functions are calculated as scattering states and normalized to unity for $R\le 10$ fm.

Figure 8

The calculated $u\left(R\right)$ of the relative wave function $u\left(R\right)/R$ of the ground state of ${}^{96}\mathrm{Mo}$ calculated in the $\alpha +{}^{92}\mathrm{Zr}$ cluster model (solid line) is compared with the harmonic oscillator wave function with $N_{\mathrm{HO}}=16$ (dashed line).

Figure 9

The occupation probability of the harmonic oscillator quanta $N_{\mathrm{HO}}$ in the ground-state wave function of ${}^{96}\mathrm{Mo}$.