Supersolidity of $\alpha$ cluster structure in ${}^{40}\mathrm{Ca}$

$\alpha$ cluster structure in nuclei has been long understood based on the geometrical configuration picture. By using the spatially localized Brink $\alpha$ cluster model in the generator coordinate method, it is shown that the $\alpha$ cluster structure has the apparently opposing duality of crystallinity and condensation, a property of supersolids. To study the condensation aspects of the $\alpha$ cluster structure a field theoretical superfluid cluster model (SCM) is introduced, in which the order parameter of condensation is incorporated by treating rigorously the Nambu-Goldstone mode due to spontaneous symmetry breaking of the global phase. The $\alpha$ cluster structure of ${}^{40}\mathrm{Ca}$, which has been understood in the crystallinity picture, is studied by the SCM with ten $\alpha$ clusters. It is found that the $\alpha$ cluster structure of ${}^{40}\mathrm{Ca}$ is reproduced by the SCM in addition to ${}^{12}\mathrm{C}$ reported in a previous paper, which gives support to the duality of the $\alpha$ cluster structure. The emergence of the mysterious $0^{+}$ state at the lowest excitation energy near the $\alpha$ threshold energy is understood to be a manifestation of the Nambu-Goldstone zero mode, a soft mode, due to the condensation aspect of the duality similar to the Hoyle state in ${}^{12}\mathrm{C}$. The duality of $\alpha$ cluster structure with incompatible crystallinity and coherent wave nature due to condensation is the consequence of the Pauli principle, which causes clustering.

Figure 1

Illustrative figures of crystallinity, condensation, and supersolidity of the $\alpha$ clusters (filled circles) proposed in ${}^{12}\mathrm{C}$ in Ref. [17] are displayed for the simplest system ${}^8\mathrm{Be}$ discussed in the text and extended to ${}^{16}\mathrm{O}$ with four $\alpha$ clusters and ${}^{20}\mathrm{Ne}$ with five $\alpha$ clusters. As the excitation energy increases vertically, the structure change occurs. In each nucleus crystallinity, condensation associated with a coherent wave, and supersolidity with both crystallinity and the coherent wave of the $\alpha$ clusters are shown. The original Ikeda diagram based on the crystallinity picture corresponds to (a), (d), (g), and (j) in each nucleus. In (b), (e), (h), and (k) of each nucleus $\alpha$ clusters are sitting in the $0s$ state of the harmonic oscillator potential with a coherent wave (broad curve). In (c), (f), (i), and (l) of each nucleus, which illustrates supersolidity, the $\alpha$ clusters are sitting in the $0s$ state of the distinct harmonic oscillator potentials separated due to the Pauli repulsion associated with a coherent wave (broad curve).

Figure 2

Illustrative pictures of ten $\alpha$ clusters (filled red circles) in ${}^{40}\mathrm{Ca}$. (a) The crystallinity picture of the $\alpha$ cluster structure of ${}^{40}\mathrm{Ca}$ with the $\alpha +{}^{36}\mathrm{Ar}$ geometrical configuration. (b) The condensation picture of the $\alpha$ cluster structure of ${}^{40}\mathrm{Ca}$ in the superfluid cluster model where the ten $\alpha$ clusters associated a coherent wave (broad curve) are trapped in the $0s$ orbit of the confining potential.

Figure 3

(a) The energy levels of ${}^{40}\mathrm{Ca}$ calculated with the superfluid cluster model with the condensation rate 6%. The zero-mode states are indicated by zm and others are BdG states. (b) The experimental low-lying energy levels of ${}^{40}\mathrm{Ca}$ [86, 87, 93]. The member states of the experimental $K=0^{+}$ and $K=0^{-}$ bands with the $\alpha$ cluster structure are indicated by the thick solid lines.

Figure 4

(a) The calculated eigenfunction (order parameter) $\xi \left(r\right)$ (dashed line) and its adjoint eigenfunction $\eta \left(r\right)$ (solid line) for the ground state of ${}^{40}\mathrm{Ca}$. (b) The calculated superfluid density distribution ${\rho }_s$ in the SCM (dashed line) and the matter density distribution $\rho$ in the OCM of Ref. [88] (solid line) for the ground state.

Figure 5

Calculated BdG wave functions of ${}^{40}\mathrm{Ca}, {\mathcal{U}}_{n\ell }\left(r\right)$ (solid lines) and ${\mathcal{V}}_{n\ell }\left(r\right)$ (dashed lines), for the $2^{+}$ ($n=0, \ell =2$) and $3^{-}$ ($n=0, \ell =3$) states.

Figure 6

The zero-mode wave functions ${\mathrm{\Psi }}_{\nu }\left(q\right)$ for the $0^{+}$ states of ${}^{40}\mathrm{Ca}$ calculated with the condensation rate 6%, $N_0=0.06N$, (a) $\nu =0$, (b) $\nu =1$, and (c) $\nu =2$. The solid lines and the dashed lines represent the real part and the imaginary part of the wave functions ${\mathrm{\Psi }}_{\nu }\left(q\right)$.

Figure 7

The condensation rate dependence of the energy levels of ${}^{40}\mathrm{Ca}$ calculated by the superfluid cluster model with the condensation rates (a) 5%, (b) 6%, (c) 7%, and (d) 8%. The three zero-mode states with $\nu =0$, 1, and 2 are indicated by zm and others are BdG states.

Figure 8

(a) The energy levels calculated with the $\alpha$ cluster model with the $\alpha +{}^{36}\mathrm{Ar}$ geometrical configuration in the OCM [88] are compared with (b) the experimental energy levels [93] and (c) the field theoretical superfluid cluster model calculation with 6% condensation rate assuming no geometrical configuration.