Bose-Einstein condensation of $\alpha$ clusters and new soft mode in ${}^{12}\mathrm{C}$–${}^{52}\mathrm{Fe}$ $4N$ nuclei in a field-theoretical superfluid cluster model

Bose–Einstein condensation of $\alpha$ clusters in light and medium-heavy nuclei is studied in the frame of the field-theoretical superfluid cluster model. The order parameter of the phase transition from the Wigner phase to the Nambu–Goldstone phase is a superfluid amplitude, square of the moduli of which is the superfluid density distribution. The zero-mode operators due to the spontaneous symmetry breaking of the global phase in the finite number of $\alpha$ clusters are rigorously treated. The theory is systematically applied to $N\alpha$ nuclei from ${}^{12}\mathrm{C}$ to ${}^{52}\mathrm{Fe}$ at various condensation rates. In ${}^{12}\mathrm{C}$ it is found that the energy levels of the gas-like well-developed $\alpha$ cluster states above the Hoyle state are reproduced well in agreement with experiment for realistic condensation rates of $\alpha$ clusters. The electric $E2$ and $E0$ transitions are calculated and found to be sensitive to the condensation rates. The profound raison d'être of the $\alpha$ cluster gas-like states above the Hoyle state, whose structure has been interpreted geometrically in the nuclear models without the order parameter such as the cluster models or ab initio calculations, is revealed. It is found that, in addition to the Bogoliubov–de Gennes vibrational mode states, collective states of the zero-mode operators appear systematically at low excitation energies from the $N\alpha$ threshold energy. These collective states, which are new-type soft modes in nuclei due to the Bose–Einstein condensation of the $\alpha$ clusters, emerge systematically in light– and medium-heavy–mass regions and are also located at high excitation energies from the ground state in contrast to the traditional concept of soft mode in the low-excitation-energy region.

Figure 1

The relation between the confining potential parameter $\mathrm{\Omega }$ and the repulsive potential $V_r$ for ${}^{12}\mathrm{C}$ is plotted for the case of $\overline{r}=3.8$ fm with $N_0=0.7N$. As shown by the vertical dashed lines, the BEC system collapses for $V_r$ smaller than the critical value around 310 MeV.

Figure 2

The energy levels of ${}^{12}\mathrm{C}$ calculated with $N_0=0.7N$ as a function of $V_r$ for $\overline{r}=3.8$ fm are compared with the observed energy levels (horizontal lines) taken from Refs. [30, 31, 55, 56, 57, 58].

Figure 3

The calculated (a) eigenfunction with zero eigenvalue $\xi \left(r\right)$, (b) its adjoint eigenfunction $\eta \left(r\right)$, and (c) radial density distribution of the condensate $r^2{\left|\xi \left(r\right)\right|}^2/N_0$ for $\overline{r}=3.8$, 3.5, and 3.2 fm with $N_0=0.7N$ in ${}^{12}\mathrm{C}$.

Figure 4

Numerically calculated BdG wave functions, ${\mathcal{U}}_{1\ell }\left(r\right)$ (solid lines) and ${\mathcal{V}}_{1\ell }\left(r\right)$ (dashed lines) for (a) $\ell =0$, (b) $\ell =2$, and (c) $\ell =4$ for $\overline{r}=3.8$, 3.5, and 3.2 fm with $N_0=0.7N$ in ${}^{12}\mathrm{C}$. In the legends of the figures, the suffix $n=1$ is omitted and only $\ell$ is given for ${\mathcal{U}}_{1\ell }$ and ${\mathcal{V}}_{1\ell }$.

Figure 5

The squares of numerically calculated wave functions of the zero-mode states, (a) $|{\mathrm{\Psi }}_0{\left(q\right)|}^2$, (b) $|{\mathrm{\Psi }}_1{\left(q\right)|}^2$, and (c) $|{\mathrm{\Psi }}_2{\left(q\right)|}^2$, for $\overline{r}=3.8$, 3.5, and 3.2 fm with $N_0=0.7N$ in ${}^{12}\mathrm{C}$.

Figure 6

The energy levels of ${}^{12}\mathrm{C}$ calculated with the parameters in Table 3 for the three condensation rates, $N_0=3.0$ (100%), 2.5 (83%), and 2.0 (67%), with fixed $\overline{r}=3.8\mathrm{fm}$.

Figure 7

The relation between the confining potential parameter $\mathrm{\Omega }$ and the repulsion potential $V_r$ for ${}^{12}\mathrm{C}$ is plotted for the case of $\overline{r}=3.8$ fm with 100% condensation ($N_0=N$).

Figure 8

The energy levels of ${}^{12}\mathrm{C}$ calculated with 100% condensation ($N_0=N$) as a function of $V_r$ for $\overline{r}=3.8$ fm are compared with the observed energy levels (horizontal lines) taken from Refs. [30, 31, 55, 56, 57, 58].

Figure 9

The energy levels of ${}^{12}\mathrm{C}$, calculated from the parameters in Table 4 with fixed 100% condensation ($N_0=N$) for $\overline{r}=3.8$, 3.5, and 3.2 fm, are compared with the experimental energy levels from Refs. [30, 31, 55, 56, 57, 58]. Vac, ZM, and BdG mean the vacuum Hoyle state, zero-mode states, and BdG excited states, respectively.

Figure 10

Numerically calculated (a) $\xi \left(r\right)$, (b) $\eta \left(r\right)$, and (c) the radial density distribution of the condensate $r^2{\left|\xi \left(r\right)\right|}^2/N_0$ with $\overline{r}=3.8$, 3.5, and 3.2 fm for100% condensation ($N_0=N$).

Figure 11

Numerically calculated BdG wave functions, ${\mathcal{U}}_{1\ell }\left(r\right)$ (solid lines), and ${\mathcal{V}}_{1\ell }\left(r\right)$ (dashed lines) for (a) $\ell =0$, (b) $\ell =2$, and (c) $\ell =4$, with $\overline{r}=3.8$, 3.5, and 3.2 fm for100% condensation ($N_0=N$).

Figure 12

The squares of numerically calculated wave functions of the zero-mode states, (a) $|{\mathrm{\Psi }}_0{\left(q\right)|}^2$, (b) $|{\mathrm{\Psi }}_1{\left(q\right)|}^2$, and (c) $|{\mathrm{\Psi }}_2{\left(q\right)|}^2$, with $\overline{r}=3.8$, 3.5, and 3.2 fm for100% condensation ($N_0=N$).

Figure 13

The energy levels of ${}^{12}\mathrm{C}$, calculated from the parameters in Table 1 with fixed $N_0=0.7N$ for $\overline{r}=3.8$, 3.5, and 3.2 fm are compared with the experimental energy levels from Refs. [30, 31, 55, 56, 57, 58]. Vac, ZM, and BdG mean the vacuum Hoyle state, zero-mode states, and BdG excited states, respectively.

Figure 14

The energy levels of ${}^{12}\mathrm{C}$ calculated with different condensation rates with $\overline{r}=3.8$ fm.

Figure 15

How the coefficient $C_{ij{i}^{\prime }{j}^{\prime }}$ of the zero-mode Hamiltonian in Eq. (29) change in the calculations of Fig. 14 with different condensation rates of the three-$\alpha$ clusters in ${}^{12}\mathrm{C}$ is displayed.

Figure 16

The $N$ dependence of $\mathrm{\Omega }$ of the confining potential, which is used in the energy-level calculations of $N=3–13$ (${}^{12}\mathrm{C}\text{–}{}^{52}\mathrm{Fe}$) in Fig. 17, is displayed by crosses. This is obtained under a constraint $\overline{r}\propto N^{1/3}$ and with fixed $V_r=400$ MeV for 100% condensation. The solid curve $5.4N^{-0.65}$ is to guide the eye.

Figure 17

The energy levels calculated for $N=3–13$ (${}^{12}\mathrm{C}\text{–}{}^{52}\mathrm{Fe}$) with 100% condensation using the $N$-dependent $\mathrm{\Omega }$ given in Fig. 16. Excitation energy is measured from the Hoyle-analog vacuum, i.e., the $N$-alpha condensate state near the $N$-alpha threshold.

Figure 18

The $N$ dependence of $\mathrm{\Omega }$ of the confining potential in case of $N_0=0.7N$, which is used in the energy-level calculations of $N=3–13$ (${}^{12}\mathrm{C}\text{–}{}^{52}\mathrm{Fe}$) in Fig. 19, is displayed by crosses. This is obtained under a constraint $\overline{r}\propto N^{1/3}$ and with fixed $V_r=400$ MeV. The solid curve $4.70N^{-0.60}$ is to guide the eye.

Figure 19

The energy levels calculated for $N=3–13$ (${}^{12}\mathrm{C}\text{–}{}^{52}\mathrm{Fe}$) in the case of 70% condensation ($N_0=0.7N$) using the interaction with $N$-dependent $\mathrm{\Omega }$ in Fig. 18. Excitation energy is measured from the Hoyle-analog vacuum, i.e., the $N$-alpha-condensate state near the $N$-alpha threshold.

Figure 20

The $N_0$ dependencies of $C_{ij{i}^{\prime }{j}^{\prime }}$ and $I$ with the parameters $V_r=403$ [MeV], $\mathrm{\Omega }=2.62$ [MeV] and for $N_0=2–14$ are shown. The two coefficients $C_{1102}$ and $C_{0202}$ that are smaller than $C_{1111}$ are not plotted.

Figure 21

The $q^4$ potential and the eigenfunctions ${\mathrm{\Psi }}_{\nu }\left(q\right)$ belonging to the eigenvalues $E_{\nu }(\nu =0,1,2)$ for ${\widehat{H}}_0^{QP}$.

Figure 22

Numerically calculated excitation energies $E_{\nu }-E_0$ of (a) ${\widehat{H}}_0^{QP}$ and (b) ${\widehat{H}}_0^{QP}+{\widehat{H}}_1^{QP}+{\widehat{H}}_2^{QP}$ are plotted for $\nu =1–5$ and $N_0=3,6,9,12$ with the parameters $V_r=403$ [MeV] and $\mathrm{\Omega }=2.62$ [MeV].