Possibility of ${}^{14}\mathrm{C}$ cluster as a building block of medium-mass nuclei

The possibility of the ${}^{14}\mathrm{C}$ cluster being a basic building block of medium-mass nuclei is discussed. Although $\alpha$ cluster structures have been widely discussed in the light $N\approx Z$ mass region, the neutron-to-proton ratio deviates from unity in the nuclei near $\beta$-stability line and in neutron-rich nuclei. Thus, more neutron-rich objects with $N>Z$ could become the building blocks of cluster structures in such nuclei. The ${}^{14}\mathrm{C}$ nucleus is strongly bound and can be regarded as such a candidate. In addition, the path to the lowest shell-model configuration at short relative distances is closed for the ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ structure, contrary to the case of the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ structure; this allows an appreciable separation distance between the ${}^{14}\mathrm{C}$ clusters. The recent development of the antisymmetrized quasicluster model (AQCM) allows us to utilize a $jj$-coupling shell-model wave function for each cluster in a simplified way. The AQCM results for the ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ structure in ${}^{28}\mathrm{Mg}$ are compared with the ones of cranked relativistic mean field (CRMF) calculations. Although theoretical frameworks of these two models are quite different, they give similar results for the nucleonic densities and rotational properties of the structure under investigation. The existence of a linear chain three-${}^{14}\mathrm{C}$ cluster structure in ${}^{42}\mathrm{Ar}$ has also been predicted in AQCM. These results confirm the role of the ${}^{14}\mathrm{C}$ cluster as a possible building block of cluster structures in medium-mass nuclei.

Figure 1

The $0^{+}$ energy curves of ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ (${}^{24}\mathrm{Mg}$, dotted line) and ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ (${}^{28}\mathrm{Mg}$, solid line). The dashed curve corresponds to the $0^{+}$ energy curve of ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ (${}^{32}\mathrm{S}$).

Figure 2

The expectation value of the principal quantum number $n$ of the harmonic oscillator for the $0^{+}$ state of the ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ configuration in ${}^{28}\mathrm{Mg}$ as a function of relative distance. The solid (red) line is for the neutrons, whereas the dashed line (blue) is for the protons.

Figure 3

Excitation energies of calculated CRMF configurations in ${}^{28}\mathrm{Mg}$ relative to a rotating liquid drop reference $AI(I+1)$, with the inertia parameter $A=0.062$. The inset shows the moments of inertia of the ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ cluster structure as a function of the spin.

Figure 4

Neutron density distributions of the configurations of interest. Panels (a), (b), and (d) show the densities of the [0,2] and [1,2] configurations obtained in the CRMF calculations, while panel (c) shows the densities of the ${}^{14}\mathrm{C}+{}^{14}\mathrm{C}$ structure obtained in the AQCM calculations. The density color map starts at $\rho =0.005\mathrm{fm}{}^{-3}$ and shows the densities in ${\mathrm{fm}}^{-3}$.

Figure 5

Neutron density distributions of the [2,2] configuration in ${}^{32}\mathrm{S}$ obtained in the CRMF calculations at spin $I=12$.

Figure 6

The energy curves for the linear chain of three ${}^{14}\mathrm{C}$ clusters (${}^{42}\mathrm{Ar}$) measured from the three-${}^{14}\mathrm{C}$ threshold as a function of the ${}^{14}\mathrm{C}\text{−}{}^{14}\mathrm{C}$ distance. The angular momentum is changed from 0 to 8.

Figure 7

The $R\text{−}\mathrm{\Lambda }$ dependence of the $0^{+}$ energy of ${}^{14}\mathrm{C}$ for the case of $\nu =0.22{\mathrm{fm}}^{-2}$.

Figure 8

The elastic form factor for the proton part of the ground state of ${}^{14}\mathrm{C}$. The size parameter $\nu$ is $0.22{\mathrm{fm}}^{-2}$ and Slater determinants with various $R$ and $\mathrm{\Lambda }$ values are superposed based on the GCM.

Figure 9

The expectation value of the principal quantum number $n$ of the harmonic oscillator for the $0^{+}$ state of ${}^{28}\mathrm{Mg}$ with the ground state configuration considered as a function of the relative distance between the quasiclusters around ${}^{16}\mathrm{O}$. Solid red and blue dashed lines are for neutrons and protons, respectively.