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Register a new userThree-$alpha$ configurations in the $0^{+}$ states of $^{12}mathrm{C}$
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Geometric configurations of three-$alpha$ particles in the ground- and first-excited $J^pi=0^+$ states of $^{12}$C are discussed within two types of $alpha$-cluster models which treat the Pauli principle differently. Though there are some quantitative differences especially in the internal region of the wave functions, equilateral triangle configurations are dominant in the ground state, while in the first excited $0^+$ state isosceles triangle configurations dominate, originating from $^8{rm Be}+alpha$ configurations.
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Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$alpha$ cluster model with a deep $alphaalpha$-potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$alpha$-Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states from weakly bound phase to a deep phase. The ground and lowest $2^+$ states of the $^{12}$C nucleus in the deep phase are created by the critical eigen states of the Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces, respectively.
The ${}^{12}mathrm{C}(alpha,gamma){}^{16}mathrm{O}$ reaction plays a key role in the evolution of stars with masses of $M >$ 0.55 $M_odot$. The cross-section of the ${}^{12}mathrm{C}(alpha,gamma){}^{16}mathrm{O}$ reaction within the Gamow window ($E_textrm{c.m.}$ = 300 keV, $T_textrm9$ = 0.2) is extremely small (about $10^{-17}$ barn), which makes the direct measurement in a ground-based laboratory with existing techniques unfeasible. Up until now, the cross-sections at lower energies can only be extrapolated from the data at higher energies. However, two subthreshold resonances, located at $E_x$ = 7.117 MeV and $E_x$ = 6.917 MeV, make this extrapolation more complicated. In this work, the 6.917 MeV subthreshold resonance in the ${}^{12}mathrm{C}(alpha,gamma){}^{16}mathrm{O}$ reaction was investigated via the ${}^{12}mathrm{C}({}^{11}mathrm{B},{}^{7}mathrm{Li}){}^{16}mathrm{O}$ reaction. The experiment was performed using the Q3D magnetic spectrograph at the HI-13 tandem accelerator. We measured the angular distribution of the ${}^{12}mathrm{C}({}^{11}mathrm{B},{}^{7}mathrm{Li}){}^{16}mathrm{O}$ transfer reaction leading to the 6.917 MeV state. Based on the FRDWBA analysis, we derived the asymptotic normalization coefficient (ANC) of the 6.917 MeV level in $^{16}$O to be (1.10 $pm$ 0.29) $times 10^{10}$ fm$^{-1}$, with which the reduced $alpha$ width was computed to be $18.0pm4.7$ keV at the channel radius of 6.5 fm. Finally, we calculated the astrophysical $S_{E2}(300)$ factor of the ground-state transitions to be 46.2 $pm$ 7.7 keV b. The result for the astrophysical $S_{E2}(300)$ factor confirms the values obtained in various direct and indirect measurements and presents an independent examination of the most important data in nuclear astrophysics.
We present a new picture that the $alpha$-linear-chain structure for ${^{12}{rm C}}$ and ${^{16}{rm O}}$ has one-dimensional $alpha$ condensate character. The wave functions of linear-chain states which are described by superposing a large number of Brink wave functions have extremely large overlaps of nearly $100%$ with single Tohsaki-Horiuchi-Schuck-Ropke (THSR) wave functions, which were proposed to describe the $alpha$ condensed gas-like states. Although this new picture is different from the conventional idea of the spatial localization of $alpha$ clusters, the density distributions are shown to have localized $alpha$-clusters which is due to the inter-$alpha$ Pauli repulsion.
Densities and transition densities are computed in an equilateral triangular alpha-cluster model for $^{12}$C, in which each $alpha$ particle is taken as a gaussian density distribution. The ground-state, the symmetric vibration (Hoyle state) and the asymmetric bend vibration are analyzed in a molecular approach and dissected into their components in a series of harmonic functions, revealing their intrinsic structures. The transition densities in the laboratory frame are then used to construct form-factors and to compute DWBA inelastic cross-sections for the $^{12}$C$(alpha, alpha)$ reaction. The comparison with experimental data indicates that the simple geometrical model with rotations and vibrations gives a reliable description of reactions where $alpha$-cluster degrees of freedom are involved.
For more than half a century, the structure of $^{12}$C, such as the ground band, has been understood to be well described by the three $alpha$ cluster model based on a geometrical crystalline picture. On the contrary, recently it has been claimed that the ground state of $^{12}$C is also well described by a nonlocalized cluster model without any of the geometrical configurations originally proposed to explain the dilute gas-like Hoyle state, which is now considered to be a Bose-Einstein condensate of $alpha$ clusters. The challenging unsolved problem is how we can reconcile the two exclusive $alpha$ cluster pictures of $^{12}$C, crystalline vs nonlocalized structure. We show that the crystalline cluster picture and the nonlocalized cluster picture can be reconciled by noticing that they are a manifestation of supersolidity with properties of both crystallinity and superfluidity. This is achieved through a superfluid $alpha$ cluster model based on effective field theory, which treats the Nambu-Goldstone zero mode rigorously. For several decades, scientists have been searching for a supersolid in nature.Nuclear $alpha$ cluster structure is considered to be the first confirmed example of a stable supersolid.
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