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Register a new userDeformation and cluster structures in $^{12}$C studied with configuration mixing using Skyrme interactions
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We report an investigation of the structure of $^{12}$C nucleus employing a newly developed configuration-mixing method. In the three-dimensional coordinate-space representation, we generate a number of Slater determinants with various correlated structures using the imaginary-time algorithm. We then diagonalize a many-body Hamiltonian with the Skyrme interaction in the space spanned by the Slater determinants with parity and angular momentum projections. Our calculation reasonably describes the ground and excited states of $^{12}$C nucleus, both for shell-model-like and cluster-like states. The excitation energies and transition strengths of the ground-state rotational band are well reproduced. Negative parity excited states, $1_1^-$, $2_1^-$, and $3_1^-$, are also reasonably described. The second and third $0^+$ states, $0_2^+$ and $0_3^+$, appear at around 8.8 MeV and 15 MeV, respectively. The $0_2^+$ state shows a structure consistent with former results of the alpha-cluster models, however, the calculated radius of the $0_2^+$ state is smaller than those calculations. The three-{alpha} linear-chain configuration dominates in the $0_3^+$ state.
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We investigate selected static and transition properties of $^{12}$C using ab initio No-Core Shell Model (NCSM) methods with chiral two- and three-nucleon interactions. We adopt the Similarity Renormalization Group (SRG) to assist convergence including up to three-nucleon (3N) contributions. We examine the dependences of the $^{12}$C observables on the SRG evolution scale and on the model-space parameters. We obtain nearly converged low-lying excitation spectra. We compare results of the full NCSM with the Importance Truncated NCSM in large model spaces for benchmarking purposes. We highlight the effects of the chiral 3N interaction on several spectroscopic observables. The agreement of some observables with experiment is improved significantly by the inclusion of 3N interactions, e.g., the B(M1) from the first $J^pi T = 1^+ 1$ state to the ground state. However, in some cases the agreement deteriorates, e.g., for the excitation energy of the first $1^+ 0$ state, leaving room for improved next-generation chiral Hamiltonians.
Although self-consistent multi-configuration methods have been used for decades to address the description of atomic and molecular many-body systems, only a few trials have been made in the context of nuclear structure. This work aims at the development of such an approach to describe in a unified way various types of correlations in nuclei, in a self-consistent manner where the mean-field is improved as correlations are introduced. The goal is to reconcile the usually set apart Shell-Model and Self-Consistent Mean-Field methods. This approach is referred as variational multiparticle-multihole configuration mixing method. It is based on a double variational principle which yields a set of two coupled equations that determine at the same time the expansion coefficients of the many-body wave function and the single particle states. The formalism is derived and discussed in a general context, starting from a three-body Hamiltonian. Links to existing many-body techniques such as the formalism of Greens functions are established. First applications are done using the two-body D1S Gogny effective force. The numerical procedure is tested on the $^{12}$C nucleus in order to study the convergence features of the algorithm in different contexts. Ground state properties as well as single-particle quantities are analyzed, and the description of the first $2^+$ state is examined. This study allows to validate our numerical algorithm and leads to encouraging results. In order to test the method further, we will realize in the second article of this series, a systematic description of more nuclei and observables obtained by applying the newly-developed numerical procedure with the same Gogny force. As raised in the present work, applications of the variational multiparticle-multihole configuration mixing method will however ultimately require the use of an extended and more constrained Gogny force.
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.
We investigate structure of $^{13}_Lambda{rm C}$ and discuss the difference and similarity between the structures of $^{12}{rm C}$ and $^{13}_Lambda{rm C}$ by answering the questions if the linear-chain and gaslike cluster states, which are proposed to appear in $^{12}{rm C}$, survives, or new structure states appear or not. We introduce a microscopic cluster model called, Hyper-Tohsaki-Horiuchi-Schuck-Ropke (H-THSR) wave function, which is an extended version of the THSR wave function so as to describe $Lambda$ hypernuclei. We obtained two bound states and two resonance (quasi-bound) states for $J^pi=0^+$ in $^{13}_Lambda{rm C}$, corresponding to the four $0^+$ states in $^{12}{rm C}$. However, the inversion of level ordering between the spectra of $^{12}{rm C}$ and $^{13}_Lambda{rm C}$, i.e. that the $0_3^+$ and $0_4^+$ states in $^{13}_Lambda{rm C}$ correspond to the $0_4^+$ and $0_3^+$ states in $^{12}{rm C}$, respectively, is shown to occur. The additional $Lambda$ particle reduces sizes of the $0_2^+$ and $0_3^+$ states in $^{13}_Lambda{rm C}$ very much, but the shrinkage of the $0_4^+$ state is only a half of the other states. In conclusion, the Hoyle state becomes quite a compact object with ${^{9}_Lambda{rm Be}}+alpha$ configuration in $^{13}_Lambda{rm C}$ and is no more gaslike state composed of the $3alpha$ clusters. Instead, the $0_4^+$ state in $^{13}_Lambda{rm C}$, coming from the $^{12}{rm C}(0_3^+)$ state, appears as a gaslike state composed of $alpha+alpha+^{5}_Lambda{rm He}$ configuration, i.e. the Hoyle analog state. A linear-chain state in a $Lambda$ hypernucleus is for the first time predicted to exist as the $0_3^+$ state in $^{13}_Lambda{rm C}$ with more shrunk arrangement of the $3alpha$ clusters along $z$-axis than the $3alpha$ linear-chain configuration realized in the $^{12}{rm C}(0_4^+)$ state.
The antisymmetrized quasi-cluster model (AQCM) was proposed to describe {alpha}-cluster and $jj$-coupling shell models on the same footing. In this model, the cluster-shell transition is characterized by two parameters; $R$ representing the distance between {alpha} clusters and {alpha} describing the breaking of {alpha} clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account starting with the {alpha} cluster model wave function. Not only the closure configurations of the major shells, but also the subclosure configurations of the $jj$-coupling shell model can be described starting with the {alpha}-cluster model wave functions; however, the particle hole excitations of single particles have not been fully established yet. In this study we show that the framework of AQCM can be extended even to the states with the character of single particle excitations. For $^{12}$C, two particle two hole (2p2h) excitations from the subclosure configuration of $0p_{3/2}$ corresponding to BCS-like pairing are described, and these shell model states are coupled with the three {alpha} cluster model wave functions. The correlation energy from the optimal configuration can be estimated not only in the cluster part but also in the shell model part. We try to pave the way to establish a generalized description of the nuclear structure.
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