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تسجيل مستخدم جديدDescription of nuclear systems with a self-consistent configuration-mixing approach. I: Theory, algorithm, and application to the $^{12}$C test nucleus
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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.
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The variational multiparticle-multihole configuration mixing approach (MPMH) to nuclei has been proposed about a decade ago. While the first applications followed rapidly, the implementation of the full formalism of this method has only been recently
completed and applied in [C. Robin, N. Pillet, D. Pe~na Arteaga and J.-F. Berger, Phys. Rev. C 93, 024302 (2016)] to $^{12}$C as a test-case. The main objective of the present paper is to carry on the study that was initiated in that reference, in order to put the MPMH method to more stringent tests. To that aim we perform a systematic study of even-even sd-shell nuclei. The wave function of these nuclei is taken as a configuration mixing built on orbitals of the sd-shell, and both the mixing coefficients of the nuclear state and the single-particle wave functions are determined consistently from the same variational principle. The calculations are done using the D1S Gogny force. Various ground-state properties are analyzed. In particular, the correlation content and composition of the wave function as well as the single-particle orbitals and energies are examined. Binding energies and charge radii are also calculated and compared to experiment. The description of the first excited state is also examined and the corresponding transition densities are used as input for the calculation of inelastic electron and proton scattering. Special attention is paid to the effect of the optimization of the single-particle states consistently with the correlations of the system. Globally, the results are satisfying and encouraging. In particular, charge radii and excitation energies are nicely reproduced. However, the chosen valence-space truncation scheme precludes achieving maximum collectivity in the studied nuclei. Further refinement of the method and a better-suited interaction are necessary to remedy this situation.
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 str
uctures 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.
Form factors for $alpha+{^{12}}$C inelastic scattering are obtained within two theoretical ($alpha+alpha+alpha$) approaches: The hyperspherical framework for three identical bosons, and the algebraic cluster model assuming the $D_{3h}$ symmetry of an
equilateral triangle subject to rotations and vibrations. Results show a good agreement, with form factors involving the Hoyle state having a slightly larger extension within the hyperspherical approach. Coupled-channel calculations using these form factors are ongoing.
The algebraic molecular model is used in $^{12}$C to construct densities and transition densities connecting low-lying states of the rotovibrational spectrum, first and foremost those belonging to the rotational bands based on the ground and the Hoyl
e states. These densities are then used as basic ingredients to calculate, besides electromagnetic transition probabilities, nuclear potentials and formfactors to describe elastic and inelastic $alpha$+$^{12}$C scattering processes. The calculated densities and transition densities are also compared with those obtained by directly solving the problem of three interacting alphas within a three-body approach where continuum effects, relevant in particular for the Hoyle state, are properly taken into account.
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.
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