Explicit inclusion of the spin-orbit contribution in the Tohsaki-Horiuchi-Schuck-R\"opke wave function

Explicit inclusion of the spin-orbit contribution in the Tohsaki-Horiuchi-Schuck-Röpke wave function

N. Itagaki, H. Matsuno, and A. Tohsaki

Phys. Rev. C 98, 044306 – Published 8 October 2018

Abstract

The Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function has been successfully used for the studies of gaslike nature of $α$ clusters in various nuclei, including the so-called Hoyle state of ${}^{12}C$ and four $α$ states of ${}^{16}O$ . In standard $α$ cluster models, however, each $α$ cluster wave function has spin zero because of its spatial symmetry and antisymmetrization effect. Thus the noncentral interactions do not contribute, and this situation is the same in the THSR wave function. In this work, the spin-orbit contribution, which is found to be quite important at short $α - α$ distances, is taken into account in the THSR wave function by combing it with antisymmetrized quasicluster model (AQCM). The application to ${}^{12}C$ is presented. The multi-integration in the original THSR wave function is carried out by using a Monte Carlo technique, which is called the Monte Carlo THSR wave function. For the nucleon-nucleon interaction, the Tohsaki interaction, which contains finite-range three-body terms and simultaneously reproduces the saturation properties of nuclear systems, the $α - α$ scattering phase shift, and the size and binding energy of ${}^{4}{He}$ , is adopted.

Received 8 June 2018
Revised 19 August 2018

DOI:https://doi.org/10.1103/PhysRevC.98.044306

Physics Subject Headings (PhySH)

Cluster models Nuclear forces

6 ≤ A ≤ 19

Nuclear Physics

Authors & Affiliations

N. Itagaki¹, H. Matsuno², and A. Tohsaki³

¹Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan
²Department of Physics, Kyoto University, Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan
³Research Center for Nuclear Physics, 10-1 Mihogaoka, Ibaraki, Osaka 067-0047, Japan

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Vol. 98, Iss. 4 — October 2018

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Figure 1
The $0^{+}$ energy convergence of ${}^{12}C$ [three $α$ 's] as a function of the number of Slater determinants superposed [ $N_{max}$ in Eq. (4)] calculated with the Monte Carlo THSR framework. The $α$ clusters are not broken [ $Λ = 0$ in Eq. (5)], and the solid, dotted, short-dashed, dashed, and dash-dotted lines correspond to $σ = 1$ , 2, 3, 4, and 5 fm in Eq. (3), respectively. The dashed line at $- 82.50 MeV$ shows the calculated three- $α$ threshold energy. The solid square at $- 92.2 MeV$ and the solid triangle at $- 84.9 MeV$ show the experimental values for energies of the ground state and three- $α$ threshold, respectively.
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Figure 2
The convergence of the root mean square (rms) matter radius for the $0^{+}$ state of ${}^{12}C$ as a function of the number of Slater determinants superposed [ $N_{max}$ in Eq. (4)] calculated with the Monte Carlo THSR framework. The $α$ clusters are not broken [ $Λ$ in Eq. (5) is zero], and the solid, dotted, short-dashed, dashed, and dash-dotted lines correspond to $σ = 1$ , 2, 3, 4, and 5 fm in Eq. (3), respectively. The solid circle at 2.35 fm shows the experimental value for the ground state deduced in Ref. [39].
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Figure 3
The $0^{+}$ energy curve of ${}^{12}C$ as a function of $σ$ defined in Eq. (3). The dotted line is for $Λ = 0$ . The solid line is obtained when $Λ$ is taken as a variational parameter to minimize the energy. The values in the parentheses show the optimal $Λ$ values for the cases of $σ = 1.0$ , 2.0, and 3.0 fm. The dashed line at $- 82.50 MeV$ shows the calculated three- $α$ threshold energy. The solid square at $- 92.2 MeV$ and the solid triangle at $- 84.9 MeV$ show the experimental values for energies of the ground state and three- $α$ threshold, respectively.
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Figure 4
The convergence of the spin-orbit energy for the $0^{+}$ state of ${}^{12}C$ as a function of number of Slater determinants superposed [ $N_{max}$ in Eq. (4)]. The solid line is for $σ = 1.0 fm$ and $Λ = 0.2$ , and the dotted line is for $σ = 2.0 fm$ and $Λ = 0.1$ .
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Figure 5
The contribution of three-body interaction terms for the $0^{+}$ state of ${}^{12}C$ as a function of the number of Slater determinants superposed [ $N_{max}$ in Eq. (4)] calculated with the Monte Carlo THSR framework. The $σ$ value defined in Eq. (3) is 2.0 fm, and $Λ$ in Eq. (5) is 0.1 (solid line) and 0 (dotted line).
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Figure 6
The absolute value of the elastic form factor ( $| F_{(q)} |$ ) for the $0^{+}$ state of ${}^{12}C$ as a function of the momentum transfer $q$ ( ${fm}^{- 1}$ ). The dotted line is for $(σ = 2.0 fm, Λ = 0.1$ ), which gives the optimal energy as shown in Fig. 3, and the solid line is the result after mixing the local minimum point (at the limit of $σ = 0 fm, Λ = 0.3$ ) based on GCM. The experimental values and other theoretical results are compared in Refs. [35, 40].
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Physical Review C

covering nuclear physics

Explicit inclusion of the spin-orbit contribution in the Tohsaki-Horiuchi-Schuck-Röpke wave function

N. Itagaki, H. Matsuno, and A. Tohsaki

Phys. Rev. C 98, 044306 – Published 8 October 2018

Abstract

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Physical Review C

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Explicit inclusion of the spin-orbit contribution in the Tohsaki-Horiuchi-Schuck-Röpke wave function

N. Itagaki, H. Matsuno, and A. Tohsaki

Phys. Rev. C 98, 044306 – Published 8 October 2018

Abstract

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