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反対称化分子動力学でテンソル力を取り扱う試 み －更に前進するには？－ A. Dote (KEK), Y. Kanada-En ’ yo （ KEK ）, H. Horiuchi (Kyoto univ.), Y. Akaishi (KEK), K. Ikeda (RIKEN) 1.Introduction 2.Requests for AMD ’ s wave function to treat the tensor force 3.Further requests found in the study of 4 He Use of wave packets with different width-parameters Importance of angular momentum projection 4.Single particle levels in 4 He 5.Summary and future plan RCNP 研究会「核力と核構造」 ‘ 04.03.22 at RCNP でもその前に …

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Introduction Why tensor force ？ 1, large contribution in nuclear forces ex) deuteron : bound by the tensor force In microscopic models, tensor force is incorporated into central force. 2, creation and annihilation of clustering structure Tensor force works well on nuclear surface more work in more developed clustering structure? 3, Development of effective interaction for AMD AMD, Hartree-Fock : without tensor force Relativistic Mean-Field approach : no πmeson How in finite nuclei? Affection to nuclear structure? Unified effective interaction for light to heavy nuclei. reduction of calculation costs

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Introduction Previous study of the tensor force † says… In the nuclear matter, the 3E effective interaction containing the tensor force is weakened. Saturation property The tensor force might be weakened in heavy nuclei rather than in light nuclei, and inside of nuclei rather than near nuclear surface. In the perturbation theory, The incorporation of tensor force into central force is sensitive to the starting energy. ???? Tensor force might favor such a structure that the ratio of the surface is large, namely well-developed clustering structure. The effective central force changes, corresponding to the nuclear structure. If the nuclear structure changes, the starting energy changes also. Therefore, the effective central force changes. Cluster Shell Tensor force Following the tensor force, each of nuclei chooses shell- or clustering-structure, which are qualitatively different from each other. Our scenario † Y.Akaishi, H.Bando, S.Nagata, PTP 52 (1972) 339

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Requests for AMD’s wave function to treat the tensor force ii. Superposition of wave packets with spin tensor force : Strong correlation between spin and space iii. Changeability of isospin wave func. tensor force : Change the charge of a single particle wave function i. Parity-violated mean field Ex.) Furutani potential ii iii i i tensor force : Change the parity of a single particle wave function

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Further requests found in the study of 4 He Akaishi potential Based on Tamagaki potential, The repulsive core of the central force is treated with G-matrix method. Tensor force : bare interaction. Interaction ： Tensor force is not incorporated into central force. 1. Wave packets with different width-parameters Gain the binding energy without shrinkage ν for S-wave ≠ ν for P-wave 2. Angular momentum projection Because of very narrow wave packets, only a little mixture of J≠0 components makes the kinetic energy increase. 3. J projection after J constraint variation 10 MeV 17 MeV 25 MeV 28 MeV 0123

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Wave packets with different width-parameters One nucleon wave function ： superposition of Gaussian wave packets with different ν’s ＋＋～＋ ・・・ ν Kinetic energy Tensor force Radius

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Why different ν’s ? vs Tensor max S×P max

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Test for different ν’s (0.9, 0.9, 0.9) (0.5, 0.5, 0.5) (0.2, 0.2, 0.2) (0.2, 0.5, 0.9) By using the wave packets with different ν’s, the binding energy can be gained without shrinkage. 4He 3 wave packets Vc×1.0 ／ Vt×3.0 Frictional cooling spin/isospin free 4He 3 wave packets Vc×1.0 ／ Vt×3.0 Frictional cooling spin/isospin free

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Result of 4 He 4 wave packets Vc×1.0 ／ Vt×2.0 Frictional cooling spin/isospin free 4 wave packets Vc×1.0 ／ Vt×2.0 Frictional cooling spin/isospin free 0. 4 wave packets with common ν ’ s － ν=0.6 － 1. 4 wave packets with different ν ’ s － ν=0.3 ～ 1.5 geometric ratio － 2. Angular momentum projection onto J=0

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Importance of angular momentum projection –details of energy gain– 〇 in case of ν = 0.3 ～ 1.5 Contributions from various J components to the kinetic energy Because of very narrow wave packets, J≠0 components have large kinetic energy. Only 1 ％ mixture of a J≠0 component increases the kinetic energy by 1 MeV.

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J projection after J constraint variation 4He 4 wave packets ν= 0.3 ～ 1.5 Vc×1.0 ／ Vt×2.0 Frictional cooling spin/isospin free J projection (VBP) 4He 4 wave packets ν= 0.3 ～ 1.5 Vc×1.0 ／ Vt×2.0 Frictional cooling spin/isospin free J projection (VBP) No constraint B.E. = -25.3 MeV Rrms = 1.32 fm J2 constraint B.E. = -28.6 MeV Rrms = 1.35 fm

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Single particle levels in 4 He What happens in the 4 He obtained by the AMD calculation? See single particle levels! Extract the single particle levels from the intrinsic state with AMD+HF method. Single Slater determinant ?

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Diagonalize the overlarp matrix How to extract single particle levels (AMD+HF) 1.Prepare orthonormal base. 2. Mimicking Hartree-Fock, construct a single particle Hamiltonian. 3. Diagonalizing h, get single particle levels. Single particle energySingle particle state : one nucleon wave function in AMD wave function

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Single particle levels 2 groups Lower: neutron-like, P(-)=15% Upper: proton-like, P(-)=19% P-state High L? High J?If 0s 1/2 +0p 1/2, J2=0.75. But J2=1.16. If S(80%)+P(20%), L2=0.4. But L2=0.83. 7% D-state?

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Summary To treat the tensor force in AMD framework, following points are needed: 1, superposition of wave packets with spin, 2, changiability of charge wave function. Other points are found to be important, by the study of 4 He: 3, wave packets with different ν’s gain the B.E. without shrinkage. ν for S-wave is rather different from that for P-wave. 4, J projection Because of very narrow wave packets, J≠0 components have large kinetic energy. 4‘, J projection after J constraint variation leads to better solution. Result of 4 He Akaishi potential, Vt×2.0.

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Summary and Future plan We have investigated the single particle levels in 4 He. Characteristics of our S.P. levels: 1, two groups (2+2) 2, 15 ～ 20% negative parity component (P-state) is mixed. 3, including some components except 0s1/2 + 0p1/2 higher L, higher J state. More detailed analysis of single particle levels. How is each component?: proton-parity +, proton-parity -, neutron-parity +, neutron-parity - Vt×2 ↓ Treat the short range part by tensor correlator method (Neff & Feldmeier) or Cut the high momentum component by G-matrix method (Akaishi-san)

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