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？－

1. でもその前に… 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) • Introduction • Requests for AMD’s wave function to treat the tensor force • Further requests found in the study of 4He • Use of wave packets with different width-parameters • Importance of angular momentum projection • Single particle levels in 4He • Summary and future plan RCNP研究会「核力と核構造」 ‘04.03.22atRCNP

2. 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. AMD, Hartree-Fock : without tensor force Relativistic Mean-Field approach : no πmeson How in finite nuclei? Affection to nuclear structure? 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 • Unified effective interaction for light to heavy nuclei. • reduction of calculation costs

3. †Y.Akaishi, H.Bando, S.Nagata, PTP 52 (1972) 339 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. Tensor force might favor such a structure that the ratio of the surface is large, namely well-developed clustering structure. The incorporation of tensor force into central force is sensitive to the starting energy. • In the perturbation theory, 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. Shell Tensor force Our scenario Following the tensor force, each of nuclei chooses shell- or clustering-structure, which are qualitatively different from each other. Cluster

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

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

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

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

8. 4He • 3 wave packets • Vc×1.0／ Vt×3.0 • Frictional cooling • spin/isospin free Test for different ν’s (0.2, 0.2, 0.2) (0.5, 0.5, 0.5) (0.2, 0.5, 0.9) (0.9, 0.9, 0.9) By using the wave packets with different ν’s, the binding energy can be gained without shrinkage.

9. 4 wave packets • Vc×1.0／ Vt×2.0 • Frictional cooling • spin/isospin free Result of 4He 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

10. 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.

11. 4He • 4 wave packets • ν= 0.3 ～ 1.5 • Vc×1.0／ Vt×2.0 • Frictional cooling • spin/isospin free • J projection (VBP) J projection after J constraint variation No constraint B.E. = -25.3 MeV Rrms = 1.32 fm J2 constraint B.E. = -28.6 MeV Rrms = 1.35 fm

12. Single particle levels in 4He What happens in the 4He 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

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

14. Single particle levels • 2 groups • Lower: neutron-like, P(-)=15% • Upper: proton-like, P(-)=19% P-state • High L? If S(80%)+P(20%), L2=0.4. But L2=0.83. 7% D-state? • High J? If 0s1/2+0p1/2, J2=0.75. But J2=1.16.

15. 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 4He: • 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 4He • Akaishi potential, Vt×2.0.

16. Summary and Future plan • We have investigated the single particle levels in 4He. • 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)