1. Satoru Sugimoto Kyoto University 1. Introduction 2. Charge- and parity-projected Hartree-Fock method (a mean field type model) and its application to sub-closed shell oxygen isotopes 3.Shell model calculation to 15,16,17 O 4. Summary Study of the Tensor Correlation in Oxygen Isotopes using Mean-Field-Type and Shell Model Methods

2. Introduction The tensor force is important in nuclear structure. There remain many open problems to be solved. How does the tensor correlation change in neutron-rich nuclei? Shell evolution (Ostuka, PRL 95 232502 (2005)). The breakdown of the magic number in 11 Li (Myo et al.) The relation to the ls splitting in 5 He (Myo et al. PTP 113 (2005) 763)

3. The correlation to be included In the simple HF calculation, 2p-2h correlations are hard to be treated. beyond mean field modelWe need to include at least 2p-2h correlations to exploit the tensor correlation.  beyond mean field model cf. Single-particle (H-F) correlation 2p-2h correlation

4. Charge- and parity-symmetry breaking mean field method Tensor force is mediated by the pion. Pseudo scalar (  ) To exploit the pseudo scalar character of the pion, we introduce parity-mixed single particle state. (over-shell correlation) Isovector (  ) To exploit the isovector character of the pion, we introduce charge-mixed single particle state. Projection Because the total wave function made from such parity- and charge-mixed single particle states does not have good parity and a definite charge number. We need to perform the parity and charge projections. Refs. Toki et al., Prog. Theor. Phys. 108 (2002) 903. Sugimoto et al., Nucl. Phys. A 740 (2004) 77; ; nucl-th/0607045. Ogawa et al., Prog. Thoer. Phys. 111 (2004) 75; Phys. Rev. C 73 (2006) 034301.

5. Results for 16 O MV1(V C )+G3RS(V T,V LS ) By performing the parity and charge projection the potential energy from the tensor force becomes sizable value. xTxT EKVVTVT V LS HF1.0-124.1230.0-354.10.0-0.9 CPPHF1.0-127.1237.1-364.24-11.7 CPPHF1.5-127.6253.9-381.6-38.3

6. V T and V LS per particle The potential energy from the tensor force has the same order in magnitude as that from the LS force. The tensor potential energy decreases as neutron numbers. p 1/2 d 5/2 d 3/2 s 1/2 p 3/2

7. Wave function （ １６ O, ｘ T =1.5) Opposite parity components mixed by the tensor force have narrow widths. It suggests that the tensor correlation needs high-momentum components. s 1/2 proton dominant

8. Mixing of the opposite parity components in single-particle states If a next j=1/2 orbit is occupied newly, the mixing probabilities of the j=1/2 orbit reduce by a blocking effect. Mixing of the opposite-parity component may affect excitation spectra of nuclei. 0p1/21s1/2

9. Shell model calculation We perform the shell model calculation including 1p-1h and 2p-2h configurations to study the tensor correlation. inclusion of narrow-width single-particle wave functions The shell model calculation can treat the correlation which cannot be treated in a mean-field-type calculation. cf. Myo et al. PTP 113 (2005) 763

10. Model space 16 O: (0p-0h)+(1p-1h)+(2p+2h) 17 O: (1p-0h)+(2p-1h)+(3p+2h) 15 O: (0p-1h)+(1p-2h)+(2p+3h) Core (hole state) (0s 1/2 ) 4 (0p 3/2 ) 6 (0p 1/2 ) 4 Harmonic oscillator single-particle wave functions Particle state Harmonic oscillator single-particle wave functions + Gaussian single-particle wave functions with narrow (half) width parameters These are ortho-normalized by the G-S method

11. Effective interaction Central force: Volkov No. 1 A.B. Volkov, Nucl. Phys. 74 ( 1965 ) 33 Tensor: Furutani force H. Furutani et al., Prog. Theor. Phys. Suppl. 68 ( 1980 ) 193 LS: G3RS. Tamagaki, Prog. Theor. Phys. 39 ( 1968 ) 91 No Coulomb force

12. 16 O HO: (1s 0d)+(1p 0f)+(2s 1d 0g) NWG: b NW = b HO /2 = 1.8 fm d-orbit: (1s 0d)+s NW +p NW +d NW f-orbit: (1s 0d)+s NW +p NW +d NW+ f NW By including single-particle orbits with narrow width parameters the correlation energy from the tensor force becomes large. EKEVCVC VTVT V LS P(CS)   CM /    HO 2   -137.0249.6-383.8-2.6-0.175%1.50 NWG (d-orbit) -136.2257.5-356.9-32.9-3.974%1.58 NWG (f-orbit) -132.5266.5-337.1-56.6-5.372%1.58

13. 17,15 O (NWG (up to f-orbit))  KE+  V C +  V T ≈0 ls-splitting nearly equals to  V LS EKEVCVC VTVT V LS P(NP)   CM /    5/2 + -132.2289.5-357.4-57.0-7.372%1.58 3/2 + -127.7288.6-356.7-56.7-2.873%1.58 diff (5/2 + - 3/2 +) -4.50.8-0.6-0.2-4.5 EKEVCVC VTVT V LS P(NP)   CM /    1/2 - -114.8250.2-301.6-55.7-7.871%1.58 3/2 - -110.3247.3-300.0-54.1-3.573%1.58 diff (1/2 - - 3/2 -) -4.52.9-1.5-1.6-4.3 15 O 17 O

14. Magnetic Moment Magnetic moments change a little in spite of the large correlation energy form the tensor force. 5/2 + 3/2 + 17 O 17 F 17 O 17 F Schmidt value-1.914.791.150.13 HO 2   -1.874.751.170.11 NWG (f-orbit)-1.894.771.150.13 1/2 - 3/2 - 15 N 15 O 15 N 15 O Schmidt value-0.260.643.79-1.91 HO 2   -0.270.653.72-1.85 NWG (f-orbit)-0.250.633.74-1.87 A=17 A=15

15. Summary We apply a mean-field model which treats the tensor correlation by mixing parities and charges in single- particle states (the CPPHF method) to oxygen isotopes. The opposite parity components induced by the tensor force is compact in size. (high-momentum component) We perform the shell model calculation up to 2p-2h states to 15,16,17 O. The tensor correlation energy becomes large by including Gaussian single-particle wave functions with narrow widths. The tensor correlation changes ls splitting and magnetic moments in 15,17 O a little in spite of its large correlation energy.