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

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 (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)

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

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/ Ogawa et al., Prog. Thoer. Phys. 111 (2004) 75; Phys. Rev. C 73 (2006)

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 HF CPPHF CPPHF

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

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

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

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

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

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

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   %1.50 NWG (d-orbit) %1.58 NWG (f-orbit) %1.58

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/ %1.58 3/ %1.58 diff (5/ /2 +) EKEVCVC VTVT V LS P(NP)   CM /    1/ %1.58 3/ %1.58 diff (1/ /2 -) O 17 O

Magnetic Moment Magnetic moments change a little in spite of the large correlation energy form the tensor force. 5/2 + 3/ O 17 F 17 O 17 F Schmidt value HO 2   NWG (f-orbit) /2 - 3/ N 15 O 15 N 15 O Schmidt value HO 2   NWG (f-orbit) A=17 A=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.