テンソル力を取り入れた平均場模型（と殻模型）による研究 Satoru Sugimoto Kyoto University

Introduction The correlations induced by the tensor force (tensor correlation) are important for structure of nuclei 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 11Li (Myo et al.) The relation to the ls splitting in 5He (Myo et al. PTP 113 (2005) 763)

Ostuka, PRL 95 232502 (2005)

テンソル力のmonopole相互作用によるESPEの変化（Ostuka, PRL 95 232502 (2005)） Ca (Z=20) n0f7/2 hole Ni (Z=28) n0g9/2 Particle 励起スペクトラム（集団性）の変化の可能性？　　 N=51 p0g9/2 Particle Sb(Z=51) n0h11/2 Particle With -0.3DN (MeV) 左端のESPEには実験データを用いてテンソル力のMonopole相互作用による変化をプロットしている。 テンソル力以外のMonopole相互作用の効果はすべての軌道に対してほぼ同じ。 実験データの傾向をよく再現。

Effect of the tensor force on the spin-orbit splitting for spin-unsaturated nuclei cases DE1g(n) DE1h(n) DE1g(p) full potential 0.76 1.60 0.99 no spin-orbit, odd force -4.10 -4.28 -3.83 no tensor even, force 1.48 2.26 1.50 no tensor forces 3.47 4.39 3.53 no non-central forces -1.39 -1.49 -1.31 neglecting contributions from the n1i13/2 and p1h11/2 levels 5.18 5.91 5.08 0.00 central all from the spin-unsaturated the n1i13/2 and p1h11/2 levels tensor, odd -1.99 -2.13 -2.03 tensor, even -0.72 -0.66 -0.49 total tensor -2.71 -2.79 -2.52 spin-orbit, odd from spin saturated levels spin-orbit, odd from spin unsaturated levels -0.32 -0.03 -0.24 total spin-orbit odd 4.86 5.88 4.84 テンソル力を含んだ相互作用を用いたHF計算。 spin-unsaturated核で通常のHF計算を行うとテンソル力はls splittingを狭める方向に働く。 Tarbutton et al., NPA 120 (1968) 1.

Effect of the tensor force on the spin-orbit splitting for spin-unsaturated nuclei 2 nucleus DE1p(n) DE1d(n) DE1f(n) DE2p(n) 12C -1.2 16O 5.8 22O 2.1 (4.0) 28Si -2.2 40Ca 4.2 (4.3) 6.6 (6.7) 48Ca 1.0 2.7 56Ni -1.5 -0.95 68Se(1f closed) 4.6 7.3 2.0 88Sr(1f2p closed) -0.091 1.3 2.9 0.57 90Zr 0.8 (1.6) 1.6 (3.1) 3.1 (5.0) 1.5 (2.0) RHF （括弧内はテンソル力を入れない計算の場合） Tarbutton et al., NPA 120 (1968) 1. spin-saturated核ではテンソル力の影響はほとんどない。 Lopéz-Quelle et al., PRC 61 (2000) 064321.

我々のこれまでの研究 平均場的計算(Charge- and parity-projected Hartree-Fock method, Sugimoto et al. NPA 740 (2004) 77, Ogawa et al. PRC 73 (2006) 034301) 自己無撞着計算 比較的重い領域に適用できる。 2粒子ー2空孔相関の重要性 (c.f. Shimizu et al. NPA 226 (1974) 282) 幅の小さい（高い運動量を持った）単一粒子軌道の重要性 (Akaishi NPA 738 (2004) 80) 殻模型的計算(Myo et al. PTP 113 (2005) 763)、AMD計算(Doté et al. PTP 115 (2006) 1069)においても上の2つの重要性が示されている。

The correlation to be included Hartree-Fock cal. (MV1(VC)+G3RS(VT, VLS)) E K V VC VT VLS 14O -89.1 199.0 -288.1 -333.2 1.1 -8.9 16O -124.1 230.0 -354.1 -418.1 0.0 -0.9 22O -156.2 354.2 -510.3 -579.6 1.8 -21.5 24O -163.2 375.0 -538.3 -612.2 1.7 -20.5 28O -176.4 424.4 -600.8 -691.0 0.1 -2.2 Single-particle (H-F) correlation In the simple HF calculation, the tensor correlation cannot be exploited. We need to include at least 2p-2h correlation to exploit the tensor correlation. beyond mean field model 2p-2h correlation

Charge- and parity-symmetry breaking mean field method Tensor force is mediated by the pion. Pseudo scalar (s) To exploit the pseudo scalar character of the pion, we introduce parity-mixed single particle state. (over-shell correlation) Isovector (t) 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. Toki et al., Prog. Theor. Phys. 108 (2002) 903. Sugimoto et al., Nucl. Phys. A 740 (2004) 77. Ogawa et al., Prog. Thoer. Phys. 111 (2004) 75. cf. Bleuler, Proceeding of the international school of physics “Enrico Fermi” 36 (1966) 464.

Application to O isotopes We calculated sub-shell closed oxygen isotopes, 14O, 16O, 22O, 24O and 28O. The spherical symmetry is assumed. Only the couplings between the same j states (s1/2 and p1/2, p3/2 and d3/2) are included. NN potential MV1 (PTP 64 (1980) 1608) for the central part. G3RS (PTP 39 (1968) 91) for the tensor and LS forces. The attraction part of the 3E part of the central force and the 3-body force are adjusted to reproduce the biding energy and the charge radius of 16O. The strength of the tt part of the tensor force is changed by multiplying a numerical factor, xT to take into account correlations like <s1/2 s1/2|VT|s1/2 d3/2> which is not included in the present calculation, effectively. (only <j1 j2|VT|j1 j2> type correlations are included in the CPPHF method.) The strength of the LS force is multiplied by 2.

Results for 16O xT E K V VT VLS HF 1.0 -124.1 230.0 -354.1 0.0 -0.9 CPPHF -127.1 237.1 -364.24 -11.7 -1.0 1.5 -127.6 253.9 -381.6 -38.3 By performing the parity and charge projection the potential energy from the tensor force becomes sizable value.

Wave function（１６O, ｘT=1.5) 4He case P(-)=16% P(p)=17% 4He case Opposite parity components mixed by the tensor force have narrow widths. It suggests that the tensor correlation needs high-momentum components.

Density

Charge Formfactor 16O The tensor correlation induce higher momentum component.

BE and Rm

V and K per particle d3/2 d5/2 p1/2 s1/2 p3/2 The potential energy of the tensor force behaves differently from those of the central and LS forces. It indicates that the tensor force affects the shell structure differently from the central and LS forces.

Mixing of the opposite parity components j=1/2 states are important for the tensor correlation. d5/2 does not contribute to the tensor correlation.

Summary We make a mean-field model which can treat the tensor correlation by mixing parities and charges in single-particle states. (the CPPHF method) The opposite parity components induced by the tensor force is compact in size. (high-momentum component) The CPPHF calculation with the spherical symmetry shows that in the oxygen isotopes j=1/2 states are important for the tensor correlation. The blocking effect changes the mixing probability.