1. Three-body cluster state in 11 B Center for Nuclear Study, University of Tokyo KAWABATA Takahiro RCNP, Osaka University H. Fujimura, H. Fujita, M. Fujiwara, K. Hara, K. Hatanaka, M. Itoh, K. Nakanishi, Y. Shimbara, A. Tamii, M. Uchida, H.P. Yoshida, M. Yosoi Department of Physics, Kyoto University S. Kishi, M. Nakamura, H. Sakaguchi, H. Takeda, S. Terashima, Y. Yasuda Department of Physics, Konan University H. Akimune Department of Physics, Kyushu University T. Wakasa Department of Physics, Osaka University Y. Fujita Yukawa Institute for Theoretical Physics, Kyoto University Y. Kanada-En’yo

2. Introduction Alpha particle clustering is an important concept in nuclear physics for light nuclei. Alpha cluster structure is expected to appear near the  -decay threshold energy. The 0  2 state at E x = 7.65 MeV in 12 C Famous 3alpha cluster state. Condensed state where three alpha particles occupy the lowest s-orbit. Dilute-gas state of alpha clusters. Similar dilute-gas-like states have been predicted in self-conjugate N = 4n nuclei. Does such a dilute cluster state exists in the other N≠4n nuclei ? T.Yamada and P. Schuck, Euro. Phys. J. A 26, 185 (2005). 12 C 0101 0202

3. GT strengths in 11 B E x (MeV) B(GT) ExperimentShell Model 0.000 (3/2  )0.345±0.0080.588 2.125 (1/2  )0.401±0.0320.782 4.445 (5/2  )0.453±0.0290.616 5.020 (3/2  )0.487±0.0290.745 8.104 (3/2  )  0.003 8.420 (5/2  )0.398±0.0310.483 GT Strengths have been measured by the ( 3 He,t) reaction. The 3/2  3 state has exotic characters. Suppressed GT strength Not predicted by the shell-model calculation Locate 100-keV below the  -decay threshold. It is interesting to study the 3/2  3 state from view of the cluster model. T. Kawabata et al., Phys. Rev. C 70, 034318 (2004).

4. Transition Strengths The IS excitation strengths should be determined by the different measurement. Nuclear transition strengths provide structural information. Alpha cluster states are mainly excited by natural-parity isoscalar transitions. Monopole (E0) and Quadrupole (E2) operators (e,e’) and  -decay measurements were extensively performed, but those are sensitive to proton parts only. IS strengths are different from the EM transition strengths in N≠Z nuclei.

5. Hadron Scattering Hadron scattering is a good probe for nuclear excitation strengths. We measured the (d,d’) reactions to investigate the cluster structure in 11 B. Simple reaction mechanism - Good linearity between d  /d  and B(ô). Selectivity for the  T = 1 and  T = 0 components. Multiple decomposition analysis is useful to separate  J . ( 3 He,t) …  T = 1 only (  T z =  1) (d,d’) …  T=0 only (  T z =0) (p,p’) …  T = 1 and  T = 0 (  T z = 0)

6. Experiment Experiment was performed at RCNP, Osaka University. 11 B(d,d’) RCNP-E207 E= 200 MeV,  lab = 0˚~25˚ - d  /d , A y, A yy B(E0; IS) and B(E2; IS) have been determined as well as B(  ).

7. Isoscalar strengths from 11 B(d,d’) Deformed potential model were used to analyze the (d,d’) data.  J  =0 +  J  =1 +  J  =2 + ExEx B(E0;IS) Exp B(  ) Exp B(E2;IS) Exp (MeV)(fm 4 ) 2.12 (1/2  1 )-------- 0.037±0.00711±2 4.44 (5/2  2 )--------??? 56±6 5.02 (3/2  2 )  9 0.035±0.0054.7±1.5 6.74 (7/2  1 )-------- 38±4 8.56 (3/2  3 ) 96±16  0.003  6 8.92 (5/2  2 )--------0.012±0.0030.4±0.3 B(  ) for 4.44 MeV was not reliably determined. 8.56 MeV is strongly excited by the monopole transition. DP model well-explains the 12 C(d,d’) result. 12 C(d,d’) is used as a reference for  J decomposition.

8. Comparison with Shell model Reasonably well-reproduced except.... M1 and GT strengths are overestimated by 20-50%. 3/2  3 state is not predicted. 3/2  3 state is inferred to be a cluster state. Measured strengths are compared with SM calculation by SFO. ExperimentSM (SFO)NCSM B(GT) B(  ) B(GT) B(  ) B(GT) 1/2  1 0.401±0.032 0.037±0.007 0.7820.0510.591 5/2  1 0.453±0.0290.6160.0320.517 3/2  2 0.487±0.029 0.035±0.005 0.7450.0470.741 3/2  3  0.003 < 0.003 5/2  2 0.398±0.031 0.012±0.003 0.4830.0250.625 JJ B(E0;IS)B(E2;IS)B(E2)B(E2;IS)B(E2) (fm4) (e 2 fm 4 )(fm 4 )(e 2 fm 4 ) 1/2  1 11±22.6±0.412.01.8 5/2  1 56±621±649.516.5 3/2  2 < 94.7±1.5  1.3 14.21.7 7/2  1 38±43.7±0.942.04.4 3/2  3 96±16  6 (9.4±0.2) 5/2  2 0.4±0.31.6±1.20.0120.014 e p eff = 1.24, e n eff = 0.23.

9. 3/2- 1/2- 5/2- 3/2- 7/2- 1 1 1 2 1 3/2- 5/2- 3 2 AMD(VAP) 1/2+ 1 5/2+ 1 3/2+ 1 5/2+ 2 7/2+ 1 1/2+ 2 3/2+ 2 7/2+ 2 11 B 5/2- 3 2  + t 3/2 － ３ shell-like 7 Li+  -like shell-like 3/2 － ２ 3/2 － １ 5/2 － 2,3 1/2 ＋ 2 1/2 ＋ 1 AMD (VAP) Calculation Y. Kanada-En’yo AMD (VAP) calculation successfully predict the 3/2  3 state with the 2  + t structure. The 5/2  2 and 5/2  3 states were described as a mixture of the SM and cluster components. Lower states have shell-like structures.

10. Comparison with AMD (VAP) AMD (VAP) successfully predicts the experimental data. E2 strength for the 5/2  2 state implies coexistence of the SM and CM wave functions. E0 and M1 strengths for the 3/2  3 states are well described by a 2  + t cluster w.f. ExperimentSM (SFO) AMD (VAP) JJ B(GT) B(  ) B(GT) B(  ) B(GT) B(  ) 1/2  1 0.401±0.0320.037±0.0070.7820.0510.430.040 5/2  1 0.453±0.0290.6160.0320.700.045 3/2  2 0.487±0.0290.035±0.0050.7450.0470.670.047 3/2  3  0.003 < 0.0030.020.002 5/2  2 0.398±0.0310.012±0.0030.4830.0250.560.039 JJ B(E0;IS)B(E2;IS)B(E2)B(E2;IS)B(E2)B(E0;IS)B(E2;IS)B(E2) (fm4) (e 2 fm 4 )(fm 4 )(e 2 fm 4 )(fm4) (e 2 fm 4 ) 1/2  1 11±22.6±0.412.01.812.32.3 5/2  1 56±621±649.516.566.519.2 3/2  2 < 94.7±1.5  1.3 14.21.772.30.02 7/2  1 38±43.7±0.942.04.434.43.6 3/2  3 96±16< 6(9.4±0.2)945.30.84 5/2  2 0.4±0.31.6±1.20.0120.0140.660.15

11. Comparison with AMD (VAP) AMD (VAP) nicely predicts the excitation strengths for the 3/2  3 state.  Evidence for the 2  + t cluster structure. Experiment AMD (VAP) JJ B(GT) B(  ) B(GT) B(  ) 3/2  3  0.003 < 0.0030.020.002 JJ B(E0;IS)B(E2;IS)B(E2)B(E0;IS)B(E0)B(E2;IS)B(E2) (fm4) (e 2 fm 4 )(fm 4 )(e 2 fm 4 )(fm 4 )(e 2 fm 4 ) 3/2  3 96±16< 6(9.4±0.2)94195.30.84 Large B(E2) value reported from the (e,e’) experiment contradicts the AMD result. E0, M1, and E2 are allowed, nevertheless, previous analysis takes only M1 and E2 into account. Re-analyze the (e,e’) data by taking E0 and M1 into account, and neglecting E2. B(E0) = 18.7  0.7 e 2 fm 4 If M n = (N/Z)M p, B(E0;IS) = 90  3 fm 4. The previous (e,e’) data do not contradict the present experimental and theoretical results. Large E0 and negligibly small M1 and E2 stregths !!

12. Analogies to the 0  2 state in 12 C The 3/2  3 state is inferred to be a 2  +t cluster state with the dilute density. Analogies between the 3/2  3 state and the 0 + 2 state in 12 C (dilute-gas- like 3  cluster state) has been observed.  Located near the  -decay threshold.  Similar monopole strengths.  Not predicted in SM calculations. Analogies suggests a dilute structure of the 3/2  3.  Large RMS radius is predicted.  r 2  1/2 = 2.5 fm  r 2  1/2 = 3.0 fm AMD (VAP) Calculation by Y. Kanada-En’yo

13. 11 B, 13 C( ,  ’) Experiment Radius of the 3/2  3 state should be experimentally determined. Analogue state for the 0 + 3 state in 12 C as well as the 0 + 2 state might exist. Similar analog states are also expected in 13 C. Recently, the 11 B and 13 C( ,  ’) reactions have been measured for further clarification. Y. Sasamoto et al.

14. Summary Isoscalar excitation strengths in 11 B are measured via the 11 B(d,d’) reaction. Experimental results are compared with the SM and AMD (VAP) calculations. –AMD (VAP) successfully predicts the 3/2  3 state with a 2  + t configuration. –Strong E0 transition for the 3/2  3 state should be an evidence of the 2  + t cluster structure. Analogous relation between 11 B and 12 C is speculated. –3/2  3 state in 11 B is inferred to be a dilute cluster state in similar to the 0 + 2 state in 12 C.

15. Dilute Character of 3/2  3 To evaluate the dilution of the 3/2  3 state, new quantity D is introduced. AMD (VAP)RGM  r 2  1/2 D D 11 B 3/2  1 2.5 0.29 11 B 3/2  2 3.0 0.42 12 C 0  1 2.5 0.21 2.40.17 12 C 0  2  0.42 3.50.57 D: Fraction of nucleon numbers in low density region with  /  0  1/5. R rms and D for 3/2  3 are... Extraordinary larger than those for the ground state, As large as those for the 0 + 2 state in 12 C.  Dilute structure of the 3/2  3 state. RGM: M. Kamimura, Nucl. Phys. A351, 456 (1981).

16. Excitation Modes in 11 B G.S. of 11 B is J  =3/2  and T=1/2. Excitation modes in 11 B are complex. To extract the IS transition strengths.... Each  J  transition must be isolated. Isoscalar and isovector transitions must be separated.

17. Macroscopic analysis for the 12 C(d,d’) reaction 12 C(d,d’) can be a reference for the  J  decomposition. 0 + and 2 + transitions are parameterized in terms of DP model. Fit the 1 + transition by the spherical Bessel function. Transition form factors in 12 C and 11 B are expected to be similar.

18. Measured Spectra ( 3 He,t) V  is strong. Spin-flip transitions are dominant. (d,d’) V 0 is strong. Non-spin-flip transitions are dominant. M1 transitions are dominant. Unpredicted by SM calculation. Non-spin-flip transition is dominant. Expected to be a cluster state. E2 transitions are dominant.

19. Proton and Neutron Quadrupole Strengths Quadruple proton and neutron transition strengths are defined by

20. B(GT) from ( 3 He,t) reactions E x (MeV) B(GT) Exp Present(p,n) 0.000 (3/2   )0.345±0.008 2.125 (1/2   )0.401±0.0320.399±0.032 4.445 (5/2   )0.453±0.029 0.961±0.060 5.020 (3/2   )0.487±0.029 8.104 (3/2   )  0.003 0.444±0.010 8.420 (5/2   )0.398±0.031 B(GT) for excited states were obtained from DWBA analysis. Present results are consistent with previous (p,n) result. Eff. Int. for DWIA were adjusted for ground-state transitions. B(GT)=0.345±0.008, B(F)=N  Z=1

21. Proton Inelastic Scattering DWIA calculation - Franey-Love NN interaction - H.O. single-particle w.f. - Strengths from (d,d’) and ( 3 He,t) No parameters adjusted. 8.92-MeV state - DWIA underestimates 50%. - Problem on linearity between B(GT) and d  /d . - Interference between IS and IV parts?? - The reason is still unclear. Describe experimental results except 8.92-MeV state.

22. Inelastic Alpha Scattering Alpha scattering is suitable to investigate molecular states. Simple reaction mechanism. Only central interaction V 0 contributes. Extract transition densities in the surface region for discrete states. Search for analog states of 0 + 3 in continuum region by means of MDA. Precise measurement of B(E2;IS) for the 5/2  2 state. Beam time is scheduled in October. M. Itoh et al.

23. Transition Strengths  -decay measurements were extensively performed, but world is not enough... M1: IV parts are dominant [(g IV s /g IS s ) 2 =28.6]. E0, E2: Sensitive to proton parts only. For light N=Z nuclei, mirror symmetry gives missing parts, but for the other nuclei..... Alpha cluster states are mainly excited by isoscalar transitions. IS transition strengths are important to study the cluster structures.