Cluster and Density wave --- cluster structures in 28 Si and 12 C---

1. Clusterand Density wave --- cluster structures in 28Siand 12C--- Y. Kanada-En’yo (Kyoto Univ.) Y. Hidaka (RIKEN) Phys. Rev. C 84, 014313 (2011) arXiv:1104.4140

2. a a a a a a a a a a a a Two- and four-body correlations in nuclear systems Cluster structures in finite nuclei gas or geometric configurations of agcluster cores matter pn pairing Dilute 3a gas a-gas 12C* Tohsaki et al., Yamada et al., Funaki et al. K-E., a-clystal? Roepche et al. 14C*(3-2) triangle Itagaki et al., Von Oertzen et al. dineutron BCS 14,15,16C* linear chain T. Suhara and Y. K-E. BEC-BCS matsuo et al.

3. Shape coexistence and clusterstructures in 28Si • Shape coexistence and cluster structure in 28Si • What is density wave • Results of AMD for 28Si structure • Interpretation with density wave

4. Shape coexistence in 28Si 7a-cluster model (1981) AMD (2005) 28Si Molecular resonance O C a-cluster Mg α D5h symmetry of the pentagon shape Energy surface Excitation energy 5－ 0+3 3－ 0+3 0+1 6 MeV K=5－ K=3－ problate 0+1 oblate δ oblate prolate Experimental suggestions(1980) oblate, prolate, exotic shapes J body-fixed axis K quanta K

5. What is density wave(DW) ? Why DW in 28Si ? DW on the edge of the oblate state Pentagon in 28Sidue to 7a-cluster SSB from axial symmetricoblate shape to axial asymmetric shape DW in nuclear matter is a SSB(spontaneous symmetry breaking) for translational invariance i.e. transition from uniform matter tonon-uniform matter D5h symmetry constructs K=0+, K=5- bands Origin of DW: Instability of Fermi surface due to correlation Correlation between particle (k) and hole (-k) has non-zero expectation value wave number 2k periodicity (non-uniform) Other kinds of two-body correlation(condensation) are translational invariant k exciton BCS

6. 2. AMD method

7. Formulation of AMD Cluster structure Wave function det Slater det. Gaussian spatial Shell-model-like states det Complex parameterZ={ } Existence of any clusters is not apriori assumed. But if a system favors a cluster structure, such the structure automatically obtained in the energy variation.

8. Energy variation and spin-parity projection Energy Variation Energy surface frictional cooling method model space (Z plane) Simple AMD Variation after parity projection before spin pro. (VBP) Variation after spin-parity projection VAP ~ Constraint AMD & superposition AMD + GCM ~

9. 3. AMD results （without assumption of existence of cluster cores）

10. AMD results Negative-parity bands Positive parity bands oblate & prolate AMD

11. Intrinsic structure K=0+, K=5- K=3- K=3- K=0+ 28Si: pentagon constructs K=0+, K=5- bands 12C: triangle does K=0+, K=3- bands

12. Features of single-particle orbits in pentagon s-orbit Consider the pentagon 28Si as ideal 7a-cluster state with pentagon configuration det p-orbit d In d=0limit Axial asymmetry axial symmetry a-cluster develops (s) π2(p) π6(sd)π2(d+f) π4 (s) π2(p) π6(sd) π6 d+fmixing results in a pentagon orbit (s) ν2(p) ν6(sd) ν2(d+f) ν4 (s) ν2(p) ν6(sd) ν6 pentagon oblate Y2+2 Y3-3 ＋ ＋ － － － － ＋ ＋ ＋ －

13. single-particle orbits in AMD wave functions Pentagon orbits d+f mixing Triangle orbits p+d mixing 5～6% Y2+2 Y3-3 ＋ ＋ － － － － ＋ ＋ ＋ －

14. SSB in particle-hole representation axial symmetry Axial asymmetry a-cluster develops (s) π2(p) π6(sd)π2(d+f) π4 (s) π2(p) π6(sd) π6 d+fmixing results in a pentagon orbit (s) ν2(p) ν6(sd) ν2(d+f) ν4 (s) ν2(p) ν6(sd) ν6 fp assumed to be HF vacuum SSB state sd lz d+f mixing pentagon orbits Wave number 5 periodicity ! Y2+2 Y3-3 The pentagon state can be Interpreted as DW on the edge of the oblate state SSB: ＋ ＋ － － － － ＋ ＋ ＋ － 6%

15. What correlation ? in Z=N system (spin-isospin saturated) 1p-1h correlation 1p-3p correlation alpha correlation (geometric, non uniform) DW fp SSB: single-particle energy loss < correlation energy gain proton-neutron coherence is important ! sd lz 28Si 12C 20C Z=N=14 Z=6,N=14 Z=N=6 oblate No SSB in N>Z nuclei becuase there is no proton-neutron coherence. DW is suppressed SSB

16. 4. Toy model of DW - Interpretation of cluster structure in terms of DW -

17. Toy model：DW hamiltonian 1. Truncation of activeorbits particle operator hole operator 2. Assuming contact interaction d(r) and adopting a part of ph terms （omitting other two-body terms） fp sd lz

18. Approximated solution of DW hamiltonian Energy minimum solution in an approximation: determination of u,v non-zero uv indicates SSB where Coupling with condensations of other species of particles: For , three-species condensation for couple resulting in the factor 3. A kind of alpha(4-body) correlation.

19. For neutron-proton coherent DW (spin-isospin saturated Z=N nuclei) Correlation energy overcomes 1p-1h excitation energy cost SSB condition For neutron-proton incoherent (ex. N>Z nuclei) SSB condition Less corrlation energy Proton DW in neutron-rich nuclei: Since protons are deeply bound, energy cost for 1p-1h Increases. As a result, DW is further suppressed at least in ground states.

20. 5. Summary

21. Cluster structures in 28Si (and 12C) • K=0+ and K=5- bands suggest a pentagon shape because of 7alpha clusters. • The clusterization can be interpreted as • DW on the edge of an oblate state, .i.e., SSB of oblate state. • 1p-1h correlation of DW in Z=N nuclei is equivalent to • 1p-3p (alpha) correlation. • n-p coherence is important in DW-type SSB. • Future: • Other-type of cluster understood by DW. • Ex) Tetrahedron 4 alpha cluster : Y32-type DW.