Cluster and Density wave --- cluster structures in 28Si and 12C--- Y. Kanada-En’yo (Kyoto Univ.) Y. Hidaka (RIKEN) Today I will report our activities on nuclear structure studies. Phys. Rev. C 84, 014313 (2011) arXiv:1104.4140

Two- and four-body correlations in nuclear systems Cluster structures in finite nuclei gas or geometric configurations of a gcluster cores matter pn pairing Dilute 3a gas a-gas a a 12C* Tohsaki et al., Yamada et al., Funaki et al. K-E., a-clystal　? Recently, further novel cluster structure has been proposed. A gas of weakly interaction alpha particle and cluster crystalization in the excited states of 12C by Tohsaki et al. and Funaki et al. and 14C by Itagaki et al, respectively. Then, one can usually speculate a alpha cluster gas or cluster crystalization in phases of nuclear matter. Roepche et al. 14C*(3-2) a triangle Itagaki et al., Von Oertzen et al. dineutron BCS 14,15,16C* linear chain a T. Suhara and Y. K-E. BEC-BCS matsuo et al.

and cluster structures Shape coexistence and cluster structures in 28Si Shape coexistence and cluster structure in 28Si What is density wave Results of AMD for 28Si structure Interpretation with density wave

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

What is density wave(DW) ? Why DW in 28Si ? DW in nuclear matter is a SSB(spontaneous symmetry breaking) for translational invariance i.e. transition from uniform matter to non-uniform matter DW on the edge of the oblate state Pentagon in 28Si due to 7a-cluster SSB from axial symmetric oblate shape to axial asymmetric shape 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 BCS exciton

2. AMD method

Formulation of AMD det Slater det. Gaussian det Wave function Cluster structure Wave function det Slater det. Gaussian spatial Shell-model-like states det Complex parameter Z={ } 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.

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

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

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

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

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=0 limit Axial asymmetry axial symmetry a-cluster develops (s) π2(p) π6(sd) π6 (s) π2(p) π6(sd)π2(d+f) π4 d+f　mixing results in a pentagon orbit (s) ν2(p) ν6(sd) ν6 (s) ν2(p) ν6(sd) ν2(d+f) ν4 oblate pentagon Y2+2 Y3-3 ＋ ＋ － － － － ＋ ＋ ＋ －

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

SSB in particle-hole representation axial symmetry Axial asymmetry a-cluster develops (s) π2(p) π6(sd) π6 (s) π2(p) π6(sd)π2(d+f) π4 d+f　mixing results in a pentagon orbit (s) ν2(p) ν6(sd) ν6 (s) ν2(p) ν6(sd) ν2(d+f) ν4 assumed to be HF vacuum SSB state fp 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%

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

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

Toy model：DW hamiltonian 1. Truncation of active orbits 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

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.

For neutron-proton coherent DW (spin-isospin saturated Z=N nuclei) SSB condition Correlation energy overcomes 1p-1h excitation energy cost 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.

5. Summary

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.