クラスターガス状態とモノポール励起関東学院大工山田泰一

Contents of my talk 1. -particle condensation in 4n nuclei 2. Hoyle-analog states in A 4n nuclei 3. IS monopole transitions to search for cluster states Dual nature of ground states IS monopole strength function of 16 O (typical case) : -cluster states at low energy mean-field-type states at higher energy 4. Summary

Introduction Cluster picture as well as mean-field picture is important viewpoint to understand structure of light nuclei. Structure of light nuclei Cluster states + Shell-model-like states Microscopic cluster models, AMD,... 8 Be=2, C=3, 16 O= C+ / 4, -condensate states in 4n nuclei: -gas-like state described by a product state of -particles, all with their c.o.m. in (0S) orbit. Typical states: C: Hoyle state (2 nd 0 + ) cluster Cluster gas Tohsaki, Horiuchi, Schuck, Roepke, Phy. Rev. Lett.87 (2001) Funaki et al., PRC (2003) Yamada et al., EPJA (2005), Matsumura et al.,npa(2004) ρ R 3.8 fm 3~ρ 5 0 0

cluster Cluster gas ρ R 3.8 fm 3~ρ 5 0 0 Condensed into the lowest orbit 70% 3 (0 S) shell-model like structure

Overlap 3 GCM Brink wf exact 0 state + 2 -gas-like nature of Hoyle state Kamimura et al., (1977): 3 RGM Uegaki et al, PTP57(1977): 3 GCM The 0 + 2 state has a distinct clustering and has no definite spatial configuration. Chernykh, Feldmeir et al., PRL98(2007) UCOM + FMD, 3 RGM About 55 components of the Brink-type wave functions are needed to reproduce the full RGM solution for the Hoyle state. Tohsaki,Horiuchi,Schuck,Roepke, PRL87(2001) Uegaki et al, PTP57(1977) THSR wave function: -condensate-type cluster wf 1 base THSR : THSR + Φ3 exact 0 state (3RGM) 99% 2 2 Funaki et al., PRC67, (2003)

THSR wave function Tohsaki, Horiuchi, Schuck, Roepke, PRL (2001) X 3 X 1 X G 2 Condensed into the lowest orbit (0 ) 3 S 3 2 2 Φ 3 ( B) = exp X ( ) 2 i φi i= 1 B B B 3 2 2 Φ3 ( B) exp X ( ) 2 i φi i= 1 B 4 8 b Φ 3 ( B ) (0 s ) (0 p ) B: parameter b: nucleon size parameter 3 (0 S) Funaki et al., PRC (2003) + Φ3 ( B) exact 0 state (3RGM) 0.9 THSR 2 3 RGM: Kamimura & Fukushima (1978) Deformation (B x,b y,b z ) 0.999 2 R 3.8 fm: alpha-gas-like structure

Occupation probabilities of -orbits in C 70% C(g.s) Hoyle state (λµ)=(04)-nature 0S-orbit SU(3) nature: confirmed by no-core shell model, Dytrych et al., PRL98 (2007) Single cluster density matrix: d r ' ρ ( rr, ') ϕ a( r ') = λϕ ( r ), () Yamada & Schuck EPJA26 (2005) with 3 OCM wf Matsumura & Suzuki, NPA739 (2004) Funaki et al., PRC (2010) with 3 THSR wf ρ(r,r') ϕ r : single-cluster orbital w.f. λ : occupation probability

-density distribution and -momentum distribution in C Density r 2 Momentum distribution 0 1 + (g.s) 0 2 + 0 2 + 0 1 + (g.s) Compact (0 1+ ) vs. Dilute (0 2+ ) 0 2 + state: δ function-like Similar to dilute atomic cond. Yamada & Schuck EPJA26 (2005) with 3 OCM wf

Single -orbits in C(0 + ) 0 1 + (g.s.) 0 2 + S-orbit (N =35%) S-orbit (N =70%) G-orbit (N =27%) D-orbit (N =35%) 2 ref.: r N0( a)exp( ar ) Large oscillation : strong Pauli blocking effect Compact structure SU(3) model: [f](λµ)=[444](04)-like structure Small oscillation: weak Pauli blocking effect Long tail: dilute structure Radial behavior: Gaussian form with a= 0.04 fm -2

cluster Cluster gas ρ R 3.8 fm 3~ρ 5 0 0 Condensed into the lowest orbit 70% 3 (0 S) shell-model like structure

Family of the Hoyle state No-core shell model calculation C P. Navr atil et al., PRC62 (2000)

Family of the Hoyle state Kurokawa, Kato, PRC71 (2005) Funaki et al., EPJA24 (2005) Yamada et al., EPJA26 (2005) Exp. for 2 nd 2+ state Itoh et al., NPA738, 268 (2004) C Freer et. al. PRC80, (2009) M. Gai et. al, (2011)

Precursor of a 3 condensate state: 3 -, 1 - Yamada et al., EJPA26, 185 (2005) 1 3 (0P) 3 : superfluid? vortex? 3a OCM: Occup. Prob. ~ 40% precursor of 3 condensate Condensed into the lowest orbit C 70% 3 (0 S)

Structure of 16 O 1. 4 condensate? 2. 4 linear chain state? 3. Monopole excitation (later) Funaki s talk

(0S ) 3 : 70%

Cluster-model analyses of 16 O + C OCM Y. Suzuki, PTP55 (1976), 1751 + C GCM M. Libert-Heinemann, D. Bay et al., NPA339 (1980) 4 THSR wf Not include + C configuration. Tohsaki, Horiuchi, Schuck, Roepke, PRL87 (2001) Funaki, Yamada et al., PRC82(2010) 4 OCM 4-gas, + C, shell-model-like configurations Funaki, Yamada et al., PRL101 (2008) Reproduction of lowest six 0+ states up to 4 threshold (15MeV)

16 O= C+ cluster model Even-parity Y. Suzuki, PTP55 (1976), 1751 Odd-parity 4 C+ EXP CAL EXP CAL C+ : molecular states

Funaki et al., PRL101 (2008) Suzuki, PTP55 (1976)

E x [MeV] Experimental data R [fm] M(E0) [fm 2 ] Γ [MeV] R [fm] 4 OCM M(E0) [fm 2 ] Γ [MeV] 0 + 1 0.00 2.71 2.7 0 + 2 6.05 3.55 3.0 3.9 0 + 3.1 4.03 3.1 2.4 0 + 4 13.6 no data 0.6 4.0 2.4 0.60 0 + 5 14.0 3.3 0.185 3.1 2.6 0.20 0 + 6 15.1 no data 0.166 5.6 1.0 0.14 over 15% of total EWSR 20% of total EWSR

4 OCM calculation Funaki, Yamada et al., PRL101 (2008) 4 cond. state C(1 1- )+(P) strong candidate 4 (0 S) 60% C(0 1+ )+(S): higher nodal C(2 1+ )+(D) C(0 1+ )+(S) shell-model-like: (0s) 4 (0p)

-particle condensates in heavier 4n nuclei than 16 O 1. THSR approach 2. Gross-Pitaevskii approach

THSR approach Energy curve of HW equation n condensate Tohsaki, Horiuchi, Schuck, Roepke, NPA738, 259 (2004)

Single- potential given via Gross-Pitaevskii approach n 2 ( n ) φ ( r ), 1 + U( r) φ ( r) = εφ ( r) Φ = i= 1 i 2 2m ( ) ( 1) ( ) v ( ) 1 n ( n 1)( n 2) ( ) ( ) ( ) 2 2 2 U r = n dr φ r 2 r, r + dr dr φ r φ r v 3 r, r, r 2 Pure bosons n states are assumed Coulomb barrier quasi-stable states Weakly interacting gas-like states condensates : up to about 10 ( 40 Ca) Yamada & Schuck, PRC 69, 024309 (2004)

Hoyle-analog states in A 4n nuclei ( 11 B and 13 C)

Search for Hoyle-analogue states in A 4n nuclei Definition of Hoyle-analogue state: A cluster-gas-like state described mainly by a product state of clusters, all with their c.o.m. in respective 0S orbits A=4n nuclei C=3 : (0S ) 3 70% 16 O=4 : (0S ) 4 60% A 4n nuclei, for example, 11 B=2+t : (0S ) 2 (0S t ) 13 C=3+n : (0S ) 3 (0S n ) Condensed into lowest orbit exit or not? 3 (0 S)

Purposes of present study Cluster structures in A 4N nuclei: 11 B ( 13 C) Hoyle-analogue states? Conditions of appearance? 11 B=2+t : (0S ) 2 (0S t ) 8 Be density 13 C=3+n : (0S ) 3 (0S n ) Which states of 11 B ( 13 C) correspond to the Hoyle-analogue? R.B. Wiringa et al., PRC(2000) What happens in 11 B ( 13 C) when an extra triton (neutron) is added into 8 Be=2 (Hoyle=3) state?

-t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive 11 B 8 Be(2) + triton 2 part in 11 B J π ( 11 B) p-orbit reduced: 1/2-,3/2 - s-orbit not so changed: 1/2 + Thus, one can expect cluster-gas-like states in 1/2+. 11 B 1/2 + : Hoyle-analogue exists or not? C 2 nd 0 + (Hoyle) S-orbit t S 1/2 -orbit t S-orbit

-n phase shifts -triton phase shifts -n potential (1) P-wave: attractive (2) S-wave: weakly attractive -triton potential (1) P-, F-waves: attractive (2) S-, D-waves: weakly attractive Parity-dependent potentials

3 2, + states in 11 B Results of OCM calculations Yamada and Funaki, PRC82 (2010)

Structure study of 11 B ++t OCM with H.O. basis Nishioka et al., PTP62 (1979) AMD calculation Enyo-Kanada, PRC75 (2007) No-core shell model Navratil et al., JPG36 (2009) Not well understood for even-odd states of 11 B

No-core shell model Navratil et al., JPG36(2009) Experimentally, three 3/2- states 7 Li+ 11 B energy levels (T=1/2)

No-core shell model Navratil et al., JPG36(2009) Not studied well for even-parity states Even-parity states ++t 7 Li+ 11 B energy levels (T=1/2)

Total wave function: 2 +t OCM Φ = Φ 11 (,3) L( B) Ac( ν, µ ) c ( ν, µ ) c, ν, µ + B Φ c, ν, µ (23,1) + (31,2) c( ν, µ ) c ( ν, µ ), Φ ( ν, µ ) =Φ ( ν, µ ) +Φ ( ν, µ ), (23,1) + (31,2) (23,1) (31,2) c c c 2 Φ ( νµ, ) = ϕ( r, νϕ ) ( r, µ ), ϕ ( r, ν) = N ( ν) r exp νr Y ( ), m r ( ij, k ) c ij λ k L m Hamiltonian ( ) H = T + V ( r ) + V ( r ) + V ( r ) + V ( r ) l s+ V ( r ) + V + V, Coul c LS Coul 2 2 + t ij + t ij ij + t ij 2t Pauli ( ij) = (23),(31) 2xe V r V r V r ar 2 Coul ( β ) ( ) (2) (2) 2 + x( ) = n exp n, + x ( ) = erf n r t t t OCM+GEM (Gaussian expansion method) V = lim λ u ( r ) u ( r ) + u ( r ) u ( r Pauli λ n n n ij n ij 2n+ < 4 2n+ < 3 ij= (23),(31) +, +t phase shifts 7 Li:3/2 -,, 1/2 -, 7/2 -, 1/2 - ), : Pauli blocking operator = : effective 2+t force π π V2 t η SU3( λµ ) : L Q SU3( λµ ) : L Q π L, Q( = 7,8) Equation of motion δ [ Φ E H Φ ] = 0

Single-cluster motions in 11 B(++t) Single-cluster density matrix: ρ ( rr, ') = Φ ( B) δ( r') δ( r) Φ ( B), alpha: triton: 2 11 1 ( G ) ( G) 11 J rn rn J 2 n= 1 ρ ( rr, ') = Φ ( B) δ( r r') δ( r r) Φ ( B), 11 ( G) ( G ) 11 t J 3 3 J d r ' ρ ( rr, ') ϕ a( r ') = λϕ ( r ), λ = 1 dr ' ρ ( rr t, ') ϕ t( r ') = λϕ t t( r ), λ = 1 t ϕ() r λ 11 Φ J ( B) : single-cluster orbital w.f. : occupation probability : ++t OCM w.f. Suzuki et al., PRC65 (2002); Matsumura et al., NPA739 (2004) Yamada et al., EPJA 26 (2005); Funaki, Yamada et al., PRL101 (2008) Yamada et al., PRA (2008), PRC(2009) triton r ( G) 3 r r G ( G) 1 ( G) 2 11 B=++t

Energy levels of 11 B 3/2- and 1/2+ states Even-parity states 7 Li+ + + t OCM + + t OCM

11 B = + + t OCM - 32 R=3.0 fm cluster state + 7 Li(3/2-) with S-wave But 7 Li part has a distorted +t structure E0 R=2.43 fm E0 shell-model-like state R=2.22 fm shell-model-like state B(E0,IS)=96±16 fm 4 B(E0,IS)=92 fm 4 T. Kawabata et al., PRC70 (2004) vs. Hoyle state: B(E0,IS)=0±9 fm 4

11 B = + + t OCM - 32 Overlap amplitude with + 7 Li(g.s) channel R=3.0 fm 1 st 1/2+ 3rd 3/2- E0 E0 R=2.43 fm R=2.22 fm 1st 3/2- B(E0,IS)=96±16 fm 4 B(E0,IS)=92 fm 4 T. Kawabata et al., PRC70 (2004) vs. Hoyle state: B(E0,IS)=0±9 fm 4 main configuration 3/2-3: + 7 Li(3/2-) with S-wave But 7 Li part has a distorted +t structure Yamada & Funaki, PRC82(2010)

11 B = + + t OCM Occupation probabilities 11 B: 3rd 3/2 - No concentration on a single orbit! R=3.0 fm NOT cluster condensate triton E0 E0 R=2.43 fm R=2.22 fm C: Hoyle state B(E0,IS)=96±16 fm 4 B(E0,IS)=92 fm 4 T. Kawabata et al., PRC70 (2004) vs. Hoyle state: B(E0,IS)=0±9 fm 4 70% S-orbit 3 (0 S)

11 B = + + t OCM Occupation probabilities 11 B: 3rd 3/2 - No concentration on a single orbit! 3 MeV R=3.0 fm NOT cluster condensate triton E0 E0 R=2.43 fm R=2.22 fm C: Hoyle state B(E0,IS)=96±16 fm 4 B(E0,IS)=92 fm 4 T. Kawabata et al., PRC70 (2004) vs. Hoyle state: B(E0,IS)=0±9 fm 4 70% S-orbit 3 (0 S)

+ states in 11 B

Structure of 1/2 + states 11 B 1/2 + : Hoyle-analogue exists or not? Overlap amplitudes with + 7 Li(g.s) channel 1 st 1/2+ S-orbit t S 1/2 -orbit 3rd 3/2- E 2+t [MeV] 5 0-5 + 1 + + (1 2 ) t 2+t OCM EXP Candidate? CAL ( T = ) 2+t 7 Li+ + 2 + 1 1st 3/2- R rms = 3.14[fm] vs. 3.00 fm for 3 rd 3/2-7 Li(3/2 - ) + : P-wave Parity doublet partner of 3 rd 3/2-

t Structure of 1/2 + states 11 B 1/2 + : Hoyle-analogue exists or not? S-orbit E 2+t [MeV] 5 0-5 + 1 + + (1 2 ) t ( T = ) 2+t 7 Li+ S 1/2 -orbit 2+t OCM EXP Candidate? CAL + 2 + 1 1/2 + : E x =.6 MeV (E 2a+t =1.5 MeV) 7 Li( 9 Be, 7 Li) 5 He T=1/2 Soic et al. NPA742 2004 7 Li( 7 Li, 11 B*)t T=1/2 11 B* + 7 Li Curis et al. PRC72 2006 10 Be(p,γ), 9 Be( 3 He,p) T=3/2 Goosman et al. PRC1 1970 Zwiegliski NPA389 1982 Ajenberg-Selove R rms = 5.98[fm] : dilute! R rms = 3.14[fm] vs. 3.00 fm for 3 rd 3/2-7 Li(3/2 - ) + : P-wave Parity doublet partner of 3 rd 3/2-

Complex-scaling method for 1/2 + with ++t OCM Bound state 1 st 1/2+ 2 nd 1/2+ exists as a resonant state. Resonant state 2 nd 1/2+ 7 Li(1/2-)+ ++t 7 Li(3/2-)+ E x (cal)= 11.9 MeV, Γ=190 kev E x (exp)=.56 MeV, Γ=210±20 kev

Occupation probabilities of cluster orbits 1 st 1/2 + 7 Li + structure 2 nd 1/2 + triton triton 52% 96% P-orbit S-orbit S-orbit Concentration on S-wave orbits! 2 nd 1/2 + : Hoyle-analogue state C(g.s) 70% Hoyle state S-orbit T. Yamada et al., EPJA 2005

Hoyle-analogue states (0S ) 3 : 70% (0S ) 4 : 60% (0S ) 2 (0S t ): 65%

, + states in 13 C What happens in C when an extra neutron (n) is added into the Hoyle state (3)? Hoyle-analogue states? 3+n OCM with GEM n S-orbit S 1/2 -orbit(0s ) 3 (S n ) n π + J =1/2

-n and -t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive 13 C C(Hoyle) + neutron p-orbit s-orbit 3 part in 13 C reduced: not so changed: J π ( 13 C) 1/2 -,3/2-1/2 + 11 B 8 Be(2) + triton p-orbit s-orbit 2 part in 11 B J π ( 11 B) reduced: 1/2-,3/2 - not so changed: 1/2 + Thus, one can expect cluster-gas-like states in 1/2+.

3 + n OCM for 13 C J=1/2-, 1/2+ : Yamada et al., Int.J.Mod.Phys. E17 (2008) reproduction of M(E0), Hoyle-analogue state energy spectra of 13 C R=5.4fm 1/2+ states Probability S -orbit C (Hoyle) + n 9 Be(1/2+) + 55% R=4.0fm 9 Be(3/2-,1/2-) + [larger than C (Hoyle) + n] 28% R=2.68fm C (g.s) + n neutron-halo-like 20%

Hoyle-analogue states (0S ) 3 (S n ): 55% (0S ) 3 : 70% (0S ) 4 : 60% (0S ) 2 (0S t ): 65%

IS monopole transition in light nuclei 1. Useful to search for cluster states 2. Dual nature of ground states 3. typical case: 16O IS monopole strength function: -cluster states at low energy mean-field-type states at higher energy

16 E (MeV) O x 15.1 15 4 14.44 MeV 14.0 13.5.05 ( Γ= 1.5 kev) 10 5 0 C 7.16 MV e 6.06 0 + 5 0 + 0 + 4 3 0 + 3 2 [11.26 M( E 0) ( Γ= =2.6 3.3 MeV)] fm (6%) + Monopole strengths M(E0) in 16 O and C Exp. 0 + 6 0 + 2 0 + 1 Strong candidate (4 condensate) C(1 - )+ (P) higher nodal C(2 1+ )+ (D) EWSR values 2 M( E 0) = 4.03 ± 0.09 fm (8%) C(0 1+ )+ (S) 2 M( E 0) = 3.55 ± 0.21 fm (3%) EWSR values E x (MeV) 15 10 5 0 14.1 10.3 7.66 3 7.272 MeV Single particle M(E0)~5.4 fm 2 4.44 C 4 + 1 0 + 3 2 + 2 0 + 2 EWSR value 2 + 1 2 M( E 0) = 5.4 ± 0.2 fm (16%) 0 + 1

Isoscalar Monopole Excitations in 16 O cluster states at low energy and mean-field states at higher energy

Isoscalar Monopole Excitation density fluctuation Typical example IS-GMR (heavy, medium-heavy nuclei) exhaust EWSR ~ 100% 208 Pb(, ) IS-GQR IS-GQR IS-GMR breathing mode E x =13.7 MeV Γ=3.0 MeV Harakeh et al., PRL(1977)

N=10 h.o.quanta 208 Pb: RPA E x =18.0 MeV N= collective motion with coherent 1p-1h states IS-GMR is well reproduced by RPA cal. Blaizot, Gogny, Grammticos, NPA(1976) N=14

Light Nuclei (p-, sd-shell,,,) Isoscalar monopole strengths are fragmented. For example, 16 O, ( C, 11 B, 13 C, 24 Mg, ) IS-monopole response fun. of 16 O(, ) (i) discrete peaks in E x 15 MeV ~20% of EWSR large M(E0) states cluster states (ii) three-bump structure : E=18, 23, 30 MeV

IS Monopole Strength Function of 16 O 16 O(, ) Exp. vs Cal. Relativistic RPA cal. Exp: histogram Lui et al., PRC 64 (2001) discrete peaks in Ex 15 MeV three bumps in 18, 23, 30 MeV Cal: real line Relativistic RPA Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV Exp. condition: E x > 10 MeV Not well reproduced by RRPA cal.

Non-rela. Mean-field calculations of IS-monopole strengths for 16 O (1) RPA calculation: 1p-1h excitations Blaizot et al., NPA265 (1976) (2) Second-order RPA (SRPA) calculations: : 1p1h + 2p2h i) Drozdz et al., PR197(1980): D1-force ii) Papakonstantino et al., PLB671(2009): UCOM iii) Gambacurta et al., PRC81(2010) : full SRPA calculation with Skyrme force

Experiment 16 O(, ) RPA calculation Isoscalar Relativistic RPA cal. Blaizot et al., NPA(1976) Exp. condition: E x > 10 MeV

SRPA(1p1h+2p2h) SRPA RPA RPA(1p1h) 0 10 20 30 40 50 E x [MeV] 0 10 20 30 40 E x [MeV] Gambacurta et al., PRC(2010) Papakonstantino et al., PLB(2009) Exp: histogram Lui et al., PRC 64 (2001) 10 20 30 40 Exp. condition: E x > 10 MeV Exp. condition: E x > 10 MeV 61

Ex [MeV] R [fm] Experiment M(E0) [fm 2 ] Γ [MeV] R [fm] 4 OCM M(E0) [fm 2 ] Γ [MeV] 0 + 1 0.00 2.71 2.7 0 + 2 6.05 3.55 3.0 3.9 0 + 3.1 4.03 3.1 2.4 0 + 4 13.6 no data 0.6 4.0 2.4 0.60 0 + 5 14.0 3.3 0.185 3.1 2.6 0.20 0 + 6 15.1 no data 0.166 5.6 1.0 0.14

0 + 2 SRPA(1p1h+2p2h) RPA(1p1h) No sharp peak! 0 + 2 SRPA RPA 0 10 20 30 40 50 No sharp peak! E x [MeV] 0 10 20 30 40 E x [MeV] Gambacurta et al., PRC(2010) Papakonstantino et al., PLB(2009) Exp: histogram Lui et al., PRC 64 (2001) 10 20 30 40 Exp. condition: E x > 10 MeV Exp. condition: E x > 10 MeV

RPA calculations: (1) Reproduction of 3-bump structure, but the energy positions are by about 3-5 MeV higher than the data. (2) No reproduction of discrete peaks at E x 15 MeV SRPA calculations: (1) Reproduction of 3-bump structure (2) Discrete peaks at E x 15 MeV are not reproduced well. In particular, M(E0: g.s 2 nd 0+) is not seen in SRPA cal. E x (2 nd 0+) = 6.1 MeV This peak should exit sharply at E x =6.1 MeV because Γ 1 ev.

Purposes of my talk What kind of states contribute to the discrete peaks? Recently, 4 OCM succeeded in describing the structure of the lowest six 0+ states up to 4 threshold (E x 15 MeV). The discrete peaks correspond to cluster states? We study the strength function of IS M(E0) with the 4 OCM framework. Funaki s talk

4 OCM calculation Funaki, Yamada et al., PRL101 (2008) 4 cond. state C(1 1- )+(P) strong candidate 4 (0 S) 60% C(0 1+ )+(S): higher nodal C(2 1+ )+(D) C(0 1+ )+(S) shell-model-like: (0s) 4 (0p)

0 + 3 0 + 4 16 O(, ) 0 + 5 0 + 6 Relativistic RPA cal. 10 20 30 40 Exp. condition: E x > 10 MeV Exp: histogram Lui et al., PRC 64 (2001) Cal: real line Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV

IS monopole strength function of 16 O within 4 OCM

Monopole Strength Function + 0 n : resonance state with E n - iγ n /2 1 SE ( ) = Im[ RE ( )] π 1 Γ /2 + + = M (0n 01 ) π n 2 2 n ( E En) + ( Γn / 2) 2 IS-monopole m.e.: Energy weighted sum rule (EWSR): n + + 2 2 ( En E1) M(0n 01) = 16 R, m rms radius of 16 O 2 16 2 1 R = 0 ( r R ) 0 16 + 2 + 1 i cm 1 i= 1

Energy weighted sum rule within 4 OCM (OCM-EWSR)

4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative motion S [ ] π Ψ( J ) = f ( ν) ϕ ( ξ, ν ) ϕ ( ξ, ν ) ϕ ( ξ, ν ) ν π u F Ψ ( J ) = 0 c 1 1 1 2 2 2 L 3 3 3 J : Orthogonality condition to Paul forbidden states Hamiltonian for Ψ ( J π ) 4 4 ( N ) (Coul) OCM = i cm + 2 + 2 i= 1 i< j H T T V (, i j) V (, i j) 4 + V ( i, jk, ) + V (1,2,3,4) i< j< k 3 4 : 4-body problem with orthogonality condition

4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative motion IS monopole matrix element rms radius of 16 nucleons where we used

Interesting characters of IS monopole operator 16 nucleons 4 4 4 2 2 ( ri+ 4( k 1) R ) 4( R cm ) k k R k = 1 i= 1 k = 1 = + internal part of each -cluster 4 16 2 2 ri R + ri R C i= 1 i= 5 relative parts acting on relative motions of 4 s with respect to c.o.m. of 16 O R R = ( ) ( ) + 3( ) C internal part of internal part of C relative part acting on relative motion of and C Decomposition into Internal part and relative part plays an important role in understanding monopole excitations of 16 O. 2 C

4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative motion EWSR of IS monopole transition in 4 OCM: OCM-EWSR OCM-EWSR where O OCM 4 2 4( R Rcm ), k k = 1 = R 16 1 0 + 2 1 ( cm ) 0 + ri R 1 i= 1 = 16 : r.m.s radius of 16O : r.m.s radius of -particle

4 OCM and OCM-EWSR Ratio of OCM-EWSR to total EWSR: 4 OCM shares over 60% of the total EWSR value. ( c.f. + C OCM : 31%) This is one of important reasons that 4 OCM works rather well in reproducing IS monopole transitions in low-energy region of 16 O. n 2 + + 2 2 ( En E1) M(0n 01) = 16 R m 2 Yamada et al.

IS monopole strength function S(E) within 4 OCM framework

Monopole Strength Function + 0 n : resonance state with E n - iγ n /2 1 SE ( ) = Im[ RE ( )] π 1 Γ /2 + + = M (0n 01 ) π n 2 2 n ( E En) + ( Γn / 2) 2 : calculated by 4 OCM Γ n E n 2 2 = Γ n(ocm) + (exp. resolution) : experimental energy of n-th 0+ state 50 kev

Exp. vs. Cal. IS monopole S(E) with 4 OCM 0 + 2 16 O(, ) 0 + 3 0 + 4 0 + 5 0 + 6 Relativistic RPA cal. 10 20 30 40 Exp. condition: E x > 10 MeV It is likely to exist discrete peaks on a small bump in E x < 15MeV This small bump may come from the contribution from continuum states of + C

Exp. vs. Cal. IS monopole S(E) with 4 OCM 0 + 2 16 O(, ) 0 + 3 0 + 4 0 + 5 0 + 6 Relativistic RPA cal. 10 20 30 40 Exp. condition: E x > 10 MeV Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

IS monopole S(E) with 4 OCM 0 + 2 Exp. vs. Cal. 0 + 3 0 + 4 0 + 5 0 + 6 Two features of monopole excitations They come from a dual nature of the ground state of 16 O Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

IS monopole S(E) with 4 OCM 0 + 2 Exp. vs. Cal. 0 + 3 0 + 4 0 + 5 0 + 6 Two features of monopole excitations They come from a dual nature of the ground state of 16 O next discuss! Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

Dual nature of ground state of 16 O mean-field character and -clustering character

Ground state of 16 O Dominance of doubly-closed-shell structure: (0s) 4 (0p) = SU(3)(0,0) Cluster-model calculations: 4 OCM, 4 THSR, + C OCM,... Mean-field calculations : RPA, QRPA, RRPA,... Supported by no-core shell model calculations: Dytrych et al., PRL98 (2007) Bayman-Bohr theorem : SU(3)[f](λµ) = cluster-model wf Doubly-closed-shell w.f, (0s) 4 (0p), is mathematically equivalent to a single -cluster w.f. This means that the ground state w.f. of 16 O originally has an -clustering degree of freedom together with single-particle degree of free dom. We call it duality of -clustering and mean-filed characters in g.s.

Bayman-Bohr theorem Nucl. Phys. 9, 596 (1958/1959) 1 det (0 4 s ) (0 p ) cm ( cm ) 16!!4! 16! [ φ R ] 1 { ( ξ,3 νφ ) } L 0( C) = φ ( ) = N0 A u40 3 J = 0!4! 16! { ( ξ } 3,3 νφ ) L ( C) = φ ( ) = N2 A u42 2 J = 0 φ 3/4 32ν 2 cm Rcm = νrcm ( ) exp( 16 ) π c.o.m. w.f. of 16 O G.S. has mean-field-type and -cluster degrees of freedom. Excitation of mean-field-type degree of freedom in g.s 1p1h states (3-bump structure) Excitation of -cluster degree of freedom in g.s + C cluster states: 2 nd 0+, 3 rd 0+ 4 16 2 2 ri R + ri R C i= 1 i= 5 R R = ( ) ( ) + 3( ) C internal parts relative part 2

4 OCM calculation Funaki, Yamada et al., PRL101 (2008) 4 cond. state C(1 1- )+(P) strong candidate 4 (0 S) 60% C(0 1+ )+(S): higher nodal C(2 1+ )+(D) C(0 1+ )+(S) shell-model-like: (0s) 4 (0p)

Monopole excitation to 0 + 2,3 and 0 + 6 state 0 + 2,3 states: C(0 1+,2 1+ )+(S) main configuration Bayman-Bohr theorem: (0s) 4 (0p) has an + C(0 1+,2 1+ ) degree of freedom relative motion (- C) is excited by IS monopole operator 0 + 6 state: 4-gas like structure Bayman-Bohr theorem: (0s) 4 (0p) has a 4 degree of freedom main configuration relative motion among 4 is excited by IS monopole operator

Bayman-Bohr theorem G.S. has a 4-cluster degree of freedom. 0 + 1 16 O(g.s.) (0s) 4 (0p) SU(3)(00) wf (N 2-,L 2- )=(4,0) = (N 3-,L 3- )=(4,0) (N 2,L 2 )=(4,0) Bayman-Bohr theorem

4 gas-like or + C(Hoyle) 0 + 6 state 4 -gas-like M(E0)=1.0 fm 2 by 4 OCM Monopole transition 0 + 1 16 O(g.s.) (0s) 4 (0p) SU(3)(00) wf (N 2-,L 2- )=(4,0) = (N 3-,L 3- )=(4,0) (N 2,L 2 )=(4,0) 4 4 4 2 ( ri+ 4( k 1) R ) + ( k k= 1 i= 1 k = 1 = R R internal part 4 ) relative part k coherent excitation Bayman-Bohr theorem cm 2

C 3 gas-like = 0 + 2 (Hoyle) state 3 4 = ( r R ) k = 1 i= 1 i+ 4( k 1) k 2 internal part portion of EWSR 2 M( E 0) = 5.4 ± 0.2 fm (16%) + 3 k = 1 4( R k R cm ) 2 relative part Monopole transition coherent excitation 0 + 1 C(g.s.) (0s) 4 (0p) 8 SU(3)(04) wf = (N 84,L 84 )=(4,0) (N 44,L 44 )=(4,0) Yamada et al., PTP0 (2008) Bayman-Bohr theorem Dominance of SU(3)(04) : no-core shell model by Dytrych et al., PRL98 (2007)

C Bayman-Bohr theorem 4 π + J = 0( C) = (0 s) (0 p) ;(, ) = (0,4), J = 0 φ λµ internal 4!4!4! 8 = N0 A u40( 2, ν) u40( 1,2 ν) φφφ ( ) ( ) ( )! ξ ξ 3 J = 0 Dominance of SU(3)(04) : confirmed by no-core shell model Dytrych et al., PRL98 (2007) This means that the ground state w.f. of C originally has an -clustering degree of freedom together with single-particle degree of free dom.

Monopole excitation to 0 + 5 and 0 + 4 state 0 + 5 state: C(1 1- )+(P) main configuration Bayman-Bohr theorem: (0s) 4 (0p) has no configuration of C(1-)+ Why this state is excited? Coupling with C(0+,2+)+ and C(Hoyle)+ configuration Coherent contribution from these configurations 0 + 4 state: C(0 1+ )+(S): higher nodal Coherent contributions from C(0+,2+)+ and C(Hoyle)+ configurations

Components of + C(L π ) channels in 0 + states of 16 O R=2.7fm SU(3)-nature (0s) 4 (0p) C(0 1+ )+(S) C(2 1+ )+(D) C(0 1+ )+(S) higher nodal C(1 1- )+(P) C(0 2+ )+(S): 4 gas-like

Two features of IS monopole excitations in 16 O IS monopole S(E): 4 OCM Two features in IS monpole exci. 0 + 2 0 + 3 0 + 4 0 + 5 0 + 6 Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA) 4 OCM Meanfield Fine structure E x < 16 MeV OK difficult 3-bump structure 16 < E x < 40 MeV difficult OK Origin: dual nature of G.S. of 16 O (1) -clustering degree of freedom (2) mean-field-type one (0s) 4 (0p) : SU(3)(00)= C+ : Bayman-Bohr theorem Huge model space is needed to reproduce S(E) in the whole energy region. cluster basis + collective basis

Dual nature of the ground states in C and 16 O : common to all N=Z=even light nuclei 20 Ne 20 Ne, 24 Mg, 32 S,.., 44 Ti,. Excitation of mean-field degree of freedom Excitation of -cluster degree of freedom + 16 O cluster states of K π = 0 + 4, 0 - bands + 16 O comp. = 80% for low spins: Q=10 quanta Almost pure + 16 O structures for low spins: Q=9 quanta

20 Ne AMD+GCM M. Kimura, PRC69, 044319 (2004)

Two features of IS monopole excitation Cluster states at low energy, Mean-field-type states at higher energy These features will persist in other light nuclei. Increasing mass number, effect of spin-orbit forces becomes stronger: some SU(3) symmetries are mixed. Goodness of nuclear SU(3) symmetry: gradually disappearing. Two features of IS monopole excitation will be vanishing. Eventually only 1p1h-type collective motions are strongly excited. Hope: Systematic experiments!!

Summary -condensation in C, 16 O, heavier 4n nuclei. Hoyle-analog states in 11 B, 13 C IS monopole tran. : useful to search for cluster states IS monopole excitations have two features: 16 O (typical) (i) -cluster type: discrete peaks in E x 15 MeV (ii) mean-field type: 3-bump structure (18,23,30 MeV) The origin: dual nature of the ground state of 16 O. (0s) 4 (0p) : SU(3)(00) = C+ : proved by Bayman-Bohr theorem Dual nature is common to all N=Z=even light nuclei クラスターガス状態 : 新しい物質形態拡がりと深さの探求 : 通常核 (4n 核 ), 中性子過剰核

クラスターガス状態と モノポール励起 関東学院大 工 山田泰一

クラスターガス状態とモノポール励起関東学院大工山田泰一