Cluster-gas-like states and monopole excitations. T. Yamada

Transcription

1 Cluster-gas-like states and monopole excitations T. Yamada

2 Cluster-gas-like states and monopole excitations Isoscalar monopole excitations in light nuclei Cluster-gas-likes states: C, 16 O, 11 B, 13 C (maybe skipped here) Funaki s talk

3 Outline of my talk 1. Introduction 2. IS monopole strengths in light nuclei Typical case : 16 O RPA and SRPA calculations, compared with Exp OCM for 0+ states in 16 O (OCM=Orthogonality Condition Model) 3. OCM energy weighted sum rule of IS monopole transition 4. IS monopole strength function with 4 OCM, compared with Exp. 6. Emphasize two features of IS monopole excitations. 7. Dual nature of 16 O ground state. 8. Summary

4 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)

5 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

6 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 at E x 15 MeV ~20% of EWSR large M(E0) states cluster states (ii) three-bump structure : E=18, 23, 30 MeV

7 IS Monopole Strength Function of 16 O 16 O(, ) Exp. vs Cal. Relativistic RPA cal Ex [MeV} Exp: histogram Lui et al., PRC 64 (2001) discrete peaks at Ex 15 MeV three bumps at 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.

8 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

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

10 Experiment 16 O(, ) SRPA (+RPA) calculation Relativistic RPA cal. SRPA RPA E x [MeV] Gambacurta et al., PRC(2010) Exp. condition: E x > 10 MeV Exp. condition: E x > 10 MeV 10 (1) Gross structure at higher energy region (Ex > 18 MeV), i.e. 3-bump structure, is reproduced by SRPA calculation. (2) Discrete peaks at E x 15 MeV are not reproduced well. In particular, the transition to 2 nd 0+ state (E x =6.1 MeV) is not seen in SRPA (+RPA) calculation.

11 Ex [MeV] R [fm] Experiment M(E0) [fm 2 ] Γ [MeV] R [fm] 4 OCM M(E0) [fm 2 ] Γ [MeV] no data no data over 15% of total EWSR 20% of total EWSR

12 Experiment 16 O(, ) SRPA (+RPA) calculation No sharp peak! Relativistic RPA cal SRPA RPA E x [MeV] Gambacurta et al., PRC(2010) Exp. condition: E x > 10 MeV Exp. condition: E x > 10 MeV (1) Gross structure at higher energy region (Ex > 18 MeV), i.e. 3-bump structure, is reproduced by SRPA calculation. (2) Discrete peaks at E x 15 MeV are not reproduced well. In particular, the transition to 2 nd 0+ state (E x =6.1 MeV) is not seen in SRPA (+RPA) calculation.

13 RPA and SRPA calculations for 16 O (1) Gross structure at higher energy region, i.e. 3-bump structure, is reproduced by RPA and SPRA calculations. (2) Discrete peaks at E x 15 MeV are not reproduced well. In particular, the transition to 2 nd 0+ state (E x =6.1 MeV) is not seen in SRPA and RPA calculations. This peak should exit sharply at E x =6.1 MeV because Γ 1 ev.

14 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). Funaki s talk We will study the IS monopole strength function with the 4 OCM.

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

17 C(2 1+ )+ C(0 1+ )+ shell-model-like (0s) 4 (0p) Funaki et al., PRL101 (2008) Suzuki, PTP55 (1976)

18 OCM (orthogonality condition model) An approximation of RGM (resonating group method) Relative motions among c.o.m. of clusters are exactly solved under an orthogonality condition arising from Pauli-Blocking effects For example of n system, S. Sato, Prog. Thor. Phys. 40 (1968) Fermion w.f N ( F ) int rel ( F ) ( F ) ( F ).: Φ = A φ χ, ( H E) Φ = 0, Φ Φ = 1, i i= 1 rel rel rel RGM eq.: ( H EN ) χ = 0, χ N χ = 1, - cluster w.f.: Φ Approximation: OCM equation: = N χ 1 1, H E Φ = 0, Φ Φ = 1, N N ( B) rel ( B) ( B) ( B) 1 1 H N N T + V (, i j) + V (, i j, k) = T + V eff eff eff 2 3 i< j i< j< k ( ) ( ) ( ) ( ) ( + eff ) Φ B = 0 with Φ B =0, Φ B Φ B = 1 T V E u F u F : Pauli forbidden states, ( B) Φ N u F = = N int 0 or A φ u 0 i F i=1 : Symmetrized w.f. with relative (Jacobi) coordinates Easy to formulate 2+t OCM and 3+n OCM based on GEM Orthogonality condition

19 Framework of 4 OCM Total w.f. : internal w.f.s of clusters relative w.f. π u F Ψ ( J ) = 0 Hamiltonian for Ψ( J π ) : Orthogonality condition arising from Paul-blocking effects: i.e. orthogonal to u F (Pauli-forbidden states) 4 4 ( N ) (Coul) OCM = i cm 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 Hamiltonian 2 potential: phase shifts Energy spectra of C(0+,2+,4+,3-,1-) Energy of 16 O g.s.

20 How to solve 4-body problem with orthogonality condition Combing Gaussian Expansion Method (GEM) and OCM S [ ] π Ψ( J ) = f ( ν) ϕ ( ξ, ν ) ϕ ( ξ, ν ) ϕ ( ξ, ν ) ν Gaussian Gaussian Gaussian c L J π u F Ψ( J ) = 0 GEM (Gaussian Expansion Method): Numerical precision is equivalent to Faddeev-Yakubousky eq. for 4 He = 4-body problem with realistic NN forces Kamimura: PRA38(1988), Hiyama, Kino, Kamimura: PPNP51(2003), Kamada et al.: PRC64(2001) GEM+OCM: Structure study of Light Hypernuclei Hiyama, Yamada: PPNP63(2009)

21 C(2 1+ )+ C(0 1+ )+ shell-model-like (0s) 4 (0p) Funaki et al., PRL101 (2008) Suzuki, PTP55 (1976)

22 E x [MeV] Experimental data R [fm] M(E0) [fm 2 ] Γ [MeV] R [fm] 4 OCM M(E0) [fm 2 ] Γ [MeV] no data no data over 15% of total EWSR 20% of total EWSR

23 Single-particle IS-monopole strength M(E0) ~ < u f r 2 u i > ~ (3/5) R 2 = 5.4 fm 2 (using R = nuclear radius = 3 fm) Uniform-density approximation for u f (r) and u i (r) u (r) = (3/R 3 ) 1/2 for 0 r R u (r) = 0 for R < r

24 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)

25 O(, ) Relativistic RPA cal 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

26 IS monopole strength function of 16 O

27 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 (total-ewsr): n ( En E1) M(0n 01) = 16 R, m rms radius of 16 O R = 0 ( r R ) i cm 1 i= 1

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

29 4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative w.f IS monopole matrix element where we used

30 Interesting characters of IS monopole operator 16 nucleons ( ri+ 4( k 1) R ) 4( R cm ) k k R k = 1 i= 1 k = 1 = + internal part of each -cluster 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 Decompositions into Internal part and relative part play an important role in understanding monopole excitations of 16 O. 2 C

31 4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative w.f IS monopole matrix element where we used

32 4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative w.f IS monopole matrix element rms radius of 16 nucleons where we used relative part internal part of clusters

33 4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative w.f. EWSR of IS monopole transition in 4 OCM: OCM-EWSR O OCM 4 2 4( R Rcm ), k k = 1 = OCM-EWSR where R ( cm ) 0 + ri R 1 i= 1 = 16 : r.m.s radius of 16 O : r.m.s radius of -particle

34 4 OCM and OCM-EWSR Total w.f. : internal w.f.s of clusters relative w.f. EWSR of IS monopole transition in 4 OCM: OCM-EWSR O OCM 4 2 4( R Rcm ), k k = 1 = OCM-EWSR where R ( cm ) 0 + ri R 1 i= 1 = 16 : r.m.s radius of 16 O : r.m.s radius of -particle

35 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 at low-energy region of 16 O. OCM-EWSR: total EWSR: n ( En E1) M(0n 01) = 16 R m 2 Yamada et al.

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

37 Monopole Strength Function with 4 OCM + 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

38 Ex [MeV] R [fm] Experiment M(E0) [fm 2 ] Γ [MeV] R [fm] 4 OCM M(E0) [fm 2 ] Γ [MeV] no data no data over 15% of total EWSR 20% of total EWSR

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

40 IS monopole S(E) with 4 OCM Exp. vs. Cal Two features in IS monopole excitations 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 Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

41 IS monopole S(E) with 4 OCM Exp. vs. Cal Two features in IS monopole excitations 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 next discuss! Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

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

43 Ground state of 16 O Dominance of doubly-closed-shell structure: (0s) 4 (0p) (λ,µ) = 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, NPA9 (1958/59) Bayman-Bohr theorem : SU(3)[f](λµ) is equivalent to a 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 mean-filed-type degree of free dom. We call dual nature of g.s.

44 Bayman-Bohr theorem 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! relative wf (S-wave) { ( ξ } 3,3 νφ ) L ( C) = φ ( ) = N2 A u42 2 J = 0 relative wf (D-wave) Nucl. Phys. 9, 596 (1958/1959) φ c.o.m. w.f. of 16 O 3/4 32ν 2 cm Rcm = νrcm ( ) exp( 16 ) π G.S. has mean-field-type and -cluster degrees of freedom. We call dual nature of g.s. : closed shell 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+ IS monopole 4 16 ( ) 2 ( ) 2 operator = r R + r R + 3( ) i i R R i= 1 i= 5 internal parts -degree of freedom C C 2 relative part

45 Bayman-Bohr theorem 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! relative wf (S-wave) { ( ξ } 3,3 νφ ) L ( C) = φ ( ) = N2 A u42 2 J = 0 relative wf (D-wave) Nucl. Phys. 9, 596 (1958/1959) : closed shell φ c.o.m. w.f. of 16 O 3/4 32ν 2 cm Rcm = νrcm ( ) exp( 16 ) π 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+ IS monopole 4 16 ( ) 2 ( ) 2 operator = r R + r R + 3( ) i i R R i= 1 i= 5 internal parts -degree of freedom C C 2 relative part

46 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)

47 Monopole excitation to 0 + 2,3 and state 0 + 2,3 states: C(0 1+,2 1+ )+ 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 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

48 Bayman-Bohr theorem : closed shell 4-cluster wf G.S. has a 4-cluster degree of freedom 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

49 4 gas-like or + C(Hoyle) state 4 -gas-like M(E0)=1.0 fm 2 by 4 OCM : monopole operator Monopole transition O(g.s.) (0s) 4 (0p) SU(3)(00) wf (N 2-,L 2- )=(4,0) = ( ri+ 4( k 1) R ) + ( k k= 1 i= 1 k = 1 = R R (N 3-,L 3- )=(4,0) (N 2,L 2 )=(4,0) internal part 4 ) k cm relative part coherent excitation Bayman-Bohr theorem 2

50 C 3 gas-like = (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 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)

51 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 3-cluster wf Dominance of SU(3)(04) in C(g.s): confirmed by no-core shell model Dytrych et al., PRL98 (2007) : SU(3) wf 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. Yamada et al., PTP0 (2008)

52 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) Monopole C(0 1+ )+(S) shell-model-like: (0s) 4 (0p)

53 Monopole excitation to and state 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 state: C(0 1+ )+(S): higher nodal Coherent contributions from C(0+,2+)+ and C(Hoyle)+ configurations

54 S 2 -factors 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

55 S 2 -factors 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

56 IS monopole S(E) with 4 OCM Exp. vs. Cal Two features in IS monopole excitations 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 Excitation to cluster states (-cluster type) Monopole excitation of mean-field type (RPA)

57 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,. : SU(3) wf relative wf (J-wave) : cluster wf 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

58 20 Ne AMD+GCM M. Kimura, PRC69, (2004)

59 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. We predict that increasing mass number, the two features will gradually be vanishing. Because, effect of spin-orbit forces becomes stronger: some SU(3) symmetries are mixed in g.s. Goodness of nuclear SU(3) symmetry: gradually disappearing. This means that dual nature of g.s is corroding with increasing mass number. Two features of IS monopole excitation will be vanishing. Eventually only 1p1h-type collective motions are strongly excited. Hope: Systematic experiments!!

60 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 B(E2): nuclear deformation (Rainwater) IS monopole excitations have two features: 16 O (typical) (i) -cluster type: discrete peaks at 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. G.S. has mean-field and -cluster degrees of freedom : (0s) 4 (0p) = SU(3)(00) = + C w.f. by Bayman-Bohr theorem Dual nature is common to all N=Z=even light nuclei Two features of IS monopole excitations will persist in other nuclei. Hope experiments.