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16 O 　における 4α 凝縮状態

16 O 　における 4α 凝縮状態. Yasuro Funaki (RIKEN) Taiichi Yamada (Kanto Gakuin) Peter Schuck (IPN, Orsay, Paris-Sud Univ. ) Hisashi Horiuchi (RCNP) Akihiro Tohsaki (RCNP) Gerd R öpke (Rostock Univ.). 原子核・ハドロン物理：横断研究会 , KEK, 19 - 21 November, 2007.

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16 O 　における 4α 凝縮状態

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  1. 16O　における 4α 凝縮状態 Yasuro Funaki (RIKEN) Taiichi Yamada (Kanto Gakuin) Peter Schuck (IPN, Orsay, Paris-Sud Univ.) Hisashi Horiuchi (RCNP) Akihiro Tohsaki (RCNP) Gerd Röpke (Rostock Univ.) 原子核・ハドロン物理：横断研究会, KEK, 19 - 21 November, 2007

  2. Appearing of cluster gas state and ``BEC’’ state in finite nuclei Energy 12 nucleons break-up threshold 12C E/A ～ 8 MeV ～100 MeV Appearing near the 3αthreshold αcluster Cluster gas 3 α break-up threshold ～10 MeV Condensed into the lowest orbit E/A ～ 1 MeV Lowest configuration of gas phase luquid A. Tohsaki et al., PRL 87, 192501 (2001). 0 MeV Lowest energy state

  3. Brink’s wave function nαcondensate wave function (THSR-w.f.) （0s）4configuration around 0s Hill-Wheeler equation (β: generator coordinate, b: fixed) Calculation of matrix elements is owing to Tohsaki’s technique

  4. First example of α condensate state in finite nuclei 3αbreak-up threshold : 7.27 MeV Hoyle state (02+state in 12C (excitation energy : 7.65 MeV) ) Indicating 3α condensate character Occupation probability of α-particle orbit Microscopic approach (3αcond. model w.f.) Semi-microscopic approach (3αOCM) The Solution of 3αRGMeq. of motion, RGM ・ Occupation probability of α-particle orbit Huge 0S occupancy (＞70 %) M. Kamimura, NPA 351, 456 (1981). is almost equivalent to the 3αcond. w.f. ・ Momentum distribution 3αcond. Delta-function-like behavior T. Yamada et al., EPJA, 26, 185 (2005). Momentum distribution of α-particle Y. F et al., PRC 67, 051306(R) (2003). Hoyle state : red line Ground state : black line

  5. First attempt to explore 4α condensate state in 16O Nucleon density distribution nαcondensate model w.f. (THSR-w.f.) Low lying 0+ levels of 16O (MeV) A. Tohsaki, H. Horiuchi, P. Schuck and G. Röpke, PRL 87, 192501 (2001). n=4 case Very dilute density. 4 αcondensate state ４αcond. w.f. Exp.

  6. First attempt to explore 4α condensate state in 16O nαcondensate model w.f. (THSR-w.f.) Low lying 0+ levels of 16O (MeV) A. Tohsaki, H. Horiuchi, P. Schuck and G. Röpke, PRL 87, 192501 (2001). Nucleon density distribution n=4 case Very much dilute density. 4 αcondensate state 13.6 MeV state, T. Wakasa, E. Ihara, M. Takashina and Y. F. et al, PLB 653, 173 (2007). ４αcond. w.f. Exp.

  7. n α cond. w. f. (microscopic) 4 α cond. w.f. hardly describes these states △ ・ ・ × ・ Motivation for 4α OCM (Orthogonality Condition Model) Low lying 0+ levels of 16O (MeV) ・Need to check the existence of 4 α cond. state in larger (4α)model space. ・4αcond. w. f. may have a difficulty to represent the 12C+α structure. 12C(2+)+α 12C(0+)+α Need to solve full 4 α problem, 4 α OCM (semi microscopic) ４αcond. w.f. Exp.

  8. Gaussian basis Adopted angular momentum channels, totally 10 channels Fully solving 4 α-particles relative motions (4αOCM) Present: Larger model space 12C+α: succeeded dilute 4α: not reproduced 4αＯＣＭ（H. O. basis） K. Fukatsu and K. Kato, PTP 87, 151 (1992). Approximately taken into account 12C+α coupled channel ＯＣＭ （H. O. basis） (K-type) (H-type) Y. Suzuki, PTP 55, 1751 (1976); 56, 111 (1976). or Added in only K-type Hamiltonian Energies from 4αthreshold Correct 12C(g.s.)+αand 4α threshold energies

  9. Energy levels, rms radii, monopole matrix elements and density distribution. Low lying 0+ levels of 16O (MeV) (MeV) Large monopole matrix element can be the evidence of cluster states (Yamada, Y.F. et el., nucl-th/0703045) Density distribution ４αOCM Exp.

  10. Reduced width amplitude for the lower states 12C(01+) +α component is dominant in surface region (～5 fm). 12C(21+) +α component is dominant in surface region (～5 fm). α＋１２Ｃ(21+) structure α＋１２Ｃ(０1+) structure Consistent with the previous Suzuki and Kato results !

  11. Reduced width amplitudes of 04+ and 05+ states obtained via 4αOCM(partial waves between α and 12C contained in 16O w.f.) Defined as ・Large amplitude in surface region(～5 fm) 　 ・developed αcluster structure ・mixing of 12C(1-)+α, 12C(3-)+αand 12C(02+)+α structures 4αcond. state obtained with THSR-w.f. Very well developed α cluster structure Furthermore, corresponds to 4 αcondensate state previously obtained usingTHSR-w.f.

  12. Reduced width amplitudes of 04+ and 05+ states obtained via 4αOCM(partial waves between α and 12C contained in 16O w.f.) Defined as 4αcond. state obtained with THSR-w.f.

  13. Single particle occupancy and single particle orbit for the 04+ and 06+ states obtained with 4αOCM (Only the S orbit (L=0)) 04+ state : the largest (n=1) 0.87/4 =22 % Not 0S orbit the second largest (n=2) 0.41/4 =10 % 0S orbit 06+ state : the largest (n=1) 2.33/4 =58 % Large 0S occupancy ! Strongly suggesting the 06+ state is 4α condensate

  14. Possible assignment of the two calculations and observations Low lying 0+ levels of 16O 4α cond. state (MeV) 4α cond. state 12C(1-)+α 12C(0+)+α(higher nodal) Γ(04+)theor ～ 0.2 MeV Γ(05+)theor< 0.05 MeV (calculated based on R-matrix theory) Γ(04+ at 13.6 MeV)= 0.6 MeV Γ(05+ at14.0 MeV)= 0.17 MeV 12C(2+)+α 12C(0+)+α All 0+ states up to 4 α threshold are simultaneously reproduced ! THSR- w.f. ４αOCM Exp.

  15. Summary ・ 4αcondensate w. f. (4αTHSR-w.f.) predicts the existence of 4α condensate state. (not as the third 0+ state but as the fourth 0+ state) Analysis by using ４αOCM（orthogonality condition model） in order to describe both 12C+α, 4αgas states and others, if any, in larger model space. 4α condensate state and other cluster states are simultaneously obtained. Large 0S-occupancey, 60 % ・Two new resonance states are obtained near the ４α threshold. One has a developed αcluster structure (Rrms～3.0 fm) in which 12C(1-)+α, 12C(3-)+α and 12C(02+)+α components are mixed. The other has a very well developed αcluster structure (Rrms～4.0 fm). 12C(0+)+α(higher nodal) ⇒ corresponding to the observed 04+and 05+ states, respectively ・Successfully reproducing the well known 12C(g.s.)+α(6.05 MeV) and 12C(2+)+α(12.05 MeV) structures, For future work, ・Analyses of condensate fraction and 4αCSM are necessary for more reliable conclusion. 　・ 4αlinear chain structure (the band head is estimated at 16.7 MeV) We are able to discuss it simultaneously with the lower structures !

  16. Assignment of the 4αcondensate state to one of the observed levels Well investigated ０2＋state ( 6.05 MeV )：　α+12C(０1＋)structure ０4＋state (12.05 MeV)：　α+12C(21＋)structure On the other hand, ０5＋state ( 13.5 MeV, Γ=0.6 MeV )：　recently observed ０6＋state ( 14.0 MeV, Γ=0.17 MeV)：　large M(E0) Both can be the candidate ! Large monopole matrix element can be the evidence of cluster states (Yamada et el., nucl-th/0703045) (α,α’) cross section Γ(03+)theor=1.5 MeV (calculated based on R-matrix theory) M(E0: (01+)theor.-(03+)theor=2.5 fm2) ) T. Wakasa, E. Ihara, M. Takashina and Y. F. et al, nucl-ex/0611021 Puzzling situation !

  17. Phys. Lett. B, 29 (1969), 306. 13.6 MeV state must have a large Monopole matrix element. Similar magnitude of cross section Asymmetric shape 　　・・・　13.6 MeV state might be hidden M(E0: 01+－ 04+ (12.05 MeV) ) = 4.03±0.09 THSR-w.f. predicts the existence of only one state around the 4α threshold. . . .

  18. 0p α-α motion with Qrel < 4 should be removed. 0s Energies from 4αthreshold Hamiltonian of 4αＯＣＭ Equation of motion Pauli blocking operator on α-α motions Pauli forbidden state: h.o,w.f. 2-body force (folding MHN force) Qrel : H.O. quantum number regarding α-αmotion Coulomb force Phenomenological 3-body force (repulsive) Phenomenological 4-body force (repulsive)

  19. 0.50 0.498590 0.326865 0.164184 0.062899 1.00 0.908864 0.815898 0.572459 0.287296 1.50 0.896297 0.336547 0.322929 0.151992 2.00 0.572353 0.881969 0.703933 0.396726 2.50 0.307156 0.978650 0.000038 0.065416 3.00 0.155720 0.866247 0.651207 0.405040 3.50 0.078273 0.676844 0.899942 0.704774 4.00 0.039853 0.489846 0.980462 0.658007 4.50 0.020752 0.338195 0.968052 0.111590 5.00 0.011094 0.227118 0.897224 0.542085 5.50 0.006096 0.150310 0.793885 0.775540 6.00 0.003442 0.098902 0.677975 0.901844 6.50 0.001995 0.065086 0.563380 0.955530 7.00 0.001185 0.043008 0.458466 0.952726 7.50 0.000721 0.028610 0.367255 0.906961

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