直交条件模型を用いた16Oにおけるαクラスターガス状態の研究 Contents ・ Introduction ・ ｎα cond. w. f. ・4α ｓｙｓtem ・４αＯＣＭ ・ Summary Yasuro Funaki (RIKEN) Taiichi Yamada (Kanto Gakuin) Akihiro Tohsaki (RCNP) Hisashi Horiuchi (RCNP) Peter Schuck (IPN, Orsay) Gerd Röpke (Rostock Univ.)

12C Energy region of cluster gas states appearing Energy Nucleon 100 MeV Nucleon gas αcluster Appearing near the 3αthreshold Cluster gas ～10 MeV Condensed into the lowest orbit luquid 0 MeV Lowest energy state

a-condensation in finite nuclei ・ 12C, 02+ (E3a=0.38 MeV) : Tohsaki et al., PRL 87, 192501 (2001) a-condensate-type w.f. （Schuck-type, microscopic) Rrms=4.29 fm ⇔ Volume(02+)/V(01+)～3　（dilute) Conjecture of dilute a condensation ・ 12C, 22+ (E3a=2.6 MeV) observed at RCNP (Itoh et al.) described well by single deformed a-cond.-type w.f. Funaki et al., EPJA, (2005). similar structure to 02+ : dilute a condensation a 0s mean field Funaki et al., PRC67, (2004)

Extremely reliable solution was obtained for the Hoyle state RGM Extremely reliable solution was obtained for the Hoyle state As for the Hoyle state, both are almost equivalent (～90 %). ３αcluster condensate model 3 α clusters occupy the same Ｓ orbit

nαcondensate wave function b ： width parameter of the internal wave function of αparticle φ(α) （size of αcluster） Ｘ： center-of-mass coordinate of αparticle Ａ： anti-symmetrizing operator acting on all of constituent nucleons β → Large ： “Bose condensed state‘’ 　　C. M. motion of ｎαclusters occupy the same S-orbit exp(-2X2/B2), forming a gas-like structure. β → ０ ： （if normalized） approaching shell model w. f. Hill-Wheeler eq.

Low lying 0+ levels of 16O ０５＋state： 0２+ ：α＋１２Ｃ(０+) 04+ ：α＋１２Ｃ(2+) (MeV) Exp. ４αcond. w.f. Low lying 0+ levels of 16O 0２+ ：α＋１２Ｃ(０+) 04+ ：α＋１２Ｃ(2+) α＋ １２Ｃ　ＯＣＭ 4αＯＣＭ ０５＋state： A candidate of ４αcondensate Ｅ＝１３．５ＭｅＶ Γ＝０．８ＭｅＶ １６Ｏ（α、α’） Ｗａｋａｓａ ｅｔａｌ. (０3＋)theor : ４αcondensed state Ｅ=14.9 MeV Γ=1.5 MeV (based on R-matrix theory)

Cross sections of (α,α’) inelastic scattering to the 4αcondensed state（13.5 MeV）（preliminary） Not bad agreement between theory and experiment The observed 0+ state (13.5 MeV) can be assigned to the 4 αcondensed state By M. Takashina(YITP）

Density（1６Ｏ） Density（1２Ｃ） 16O 01+ 02+ 03+ Rr.m.s(fm). 2.50 3.12 4.19 D 0.16 0.31 0.68 12C 01+ (RGM) 02+ (RGM) Rr.m.s(fm). 2.40 (2.40) 3.83 (3.47) D 0.18 (0.18) 0.68 (0.58) Defined by En’yo-san How much ingredients are in low density region.

4αＯＣＭ（orthogonality condition model） Total wave function 4αＯＣＭ（H. O. basis） Gaussian basis 1 2 3 4 + ・・・ K-type H-type Adopted angular momentum channels

4αＯＣＭ（orthogonality condition model） Hamiltonian 0s 0p Remove Pauli forbidden states between 2a parts with the relative h.o. quanta Qrel <4 Pauli blocking operator for 2a parts Equation of motion Pauli forbidden state: h.o.w.f Cal.(from 4α threshold) Exp.(from 4α threshold) 12C(g.s.) -7.27 MeV -7.28 MeV 16O(g.s.) -14.38 MeV -14.44 MeV

Convergence of 0+states of 16O チャンネル 数 + H-type + K-type

0+states of 16O obtained by ４αOCM （preliminary） (MeV) Exp. 4αOCM Bound state approximation Possible mixture of spurious continuum states

Identification of resonances （preliminary） 0+state as a function of δ Measured from 12C+α threshold Identification of resonances　（preliminary）

Identification of resonances （preliminary） 0+state as a function of δ Measured from 4α threshold Identification of resonances　（preliminary）

0+states of 16O obtained by ４αOCM （preliminary） (MeV) Exp. 4αOCM Bound state approximation Possible mixture of spurious continuum states

Summary and future study ・ The recently observed 05+state can be assigned to the 03+state which 4αcondensate w. f. (microscopic model) gives. （Energy, width and (α、α’)cross section（preliminary）） 4αcond. w. f. may have a difficulty to represent the 12C+α structure. Analysis by using ４αOCM（orthogonality condition model） in order to describe both 12C+αand 4αgas states. ・ The number of resonances can be determined by introducing a pseudo potential. ・ 3 states with 12C+α resonance are obtained. Two of them may be considered to have 12C(0+)+α, 12C(2+)+α structures, respectively. ・Resonance state obtained near the ４αthreshold may correspond to the 4αgas state.

Rr.m.s. dependence of transition density and inel. form factor （12C） Inel. form factor (linear scale) Transition density Absolute value of form factor largely depends on the r.m.s. radius of the excited state. Possibility of measuring r.m.s radii of excited states. Corresponding r.m.s radius Rr.m.s. are shown in the parenthesis. Units are all in fm.

Rr.m.s. dependence of transition density and inel. form factor （1６Ｏ） Inel. form factor (linear scale) Transition density Similarly to the case of 12C, large Rr.m.s. dependence can be seen. monopole matrix elements 12C : M(０1＋→ ０2＋ ) =6.5 fm2 1６Ｏ : M(０1＋→ ０3＋ ) =2.5 fm2 Corresponding r.m.s radius Rr.m.s. are shown in the parenthesis. Units are all in fm.