1. 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.)

2. 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

3. a-condensation in finite nuclei
・ 12C, 02+ (E3a=0.38 MeV) : Tohsaki et al., PRL 87, (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)

4. 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

5. 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.

6. 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)

7. 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）

8. 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
(RGM)
(RGM)
Rr.m.s(fm).
2.40 (2.40)
(3.47) D
0.18 (0.18)
(0.58)
Defined by En’yo-san
How much ingredients are in low density region.

9. 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

10. 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.)
MeV
MeV

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

12. 0+states of 16O obtained by ４αOCM （preliminary）
(MeV)
Exp.
4αOCM
Bound state
approximation
Possible mixture of spurious continuum states

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

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

15. 0+states of 16O obtained by ４αOCM （preliminary）
(MeV)
Exp.
4αOCM
Bound state
approximation
Possible mixture of spurious continuum states

16. 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.

17. 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.

18. 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.