Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 48 (2015)

Hoyle band and α condensation in 12 C
Y. Funaki∗1
The Hoyle state, the second J π = 0+ state at 7.65
MeV in 12 C, is a typical example of cluster states andhad a long history since it was predicted by F. Hoyle
and subsequently observed by Cook et al. as a key
state in the synthesis of 12 C in stellar evolution. In thelast decade, the aspects of the α condensate, in which
α clusters occupy an identical S-orbit, has attracted
great interest since the so-called Tohsaki-HoriuchiSchuck-Röpke (THSR) wave function1) , which has the
3α condensate character, was shown to be equivalent
to the Hoyle state wave function obtained by solving
the equations of the full 3α resonating group method(RGM) or generator coordinate method (GCM)2) . In
addition to the Hoyle state, the nature of the otherpositive-parity excited states were recently highlighted
by many experiments3–6) .
In this report, we investigate the structures of thepositive parity excited states above the 3α thresholdby using an extended version of the THSR wave function, which includes the 3α condensate and 8 Be + α
asymptotic conﬁgurations, with a treatment of resonances7) . In particular, we focus on the structures of3–5)
6)
the “Hoyle band” states as well as the 2+and 4+
22states, which were recently observed above the Hoyle+state, in addition to the structures of the 0+
3 and 04states, which were also quite recently identiﬁed in experiment3) .
7

experiment, which gives a support to the rotational
picture. In fact, we found the very strong E2 transi+tions inside the Hoyle band B(E2; 4+2 → 22 ) = 591
44++22e fm and B(E2; 22 → 02 ) = 295 e fm .
On the other hand, the J π = 0+ band head in experiment seems to be fragmented into the Hoyle state andthe 0+
3 state, and the calculated levels also have a similar tendency concerning the B(E2) transition from the2+2 state, the Hoyle state being located slightly below
and the 0+
3 state slightly above the J(J + 1) line. Ac+
cordingly, the transition between the 2+2 and 03 states++2is also very strong B(E2; 22 → 03 ) = 104 e fm4 .
This suggests that the Hoyle band, especially inwhat concerns the 0+ band-head state, cannot be considered a simple rotational band. This results fromthe fact that the 3α condensate structure in the Hoyle
state is not the same as the usual 8 Be(0+ ) + α rotation, in which the remaining α cluster orbits outside
the 8 Be core. Namely, in the Hoyle state, the remaining α cluster also orbits inside the 8 Be core, and independent 3α-cluster motion in an identical 0S-orbit
is realized. Consequently, the Hoyle state gains extra binding energy, and hence its energy position isconsidered to be pushed below the J(J + 1) line, asshown in Fig. 1. The same eﬀect is also argued tooccur in the study of 16 O8,9) , in which the 4α condensate is identiﬁed as a “complete condensate” and the12C(0+2 ) + α state as a “local condensate”. Because of
the existence of the “complete condensate”, a higher
0+ excited state, which is shown to have a prominent8Be(0+ ) + α structure7,10) with the remaining α cluster orbiting outside the 8 Be core, appears as a higher
nodal state, the 0+
3 state excited from the Hoyle statewith a very strong monopole transition strength calcu2+lated to be M (E0; 0+2 → 03 ) = 35 fm .

(4++))exp.
(4exp.

6th. [MeV]
E−E
E−E3th.
3αα [MeV]

5
4+(0+44 )exp.
exp.

(2++))exp.
(2exp.

3++exp.
2 (033 )exp.

References

1+(0+22 )exp.
exp.

1) A. Tohsaki et al. Phys. Rev. Lett. 87, 192501 (2001).
2) Y. Funaki et al. Phys. Rev. C 67, 051306(R) (2003).
3) M. Itoh et al. Nucl. Phys. A 738, 268 (2004); M. Itohet al. Phys. Rev. C 84, 054308 (2011).
4) M. Freer et al. Phys. Rev. C 80, 041303(R) (2009).
5) W. R. Zimmerman et al. Phys. Rev. Lett. 110, 152502(2013).
6) M. Freer et al. Phys. Rev. C 83, 034314 (2011).
7) Y. Funaki arXiv: 1408.5855.
8) S. Ohkubo and Y. Hirabayashi Phys. Lett. B 684, 127(2010).
9) Y. Funaki et al. Prog. Theor. Phys. Supple. 196, 439(2012).
10) C. Kurokawa and K. Katō Phys. Rev. C 71, 021301(2005); Nucl. Phys. A 792, 87 (2007).

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J(J+1)
J(J+1)

12122020++Fig. 1. The observed energy levels for the 0+
3 , 04 , and 22+ 5)+ 6)
3)states in Ref. , and the 22and 42
states are denoted by black circles and black squares, respectively.
The calculated energy levels for the five states are denoted by red diamonds.

In Fig. 1, the calculated energy levels are plotted asa function of J(J + 1), together with the experimental++data. We can observe that the 0+
2 , 22 , and 42 statesroughly follow a J(J +1) trajectory both in theory and∗1

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