Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 49 (2016)

Excited states above the Hoyle state
Y. Funaki∗1[MeV]

In 12 C, there exists besides the Hoyle state (the second 0+ state of 12 C) a number of other α gas states
above the Hoyle state that one can qualify as excitedstates of the Hoyle state. For a description of thosestates, it is useful to adopt a generalized THSR wave
function1) , as the so-called THSR wave function2,3) isknown to be the best to describe the Hoyle state4) .
The part of the 3α THSR wave function that contains
the c.o.m. motion of the α particles contains two Jacobi coordinates ξ 1 and ξ 2 . As a natural extension of
the original THSR wave function, it is possible to associate two diﬀerent width parameters B1 , B2 with thetwo Jacobi coordinates. In this case the translationallyinvariant THSR wave function has the following form:)][(1 24 2thsr
Ψ3α = A exp −ξ −ξ φ1 φ2 φ3 . (1)
3B12 1 B22 24+
4+

4+
44+
36.0

112

568

2+
34.0
2.0

601

4+
13044+295591

390+
3104

5.1

520+
4(2+)
0+2+

60

270+

2+
20+0.73

11.7

−2.0

2+
40+
20.0

−4.0

26 1229535

2.60.97

9.50+
1−8.0

2+
5.4

6.4

2+
1−6.0

With this type of generalized THSR wave function, one
can get a much richer spectrum of 12 C. Axial symmetry
has been assumed and the four B parameters are taken
as Hill-Wheeler coordinates. In Fig. 1, the calculated
energy spectrum is shown. One can see that besides
the ground state band, there are many J π states obtained above the Hoyle state. All these states turnout to have large rms radii (3.7 ∼ 4.7 fm), and therefore can be considered as excitations of the Hoyle state.
The Hoyle state can thus be considered as the “groundstate” of a new class of excited states in 12 C. In par+ +ticular, the nature of the series of states (0+2 , 22 , 42 )++and the 03 and 04 states have recently been widely
discussed from the experimental side5–9) .
In Fig. 1, the E2 transition strengths between Jand J ± 2 states and monopole transitions between 0+
states are shown with corresponding arrows. We notethe very strong E2 transitions inside the Hoyle band,+++24B(E2; 4+2 → 22 ) = 591 e fm and B(E2; 22 → 02 )+2
4
= 295 e fm . The transition between the 22 and 0+
3+
2
4
states is also very large, B(E2; 2+2 → 03 ) = 104 e fm .
There have been attempts to interpret the Hoyle bandas a rotational band of a spinning triangle as this was
successfully done for the ground state band10) . However, the situation may not be as straightforward as+
it seems11) . Since the two transitions 2+
2 → 02 and++22 → 03 are of similar magnitude, no clear band headcan be identiﬁed. Whether they can be qualiﬁed as
members of a rotational band or of a vibrational bandor a mixture of both is an open question. One should
also realize that the 0+
3 state is strongly excited fromthe Hoyle state by monopole transition whose strengthis obtained from the extended THSR calculation to be++2
M (E0; 0+3 → 02 ) = 35 fm . Therefore, the 03 state∗1(2+)

8.0

2.4

7.81

0+
Exp.

Fig. 1. Spectrum of 12 C obtained from the extended THSR
approach in comparison with experiment.

seems to be a state where one α particle has been liftedout of the condensate to the next higher S level with anode. This is conﬁrmed by the S 2 factor, which is calculted to be close to unity for the 12 C(0+1 )+α(S) chan+nel. We can also conclude that the 2+
3 and 43 states,+
together with the 03 state, form the higher nodal rotational band, from the Reduced Width Amplitude anal++ysis. On the other hand, the 0+
4 , 24 and 44 states all
have the largest contribution from the 8 Be(2+ ) + α(D)channel, indicating that these states are built out of anα particle orbiting in a D-wave around a (correlated)
two α pair, also in a relative 0D state.
References
1) Y. Funaki et al., Prog. Part. Nucl. Phys. 82, 78 (2015).
2) A. Tohsaki et al., Phys. Rev. Lett. 87, 192501 (2001).
3) Y. Funaki et al., Prog. Theor. Phys. 108, 297 (2002).
4) Y. Funaki et al., Phys. Rev. C 67, 051306(R) (2003).
5) M. Itoh et al., Nucl. Phys. A 738, 268 (2004).
6) M. Itoh et al., Phys. Rev. C 84, 054308 (2011).
7) M. Freer et al., Phys. Rev. C 80, 041303(R) (2009).
8) W. R. Zimmerman et al., Phys. Rev. Lett. 110, 152502(2013).
9) M. Freer et al., Phys. Rev. C 83, 034314 (2011).
10) D. J. Maŕin-Lámbarri et al., Phys. Rev. Lett. 113,012502 (2014).
11) Y. Funaki, Phys. Rev. C 92, 021302 (2015).

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