原子核におけるα粒子凝縮状態の研究

Yasuro Funaki (Hiyama lab., RIKEN)

素核宇融合による計算基礎物理学の進展
素核宇融合
よる計算基礎物理学 進展－ミクロとマクロのかけ橋の構築－@合歓の里, 2011年12月3日－5日.

Prediction of cluster states in light nuclei (Ikeda Diagram)

The most tightly
bound light cluster
αparticle (quartet)

E/A ～7 MeV E＊～20 MeV
stiff

The most elemental
subunit in nuclear
cluster structures.
Pair (deuteron)is less bound innuclear system.

Classified according to the Threshold Rule.
K. Ikeda et al., PTP suppl. Extra num., 464 (1968).

E/A ～1 MeV

Typical mysterious 0+ states in nuclear structure problem
02+ state of 12C (Hoyle state) indispensable to 12C production in stars

Ab initio non-core shell model calculation

02+ , 22+ , 03+states : missing
02+ state: excitation energy
is not lower than 20 MeV

The typical excited states
which resist a shell model
description

P. Navratil et al., PRL 84, 5728 (2001).

First example of alpha cond. state
Clt gasCluster
αclusterRrms ≈ 3.8
3 8 ffm
Condensed into the lowest orbit（lowest config. of gas)

0 3～0 5excitation

0dissolution

Rrms ≈ 2.42 4 ffm

shell-model –likestructureFirst example of α condensate state in finite nuclei
RGM (Full 3α) vs 3αcond. (3αconfined in 0S orbit)

A: antisymmetrizer acting on 12 nucleons
RGM w.f.

M. Kamimura,, NPA 351,, 456 ((1981).
)

Exp.

RGM

Energy (MeV)

7.65

7.74

αdecay width (eV)8.7±2.7 7.7M(02+ → 01+) (fm2)

5.4±0.2 6.7

B(Ｅ２; 02+ → 21+) (e2 fm4)

13±4

5.6

First example of α condensate state in finite nuclei
RGM (Full 3α) vs 3αcond. (3αconfined in 0S orbit)
The Solution of 3αRGM eq. of motion is almost equivalent to the 3αcond. w.f
The full ３α problem gives the 3αcondensate w.f. as its solution!
RGM w.f.

M. Kamimura,, NPA 351,, 456 ((1981).
)

３αcond. w.f.

Xi : com coordinate of the i-thth 
X G : total com coordinate
Y. F. et al., PRC 67, 051306(R) (2003).

B

A: antisymmetrizer acting on 12 nucleonsFirst example of α condensate state in finite nuclei
RGM (Full 3α) vs 3αcond. (3αconfined in 0S orbit)
The Solution of 3αRGM eq. of motion is almost equivalent to the 3αcond. w.f
The full ３α problem gives the 3αcondensate w.f. as its solution!
RGM w.f.

M. Kamimura,, NPA 351,, 456 ((1981).
)

３αcond. w.f.

Xi : com coordinate of the i-thth 
X G : total com coordinate
Y. F. et al., PRC 67, 051306(R) (2003).

B

3αclustering also appears starting
without assumption of α’s by FMD & AMD
M. Chernykh, T. Neff et al., PRL 94, 032501 (2007).
Y. Kanada-En’yo, PTP 117, 655 (2007).

Electron Scattering Data (01+→ 02+)

© T. Neff

M Chernykh.
M.
Chernykh et alal., PRL 98
98, 032501 (2007)

Very nice reproduction by THSR w.f.
w f (BEC)``BEC’’ from Y.F.et al., EPJA 28, 259(2006)

Direct information of alpha condensation for the Hoyle state
via 3αOCM(Orthogonality Condition Model)

 dr '  (r , r ') (r ')    (r )Momentum distribution of α-particle
α particle

Occupation probability of single α-orbit
α orbit
12C(g.s)

Hoyle state : red line
Ground state : black lineδ function –like peakaroundd zero momentumt
T. Yamada and P. Schuck, EPJA 26, 185 (2005).

SU(3)()=(04)-nature

Hoyle state70%
0S-orbit``gas phase’’ in finite nuclei
Energyexcitation energy
14.2 MeV
7.2 MeV19.2 MeV

Infinite nuclear matter(low density) 0 /5
Crust of neutron star？

nα threshold energy

0 3～0 5 Cluster gas

dissolution

excitation

012C

Quantum liquid

16O

20Ne

Investigation
Iti ti ini
heavier nuclei
than 12C

Analogue
g to the Hoyley state in 16O?

Energy16O
4αchain？
4αbbreakupkthreshold

To describe 4αcluster statesGas～ 14 MeV

×mean field model× 12C+α cluster model○ 4α cluster model

12C+α configurations
12C+α breakup～7 MeV

0 MeVthreshold

LiquidVery important : simultaneous descriptionof lower states(g.s.、12C+α)Fully solving 4 α-particles relative motions (4αOCM)
Present: Larger model spacem (r, )  N ( )r  exp   r 2  Ym (r)GGaussiani basis
b i (GEM)

E. Hiyama et al. Prog. Part. Phys. 51, 223(2003).

Approximately taken into account(K-type) 3 (r3 , 3 )

 (r1 , 1 )(H-type) 3 (r3 , 3 )and1

 (r2 , 2 )2

 (r1 , 1 ) (r2 , 2 )2

1

Adopted angular momentum channels: [[l1,l2], l3] (l3+ll2+ll1 ≦8) (up to now, ≦5)
Including l3, l2, l1 =4

Total w.f.

OCM ( J k )   Al(1k,l)2 ,l12 ,l3 (1 , 2 , 3 )Sˆ  1 (r1 ,1 ),)  2 (r2 , 2 ) , 3 (r3 , 3 ) Jl12l Al(1k,l)2 ,l12 ,l3 (1 , 2 , 3 ) : Determined by diagonalizing HamiltonianHamiltonian of 4αＯＣＭ
H  T   V2 (rij )  V2Coul ( rij )   V3  V4  VPaulii j

Pauli blocking operator on α-α motions
VPauli  lim      un (rij ) un (rij )2 n    4 ij

Pauli forbidden state: h.o,w.f.
2-body force (folding MHN force)V2 (r )   Vn(2) exp   n(2) r 2

Energies from 4αthreshold
Exp (MeV)
Cal (MeV) Exp.
Cal.

nCoulomb force
Coul2

V

4e 2(r ) erf  ar r12V3  VV(3)222exp
(rrr  ij jk ki ) C(21+)

－4.88
4 88

－2.84
2 84

C(41+)

2.06

6.43

C(02+)

0 70
0.70

0 38
0.38

 =0.15 fm

O(g.s.) －14.2－14.44-2

Phenomenological 4-body force (repulsive)

V4  V (4) exp    (r122  r132  r142  r232  r242  r342 ) V (4)  12000 MeV,  =0.15 fm-21212i j k 87.5 MeV,－7.2812Phenomenological 3-body force (repulsive)(3)

C(g.s.) －7.32

16V3 , V4 7V2
100

0+spectra, rms radii, monopole matrix elements

E x (MeV)

4α cond. state4

151212C(1-)+α(P)C(0+)+α(S) (higher nodal)
12C(2+)+α(D)10  1 2 C (0 1+ )12C(0+)+α(S)504+ state: T. Wakasa, Y. F. et al.,
PLB 653, 173 (2007).
Y. F. et al., PRL101, 081502 (2008).

0Exp.

４αOCM

α+12C OCM
by Y. Suzuki

0+spectra, rms radii, monopole matrix elements

Large monopole matrix element canbe the evidence of cluster states.
T. Yamada, Y. F. et al., PTP120, 1139 (2008).

Experimental
data
M(E0)
[f 2]
[fm
[M V]
[MeV]

4 OCM
R
[f ]
[fm]

M(E0)
[f 2]
[fm
[M V]
[MeV]

Ex
[M V]
[MeV]

R
[f ]
[fm]

0+1

0.00

2.71

0+2

6.05

3.55

3.0

3.9

0+3

12.1

4.03

3.1

2.4

0+4

13.6

no data

0.6

4.0

2.4

0.60

0+5

14 0
14.0

33
3.30 185
0.185

31
3.1

26
2.6

0 20
0.20

0+615.1no data

0.166

5.6

1.0

0.14

2.7

over 15%of total EWSR

20%of total EWSR

S-factor : 12C+α and 8Be+8Be components for the 06+ state
4!   r  r '12
ˆ
r'()
(C())
(0r Y IL,J 0 (r)  r YI
 OCM k )OCM
L
3!1!  rr '0

S (J  0)= dr rYIL,J 0(r)2
IL2Analogue
g to the Hoyley stateα+12C(Hoyle) configuration is dominant.
12C(Hoyle): 3αcondensate

abscissa: Spin of 12C
4αcondensateMomentum distributions of the α particles

ri : coordinate of the i -th  particle
X G : coordinate of total center-of-mass

BEC in the atomic world

JILA BEC Homepage

80 nK

Nvyvx
06+ : delta-function-like peak at zero momentum
4αcondensate state character.

de Broglie w.l.



180 nK
T=1.6μK2
k2

 20 fm8

2Eexc. =J(J 1)
2MI

8

Rotational bands of Hoyle + alpha, Be+ Be ?

J=4

L=0J 1J=10.68

J=20.56

J=30.92

0.88
1.28

1.12num.: S2-factor of alpha + Hoyle

Y.F. S. Ohkubo et al., in preparation.

82Eexc. =J(J 1)
2MI

8

Rotational bands of Hoyle + alpha, Be+ Be ?

J=4L=0J 1
J=10.68

J=20.56

J=30.92

0.88
1.28

1.12

2.16+06 state : energy gain due to condensation
Rotating alphap dropped
pp at the lowest orbit.
Local condensate → complete condensate
Y.F. S. Ohkubo et al., in preparation.

num.: S2-factor of alpha + Hoyle
Momentum inertia is reduced.
Signature of superfluidity ??

Rotational band of Hoyle + alpha

Hoyle + alphaalpha, 22-bodybody scattering solutions.
solutions
Momentum inertia is reduced.
Signature of superfluidity ??

S. Ohkubo and Y. Hirabayashi, PLB 684, 127 (2010).

Summary
Investigation of loosely bound alpha gas states in heavier nuclei than 12C.

・ More α-particle condensate states very likely to exist.
Analogue state in 16O to the Hoyle state (found with 4αOCM calc.)
as the sixth 0+ state
Assigned to 15.2 MeV state?
More experimental information is needed.
needed・ Hoyleoy e analogs
a a og foro non-zerooe o spinp statesa e area e promising.
p og.
likely Hoyle + alpha rotational band
sign of condensate
Problem is continuum mixing
O goingOn
i issue:ibeyondbd boundbd statet t approximation
i ti
4-alpha CSM (Complex Scaling Method) with T2K-Tsukuba (up to 512cpu’s)Thanks
to my Collaborators
Taiichi Yamada (Kanto Gakuin Univ.)
Univ )
Hisashi Horiuchi (RCNP)
Akihiro Tohsaki (RCNP)P t SPeterSchuckh k (IPN(IPN, OOrsay))
Gerd Röpkep (Rostock(Univ.))
Masaaki Takashina (RCNP)
Tomotsugu Wakasa (Kyushu Univ.)
Wolfram von Oertzen (HMI(HMI, Berlin)
Shigeo Ohkubo (Kochi women Univ.)and for your attention.