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Container picture for cluster structures of 12C
Bo Zhou (周波)
Nanjing University / Hokkaido University
1
Collaboration with
H.Horiuchi(RCNP), A.Tohsaki(RCNP), Zz.Ren(Nanjing Univ.),
Y.Funaki(RIKEN)
9th Japan-China Joint Nuclear Physics Symposium (JCNP2015)
2015-11-08@RCNP
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Outline
1. Nonlocalized clustering and Container picture
4. Summary and Prospect
2
2. The ground state of 12C in the container picture
3. The excited 0+ states of 12C
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Cluster structures in light nuclei
Single-particle motion Cluster motion
Two picturesin light nuclei
R. B. Wiringa, et al., PRC 62, 014001(2000)
Monte Carlo for the ground state density of 8Be
The relative motion of clusters become a very important
freedom.
𝜶+𝜶
No-core nuclear shell model calculation
P. Navrátil, et al, Phys. Rev. Lett. 84, 5728 (2000)
+
2
+
2
+ +
3 4, , 0 ,00 2 vanish
-
4
Traditional pure cluster models：
Resonating Group Method (RGM): Full solution for relative wave
function 𝝌 𝝃𝟏, … , 𝝃𝒏−𝟏
Generator Coordinate Method (GCM): Superpose the basis (Brink)
wave functions
Orthogonality Condition Model(OCM): Approximate treatment of
antisymmetrization
Ψ = 𝒜{𝝌 𝝃𝟏, … , 𝝃𝒏−𝟏 𝜙1 …𝜙𝑛}𝐻 = 𝑇 − 𝑇𝐺 + 𝒊>𝒋
𝑨
𝑽𝒊𝒋
Nuclear Cluster Models
)( Be8 )(3C12 O)e(1620 N
Three typical cluster structures,
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New cluster wave function
Nuclear Cluster Models
𝑹𝟏 𝑹𝟏 𝑹𝟐𝑹𝟏
Three typical cluster structures,
𝚽𝐓𝐇𝐒𝐑(𝛃)= 𝐝𝟑 𝐑𝟏. . . 𝐝𝟑𝐑𝐧𝐄𝐱𝐩[−
𝐑𝟏𝟐+. . . +𝐑𝐧
𝟐
𝛃𝟐]𝚽𝐁𝐫𝐢𝐧𝐤(𝐑𝟏,. . . ,𝐑𝐧)
∝ 𝝓𝑮 𝓐{ [
𝐢=𝟏
𝐧
Exp(−𝟐 𝐗𝐢 − 𝐗𝑮
𝟐
𝐁𝟐) 𝝓(𝜶𝒊 ]}
𝐁𝟐 = b𝟐+𝟐𝛃𝟐𝛟(𝛂) ∝ 𝐞𝐱𝐩[− 𝟏≤𝐢
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New cluster wave function
Nuclear Cluster Models
)( Be8 )(3C12 O)e( 1620 N
Three typical cluster structures,
𝚽𝐓𝐇𝐒𝐑(𝛃)= 𝐝𝟑 𝐑𝟏. . . 𝐝𝟑𝐑𝐧𝐄𝐱𝐩[−
𝐑𝟏𝟐+. . . +𝐑𝐧
𝟐
𝛃𝟐]𝚽𝐁𝐫𝐢𝐧𝐤(𝐑𝟏,. . . ,𝐑𝐧)
∝ 𝝓𝑮 𝓐{ [
𝐢=𝟏
𝐧
Exp(−𝟐 𝐗𝐢 − 𝐗𝑮
𝟐
𝐁𝟐) 𝝓(𝜶𝒊 ]}
𝐁𝟐 = b𝟐+𝟐𝛃𝟐𝛟(𝛂) ∝ 𝐞𝐱𝐩[− 𝟏≤𝐢
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0 𝟎+𝟐+
𝟒+
𝟔+
𝟖+
1.63
4.25
8.78
11.95
𝟏−𝟑−
𝟓−
𝟕−
5.79
7.16
10.26
13.74
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10K
7
±
20Ne=𝜶+16O
10K
Localized Vs Nonlocalized
Inversion doublet rotational bands in 20Ne
H. Horiuchi and K. Ikeda, PTP40, 277 (1968)
𝑃𝐽𝜋𝓐{exp[−
8 𝒓 − 𝑺 2
5𝑏2]𝜙(𝛼 𝜙(16O }
Energy curves of 20Ne in the Brink model
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0 𝟎+𝟐+
𝟒+
𝟔+
𝟖+
1.63
4.25
8.78
11.95
𝟏−𝟑−
𝟓−
𝟕−
5.79
7.16
10.26
13.74
-
10K
8
±
20Ne=𝜶+16O
10K
Localized Vs Nonlocalized
Inversion doublet rotational bands in 20Ne
H. Horiuchi and K. Ikeda, PTP40, 277 (1968)
𝑃𝐽𝜋𝓐{exp[−
8 𝒓 − 𝑺 2
5 𝑏2 + 2𝜷2]𝜙(𝛼 𝜙(16O }
Energy curves of 20Ne in the Brink model
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9
𝑃𝐽𝜋𝓐{exp[−
8 𝒓 − 𝑺 2
5 𝑏2 + 2𝜷2]𝜙(𝛼 𝜙(16O }
}
B. Z,Y. Funaki,H.Horiuchi, Zz Ren et al.
PRL.110,262501(2013)
0 𝟎+𝟐+
𝟒+
𝟔+
1.33
4.25 𝟏−
𝟑−
𝟓−
4.67
7.00
-
10K
±
20Ne=𝜶+16O
10K
Inversion doublet rotational bands in 20Ne
0.9929
0.9879
0.97750.9998
0.9987
Localized Vs Nonlocalized
Energy curves of 20Ne with different width of Gaussian relative
wave functions
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Brink cluster model
20𝐍𝐞(𝜶+𝟏𝟔𝐎)
𝑹
𝟏𝟔𝐎𝜶 The clusters make the localized motion confined by the
inter-cluster distance parameter R.
Container picture
The clusters make the nonlocalized motion in acontainer whose
size is described by parameter 𝜷
𝜷
Single THSR wave function≈ Superposed Brink wave functions
𝓐{exp[−8 𝑟2
5 𝒃2 + 2𝜷2]𝜙(𝛼 𝜙(16O }
𝓐{exp[−8 (𝒓 − 𝑹 2
5𝒃2]𝜙(𝛼 𝜙(16O }
Container picture
B.Z, Y. Funaki et al., PRC89 (2014). Y. Funaki et al., PPNP82,78
(2015).
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Brink cluster model
20𝐍𝐞(𝜶+𝟏𝟔𝐎)
𝑹
𝟏𝟔𝐎𝜶 The clusters make the localized motion confined by the
inter-cluster distance parameter R.
Container picture
The clusters make the nonlocalized motion in acontainer whose
size is described by parameter 𝜷
𝜷
Single THSR wave function≈ Superposed Brink wave functions
𝓐{exp[−8 𝑟2
5 𝒃2 + 2𝜷2]𝜙(𝛼 𝜙(16O }
𝓐{exp[−8 (𝒓 − 𝑹 2
5𝒃2]𝜙(𝛼 𝜙(16O }
Container picture
B.Z, Y. Funaki et al., PRC89 (2014). Y. Funaki et al., PPNP82,78
(2015).
Y. Funaki et al., PPNP82,78 (2015).
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Ground state of 12C in the container picture
12
12C(3𝛂) | 𝜱𝒉𝒐𝒚𝒍𝒆𝐓𝐇𝐒𝐑 |𝑹𝑮𝑴/𝑮𝑪𝑴 |𝟐= 0.993
| 𝜱𝒈𝒓𝒐𝒖𝒏𝒅𝐓𝐇𝐒𝐑 |𝑹𝑮𝑴/𝑮𝑪𝑴 |𝟐= 0.93
[4] Y. Funaki et al., PRC 67, 051306(2003)
| 𝜱𝒈𝒓𝒐𝒖𝒏𝒅𝐓𝐇𝐒𝐑 |𝑹𝑮𝑴/𝑮𝑪𝑴 |𝟐= 0.993
20Ne(𝛂+ 16O)
[1] H.Horiuchi, Prog.Theor.Phys. 51,1266 (1974);
53,447(1975)(OCM)
[2] Y.Fukushima and M.Kamimura in Proceedings of the
International Conference on
Nuclear Structure (1977). M.Kamimura, Nucl.Phys.A
351,456(1981)(RGM)
[3] E.Uegaki, S.Okabe, Y.Abe and H.Tanaka, Prog.Theor.Phys.
57,1262(1977);
59,1031(1978); 62,1621(1979) (GCM)
Microscopic cluster calculations for 12C,
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13
The 2𝜶+𝜶 THSR wave function
𝜷1 = (𝛽1x = 𝛽1𝑦 , 𝛽1𝑧
𝜷2 = (𝛽2x = 𝛽2𝑦 , 𝛽2𝑧
β1β2β0
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Microscopic Hamiltonian of 12C
The Hamiltonian of 12C in this work can be written as:
14
The effective nucleon-nucleon potential part is taken a Gaussian
form, which is expressed as:
Force1: Uegaki et al. GCM, {Volkov1+M=0.575+b=1.41 fm}E. Uegaki,
et al. PTP 57, 1262, 1977.
Force2: Kamimura et al. RGM, {Volkov2+M=0.59+b=1.35 fm}M.
Kamimura, Nucl. Phys. A351, 456, 1981.
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15
Contour map of the energy surface of the ground state of 12C in
the two-parameter space β1x = β1y = β1z and β2x = β2y = β2z . Force
1 potential parameter is used.
The optimum wave function can beobtained by variational
calculations.
No improvement compared with the deformed one-beta THSR wave
function
𝛽1x= 𝛽1y = 1.5 fm
𝛽1z = 0.1 fm
𝛽2x= 𝛽2y = 0.1 fm
𝛽2z = 3.2 fm
Emin =-87.28 MeV
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16
PotEmin( 𝛃𝟎 )[3]
Emin(𝛃𝟏,𝛃𝟐)
Full solution(GCM/RGM)
GCM
( 𝛃𝟎)[3]GCM
(𝛃𝟏 ,𝛃𝟐)Squared overlap
F1 -86.09 -87.28 -87.92 [2] -87.81 -87.98 0.975
F2 -87.68 -89.05 -89.4 [1] -89.52 -89.65 0.978
Squared overlap: < Φ𝒎𝒊𝒏 𝜷𝟏 ,𝜷𝟐 Φ𝑮𝑪𝑴(𝜷𝟏 ,𝜷𝟐) > |𝟐
β1β2β0
In the container picture, 2𝜶 correlation is important for the
ground state of 12C.
The single THSR wave function is almost equivalent to the
RGM/GCM wave function.
B.Z, Y. Funaki et al, PTEP. 2014, 101D01
[1] Y.Fukushima and M.Kamimura in
Proceedings of the International Conference
on Nuclear Structure (1977). M.Kamimura,
Nucl.Phys.A 351,456(1981)(RGM)
[2] E.Uegaki, S.Okabe, Y.Abe and H.Tanaka,
PTP 57,1262(1977); 59,1031(1978);
62,1621(1979) (GCM)[3] Y. Funaki et al., PRC 67,
051306(2003)
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Rich cluster structures for the 0+ states in 12C
01+ (0)
02+ (7.6)
03+ (9.0)
04+(10.6)
M. Itoh, et al., PRC84, 054308 (2011)
0+
(10.3)
17
Γ ≈ 1.42 MeV
Γ ≈ 1.45 MeV
Γ ≈ 8.5×10−6 MeV
OCMK : C. Kurokawa and K. Kato, PRC 71, 021301(2005); NPA 792,
87 (2007).OCMo : S. Ohtsubo, Y. Fukushima, M. Kamimura, and E.
Hiyama,PTEP,2013, 073D02.
Gas-like cluster structure
+Shell-model state Compact cluster state
Broad resonance states
β1β2
-
18exp
04+
02+
03+
01+
AMD FMD RGM GCM OCM+CSMY.K-En’yo T. Neff M.Kamimura
S.OhtsuboE.Uegaki
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19
Contamination from continuum states
GCM bound-state approximation
No available for the broad resonance states !
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GCM-THSR + Radius Constraint Method
The THSR wave function,
Radius Constraint Method,
20Y. Funaki, et al., Prog. Theor.Phys.115, 115 (2006).
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We firstly choose a very large basis and then select the basis
one by one. The selected rule is that the chosen basis should give
the deepest energy for the ground state and also deepen the
energies for the 03
+ and 04+ states.
21
Cut off parameter and basis wave function problems
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22
01+ 02+ 03+ 04+
Energy(exp.MeV) 0.0 7.65 9.04 10.6
Energy(Cal. MeV)Radii (Cal. fm)
0.0 2.39
7.783.79
9.744.94
11.534.12
Ohtsubo et al. 0.0 8.05 8.09 11.89
Kurokawa et al. 0.0 8.05 8.95 11.87
M(01+ -->02+) M(01+ -->03+) M(01+ -->04+) M(02+
-->03+) M(02+ -->04+) M(03+ -->04+)
Cal. 5.00(exp. 5.4)
4.89 4.86 36.79 1.28 13.08
The calculated monopole matrix elements M(E0) in units of
e2fm4
Very large radius for 03+ state , even compared with 02
+ state
Very large monopole transition strength from 02+ state to 03
+ state
GCM results for energies and r.m.s (Rcut=6 fm)
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The orthogonal operator can be constructed as follows,
Y. Funaki, et al. Phys. Rev. C 67, 051306 (2003).
23
Single THSR wave function for the Hoyle state
One-beta 3𝜶 THSR wave function,
EGCM = -81.79 MeV Emin = -81.55 MeV
Two-beta 2𝜶+𝜶 THSR wave function,
EGCM = -81.87 MeV Emin = -81.79 MeV
The ground state and Hoyle state of 12C can be described in a
very exact way by the single 2𝜶+𝜶 THSR wave function.
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24
The single 03+ THSR wave function
Another orthogonal operator can be constructed as follows,
The contour plot for the state by the variational calculations
of β1 and β2using the constructed single 03
+ THSR wave function
the 03+ state ？
Search for the local minimumenergy point in deformed two-beta
parameter space.
Emin= -80.74 MeV
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25
The single 03+ THSR wave function
Another orthogonal operator can be constructed as follows,
Search for the local minimumenergy point in deformed two-beta
parameter space.
Emin= -80.74 MeV
01+
02+
03+
reduced width amplitude
r [fm]
r y
(r)
[fm
-1/2
]
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1. A new container picture for cluster structures was
proposed.
2. 2α correlation in the ground state of 12C is important.
3. The 03+ and 04
+ states in 12C are confirmed in the microscopic model. The
03
+ state can be considered as a breathing mode of the Hoyle
state.
Summary
263α + α cluster structure of 16O in the container picture
β1
β2
β3
Future:
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Thanks for your attention !
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