Transcript 複素レンジガウス基底関数を用いた Complex Scaling Method

大坪慎一 (福大理)
福嶋義博 (福大理)
上村正康 (理研仁科セ)
肥山詠美子(理研仁科セ)
RCNP研究会「核子・ハイペロン多体系におけるクラスター現象」,
KGU関内メディアセンター, 2013年 7月26 ～ 27日.
∗ Complex Scaling Method (CSM) において,
複素レンジガウス基底関数が, 実数レンジガウス基底関数
よりも有効に機能するかを調べた.
∗ 例として, 12C の ３α-cluster resonant states を
OCM で行った.
∗ C. Kurokawa and K. Katō
• Phys. Rev. C71 (2005) 021301.
•
Nucl. Phys. A792 (2007) 82.
主に, New 0+
3
OCM : Orthogonality Condition Model ,
直交条件模型
3
𝐻 =
3
𝑡𝑖 − 𝑇𝐺 +
𝑖=1
•
•
•
•
•
𝑡𝑖
𝑇𝐺
𝑉𝛼𝛼
𝑉3𝛼
𝑉Pauli
𝑉𝛼𝛼 𝑟𝑖 + 𝑉3𝛼 𝑟1 , 𝑟2 , 𝑟3 + 𝑉Pauli
𝑖=1
: kinetic energy of i–th cluster
: kinetic energy of the center of mass
: 𝛼-𝛼 potential, Schmit-Wildermuth
: 3𝛼 potential
: pseudo potential representing the Pauli principle
between 𝛼-clusters
変換 :
𝑟𝑗 → 𝑟𝑗 𝑒 𝑖𝜃
𝑅𝑗 → 𝑅𝑗 𝑒 𝑖𝜃
scaling angle : 𝜃
𝐻 → 𝐻 𝜃 = 𝑈 𝜃 𝐻𝑈(𝜃)−1
𝑈(𝜃)𝑟𝑗 = 𝑟𝑗 𝑒 𝑖𝜃
𝐻 𝜃 −𝐸 𝜃
Jacobi 座標
𝛹 𝜃 =0
共鳴状態
𝐸𝑟𝑒𝑠
𝛤
= 𝐸𝑟 − 𝑖
2
𝐸𝑟 : resonance energy
𝛤 : decay width
scaling angle : 0 ≤ 𝜃 <
𝜋
4
C. Kurokawa and K. Katō, Phys. Rev. C71, 021301(2005)
scaling angle : 𝜃 = 16°
• by ACCC + CSM
New 0+
3
𝐸0+3
Fig.1
in Phys. Rev. C71, 021301(2005).
1.48
= 1.66 − 𝑖
2
• M. Itoh, et. al,
Phys .Rev. C84,
054308(2011).
New 0+
3
𝛤
𝐸0+3 = 𝐸𝑟 − 𝑖
2
1.45
= 1.77 − 𝑖
2
𝐻 𝜃 −𝐸 𝜃
𝛹 𝜃 = 0,
𝛾max
𝛹 𝜃 =
𝐶𝛾 𝜃 𝛹𝛾
𝛾=1
𝛾 ≔
𝑛𝑙, 𝑁𝐿, 𝐽𝑀
𝛹𝛾 = 𝛷𝛾 𝒓1 , 𝑹1 + 𝛷𝛾 𝒓2 , 𝑹2 + 𝛷𝛾 𝒓3 , 𝑹3
𝛷𝛾 𝒓𝑗 , 𝑹𝑗
= 𝜙𝑛𝑙 𝑟𝑗 𝜓𝑁𝐿 𝑅𝑗 𝑌𝑙 𝒓𝑗 ⊗ 𝑌𝐿 𝑹𝑗
𝐽𝑀
𝑙
• 𝜙𝑛𝑙 𝑟 ~ 𝑟 exp −
𝑟 2
𝑟𝑛
geometric progression :
𝑟𝑛 = 𝑟1 𝑎𝑛−1
∗ ガウス型基底関数の利点
• 短距離相関や長距離 tail の記述に適している
• ３体系以上の座標変換や行列要素の積分に適している
∗ ガウス型基底関数の不得意
• 高励起状態
• 入射エネルギーの高い散乱状態
 Complex-range Gaussian Basis Function
•
•
+𝜔
𝜙𝑛𝑙
−𝜔
𝜙𝑛𝑙
~ 𝑟 𝑙 exp − 1 + 𝑖 𝜔
𝑟 2
𝑟𝑛
~ 𝑟 𝑙 exp − 1 − 𝑖 𝜔
𝑟 2
𝑟𝑛
• 金城 隆志, 九州大学 修士論文 (2000)
• E. Hiyama, Y. Kino, and M. Kamimura,
Prog. Part. Nucl. Phys. 51 (2003) 223.
𝑛−1
𝑟
=
𝑟
𝑎
geometric progression : 𝑛
1
•
cos
𝜙𝑛𝑙
•
𝜙𝑛𝑙sin
~ 𝑟 𝑙 exp −
𝑟 2
𝑟𝑛
~ 𝑟 𝑙 exp −
𝑟 2
𝑟𝑛
cos 𝜔
𝑟 2
𝑟𝑛
sin 𝜔
𝑟 2
𝑟𝑛
l = 0, 2
𝜋 3
𝜔= , 𝜋
5 5
3
2𝑛 + 𝑙 +
ℏ𝜔
2
𝑙=0
ℏ𝜔 = 1
𝟐𝒏
Eigenvalue −𝟏. 𝟓
12
12.00000007
16
16.00000068
20
20.00000510
24
24.00003082
28
28.00016387
32
32.00056547
36
36.00439902
40
40.00477577
44
44.18678726
48
48.41380294
n = 18
New 0+
3 by C. Kurokawa and K. Katō
1.48
𝐸0+3 = 1.66 − 𝑖
2
Our result
𝐸0+3
1.68
= 0.79 − 𝑖
2
C. Kurokawa and K. Katō, Phys. Rev. C71, 021301(2005)
Analytical Continuation in the Coupling Constant (ACCC) + CSM
𝐻 → 𝐻 + 𝑉aux 𝑟1 , 𝑟2 , 𝑟3
𝑉aux 𝑟1 , 𝑟2 , 𝑟3 = 𝛿 exp − 𝜇 𝑟12 + 𝑟22 + 𝑟32
Fig.2 upper
in Phys. Rev. C71, 021301(2005).
New 0+
3 by C. Kurokawa and K. Katō
1.48
𝐸0+3 = 1.66 − 𝑖
2
Our result
𝐸0+3
1.68
= 0.79 − 𝑖
2
Present work
Kurokawa and Katō
Experimental Data
𝐽𝜋
01+
21+
0+
2
0+
3
2+
2
0+
4
2+
3
2+
4
0+
5
𝐸𝑥
0.00
4.32
8.05
8.09
9.54
11.89
12.47
15.67
21.60
𝐸𝑟
- 7.30
- 2.98
0.75
0.79
2.24
4.59
5.15
8.36
14.3
𝛤
----0.0088
1.68
1.2
1.0
1.8
4.3
1.7
𝐸𝑥
0.00
4.31
8.05
8.95
9.57
11.87
12.43
15.93
21.59
𝐸𝑟
- 7.29
- 2.98
0.76
1.66
2.28
4.58
5.14
8.64
14.3
2+
5
22.70
15.3
1.8
22.39
15.1 1.2
---
---
---
2+
6
24.70
17.4
8.0
24.89
17.6 6.0
---
---
---
𝛤
----0.0024
1.48
1.1
1.1
1.9
3.9
1.5
𝐸𝑥
𝐸𝑟
𝛤
0.00000 -7.2747
--4.443891 -2.8358
--7.65420
0.3795 8.5 × 10−6
9.04(9)
1.77
1.45(18)
9.84(6)
2.57
1.01(15)
10.56(6)
3.29
1.42(8)
11.16(5)
3.89
0.43(8)
15.44(4)
8.17
1.5(2)
-------
Present work
𝐽𝜋
𝐸𝑥
4+
12.25
4+
𝐸𝑟
Kurokawa and Katō
Experimental Data
𝛤
𝐸𝑥
𝐸𝑟
𝛤
𝐸𝑥
𝐸𝑟
𝛤
4.96
2.2
---
---
---
---
---
---
13.91
6.61
0.20
14.11
6.82
0.24
14.083
6.808
0.258
4+
18.92
11.62
8.0
---
---
---
---
---
---
4+
19.53
12.23
2.2
20.39
13.1
3.4
---
---
---
4+
24.41
17.11
6.3
---
---
---
---
---
---
C. Kurokawa and K. Katō, Nucl. Phys. A792, 82(2007).
M. Itoh, et. al, Phys .Rev. C84, 054308(2011).
F. Ajzenberg-Selobe, Nucl. Phys. A506, 1(1990),
𝐽𝜋
𝑉3𝛼
𝑉3𝛼 𝑟1 , 𝑟2 , 𝑟3 =
exp − 𝜇 𝑟12 + 𝑟22 + 𝑟32
𝜇 = 0.15 [fm−2 ]
0+
𝑉3𝛼
other
= 31.7 [MeV]
𝐽𝜋
𝑉3𝛼
= 63.0 [MeV]
4+
𝑉3𝛼
= 150.0 [MeV]
= 31.7 [MeV]
𝑱𝝅
𝐽𝜋
𝑉3𝛼
2+
𝑉3𝛼
[MeV]
𝟎+
31.7
𝟐+
63.0
𝟒+
150.0
Others
31.7
Present work
Kurokawa and Katō
𝐽𝜋
𝐸𝑥
𝐸𝑟
𝛤
𝐸𝑥
𝐸𝑟
𝛤
31−
8.79
1.49
2.1 × 10−3
8.80
1.51
2.0 × 10−3
11−
10.98
3.68
0.35
10.94
3.65
21−
12.10
4.80
0.57
11.97
41−
13.51
6.21
0.45
12−
15.40
8.10
32−
17.25
22−
Experimental Data
𝐸𝑥
𝐸𝑟
𝛤
9.641(5)
2.366
3.4 × 10−2
0.30
10.844(16)
3.569
0.315(25)
4.68
0.42
11.828(16)
4.553
0.260(25)
12.45
5.16
0.12
13.352(17)
6.077
0.375(40)
5.20
---
---
---
---
---
---
9.95
0.45
18.29
11.0
0.5
---
---
---
17.80
10.50
5.40
16.60
9.31
4.65
---
---
---
51−
18.66
11.36
0.29
18.69
11.4
0.3
---
---
---
42−
18.97
11.67
2.93
---
---
---
---
---
---
52−
19.47
12.17
6.37
---
---
---
---
---
---
33−
19.49
12.19
3.67
---
---
---
---
---
---
43−
23.15
15.85
2.34
---
---
---
---
---
---
13−
23.58
16.28
7.79
---
---
---
---
---
---
53−
23.70
16.40
13.27
---
---
---
---
---
---
∗ Complex Scaling Method (CSM) において,
複素レンジガウス基底関数も有効に機能する.
arXiv:1302.4256
Prog. Theor. Exp. Phys. (2013) 073D02.