原子核集団現象への平均場アプローチ
第４回 大振幅集団運動理論

日野原 伸生
筑波大学計算科学研究センター
原子核理論研究室

Aug. 8, 20182018年度原子核三者若手夏の学校

講義計画
p 1. イントロダクション

p 平均ポテンシャルと一粒子運動・集団運動p 対相関p 2. 平均場理論

p Hartree-Fock, Hartree-Fock-Bogoliubov
p 原子核密度汎関数理論

p 3. QRPA

p RPA, QRPA
p 線形応答理論
p 有限振幅法

p 4. 大振幅集団運動の理論

p ボソン展開法
p 生成座標法
p 時間依存Hartree-Fock-Bogoliubov
p 自己無撞着集団座標法

大振幅集団運動
p 変形相転移・遷移核

粒子数とともに形の連続的変化
球形・変形：QRPA(小振幅近似)で記述可能
遷移領域：平均場の形ゆらぎが大きい

原子核の理論(市村、松柳、坂田)より

大振幅集団運動
p 非調和振動球形核

変形核(非調和γ振動)

1フォノン状態
2フォノン状態
2フォノン状態
1フォノン状態
76Se

78Se

Lie and Holzwarth, Phys. Rev. C 12,1035(1975)

Kotolinski et al. Nucl. Phys. A 517 365 (1990)

大振幅集団運動
p 変形共存平均場解が同じ原子核で共存
p 核図表で隣の原子核と変形度が突然変わる
p 励起状態に異なる変形状態が現れる場合
p 変形状態は固有状態ではない→変形の混合

変形の混合
変形ポテンシャル量子トンネル現象による変形の混合

before mixing
4+
2+
0+

5

4+
2+
0+

after mixing
4+
4+
2+
0+

エネルギーレベルが不規則に

2+0+変形共存原子核の実験データ

p 陽子過剰核Se 同位体でのオブレート・プロレート変形共存
68Se

実験データ

理論計算：Skyrme HFB

Yamagami,et al., NPA693 (2001) 579
Fischer et al., Phys.Rev.C67 (2003), 064318.

2つの回転バンド
I(I+1)規則からの大きなずれ

集団運動の理論
小振幅の理論p QRPAあるいは線形応答理論
p 平均場の微小変化(２準粒子励起など)を取り扱う
変形共存・変形相転移などは取り扱えない

大振幅の理論

p 摂動論(QRPA+高次効果) (second QRPA,ボソン展開など)
p 非摂動論p 生成座標法(GCM)
p 時間依存平均場理論

ボソン展開法
QRPA: 2準粒子(phペア)で励起フォノンを記述

2準粒子(2フェルミオン演算子) ~ ボソン[Jˆz , Jˆ+ ] = Jˆ+[Jˆ+ , Jˆ ] = 2Jˆz
例：角運動量演算子<latexit sha1_base64="6OPuoikxFYDX14a13j7Yxh3ED5I=">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</latexit><latexit sha1_base64="DtNQ2NcSOiK6qc+T2oBmw0H6meM=">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</latexit>

ボソン演算子を導入<latexit sha1_base64="iyScgaed1h/w7slvx86aJmvLVh0=">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</latexit><latexit sha1_base64="dpsVap/iNDibBtrt1g70fGZnz9Y=">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</latexit>

Holstein-Primakoff表示prB̂|0) = 0 ボソン空間の真空[B̂, B̂ † ] = 1
Dyson表示
p

1 †
B̂ B̂
2I

(J+ )B = 2I B̂ † 1rp
1 †(J )B = 2I 1
B̂ B̂ B̂2I
(Jz )B = I + B̂ † B̂

✓

1 †
B̂ B̂
2I

(J+ )B = 2I B̂ † 1p(J )B = 2I B̂
<latexit sha1_base64="eGlWhCw5cVHrvkifI+EpQv7yc6I=">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</latexit>(Jz )B =Schwinger表示◆

I + B̂ † B̂(J+ )B =B̂ † Â(J )B =Â† B̂
1(Jz )B = (B̂ † B̂2Â† Â)<latexit sha1_base64="hQBPEbaCAFVxJQZiS31TSbh6d/w=">AAADDnichVFNS9xQFL2JVu3U6qgIQqENDsqIONyI0CIImm6kK79GBaPhJb4Zg5kkJG8GNOQH1JU7F64UXIh7XZUuuvEPdOFPKF24sNBNF95JAmpt9YXknXveOfflcE3fsUOBeCXJLa0v2to7XuZedb7u6s739C6HXj2weNnyHC9YNVnIHdvlZWELh6/6AWc10+Er5vbH5vlKgweh7blLYsfn6zVWde2KbTFBlJH/XPxkjI4YmjI1rOhbTERavKFvsmpazMSKrudIMnZfMnNfosWpYjdTKHolYFakxtF4XPxHSy1Wxh60yfCIkS9gCZOlPAZqBgqQrTkvfwE6bIIHFtShBhxcEIQdYBDSswYqIPjErUNEXEDITs45xJAjb51UnBSM2G36Vqlay1iX6mbPMHFbdItDb0BOBYbwO57iDV7iGf7AP//tFSU9mv+yQ7uZerlvdO8NLP5+1lWjXcDWnesJh0nqpzMJqMCHJItN2fyEaaa00v6N3YObxcmFoWgYj/En5TvCK/xGCd3GL+tkni8cQo4GpP49jsdgebykYkmdnyhMa9moOuANDEKR5vEepmEW5qBM915L/dJb6Z28L5/LX+SvqVSWMk8fPFjy5S06dL8A</latexit><latexit sha1_base64="RpXF4ZIDFl3niTvD/N+PdzuwuQw=">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</latexit>

演算子はエルミートではない演算子はエルミート
角運動量の交換関係を満たす 角運動量の交換関係を満たす
有限個のボソンで表現
無限個のボソンが必要(角運動量の場合は高々2I)

二種類のボソン[Â, Â† ] =1
[Â, B̂] =[Â, B̂ † ] = 0<latexit sha1_base64="bXO0HG1uSihJhXqGfRDxhZn5j/Y=">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</latexit>

ボソン空間：|0), B̂ |0), · · · (B̂ ) |0), · · · で|IM>を表現†† i<latexit sha1_base64="7sEIcmTdyRJcgcxBnO4qc+DHBcg=">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</latexit>

Schwinger表示の場合は二次元ボソン空間

ボソン展開法
ボソン[B̂µ , B̂µ† 0 ] =B̂µ |0) = 0<latexit sha1_base64="m/J464PTdO6LWNqjGq48D32ILPo=">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</latexit>µµ0<latexit sha1_base64="aAExM8s1qILfrh/d5QwSvefPZ2c=">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</latexit>[B̂µ , B̂µ0 ] = 0<latexit sha1_base64="YyZUvgtkhyeUIfkMBr4Q8XeSf8c=">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</latexit>

Belyaev-Zelevinsky型ボソン展開
交換関係が保存するようにフェルミオン対 ボソンの写像を作る
Marumori型ボソン展開
行列要素が保存するようにフェルミオン対 ボソンの写像を作る

ÔB =<latexit sha1_base64="w5XtQ0TmvOOrsHp/mXNDZ38oWQc=">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</latexit>

Dyson型ボソン展開

X
nn0

hn|ÔF |n0 i|n)(n0 |Dyson表示の一般形を用いる。
フェルミオン対としては例えば集団的なものを使う(TDA)Ô⌫† =<latexit sha1_base64="Jv8YPi4khEOOX5PRVBtDmzvDdYo=">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</latexit>

X
ph

⌫ † †
Cph
d p bhボソン展開法の応用例
Kishimoto-Tamuraの方法

Sakamoto, Phys. Rev. C 93, 034315 (2016)

生成座標法(generator coordinate method)
平均場(HFB状態)を重ね合わせて状態を作る|<latexit sha1_base64="/+RH2hoyA1dvBLDFgclQqokVxzg=">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</latexit>

↵i =

Z

dqf↵ (q)| (q)iq: 生成座標f(q): 重み関数(weight function)|Φ(q)>：GCMの基底波動関数(平均場)

生成座標とは平均場を特徴づける量| (Z)i = e<latexit sha1_base64="qcq384x5+X/yPitKGoklzdKpf7c=">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</latexit>

i

P

† †
µ<⌫ Zµ⌫ ↵µ ↵⌫

|

HFB i

(Thoulessの定理)

例えばZを生成座標とすれば任意のSlater行列式が表現できる
一般的には少数の”集団的”で重要なものに限定
例：四重極変形の期待値

q20 = h (q20 )|Q̂20 | (q20 )i<latexit sha1_base64="YsReWCW/UUgCX2+1t6goE4QrqhI=">AAACwnichVHLTttQED24lNLwCnRTqZuoEQg20ThCKkJCitouukyAABJGkW0uyYUb29g3kVInP9Af6KKrtuqi6gf0A7opWxALPqFiSaVuuujEsVQeAsay75kzc+b6aJxAyUgTnQ0ZD4YfjjwafZwZG5+YnMpOz2xEfit0RdX1lR9uOXYklPREVUutxFYQCrvpKLHpHLzq1zfbIoyk763rTiB2mnbdk3vStTVTtezKYS0uUi+3krOU7dWVsMoNOT8gF7pWw9ZxpZdk3csVK0yaa9k8FSiJ3E1gpiCPNMp+9jss7MKHixaaEPCgGSvYiPjZhglCwNwOYuZCRjKpC/SQYW2LuwR32Mwe8LfO2XbKepz3Z0aJ2uVbFL8hK3OYpVP6Shf0k77RL/p766w4mdH/lw6fzkArgtrUu6drf+5VNfnUaPxX3aFwuPtuTxp7WEq8SPYWJEzfpTuY3377/mJteXU2nqNPdM7+PtIZ/WCHXvu3+6UiVj8gwwsyr6/jJtgoFkwqmJXFfOlluqpRPMNzzPM+XqCENyijyvd+xhGOcWK8NvaNQyMatBpDqeYJroTR/QcHX6fO</latexit>

拘束付きHFB状態として|Φ(q)>を計算
生成座標自体は理論から決定できない

生成座標法
重み関数f(q)の決定：変分により決定<latexit sha1_base64="oRyqfuXqCk3R4e3qGuEGKlmiUA8=">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</latexit>Zh |Ĥ| i=0
h | i

dq 0 h (q)|Ĥ| (q 0 )if↵ (q 0 ) = E↵<latexit sha1_base64="2H+NioJmxDd6MCcanUW/uxbBR0g=">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</latexit>

Norm Kernel (overlap)<latexit sha1_base64="FacqGVVE7kQqVIDAX4gIPi2XN+g=">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</latexit>Z

dq 0 h (q)| (q 0 )if↵ (q 0 )Hill-Wheeler方程式

N (q, q 0 ) = h (q)| (q 0 )i

GCMの基底は直交していない

Hamiltonian KernelH(q, q 0 ) = h (q)|Ĥ| (q 0 )i
<latexit sha1_base64="DjV9Ai1v51u4Iy0fe4o1D7OuiFE=">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</latexit>

基底の直交化(Norm Kernelの対角化)
Z<latexit sha1_base64="AdJwP3+LBF4ownF6u5gXqn057ro=">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</latexit>

固有ベクトルX<latexit sha1_base64="ADrx7ELZM8JljHdDSxQcSSPN7bg=">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</latexit>N<latexit sha1_base64="FKiTZ6xzqhYetkoQ0H0XEPADyfY=">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</latexit>

1/2

0(q, q ) ⌘Xkdq 0 N (q, q 0 )uk (q 0 ) = nk uk (q)uk (q)u⇤k (q 0 ) =

(q

0

q)

k

uk (q)

Znk:ノルムの固有値
dqu⇤k (q)uk0 (q) =kk0<latexit sha1_base64="DonwWlJwdHWf43kBRO7L2aOfLqc=">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</latexit>pnk u⇤k (q 0 )N<latexit sha1_base64="2UfTY4Ud/vIlFBk3f5pIx8cWHpU=">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</latexit>

1/2

0(q, q ) ⌘Xk1
uk (q) p u⇤k (q 0 )nk生成座標法Hill-Wheeler方程式

Z<latexit sha1_base64="sq6KqFu0Cvq9Tlz1BAfRVJn0w8k=">AAACynicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQu4xmTmlSikFKorVMckJ+YoeNRqFOoUqmsqpMVna4BoWwXX+GwFNFV+aKriBZQN9AzAQAGTYQhlKDNAQUC+wHaGGIYUhnyGZIZShlyGVIY8hhIgO4chkaEYCKMZDBkMGAqAYrEM1UCxIiArEyyfylDLwAXUWwpUlQpUkQgUzQaS6UBeNFQ0D8gHmVkM1p0MtCUHiIuAOhUYVA2uGqw0+GxwwmC1wUuDPzjNqgabAXJLJZBOguhNLYjn75II/k5QVy6QLmHIQOjCoyMJqBq/n0oY0hgswH7JBPqtACwC8mUyxPyyqumfg62CVKvVDBYZvAb6b6HBTYPDQB/mlX1JXhqYGjSbgQsYQYbo0YHJCDPSMzTQMww0UXZwgkYVB4M0gxKDBjA+zBkcGDwYAhhCgfauZrjIcIvhNpMPUxFTJVM1RCkTI1SPMAMKYGoBAGyJpx4=</latexit>

dq 0 H(q, q 0 )fk (q 0 ) = EkZ

dq 0 N (q, q 0 )fk (q 0 )Norm Kernelを左辺に移動
gk (q) =<latexit sha1_base64="7HTFZl1K8d8F33Yn1ZqIzzGHMy8=">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</latexit>Z

dq 0 N 1/2 (q, q 0 )fk (q 0 )Norm Kernelにゼロ固有値があるとN-1/2は発散。
小さい固有値は和から除外(natural state)natural stateで張られる空間(集団空間)でのHamiltonian
Z
X
⇤
0
0
u
(q)
u
(q
)
k
(Hcoll )kk0 gk0 ↵ = Ek gk↵(Hcoll )kk0 = dqdq 0 pk H(q, q 0 ) p
nk
nk 0
k0
<latexit sha1_base64="naIS+cg6aLktwFghxYlz9kwXwgc=">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</latexit><latexit sha1_base64="XgmFlEl/cYnA8/3g1mPjX/+x1G0=">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</latexit>

X gk↵f↵ (q) =p uk (q)nk<latexit sha1_base64="f1iwlZqOFEXYIr+PoC3hmtVxN0M=">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</latexit>k

集団波動関数 g↵ (q) =<latexit sha1_base64="MT7FK+JfhtQRySB0QB3yK2X8GhY=">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</latexit>

確率の解釈ができるのは集団波動関数g(q)重み関数f(q)は物理的意味のある振る舞いをしないX

gk↵ uk (q)kp GCMで生成座標について連続極限を取ると問題(ゼロ固有値が無限に出てくる)
p 通常は積分を離散化、ノルムの固有値が小さくなりすぎないようにすることが多い
p 複素生成座標を用いないと重心運動が記述できない

量子数射影GCM基底に使う平均場が対称性を破っている場合量子数射影でZ対称性を回復させる|↵i =dqf↵ (q)| (q)i量子数射影はGCMの特殊な場合に対応(平均場の重ね合わせ、重み関数は決まったものを用いる)<latexit sha1_base64="/+RH2hoyA1dvBLDFgclQqokVxzg=">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</latexit>

粒子数射影演算子

P̂ N =<latexit sha1_base64="2yqzmm2looEY/kwsIvReZaP7Ix4=">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</latexit>

1
2⇡

Z 2⇡

d ei (N̂N)

0i = P̂ N P̂ Z | i
粒子数の固有状態Z
2J+
1
角運動量射影演算子 P̂ J =
J⇤
d⌦D
MK
M
K (⌦)R̂(⌦)
8⇡ 2
N,Z

|
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sha1_base64="5WANmAVOYmCso2uXgEvMMxzmcIg=">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</latexit>
sha1_base64="wdxgX7KiIsAEUHqZW0VBk45C2CM=">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</latexit>

Euler角i↵J i J i J
回転の演算子 R̂(⌦) = e z e y e zˆ⌦ = (↵, , )<latexit sha1_base64="osTIM9OTAf9D22uCjhba+oyBL4Y=">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</latexit>ˆ<latexit sha1_base64="WEuhHPMddXlGJSyKzUotl8ZD/z4=">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</latexit>

WignerのD関数(行列)J⇤DM
K (⌦) = hJM |R̂(⌦)|JKi<latexit sha1_base64="qYXHzhyaYeGfgfDB3vcm3dl2GzA=">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</latexit>

J
JM K i = P̂M K ||<latexit sha1_base64="tfr4/IPiVHWvaTOljcsB5D+GZ98=">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</latexit>

GCM基底としては|<latexit sha1_base64="Vf67KeArod2IefET7D07xGwYST4=">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</latexit>i

角運動量の固有状態

NZ
J
N Z
JM K i = P̂M K P̂ P̂ |iˆ

GCMの計算例
生成座標：β(軸対称四重極変形)
GCM基底：Skyrme HF+BCS

集団波動関数0+2+
4+Bender et al. Phys. Rev. C 69, 064303 (2004)

GCMの計算例
生成座標：β、γ(四重極変形)
Gogny HFB
集団波動関数g(β,γ)

励起エネルギー

Rodriguez and Egido, Phys. Rev. C 81, 064323 (2010)

GCMの計算例
生成座標にpairingを用いた計算(Gogny)

Vaquero et al., Phys. Lett. B 704, 520 (2011)

クラスター間の相対距離を生成座標に用いた計算(AMD+GCM)

Taniguchi et al., Prog. Theor. Phys. 112, 475 (2004)

GCMの計算例
生成座標にneutron-proton pairingを入れたもの(schematic model)
集団波動関数g(β、φ)

二重ベータ崩壊の原子核行列要素(76Ge→76Se)

アイソスカラー型対生成演算子

アイソスカラー型対の生成座標

NH and Engel, Phys. Rev. C 91, 031301(R) (2014)

集団的な自由度
p 集団座標(GCMでは生成座標)p 系の集団的振る舞いを表す少数のパラメター
p 変形のパラメター (β2,γ, β3, …), 対ギャップ, etc
p 集団経路p 集団座標で書かれたスレーター行列式のセット
オブレート・プロレート変形共存
オブレート変形

対称軸:y?

非軸対称変形経路？
球形

軸対称変形経路？
集団経路を理論から決められないだろうか？

γβ 対称軸:z

プロレート変形集団的な自由度

自発的核分裂 量子力学的トンネル現象で原子核が２つに分裂

重い核の厳密計算は不可能：平均場理論が唯一の微視的理論

対称核分裂vs非対称核分裂

分裂片の陽子数分布

K-H Schmidt et al., Nucl. Phys. A 665, 221 (2000)

Moller et al., Nature 409, 785 (2001)

変形の自由度(八重極変形β3)沿いの性質が分裂片の対称・非対称分布に影響

集団経路→核分裂経路
集団経路を理論から決められないだろうか？

集団的な自由度
264Fmの自発的核分裂

Sadhukhan, et al., Phys. Rev. C 88, 064314 (2013)

作用積分を最小化するように集団経路を決定

集団経路はポテンシャルの形だけで決まるわけではない
集団質量の評価も重要時間依存平均場理論
時間依存平均場理論(TDHF(B))
時間依存変分原理<latexit sha1_base64="psSNf4ERG5+Yh0pyEKw7sAi7Bsw=">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</latexit>@h (t)|i~@t

Ĥ| (t)i = 0

Y †| (t)i =ai (t)|0i

HF軌道または準粒子が時間依存i<latexit sha1_base64="zKi4ipyUdB2YoxJ8e7XSN5Qznqw=">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</latexit>

一般化密度行列

R(t) =<latexit sha1_base64="n3VImQuTkNNio6q6WISjPJH45XM=">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</latexit>✓⇢(t)⇤ (t)

(t)1 ⇢⇤ (t)◆=⇢ij (t) = h (t)|c†j ci | (t)iV ⇤ (t)U ⇤ (t)◆V T (t)<latexit sha1_base64="GmbT9uAx/k4Aj4vcEmECJqeh7Ys=">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</latexit>Uµ (t)Vµ (t)◆U T (t)H=<latexit sha1_base64="8eHYjiKCVaFg2Y/ZAeSRvaFdQ0o=">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</latexit><latexit sha1_base64="OP5VLAmRDeXHxyxOkgsSBy7r+yU=">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</latexit>✓µ<latexit sha1_base64="vnxHybpsqlha9iI68Vgx6oXBaG8=">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</latexit>@i~ R(t) = [H, R(t)]
@t
@
i~
@t

<latexit sha1_base64="jDUXNujMoim1wTvuNWPlSeaEg70=">AAACt3ichVHLShxBFD22eejk4cRsBDdNBsVshtsSyAMCEjdZ+hoVbGmq25qZwuruSnfNgBnzA/kBBVcJZBHyAe6yySL+gAs/IWRpIJsscrunQRKJuU13nTp1zq0+3NBolVuisxFn9Nr1GzfHxmu3bt+5O1G/N7mep70skq0o1Wm2GYpcapXIllVWy02TSRGHWm6Eu4vF+UZfZrlKkzW7Z+R2LDqJaqtIWKaC+vy+b7pqzj70M5F0tHSfu77J0p3Aj3uuL7TpigKyYJ8qSVBvUJPKci8DrwINVLWU1o/hYwcpIvQQQyKBZawhkPOzBQ8Ew9w2BsxljFR5LvEGNfb2WCVZIZjd5W+Hd1sVm/C+6JmX7ohv0fxm7HQxQ6f0kc7phD7RN/r1z16DskfxL3u8hkOvNMHE26nVn/91xbxadC9cVzhCVl+dyaKNJ2UWxdlMyRQpo2H//uuD89VnKzODWXpP3znfOzqjL5ww6f+IPizLlSPUeEDe3+O4DNbnmx41veVHjYUX1ajGMI0HmON5PMYCXmIJLb73EJ/xFSfOUydw2k53KHVGKs99/FHOq98XvaPV</latexit>↵µ (t)|0iij (t) = h (t)|cj ci | (t)i

<latexit sha1_base64="vO0Yzsa1SxZg2sRhWcccx8bYPEA=">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</latexit>

TDHFB方程式✓| (t)i =Y✓Uµ (t)=H
Vµ (t)◆

粒子数の拘束項-λNは出てこない

✓

h[⇢]
⇤
[]

[]
h⇤ [⇢]◆

時間依存平均場理論
TDHFB方程式の性質
定常解はHFBi~<latexit sha1_base64="OP5VLAmRDeXHxyxOkgsSBy7r+yU=">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</latexit>@
R(t) = [H, R(t)]@t

時間発展：初期値問題(依存)運動の定数di h (t)|F̂ | (t)i = h (t)|[F̂ , Ĥ]| (t)idt<latexit sha1_base64="vCHoZ+KH97P4cRKJ7Hg6t+PDK8s=">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</latexit>

古典系との対応

エネルギー<H>, Hamiltonianと可換な量は時間発展中保存→トンネル現象はTDHFBでは起きない→粒子数拘束は時間発展中いらない

TDHFの一粒子軌道1| (t)i = p det[ 1 (t) 2 (t) · · ·n!

n (t)]<latexit sha1_base64="Klsh5bZTcl0z++m1rYLg7x5YoO8=">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</latexit>i<latexit sha1_base64="mZwFDzjVGunxdHy6V+XuTPdrG0Y=">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</latexit>@@ti (t) =@H@ i⇤i<latexit sha1_base64="jP/dArDKHVJKTFlowf4UdyWkH80=">AAAC3XichVE9b9RAFJyYr+QC5CBNJBqLU1AoOD0jJBASUpQ0KfPBJSfFwVove2EVn23ZeycFy2WaCNGmSJVIFIgfQEdDkz+QIj0NQqQJEg0F72zzGRGeZe/s7Mxbj54fBzo1REdD1rnzFy5eGh6pjV6+cnWsfu36chr1EqlaMgqipO2LVAU6VC2jTaDacaJE1w/Uir8xOzhf6ask1VH42GzGaq0r1kPd0VIYprx6W7udRMjMjUVitAjyn8g2uRun2tNPXJGaKXPbfmTfsUu1/UOUuZKVc/lvttKUe/UGNako+zRwKtBAVfNR/S1cPEUEiR66UAhhGAcQSPlZhQNCzNwaMuYSRro4V8hRY2+PVYoVgtkN/q7zbrViQ94PeqaFW/ItAb8JO21M0iG9phM6oDf0kb79s1dW9Bj8yyavfulVsTe2PbH09b+uLq8Gz365znD4rD47k0EHD4osmrPFBTNIKcv+/ec7J0sPFyezW7RPnzjfHh3Re04Y9r/IVwtqcRc1HpDz9zhOg+W7TYeazsK9xvRMNaph3MBNTPE87mMac5hHi+99hw/4jGPLs7asF9bLUmoNVZ5x/FHWzndHYrSc</latexit>@@t⇤i (t) =@H
@ i

古典系Hamiltonの運動方程式と同じ形式(ψ,ψ*)は正準変数になっている

小振幅近似：RPAまたはQRPAに帰着

TDHF(B)の計算例
Skyrme SLy5

Sekizawa and Yabana, Phys. Rev. C 88, 014614 (2013)

Gogny TDHFB 20O+20O

Hashimoto and Scamps, Phys. Rev. C 94, 014610 (2016)
Scamps and Hashimoto, Phys. Rev. C 96, 031602(R) (2017)

大振幅集団運動理論
集団座標を理論で決められないか？
時間依存変分原理<latexit sha1_base64="psSNf4ERG5+Yh0pyEKw7sAi7Bsw=">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</latexit>

Marumori et al., Prog. Theor. Phys. 64, 1294 (1980)
Matsuo, Prog. Theor. Phys. 76, 372 (1986)@h (t)|i~@t

Ĥ| (t)i = 0系の時間変化：集団変数の時間変化 | (t)i = | (q(t), p(t), '(t), n(t))i = e i'(t)Ñ | (q(t), p(t), n(t))i<latexit sha1_base64="oftNS/Jv9GW89fBLNlqTDtYuakw=">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</latexit>Ñ = N̂ = N0(q,p): 1次元の集団座標・集団運動量(φ,n): ゲージ空間での回転角、粒子数ゆらぎ(pairingがある場合)<latexit sha1_base64="JtzyuvybFWA3NbrjoPC/lmB7ohM=">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</latexit>

自己無撞着集団座標法(self-consistent collective coordinate method)
SCC基本方程式I: 集団経路の方程式
h (q, p, n)|Ĥ

i

✓

@H @
@p @q

SCC基本方程式II: 正準変数条件<latexit sha1_base64="chLESGuMwKV7ZLy+pCdC9EJzC78=">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</latexit>@@S| (q, p, n)i = p +@q@q@Sh (q, p, n)|Ñ | (q, p, n)i = n +@'h (q, p, n)|i<latexit sha1_base64="F1c+hpu9AdLeLt+g4UPamz88ArU=">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</latexit>

SCC基本方程式III: 集団Hamiltonian<latexit sha1_base64="MbjzMGLW+VUEO1VvrMEmUO/fjUo=">AAAC5XichVHNahRBEK6MRuP4k41ehFwal0jEsNSIEBECQS85hSTrJoFMWHsmvZsmvT1tT+9CMtkXELxJPHgQBQ/iA/gAQsgLeMgjiN6iySUHa2cHogZjDT399Vf1VfdHRUbJ1CHuD3jnzg9euDh0yb985eq14dLI9cU0adtY1OJEJXY54qlQUouak06JZWMFb0VKLEUbj3v5pY6wqUz0E7dpxGqLN7VsyJg7ouqlp6HiuqkEC826HH82YSb0HbbNmB86qdZENtv1t39PhbZfPsV8fZeFDcvjLDTcOskVq3ZPcNjhlnTdeqmMFcyDnQZBAcpQxFxS+gQhrEECMbShBQI0OMIKOKT0rUAACIa4VciIs4RknhfQBZ+0baoSVMGJ3aB/k04rBavp3OuZ5uqYblG0LCkZjOEX/IAHuIcf8Sse/7NXlvfovWWT9qivFaY+/Pxm9ei/qhbtDtZPVGcoIqo+25ODBjzIvUjyZnKm5zLu9+9svTqoPlwYy27jO/xG/t7iPn4mh7rzI34/LxZeg08DCv4ex2mweK8SYCWYv1+eflSMaghG4RaM0zwmYRpmYA5qdO8ufIefcOg1vRfeS2+nX+oNFJob8Ed4b34BUVa0vw==</latexit>

@H @
1 @H
+
Ñ
@q @p
i @n

h (q, p, n)|<latexit sha1_base64="N5W/3EyRxL3o/L0LJxhksh0gB88=">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</latexit>h (q, p, n)|<latexit sha1_base64="YcHCqwxhf/P8gMk8vpkediB+3/I=">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</latexit>◆| (q, p, n)i = 0

1 @| (q, p, n)i =
i @p

1 @
| (q, p, n)i =
i @n

@S
@p
@S
@n

H(q, p, n) = h (q, p, ', n)|Ĥ| (q, p, ', n)i = h (q, p, n)|Ĥ| (q, p, n)i<latexit sha1_base64="4BZGoLpz/Ugl7cGgy6j1WDsJ8L8=">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</latexit>

SCC基本方程式の解法(η*, η)展開法 (Marumori)| (q, p, n)i = eiĜ(q,p,n) | HFB i<latexit sha1_base64="I0KdJF7d55D38PP3sfgqFK6qFZw=">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</latexit>(η*,η,(n))に関して4次までG, Hを展開: 非調和振動の記述が可能 (η=q+ip)
p 球形核の2-フォノン状態: 100-104Ru, 76-78Se (Aiba et al.)p 非調和γ振動 164-168Er (Matsuo et al.)p Backbending phenomena (Shimizu et al.)p 回転運動 (Terasaki et al., Kaneko)p 球形-変形相転移 148-152Sm (Yamada et al.)(η*, η)-展開法 では1つの平均場(HFB状態)を仮定
平均場ゆらぎの大きな現象の記述は困難
p Q,P: 集団変数のシフト演算子
p 小振幅近似でRPAを含んでいる。
p 集団変数は正準変数である。

断熱的大振幅集団運動理論
断熱近似
p pの2次までで理論を展開
p qに関しては展開なし
p 集団経路を決定することが目的

ボソン展開・(η*,η)展開
p 摂動展開p q, pの両方に関して展開ˆ

ipQ̂(q)+in⇥(q)
|
(q,
p,
n)i
=
e
| (q)iTDHFB状態の断熱展開<latexit sha1_base64="NRzcAfPJgYPoFpNOzDLIt3sHJFU=">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</latexit>

Q̂(q) =X⇥

Qµ⌫ (q)↵µ† ↵⌫† + Q⇤µ⌫ (q)↵⌫ ↵µµ<⌫<latexit sha1_base64="gTMGDRnSMmUURRJNnhis3aVxIZ0=">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</latexit>

ˆ
⇥(q)
=i

X⇥

⇥µ⌫ (q)↵µ† ↵⌫†⇥⇤µ⌫ (q)↵⌫ ↵µ

µ<⌫

⇤

⇤

準粒子はqに依存して定義 ↵µ (q)| (q)i = 0<latexit sha1_base64="KbUFehoD0xM0EevDtPsYvmRTebU=">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</latexit><latexit sha1_base64="9v6LaJzpkMGpZ0BN3ll9FQkv9d8=">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</latexit>

集団Hamiltonianの断熱展開
1
H(q, p, n) = V (q) + B(q)p2 + (q)n2
V (q) = H(q, p, n)|p=n=0 = h (q)|Ĥ| (q)i<latexit sha1_base64="u0WkWT6HRnEx6IxiimRtTvwsNpM=">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</latexit>

集団ポテンシャル
集団質量(逆数)<latexit sha1_base64="6bDT7WBB5c0hh66F4FxmceqqduM=">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</latexit>@ 2 H(q, p, n)B(q) ==@p2
p=n=0<latexit sha1_base64="HT0hhzkmO7I9BhmpmgJ6sft88QY=">AAADDHichVHPaxNBFH67/qrxR2N7EVRYDJUEYpgNQkUIlHjJsWlNW8ikYXY7SYZOdqezk0Dd7Fnw4NWDJwUP4lXwIJ4E8R/w0D9BxFMFLx582Sz+Kta37L5v3nvfN/vxPCVFZAg5sOwTJ0+dPjN3Nnfu/IWL8/lLCxtRONI+b/mhDPWWxyIuRcBbRhjJt5TmbOhJvunt3p32N8dcRyIM7pl9xTtD1g9ET/jMYKmbf1Av7pWcmkN7mvkxVUwbweR21Ympz6TTSIp7ZVUOSsnPnqO2qwmti/6kG6taUCMJ0m9SyYK+5A5VA4GKk3abDpiJG0k5zU3UKXV+x5NskuqU2M0XSIWk4RwFbgYKkMVqmH8NFHYgBB9GMAQOARjEEhhE+LTBBQIKax2IsaYRibTPIYEcckc4xXGCYXUXv308tbNqgOepZpSyfbxF4quR6cAS+UhekEPygbwkn8j3f2rFqcb0X/YxezMuV935h5fXv/2XNcRsYPCLdQzDw+njPRnowe3Ui0BvKq1MXfoz/fH9x4frd9aW4hvkGfmM/p6SA/IOHQbjr/7zJl97AjlckPv3Oo6CjWrFJRW3eauwUs9WNQdX4DoUcR/LsAINWIUW3vvFWrSuWtfsR/Yr+439djZqWxlnEf4I+/0P/EbBbA==</latexit>

化学ポテンシャル(q) =<latexit sha1_base64="qBThHt/zrSF7d8bE5049qb0cnq0=">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</latexit>h (q)|[[Ĥ, Q̂(q)], Q̂(q)]| (q)i@H(q, p, n)ˆ= h (q)|[Ĥ, i⇥(q)]|
(q)i
@n
p=n=0

ASCC法
集団経路の方程式を集団運動量pの2次までで展開
Moving-frame HFB方程式 (pのゼロ次より)<latexit sha1_base64="ox2HxP9PNRqLLoW6oCkVTGY7Ayc=">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</latexit>

moving-frame (動座標系での) Hamiltonianh (q)|ĤM (q)| (q)i = 0

ĤM (q) = Ĥ(q)N̂<latexit sha1_base64="TvMUw4F3LR7IPRe4mLBKt3h/3zM=">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</latexit>

@V
Q̂(q)
@q

Moving-frame QRPA方程式 (pの1,2次より)
h (q)|[ĤM (q), Q̂(q)]<latexit sha1_base64="U7l0dBIzbKgXCI58Rs+ih1ehJ+c=">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</latexit><latexit sha1_base64="bvW1fsjtmBfkSYixDs5l2u06yS4=">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</latexit>h (q)|[ĤM (q), P̂ (q)]C(q) =

1
B(q)P̂ (q)| (q)i = 0
i

iC(q)Q̂(q)1@V[[ĤM (q),Q̂(q)], iQ̂(q)]
2B(q)
@q

i

@
Ñ | (q)i = 0@q@2V
1 @B @V+2@q
2B(q) @q @q<latexit sha1_base64="woKAPAfV838dl/woPpOrPoGxRLQ=">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</latexit>h (q)|Q̂(q)| (q)i =0@
P̂ (q)| (q)i = i | (q)i@qh (q)|P̂ (q)| (q)i =0<latexit sha1_base64="dF+wJEsFS4xogmoeYWbxl93uSVM=">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</latexit>h (q)|Ñ | (q)i =0
ˆh (q)|⇥(q)|(q)i =0

正準変数条件(pの1次までで展開)
h (q)|[Q̂(q), P̂ (q)]| (q)i =i<latexit sha1_base64="s1keuBKLfXc4Kk8w3eN6GfyXPMU=">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</latexit>h (q)|[Ñ , P̂ (q)]| (q)i =0

ˆh (q)|⇥(q),Ñ ]| (q)i =iˆh (q)|Q̂(q), ⇥(q)]|(q)i =0h (q)|@ Q̂
| (q)i =@q

1

Matsuo, Nakatsukasa, Matsuyanagi, Prog. Theor. Phys. 103, 959 (2000)<latexit sha1_base64="8iApwQyWZBqPavwVQi/pUEvVuso=">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</latexit>

ATDHF理論
1970年代にいろいろなバージョンが提案される
簡単なポテンシャルモデルでの比較計算

Rowe and Basserman, Can. J. Phys. 54, 1941 (1976)
Marumori, Prog. Theor. Phys. 57, 112 (1977)
Goeke, Reinhard, Rowe Nucl. Phys. A 359,408 (1981)

Villars, Nucl. Phys. A 285, 269 (1977)

Goeke and Reinhard, Ann. Phys. 112, 328 (1978)
Baranger and Veneroni, Ann. Phys. 114, 123 (1978)

ATDHFBのレビュー：G. Do Dang, A. Klein and N.R. Walet, Phys. Rep. 335 (2000), 93.

一次元集団的自由度の決定

ASCC方程式

Moving-frame QRPA eq.

Moving-frame HFB eq.
Double iteration for each q

moving-frame QRPAによる小振幅振動

potential energy

・・・・・・
大振幅振動運動|φ(q=δq)>

HFB state (prolate)δqmoving-frame HFB state (振動してる状態)|φ(q=0)> 変形ポテンシャル極小点(平均場解)

p 平均場計算(moving-frame HFB)と小振幅振動計算(m-f QRPA)を自己無撞着に実行
p 系が最も集団的な方法を決定、小振幅振動によって進め、最終的に大振幅になる
p 得られた経路は四重極演算子など既知の演算子の期待値としては求まらない

ASCCの計算例(68Se)

NH, Nakatsukasa, Matsuo, Matsuyanagi, Phys. Rev. C 80, 014305 (2009)

Pairing + Quadrupole Modelq~1.75β-γ四重極変形曲面に射影した
集団経路

M(q) (MeV-1)

q

q=0

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

10 20 30 40 50 60γ(q)0

10 20 30 40 50 60
γ(q)

60
50
40
30
20
10
0

p オブレート・プロレート変形共存の最適集団経路
p 非軸対称変形の自由度がオブレートとプロレートの平均場をつなぐ
p もう一つの自由度：集団質量(座標のスケーリング)

ASCC法の四重極集団運動への応用
完全自己無撞着な２次元集団部分空間計算：まだ誰も行っておらず困難
一般化された５次元四重極集団Hamiltonian

Próchniak and Rohoziński, J. Phys. G36, 123101 (2009)

四重極の５つの自由度(変形パラメターβ, γと3つのオイラー角Ω)をすべて取り入れた
最も一般的な断熱的古典Hamiltonian
小振幅近似：四重極振動、回転、β 、γ 振動

V(β, γ) 集団ポテンシャル
D(β, γ) 振動の慣性質量
J(β, γ)回転の慣性モーメントp 集団的な自由度：四重極の5つの自由度
p 慣性質量の計算：ASCCを利用(CHFB+LQRPA)

NH, K. Sato, T. Nakatsukasa, M. Matsuo, K. Matsuyanagi, Phys. Rev. C 82, 064313 (2010)

V(β,γ): 拘束HFB方程式で計算
J(β,γ): 拘束HFB状態でのThouless-Valatinの慣性モーメント(QRPA)
D(β,γ): 拘束HFB状態でQRPA
集団Hamiltonianの再量子化
Ĥcoll<latexit sha1_base64="QABL2BKuIb5IbS0fr3B/gZYZu4g=">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</latexit>

IM ↵ (

Kumar and Baranger, Nucl. Phys. A 92, 608 (1967), , ⌦) = EI↵IM ↵ (, , ⌦)

励起エネルギー、電磁遷移、電気四重極モーメント、波動関数など

CHFB+LQRPAの計算例(変形共存)

1-dim ASCC

2-dim ASCC (LQRPA)
5D collective Hamiltonian

Experimental data( ) .... B(E2) e2 fm4

EXP：Fischer et al., Phys.Rev.C67 (2003) 064318.
B(E2;21+→01+) Obertelli et al, Phys.Rev.C80 (2009)031304(R)effective charge en = 0.4,

ep = 1.4 is used for B(E2)

68Se 変形混合

β4 x collective wave function squared

p 波動関数は非軸対称変形領域に広がる
p 主要成分 基底回転バンド: オブレート変形, 励起回転バンド：プロレート変形
p 02+ , 23+ 状態: 大振幅な γ 振動と β-振動の混ざったもの78Se 非調和振動(球形核)|0,2,4+>NH et al., Mod. Phys. Lett. A25 , 1796 (2010)

|2+>
|0+>

2 phonon state
1 phonon state
76Se

78Se

Lie and Holzwarth, Phys. Rev. C 12, 1035 (1975)

p ボソン展開法など摂動論で記述可能
p 5次元集団Hamiltonianで2-フォノン状態、３フォノン状態の順番を再現

中性子過剰Mg同位体での変形相転移, 0+状態
球形からプロレート変形への転移 N=20
30Mg

32Mg

NH, et al, Phys. Rev. C84, 061302 (2011)

34Mg

励起 0+ 状態の発見

30Mg: 1789keV Phys. Rev. Lett. 103 , 012501 (2009)
32Mg: 1058keV Phys. Rev. Lett. 105 , 252501 (2010)

Gogny HFB

Rodríguez-Guzmán et al., Nucl. Phys. A 709 201 (2002)

0+状態での変形混合は？

中性子過剰Mg同位体での変形相転移、0+状態
励起バンド

基底バンド

2+
2+
0+
K=01 band

0+

K=02 band

B(E2) ratio = 1
if the two bands havesame intrinsic structure中性子過剰Mg同位体での変形相転移、0+状態
30Mg

集団波動関数二乗

30Mg

32Mg

プロレート変形状態

34Mg

Probabilityβ-振動

30Mg: 球形の基底0+状態、 変形励起0+ 状態
32Mg: 大振幅の変形ゆらぎが起こっている

32Mg
34Mg

まとめ
p 原子核大振幅集団運動
p 非調和振動・変形転移
p 変形共存p 大振幅集団運動の理論
p ボソン展開法
p 生成座標法
p 時間依存平均場理論
p 自己無撞着集団座標法
p 5次元集団Hamiltonianによる四重極集団運動