原子核集団現象への平均場アプローチ

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. 7, 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

参考書・参考文献
教科書
p Ring and Schuck: The Nuclear Many-Body Problem, Chap. 5-12
p 高田・池田 原子核構造論(朝倉書店)
p Nilsson and Ragnarsson: Shapes and Shells in Nuclear Structurep Bring and Broglia, “Nuclear Superfluidity Pairing in Finite Systems”
レビュー論文Bender, Heenen, and Reinhard, Rev. Mod. Phys. 75, 121 (2003)“Self-consistent mean-field models for nuclear structure”
Nakatsukasa, Matsuyanagi, Matsuo, and Yabana, Rev. Mod. Phys. 88, 045004 (2016)“Time-dependent density-functional description of nuclear dynamics”

原子核はどんな形をしていますか
原子核：陽子と中性子(フェルミ粒子)からなる
Schrödinger方程式に従う

Ĥ|N, Z, J, ⇡, ki = EN,Z,J,⇡,k |N, Z, J, ⇡, ki
原子核の持つ量子数

N: 中性子数
Z:陽子数
J: 全角運動量(と3軸成分M)π: パリティ

量子数：Hamiltonianと演算子が交換・・エネルギー固有状態での同時固有状態[Ĥ, Jˆk ] = 0(k=1,2,3)角運動量がよい量子数→原子核のエネルギー固有状態は回転対称性を保つ

原子核は回転対称(丸い)平均場理論
量子力学系としての原子核
核子は他の核子それぞれと強い相互作用や電磁気力で複雑な相互作用

実際はそんなに原子核内核子は相互作用をしない(媒質効果)
核内核子の平均核子間距離：2 fm 程度(原子核の飽和性より)
核内核子の平均自由行程〜

原子核の直径程度(半径：R=1.2 fm * A1/3)

パウリ排他律により散乱される終状態が禁止されているため

平均場理論
量子多体問題の一体問題への置き換え
Hartree-Fock-Bogoliubov (HFB)
密度汎関数理論(density functional theory, DFT)

原子核内核子は他の核子が作る
平均ポテンシャルと相互作用

平均ポテンシャルによるSchrödinger方程式解(平均ポテンシャル内の一粒子運動)

平均ポテンシャル Γ(r)核子の配位(あるいは密度)から平均ポテンシャルを更新

多体問題の自己無撞着解

すべての核子の軌道

球対称平均ポテンシャル
平均ポテンシャルからどのようなことがわかるか
原子核内の核子の従うSchrödinger方程式

Ĥ (r) =<latexit sha1_base64="cDbHtHhGX63an1y2TE4OGczgM2A=">AAAC73ichVHLahRBFL3TvuL4yKgbIZvCIRIRh9uDoAhC0AhZ5uFMAlPjUN3WzJSpflhdMxCb/gF/IAshmEAWwQ/wA9y4dOMinyAuI7oRkjs9jZIE422669Spe0714XqxVolF3Cs5Z86eO39h4mL50uUrVycr1643k2hgfNnwIx2ZVU8kUqtQNqyyWq7GRorA03LFW3s6Ol8ZSpOoKHxu12PZDkQvVF3lC0tUp/KK94VN5zMeJ2qGe0FqsjvsMeNadm2L3WO8a4Sf8r4nzIt6ltaDjPE5qa1gd1nzj4Ab1evbNjvm8uwI0alUsYZ5sZPALUAVilqIKh+Bw0uIwIcBBCAhBEtYg4CEnha4gBAT14aUOENI5ecSMiiTdkBdkjoEsWv07dGuVbAh7UeeSa726RZNryElg2n8iru4j5/xA37D3//0SnOP0b+s0+qNtTLuTL69ufzrv6qAVgv9v6pTFB51n57JQhce5lkUZYtzZpTSH/sP32zsLz9amk5v4zZ+p3xbuIefKGE4/OHvLMqld1CmAbnHx3ESNOs1F2vu4v3q7JNiVBMwBbdghubxAGZhHhagQfd+gZ9wUALntbPhbDrvx61OqdDcgCPl7B4C1jq3qA==</latexit>



~2
2m

+ V (r)ポテンシャルが球対称な場合：変数分離<latexit sha1_base64="QxoOF00xFV2W4QFQxswOUa4BovE=">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</latexit><latexit sha1_base64="uGgUzFjup1zCHeHO6ngd1roAI6c=">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</latexit>(r) = Rl (r)Ylm (✓, ')~2 1 @ 2L̂2r++ V (r)
2m r @r2
2mr2

Ĥ =

L̂2 =

<latexit sha1_base64="ykj81nMVAEwI09YeQ6+wdmXENn4=">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</latexit>(r) = E (r)

~2



1 @
sin ✓ @✓

✓

@
sin ✓
@✓

◆

+

1 @2
sin2 ✓ @'2

角度部分：球面調和関数(l:軌道角運動量量子数、m:磁気量子数)
L̂2 Ylm (✓, ') = ~2 l(l + 1)Ylm (✓, ')<latexit sha1_base64="LObVGsneYgDz7/lzvkLoulZjDiA=">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</latexit>

動径部分
<latexit sha1_base64="xD+r/cUUOnnAwdvk4rl8vp1gZIg=">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</latexit>~2 1 d 2~2 l(l + 1)r+
+ V (r) Rl (r) = ERl (r)2m r dr22mr2

３次元調和振動子
平均ポテンシャルの例：３次元調和振動子

V (r) =
<latexit sha1_base64="XkoAd6Qal2qjhfvb+eJpnLmHvCQ=">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</latexit>



動径部分

~2 1 d 2~2 l(l + 1)r+
+ V (r) Rl (r) = ERl (r)2m r dr22mr2<latexit sha1_base64="xD+r/cUUOnnAwdvk4rl8vp1gZIg=">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</latexit>⇢=<latexit sha1_base64="usgLX+zJKoGq4kFQT3dClye2ekE=">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</latexit>

r
1
2

n=
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m!02 r2
2

Rl (r) = C⇢l F

m!0
r
~

✓

<latexit sha1_base64="QWtWRog0QRBQoojxC+nchOyfUvE=">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</latexit>

E
~!0

l

3
2

✓

◆
2
3 2
n, l + ; ⇢ e ⇢ /2
2

合流型超幾何関数✓◆
3
E = ~!0 2n + l +2◆<latexit sha1_base64="3e6SStjKsTPQfWJAQrU51TEYL6w=">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</latexit>

N = 2n + l :主量子数(principal quantum number):動径量子数(radial quantum number)<latexit sha1_base64="pyk33kYqx8Cosg/s7NzftUFeoRc=">AAACjXichVHLSsNAFD3Gd3206kZwEyyKIJSboiiiIrrQlfiqLdYiSZxqMC+StKDFH3AtuBAFBRfiB/gBbvwBF36CuFRw48LbNCAq6h1m5syZe+7M4WquafgB0WOdVN/Q2NTc0hpra+/ojCe6utd9p+TpIqM7puPlNNUXpmGLTGAEpsi5nlAtzRRZbW+uep8tC883HHst2HdFwVJ3bKNo6GrA1MaiPCWnbXlYNrcSSUpRGPJPoEQgiSiWnMQtNrENBzpKsCBgI2BsQoXPIw8FBJe5AirMeYyM8F7gEDHWljhLcIbK7B6vO3zKR6zN52pNP1Tr/IrJ02OljAF6oGt6oXu6oSd6/7VWJaxR/cs+71pNK9yt+FHv6tu/Kov3ALufqj8UGmf/7SlAEeOhF4O9uSFTdanX6pcPTl5WJ1YGKoN0Sc/s74Ie6Y4d2uVX/WpZrJwixg1SvrfjJ1hPpxRKKcsjyZnZqFUt6EM/hrgfY5jBApaQ4XdtHOMM51JcGpUmpelaqlQXaXrwJaT5D+y+khM=</latexit>n<latexit sha1_base64="1BKtllaTHKB0PmbyiVAY4JlTeG0=">AAAChHichVHLSsNAFD2N7/po1Y3gplgqLqTc+EBxIUU3LvuwtlCLJHGsoWkSkrSgxR/QreLClYIL8QP8ADf+gIt+grhUcOPC2zQgKuodZubMmXvuzOGqtqG7HlErJHV19/T29Q+EB4eGRyLR0bEt16o7mshrlmE5RVVxhaGbIu/pniGKtiOUmmqIglpdb98XGsJxdcvc9A5sUa4pFVPf0zXFYypj7kTjlCQ/Yj+BHIA4gkhb0TtsYxcWNNRRg4AJj7EBBS6PEmQQbObKaDLnMNL9e4EjhFlb5yzBGQqzVV4rfCoFrMnndk3XV2v8isHTYWUMCXqkG3qhB7qlJ3r/tVbTr9H+ywHvakcr7J3I8UTu7V9VjXcP+5+qPxQqZ//tycMeln0vOnuzfabtUuvUbxyev+RWsonmNF3RM/u7pBbds0Oz8apdZ0T2AmFukPy9HT/B1lxSpqScWYin1oJW9WMSU5jhfiwhhQ2kked3BU5wijOpV5qV5qXFTqoUCjTj+BLS6geueo/l</latexit><latexit

第一種合流型超幾何関数(Kummer関数)
az
a(a + 1) z 2
F (a, c; z) ⌘ 1 +
+
+ ···
c 1!
c(c + 1) 2!
✓◆n!(l + 12 )! l+1/2 2
3 2
Fn, l + ; ⇢ =L(⇢ ) Laguerre陪多項式
2(n + l + 12 )! n<latexit sha1_base64="Hbf/8KD78NktuE+JC9MVdXuahJ8=">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</latexit><latexit sha1_base64="qZ4/w67qHTw5myQQ3shejUSXbCQ=">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</latexit>

調和振動子ポテンシャルの波動関数は遠方でexp(-r2)で小さくなるE = ~!0<latexit sha1_base64="Fq0FIlTYzu5J1wzphrP96/mvQ18=">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</latexit>✓

32n + l +2◆✓

３次元調和振動子

3= ~! N +2◆

N = 2n + l : 主量子数(principal quantum number)n: 動径量子数(radial quantum number)l: 軌道量子数(orbital quantum number)

縮退度：(N+1)(N+2)
lごとの縮退度: 2(2l + 1)
n=0, l=5 (1h) n=1, l=3 (2f) n=2, l=1 (3p)
N=5
42
70
40

N=4
N=3

20
N=2

~!

8

~!

2

<latexit sha1_base64="iPZQCN4+HXWoq2XdghmtB4N/PnE=">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</latexit><latexit sha1_base64="iPZQCN4+HXWoq2XdghmtB4N/PnE=">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</latexit>

30

22

14

6

n=0, l=4 (1g)n=1, l=2 (2d)n=2, l=0 (3s)

10

2

18

20
12

N=1

6

N=0

2

n=0, l=3 (1f)n=1, l=1 (2p)146n=0, l=2 (1d)n=1, l=0 (2s)102n=0, l=1 (1p)6n=0, l=0 (1s)2p エネルギーは主量子数だけで決まるppp

シェル構造(液滴模型ではでない量子力学系の性質)
縮退度の2はスピン自由度から
魔法数：2, 8, 20, 40, 70,... (閉殻構造をとる数)

調和振動子ポテンシャルでの問題
・3次元調和振動子での問題１：lの大きな軌道のエネルギーが高すぎる
修正１：調和振動子にl2に比例するポテンシャルを追加

~!0 µ(l2<latexit sha1_base64="SV3vaONncB/yC1Pru8RQzOH+VG8=">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</latexit>

hl2 iN )修正２：井戸型ポテンシャル

lの大きな軌道は遠心力により原点から離れたところに存在

修正３：現実的なポテンシャル(Woods-Saxonポテンシャル)
V0
VWS (r) =1 + exp[(r R)/a]<latexit sha1_base64="xdfFXKTGj1Z7l33El0XCQQPEyCo=">AAACvXichVFNaxNRFD0dtdZo21Q3gpuHoSVFGu+UQosgFt24bBOTFDJhmBlf0qHzxZtJ2jrMH/APuBAEhS6KP8AfoAs3unPRnyAuK7hx0ZvJQKnF9j7ee+edd89973DtyHPjhOhoQrty9drk9akbpZu3pmdmy3O3W3E4UI5sOqEXqi3biqXnBrKZuIkntyIlLd/2ZNveeTa6bw+lit0weJHsR7LrW/3A7bmOlTBlltdaZmooX7QbWdWw/VRli+KxEEvC6CnLSUXLpCwVunggDLkXdURVLdUXH1rdzCxXqEZ5iPNAL0AFRWyE5U8w8BIhHAzgQyJAwtiDhZhHBzoIEXNdpMwpRm5+L5GhxNoBZ0nOsJjd4bXPp07BBnwe1YxztcOveDwVKwXm6Qcd0jF9pY/0k/7+t1aa1xj9ZZ93e6yVkTn7+m7jz6Uqn/cE26eqCxQ2Z1/sKUEPa7kXl71FOTNy6YzrD1+9OW48qs+nC/SBfrG/93REn9lhMPztHGzK+luUuEH6v+04D1rLNZ1q+uZKZf1p0aop3MN9VLkfq1jHc2ygye++wxd8w3ftiSY1TwvGqdpEobmDM6HtngCPiqNM</latexit>

30

Harmonic Oscillator
fnite wellWoods-Saxon

20

34
20

Energy (MeV)

10
0
-10
-20
-30
-40
-50
0

2

4

6
r (fm)

8

10

Nilsson and Ragnarssonより

スピン-軌道力 (LS力)・3次元調和振動子での問題２：魔法数が説明できない(28など)

VLS = W (r)l̂ · ŝ<latexit sha1_base64="jhIGR1YRkHnA+xwCA4Rz9vqGFog=">AAACsnichVHLShxBFD22iY+J0VE3QjaNg6KLDLdFUARBzCaLLNRxZkRHhu6y1MbqB901A9r0D7jNIousEsgi5ANCtnHjD7jwE8SlgpssvNPTIIlEb1FVp07dc6sO1wmVG2uiyx6j98XLvv6BwcKrodfDI8XRsVoctCIhqyJQQbTl2LFUri+r2tVKboWRtD1Hybpz9K5zX2/LKHYDf1Mfh3LXsw98d98VtmaqWXxbayYfKqm5bNZnolmzcWjrpOF4iUpTsyH2Av1AxWnaLJaoTFmYj4GVgxLyWAuKP9HAHgIItOBBwodmrGAj5rEDC4SQuV0kzEWM3OxeIkWBtS3OkpxhM3vE6wGfdnLW53OnZpypBb+ieEasNDFFF/SdbuicftAV/flvrSSr0fnLMe9OVyvD5sjpROXuWZXHu8bhg+oJhcPZT3vS2Mdi5sVlb2HGdFyKbv32yaebytLGVDJNX+ma/X2hSzpjh377VnxblxufUeAGWf+24zGozZUtKlvr86WV1bxVA3iDScxwPxawgvdYQ5Xf/Yhf+I0zY97YNmxDdFONnlwzjr/CUPdUKKHw</latexit>

スピンー軌道力l̂ · ŝ =<latexit sha1_base64="CeNEOARvFqDolFFzV7qBj8Bh/3w=">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</latexit><latexit sha1_base64="OxrF61yR2FOU/bfHTVfwpzqLWLQ=">AAACsHichVFNSxtRFD1O60fj11g3hW5Cg6UQCHdCoSIURDcu/UoMGBlmxpdkzHwx8xKww/wBd1110VWFLkp/gNsWN/4BF/kJpUsFNy68MxmUKto7vHn3nHfPfe9wzcCxI0k0GFGePR8dG594UZicmp6ZVede1iO/F1qiZvmOHzZMIxKO7YmatKUjGkEoDNd0xI7ZXU3Pd/oijGzf25aHgdhzjbZnt2zLkEzparnZMWTcNN34IEmKH4u30GFYvoNRkhR0tUQVyqL4MNHypIQ81n31BE3sw4eFHlwIeJCcOzAQ8bcLDYSAuT3EzIWc2dm5QIICa3tcJbjCYLbL/zaj3Zz1GKc9o0xt8S0Or5CVRSzQOf2gCzqjn/SHrh/tFWc90rcc8m4OtSLQZ49ebV39V+XyLtG5Uz2hMLn6aU8SLSxmXmz2FmRM6tIa9u9/+nKxtbS5EL+lY/rL/r7RgE7Zode/tL5viM2vSAek3R/Hw6RerWhU0Tbel5ZX8lFN4DXe4B3P4wOWsYZ11PjezzjBL/xWqkpD0RVjWKqM5Jp5/BPKwQ2Wf6Gj</latexit>

1 2
(ĵ
2

l̂2

ŝ2 )

ĵ = l̂ + ŝ 全角運動量(軌道とスピンの合成)
1hl̂ · ŝi =j(j + 1)
2<latexit sha1_base64="zGdDDM4n10Hut2SvJvaoRlj4eoc=">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</latexit>

lとsが平行なら引力 (W(r)<0)
反平行なら斥力

l(l + 1)8><l
3
= 2l+14>:
2

j = l + 12
j=l

1
2

<latexit sha1_base64="egHv6Mx5A+7rx5PsniHd/oebmjo=">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</latexit>

スピンー軌道力がないと l, lz, s, szが量子数
スピンー軌道力でj,jz,l,sが量子数
スピンー軌道力は波動関数の形を変えない

Nilsson and Ragnarssonより

魔法数を説明 (Mayer and Jensen 1963年ノーベル物理学賞)

原子核の変形

原子核の変形(回転対称性の破れ)ーー平均ポテンシャルの変形(回転対称性の破れ)
原子核の密度を球面調和関数で展開 2
3
λ=0:
λ=1:
λ=2:
λ=3:
λ=4:

R(✓, ') = R0 41 +Xµ↵⇤ µ Y µ (✓, ')5半径を変えるのみ(飽和性のためあまり半径は変わらない)
双極子、球形のまま重心の位置がずれる
四重極変形、原子核の変形で最も重要
八重極変形、パリティ対称性を破る
十六重極変形、核分裂などでは重要

四重極変形：５つの自由度 (μ=-2,-1,0,1,2)
3つは軸のとり方の自由度、a1=a-1=0ととれば(主軸系)a2=a-2, a0の２つ。
軸対称四重極変形度β非軸対称変形度γ

a0 = cos1
a2 = a 2 = p sin
2

γ=120
対称軸:x

オブレート変形

γ=60 対称軸:y
オブレート変形γ

球形
β=0

プロレート変形
β

γ=0
対称軸:z

基底状態での四重極変形
核図表
緑色：球形(β=0)、

赤：プロレート変形(β>0)

青：オブレート変形(β<0)

軸対称HFB計算：Stoitsov et al., Phys. Rev. C 68, 054312 (2003)

平均場計算では多くの原子核は変形→四重極相関は強い引力変形調和振動子ポテンシャル
軸対称変形(z軸周りで回転対称)1
V (r) = m!02 r2
2

V (r) =

12m[!?
(x2 + y 2 ) + !z2 z 2 ]2✓<latexit sha1_base64="3R71LunAHXKmbZA5bYvDRADj06I=">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</latexit><latexit sha1_base64="XkoAd6Qal2qjhfvb+eJpnLmHvCQ=">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</latexit>2"3!z = !0 (") 1<latexit sha1_base64="nOKhXmHJk6OalTzA+/9c7b6UfpY=">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</latexit>"=<latexit sha1_base64="E8zRPSfj3p+05W69mDScLMgQDWg=">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</latexit>

球形付近での振る舞い(ε<<1)
Hamiltonianに対して1次の摂動項

!?

!z

✓

1
!? = !0 (") 1 + "3<latexit sha1_base64="y0rhIqUc7A8buC77EkfMk4E1+zY=">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</latexit>2

2z 2 ) =<latexit sha1_base64="v/OJyXk+Vbn7ME1r4Itle3Qv0g8=">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</latexit>2
M !02 "
3

j軌道は2j+1重に縮退→摂動項によって縮退が解ける
hN lj⌦|<latexit sha1_base64="aiCXAXkBKaKK53jNPhTuU9vUI94=">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</latexit>

3⌦2 j(j + 1)
"r Y20 (✓, ')|N lj⌦i ⇠ "hr ij(j + 1)
2

2

f7/2

Ωの大きさでエネルギーが変わる
Ωの順位は縮退
Ω：角運動量の第三成分(jz)◆distortion parameter!0

1"M !02 (x2 + y
3

◆

r

4⇡ 2
r Y20 (✓, ')
5

Ω=7/2
Ω=5/2
Ω=3/2
Ω=1/2

oblate0εprolate変形した平均ポテンシャル
変形が大きいときの振る舞い(漸近的な振る舞い)~2m 2 2
変形調和振動子の場合 Ĥ =+ [!?
(x + y 2 ) + !z2 z 2 ]
2m
2
r
r
r
⇠ =⇢ cos '
m!?
m!?
m!z
⌘=y
⇠=x
⇣=z
~
⌘ =⇢ sin '~~<latexit sha1_base64="OS3tAWvzDj8QjJnmv9zlZTwT22M=">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</latexit><latexit sha1_base64="Mp0zdjBsSyQjSr7LoS/6bxfmPQ4=">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</latexit><latexit sha1_base64="xQs7mczL4fXuxYeJuLdxlCckGV4=">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</latexit><latexit sha1_base64="EVhp4cskuwPz15pYcE2QWv41WlE=">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</latexit>



1
~!?
2

<latexit sha1_base64="ms4jBYpbXBu8LNfgiHuAtwvadHA=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="8t1wxr9WcBoF0grpoN4Lt98a7yY=">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</latexit>
sha1_base64="Q0jcDkcoyqQuJFcPMXhJ+EDYU7s=">AAAD5XichVFNaxNBGH63q7auH031IngZDJWUNmE2F0UQiiJ47IdpC5kYZtdJduhmd5idBJqQPyB4Ez14EAUP4g/wBwjiH/DQnyB6qx8XD777QbQtjbPszDPvPM8z8/B6KpSJoXTfmrFPnT4zO3fWOXf+wsX50sKlrSTua180/DiM9Y7HExHKSDSMNKHYUVrwnheKbW/3bnq+PRA6kXH0wOwp0erxbiQ70ucGS+0Fq81C0TFNwjqa+yN3PKqPWeBxTVjcE13eZkpo5WSkSnVCYjqIx4WGKa6N5OF4grLTdJpKIE6VHDJ8WD9qiZUJJmzAtQpkylomOd9hWnYDs0ScZeL8k4BkEYoEw+L1pDrVfSgML7xzSCbmOWgRphJZSW9eKd6yklGXyG1y76SzdqlMazQb5DhwC1CGYqzFpffA4BHE4EMfeiAgAoM4BA4Jfk1wgYLCWgtGWNOIZHYuYAwOavvIEsjgWN3FuYu7ZlGNcJ96Jpnax1tC/DUqCSzSz/QtPaCf6Dv6hf4+0WuUeaRv2cPVy7VCtecfX9n89V9VD1cDwV/VFIWH7OmZDHTgZpZFYjaVVdKUfu4/GD4/2Ly1sTi6Tl/Tr5jvFd2nHzBhNPjuv1kXGy/AwQa5R9txHGzVay6tueu0vHqnaNUcXIVrUMF+3IBVuA9r0ADf+mh9s35YP+2u/cR+aj/LqTNWobkMh4b98g/rtxRD</latexit><latexit sha1_base64="J+fxygKXOMNJunWCl9ZBFXIIV10=">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</latexit>

@2
@'2

~!z
<latexit sha1_base64="mOWccud8u61zyHShI0++OXU93sg=">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</latexit><latexit sha1_base64="BEXa7wSVQNi/8iDc4qN3DAHQ69E=">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</latexit>

1 @2
+ ⇢2
2
2
⇢ @'

1 @ @
⇢
⇢ @⇢ @⇢

変数分離

✓

1
~!?
2

✓

◆

<latexit sha1_base64="BsPL6wPqVJZlGx0m7uaOpCEWTXw=">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</latexit>

1
+ ~!z
2

✓

@2
+ ⇣2
2
@⇣

◆

(⇢, ', ⇣) = E (⇢, ', ⇣)(⇢, ', ⇣) = U (⇢)Z(⇣) (')<latexit sha1_base64="WTDspRDlzo3YsMlENCuVHO2D89M=">AAACvHichVFNSxtRFD1OrU1jNWndFLoJBksCEu6IkFJoCXbTpV9RMSNhZnyaR+aLmZeBJPQP9A+4EBcWuij9Af0Bbrqoyy7yE0qXKbhx4c1kQNpQvcObd9+559z3DtcKHBkposGU9mD64cyjzOPs7JO5+Vz+6bOdyO+EtqjbvuOHe5YZCUd6oq6kcsReEArTtRyxa7Xfjeq7sQgj6XvbqhuIA9c89uSRtE3FUDNfNYJIloyw5S8bsRkGLbls9IQyy4U3hXqCl/dLY8TgYikllZv5IlUoicJkoqdJEWms+/lvMHAIHzY6cCHgQXHuwETEXwM6CAFjB+gzFnImk7rAB2RZ22GWYIbJaJv/x3xqpKjH51HPKFHbfIvDK2RlAUv0k77QkL7TV/pF1//t1U96jN7S5d0aa0XQzH18vnV1r8rlXaF1q7pDYTH7bk8KR3iVeJHsLUiQkUt73D/unQy3Xm8u9V/SJ/rN/s5pQBfs0Iv/2J83xOYpsjwg/d9xTCY7KxWdKvrGarG2lo4qgxdYRInnUUUN77GOOt97hgv8wKX2VjvU2po7pmpTqWYBf4UW3wBGHaUH</latexit>(') = ei⇤'

= ⇤2<latexit sha1_base64="sjqB0oas1q7boiSZn/95kx7wHjA=">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</latexit>◆@22+ ⇣ Z(⇣) = Ez Z(⇣)@⇣ 2

Ez = ~!z<latexit sha1_base64="BzazEAB2K5O3Kru9LyzIzCVTSyE=">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</latexit>

✓

1
nz +
2

◆

2

Znz (⇣) = Hnz (⇣)e ⇣ /2
<latexit sha1_base64="L+QrjJg9gZBNxVI931KwHPQhKgQ=">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</latexit>

ζ(z)方向は1次元調和振動子と同じ形→Hermite多項式

✓

◆
1 @ @
⇤2
2
⇢
+ 2 + ⇢ U (⇢) = E? U (⇢)⇢ @⇢ @⇢⇢|⇤|U (⇢) = ⇢

e

⇢2 /2

Wn⇤⇢ (⇢) = F ( n⇢ , |⇤| + 1; ⇢2 ) =W (⇢)<latexit sha1_base64="vXHjTvqCd39A0v9HpWvmKUIRH98=">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</latexit>n⇢ !|⇤|!
L ⇤ (⇢2 )(n⇢ + |⇤|)! n⇢

動径(ρ)方向はLaguerreの陪多項式nρ=0,1,2 ...

E? = ~!? (2n⇢ + |⇤| + 1) = ~!? (n? + 1)
<latexit sha1_base64="PWA1//9LRUJilvitCQ6iL60FGJI=">AAAC3XichVE9SxxRFD1OotGN0dU0gTSDi7IhsNyRQIIgiBKwSOFHVhccGd6Mz93B+eLN7IJZLdNISGthZSBFyA9IZ2PjH7CwtwlBGwUbC+/MDkhiYu4w884775z75nDtyHPjhOikS3vwsLvnUW9f4XH/k4HB4tDwUhw2lSOrTuiFqmaLWHpuIKuJm3iyFikpfNuTy/bGTHq+3JIqdsPgfbIZyVVf1AN33XVEwpRVrL21zEiqSJ/UzYYtlBn6si5yrqyPB5apGqH+Ut8y33HXNbHF2HjxN3lZD3JfqrCKJapQVvpdYOSghLzmwuIPmFhDCAdN+JAIkDD2IBDzswIDhIi5VbSZU4zc7FxiGwX2NlklWSGY3eBvnXcrORvwPu0ZZ26Hb/H4VezUMUrH9I0u6Ii+00+6/mevdtYj/ZdNXu2OV0bW4M6zxav/unxeEzRuXfc4bFbfnynBOt5kWVzOFmVMmtLp9G992L1YnFgYbY/RF/rF+fbphA45YdC6dL7Oy4U9FHhAxp/juAuWxisGVYz5V6Wp6XxUvXiOEZR5Hq8xhVnMocr3HuAUZzjXLO2j9kn73JFqXbnnKX4rbfcGB8SwHQ==</latexit>

軌道のエネルギー

E(nz , n? ) = ~!z<latexit sha1_base64="g54J3KJusGTZtjvipe03wrHwJx0=">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</latexit>

✓

1
nz +
2

◆

+ ~!? (n? + 1)

n? = nx + ny
<latexit sha1_base64="5lCMphPj4F3RyGI6GbGaDQqVHcc=">AAACmXichVHLSsNAFD2Nr1pf9bEQ
sha1_base64="5lCMphPj4F3RyGI6GbGaDQqVHcc=">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</latexit>

一粒子エネルギーの変形度依存
球形"=0!z = !? = !0<latexit sha1_base64="Xs4fUpI2ufq/bb60yVJvsU4cXOg=">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</latexit><latexit sha1_base64="df5C+l7g2NTUTxVDVQJyzxN5AqU=">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</latexit>

E(nz , n? ) = ~!0<latexit sha1_base64="LibII8fWGDa/WRBNI5yfzTMWuaA=">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</latexit>✓3nz + n? +2◆

主量子数ごとに順位が縮退(魔法数の出現)超変形 !? : !z = 2 : 1<latexit sha1_base64="082qEhAvNIsaOpX2kXC5htLff+0=">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</latexit><latexit sha1_base64="Zwqk7gaR7GTaiUx421wtmAlnkbU=">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</latexit>" = 0.6!? = 2!z =<latexit sha1_base64="DGYQGM8nYtjk8qZWaUH728Da3Zs=">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</latexit>

6
!0
5

✓
◆
3
5
E(nz , n? ) = ~!0 nz + 2n? +52<latexit sha1_base64="Y/YAtiXbO09pxt4GOVhj4ZhwkTM=">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</latexit>

再び多くの準位が縮退(超変形魔法数の出現)
Ragnarsson et al., Phys. Rep. 45, 1 (1978)

漸近量子数
スピンー軌道力とL2項は変形調和振動子の基底関数では対角的ではない
変形度の大きい極限では無視できる→変形調和振動子基底の近似はよい

軸対称変形の漸近量子数(asymptotic quantum number)|N nz ⇤⌦i<latexit sha1_base64="9aUmg5XobAJwHfxdM5CyUOv4tQk=">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</latexit>

N: 主量子数、 nz: z方向のノード数、Λ: 軌道角運動量のz成分
Ω：角運動量のz成分⌦=⇤+⌃(π：パリティ)<latexit sha1_base64="ycv5Zo1ygJO+w54w67wVyqkvsBo=">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</latexit>

Nilssonレベルの表記： Ωπ[ N nz Λ ]軸対称変形での量子数：Ω, πのみ|⌦, ki =X

C(N ljk)|N lj⌦i =

N lj

X

N nz ⇤k

C 0 (N nz ⇤k)|N nz ⇤⌦i

変形一粒子状態は球形基底でもNilsson基底でも展開できる<latexit sha1_base64="R0Ke2sc12WzH+dYSCmif4oxhY8c=">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</latexit>

球形での量子数|N lj⌦i<latexit sha1_base64="y2T55g6o1r57QUGXCR1NUS1jkHA=">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</latexit>

Nilsson図

Table of Superdeformed bandより量子数が同じ軌道は相互作用して交差しない

球形核の集団振動運動
原子核内の核子の運動：平均ポテンシャルによって作られる一粒子軌道
一粒子運動は平均ポテンシャルの性質に影響を及ぼす
集団運動：コヒーレントな一粒子運動による平均ポテンシャル全体の運動
平均ポテンシャルの表面振動(形の集団運動)
球形からのずれαは小さい
表面振動の量子化 b† µ =<latexit sha1_base64="uRoBEfQUle56izWWuhy2J9b2FD4=">AAADJ3ichVFLaxRBEK4ZX+v6yKoXwcvgEvHiUhMCCYIQ4sVjHm4SSMehp7d3t0nPw57ehTjMH/APeMgpAQ8hPyA/wIs3T4o5ehSPEbx4sHZ2VJLFWMNMV31V3zf9UWGqVWYRjx33wsVLl6/UrtavXb9xc6px6/ZalgyMkG2R6MRshDyTWsWybZXVciM1kkehluvh9tNRf30oTaaS+LndSeVWxHux6irBLUFBYz98wTq8F+RME6nDWTQovCcey14am3usa7jIF4Oq6bEkkj3+uyzyGdYPuSkKj3Gd9vlplUcVXRV5JTce9/7onZEjmVRNXidoNLGFZXiTiV8lTahiKWkcAYMOJCBgABFIiMFSroFDRs8m+ICQErYFOWGGMlX2JRRQJ+6ApiRNcEK36dujarNCY6pHmlnJFvQXTa8hpgfT+BEP8ATf4yF+xZ//1MpLjdFddugMx1yZBlOv767++C8rotNC/y/rHEZI0+d7stCF+dKLIm9piYxcirH+8NWbk9XHK9P5A9zHb+RvD4/xHTmMh9/F22W5sgt1WpB/dh2TydpMy8eWvzzbXFisVlWDe3AfHtI+5mABnsEStEE4NaflzDnz7oH7wf3kfh6Puk7FuQOnwv3yC6cYz+Q=</latexit>r

2R(✓, ') = R0 41 +3

X↵⇤ µ Y µ (✓, ')5

µ

X 1
1X
2
Hcoll =|⇡ µ | +C |↵ µ |22B2µ<latexit sha1_base64="sqfF9V/ikmuEvuOoo+g63vSY3dA=">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</latexit>

B !
↵ µ
2~

p

µ

i
⇡ † µ 表面振動のフォノン演算子
2~B !

0+,2+,3+,4+,6+

r

C
B

~!

✓

! =
<latexit sha1_base64="p1d9YDLBoXPBsDUCMP6IvA6S0VE=">AAACuHichVHLShxBFD22STRjEkfdBLIZMihZDbdVUARBdJOlj4wKtozVbc1Y2C+rawa0mR/wB0SyUnAh+YAss8jGfIALP0GyNJCNi9zp6RiixNyiqk6duudWHa4b+yoxRFc9Vu+Tp8/6+p8XBl68fDVYHBpeTaKm9mTVi/xIr7sikb4KZdUo48v1WEsRuL5cc3cXOvdrLakTFYUfzH4sNwPRCFVdecIwVStOOFEgG6Lm+KzZFqXZkpPsaZM6dS28dOE3307n72C7VixThbIoPQR2DsrIYzEqfoaDbUTw0EQAiRCGsQ+BhMcGbBBi5jaRMqcZqexeoo0Ca5ucJTlDMLvLa4NPGzkb8rlTM8nUHr/i89SsLGGULumcbuiCPtE13f6zVprV6Pxln3e3q5VxbfDw9crP/6oC3g12/qgeUbic/bgngzqmMy+KvcUZ03Hpdeu3Do5uVmaWR9MxOqXv7O+ErugrOwxbP7yzJbn8EQVukH2/HQ/B6njFpoq9NFmem89b1Y83eIt33I8pzOE9FlHld4/xBRf4Zs1YW1bDUt1UqyfXjOCvsPQvkaulXQ==</latexit>

Ĥcoll =<latexit sha1_base64="X0l9Ia6fWuMGqop9yNMCgOwKi60=">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</latexit>Xµ0+,2+,4+1b† µ b µ +
2

◆

~!2
<latexit sha1_base64="Ng9ZKfRBYr0AyWVfVfXkZkOPT8c=">AAACkHichVG7SgNBFD1Z3/EVtRFsxBCxCndFUGyM2ohVYowJJCHsrmNc3Je7m0AM/oA/YGGjgoX4AX6AjT9g4SeIpYKNhTebBdFgvMPMnDlzz505XNUxdM8neo5IPb19/QODQ9HhkdGx8djE5J5n11xN5DTbsN2CqnjC0C2R83XfEAXHFYqpGiKvHm227vN14Xq6be36DUeUTaVq6Qe6pvhMlUuHquKWbFNUlcpiJRanJAUx2wnkEMQRRtqO3aOEfdjQUIMJAQs+YwMKPB5FyCA4zJXRZM5lpAf3AqeIsrbGWYIzFGaPeK3yqRiyFp9bNb1ArfErBk+XlbNI0BPd0hs90h290OeftZpBjdZfGryrba1wKuNn09mPf1Um7z4Ov1VdFCpnd/fk4wArgRedvTkB03KptevXT87fsqs7ieY8XdMr+7uiZ3pgh1b9XbvJiJ0LRLlB8u92dIK9xaRMSTmzFE9thK0axAzmsMD9WEYKW0gjx+8e4xyXuJImpRVpTVpvp0qRUDOFHyFtfwEikJTe</latexit>[b†2 b†2 ]LM |0)<latexit sha1_base64="9RiU3IxFRWzMdPO+hulV4fjpsKk=">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</latexit>

~!2

2+

<latexit sha1_base64="Ng9ZKfRBYr0AyWVfVfXkZkOPT8c=">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</latexit>~!2b†2µ |0)<latexit sha1_base64="M99a+P0XLe8Ilf/wbkvk+Ypc28s=">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</latexit><latexit sha1_base64="Ng9ZKfRBYr0AyWVfVfXkZkOPT8c=">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</latexit>

0+|0)<latexit sha1_base64="ejH+s2onwfF1FVrt34pJ/JikLpQ=">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</latexit>

変形核の集団運動
球形核での四重極振動のHamiltonianH=<latexit sha1_base64="Za5Am0TshNGD+7glmCP9sUOSUso=">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</latexit>

1 X
1 X
B
|↵˙ µ |2 + C|↵µ |222µµ

座標変換(変形パラメターとEuler角)↵µ ! , , ⌦<latexit sha1_base64="qD8dv+5IeSyIu44d9p81jxLHK5s=">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</latexit>

H = Trot + Tvib + V ( , )<latexit sha1_base64="dOj6AUI6YzImcfKCBVGRmUo6UMo=">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</latexit>

3

X
回転の運動エネルギー Trot = 1
Jk !k22<latexit sha1_base64="I9+JBZuT/Nqj37dNRPOMEM1sn3g=">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</latexit>

慣性モーメントk=1

振動の運動エネルギー Tvib = 1 B( ˙ 2 +2

Jk = 4B 2 sin2

2 2

˙ )

<latexit sha1_base64="L/88HPev+VkJhFxV9F/qCQSsh5U=">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</latexit>

✓

2
⇡k
3

◆

<latexit sha1_base64="Gy2V+rKqCxyLtZuhB5YiePYZfEU=">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</latexit>

軸対称変形核の場合の量子化
対称軸周りの回転は禁止(J3=0)、J1=J2
3
XIˆk2
1 ˆ2 ˆ2
1 ˆ2
Ĥrot =
回転
Ĥ
=
(
I
+
I
)
=
(I
rot
1
2
2Jk
2J
2J
k=1

Iˆ32 )<latexit sha1_base64="FNDgcAUSwNV8ecJ79dRZJ04WG5g=">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</latexit><latexit sha1_base64="BiBJG5Jhj9yIDSfHWwkYyDdXJUw=">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</latexit>

β振動

1 @2
1
Ĥ =
+
C (
2B @ 2
2

1
1 @ @
Ĥ =
2B 02
@ @

0)

2

<latexit sha1_base64="zE+yiPQBJte6dGCoB03liRvG3Dc=">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</latexit>

γ振動

K2
1
+
C
4 2
2

2

対称軸周りの回転はγ振動と結合<latexit sha1_base64="9oP5sTiPyuG+lOhDHmN33qJbNJQ=">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</latexit>

球形核の四重極振動を出発点にしているのでBの扱いは正確ではない

軸対称変形核のスペクトル~2
E(I, K, n , n ) =[I(I + 1)
2J✓

1
K 2] + n +2<latexit sha1_base64="skn71HdVV8D1dYuRMEZtPscCBU4=">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</latexit>

回転励起

◆

~! + (n + 1)~!

β振動

γ振動

変形して初めて現れる自由度：回転運動
球形核は回転の自由度がない(量子系では回転させても状態が変わらない)

6+
6+
4+
2+
0+

(nβ,nγ)=0

4+
2+
0+

5+

6+

4+

4+

5+
4+

2+
0+

3+
2+

β band γ band

Casten, “Nuclear Structure from a simple perspective” より

第一2+励起状態と第一4+励起状態のエネルギー比(R4/2)
球形核では2, 変形核では10/3~3.3
基底状態の変形の情報は励起状態からわかる

γ bandγ band対相関
対相関：時間反転対称な軌道に対で入った核子は引力相互作用を受ける
これまでの平均ポテンシャルには入らない相関(残留相互作用)

錫同位体の１中性子の分離エネルギー

Sn(N,Z) (MeV)

対相関の実験的証拠
18
偶々核と奇核で束縛エネルギーに差(odd-even staggering)1614(奇核では最後の粒子がペアを組めないため
12近くの偶々核より対相関分だけエネルギーを損する)
10
1中性子の分離エネルギー
Sn (N, Z) = B(N, Z)

B(N

1, Z)

<latexit sha1_base64="wEWnZpr1Ypytfo/ECK8AeTIJDrg=">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</latexit>
sha1_base64="dO1RVWgbv0QBXFnWGIM/DJuxBX8=">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</latexit>
sha1_base64="G/xB5J7tl/9e8KXsKeJf8wtsrVY=">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</latexit>
sha1_base64="nNGyvmhRMIH/nJRMpJ3GBEJ9aAA=">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</latexit>

B:束縛エネルギー

E(N, Z) =対ギャップ(いろいろな定義)<latexit sha1_base64="p3yW2uL0Cq+p+E0Yk6gJ19aXB1c=">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</latexit>

2Δ

8
6
4
2
0
50 55 60 65 70 75 80 85 90
Nn (N, Z) =1
[Sn (N + 1, Z)
4

2Sn (N, Z) + Sn (N

(
(3)
(N,
Z)
=
n

1)N +1[B(N2

2B(N, Z) + B(N + 1, Z)]<latexit sha1_base64="8tQ0WimkowcKwQn2veQRohPi+2M=">AAACyXichVFNS9xQFD2mVu201Wm7KbgJDhaLznAjQkUQpHVREEQdR8WZYUjiGw1mkpi8GaphVt31D7joqgUX2h/QH+DGpZsupv+guFRw04U3HyAq6g3JO/fce+7L4RqebQWSqNOlPOl+2tPb9yzz/MXL/oHsq9crgdv0TVEyXdv11ww9ELbliJK0pC3WPF/oDcMWq8b2p6i+2hJ+YLnOstz1RLWhbzpW3TJ1yVQtO1uZFbbUa87I/Nj6e3VazVfqvm6GWjucaJdVtRhVRrWolh9PsgirownOR5VqLZujAsWh3gVaCnJIY8HN/kYFG3BhookGBBxIxjZ0BPyUoYHgMVdFyJzPyIrrAm1kWNvkLsEdOrPb/N3krJyyDufRzCBWm3yLza/PShXD9IcO6ZxO6Bf9o//3zgrjGdG/7PJpJFrh1Qa+vS1ePqpq8Cmxda16QGFw98OeJOqYjL1Y7M2Lmcilmcxv7e2fF6eWhsN39JPO2N8P6tAxO3RaF+bBolj6jgwvSLu9jrtgZbygUUFbnMjNfExX1YdBDGGE9/EBM/iMBZT43iOcooO/ypyyo3xR9pJWpSvVvMGNUL5eAe54pSA=</latexit><latexit sha1_base64="W1Cq/bWQCTg4vNXhH4xa80Rj4Lk=">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</latexit>

B(N, Z)<latexit sha1_base64="cUCrrqVGnwhf5jgT16XAIsVdfDU=">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</latexit>

Sn

1, Z)

p
⇡ 12/ A MeV

対相関は平均ポテンシャルに対ポテンシャル入れる拡張で扱える1, Z)]

準スピン形式
対相関を記述するHamiltonian Ĥ = "N̂GP̂ † P̂ の厳密解

<latexit sha1_base64="32lYpLlTSmKk15HkaBwIyphpM0c=">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</latexit>

一準位の模型、縮退度は2Ω
g9/2とすれば2Ω=2j+1=10N̂ =<latexit sha1_base64="AjISf3eqp7vhoEkMT1BcbD18iXs=">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</latexit>P̂ † =<latexit sha1_base64="1v40XPJylL3cScbg6rfAc0hm6ZI=">AAACv3ichVHLShxBFD12YjQTE8dkE3DTOChZDbdDIEEQTNxkl/ExKtjaVLdlT2G/6K4ZmDTzA/kBF9kkgSzED/ADBMku2bjwEyRLA9lkkTs9LaISc5vqOnXuPbfqcN0kUJkmOh0y7twdvjcyer/yYOzho/HqxOPVLG6nnmx6cRCn667IZKAi2dRKB3I9SaUI3UCuubsL/fxaR6aZiqMV3U3kZij8SO0oT2imnOqs3RI6b/S27G3hm3OmnbVDJ1dzFjPvQukL0ytSjroAue2KNFe9nlOtUZ2KMG8CqwQ1lNGIq4ewsY0YHtoIIRFBMw4gkPG3AQuEhLlN5MyljFSRl+ihwto2V0muEMzu8t/n00bJRnzu98wKtce3BLxSVpqYphPap3P6Rgd0Rn/+2SsvevTf0uXdHWhl4ox/eLr8+7+qkHeN1qXqFoXL1bd70tjBq8KLYm9JwfRdeoP+nfd758uzS9P5DH2hn+zvM53SETuMOr+8r4ty6SMqPCDr+jhugtXndYvq1uKL2vybclSjmMQUnvE8XmIeb9FAk+/9hGN8xw/jteEbkZEMSo2hUvMEV8Lo/gUhqKeL</latexit>

準スピン形式Ŝ+ = P̂ †<latexit sha1_base64="xNvO/D/e2vHGK76ocCb8ZAF/AxI=">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</latexit>⌦X(c†i ci + c†ī cī )

i=1
⌦
X
i=1

<latexit sha1_base64="NCDcqimm0opWh3mxKhlYVF3Q7Vw=">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</latexit>

対生成演算子c†i c†ī

ibar: iの時間反転状態

1= P̂
Ŝz = (N̂2
[Ŝ± , Ŝz ] = ±ŜzŜ

粒子数演算子c†j m̄ = ( 1)j m c†j, m<latexit sha1_base64="tqxJrBK7LxARwLAJWMoZehw4BJQ=">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</latexit>⌦)<latexit sha1_base64="T1rG8QViX9nCWi7x/6qd3jBFyZk=">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</latexit>[Ŝ+ , Ŝ ] = 2Ŝz<latexit sha1_base64="J5Mcl+kLkAKYWGgYwDef0/vN3+U=">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</latexit><latexit sha1_base64="k5jsKSS1CCZb8Kn5N96OGWwojz8=">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</latexit>

Sˆ |S, Sz =Sz=-S状態での粒子数v: seniority

Si = 0<latexit sha1_base64="V3yZ41ciOOrimBiqCpSTNmw54Zc=">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</latexit>

Sz =<latexit sha1_base64="ydzLoQJPtFsmtLWuXs4SEV/tZk4=">AAACpnichVHLSiNBFD227/iKuhmYTWtQMgvDbREUQRBn42pGjTGCkUx3W4mN/aK7E4hN1oI/4MKVgguR2ZoPcOMPuPATxKWCGxfedBpERb1FVZ06dc+tOlzNNQ0/ILptk9o7Oru6e3oTff0Dg0PJ4ZEN36l4usjpjul4m5rqC9OwRS4wAlNsup5QLc0UeW3vd/M+XxWebzj2elBzxballm2jZOhqwFQxOZYt7i/IU1l5QS6UPFUPlXo4XU9X5Sm58NcSZfVXMZmiDEUhfwRKDFKIY8VJNlDADhzoqMCCgI2AsQkVPo8tKCC4zG0jZM5jZET3AnUkWFvhLMEZKrN7vJb5tBWzNp+bNf1IrfMrJk+PlTIm6IbO6YGu6YLu6PnTWmFUo/mXGu9aSyvc4tDhj+zTtyqL9wC7r6ovFBpnf+0pQAlzkReDvbkR03Spt+pX948esvNrE+EkndI9+zuhW7pih3b1UT9bFWvHSHCDlPft+Ag2pjMKZZTVmdTiUtyqHvzEONLcj1ksYhkryPG7B/iPSzSktPRHykn5VqrUFmtG8Sakfy+GN5sy</latexit>

Ĥ = "(2Ŝz + ⌦)<latexit sha1_base64="Au7lw48PoSQ5jHuoYnm0+AqDAXY=">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</latexit>

G[Ŝ 2S=

1(v
2Ŝz (Ŝz

角運動量と同じ代数(SU(2))

v: 対を組んでいない粒子の数

⌦)

1)]

E(N, v) = "N<latexit sha1_base64="Ws9oA9Euwsrl0Khg3oAJCxJepGA=">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</latexit>

1
G(N
4

v)(2⌦

N

v + 2)

Nが偶数のとき。基底状態ではv=0
Nが奇数のとき。基底状態ではv=1(すべての粒子が対を組む)
G
G
E(N, 0) = "N
(2N ⌦ N 2 + 2N )E(N, 1) = "N(N 1)(2⌦ N + 1)441
対相関力からodd-even staggeringが出る
n (N ) = G(2⌦ + 1)
4<latexit sha1_base64="/2X0m8dXh5KyM0KiZT5m1nzSgx4=">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</latexit><latexit sha1_base64="uenXNf65A1Lh7rfxL5exXPHlymQ=">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</latexit><latexit sha1_base64="3WMNKmOVV6CmJ6gy1e/MV/82KUs=">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</latexit>

対相関の集団運動
対振動
通常相原子核からのpair transfer→対フォノンの励起

Epair (N ) = |N

N0 |~!vib

対回転
超伝導相でゲージ対称性の破れ
対ポテンシャルのゲージ空間での回転運動

1
Epair (N ) =(N
2JN

N0 ) 2

Brink and Broglia “Nuclear Superconductivity”
review: Broglia et al., Phys. Rep. 335, 1(2000)

まとめ
p 原子核の形
p 厳密には原子核(有限孤立量子系)変形しない(回転対称性は破れない)
p 平均ポテンシャルを導入すると変形しうる
p 平均ポテンシャル
p 球対称調和振動子ーシェル構造
p スピンー軌道力ー魔法数p 軸対称変形調和振動子ー変形シェル構造・Nilsson軌道・超変形
p 平均ポテンシャルの集団運動
p 球形核：表面振動の量子化
p 変形核：振動運動(β、γ振動)と回転運動
p 対相関
p 準スピン形式
p 対相関の集団運動：対振動と対回転