原子核系における大振幅集団運動の非経験的記述
東北大学理学研究科

松本 萌未
共同研究者：谷村雄介、萩野浩一
The world of mathematical sciences 2023/8/24@IPMU

原子核のかたち
原子核ってどんな形をしていると思いますか？

たんぽぽの種（cf. 石原さん）

陽子と中性子原子核のかたち
原子核ってどんな形をしていると思いますか？
実は、原子核って必ずしもまんまるじゃない！
原子核の密度分布の理論計算

陽子と中性子原子核の集団運動
原子核系では、大振幅の集団運動が重要となる現象が特徴的
形が大きく変化する！

変形共存現象核分裂
我々の目標:

集団運動を微視的に記述する理論手法の開発

全ての陽子・中性子を動力学的自由度として扱う

J. Dobaczewski, J. Phys.: Conf. Ser. 312 092002 (2011). / A. N. Andreyev. et al. Nature 405, 430 (2000).

集団運動の微視的理論| i=| i<latexit sha1_base64="UxLzuwEQ/Sh6+6wOs9Bozmr0JL0=">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</latexit>: 単一のSlater 行列式 (Hartree-Fock近似)

変形などの揺らぎを表現できない集団運動

集団運動の微視的理論| i=| i<latexit sha1_base64="UxLzuwEQ/Sh6+6wOs9Bozmr0JL0=">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</latexit>: 単一のSlater 行列式 (Hartree-Fock近似)

変形などの揺らぎを表現できない
集団運動

生成座標法

異なる変形の状態を重ね合わせて重みを決定
基底: 事前に用意

X
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X X X
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2a =
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重みf を変分原理に基づいて決定

変形度（伸び）のパラメータQ2aai + · · ·<latexit sha1_base64="980HV1H4Um25xOXbFgoQ+2cmXL4=">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</latexit>

経験則に基づいて基底を準備

集団運動の微視的理論
生成座標法

異なる変形の状態を重ね合わせて重みを決定
基底: 事前に用意

X
X
X X X
fa4 |
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2a =
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sha1_base64="YfseJ6SIYm84otIO7JRnCtWtpmQ=">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</latexit>aaa重みf を変分原理に基づいて決定a

変形度（伸び）のパラメータQ2aai + · · ·<latexit sha1_base64="980HV1H4Um25xOXbFgoQ+2cmXL4=">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</latexit>

経験則に基づいて基底を準備

我々の新手法 重みと基底を同時に変分で決定

X
X
X X X
fa4 | ?a i + · · ·| ?a i +f
i =| i =| ffai1 |=f?aa|i |+f| ?a i |+ffia3 =
2a =
a ifi
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a

a

a

a

a

大振幅集団運動の非経験的な記述→“最適”な集団運動とは？

手法
試行関数：

一粒子状態 (正規直交系)：Hamiltonian kernel

エネルギー：：Norm kernel<latexit sha1_base64="BF+eE9f+8LHjDe7lCdVWeVB/3TI=">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</latexit>

h
(ab)
⇢ ↵ =

| bi
：遷移密度行列h a| bi†a | a↵ a：EDF：HF Hamiltonian

手法
エネルギーの勾配を計算

Shimizu et al. , PTEP 2012, 01A205

・一粒子状態：Hamiltonian kernel

・重み関数：Norm kernel<latexit sha1_base64="BF+eE9f+8LHjDe7lCdVWeVB/3TI=">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</latexit>

h
(ab)
⇢ ↵ =

(=0 Hill-Wheeler eq.)
エネルギーの極小値を共役勾配法で探索| bi：遷移密度行列h a| bi†a | a↵ a：EDF：HF HamiltonianSet up
28Si の基底状態の計算

軸対称 / 反転対称性を課した計算• Skyrme 相互作用（time-odd部分を無視）
SIII

M. Beiner et al., Nucl. Phys. A 238, 29 (1975).

• クーロン力はなし
初期状態のSD: 異なる変形度の変形Woods-Saxonポテンシャルにより決定

結果 28Si| i=| i<latexit sha1_base64="UxLzuwEQ/Sh6+6wOs9Bozmr0JL0=">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</latexit>

Hartree-Fock<latexit sha1_base64="0fTyWOQdkgjnCnpmmjD/osyF4AM=">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</latexit>

生成座標法（従来手法）

| i=

X

fa |

X

fa |

a
基底の選び方：“通常行われる方法”で選択（次のスライドで詳しく）本手法（基底も最適化）<latexit sha1_base64="uGsyHH3zuHPndqxKFHtYjgDje+0=">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</latexit>

| i=

a

-272.63 MeV
1.3MeVai-273.95 MeV
1MeV

ai

-275.00 MeV“通常行われる方法”では集団運動の記述は十分でない！

結果 28Si“通常行われる方法”
色々な変形度に対し、その変形度であるようなエネルギー最小の状態を基底として重ね合わせる<latexit sha1_base64="ZQnh5+6K/VlsfZ4BQca7fhue178=">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</latexit>

Q =

Z

z

3

2

d r r Y 0 (r̂)⇢(r)

Q2 < 0パンケーキ

Q2 = 0球形

Q2> 0
ラグビーボールQ2

結果 28Si“通常行われる方法”
色々な変形度に対し、その変形度であるようなエネルギー最小の状態を基底として重ね合わせる-245

Energy (MeV)

-250

CHF
OptGCM

青い線：従来手法
赤い線：本手法

-255
-260
-265
-270
-275
-50 -40 -30 -20 -10 0 10 20 30
Q2 (fm2)“最適”な基底は、
その変形度の基底状態ではない！

結果 28Si得られた基底でのQ4（次に高次のモーメント）を計算してみた
800

-245

600
200
0
-200

Energy (MeV)

Q4 (fm4)

400

-250

CHF
OptGCM

-255
-260
-265

-400

-270

-600
-50 -40 -30 -20 -10 0 10 20 30
Q2 (fm2)-275-50 -40 -30 -20 -10 0 10 20 30
Q2 (fm2)

Q2・Q4両方を考慮してエネルギー最小の状態を基底として取っても、最適化で含まれたような励起状態は含まれない。

まとめ
ü 基底の変分を伴う生成座標法を開発→ 経験によらない集団運動の記述が可能にü

28Siの解析では、高い励起エネルギーの状態も基底として得られた

ü 大振幅集団運動の記述は、従来仮定していた方法よりも複雑なことを考える必要がありそう

これからやりたいこと＊ 集団励起状態の解析＊ 既存の集団運動の模型計算との関係の議論