マス・フォア・インダストリ研究 No.26

材料科学における
幾何と代数IV

編集

松谷 茂樹
緒方 勇太
落合 啓之
加葉田雄太朗
佐伯
修
濵田 裕康
松江
要

Institute of Mathematics for Industry
Kyushu University

About the Mathematics for Industry Research
The Mathematics for Industry Research was founded on the occasion of the certificationof the Institute of Mathematics for Industry (IMI), established in April 2011, as a MEXT
Joint Usage/Research Center – the Joint Research Center for Advanced andFundamental Mathematics for Industry – by the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) in April 2013. This series publishes mainly proceedingsof workshops and conferences on Mathematics for Industry (MfI). Each volume includes
surveys and reviews of MfI from new viewpoints as well as up-to-date research studies
to support the development of MfI.
October 2022
Kenji Kajiwara
Director
Institute of Mathematics for Industry

Off-Shell Mathematical Science Toward System’s Analysis and Designing
Mathematics for Industry Research No.26, Institute of Mathematics for Industry, Kyushu University
ISSN 2188-286X
Editors: Shigeki Matsutani, Yuta Ogata, Hiroyuki Ochiai, Yutaro Kabata, Osamu Saeki, Kaname Matsue,
Hiroyasu Hamada
Date of issue: 28 November 2023
Publisher:
Institute of Mathematics for Industry, Kyushu University
Motooka 744, Nishi-ku, Fukuoka, 819-0395, JAPANTel +81-(0)92-802-4402, Fax +81-(0)92-802-4405
URL https://www.imi.kyushu-u.ac.jp/
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TEL +81-(0)92-531-7102 FAX +81-(0)92-524-4411

材料科学における幾何と代数 IV

編集

：

松谷

茂樹

緒方

勇太

落合

啓之

加葉田雄太朗
佐伯

修

濵田

裕康

松江

要

はじめに

本研究集会 II は，研究集会 II「結晶のらせん転位の数理」（2016 年）
，研究集会 I「結晶の界面，転位，構造の数理」（2017 年）
，研究集会 II「結晶の転位の先進数理解析」（2018年），研究集会 II「結晶の界面，転位，構造の先進数理解析」（2019 年）
，研究集会 II「材料科学における幾何と代数 I」（2020 年）
，研究集会 II「材料科学における幾何と代数 II」（2021年）研究集会 I「材料科学における幾何と代数 III」
（2022 年）の継続と位置付けられる研究会である．
2020 年から実施している「材料科学における幾何と代数 I,II,III」に引き続き，本研究会
は材料科学と幾何学や代数学との交流を目指した．この背景には，1)技術の発展により産業界では求められる仕様が大きく変貌したこと，2)観測装置や材料の製造装置・方法が発展し，
例えば原子レベルでの構造の乱れの観測や，原子レベルでの材料の制御などが可能となったこと，3)それらにより，従来材料科学で使われてきた数学だけでは表現できていない新たな観測事実や現象が生じていることがある．現在，科学・技術の言葉として，より高度な数学が望まれている．解析分野においては，既に材料科学者と数学者の交流が行われているようであるが，幾何学や代数学では，交流は限られたものとなっている．そこで，幾何学，代数学的手法に関わる数学者を迎えて，材料科学の研究者と議論する場を提供し，相互理解のきっかけを得ることが本研究集会の目的とした．
本研究会では 46 名の参加者を得，次に示す方々に講演をして頂いた．l) オープニングとして， 2022 年の研究集会「材料科学における幾何と代数 III」の成果である「A novel
symmetry in nanocarbons: pre-constant discrete principal curvature structure 」
Kabata,Matsutani, Noda, Ogata, Onoe の概要を述べ、今回のテーマについて，松谷茂樹（金沢大学）．

2) 曲がった量子系の量子効果を発言する新規ナノカーボンでの実験的実証について，尾上順氏（名古屋大）． 3) 尾上氏が対象としているナノカーボンの第一原理計算による考察について，野田祐輔氏（岡山県立大学）. 4) III の成果でもある新たな離散幾何
学的対称性に関して(arXiv:2306.15846)， 緒方勇太氏（京都産業大学）， 5) カーボングラ
フに関わる離散力学系のソボレフ不等式に関して，山岸弘幸氏（都立産技高専）， 6) カー
ボングラフに関わる分子力学系としての安定性と離散幾何に関して，雷霄雯氏（東工大），
7) C60 が持つ群論的な高次の対称性について，落合啓之氏(IMI)、 8) 材料科学と数学の共同研究のあり方や成果に関して，中村振一郎氏（熊本大）．
また、講演の後の質疑応答やフリーディスカッションなどにより，今回も，本研究集会の目的は達成されたと考えている．

－i－

材料科学においては，近年，急速に必要となっている幾何・代数の材料科学への適応に関わる研究の更なる加速が期待される．本研究会はその礎・足場として期待に応えるものとなったと考え，ここに報告をする．
組織委員代表 松谷茂樹
2023 年 10 月 30 日

組織委員
松谷茂樹

金沢大学

緒方勇太

京都産業大学落合啓之

九州大学

加葉田雄太朗 長崎大学

－ii－

佐伯修

九州大学

濵田裕康

佐世保高専

松江要

九州大学

Contents
はじめに ・ ・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・

iプログラム・ ・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・ v
集合写真・ ・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・ vii
オープニング：幾何，代数，ナノカーボン(On a theme in the workshop, Geometry and slgebra in material science IV)
松谷茂樹 , Shigeki Matsutani (Kanazawa University) ・・・・・・・・・・・・・ 1
１次元凹凸 C60 ポリマーを用いた幾何曲率効果の理論予想と実験的検証(Theoretical prediction and experimental demonstration of geometry-driven
curvature effects using 1D uneven structured C60 polymer)
尾上順 , Jun Onoe (Nagoya University)・ ・・・・・・・・・・・・・・・・・ 20
ピーナッツ型フラーレンポリマーのエネルギー的安定性の第一原理的考案(A first-principles consideration of energetic stability of peanut-shaped)
fullerene polymer
野田祐輔 , Yusuke Noda (Okayama Prefectural University)・・・・・・・・・・・ 37
格子欠陥の階層性に着目した低次元ナノ炭素材料の数理解析(Mathematical analysis of low-dimensional nanocarbon materials focusing on thehierarchy of lattice defects)
雷霄雯 , Xiao-Wen Lei (Tokyo Institute of Technology)・・・・・・・・・・・・・ 52
C60 の数理 I, II(Math behind C60 I , II)落合啓之 , Hiroyuki Ochiai (Kyushu Univ ersity)・ ・・・・・・・・・・・・・・ 69
境界値問題のグリーン関数とソボレフ不等式の最良定数(Green function of boundary value problem and the best constant of Sobolev inequality)
山岸弘幸 , Hiroyuki Yamagishi (Tokyo Metro. Col. of Ind. Tec)・・・・・・・・・ 112
On discrete constant principal curvature surfaces
緒方勇太 , Yuta Ogata (Kyoto Sangyo University)・ ・・・・・・・・・・・・・ 142
自然界にある分子と材料の振る舞いを決めている数理と計算科学(Mathematical and computational science governing the functions of molecules andmaterials in nature)
中村振一郎 , Shinichiro Nakamura (Kumamoto University)・ ・・・・・・・・・ 152－iii－

IMI Workshop II: 材料科学における幾何と代数 IV(Geometry and Algebra in Material Science IV)

九州大学マス・フォア・インダストリ研究所
ハイブリッド 研究会 (2023 年 9 月 4 日 (月)-5 日 (火))1Program

9 月 4 日 ( 月)
12:45-13:05
13:10-14:10

松谷茂樹 (金沢大学)

尾上順（名古屋大学）

１次元凹凸 C60 ポリマーを用いた幾何曲率効果

14:25-15:25

野田祐輔（岡山県立大学）

ピーナッツ型フラーレンポリマーのエネルギー的安定性

15:40-16:40
16:55-17:55

雷霄雯(東京工業大学)
落合啓之 (九州大学)

格子欠陥の階層性に着目した低次元ナノ炭素材料の数理解析

山岸弘幸

境界値問題のグリーン関数とソボレフ不等式の最良定数

9 月 5 日 ( 火)
9:50-10:50
11:05-12:05
12:05-14:05
14:05-15:05
15:20-16:20
16:20-16:25(都立産業技術高等専門学校)
緒方勇太（京都産業大学）
昼休憩
落合啓之 (九州大学)
中村振一郎（熊本大学）
クロージング

オープニング／幾何と代数とナノカーボン材料の理論予想と実験的検証の第一原理的考察

C60 の数理 I

On discrete constant principal curvature surfaces
C60 の数理 II

自然界にある分子と材料の振る舞いを決めている数理と計算科学－v－1

IMI Workshop II: Geometry and Algebra in Material Science IV
Hybrid conference (Zoom & West W1-D-413 room in Kyushu University)
September 4 (Mon) - 5 (Tue), 2023

September 4 (Mon)12:45-13:05 Shigeki Matsutani (Kanazawa Univ.)

Opening: Geometry, algebra and nanocarbon

13:10-14:10

Theoretical prediction and experimental demonstration

Jun Onoe（Nagoya Univ）

of geometry-driven curvature eﬀects using 1D unevenstructured C60 Polymer
A ﬁrst-principles consideration of energetic stabilityof peanut-shaped fullerene polymer

14:25-15:25

Yusuke Noda (Okayama Pref. Univ)

15:40-16:40

Xiao-Wen Lei (Tokyo Inst. Technology)

16:55-17:55

Hiroyuki Ochiai (Kyushu Univ.)

Mathematical analysis of low-dimensional nanocarbon
materials focusing on the hierarchy of lattice defects
Math behind C60 I

September 5 (Tue)
9:50-10:50 Hiroyuki Yamagishi(Tokyo Metro. Col. of Ind. Tech.)Green function of boundary value problem and the best
constant of Sobolev inequality

11:05-12:05

On discrete constant principal curvature surfaces

12:05-14:05
14:05-15:05
15:20-16:20
16:20-16:25

Yuta Ogata（Kyoto Sangyo Univ.）
Lunch
Hiroyuki Ochiai (Kyushu Univ.)

Math behind C60 II
Shinichiro Nakamura（Kumamoto Univ.） Mathematical and computational science governing thefunctions of molecules and materials in nature
Closing2－vi－

－vii－IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, JapanOn a theme in the workshop,
Geometry and Algebra in Material Science IV
Shigeki MATSUTANIKanazawa University
I will give an overview of our workshops ”Geometry and Algebra in Material Science
I-IV” and describe the theme of this meeting based on [1].
[1] Y. Kabata, S. Matsutani, Y. Noda., Y. Ogata, and J. Onoe, A novel symmetry innanocarbons: Pre-constant discrete principal curvature structure,
arXiv:2306.15839, 2023.

－1－※‿※‒⁉⁡⁤⁝⁥⁚⁡⁢‒※※
஬૰ᅹ‫↚ܖ‬ⅹↀ↺ˊૠ↗࠹˴※⁈ʋ߸‫˙ ܖٻ‬ᣃ⇓∉∙⇷⇟ ⇌⇎⇟⇮‣ӭ᫾ ‶౐…᨞ ※‿※⇐∞⇭⇉⇮∐⇈∆≋⁉‣‟‶‟…‣‥≌≕≓≕≖࠰≜உ≗ଐ≋உ≌∞ ≘ଐ≋້≌≜உ≗ଐ ‣․‬‧•᧏‫ڼ‬․•‣ ࠰ᄂᆮᨼ˟※※Ⅴኽ୒↝↸↊⇂᠃ˮ↝ૠྸⅥ․•‣ ࠰ᄂᆮᨼ˟※Ⅴኽ୒↝မ᩿≏᠃ˮ≏ನᡯ↝ૠྸⅥ․•‣‪࠰ᄂᆮᨼ˟※※Ⅴኽ୒↝᠃ˮ↝έᡶૠྸᚐௌⅥ․•‣‫࠰‫‬

ᄂᆮᨼ˟※※Ⅴኽ୒↝မ᩿≏᠃ˮ≏ನᡯ↝έᡶૠྸᚐௌⅥ․•․•࠰ᄂᆮᨼ˟※※Ⅴ஬૰ᅹ‫↚ܖ‬ⅹↀ↺࠹˴↗ˊૠ※Ⅵ․•․‣࠰ᄂᆮᨼ˟※※Ⅴ஬૰ᅹ‫↚ܖ‬ⅹↀ↺࠹˴↗ˊૠ※※Ⅵ․•․․࠰

ᄂᆮᨼ˟※Ⅴ஬૰ᅹ‫↚ܖ‬ⅹↀ↺࠹˴↗ˊૠ※※※Ⅵ․•․‥࠰

ᄂᆮᨼ˟※Ⅴ஬૰ᅹ‫↚ܖ‬ⅹↀ↺࠹˴↗ˊૠ※⁈Ⅵ
2－2－

ʻ‫ׅ‬Ʒᄂᆮ˟ƷᏑ୎
“A novel symmetry in nanocarbons:Pre-constant discrete principal curvature
structure”
arXiv:2306.15839, 2023
Y. Kabata, S.M., Y. Noda., Y. Ogata, J. Onoe

Background
1. Development of Carbon manufacturing
technology
Onoe et al. have generated and studied the C60
polymers extending linearly from C60 thin films.

‫؞‬Onoe, Nakayama, Aono, Hara,
Appl. Phys. Lett., 2003.
‫؞‬Ueda, Ohno, Noguchi, Ishii, Onoe, J. Phys. Chem. B, 2006.
‫؞‬Onoe, Ito, Shima, Yoshioka, Kimura, Eur. Phys. Lett., 2012.
‫؞‬Beu and Onoe, Phys. Rev. B, 2006.

－3－

Background
1. Development of Carbon manufacturing
technology
Onoe et al. have generated and studied the C60
polymers extending linearly from C60 thin films.
Image data of electro microscopy of the C60 polymers.

›⁛⁙⁚‒⁤⁗⁥⁡⁞⁧⁦⁛⁡⁠‒‵⁤⁫⁡‟⁆‷‿‿⁓⁥⁧⁖⁓‟⁁⁠⁡⁗‟⁋⁓⁥⁧⁖⁓„‒‵⁓⁤⁔⁡⁠‒‪‣„‒‪…․‒‚․•‣‧‛‿⁓⁥⁧⁖⁓‟⁋⁓⁥⁧⁖⁓‟⁁⁠⁡⁗„‒‵⁓⁤⁔⁡⁠‒‫†‛ ‣•․‚‒ ‣‥‒„ ‫‬Background
1. Development of Carbon manufacturing
technology
Onoe et al. have generated and studied the C60
polymers extending linearly from C60 thin films.
Image data of electro microscopy of the C60 polymers.

›⁛⁙⁚‒⁤⁗⁥⁡⁞⁧⁦⁛⁡⁠‒‵⁤⁫⁡‟⁆‷‿‿⁓⁥⁧⁖⁓‟⁁⁠⁡⁗‟⁋⁓⁥⁧⁖⁓„‒‵⁓⁤⁔⁡⁠‒‪‣„‒‪…․‒‚․•‣‧‛‿⁓⁥⁧⁖⁓‟⁋⁓⁥⁧⁖⁓‟⁁⁠⁡⁗„‒‵⁓⁤⁔⁡⁠‒‫†‛ ‣•․‚‒ ‣‥‒„ ‫‬－4－

Background
1. Development of Carbon manufacturing
technology
Onoe et al. have generated and studied the C60
polymers extending linearly from C60 thin films.

‫؞‬Onoe, Nakayama, Aono, Hara,
Appl. Phys. Lett., 2003.
‫؞‬Ueda, Ohno, Noguchi, Ishii, Onoe, J. Phys. Chem. B, 2006.
‫؞‬Onoe, Ito, Shima, Yoshioka, Kimura, Eur. Phys. Lett., 2012.
‫؞‬Beu and Onoe, Phys. Rev. B, 2006.

Background
2. Development of discrete surface theory
Kotani, Naito, Omori (KNO), Comp. Aided Geome. Design 2017

Let a trivalent oriented graph G embedded
into E3, ₙ: G Ѝ E3
For each vertex ₙ(x),the triangle with thevector v1, v2, and thenormal vector n(x) is
defined by the figure:－5－

Background
2. Development of discrete surface theory
Kotani, Naito, Omori (KNO), , CAGD 2017

The discrete first fundamental form

The discrete second fundamental
form

Background
2. Development of discrete surface theory
Kotani, Naito, Omori (KNO), , CAGD 2017

By the discrete first and the second
fundamental forms, we define
the discrete mean curvature,and the discrete Gauss curvature,

Note that these values don’t depend on the choiceof v1 and v2, and thus are geometrical values.
－6－

Geometrical analysis of C60 polymers
Motivation:

The shape of C60 polymers reminds
us of Delaunay surfaces.

Delaunay surface,constant mean curvature surface

Geometrical analysis of C60 polymers
The shape (C-configurations) of the C60
polymers can be determined by the firstprinciples computations:Noda, Ono, Ohno, J. Phys. Chem. A 2015.
－7－

Geometrical analysis of C60 polymers
The shape (C-configurations) of the C60
polymers
canminimum
be determined
Severallocal
pointsbyofthe
thefirst
principles computations:semi-stableconfigurations exist.

Noda, Ono, Ohno, J. Phys. Chem. A 2015.

Geometrical analysis of C60 polymers
The shape (C-configurations) of the C60
polymers can be determined by the firstprinciples computations:

Several local minimum points of thesemi-stable configurations exist.
We basically focus on the minimal one

Noda, Ono, Ohno, J. Phys. Chem. A 2015.
－8－

Geometrical analysis of C60 polymers
Problem:To find whether the shape of the C60
polymers has constant mean curvature
given by KNO-scheme!

C60 polymer

Delaunay surfaces:

constant mean curvature surface

Geometrical analysis of C60 polymers
Problem:To find whether the shape of the C60
polymers has constant mean curvature!
Number of C
⃼-1

⃼-1

0.06

0.25

0.04

0.2

0.02
0
-0.02

0.15
0

10

20

30

40

50

60

-0.06

0

-0.08

Smallerяlarger

0.1
0.05

-0.04

Number of C

Gauss curvature

－9－

0

10

20

30

40

50

60

Number of C

mean curvatureGeometrical analysis of C60 polymers
Problem:To find whether the shape of the C60
polymers has constant mean curvature!
Number of C
⃼-1

⃼-1

0.06

0.25

0.04

0.2

0.02
0
-0.02

0.15
0

10

20

30

40

50

60

0.05

-0.04
-0.06

0

-0.08

Smallerяlarger0.1

0Number of C

Gauss curvature

10

20

30

40

50

60

Number of C

mean curvature

Geometrical analysis of C60 polymers
Problem:To find whether the shape of the C60
polymers has constant mean curvature!
The answer is No!
⃼
But ……………

Number of C

-1

⃼-1

0.25

0.06
0.04

0.2

0.02
0
-0.02

0.15
0

10

20

30

40

50

60

-0.06
-0.08

Smallerяlarger

0.1
0.05

-0.04

0

Number of C

Gauss curvature

－10－

0

10

20

30

40

50

60

Number of C

mean curvatureGeometrical analysis of C60 polymers
2. Revision of discrete surface theory
The mean curvature,and the Gauss curvature,
naturally induce the discrete
principal curvature|k1|>|= k2|

Geometrical analysis of C60 polymers
Problem:To find whether the shape of the C60
polymers has constant mean curvature!
st principal curvature
No,
but
the
1
Number of C
is almostconstant!
⃼
-1

0.4
0.3

k1

0.2
0.1
0.0
-0.1 0

10

-0.2
-0.3

Smaller

20

30

40

50

60

k2
k1

k2

Number of C

We call itDistribution
a pre-constantof thediscrete
principal curvatureprincipal curvature surface!

яlarger－11－

Geometrical analysis of C60 polymers

Noda, Ono, Ohno, J. Phys. Chem. A 2015.

Geometrical analysis of C60 polymers
FP5N

0.4

0
0

20

40

60

-0.5
k1

-0.2 020

k1-0.4

40

60

0

0

20

40

k1

FP6O

k2

k1

0
40
k1

60

0

20

40

60

0

0

-0.5
k1

k2

20

40

k1

FP6E

FP6F
0.5

0.5

0

0

0

0

20

40

60

20

40

60

-0.5
k1

k2

k1

FP6I

0

20

40
k1

k2

60

0
-1

0

20

40

60

k1

0

40

60

k2

20

k2

40

60

k1

60
k2

0.5
0
0

20

40

60

0

20

40

-0.5
k1

－12－

k2

T3

-0.5
k1

0

FP6K

40

20

-0.5

0
20

k2

FP6J

0.5

0

60

k1

k1

k2

-0.5

k2

1

40

0.5

FP6C

1
0

0

0.6
0.4
0.2
0
-0.2 0
-0.4

FP3F

0.5

-0.5

60

-0.5

k2

20

FP5K

0.5

20

0

-0.4

k2

FP5L

0.5

0

0.2
-0.2 0

FP4J

0.2

-1

60

-0.5

k2

0.4

FP4L
0.6

0.5

0.2

0

-0.2

FP4K

0.4

0.5

0

Light blue shows the 1st principal curvature!

FP6L

k2

k1

k2

60

Geometrical analysis of C60 polymers
FP5N

0.4

0
0

20

40

60

-0.2 0

20

60

0

0

20

k1

k2

k2

k1

0

20

40

60

0

20

40

60

0

0

-0.5

k1

k2

20

40

0

20

40

0.5

0.5

0

0

60

-0.5

20

40

60

-0.5
k1

k2

k1

FP6I

0

0.6
0.4
0.2
0
-0.2 0
-0.4

20

40
k1

60

0

20

k2

40

60

40

k2

20

40

k1

60

k2

FP6J

20

40

60

k1

60

k2

T3
0.5
0

0

20

40

-0.5
k1

0

FP6K
0

20

k1

0

k2

0.5

0

60

-0.5
k1

0
-1

k2

-0.5

k2

1

40

0.5

FP6C

1
0

0

60

-0.5

k2

k1

0

20

All of the stable and semi-stableconfigurations of the C60 polymers are
FP6E
FP6F
FP3F
pre-constantdiscrete principal
curvature surfaces!
k1

0.5

0

FP5K

0.5
0

0.2

-0.4

k2

FP5L

0.5

0.2

-1

60

-0.2 0

FP4J

0.4

40

-0.5

FP6O

0

40

-0.4
k1

FP4L
0.6

0.5

0.2

-0.5

-0.2

FP4K

0.4

0.5
0

Light blue shows the 1st principal curvature!

FP6L

k2

60

0

20

40

-0.5
k1

k2

k1

k2

Geometrical analysis of C60 polymers

nanotubes

C60,
k1

k1

k1

C60, and nanotubes have the constantfirst principal curvature.
－13－

60

Geometrical analysis of C60 polymers
k1
k2

C60-dimer
C60-dimer also has the pre-constant firstprincipal curvature.

Geometrical analysis of C60 polymers
Concerned nanocarbons have the novel
symmetry, i.e., the pre-constant discrete
principal curvature including constantprincipal curvature cases.

－14－

Geometrical analysis of C60 polymers
Why?
Geometrical answer is not clear!
In continuum picture, constant principal curvaturesurfaces have the center axes.

Center axis is given by -k1-1 ¼ the normal vector, n.
Not tube type shape cannot havethe constant principal curvature.

Geometrical analysis of C60 polymers
Why?
In discrete picture, the set of the “center axis” givenby “-k1-1 ¼ the normal vector n from each vertex”
does not form a curve in general.

We call it the center axisoid.

－15－

Geometrical analysis of C60 polymers
Why?
Due to “not strictly constant” principal curvature,
discreteness, center axisoid, and not-strictly equibond length, we have a non-trivial pre-constantprincipal curvature surface:

Geometrical analysis of C60 polymers
Why?
The center axisoid, discrete analogue of the center
axis, is not on a curve in general, though, in continuum

picture, constant principal curvature surfaces have the center axes.

It is a profound problem what is a constant discrete
principal curvature surface in discrete differential
geometry!

continuum picture vs discrete picture－16－

Geometrical analysis of C60 polymers
Why?
Physically, it may be related to the overlap
integrals of the wave functions as the curved
effect.

Flat

vsCurvesGeometrical analysis of C60 polymers－17－

Geometrical analysis of C60 polymers
Average of the 1st principal curvature isirrelevant to the total energy of the firstpriciples computations.

Geometrical analysis of C60 polymers
Average of the absolute value of the graphlaplacian,of the first principal curvatureisstrongly relevant to the total energy in the
1st principles computation!

－18－

মଢ଼஢ভभਖ਻ਡᵏᵌ ბႸẲẺἜἠỽὊἮὅỊẴỔềᩉ૝ɼ୺ྙಒɟ‫୺ܭ‬᩿ểễẾềẟỦ
ẮủỤỉཋྸႎẆૠ‫ܖ‬ႎẆᏑ୎Ị˴ẦᾎὉ᬴ܱႎཋྸႎễբ᫆ẦᾎὉᇹɟҾྸᚘምႎễբ᫆ẦᾎὉӞχЎ‫܇‬ѣщ‫ܖ‬ể࠹˴ỉբ᫆ẦᾎὉᵡᵔᵎỉ፭ᛯႎễࣱឋẦỤỉࢨ᪪ỊẝỦẦᾎίᵡᵔᵎỉ‫ݣ‬ᆅࣱỉ፭ỉᢿЎ፭ểɳᡶ‫ݣ‬ᆅ፭ỉࢨ᪪ᾎὸὉᾂദЩἂἻἧỉщ‫ܖ‬ႎբ᫆ẦᾎὉࣇЎ࠹˴ႎễբ᫆ẦᾎᾁώẸờẸờ஬૰ᅹ‫ܖ‬ỆấẟềẆᇹɟҾྸểૠ‫ܖ‬ửᗡӳẴỦẮểỂૼẺễႆᙸửẴỦểỊỄỉợạễẮểẦᾎ－19－

IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, Japan

Theoretical prediction and experimental
demonstration of geometry-driven curvature eﬀects
using 1D uneven structured C60 Polymer
Jun ONOE
Nagoya University
We have reported that 1D C60 polymer with a concavo-convex periodic curved
structure [1-3] formed from electron-beam-irradiation (3-7 keV) of a C 60 ﬁlm exhibits
physical properties arising from 1D metal [4]. The behavior of the free electrons on
the curved surface is characterized by the Hamilton operator of the following equation(quantum mechanics of submanihold).




2
1  ∂ √ ij ∂ℏ22+ (h − k)gg
H=− ∗ √
2m
g i,j=1 ∂q i
∂q jHere, g = det[gij ] represents the metric tensor. The ﬁrst term is an operator corresponding to the kinetic energy of electrons, and the second term consisting of the mean
curvature h and the Gaussian curvature k appears like a scalar potential (the second
term does not appear in the 1D plane surface). It has been unclear whether or not thiscurvature term aﬀects the behavior of electrons since 1950s. We theoretically predict
the eﬀect of the geometric curvature term on the electronic behavior of the above 1DC60 polymer [5] and then experimentally demonstrate it [6]. In this talk, I will introduce the quantum mechanics of submanihold of the 1D C 60 polymer when compared tothe other nanocarbons such as fullerenes, nanotubes, and grephene.

References[1] J. Onoe et al., Appl. Phys. Lett. 82, 595 (2003).
[2] A. Takashima, J. Onoe, and T. Nishii, J. Appl. Phys. 108, 033514 (2010).
[3] H. Masuda, H. Yasuda, and J. Onoe, Carbon 96, 316 (2016).
[4] H. Shima and J. Onoe, The Role of Topology in Materials (S. Gupta and A.
Saxena eds.), Springer, Chap. 3, pp. 53-84 (2018) and references therein.
[5] H. Shima, H. Yoshioka, and J. Onoe, Phys. Rev. B 79, 201401 (R) (2009).
[6] J. Onoe, T. Ito, H. Shima, H. Yoshioka, and S. Kimura, Europhys. Lett. 98, 27001(2012) [Press release]－20－

１次元凹凸C60 ポリマーを用いた
幾何曲率効果の理論予想と実験的検証

One dimensional peanut-shaped C60 polymer
Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120

UNESCO World Heritage

stable isomer predicted using the general Stone-Wales rearrangement.

The area

Jun Onoe*colored in sky blue represents a Riemannian curved space in which !-electrons move
one-dimensionally. The !r denotes the degree of uneven deformation.

Department of Energy Science and Engineering
Nagoya University*E-mail: j-onoe@energy.nagoya-u.ac.jp

Topology-induced geometry in graphitic carbon network

Side view

Top view

O.V. Yazyev and S.G. Louie: PRB 81, 195420 (2010)

M. Terrones: ACS Nano 4, 1775 (2010)

The structure of two buckled Stone-Wales
defects in graphene.

J. Ma et al.: PRB 80, 033407 (2009)

K. Balasubramanian et al., Small 1, 180 (2005)

Only six pentagon insertion is allowed forcapping a CNT regardless of its tube radiusand chirality (Gauss-Bonnet theorem).
K. Suenaga et al., Nat. Nanotech. 2, 358 (2007)－21－

Y. Saito et al., JJAP 39, L271 (2000)

How do we produce 1D uneven (凹凸) C60 polymer ?
Carbon 81, 842 (2015).

Electron beam (3-10 kV)
1015–1017 cm-2s-1

5-1000 nm

CsI/Si/Cu/Mica/SiO2 substrates

C60 ﬁlm: insulator (108–1014 Ωcm)/soluble

1D peanut-shaped C60 polymer: metal (<10–2 Ωcm)/insoluble
Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120
stable isomer predicted using the general Stone-Wales rearrangement.

The areacolored in sky blue represents a Riemannian curved space in which !-electrons move

FT-IR

MCT
detector

The cross-linked
structure
the degree of uneven deformation.
one-dimensionally. The !r denotesof P08 C120 isomer

E-gun
K-cell (C60)

In situ UHV FT-IR apparatus

FIG. 6. Takashima et al.

APL 82, 595 (2003); APL 85, 2741 (2004); PRB
72, 155416 (2005); PRB 74, 195426 (2006); JPCB
110, 22374 (2006); PRB 75, 233410 (2007); APL
92, 094102 (2008); JAP 104, 103706 (2008); PRB
79, 201401 (R) (2009); JAP 108, 033514 (2010);
APL 97, 241911 (2010); EPL 98, 27001 (2012);
JPCM 24, 175405 (2012); APL 104, 113301
Fig. 1 Thermodynamically stable 24 C120 isomers derived from the general Stone-Wales(2014); PRB 90, 155435 (2014); Carbon 81, 842
Optimized structure of GSW C isomers(2015); Carbon 96, 316 (2016); Carbon 152, 882
rearrangement. Where, nth C120 isomer is denoted as Pn. 120(2019); Adv. Sustain. Sys. 5, 2000156 (2021)
obtained using Gaussian09

The formation processes of the 1D 凹凸 C60 polymer

GSW reaction modelFig. 6 Irradiation-time evolution of in situ IR spectra of an EB irradiated C60 film and theJ. Phys, D: Appl. Phys. 2012

intermediate structure of the exotic fullerene polymer [8].

－22－Fig. 6 _NkD`TF
08>;=iC Fig. 5 Z
2Ghb  ?#Z  

GSW e[',+1D structure evidenced using cryo-HRTEM
FFT image

FFT

FFT

FFT

FFT
(b)HCP-based model(monoclinic)

three equivalent directions for 1D polymerization
Carbon 81, 842 (2015).

Carbon 96, 316 (2016).

cryo-TEM (Osaka Univ.)

フラーレン固体の光重合

UHV-STM/STS装置

PRB 1996, PRB 1997, PRL 1997
Surf. Rev. Lett. 1999

AIP Adv. 10, 085212 (2020) [Editor’s pick]

－23－

－24－

2a）。一方，絶縁体相にはエネルギーギ J. Appl. Phys. 108, 033514 2010
する。このとき完全に理想的な一次元電子はゆらぎの効
033514-2 に達する（Fig.
Takashima, Onoe, and Nishii
果が大きく，CDW 形成は抑制される。したがってパイエルス転移が観測される温度 T p では，一次元電子鎖間

Physical properties arising from 1D metal
FIG. 2. Schematic illustration of the structural region considered for obtaining theoretical IR spectra in the case of the P04 isomer. The region marked
by the dashed rectangle shows the common capped structure for all theisomers, thus omitting the IR modes originating from this region for comparison with the experimental IR spectra.

Tomonaga-Luttinger liquid (TLL) states

Peierls transition

FIG. 1. Color online Schematic illustration of C120 stable isomers derived
from the GSW rearrangement. The nth C120 isomer is denoted as Pn.

for 1D C60 polymers,19 individual vibrational modes are
uniquely assigned to their individual local structures. This
allows us to examine the theoretical IR spectra of the C120
isomers regarded as the minimum(a) unit of their corresponding(b)
The present apparatus used for measuring in situ IR
1D polymers by excluding the common structuresimilar
104 toρ(ω)∝|ω| αspectroscopy consists of an ultrahigh vacuum UHV chamρ(ω)∝|T| αthat of C60 at their edges. Figure 2 shows the schematic−7
T=350 K
ber a base pressure: 2  10 Pa, a Fourier-transform IR
Fig.
structure of the P04 C120 isomer taken from
K 1. Since theT =150spectroscope Mattson Research Series, a Knudsen cell K250
K
region marked by the dashed rectangle overlapps with a C60
ΔE=12 meV
cell, a molecular turbo pump combined with a rotary pump
Peak (c)!
cage structure, it 150
canK be excluded
when the corresponding
Res. Limitω0.66
Balzers, an EB
gage
33.1gunK!Omegatron, a partial pressure
103 T 0.59
0.66
1D P04 polymer is considered. As a result, the regionAnelva, and a helium He cryostat Iwatani. ω
Details of
30 K
125
K
markedbythesolidrectangle,
which
includesthecommon
40.0
K!described elsewhere.15,16 C60 films 60
this system have
been
0.1
0.01
30
100
and its correcross-linked structure
P04 C120 (eV)85 K between the
Binding Energy
nm thick were50.0
formedon cesium iodide CsI substrates
20
Temperature(K)C60-polymer
sponding 1D polymer, is obtained. In this way, we consid(color online). Schematic illustrations of Peierls
Fig. and
2. K!
mm in diameter2 (color
mm thickonline).
by thermal Schematic
evaporation of illustrations of30 photoK
2Δ=
3.52kBTquasi-1D
only from the cross-linked strucp 〜 15 meV (Tp 〜50 K)α)the IR modes arising(Heered
IIT＞T
60.0 K!induced
99.98% purein the dynamics
K-cellhν=40.8eV
at 673 at
K (a)transition inmetals at (a) T＞T p, (b) T＜TC60p. powder Matsubo,
carrierp, (b) T＜T p.
ture of all the C120 isomers, and compared these with thefor 3 min after70.0
residual
powder were0.2
EF-0.2
K! organic solvents in the0.4
experimental
ones.
removed at 473 k for 2 h in the same UHV chamber. SubseBinding
Energy(eV)
To obtain the theoretical IR spectra due to the cross80.0 K!the in situ IR spectra of the C60 films
quently, we measured
linked structure of all the C120 isomers, we replaced eachAppl. Phys. Lett. 92, 094102 (2008).
before EB-irradiation and after EB-irradiation 3 kV, 0.5
stickpeak with a Lorentzianfunctionwith a(2012).
full width at half
90.0 K!
Europhys.
Lett. 98,
27001
mA for 1, 3, 5, 10, and 25 h. All the IR spectra were remaximum FWHM corresponding to that of the intense and−1corded with a100.0
resolution
K! of 4 cm and with 5000 scans narrow
Phys.IRRev.
B 79, 201401(2009).
for the25 h EBpeak appearing
at 565 cm−1(R)S / N  104. The substrate temperature was controlled in theirradiated C60 film.
range of 50–300 K within a variation of 0.1 K, using the
He cryostat as well as a heating system.
IV. RESULTS AND DISCUSSION

Transmittance [arb. unit]!

Fig. 1.

1125

1115

1105

Charge-Density-Wave (CDW) phonon
modeIII. THEORETICALCALCULATIONS

A. IR spectra
1D
van Hove Singularity

We have carried out first-principles density-functional
Peak (g)!
Peak (c)!
calculations of IR spectra for all C120 stable isomers derived
33.1 K!
from the GSW rearrangement, using the GAUSSIAN03
33.1
K!
package.17 In all
calculations,
we employed a 6–31g d basis
40.0 K!
set and the Perdew–Burke–Ernzerhof exchange-correlation
50.0 K!
40.0 K!
these conditions have been shown topotential,18 because
071301-9
Wu, Yu, and Shenprovide good 50.0
agreement
K! in the energy gap and IR spectra
60.0 K!
between computational and experimental results for C60.12,19
70.0 K!
For example, theenergy60.0
K! gap of C60 was estimated to be 1.67
eV, which is in an excellent agreement with the values of
80.0 K!
70.0 K!experimentally.20
1.6–1.85 eV obtained
Figure 1 shows a schematic illustration of thermody90.0 K!
80.0
K!isomers derived from the GSW rearnamically stable
C120
100.0 K!
rangement, where
90.0theK!nth C120 isomer is denoted as Pn. For
example, the firstC
isomer with a dumbbell-shaped cross120
100.0 K!
linkage is named P01, while the last C120 isomer is P24.
1D infinite structure of each isomer should be
1115
1105
1252
1245Ideally, the1238
considered. However, unlike the estimation of the energy gap

Figure 3 shows the time evolution of the IR spectra of an

CsI substrate

Transmittance [arb. unit]!

Transmittance [arb. unit]!

1125

J. Appl. Phys. 108, 071301 2010

1340 cm–1

FIG. 3. Irradiation-time evolution of in situ IR spectra of an EB irradiatedC60 film.

Downloaded 08 Aug 2010 to 131.112.115.47. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

Peak (g)!

J. Appl. Phys. 108, 033514 (2010)

Appl. Phys. Lett. 97, 241911 (2010).

Transmittance [arb. unit]!

Int(EF)/Int(0.5eV)Intensity (a.u.)

Intensity (arb. units)

II. EXPERIMENTS

J. Phys.: Condens. Matter 24, 175405 (2012).

33.1 K!
40.0 K!

50.0 K!
60.0 K!
70.0 K!

Novel properties different from those of the other nanocarbons
80.0 K!
90.0 K!
100.0 K!

1252

1245

FIG. 5. Scanning electron micrographs of a and b HOPG islands, and cand d HOPG plates on Si001 substrates. Reprinted with permission fromX. Lu et al., Nanotechnology 10, 269 1999, Copyright 1999, IOP Publishing Ltd.

1238

Tomonaga-Luttinger
liquidstatesFIG. 4. Color online AFMimages(TLL)
HOPG. cand d Are the highIntensity (arb. units)magnification images of the portions indicated in a as C and D, respectively. Reprinted by permission from Macmillan Publishers Ltd, Nature,
Hiura et al., 367, 148 1994, Copyright 1994.

T =150 K

0.66

Int(EF)/Int(0.5eV)Intensity (a.u.)

betweenthe
T=350
K tip and sample when scanning selected areas.
(a)(b)that the folding/unfolding
Energeticsuggested
250considerationsK
104
ρ(ω)∝|ω|ρ(ω)∝|T| αvibration.
process 150
canK be accounted
for αby thetip-sample
ω0.66
In allexperiments,
one
must firstlocatestep
edges
with anFigure1. Schematic illustrationof a one-dimensional
metallic
C polymer
125these
K
ΔE=12 thus
meV
using AFMa ωlarge structuresamplesurface area andthe
85 K overuneven peanut-shaped
similar to the cross-linked structure ofRes.
the Limit
P08 C
C60-polymer
103 Furthermore,
T 0.59entireprocessisnotwell-controlled.
it
is
also
30
K
hν=40.8eV (He IIα)stable isomer predicted using the general Stone-Wales rearrangement. The areadifficultto-0.2obtain large
size 0.01graphene 30sheets100 using this
0.1
0.2
EF
0.4
Binding Energy (eV)Temperature(K)colored in sky blue represents
space works
in which !-electrons
method.
overa Riemannian
these curved
early
wasmoveto first
Binding Energy(eV) An improvement
pattern the graphite
into The
small
followedby tearing
one-dimensionally.
!r denotesislands,
the degree of uneven
deformation.
118
The patterning of HOPG into
using theLett.
AFM
or 27001
STM tip.
Europhys.
98,(2012).
islands– 40201401
m was(R)achieved
Phys.
Rev.ofB279,(2009).through the combinationof a 200-nm-thick SiO2 mask and oxygen plasma etching.
The subsequent manipulation using AFM tip results in thedisplacement of graphite plates from the original island. The
thickness of the graphite plates obtained in this work was
about 100 nm Fig. 5. Zhang et al.119 succeeded in reducing
the graphite plate thickness by first transferring the detachedgraphite island to a micromachined silicon cantilever, andthen use the mounted graphite block on the cantilever as thetip of an AFM to scan it over a SiO2 / Si substrate. By doingso, graphite plates with a thickness of 10 to 100 nm havebeen obtained. These thin graphite plates were found to exhibit clear field-effect.
Instead of attaching the graphite island onto the cantilepatternedgroup,
HOPGDelft)square mesasver, Novoselov
al.6 pressed
HR-STM
image of etSWCNT(Dekker’s5 m in height and 20 m to 2 mm in lateral size against
m-thick
layer
of
a
fresh
wet
photoresist
spun over aa 1-S.
Iijima, Nature 354, 56 (1991)
glass substrate. After baking, the mesas were cleaved off the
HOPG sample and attached to the photoresist layer. The subsequent repeated peeling using a scotch tape led to only thinflakes left in the photoresist. These flakes were then releasedin acetone. When a SiO2 300 nm / Si n+-doped wafer
was dipped in the solution and then washed in water and
60

120

30 K

Nanotubes－25－

Fullerenes

propanol, some flakes became captured on the wafer’s surface. The thick flakes were further removed through ultrasound cleaning in propanol. Thin flakes d  10 nm were
found to attach strongly to SiO2, presumably due to van der
Waals and/or capillary forces. By using this method,graphene sheets as thin as one atomic layer have been obtained Fig. 6. Ever since this work, mechanical exfoliation
has become the method of choice for producing graphenewith highest quality. Many variations in original exfoliation
techniques have been developed and applied to different
types of graphites. Although the mechanical exfoliation technique has been improved significantly, its primary drawbacks
still remain. Its low-productivity does not allow synthesis of
Kroto quantities.
et al., Nature318,
162(1985)
grapheneH.W.
in large
It isalso
incompatible
withstandard Si 1996
processes.
might be overcome byNobelThe
prizeformer
in chemistrychemical exfoliation and CVD, while the latter may be
avoided by using epitaxial growth.
2. Chemical exfoliation

Like mechanical exfoliation, chemical exfoliation ofgraphite is also an old technique. The primary advantage ofGraphenes

K.S.online
Novoselov
et al.,films
Science
306, by
666mechanical(2004). exfoliation.
FIG. 6. Color
Grapheneobtaineda Photograph of a graphene flake with a thickness of 3 nm placed on top of
Nobel
prize
an oxidized Si wafer. 2010b AFMimageof 2in Physics2 m2 area of the flake in anear its edge dark brown, SiO2 surface; orange, 3 nm height above the SiO2
surface. c AFM image of SLG central area. d SEM image of a few
layer graphene device e Schematic view of the device in d. From Novoselov et al., Science 306, 666 2004, Reprinted with permission from
AAAS.

Downloaded 18 Nov 2010 to 131.112.115.47. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

－26－

Vector potential（AB effect）
Magnetic ﬁeld lines

Vector potential

Magnetic ﬁeld
Current
Current

Vector potential A was thought to be introduced
mathematically, but its physical meaning was
given by Aharonov and Bohm.

Y. Aharonov and D. Bohm: Phys. Rev. 115, 485 (1959)

Aharonov-Bohm (AB) effectcoilshieldphase shift

magneticﬁeldphase difference = 0

superconductive coilinterference
fringesphase difference = 1/2 λElectron-beam holography
A. Tonomura et al.: Phys. Rev. Lett. 56, 792 (1986)

Prediction of the geometric curvature effects on
Tomonaga-Luttinger liquid (TLL) states
2-3 D metal

Fermi Liquids (FL)CNT

1D metal
Tomonaga-Luttinger Liquids (TLL)

Power-law dependence

α<1

島 弘幸・小野頌太・

膜を考えます。次に，この膜の形に沿った量子井戸型ポ電子

テンシャルで膜を上下から挟むことにより，膜の内部にてい粒子を閉じ込めます。その後，ポテンシャル障壁の高さ

証でを十分高くし，膜厚ゼロの極限を取ります。すると，曲即し

面接線方向に運動する粒子の有効ハミルトニアンが，ある曲線座標系（q 1，q 2）を用いて以下の形に求まりまδr [Å]す 16)。

experimentally determined using
photoelectron spectroscopy

6

2



r





本

 1 2  g g ij j ＋(h 2−k) （ 1 ）Ĥ ＝−6
2 m *  g i, j＝1 q iq表面

ここで m * は粒子の有効質量，g ij は行列［g ij］の逆行列

は，成分，g は行列［g ij］の行列式を表します 。h と k は状態

それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であ

クー

11)

Phys. Rev. B 79, 201401 (R) (2009).

り，曲面上の各点における曲がり具合を表します。h とず，

k は g ij の関数ですので，結局曲面の形さえ決まれば，－27－ その形状を記述する計量テンソル g ij により Ĥ が一意に成さ決まります。

のよPrediction of the geometric curvature effects on
Tomonaga-Luttinger liquid (TLL) states

CNTρ(T) ∝ |T|α
ρ(E) ∝ |Ε – EF|α

CNTs: α = 0.5

島 弘幸・小野頌太・

膜を考えます。次に，この膜の形に沿った量子井戸型ポ電子

テンシャルで膜を上下から挟むことにより，膜の内部にてい粒子を閉じ込めます。その後，ポテンシャル障壁の高さ

証でを十分高くし，膜厚ゼロの極限を取ります。すると，曲即し

面接線方向に運動する粒子の有効ハミルトニアンが，あPower-law dependence

る曲線座標系（q 1，q 2）を用いて以下の形に求まりまδr [Å]す 16)。

α<1

2



r





表面

Phys. Rev. B 79, 201401 (R)ij(2009).
ここで m * は粒子の有効質量，gは行列［g ij］の逆行列

は，成分，g は行列［g ij］の行列式を表します 。h と k は状態

それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であクーり，曲面上の各点における曲がり具合を表します。h とず，

k は g ij の関数ですので，結局曲面の形さえ決まれば，その形状を記述する計量テンソル g ij により Ĥ が一意に

成さ

11)

決まります。
Evidence for the geometric curvature effects
on electronic properties

式（ 1 ）かっこ内の第一項は，通常の Schrödinger 方(b)ρ(ω)∝|ω| α
T =150 K

uneven peanut-shaped structure similar to the cross-linked structure of the P08 C120

103

T=350 K
250 K
150 K

ω0.66

125 K
85 K

C60-polymerhν=40.8eV (He IIα)0.2

T 0.59

ΔE=12 meVRes. Limit

程式で目にするラプラシアン演算子  の曲線座標表示

作用が決座標系（例えば極座標）を採用すれば平面系でも現れるや直ものなので，系の曲面性と直接は関係しません。代わりて，に式（ 1 ）で注目すべきは，h 2−k で表された新しいス

こ

カラーポテンシャル項の存在です。これがまさに曲面のTLL

幾何曲率に起因する有効電場ポテンシャルであり，後述

リマ

する曲率誘起型のナノ物性変化を駆動する役割を果たし

ャリ

ます。

って

30 K

EF

Binding Energy (eV)-0.2

one-dimensionally. The !r denotes the degree of uneven deformation.

30
100
Temperature(K)ここThe areacolored in sky blue represents a Riemannian curved space in which !-electrons move

0.1
0.01
Binding Energy (eV)Intensity (arb. units)ρ(ω)∝|T| αstable isomer predicted using the general Stone-Wales rearrangement.

ω0.66
30 K

0.4

104

Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anInt(EF)/Int(0.5eV)Intensity (a.u.)

(a)

のよ

です 11)。見た目は多少複雑ですが，この表示自体は曲線2δr = 1.4 Å本 1 2  g g ij j ＋(h 2−k) （ 1 ）Ĥ ＝−6
2 m *  g i, j＝1 q iqH. Ishii, H. Yoshioka et al., Nature 426, 540 (2003).

In situ high-resolution PES

6good agreement

ρ(T) ∝ |T|α
ρ(E) ∝ |Ε – EF|α凹凸periodic: α 〜 0.6

Fig. 6 に は そ の 例 と し て，正 弦 波 形 状 の 凹 凸 薄 膜（Fig. 3）が生み出す有効ポテンシャル U(x)の空間分布数aを示しました 18)。凹凸振幅 a の増加に伴い，凹部と凸部数aに対応する点 x＝lp/g(l＝0, :1, …)での U(x)の下向きづけ

ピークが，徐々に鋭くなる様子がわかります。こうした

CNT: α 〜0.5正弦波曲面の他にも，これまで様々な曲面構造におけるδr [Å]
Phys. Rev. B 79, 201401 (R) (2009).

Europhys. Lett. 98, 27001 (2012).

－28－

立場

テンシャルで膜を上下から挟むことにより，膜の内部にて粒子を閉じ込めます。その後，ポテンシャル障壁の高さ

証を十分高くし，膜厚ゼロの極限を取ります。すると，曲

即

Summary& Perspectives
面接線方向に運動する粒子の有効ハミルトニアンが，ある曲線座標系（q，qgeometry）を用いて以下の形に求まりまA potential driven by a periodic
uneven (凹凸)has been curious to16)
affect electronic properties
ofmaterials
since
1950s.
す 。
122 1 2 ̂ g g ij j ＋(h 2−k) （ 1 ）H＝−
6
2 m *  g i, j＝1 q i
q



r

*





ここで mcurvature
は粒子の有効質量，gは行列［gij］の逆行列First observation of geometriceffects on electron
behaviors
655
11)(Quantummechanics
in submanihold)
成分，g
は行列［g
。h と k はij］の行列式を表します

・平 久夫

ij

子状態 19, 20)とその電子透過率への影響 21, 22)が議論され

表

は

状

それぞれ曲面の平均曲率・ガウス曲率と呼ばれる量であクできるか，そのための実験手段の提案と具体的物質にり，曲面上の各点における曲がり具合を表します。h とずした物性推算が重要であると著者は考えます。

成います。今後はこうした理論的予言をどう実験的に検k は g の関数ですので，結局曲面の形さえ決まれば，
Discrete Geometry Analysis
6．擬一次元導体の朝永-Luttinger 液体におけその形状を記述する計量テンソル g ij により Ĥ が一意に
5-9 membered rings get together to formija curved surface

る曲率効果

の

決まります。

本節と次節では，前述の曲率誘起ポテンシャルがナノ式（ 1 ）かっこ内の第一項は，通常の Schrödinger 方
Geometric quantities，擬一次元凹凸電子系の朝永-Luttinger 液体（TLL）
程式で目にするラプラシアン演算子  2 の曲線座標表示
態です。一般に一次元性の強い電子系では，電子間のquantitative or qualitative
correlation ?
です 11)。見た目は多少複雑ですが，この表示自体は曲線が，フェルミ準位 E F の近くで電子-ホール対が多数形座標系（例えば極座標）を採用すれば平面系でも現れるやされます。その結果，系の一電子状態密度 n(w)は次てPhysical quantities

面物性に顕在化する例を二つ示します。まず一つ目

Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120
stable isomer predicted using the general Stone-Wales rearrangement.

The areaーロン相互作用のために基底状態がフェルミ縮退せcolored in sky blue represents a Riemannian curved space in which !-electrons move
one-dimensionally. The !r denotes the degree of uneven deformation.

ものなので，系の曲面性と直接は関係しません。代わり

ようなべき乗則に従います 23)。
n(w){*w−E F* a，a＝(K＋K −1)/2−1

現代幾何学と物質科学 2（ 2 ）に式（ 1 ）で注目すべきは，h −k で表された新しいスΤΩκςΟοΫͰͳ͘ͱ΋ɼԁ౵‫ܕ‬ͷΧʔϘ
こで現れる指数 a は
਺ֶTLL
Libre指数と呼ばれ，電子間相互ɿ2020 ೥ 9 ݄͜ͱ͕஌ΒΕ͍ͯ·͢ɽඌ্Β͸ߋʹɼΧʔϘͯৼΔ෣͏͜ͱ͕஌ΒΕ͍ͯ·͢ɽ̍࣍‫ܥݩ‬ͷ
用の強さを表す TLL パラメータ K(＞0)によりその値ϯωοτϫʔΫΛߟྀͯ͠ɼ̎ͭͷ C60 ʹର͢·͢ɽ
ʯ[6]ʢஶऀ༁ʣͱॻ͖·ͨ͠ɽΔ݁߹ͨ͠෼ࢠʹର͢ΔୈҰ‫ݪ‬ཧ‫ʹࢉܭ‬ΑΔߏ

୸ૉ‫͔ܥ‬ΒͳΔԽ߹෺͸ɼ‫ز‬Կֶ‫ܗ‬ঢ়ͱ௚઀଄‫ࢉܭ‬2 ͱɼ੺֎෼ޫεϖΫτϧͷ‫ٵ‬ऩεϖΫτͭͳ͕Γ๛͔ͳ෺ཧత‫ݱ‬৅Λఏ‫͢ڙ‬Δ͜ͱ͕஌

24, 25)
幾何曲率に起因する有効電場ポテンシャルであり，後述͜ͱͰ͢ɽͦͷͨΊɼిࢠঢ়ଶ͸ɼ̏࣍‫ۭؒݩ‬
決まります。実験的には，カーボンナノチューブΛධՁ͠ɼδr = 0.14 nm ͱͳΔ͜ͱΛಉఆ͠· ֶɾ‫ز‬Կͱͷ৽ͨͳ࿈‫ܞ‬ͷεςʔδʹҠߦͨ͠ಛ௃͸ͦͷྔࢠత༳Β͕͗‫ۃ‬Ίͯେ͖͍ͱ͍͏
ͰͷϑΣϧϛྲྀମͱ͸ҟͳΔேӬɾϥοςΟϯ

લճɼJensen-Koppe, da Costa (JKdC) ͕ࣔϧʹΑΔൺֱʹΑΓɼର৅ͱͨ͠ 1DEC ͷߏ଄ΒΕ͍ͯ·͢ɽ[7] Ͱඌ্ɼౡ͸ɼ෺࣭Պֶ͸਺ͨ͠ [2]ɽߋʹɼిࢠͷ೾ಈؔ਺ͷ෼෍ʹΑΓ෼ͱͯ͠ɼ਺ֶͱ෺ཧͷߋͳΔ‫ʹྗڠ‬ΑΓ৽ͨͳ

26)
δϟʔྲྀମͱͳΓ·͢ɽ
直鎖重合型の C 60 ポリマーなどの擬一次元系についͨ͠෦෼ଟ༷ମͷྔࢠྗֶ͕਺ཧ෺ཧతʹͱͯ‫ʹۃ‬ΑΔిࢠঢ়ଶ΁ͷӨ‫͕ڹ‬ແࢹͰ͖Δ͜ͱ΋ ෺ཧֶΛߏங͢Δ͜ͱΛఏҊ͍ͯ͠·͢ɽ
する曲率誘起型のナノ物性変化を駆動する役割を果たし͜ͷ 1DEC Ͱ΋ࣗ༝ిࢠ͕େҬతʹଘࡏ͠ɼ΋ࣗવͳର৅Ͱ͋Δ͜ͱΛࣔ͠·ͨ͠ɽͱ͸͍ͦͷઈରԹ౓ྵʹ͓͍ͯ΋ϑΣϧϛΤωϧΪʔ，この TLL 指数 a͑ɼͦͷਖ਼౰ੑ͸࣮‫ݧ‬తʹ‫͞ূݕ‬ΕΔ΂͖Ͱ͢ɽの値が計測されています。

ます。

EF ʹ͓͚Δిࢠঢ়ଶີ౓ n(ω) ͕ෆ࿈ଓͰ͸ͳͦΕ͕෺ཧֶͷԦಓͰ͢ɽͦΕΛߦͬͨͷ͕ඌ

ここで疑問となるの が，擬 一 次[1,2]元系 の ͘ɼ࿈ଓతͳ
幾何曲率と
ɽ͜Ε͸ը
্ɾҏ౻ɾౡɾ٢Ԭɾ໦ଜͰ͢
‫ظ‬తͳ͜ͱͰ͢ɽࢲࣗ਎͸ͦͷ಺༰Λ‫׬‬શʹ೺n(ω) ∝ |ω − EF |α

L 指数 a との相関の有無です。例えば上記のC 60 ポѲ͢ΔྗΛ࣋ͪ͋Θ͍ͤͯ·ͤΜ͕ɼ[1-4] ʹैͱͳΓ·͢ɽ

͍ͦͷ࣮‫ݧ‬ͷ಺༰Λ֓‫͢·͍ࢥͱ͍ͨ͠؍‬ɽࣔ͠ɼͦͷ͜ͱʹΑΓɼேӬɾϥοςΟϯδϟʔҰ࣌‫ؔʹڀݚظ‬Θͬͨऀͱͯ͠ɼ෦෼ଟ༷ମ

ྲྀମͷύϥϝʔλ α ͷมҐ͸७ਮʹ‫ز‬Կֶతޮͷྔࢠྗֶ͕৽ͨͳύϥμΠϜΛఏࣔ͢Δֶ໰

ՌͰ͋Δ͜ͱΛࣔ͠·ͨ͠ɽ෼໺ͱͯ͠੒௕͢Δ͜ͱΛ‫ͯͬئ‬΍Έ·ͤΜ3 ɽ͜ͷిࢠʹؔΘΔ࣮‫ͱݧ‬ཧ࿦‫ʹࢉܭ‬ΑΔධՁ

松谷茂樹 教授（金沢大学）

ࢀߟจ‫ݙ‬[1] ඌ্ॱ, ిࢠ෺ੑʹ͓͚ΔϦʔϚϯ‫ز‬ԿֶޮՌɹ
ද໘Պֶ 34 (2013) 38-43.
[2] J. Onoe, T. Ito, H. Shima, H. Yoshioka andS. Kimura, Observation of Riemannian geometriceffects on electronic states EPL (Europhys. Lett.)
98 (2012) 27001.
[3] H. Shima, H. Yoshioka and J. Onoe, Geometrydriven shift in the Tomonaga-Luttinger exponentof deformed cylinders, Phys. Rev. B 79 (2009)201401(R).
[4] ౡ߂޾ɼখ໺ᰌଠɼฏ‫෉ٱ‬ɹ‫ۂ‬໘ྔࢠ‫ܥ‬ͷ‫ͱૅج‬෺࣭Պֶ΁ͷద༻ɹද໘Պֶɹ 30 (2009) 652-658.
[5] H. Masuda, H. Yasuda, J. Onoe, Twodimensional arrangement of one-dimensional polymerized C60 evidenced by high-resolution cryotransmission electron microscopy, Carbon 96(2016) 316-319.
[6] J. von Neumann, The Mathematician, The
Works of the Mind edited by R. B. Heywood, Univ.
Chicago Press 1947.
[7] H. Shima and J. Onoe, Chapter 3ɿTopologyInduced Geometry and Properties of Carbon
Nanomaterials, in The Role of Topology in Materials Springer 2018ɼS. Gupta, A. Saxena ฤɽ

Fig. 6 に は そ の 例 と し て，正 弦 波 形 状 の 凹 凸 薄 膜
஋ δr = 0.16 nm ͱɼ੺֎෼ޫεϖΫτϧʹΑ

Δߏ଄ͷධՁ஋Ͱ͋Δ δr = 0.14 nm ͷ྆ऀ͕΄΅Ұக͢Δ͜ͱͰɼిࢠ͕‫ز‬Կֶ͔Βड͚ͨ

マーの包絡面は一軸性の周期凹凸曲面であり，そのキඌ্Β͸਺चঢ়ʹ C Λ࿈ͶٖͨҰ࣍‫࠯ݩ‬Λ ɹ ౡɾ٢ Ԭɾඌ ্ ͸（Fig. 3）が生み出す有効ポテンシャル U(x)の空間分布

60ޮՌɼͦΕ͸ JKdC ͷ࿈ଓͨ͠Ԝತ‫ۂ‬໘Ϟߏ੒͠ɼͦͷిࢠঢ়ଶͱߏ଄ʹ͍ͭͯ‫࡯؍‬ɼߟ࡯

リアは原子シート内をほぼ自由に運動することがわか͠·ͨ͠ɽC ͸௚‫ ܘ‬0.71 nm ఔ౓ͷαοΧʔ σϧͷߏ଄Λ༻͍ͯɼޮՌͰ͕͢ɼͦΕ͕࣮

60

1DEC ͷ α ͕‫ز‬Կֶূ͞ΕͨͷͰ͢ [1-4]ɽ

nm) ʹ 3 keV, 0.5 mA ͷిࢠઢΛ 16 ࣌ؒɼরࣹ ͱΛղ໌͠·ͨ͠ [3]ɽJKdC ʹԊͬͯɼδr Λ
͢Δͱ C60
60 ͕࿈ͳͬͨ 1 ࣍‫ݩ‬తͳ C60 ϙϦϚʔ௕Ԭ [III1] ͷߟ࡯͔Βɹ JKdC [I1,I2] ஑ాɾ。凹凸振幅 a の増加に伴い，凹部と凸部ています 27)。よって系の集団励起状態を曲面量子系の௒ߴਅۭԼͰ࡞੡ͨ͠ C ബບ (ບް 20ʙ30 ‫ܗ‬ঢ় δr ͱؔ࿈͢Δ͜ を示しましたϘʔϧ‫ܕ‬ͷ෼ࢠͰ͢ɽ

18)

60

場から解析すれば，C ポリマーの凹凸構造と TLL 指ͦΕͧΕ 40 ೥ɼ30 ೥,に対応する点 x＝lp/g(l＝0, :1, …)での U(x)の下向きேӬৼҰ࿠

20 ೥Λ‫ͯܦ‬ɼφϊΧʔ

ʢ1906-1979)
a との関係性を新たに見出す可能性があります。指ಁա‫ݦࢠిܕ‬ඍ‫ڸ‬ʣͷ‫ʹ࡯؍‬ΑΔͱ਺ඦ nm ఔ ͳ‫ܗ‬ঢ়ʹΑΓఆ·ΔϘϯͷࡐྉՊֶͷൃల

࣋ͭਤ 1 ͷ‫ز‬Կֶత͕ੜ੒͞Ε·͢ɽͦͷ௕͞͸ cryo-TEMʢ௿Թ

ピークが，徐々に鋭くなる様子がわかります。こうしたϥοςΟϯδϟʔྲྀମͱ͜ͷ‫ز‬ԿޮՌΛ݁ͼͭδϟʔྲྀମͱͷ༥߹
ʢ1903-1957)
a の値により TLL 状態の各種物理量の特異性が特徴౓Ͱ͋Γ [5]ɼ‫ܗ‬ঢ়͸ԼਤͷΑ͏ʹͳ͍ͬͯ·͢ɽෛͷϙςϯγϟϧͷޮՌΛՃຯͨ݁͠ՌͰ͢ɽ

ͱɼேӬɾϥοςΟϯ

͚ͨ͜ͱͰɼJKdC ͷཧ࿦͸େ͖͘ඈ༂ͨ͠ͷͱ͍͏ཧ࿦ͷඈ༂ʹΑͬͯɼ͍ͭʹͦΕ͕࣮ূϑΥϯɾϊΠϚϯ

Ͱ͢ [4]ɽͦͷґଘੑ α(δr) ͸ର৅ͱ͢Δ۠ؒͰ
けられる 23)ことを考えると，凹凸構造との相関関係の͞Ε·ͨ͠ɽӶ͍ண‫͍ਂͱ఺؟‬ಎ࡯ͷࣀ෺ͱߟ
正弦波曲面の他にも，これまで様々な曲面構造における͸࿈ଓͰ୯ௐ૿ՃͱͳΓ·͢ɽ͑·͢ɽଞํɼߴ෼ղೳࢵ֎ޫిࢠ෼ޫ‫ܭ‬ଌʹΑͬͯਤ 1: 1D ΤΩκνοΫφϊΧʔϘϯͱ δr [1]͜ͷ̍࣍‫ݩ‬తφϊΧʔϘϯΛ 1D ΤΩκνο
ΫφϊΧʔϘϯͱඌ্Β͸‫ݺ‬ΜͰ͍·͢1 ɽ͜͜ޫ‫ٵ‬ऩεϖΫτϧͷ‫౓ڧ‬ͷґଘੑΛ‫͢࡯؍‬Δͱɼ

ϑΥϯɾϊΠϚϯ͸ʮ਺ֶ͸‫ʹݧܦ‬ଈͨͦ͠α Λ௚઀ɼධՁ͢Δ͜ͱ͕Ͱ͖·͢ɽඌ্Β͸ͦͷൃੜͷ‫͔఺ݪ‬Βԕ͘཭Εɼʮ‫࣮ݱ‬ʯ͔Βဃ཭͠ΕΒͷ‫ʹ࡯؍‬ΑΓ 1DEC ͷ৔߹ɼα = 0.6 Ͱ͋ͨୈೋɼୈࡾੈ୅ʹͳΔͱɼக໋తͳ‫ؕʹػة‬Δ͜ͱΛ࣮‫ݧ‬తʹࣔ͠·ͨ͠ɽؔ਺ α(δr) ͷ୯Γ·͢ɽʜʜ‫ͱػة‬͸ຎࡲΛආ͚ͨ҆௚ͳൃలௐ૿ՃੑʹΑΓ࣮‫݁ݧ‬Ռͷ α = 0.6 ͸ δr = 0.16

Ͱ͸ 1DEC ͱུ͠·͢ɽδr ͕ͦͷ࿪Έ۩߹ͷࢦnm ʹରԠ͢Δ͜ͱ͕൑Γ·ͨ͠ɽඪͱͳ͍ͬͯ·͢ɽ͜Εʹରͯ͠ɺ੺֎ͷ෼ޫ͞ΕͨޫΛরࣹ͠ɼ

1 ϛϧφʔͷΤΩκνοΫ‫ٿ‬໘ʹΠϯεύΞʔ͞Εͯɼ໋ͦͷ‫ٵ‬ऩεϖΫτϧΛ‫ݟ‬Δ͜ͱͰɼߏ଄͕൑Δ

໊͞Εͨͱͷ͜ͱͰ͢ɽ

Ͱ͋Γɼ‫఺ݪ‬Λ๨ΕΔ͜ͱʹΑΔֶ໰ͷແҙຯ(·ͭͨʹ͖͛͠

2 ‫͔ͭز‬ͷԾఆ͸ೖΓ·͕͢ɼྔࢠྗֶʹैͬͯɼ‫ݧܦ‬ɹɹɹɹɹɹʗۚ୔େֶ ిࢠ৘ใ௨৴ֶ‫)ܥ‬తͳύϥϝʔλΛར༻͢Δ͜ͱͳ͘ɼԽ߹෺ͷߏ଄ʢ‫ܗ‬ঢ়ɼ
3 ँࣙɿ͜ͷ‫ࣄه‬Λॻ͘ʹ౰ͨͬͯɼඌ্ॱ‫ڭ‬त͔Β༷ʑߏ଄ͷ‫ݻ‬༗ৼಈͳͲʣ΍ిࢠঢ়ଶʢ೾ಈؔ਺ͷҐஔతɼΤωϧΪʔత෼෍ʣΛܾఆ͢Δ‫ํࢉܭ‬๏Ͱ͢ɽͳ͜ͱΛ͝‫ڭ‬त௖͖·ͨ͠ɽ

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現代数学9月号（2020）

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week ending
10 APRIL 2015

PHYSICAL REVIEW LETTERS

PRL 114, 146401 (2015)

Mobility of Holstein Polaron at Finite Temperature: An Unbiased Approach

Geometric shape effects on electron-phonon coupling strength1

4
B 90, 155435A.
(2014)A. S. Mishchenko,1,2 N. Nagaosa,1,3PHYSICAL
G. DeREVIEWFilippis,de Candia,4,5 and V. Cataudella4

RIKEN Center forUnified
Emergent
Matter Science (CEMS), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
2understanding of the electron-phonon coupling strength for nanocarbon allotropes
NRC “Kurchatov Institute”, 123182 Moscow, Russia
3
Shota Ono,1,* Yasunori Toda,2 and Jun Onoe3
Department of Applied
Physics, The Universityof Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan
Department of Physics, Graduate School of Engineering,Yokohama National University, Yokohama 240-8501, Japan
4
Division of Applied
Physics, Facultyof Engineering, Hokkaido
University,
Sapporo 060-8628,
SPIN-CNR and Dipartimento
di Fisica,
Università
di Napoli
Federico
II, Japan
I-80126 Napoli, Italy
DepartmentofPhysicalScienceandEngineering,
Nagoya University, Nagoya 464-8603, Japan
5
INFN, Sezione di Napoli,
Universitario
di Monte
S. Angelo,
Napoli, Italy(ReceivedComplesso
8 April 2014; revisedmanuscript received 3 September
2014; published
21 October I-80126
2014)
25 June2014;
published
7 0.006)
April
2015)
The(Received
electron-phonon coupling
constant(λ) of carbon
nanotubes (λ =is muchsmaller than that of1

2

3

alkali-metal-doped C60 crystals (λ = 0.6). This difference may be due to the shape of2theπ kπ -electron conjugated= the shape effects
Tsystem: the former has a flat, whereas the latter has a sphere. In order τto confirm
in λ, we# Holstein
We present the first unbiased
results for the mobility μ of a one-dimensionalpolaron obtained
3λ !"C ωpolymer$
that has anhave examined the magnitude of λ for a one-dimensional (1D) uneven peanut-shapedusing Carlo methods inintermediate shape ofcombined
the π -electron conjugatedsystem between a carbonand
nanotubeand a C system,by numerical analytic continuation
with diagrammaticworldline
Montefemtosecond (fs) time-resolved pump-probe spectroscopy, because it can be expected to have an intermediate
the thermodynamic limit.
We
have
forof fs-transient
the first
timeobtained
severaldistinct
value ofλ between
them.identified
Theoretical analysis
refractivity
experimentallyfound theregimes in the λ-Tmagnitude of λ of the 1D C polymer film to be 0.02 as our expectation. This indicates that the shape of theplane including a band conduction
region,incoherent
region,an activated hopping region, and a
π -electron conjugatedsystem affects
the magnitude of metallic
λ for nanocarbon
allotropes significantly.
week ending
H Y Salthough
IPACS
C Anumber(s):L R mobilitiesEV
I E W 63.20.kd,
LE
T78.40.Ri,
T meanE R S78.47.jp
high-temperature saturation
region.
We observePthat
andfree paths at different
10 APRIL 2015
PRL 114,146401(2015)
DOI: 10.1103/PhysRevB.90.155435
78.67.Ch,
limit of ourvalues of λ differ by many orders of magnitude at small temperatures,Resolution
their values
at T larger than the(pulse duration)
I. INTRODUCTION
close to that of the P08 Claser
120 isomer (obtained via GSW) [17].
bandwidth become
very
close
to
each
other.
molecules
gives
In otherwords,Temperature:
the coalescence among
nCUnbiased
Mobilityof
Holstein
Polaronat
FiniteAn
Approach
60
Electron-phonon (e-ph) interaction is one of the most
B

e−ph

2

L

2

60

60

60

key ingredients for understanding the electronic properties
1,2

rise to form a 1D (C60 )n polymer (n is a positive integer).
In
1,3 case of n = 2, there
4 has been reported
4,5 to be 24 C120

4

A. S.
Mishchenko,G.
Deisomers
Filippis,
A. de
Candia,and AND
V.02.70.Ss,
Cataudellaof nanocarbon allotropes,
becauseit is related N.
to Nagaosa,
many
SHOTA
ONO,
YASUNORI
TODA,
JUN
ONOE
DOI: 10.1103/PhysRevLett.114.146401
PACS
numbers:
72.20.Fr
structural
obtainedand one of whichisLattice Temperature[K] via GSW,71.38.-k,
PHYSICAL
1 REVIEW B 90, 155435 (2014)

AND JUN ONOE

π-electron conjugated system

upling constant λ of nanocarbon
nding between C60 is also shown.

physical phenomena suchas superconductivity[1,2], chargeRIKEN
Center for Emergent
Matter Science(CEMS),
2-1 Hirosawa,
Wako, Saitama
351-0198,
equivalent
to the shortest
capped (5,5)CNT[20]. Because
the Japan
density-wave (CDW) [3], electron transport [4], 2and
NRCoptical“Kurchatov
Institute”, 123182
Moscow,
hasRussia
an intermediate shape
1D peanut-shaped
C60 polymer
3
coupling
constant
λ [7]has been
λ of nanocarbon
TABLE
I.
Electron-phonon
coupling
constant(a) responses(b) [5,6]. The(c)e-ph(d) of(e)
Department
Applied
Physics,
The UniversityofTokyo,
7-3-1
Hongo,
Bunkyo-ku,
Tokyo113, Japanof a π-electron conjugated system between (5,5)CNT and
4determined for those systems
so far [8–13]:
For example,di Fisica,allotropes.
The
kindof chemicalbonding
between
SPIN-CNR
and Dipartimento
Università
II, I-80126
Napoli,
Italy C60 is also shown.
it Napolicanbe Federico
expectedthatthe
1D polymer
pristinesolid C60 ,diλ  0.6 for A3 C60 [9] and λ5  0.1 for A1 C60 [10] with A = K,††
has Universitario
an intermediate divalue
between0.006 and
0.6. In Napoli,
addition, Italy
INFN, Sezione di Napoli, Complesso
Monte
S. Angelo,
I-80126
Rb, Cs, while λ  0.006 for single-walled carbon nanotubes
kpolymer
k exhibits
fascinatingsuch asthe 1D 2014;(Received
published
7 April2015) phenomena
C60C60compoundλBonding (C60 -C60 )(CNTs) [12] and λ  5.4 × 10−4 for multiwalled CNTs[13]. 25 Junegeometricalcurvature effects on Tomonaga-Luttinger liquid
What causes the large
in theλ for the mobility μ of a one-dimensional Holstein polaron obtained k
Wedifference
present thefirstmagnitudeunbiased ofresultsstates [21], CDW transition at 50 K [22,23], and anomaly in the(a) A3 C60 (A = K, Rb)
between CNTs and C60 systems? When both systems are
0.6
van inder Waals
resistivity
temperatures lower
50 K(0.5)[24].
In particular,
by numerical analytic continuation combined
withat diagrammaticand thanworldline
Monte
Carlo methods
compared to each other, the former CNTs have a flat-shaped(b)latterthealsoseveral
related tothe e-phregimes
interaction
C
0.3
Na4 twothermodynamic
limit. We have identified
for60properties
the first are
time
distinct
inSingle
the λ-TC-C σ bond
π-electron conjugatedthe
systemalong the electron-conductiveandthus
inspireus
toexaminethemagnitude
of
λ
for
the
1D
(c)incoherent= K, Rb)∼0.1 hopping2region,+ 2 cycloadditionalbond
AC60 (A metallic
conduction
region, an activatedand asphericallyshapedregion,
direction, whereas theplanelatterincluding
C60 have aa band
C60 polymers.
π-electronThis high-temperaturesuggeststhat theshapeof π-electron(d)In1D
saturation
region.
We observe
that
mean
at
different GSW bondpolymer
Coalesced
FIG. 4. (Color
online)one.
Schematic
illustrationof nanocarbon
al60 althoughtheC
present
work0mobilities
we report andon0.02the λ freevaluepathsof theconjugationinfluencesthemagnitudeof
λfornanocarbon
valuesof λ viadifferby manyorders of magnitude
at peanut-shaped
small temperatures,
theirvaluesat T larger
connected(a) vander Waals
lotropes: two C60 between which are(e) unevenby examining
the than the –1DSingle-walled
CNTC60 polymer
0.006
were coalesced
to form
allotropes.
Given
that C(c)60 moleculesinteraction, (b)single C-C
σ -bond,2 + 2 cycloadditionalbond,temperaturedependenceof the relaxation
time −4of photoexcited
bandwidth
become very close
to each other.
–
Multiwalled
CNT
5.4×10a structure like CNTs via the generalized Stone-Wales (GSW)(d) 1D C60 polymer having coalesced GSW bond, and (e) singleelectrons in the polymer, and confirm that the shape of πrearrangement [14], we could examine the shape effects onelectron conjugation affectsPACS
the magnitude
λ for nanocarbon
DOI: 10.1103/PhysRevLett.114.146401
numbers:of71.38.-k,
02.70.Ss, 72.20.Fr
walled CNT.
†
the magnitude of λ and thus confirm the above speculation.
allotropes.
kk k
Although it was reported that C60 molecules react with each
In Sec. II we describe briefly experimental details. In Sec. III
other to form a polymer inside a CNT [15], it is reasonable
V =measured
300 meVforreflectivity
C6 H6 and
V and
= 50
meV for C60 , though a
we showtransientchangesconfirm
deformationto coupling,respectively.
Unfortunately,
calcula††considerthatthe value of
λmotionin the Cofpolymerthus
Introduction.—Theaquantum
particleina60(bkthe) witharerelationelectron(phonon)operators
in theto
Here,
ck of
2 a
that theseresultsare consistent
the previously
measured
may beattributed
slightdeviation
inthe
C60creation
tions both ofthe phonon
spectramodified
and ofbythethee-phcoupling
formedis significantly
exterior
tubes: Theis an issue
backgroundof quantum
phonons(polaron)one [22].ofFromstatethese ofdatamomentum
we estimate k.
theThee-ph dispersion
coupling
εk ¼the−2tinteractioncosðkaÞ
constant forshape
all theeffects
modelsa largeamountof computaeffectdue
to itsInsphericalshape. Sinceon require
λ are notdirectlyobservedin the systems.
fundamental
importance[1]. Historically,
it is constantthecurvature
firstusing astems
theoreticalmodel.
Sec. IV we confirm
fromanearest-neighbor
hoppingonalinearlatticerecentdiscoveryof a one-dimensionaltional time, Fortunately,
owing to a thelarge
numberof carbon
atoms in a unit (1D)
areEinsteingovernedby thevan deradjacentC
polymerfromthethat between
the estimatedlengthof1D
the
1D60Cpolymers
60 a,condensed-matterproblem
where
theconfirm
quantumwithlatticeconstant
and theopticalphonons
enablesus to
the field theory
peanut-shaped C60 polymer
cell. This isunevenin progress.
magnitude
e-ph coupling is quite
similar tothe previous can be regarded as ainteraction,polymers
has
been
successfully
appliedExperimentally,a Waals
smallof the
60 term
speculation
described
above.
The C
last
describes the local EPC. All
have energytheω0c. 1D
Alternatively, electron transport measurements[2].
wouldbedata obtained by the resistivity measurement[24]. We also
hassynthesized
fromsemiconductors
electronThenumber
1D Ctheofcarriers
in insulators
andintro60 polymer
molecular
crystal.
It
is reasonable
tothe
setCBrillouinN(E10/eV/spin,
sums
over
momentaare ofoverzone,
and weuseful to investigate
natureof beenthee-phinteractioninλF ) 
λcomparethe magnitudeof thee-phcoupling
the
1D
60
films
beam
(EB) size
irradiationof orpristine
C60byFiniteone-ph1DVRH
duced
by dopingexcitedlightviaarethekey
playerspolymer
in manytakethenumber→ is∞. The charge current
low-dimensional
systems.
Ineffects
fact,theinteraction
inGSW
with thatof othernanocarbon
InNSec.
becauseN(Eofthe
1D allotropes.
CofpolymerF ) total
60 sites
PV slightly† smaller thanas shownproperties
transformations
betweenadjacent where
C60 molecules,
importantphenomena,thetransportof
we conclude ouroperator
the metallicinCNTs
beenthus
studiedusing anthat
atomicforceeffectssinðkaÞck ck , whereeof the model is ĵ ¼ 2eat
thatisof Apaper.
Fig. 1 has[16–19],it is expected
the shape
3 C60 [39]. Then we evaluate kV = λ/N(EF ) to bethesecarriers
are influenced
by polaroneffects [3].
microscopeonasλ an
probe[4].systems
A similar
techniqueisthe electron
Theofstrength
the EPCλ Thisλ is measured
canelectricalbe observedin suchbecauseof no exterior
2 meV,
which
givescharge.
a valueNλπ =of900.
Since
there are
also
the
case
of
doped
Mott
insulators
where
the
role
of
would enableustoclarifythenatureofthee-phinteraction
2
II.
EXPERIMENTS
tube. The 1D C60 polymer has a cross-linked structure roughly
by the dimensionless coupling constant λ ¼ g =ð2tω0 Þ,
60πin electrons
per theoneborderline
C60 , thebetween
value ofNπ = 900
suggests
electron-phonon
to be consideredwhile we coupling
postulate (EPC)that thehas
naturein the 1D C60 polymers,
the weakand strongMetallic 1D which
uneven sets
peanut-shaped
C60 polymers wereof the interactionrelation
has an intermediatecharacter between
solid
to high-temperature
superconductivity
[4].
that
the
1D
C60regime
polymerconsistof 15ForCone
60 molecules
formedfrom
3-kVelectron-beamirradiation
a 500-nm-thickofoftheorder of unity.
polaron,coupling
at λc may*
CNTs.
C60 and single-walled
There
is no exact solution of the polaron problem
shota-o@ynu.ac.jp
pristineCdepositedon a cesium
iodide (CsI)
substrate
60 filmwe
atand
short,
onintroduce
whichatheconducting
electronmoveTÞ ¼
σ ðω;without
TÞ=edynamic
mobilityμ ðω;can(a)Introduction.—The
motion of a quantum particle in a
Here, c (b ) are electron (phonon) creation operators in the
Bonding (C -C )backgroundofquantum
phonons(polaron)
is
an
issue
of
state of momentum k. The dispersion ε ¼ −2t cosðkaÞ
van der Waals
Single C-C σ bondfundamental
importance [1]. Historically, it is the firststemsfrom a nearest-neighbor hopping on a linear lattice
2 + 2 cycloadditional bondcondensed-matterproblem where the quantum field theory
Coalesced GSW bondwith lattice constant a, and the Einstein optical phonons
has been ––successfully applied [2]. Experimentally, a small
have energy ω . The last term describes the local EPC. All
number of carriers in insulators and semiconductors introsums over momenta are over the Brillouin zone, and we
ducedby
dopingorexcited
by
light
are
key
playersinmanytake the total number of sites N → ∞. The charge currentFIG. 4. (Color o= 50 meV for C60 , though aPlotropes: two C60 beimportantphenomena,
where the transport properties of
to atheC60 may be attributedsinðkaÞc c , whereeoperator of the model is ĵ ¼ 2eat
interaction,(b) sing
al shape. Since the interaction
these carriers are influenced by polaron effects [3]. This is(d) 1D C60 polyme
ers are governed by the van deris the electron charge. The strength of the EPC is measured
olymers
as adoped Mott insulators where the role ofalso canthebe regarded
case ofCNT.
by the dimensionless coupling constant λ ¼ g =ð2tωwalled0 Þ,
e to set N(EF )  10/eV/spin,electron-phononhas to beinconsideredin
olymeris slightly smaller than coupling (EPC)
Anomalytheresistivity
attemperatures
below
50
K
which sets the borderline between the weak- and strongvaluate
V = λ/N(EF ) to be
relation
to high-temperature
superconductivity [4].
deformation cou
coupling regime at λ of the order of unity. For one polaron,f Nπ = 900. Since there areis noexact solution of the polaron problem andtions both of thevalueThereof Nπ = 900
suggests
ðω;
TÞ
¼
σ
ðω;
TÞ=e
we
introducethedynamic
mobility
μ40
consistof of
15 Cthe60 molecules
constant for all th
mostanalysis
is
based
on
approximate
methods[1],
as a quantity related to the optical conductivity (OC)g electron can move without
tional time, owing
rangingfromvariational[5]
andperturbative
approaches
h of tube consisting of 15 C60
σ ðω; TÞ. The static mobility μ ðTÞ ¼ μ ðω → 0; TÞ is justcell. This is in pro
h the mean hopping distance[6] to the momentum average approximation for the
1D C60 polymer film [24].
the quantity measured in standard transport experiments.Alternatively,
30
mationof the λfunction
value. From [7]. It is only recently that numericallyGreen’s
useful to investig
π , the longer polymer has the
low-dimensional
exactsolutions
ould
be verified
by using otherfor the ground-state energy and opticalλλthe metallic CNT
most of the analysis is based on approximate methods[1],scattering.
mica
substrates instead of
as a quantity
relatedto theofopticalconductivity
(OC)
any
In
fact,
the
lengthtube
consistingof
15
C
conductivity
atCsIzero temperature
were
obtainedbydia60
1098-0121/2014/90(15)/155435(5)155435-1©2014 American Physical Society
rangingfrom variational [5] and perturbative
approaches
microscope as an
ðω;
TÞ.
The
staticmobility
μ
ðTÞ
¼
μ
ðω
→
0;
TÞ
is
just
σ
20
V. SUMMARYλλλis in coincidence with the mean hopping distancegrammatic
Carlo simulations[8–13],
variational[6] to the momentum
averageapproximation
formoleculestheIncoherent
metal
?
ding
may provide us aMonteunifiedwould enable us
measuredin standardtransport
experiments.
of 8–13thenmquantity
obtainedfor the1D C60polymerfilm [24].
In summary, we
have examined
carrierrelaxation
dynamics
Green’sfunction[7].
It
is only
recentlythat numerically
ects
on
the
magnitudeofλ(1D
Holsteinpolaronmodel)[14–16] and exact of[17]density-matrix
in the 1D C60 pousing femtosecond
time-resolved
the 1D diagonalization,
C60 polymers,
This strongly supports our estimation of the λ value. Frommarizes the value of λ forexact solutions
for the ground-stateenergy and opticalof the interaction
pump-probe
andatobtained
theexpansion
e-ph coupling
renormalization[18],spectroscopy,
and Chebyshev
alongwith the classificationgroupconductivity
zero temperature
were obtained bythe
dia-relation λ = 1800N(EF )/Nπ , the longer polymer has theconstant(λ) to begrammatic
0.02, whichMontesupports
our speculation
C60 and single-wa
djacent C60 molecules. Since
10
Carlosimulationsthat[8–13], variational
smaller e-ph coupling, which would be verified by using other[19],
showing
the thelimitsof [14–16]
theC60different
approximate
polymer
has anintermediate
value density-matrix
1D peanut-shapedding
can be ordered
as follows:
and exact[17]
diagonalization,
1D
VRH
samples prepared on such as mica substrates instead of CsIand CNTs on thearebasis evenof the shape of aof between Ax C60propertiesbond< 2 + 2 cycloadditional
schemes.
Finite temperature
renormalization group[18], andmore
Chebyshev expansion
π-electron conjugated system. This implies that the magnitude
substrates [28].
alesced GSW bond (not only[19], constantshowingthelimits problems,
ofallotropes
the different
approximate
difficulttoanalyzedue
toseveraltechnical
of
an
e-ph
couplingfornanocarbon
ist also C-C π bonds) [see also
The concept of chemical bonding may provide us a unified
schemes.
Finiteoftemperature
properties are even moresignificantly
affectedby the shapea π-electron conjugated0
at aandstrongerchemicalbonding
In summary, w
more than 6 decadesofefforts
tounderstandtheanalyzedue
to several
technical
problems,
understanding of the shape effects on the magnitude of λsystem.0We believethat tothisfinding
becomes
usefulfor35
5 difficult
10
15
20
25
30
ociation with an enhancement
and 10
more
than
6 20
decades
effortsa unifiedof electron-phonon
interaction
forto
0 understanding
15
25
30TÞ,
35understandforthenanocarbons. Table I summarizes the value of λ for of the 1D C60 p
temperatureof 5polaronmobility
μofλ ðω;bilitybetween adjacentdependence
C60-11000/T[K ] of polaron mobility μλ ðω;TÞ,temperaturedependence
pump-probe spec
nanocarbon allotropes.
nanocarbon allotropes, along with the classification
mall value of V and λ, as
ranging from alreadyhistoricranging
papersthe[2,20–27] tovarious
from [2,20–27]
already historic topaperstheconstant (λ) to b
polymers. The single-walledmolecules.
Sinceofchemicalbonding
betweenadjacentC
60
modern studies[3,28–35], behaviors
established different behaviors
remecase [Fig. studies4(e)].
modern[3,28–35],
different" 2Lestablished% which,the 1D peanut-shL:thelengthofthe1D
disorderedsystemACKNOWLEDGMENTS
thestrengthofthechemicalbondingcanbeorderedasfollows:
however,donotexhausttheregimes
exhibitedbyr any lattice model for the !
1D! exp
ofbetween Ax C6$'
which,
however,donotexhausttheregimes
exhibitedbyFIG.
1(coloronline).
Transport
regimesofpolaron.
Schematic
van
der
Waals
<
single
C-C
σ
bond
<
2
+
2
cycloadditional
ξ:electron
localization
lengthmaterials
where
EPC
is
relevant.
To
discussthem,
wey, we have already employed
We thank Professor K. Ohno, Mr. Y. Noda (Yokohama#is "University),& consider
phase
diagramshowing
fourregimes
different
regimesof (not
polaronπ-electron conjug
FIG.bond
1 (color
online).
Transport
ofbondpolaron.
SchematictheA.one-dimensional
Holsteinpolaron model
[20]
(two
C-C
σ bonds)<thecoalesced
GSW
onlyusing the EPC
1D
C60 polymers,where
National
and Dr.
Takashima (Aoyama
Gakuinmaterials
relevant.
To
discuss
we[E. N.
Economou,
PRBthem,
31, 6172 (1985)]
mobility
μ
in
the
plane
of
λ-T
(λ:
EPC
strength,T:temperature).
of an e-ph coup
X
X discussions. A part of theg present
uld be theoretically interesting
University) for their helpful
more
than
two
C-C
bonds
π bonds)[see also
phase
diagramshowingtheis tbut
four
regimesof polaron
Here,
the
unit ofσenergy¼ 1,also
anddifferent
ωC-C
considerthe forone-dimensional
Holstein
0 is the phonon frequency.
ffiffiffiffi[20]ðεkpolaroncby†k cakGrant-in-Aidþ ω0 b†kmodelbk Þ þ
c†k−q ck ðb†q þ b−q Þ:
H supported¼
ion
matrix elements
thework was financiallyforp
Scientificsignificantly affec
Figs.
4(a)–4(d)],
it
can
be
said
that
a
strongerchemicalbonding
Arrowsshow
thedirection
of
shift
of
the
borderlines
betweenN
mobility
μ
in
the
plane
of
λ-T
(λ:
EPC
strength,T:temperature).
kk;qdels. If the X
matrix elements
Research on Innovative Areas(Grant No. 21200032 ) fromsystem. We belidifferent
regimesNwhen
thephonon frequency
decreases.
g X
inassociation
withanenhancement
gives
rise
to
a
largerπ
ustic phonons are strong,
the Ministryof Education,
Culture,† the†† Sports, and† Technology
Here, the unit of energy is t ¼ 1, and ω0 is the phonon frequency.
ðεk cand/or
ω0 bk bk Þ þ pffiffiffiffi
ck−q ck ðbq þ b−q Þ:¼ to polara unified unders
be H
attributedin the electron hopping probability between adjacent C60
k ck þ (MEXT).
Arrowsshow the direction
shiftof theN k;qnanocarbon allotr
kS. Mishchenko
et al.,
PRL
114,
146401(2015)
molecules.
in ©ofa 2015small
value
ofborderlines
V andλ, asbetween
0031-9007=15=114(14)=146401(5)146401-1 ThisA. results
AmericanPhysicalSociety
60

Resistivity [!cm]60

Guéron, B. Reulet, I. I. Khodos,
L. Vaccarini, and H. Bouchiat,1).

different
regimesfrequency
decreases.
The single-walled
described
abovewhen
for thethe
1D phonon
C60 polymers.

[2] M. J. Rosseinsky, A. P. Ramirez, S. H. Glarum, D. W. Murphy,
R. C. Haddon, A. F. Hebard, T. T. M. Palstra, A. R. Kortan, S. M.
Zahurak, and A. V. Makhija, Phys. Rev. Lett. 66, 2830 (1991).

(or multiwalled) CNT is one extreme case [Fig. 4(e)].
Although we did not consider any lattice model for the 1D
146401-1 C60 polymer in the present© 2015
AmericanPhysicalstudy, wehave already
employedSocietyWe thank Pro
Figure 1. Schematic illustration of a one-dimensional metallic C polymer with anseveral kinds of models for the 1D C60 polymers, using the
National Universi
GSW transformation [40]. It would be theoretically interesting
University) for th
uneven peanut-shaped structure similar to the cross-linked structure of the P08 C－30－
to investigate the e-ph interaction matrix elements for thework was financia
stable isomer predicted using the general Stone-Wales rearrangement. The area
excited electrons in those models. If the matrix elements
Research on Inno

Appl. Phys. Lett. 104, 113301 (2014)0031-9007=15=114(14)=146401(5)155435-460

120

Application to environmental issues: CO2 immobilization
COMMUNICATION
www.advsustainsys.com

Immobilization of CO2 at Room Temperature Using theSpecific Sub-NM Space of 1D Uneven-Structured C60
Polymer Film
Masato Nakaya, Yasutaka Kitagawa,* Shinta Watanabe, Rena Teramoto, Iori Era,
Masayoshi Nakano, and Jun Onoe*

conductivity or high heat-resistance, electrochemical, or heat treatments can beutilized to promote chemical reactions intheir nanospaces.
We have investigated 1D uneven-a
structured (peanut-shaped) C60 polymer(Figure 1b) by electron-beam (EB) irradiation of C60 films (Figure 1a) underultrahigh vacuum (UHV) conditions.[9–16]
The polymer film has a hexagonal-closed
pack (hcp) structure (the left panel of
Figure 1b),[12] resulting in formation of
sub-nm spaces with a sizes of ≈0.3 and0.6 nm periodically arranged (the rightpanel of Figure 1b). In addition, the 1D
polymer exhibits physical properties(b)
arising from 1D metal[13–16] and a highheat-resistance[17] comparable to that ofcommercially available polyimides. These findings suggest
Immobilization and reuse of CO2 are urgent tasks not onlythat the robust sub-nm space can act as a specific reaction field
for suppressing global warming but also for sustaining or
similar to those of conventional porous materials.[1–8] We report
improving our global environment. One possible way toachieve these processes is to use porous materials, such as zeohere the immobilization of CO2 at room temperature (RT) inlites and metal–organic-frameworks that include nanospaces.
the sub-nm space of the 1D polymer film by its reaction with
These materials have been shown to exhibit storage, exchange,
H2O molecules, despite the fact that the reaction cannot occur
separation, and reaction field when molecules and/or ions are
at RT in the gas phase (the activation energy, Ea: ≈2 eV).
incorporated,[1] and have been extensively applied to materials
Figure 2A shows the change in IR spectra of the 1D C60
synthesis, medicine, electronics, catalysis, energy storage, andpolymer film (Figure S1, Supporting Information) before andenvironmental engineering.[1–8] Accordingly, immobilization
after exposure to various gases. As shown in Figure 2A(c), no significant changes in the IR spectra before and after N2-exposureand reuse of environmentally hazardous compounds, suchas CO2, NOx, or SOx using these porous
materials
will play ionwere observed. In the case of exposure to dry air (Figure 2A(d)),
CO32–: carbonate
online) Schematic illustration of nanocarbon
al-in solving our environmental
although the peak at 565 cm−1 (radial motion) remained almost
key roles
and energy issues.[2,3]that these nanoporous materials exhibit high electricalunchanged, IR peaks at 1182 (radial and in-plane mixed motion)etween which are connected via (a) van derGiven
Waals
and 1340 cm−1 (in-plane motion) originating from the convex andgle C-C σ -bond, (c) 2 + 2 cycloadditional bond,concave portions, respectively, of the 1D polymer became broadProf.
M.
Nakaya,Dr.
S.
Watanabe,
Prof.
J.
Onoe
ened.
This indicates that the in-plane motion included in thoseer having coalesced GSW bond, and (e)Department
single-of Energy Science and Engineering
IR peaks is aﬀected by O2 adsorption onto the polymer, whereas
Nagoya University
theradialmotion is insensitive to O2 adsorption. In contrast,
Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
more remarkable changes in the IR spectra were observed afterE-mail: j-onoe@energy.nagoya-u.ac.jp
exposure to atmospheric air with 14% humidity (Figure 2A(e)).
Prof. Y. Kitagawa, R. Teramoto, I. Era, Prof. M. Nakano
IR peaks at 565 and 1340 cm−1 for the 1D polymer were drastiDepartment of Materials Engineering Science
Osaka University
cally weakened when compared to those of Figure 2A(d). In addiMachikaneyama, Toyonaka, Osaka 560-8531, Japantion, four new IR peaks appeared at 523, 667, 878, and 1370 cm−1.
E-mail: kitagawa@cheng.es.osaka-u.ac.jp
Model The ORCID identification number(s) for the author(s) of this article Since atmospheric air consists of not only O2/N2 molecules but
also CO2/H2O molecules, the diﬀerence in IR spectra between
can be found under https://doi.org/10.1002/adsu.202000156.
Figure 2A(d,e) is caused by both CO2 and H2O. However, the
new IR peaks could not simply be assigned to IR modes of free
DOI: 10.1002/adsu.202000156

Immobilization and reuse of CO2 are urgent tasks for sustaining the global
environment. Here it is demonstrated that CO2 can be immobilized by reaction with H2O at room temperature (RT) in the specific sub-nm space of 1D
uneven-structured C60 polymer (1D polymer) film, even though the reaction
does not occur at RT in the gas phase (activation energy: 2 eV). First-principles calculations reveal that the CO2 molecule is stacked as a bridge between
the concave portions of adjacent 1D polymer frameworks via locally induced
Coulomb interactions. Such induced charge polarization activates CO2
both by weakening its double-bonds and by lowering its lowest unoccupied
PHYSICAL REVIEW B 90, 155435molecular(2014) orbital energy due to the bending motion. In addition, the flexible
π orbitals of the 1D polymer provide the optimal Coulomb field to stabilize
the transition state of the CO2 + H2O reaction remarkably. These factors work(c)(d)(e) together to make the reaction possible at RT.

E = 2.0 eV

Adv. Sustain. Sys. 5, 2000156 (2021)

Reaction Mechanisms

upling, respectively. Unfortunately, calculae phonon spectra and of the e-ph coupling
he models require a large amount of computag to a large number of carbon atoms in a unit
ogress.
Adv. Sustainable
electron transport measurements would
be Syst. 2020, 2000156
gate the nature of the e-ph interaction insystems. In fact, the e-ph interaction in
Ts has been studied using an atomic force
n electrical probe [4]. A similar techniqueto clarify the nature of the e-ph interaction
δ– δ+ δ–olymers, while we postulatethe δ+
nature O=C=O
凸: thatδ–, 凹:
n has an intermediate character between solid
alled CNTs.
V. SUMMARYδ–δ+

2000156 (1 of 5)© 2020 Wiley-VCH GmbHLUMO

Electron density
Convergence condition: 10–4

δ–
δ–

δ– δ+

we have examined carrier relaxation dynamics
δ–δ–
polymers, using femtosecond time-resolvedδ+
ctroscopy, and obtained the e-ph coupling
be 0.02, which supports our speculation that
haped C60 polymer has an intermediate value
60 and CNTs on the basis of the shape of a
Pinned by Coulomb interactions
gated system. This implies
that the magnitude
pling constant for nanocarbon allotropes is
cted by the shape of a π-electron conjugatedieve that this finding
for Coulomb interactions due to mirror image effects increase positive and negative charges(1)becomes
Locally useful
inducedstanding of electron-phonon
interaction
foron theC (1D polymer)
and O (CO2) atoms, respectively, which play roles not only in weakening the CO2
ropes.
double-bond slightly but also in stabilizing the TS of the CO2 + H2O reaction.
(2) In addition, the O atom of H2O easily approaches the C atom of CO2 when activated by the bending motion.
ACKNOWLEDGMENTS

ofessor K. Ohno, Mr. Y. Noda (Yokohamaity), and Dr. A. Takashima (Aoyama Gakuin
heir helpful discussions. A part of the present
ally supported by a Grant-in-Aid for Scientificovative Areas (Grant No. 21200032 ) fromAdv. Sustain. Sys. 5, 2000156 (2021)－31－

CO2 activation methods
Conventional
・ Radiochemical
・ Thermochemical
・ Biochemical
・ Photochemical
・ Electrochemical
・Catalysts

1D凹凸C60 Polymer
CO2 pinning
Angular vibration: LUMO energy decreases

B. Khezri et al., J. Mater.
- Chem. A 5, 8230 (2017)

Reduction
To add electrons to LUMO
(antibonding orbital)
Weakening O=C=O bonds

Not reduction
LUMO electron afﬁnity up↑
Increasing electron withdrawing

N-doped 1D 凹凸 C60 polymer exhibiting CO2 reduction efﬁciently

Several kinds of substitution position

W. Zhou, H. Shen, Q. Wang, J. Onoe, Y. Kawazoe, and P. Jena, Carbon 152, 241 (2019)－32－

Curvature-driven new functional nanocarbons(theoretical prediction)

2D exotic-nanocarbons

3D exotic-nanocarbons

three C atoms has an
unpaired electron
Uneven curved surface makes the Diraccone inclined to form an intense ﬂat band at
the Fermi level (like Pt)

Y. Noda et al., Phys. Chem. Chem. Phys. 16, 7102 (2014)

Concave surface induces
magnetic properties

N. Park et al., Phys. Rev. Lett. 91, 237204 (2003)

PNAS 112, 2372 (2015)

Negative Poisson’s Ratio

Structural model made using a 3D printer (given by Prof. Kawazoe)

top

middle

bo-om

－33－

次元

Mapping of nanocarbons using Gaussian curvature (k) and dimension (D)???

fullerene

Nanodiamond

2D 凹凸 carbon sheet

1D 凹凸 carbon tube

graphenenanotube

k 0

unidentiﬁed

Penta-graphene

Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120
stable isomer predicted using the general Stone-Wales rearrangement.

The areacolored in sky blue represents a Riemannian curved space in which !-electrons move
one-dimensionally. The !r denotes the degree of uneven deformation.

1D 凹凸 carbon tube

2D 凹凸 carbon sheet

Mackay crystal

Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120

01stable isomer predicted using the general Stone-Wales rearrangement.

The areacolored in sky blue represents a Riemannian curved space in which !-electrons move
one-dimensionally. The !r denotes the degree of uneven deformation.

2

D

3

What happens twice happens three times !?
Japanese Maxim

Discovery of new novel materials
based on (1) knowledge (literature), (2) experience, and (3) intuition/inspiration

Serendipity－34－

“Beyond serendipity”

機能発現

結び目理論による液晶コロイド形態解析

ナノ集積機能システム

構造計測

物理情報

光機能計測

タンパク質
フォールデイング機構の解明
生体機能の解明
創薬

電子機能計測電子・スピン・フォノン

新炭素物質相発
分子素子
新触媒

曲面量子論

物性理論

熱物性

機械学習・情報科学

光学特性
熱物性

生理活性

物質科学的数理モデル

導電性
磁性

量子化学

特異多面体
ナノコロイド機械特性

Figure 1. Schematic illustration of a one-dimensional metallic C60 polymer with anuneven peanut-shaped structure similar to the cross-linked structure of the P08 C120
stable isomer predicted using the general Stone-Wales rearrangement.

例：3配位

The areacolored in sky blue represents a Riemannian curved space in which !-electrons move
one-dimensionally. The !r denotes the degree of uneven deformation.

.....
.....

ホールデイング
特性

数理モデル
（点と線）

トポロジー

触媒活性

機械特性

化学情報
本数（配位数）方向（結合角）曲面ナノカーボン

光学特性

高分子

支点

離散幾何

多様体
結び目理論

公募班

吸蔵特性

接合点
超分子

高分子

熱物性
光学特性

特異空間場
多孔性物質

生理活性
トポロジカル高分子

機械特性

導電性

創薬

画期的な触媒
生体機能模倣無機固体材料
DNA組替・複製機構の
全容解明

－35－

伊達な大学院「TANSO：古くて新しい炭素材料」
講義配信中

https://brainnavi-online.com/contents/category/bacademy☜ここから新規登録（無料）すると視聴可能となります。

Acknowledgments
Onoe Lab.

Masato Nakaya, associate professor
Shinta Watanabe, research associate (Tokyo Inst. Technol.)
Takuya Kawai (Former master student)
Shintaro Umeda (Former master student)
Shu Hosono (Former master student)
Kentaro Yashiro (Former master student)

Collaborators

Prof. Yasutaka Kitagawa (Osaka Univ.)
Prof. Kaoru Ohno (Yokohama National Univ.)
Prof. Yusuke Noda (Okayama Prefect. Univ.)
Dr. Tomonobu Nakayama (NIMS, WPI-MANA)
Prof. Hiroyuki Shima (Yamanashi Univ.)
Prof. Hideo Yoshioka (Nara Women’s Univ.)
Prof. Shin-ichi Kimura (Osaka Univ.)
Prof. Takahiro Ito (Nagoya Univ.)
Prof. Qian Wang (Peking Univ.)
Prof. Yoshiyuki Kawazoe (Tohoku Univ.)
Prof. Hirohide Yasuda (Osaka Univ.)
Dr. Hideki Masuda (JEOL)
Prof. Shigeki Matsutani (Kanazawa Univ.)
Prof. Yutaro Kabata (Nagasaki Univ.)
Prof. Yuta Ogata (Kyoto Sangyo Univ.)

Research grants－36－

IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, JapanA ﬁrst-principles consideration of energetic
stability of peanut-shaped fullerene polymer
Yusuke NODA
Okayama Prefectural University
In a previous experimental study, it was reported that a peanut-shaped fullerenepolymer (PSFP) with a pseudo-one-dimensional structure can be formed by electronbeam irradiation to C60 thin ﬁlms. However, atomic-level structure of the PSFP is still
unclear, and various PSFP models have been proposed in previous theoretical studies.
In this study, we performed ﬁrst-principles calculations based on density functional
theory, and evaluated energetic stability of various PSFP models. The relationshipbetween the energetic stability and geometrical structure of the PSFP was revealed bycomputational results.

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+1;¨µ©R4·I¤Þ#:gD¬¼ÊÖ
½Û·µÜ¼ÊÖ½Û¬'hF´t¦Ý

－44－



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,&+7&*Or=

 
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,&+7&*ÍÅØÁÐÖ
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  !EF#/NEGRWguZz ,&+7&*¼ÊÖ½Û¬z!·|E§Þ¼ÊÖ½Û·u¡µ¼ÊÖ½Û

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ßLl¼ÊÖ½Û

－45－



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':O";S&HRdz` xÚ+N +<[
 ÓÉÄÇÂÖÜW+Ì¾ÇÖÝ 
$aZ_eM3Ü ÝÁÕÀØÜÃ¹ÑÏØÈZÝ$aZ_e% >5AEF
 O
ÏÆÖ[ \

5

}0,+#7)#/ ÜÝ'L·
~dxyÜÙGÝ
!O

－46－



 >5AEFÏÆÖ¬$ÏÆÖß++ß¬ N
dx+Nß ÜÂÖ£´Ý ß(ÌÖÜ6Ý
Nß!Oß T  Oß T g,09?Rh >5AEFÏÆÖÜ) &Ýß Þß Þ'ß Þß －47－

 >5A/27A3D

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EF(1)M"4M >5A/27A3D

WguZz ?IRj$hWguZz#
g

－48－

?IRj$ h







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EF



;jc^eg h<jc^eg h:jc^eg h

 

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 

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27A3D.,/--+6gW`eYf (1)g8hMh)MB@OKiW`eYfN&FQG@g FQG@h)MB@OKiW`eYfN FQG@g FQG@h+6gW`eYf (1)M"4Mh)$)M"4MB@OKi
W`eYfN&FQG@g FQG@h)$ )M"4MB@OKi
W`eYfN FQG@g FQG@h

－49－



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－50－

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 1(''- &'!.647.$/9257
38 647+"06471&' 9@BCAE7H%!-9D%!9. ,.1/$-&'os        osÜÝ*" µÜ t*- MJÝ4" % Ü*- # osG MJÝ

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－51－



IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, Japan

Mathematical analysis of low-dimensional
nanocarbon materials focusing on the hierarchy oflattice defects
Xiao-Wen LEITokyo Institute of Technology
In this study, the objective is to obtain knowledge that will lead to the establishmentof a fundamental theory for designing curved surfaces, focusing on the representationof the mechanical properties of graphene sheets (GS) using diﬀerential geometry. We
create several stable structure models of GS with diﬀerent lattice defects using molecular dynamics and perform mathematical analysis of their geometric and mechanicalproperties. For the geometrical properties, we measure the mean curvature and Gaussian curvature of the curved surfaces, and for the mechanical properties, we analyze theinternal stress distribution and potential energy distribution. Changes in mechanicalproperties due to changes in structure and arrangement (especially hierarchy of latticedefects) are also considered. Furthermore, the obtained results will be used to explore
the interrelationships between the geometrical and mechanical properties, which willfurther pioneer the mathematical analysis of nanomaterial mechanics.

－52－

Tokyo Techત৕ಳีभమಽਙपା৯खञ
଩ઃ੪ॼঀຍಞ౫મभਯ৶ੰෲ
ैःィ⟬ຊᏛ䞉ᮦᩱຊ
Ꮫศ㔝䛷⌮ㄽィ⟬䞉䝅䝭䝳䝺䞊䝅䝵䞁䜢୰ᚰ䛻䛩䜛

खॆअव॒

໹ ℙ䒎

ূ਎ੵ঵প৾‫؞‬੟ସ৶ੵ৾੹
IMI ଢ଼஢ૐভII:౫મఐ৾पउऐॊ࿥୦ध৻ਯIV
ഥਮ5ফ9া4঩‫ع‬5঩㻝
ృପপ৾IMI ड़‫ॹش‬ॕॺজ॔঒ क़ग़५ॺ1 ಀைD༠4మ413ಀ஼

ଢ଼஢৽ഄTokyo Tech
T
Te
ecch
h

ຊᏛ䝅䝭䝳䝺䞊䝅䝵䞁䜢⏝䛔䛶䚸ᵓ㐀䛸≉ᛶ䛾㛵ಀ䜢᫂䜙䛛䛻䛧䚸 ᶵ⬟䜢Ⓨ⌧䛩䜛ᵓ㐀タィ䛻ᙺ❧䛶䜛•

2006-2009ᶫ䛾᭱㐺໬タィ•

2009-2013⌮᝿ⓗ䛺䝘䝜Ⅳ⣲ᮦᩱ䛾ືⓗ≉ᛶ•

2013-2016䝅䞁䜽䝻ᆺLPSOᵓ㐀㔠ᒓᮦᩱ䛾ຊᏛ≉ᛶ

•

2014-2017పḟඖ䝘䝜Ⅳ⣲ᮦᩱ䛾᱁ᏊḞ㝗ᣲື䛾ゎ᫂•

2017-2020పḟඖ䝘䝜Ⅳ⣲ᮦᩱ䛾⮬Ⓨ᭤⋡ᙧᡂᵓ㐀䛾ゎ᫂

•2020-2023
పḟඖ䝘䝜Ⅳ⣲ᮦᩱ䛾ᚤศᗄఱᏛ䛸ຊᏛᶵ⬟䛾⼥ྜタィㄽ

•2021-⌧ᅾᗄఱᏛ䛸ຊᏛ⼥ྜ䛻ᇶ䛵䛟ᅇ఩ไᚚ䛻䜘䜛ᮦᩱᶵ⬟タィ

㻞
－53－

৔ઍ

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मगीप

• ଩ઃ੪ॼঀຍಞ౫મ
• ત৕ಳีଢ଼஢৔ઍ• ॢছইख़থ३‫ॺش‬भ૗஄્ਙ• ढ़‫ش‬঎থॼঀॳগ‫ش‬ঈभ૗஄્ਙ

उॎॉप㻟ॼঀຍಞ౫મ৅ৄऔोञଡୗभ੗஘ਙ

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oky
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2022ᖺ䚸ྡྂᒇ኱Ꮫ䝖䝷䞁䝇䝣䜷䞊䝬䝔䜱䝤⏕࿨ศᏊ◊✲ᡤ䛾ఀ୹೺୍㑻ᩍᤵ䜙䛻䜘䜚䚸Ⅳ⣲䛾䝯䝡䜴䝇䛾㍯䛂䝯䝡䜴䝇䜹䞊䝪䞁䝘䝜䝧䝹䝖䛃䛾ୡ⏺ึ䛾ྜᡂ䛻ᡂຌ䛧䛯䚹

2004

Graphene sheets (GSs)䠖
䜾䝷䝣䜵䞁䝅䞊䝖
䠄Novoselov et al. 2004䠅 2-Dimension
2010ᖺ䝜䞊䝧䝹≀⌮Ꮫ㈹」㞧䛺ᗄఱᏛᵓ㐀䜢䜒䛴᪂䛯䛺䝘䝜䜹䞊䝪䞁ᮦᩱ䛾㛤Ⓨ䛻㐨䜢䜂䜙䛟⏬ᮇⓗ䛺ᡂᯝ
https://www.itbm.nagoya-u.ac.jp/ja/research/2022/05/post-43.php

1991

Carbon nanotubes (CNTs)䠖
䜹䞊䝪䞁䝘䝜䝏䝳䞊䝤䠄Iijima 1991䠅 1-Dimension

C60䠖䝣䝷䞊䝺䞁
1985 䠄Kroto et al. 1985䠅 0-Dimension
1996ᖺ䝜䞊䝧䝹໬Ꮫ㈹－54－㣤ᓥ⃈⏨ᩍᤵ 㐲⸨Ᏺಙᩍᤵ
䠄ಙᕞ኱䠅
䠄ྡᇛ኱䠅

㻠

ॼঀຍಞ౫મभ્ਙ‫؞‬ਃચ‫ૢ؞‬৷ཎᏊ㛫ຊ㢧ᚤ㙾
Atomic Force Microscope
䠄AFM䠅

䝲䞁䜾⋡
1TPa

ᶵ⬟ᛶ

≉ᛶ

Conductive
material྾཰ᛶᘬᙇᙉᗘ
䠚50GPa CNTs

᣺ື࿘Ἴᩘ
~THz

䝩䝖䝹䝭䝛
䝉䞁䝇

䝷䝬䞁
ศගᛶ

ᶵᲔຊᏛ≉ᛶ
ᶵ
ᲔຊᏛ≉

ගᏛ≉ᛶ
ග
Ꮫ≉

Nanodevice

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⇕≉ᛶ

Energystorage

Thermalmaterials㟁Ẽ≉ᛶNanosensor

䝞䝸䝇䝔䜱䝑䜽
⇕ఏᑟఏᑟ㔠ᒓ≉ᛶ
䡚2000W/m䞉K

Nanomembrane༙ᑟయຊᏛ≉ᛶ䛸ᙧែ䛻ὀ┠৔ઍ

Biomedical
application
Field emission 㻡

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मगीप

• ଩ઃ੪ॼঀຍಞ౫મ
• ત৕ಳีଢ଼஢৔ઍ• ॢছইख़থ३‫ॺش‬भ૗஄્ਙ• ढ़‫ش‬঎থॼঀॳগ‫ش‬ঈभ૗஄્ਙ

उॎॉप
㻢
－55－

ત৕ಳี

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TechT h੥အಳีभரథ
ਡಳีਉ৕૬ሠ‫ؚ‬ત৕৑ਉ৕‫ؚ‬
ਂෞ੟ਉ৕

଍ಳี

ૡਜ਼‫ؚ‬ਡಳีभ଍૾ଦഔ

એಳี੥အ၄ੀ‫ؚ‬൨အ୆ੀ‫ؚ‬஋ಽಳี‫ؚ‬ಗਜ਼ৼ୆ੀ‫ؚ‬ੀએ‫਀ؚ‬એ

৬஋ಳี

㼇㻝㼉

ෲল੟‫ؚ‬ਸ਼2ৼ‫ؚ‬સो‫ؚ‬঎ॖॻਸઢ෎੘ ‫؜‬ૡਜ਼ো୅૛‫؜‬፻෾ඬ‫؜‬1999㻣㼇㻝㼉 ஑ᕞ኱Ꮫ䜹䞊䝪䞁䞉䝙䝳䞊䝖䝷䝹◊✲ᡤ㼇㻞㼉㻌 䜰䜲䜰䞊䝹ᢏ⾡⪅◊✲◊✲ᡤ㼇㻞㼉ঽேभরदभત৕ಳี
᳜≀

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䝃䝪䝔䞁

䝖䜴䝰䝻䝁䝅

https://www.flickr.com/photos/corematerials/4599222126

https://almerja.com/reading.php?idm=115418

ື≀

㌿఩
ᅇ఩

䝅䝬䜴䝬
－56－

https://ja.wiktionary.org/wiki/Zebra

㻌㻌㻌㻌㻌

㻤

౫મরभત৕ಳี

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㔠ᒓ
䝘䝜Ⅳ⣲ᮦᩱ

https://quizlet.com/87802654/ ᅵᆅ䛾䛴䛟䜚䛸ኚ໬(https://qiita.com/tktk0430/items/ef15f9ce15c4aa79881d)(J. Kotakoski, et. al, Phys. Rev. B, 2011)

䝬䜽䝻

䝯䝋

䝭䜽䝻
ᅇ఩

ᅇ఩
ࣔࢹࣝ

ᅇ఩࠿ࡽᒾ▼ࢆኚᙧࡉࡏࡿᶵᵓࡢゎ᫂(Cordier, et. al, Nature, 2014)ᅇ఩ไᚚ䛻䜘䜛㧗ᘏᛶ䞉㧗ᙉᗘ䛾ᵓ㐀ᅇ఩䛛䜙ᒾ▼䜢ኚᙧ䛥䛫䜛ᶵᵓ䛾ゎ᫂䝖 䝫 䝻 䝆 䜹 䝹 ⤯ ⦕ య䛻䛚䛡䜛ᅇ఩䛻ᤕᤊ䛥䜜䛯ศᩘ㟁Ⲵࢺ࣏ࣟࢪ࢝ࣝ⤯⦕య࡟࠾ࡅࡿ
ᅇ఩࡟ᤕᤊࡉࢀࡓศᩘ㟁Ⲵ
etc.

㻥

(Poo, Jiang, et. al, Nature, 2021)ॼঀຍಞ౫મभଡୗ

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஬ဨ⎔

⏕ᡂ
㧗
እຊ
୐ဨ⎔

௓ᅾ≀
䛺䛹
⌮᝿ⓗ䛺䜾䝷䝣䜵䞁䝅䞊䝖

஬୐ဨ⎔
䛾⤌䜏ྜ
䜟䛫
StoneWalesḞ㝗භဨ ⎔ 䛾 Ⅳ ⣲ ཎ Ꮚ 䛜つ๎ⓗ䛻㓄ิ䛩䜛㻝㻜
－57－ત৕ಳีभమಽਙ

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oky
kyo Teky
TechT h໹ℙ䒎‫؜‬౫મ‫؜‬2023 ফ 72 ඕ 8
ಀ p. 568-573

㻝㻝

৔ઍ

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मगीप

• ଩ઃ੪ॼঀຍಞ౫મ
• ત৕ಳีଢ଼஢৔ઍ• ॢছইख़থ३‫ॺش‬भ૗஄્ਙ• ढ़‫ش‬঎থॼঀॳগ‫ش‬ঈभ૗஄્ਙ

उॎॉप
㻝㻞
－58－

ශએਝੑ

䝗䞊䝮ᆺ
Tokyo
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oky
kyo TekyTechT hపḟඖ䝘䝜Ⅳ⣲ᮦᩱᵓ㐀䛟䛥䜃ᅇ఩䠄ṇ䠅㞳ᩓ᭤㠃タィ㸦஧ḟඖཎᏊⷧ⭷㸧㠡ᆺᖹ㠃• ࣑࣭࣑࢟ࣜ࢞࢜ࣜ࢞
• ศᏊືຊᏛ኱つᶍィ⟬
᏶඲᱁Ꮚ

ṇ䛾ᅇ఩

㻌㻌㻌㻌㈇䛾ᅇ఩䛟䛥䜃ᅇ఩䠄㈇䠅ṇ䛾㌿఩㓄ิ 㻌㻌㈇䛾㌿఩㓄ิ

ᅇ఩཮ᴟᏊ

䜻䝸䜺䝭
䜸䝸䜺䝭

ศᏊືຊᏛ䝅
䝭䝳䝺䞊䝅䝵䞁

໹ℙ䒎‫؜‬଩ઃ੪ॼঀຍಞ౫મपउऐॊત৕ಳีभਯ৶ੰෲ‫؜‬౫મ‫؜‬2023 ফ 72 ඕ 8 ಀ p. 568-573

৚ਜ਼पेॊଡୗ૗஄㻝㻟• એਗ૗஄म଩ઃ੪౫મपਠोॊ
ঘॽ‫ش‬ॡऩਙସ
Tokyo
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• ী৕৿ৡ৾ेॉ३঑গঞ‫ش‬३ঙথदऌ
ञශએ॑ঢ়ਯ஄द੺๚घॊऒधऋ૭ચ

䝕䜹䝹䝖ᗙᶆ

෇ᰕᗙᶆ ‫ = ݖ‬C ‫)ߠ(݂ݎ‬᏶඲䛺෇㗹ᙧ䚸ゅᗘ䛻౫Ꮡ䛧䛺䛔‫ = ݖ‬0.65‫ݎ‬

ษ䜚ฟ䛧
ṇ䛾ᅇ఩
䠄஬ဨ⎔䠅

ߨ
ᤄ䛧㎸䜏
‫ = ݖ‬0.ͷͶ‫ ݎ‬sin 2ߠ െ2㈇䛾ᅇ఩ Ἴ䜢᭷䛩䜛䝫䝔䝖䝏䝑䝥䝇≧䚸ゅᗘ䛻౫㻝㻠䠄୐ဨ⎔䠅 Ꮡ䛧䚸࿘ᮇⓗ䛻᭤㠃䛾ᙧ≧䛜ኚ໬䛩䜛－59－ॼঀৡ્৾ਙ

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ຊ

䜶䝛䝹䜼䞊᭱ᑠ໬
ึᮇ㓄⨨

Ᏻᐃᵓ㐀

ᩘ್ィ⟬䜰䝹䝂䝸䝈䝮䠖ඹᙺ䛣䛖㓄ἲ䠄conjugate gradient method䠖CGἲ䠅ᘧ䛾ㄝ᫂䜢┬␎䛣䜜䛺䜙ศ䛛䜛᭱㐺໬ᩘᏛ ʊᇶ♏ཎ⌮䛛䜙ィ⟬ᡭἲ䜎䛷ʊ䠈㔠㇂೺୍ⴭ䠈ඹ❧ฟ∧䠈2005
https://qiita.com/Dason08/items/27559e192a6a977dd5e5࿥୦્৾ਙ‫؟‬ශ૨ੑ઴
ᚤศᗄఱᏛ
ᚤศᗄఱ

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ᙧ

୺᭤⋡݇ଵ, ݇ଶ᭤㠃ෆ䛾᭤䛜䜚ලྜ䛾 ᐃ௞ ା௞䚓ᖹᆒ᭤⋡‫ = ܪ‬భ మଶ᭤㠃ᙧ≧䛾 ᐃ䚓䜺䜴䝇᭤⋡‫݇ = ܭ‬ଵ ݇ଶ୍⯡᭤⥺ᗙᶆ⣔䛻䜘䜛஧ḟඖཎᏊⷧ⭷ᙧ≧䛾⾲⌧

㻝㻡

㞳ᩓ᭤㠃

䛟䛥䜃ᅇ఩䠄㈇䠅

ᚤศᗄఱᏛ࡟ࡼࡿ㞳ᩓ᱁Ꮚᵓ㐀ࡢ⮬Ⓨ᭤⋡ࡢ⾲⌧X.-W. Lei*, S. Shimizu, J.-X. Shi, Nanomaterials, 2022, 12(6), 903.

－60－

㻝㻢

৚ਜ਼॑થघॊGS‫؟‬এॸথ३কঝग़ॿঝॠ‫ش‬

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TechT hএॸথ३কঝग़ॿঝॠ‫ش‬ṇ䛾ᅇ఩䠄஬ဨ⎔䠅
10ିସ

െ10ିସ eV㈇䛾ᅇ఩䠄୐ဨ⎔䠅
10ିସ

െ10ିସ eV㻝㻣৚ਜ਼॑થघॊGS‫؟‬਴಑ශ૨धफ़क़५ශ૨
ṇ䛾ᅇ఩䠄஬ဨ⎔䠅

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㈇䛾ᅇ఩䠄୐ဨ⎔䠅

ᖹ
ᆒ
᭤
⋡

ྠᚰ෇

࿘ᮇᛶ

䜺
䜴
䝇
᭤
⋡

㻝㻤
－61－

ग़ॿঝॠ‫ش‬धශ૨भঢ়બ

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T hGraphene sheet

Helfrich-Hamiltonian⌮ㄽ䠄䝁䝻䜲䝗䛾≀⌮Ꮫ䠅ത݂௖ = 2݇ ‫ ܪ‬െ ܿ଴ ଶ + ݇‫ܭ‬Stone
ne-Wales defectne݇㻦㻌᭤䛢๛ᛶ⋡‫ܪ‬㻦㻌ᖹᆒ᭤⋡ܿ଴ 㻦⮬Ⓨ᭤⋡݇ത 㻦㻌䜺䜴䝇๛ᛶ‫ܭ‬㻦㻌䜺䜴䝇᭤⋡5-7 defect5-

Pentagon defectᚤศᗄఱᏛ࡟ࡼࡿ㞳ᩓ᱁Ꮚᵓ㐀ࡢ⮬Ⓨ᭤⋡ࡢ⾲⌧

Heptagon defect

X.-W. Lei*, S. Shimizu, J.-X. Shi, Nanomaterials, 2022, 12(6), 903.

• ᖹᆒ᭤⋡‫ߢ = ܪ‬ଵ + ߢଶ Τ2• ࢞࢘ࢫ᭤⋡ ‫ߢ = ܭ‬ଵ ߢଶ

ীഘ

㻝㻥

䜶䝛䝹
䜼䞊

ᖹᆒ
᭤⋡

䜺䜴䝇
᭤⋡

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-7.20eV

0.15290.0100-7.45eV-0.1557-0.0010-7.20eV

0.00810.0100-7.45eV-0.2899-0.0100-7.20eV

0.00810.0832-7.45eV-0.2899-0.1345-7.20eV

0.00790.0005-7.45eV-0.0729

Stone
ne-Wales defectne

5-7 defect5-

Pentagon defectHeptagon defect

－62－

㻞㻞㻜

-0.0001

ශએ೐੅

Tokyo
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⣬ᶍᆺึᮇཎᏊ㓄⨨

Ᏻᐃᵓ㐀≧ែ

䛟䛥䜃䛾ᤄධ
⌮᝿ⓗ䛺㻳㻿䜢ᶍ䛧䛯
භゅᙧ᪉║⣬⤌䜏ྜ䜟䛫䛶พฝᙧ≧䜢ᣢ䛯䛫䛯䝰䝕䝹㠡ᙧ㈇䛾Ꮩ❧ᅇ఩

஺㏻ᶵჾ䚸㐠ື㐨ල䛺䛹⾪ᧁ྾཰⬟ຊ

᭤⋡Ⓨ⏕᭤㠃䛾⥔ᣢࢫ࣐࣮ࢺᮦᩱ࣭ᵓ㐀ࡢ㛤Ⓨ㻞㻝ᡂᙧᛶ䛾ྥୖ

ශએ೐੅ࢫ࣐࣮ࢺᮦᩱ࣭ᵓ㐀ࡢ㛤Ⓨ

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⮬Ⓨ᭤⋡Ⓨ⏕
᭤㠃ࡢ⥔ᣢᡂᙧᛶࡢྥୖ

⾪ᧁ྾཰⬟ຊ

㞳ᩓ஌ࡾ≀ࠊࢫ࣏࣮ࢶࡢ㜵ල࡞࡝ࡢ⪏⾪ᧁᛶ

㐃⥆

ศᏊືຊᏛ

㐃⥆యຊᏛ

ᅇ఩䛻䜘䛳䛶GS䛾≉ᛶ䜢ไᚚ䛷䛝䜛ྍ⬟ᛶᅇ఩䛾ᑟධ䛻䜘䜛᪂䛯䛺⾪ᧁ≉ᛶ䛜௜୚䛥䜜䛯GS䛾ᵓ㐀ゎ᫂䜢┠ᣦ䛩
－63－

㻞㻞

ਃચ౫મ

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㐠ື᪉⛬ᘧ
‫ܦ‬

߲ ସ ‫ݔ ݖ‬, ‫ݕ‬, ‫ݐ‬߲ ସ ‫ݔ ݖ‬, ‫ݕ‬, ‫ݐ‬߲ ସ ‫ݔ ݖ‬, ‫ݕ‬, ‫ݐ‬߲ ଶ ‫ݔ ݖ‬, ‫ݕ‬, ‫ݐ‬+ 2‫ܦ‬+‫ܦ‬+ ߩ݄= ‫ݔ ݍ‬, ‫ݕ‬, ‫ݐ‬
ସ
ଶ
ଶ
ସ
߲‫ݔ‬
߲‫ݕ߲ ݔ‬߲‫ݕ‬߲‫ ݐ‬ଶ

྾཰䜶䝛䝹䜼䞊௧௧ߖ ‫ = ݐ‬න ‫ = ݓ݀ ߬ ܨ‬න ‫ݓ ߬ ܨ‬ሶ ߬ ݀߬଴଴⌮᝿ⓗ䛺GS

Y. Tomioka
Y
Tomioka, T.
T Natsuki,
Natsuki J.-X.
J -X Shi,
Shi X
X.-W.
-W Lei*
Lei*, Nanomaterials
Nanomaterials, 2022
2022, 12(3),12(3) 436.
436

৔ઍ

㻞㻞㻟

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मगीप

• ଩ઃ੪ॼঀຍಞ౫મ
• ત৕ಳีଢ଼஢৔ઍ• ॢছইख़থ३‫ॺش‬भ૗஄્ਙ• ढ़‫ش‬঎থॼঀॳগ‫ش‬ঈभ૗஄્ਙ

उॎॉप
㻞㻠
－64－

ශએਝੑ

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T hJ. H. Lehman, M. Terrones, E. Mansfield, K. E. Hurst, Evaluating thecharacteristics of multiwall carbon nanotubes, Carbon, 49, (2011), 2581-2602

A. Hashimoto, K. Suenaga, A. Gloter, K. Urita, S. Iijima, Direct evidence foratomic defects in graphene layers, Nature, 430, (2004), 870-873ཎᏊ㛫ຊ㢧ᚤ㙾ࡢ᥈㔪[1]㟁Ꮚ࣮࣌ࣃ࣮[2]⮬ᕫ⤌⧊໬ࢆ฼⏝ࡋ
ࡓ᭤ࡆCNTࡢᐇ⌧[3]

ࢼࣀⅣ⣲ᮦᩱ
ࡢከᵝᛶ[4]㻞㻡[1] jst.go.jp/sentan/result/pdf_file/result001.pdf [2] nanoflex.jp/public-j/research.htm l[3] https://www.azonano.com/news.aspx?newsID=38568 [4] https://pubs.acs.org/doi/10.1021/acs.accounts.1c00505

CNT䛻㌿఩㓄ิ䜢ᑟධ

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䐟 Stone-Wales䠄SW䠅Ḟ㝗䛜⏕ᡂ
䝰䝕䝹䛾౛
⤖ྜ䛜
90ᗘᅇ㌿

䙶ศ㞳䛧䛶㌿఩ᑐ䛜⏕ᡂ

5-7Ḟ㝗䠄㌿఩䠅

㌿఩ᑐ

㻞㻢
－65－

ᙧ≧ኚ໬

㌿఩㓄ิ

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ᅇ఩཮ᴟᏊ

ࣂ૚

ຊ

(6,6)

ᙧ

᭤ࡆゅᗘ㸻 ࣂ૚  ࣂ૛ࣂ૛1࿘ศࡢཎᏊࡢᖹᆒᗙᶆ

(10,2)-(2,10)

ৡध஄भঢ়બ

㻞㻣
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ऱङाग़ॿঝॠ‫ ش‬U૗஄पৌखथഩਙ৬प໮इैोॊग़ॿঝॠ‫ش‬ශ૨઄ྫྷ[࣋]धभঢ়બ
৴ਢ৬৶૛ૄ‫ق‬৞࿞‫ك‬‫ق‬৞࿞‫ك‬Uٙ૗஄৏भPE - ੂ਋ଡୗभPE௟‫ܯ‬ଶ଴ ʹ‫ܫܧ‬ܷ=න

11= ‫ܫܧ‬
2ߩ

MS३঑গঞ‫ش‬३ঙথଶȉ݈

1 ߨ(݀ ସ െ ݀ଵ ସ ) 1
= ‫ܧ‬
2
64
ߩ

૗஄खञCNTभ
3ਡ॑ৢॊ৞॑੷ीॊଶȉ݈ශ૨઄ྫྷߩ॑଑ল

ْCNTभখথॢ૨ E‫؟‬1TPaٓ

㻞㻤
－66－

ऱङाग़ॿঝॠ‫ش‬

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୬ः

಍ः

• ৶૛ૄध३঑গঞ‫ش‬३ঙথधुपੜਸൊ਱॑ંघ
• ઉྫྷऋ৵औःऺन‫ؚ‬ൊऌम৵औऎ‫ؚ‬૗஄खृघः
ऒधऋীऊढञ‫؛‬

৔ઍ

㻞㻥

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मगीप

• ଩ઃ੪ॼঀຍಞ౫મ
• ત৕ಳีଢ଼஢৔ઍ• ॢছইख़থ३‫ॺش‬भ૗஄્ਙ• ढ़‫ش‬঎থॼঀॳগ‫ش‬ঈभ૗஄્ਙ

उॎॉप
㻟㻜
－67－

उॎॉप⋇

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• રोञ્ਙ॑થघॊ‫؛‬• ളහऩ࿥୦৾ଡୗ॑ुणৗञऩॼঀ
ढ़‫ش‬঎থ౫મभ৫৅ऋ਋ୄऔोथःॊ‫؛‬¾ ત৕ಳี

https://www.itbm.nagoya-u.ac.jp/ja/research/2022/05/post-43.php• ৚ਜ਼॑କਡधघॊત৕ಳีभమಽਙऋँॊ‫؛‬• ෩खःએਗ૗஄पेढथ‫ؚ‬ਕઃ੪৓ऩශએऋ஄ਛऔोॊ‫؛‬

㻟㻝

उॎॉप⋈

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¾ GSधCNTभ૗஄્ਙ

• ત৕ಳีऋથघॊघॊॼঀຍಞ౫મශએपणःथ‫ؚ‬ী৕ಯৡ৾दभএॸথ३কঝग़ॿঝॠ‫॑ش‬௬੼‫ؚ‬๶ী࿥୦৾
दශ૨॑௬੼खञ‫؛‬• ત৕ಳีभரథपૢगञශ૨धત৕ಳีଦ઼ऊै஄ਛऔ
ोॊශએ஄૾ऋ੒೾૭ચपऩॊध਋ୄऔोॊ‫؛‬

㻟㻞
－68－

IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, Japan

Math behind C60 I, II
Hiroyuki OCHIAI
IMI, Kyushu University
In this two slot, I will review the article of B. Kostanthttps://www.ams.org/notices/199509/kostant.pdf
with a fascinating title “The Graph of the Truncated Icosahedron and the Last Letterof Galois” appeared in Notice of American Mathematical Society in 1995 July. The
organizer of this workshop noticed an intimate relation and suggested me to introduce
this paper for broad audience. A topic in this paper is a graph coming from objects like
C60 , whose symmetry is described in groups. Several special groups play an important
role.

－69－

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－111－IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, Japan

Green function of boundary value problem and thebest constant of Sobolev inequality
Hiroyuki YAMAGISHITokyo Metropolitan College of Industrial Technology
The Sobolev inequality shows that the supremum of a function is estimated from above
by a constant multiple of the potential energy. We have found the best constant andfunction, which attain the equality. In the background, there is a boundary value problem of the linear diﬀerential equation corresponding to a bending problem of a stringor a beam. The solution is expressed by using the Green function. The Green functionis the reproducing kernel for a suitable set of a Hilbert space and an inner product.
Applying the Schwarz inequality to the reproducing relation, we have the Sobolev inequality. The best constant and function of the inequality are also expressed by using
the Green function. As an application, we consider the discrete version of the Sobolev
inequality corresponding to a classical mechanical model of the carbon molecular C 60
fullerene. The discrete Sobolev inequality shows that the square of the maximum of
the deviation is estimated from above by constant multiples of the potential energy.
In the background, there is a bending problem of a classical mechanical model. The
solution is expressed by using the Green matrix. Using the Green matrix, we have thebest constant and the vector, which attain the equality. It is considered that the best
constant represents the rigidity of the mechanical model.

－112－境界値問題のグリーン関数とソボレフ不等式の最良定数
山岸 弘幸（都立高専）
亀高 惟倫（阪大）

渡辺 宏太郎（防衛大）永井 敦（津田塾大）
武村 一雄（日大）
關戸 啓人（京大）
2023/08/31 11:14

ٛછආງஂٜ

ॢজ‫ش‬থঢ়ਯ‫ق‬นৈኙ຃‫ె؞‬੩ ྯ‫؞‬৛഍൞ౘ ુ෩‫ْك‬ਯ৾ٓ঳ण৐पರॊછආ৯ஈභো੧৔ও‫ش‬ঝঐफ़४থ

१ॖॺঐॵউhomeॢজ‫ش‬থঢ়ਯ

นৈኙ຃

*UHHQIXQFWLRQऊीञऊेखभॉ

পଚপ৾੡༁ઇ౸
นৈኙ຃‫؞‬৶৾೗૒ஸি྆প৾ઇ౸ ೗૒‫ق‬ਯ৶ఐ৾‫ె ك‬੩ ྯ‫؞‬ূ਎੃য়ਓ঵ૼ୒ৈಉ௧୅৾ૅၳઇ౸
૒‫ق‬৶৾‫ك‬೗৛഍൞ౘુ෩

‫ڱڶ‬ਖ਼‫ط‬ᅚ‫ط‬৒੼৞‫ق‬ম৬
৞ٔઘ٫‫طك‬ফা৅ষ
,6%1 &

$PD]RQ௫ଠঈॵॡ५
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চॻং३‫؞ॺॵॻ؞‬॥঒
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ਕ੄ൡછ૲
થๆൡ
7687$<$

๶ী্ஙૄभ୆ੀகਖ਻भরੱ॑ऩख‫ؚ‬ૢ৷঱ु੎ਏऩॢজ‫ش‬থঢ়ਯपणःथ‫ؚ‬প৾‫ڮؚڭ‬ফেभ๶ী஋ী৾भੴ௙॑৐઀प
ੰହघॊ‫؛‬মછदम‫ؚ‬ල౐ऊण੎ਏऩਖ਻प଒৒
ख‫ॢؚ‬জ‫ش‬থঢ়ਯभ఻৬৓ऩଡਛ১॑஽स
ॊ‫؛‬ਸ਼‫ڭ‬ฆदम మଞ๶ী্ஙૄभ୆ੀக
ਖ਻दँॊໃभञॎाਖ਻‫ؚ‬ਸ਼‫ڮ‬ฆदमమଞ๶ী্ஙૄभ୆ੀகਖ਻दँॊ༈भञ
ॎाਖ਻‫ؚ‬ਸ਼‫گ‬ฆदमమ౐ෞ஑ଞ๶
ী্ஙૄभ୆ੀகਖ਻॑੶஽घॊ‫؛‬ਸ਼‫ڰ‬ฆदम మ୷ী্ஙૄभ୆ੀகਖ਻दँॊ௞
ങໃभञॎाਖ਻‫ؚ‬ਸ਼‫ڱ‬ฆदमమ౐ෞ஑୷ী্ஙૄभఢ਋୆ੀகਖ਻‫ؚ‬ਸ਼‫ڲ‬ฆदमਫ੗એ৬ृ
ইছ‫ش‬ঞথभञॎा
ਖ਻॑ඞअ‫؛‬ਈ৏भਸ਼ฆपउःथम
ઃ੪ঘ‫ش‬ॡজॵॻ૬৑৸৬दभৈమᓳ৞஑
ု๶ী্ஙૄभॢজ‫ش‬থঢ়ਯपणःथ஽स

ফেऽो‫؛‬ফ਎੃প৾৶৾৖ਯ৾ఐ෮
঵‫؛‬পଚপ৾ઇ౸ऩन॑ഄ
ભ‫؛‬਌ऩ෩છपِశ଍஑ု
๶ী্ஙૄّ‫ق‬ਓ঵௕છ‫ك‬ऩनऋँॊ‫؛‬ె੩ ྯऩऋःँणख

ফઁਣ਩पেऽो
ॊ‫ূ؛‬਎প৾ੵ৾৖෮঵‫ؚ‬ূ਎প৾প৾੹ਯ৶ఐ৾ଢ଼
஢ఐ೗૒ୖஙఊവ‫؛‬পଚপ৾ஃু‫ؚ‬঩মপ৾௧ભ൥
ప‫؞‬ၳઇ౸‫؞‬ઇ౸ऩन॑৽
थਠ૙‫؛‬਌ऩ෩છपِ੦ຊ
ਯِّ৾৶ੵ௺भਯ৶଍஄
৻ਯِّ૭஋ী௺भૢ৷ਯ
৶ّ‫ق‬ਰ঱ુ෩‫ؚ‬፻෾ඬ‫৾ੵِؚك‬੦ຊ๶ী஋
ীِّੵ৾੦ຊ๶ী্ஙૄ
ਸ਼‫ڮ‬ගّ‫ق‬ਰ঱ુ෩‫ؚ‬१ॖग़থ५঺‫ك‬ऩनऋँॊ‫؛‬৛഍൞ౘृऽऍखऱौॅऌ

ফশ৙਩पেऽो
ॊ‫؛‬স૕প৾৶৾৖෮঵‫ؚ‬পଚপ৾প৾੹੦ຊੵ৾ଢ଼
஢ఐ೗૒৏਋ୖஙఊവ‫ূ؛‬਎੃য়ਓ঵ૼ୒ৈಉ௧୅৾
ૅஃઇऩन॑৽थਠ૙‫؛‬௧୅ী৙म๶ী্ஙૄ૛‫؛‬਌ऩ෩છपِਉ৶धਠ଴ّ
‫ુق‬෩‫ؚ‬သ௯ை‫ك‬ऩनऋँॊ‫؛‬‫ق‬ੲਾमੂග໷ষৎभुभऊै঳৖
ఊਫखथःऽघ‫ك‬ॊ‫؛‬ॶॖ‫ॺش‬

01

https://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1597-9.htm

੦ຊਯ৾
1/3

－113－

微分方程式の境界値問題
解の積分核がグリーン関数
グリーン関数 = 再生核
ソボレフ不等式
最良定数はグリーン関数の対角成分の最大値
最良関数はグリーン関数の断面

高階楕円型偏微分方程式とソボレフ不等式の最良定数⎛⎞∂⎜ x1 ⎟
⎟
∇=⎜
⎝
⎠
∂xN

..

2 + ∂2 + · · · + ∂2
Δ = ∇ · ∇ = ∂x
x2
xN
1

－114－

P (z) =

M
−1 
j=0

M


z + a2=
pM −j z jj
j=0

0 < a0 < a1 < · · · < aM −1
p0 = 1,22
p1 = a2
0 + a1 + · · · + aM −1,
··· ,
2
2
pN = a2
0a1 · · · aM −1

BVP
2M > N (N = 1, 2, 3, · · · , 2M − 1)
P (−Δ)u = f (x) (x ∈ RN )�
G(x − y)f (y)dy (x ∈ RN )u(x) =
RN








a2i
M
−1
j
(−1)






Gj (x)
0≤i≤M −2, 0≤j≤M −1
G(x) =






a2i
j




0≤i,j≤M −1
 ∞
2
−aj t
Gj (x) =e
H(x, t)dt(0 ≤ j ≤ M − 1, x ∈ RN )0

H(x, t) =



1
4πt

N
2

|x|2

e− 4t

(x ∈ RN , 0 < t < ∞)H = W M (RN )(u, v)H =


RN

M
 �
2 �

j=0




pM −2j Δj u(x) Δj v(x) +� M2−1 �



j=0



 

pM −2j−1 ∇Δj u(x) · ∇Δj v(x) dx

�u�2H = (u, u)Hu(y) = (u(·), G(· − y))H

－115－

⎞2

⎛

⎝ sup |u(y)|⎠ ≤ C�u�2 ≡Hy∈RN� �M �
�2���pM −2j �Δj u(x)�2+
C
N
R
j=0
� M2−1 ��
j=0

pM −2j−1����∇Δj u(x)�2 dx

C0 = max G(y − y) = G(0)y∈RNu(x) = G(x − y0)

N = 2n + 1
G(0) =(n = 0, 1, 2, · · · , M − 1)���(−1)M −1+nΓ(1/2) ��
2(4π)nΓ(n + 1/2) ���N = 2n + 2
G(0) =
��(−1)M +n ���(4π)n+1n! ��

�
�
�
�
�
�
�
�

a2i
j
a2n−1
j

�
�
�
�
�
�
�

2
a2n
j log(aj )

�
� ��
�
�
�

a2i
j

0≤i≤M −2, 0≤j≤M −1(n = 0, 1, 2, · · · , M − 2)
a2i
j

⇒

u(x) =

�

R

�
� ��
�
�
�

a2i
j

0≤i≤M −2, 0≤j≤M −1

G(x) =

G(x − y)f (y)dy

�
�
�
�
M
−1
(−1)
�
�
�

Gj (x) =12aj

e−aj |x|(x ∈ R)

a2i
j
�
�
�
�
�
�

Gj (x)
a2i
j

�
�
�
�
�
�

�
�
�
�
�
�
�

0≤i≤M −2, 0≤j≤M −10≤i,j≤M −1(0 ≤ j ≤ M − 1, x ∈ R)

－116－

0≤i,j≤M −1⇒

BVP M = 1, 2, 3, · · · ,N = 1 D = d/dx����−D 2 + a20 · · · −D 2 + a2M −1 u = f (x) (x ∈ R)
�

�
�
�
�
�
�

�
�
�
�
�
�

0≤i,j≤M −1

�sup |u(y)|

y∈R

�2

C0 = G(0) =(−1)M −1
2a0 · · · aM −1·≤ C�u�2H ≡ C

�
�
�
�
�
�
�

u(x) = G(x − y0)���2
pM −j �u(j)(x)� dx

R j=0

a2i+1
j
�
�
�
�
�
�

M
�

1
a2i
j

(u ∈ H = W M (R))

�
�
�
�
�
�
�
0≤i≤M −2, 0≤j≤M −1
�
�
�
�
�
�
0≤i,j≤M −1

BVP M = 2,
N = 2, 3
�
�
�
�
2
2
2
Δ − a0 + a1 Δ + a20a21 u = f (x)�u(x) =

�RN��
G(x) = − ��G(x − y)f (y)dy(x ∈ RN )(x ∈ RN )����
1
1 �� �� 1 1 ��2
2
G0(x) G1(x) � � a0 a1 �

⎧
1
⎪
⎪
⎪
K0(aj |x|)⎨2π
Gj (x) =1⎪−aj |x|⎪⎪⎩ 4π|x| e(N = 2)(j = 0, 1)(N = 3)�2�≤ C�u�2H ≡sup |u(y)|
y∈R
�
�
�
C
|Δu(x)|2 + (a20 + a21) |∇u(x)|2 + a20a21 |u(x)|2 dx
RN⎧⎪
1 log(a21) − log(a20)⎪⎪⎨4πa21 − a20
C0 = G(0) =
⎪
1
1
⎪
⎪
⎩
4π a0 + a1
u(x) = G(x − y0)－117－(u ∈ H = W 2(RN ))(N = 2)(N = 3)

糸のたわみ問題とソボレフ不等式の最良定数



























q










f (x)


u(x)


:

荷重密度

u(x)

:

糸のたわみ

q = a2

:

バネ定数

f (x)

0<a<∞

IBVP
⎧�
�
⎨ ∂ 2 − ∂ 2 + a2 u = f (x, t)xt⎩ IC, BC(Ω)
t → +∞

BVP�−u�� + a2u = f (x)BC(Ω)(x ∈ Ω)－118－(x ∈ Ω, 0 < t < ∞)

BVP(P; a)�−u�� + a2u = f (x)�(0 < x < L)u(i)(L) − u(i)(0) = 0

u(x) =

� L(i = 0, 1)

G(P; a; x, y) f (y) dy0

G(P; a; x, y) =ch(a(|x − y| − L/2))
2a sh(aL/2)

BVP(m, n; a)�−u�� + a2u = f (x)(m)u(0) = u�u(x) =

� L(n)(0 < x < L)(0) = 0 (m, n = 0, 1)

G(m, n; a; x, y) f (y) dy0G(m, n; a; x, y) =Km(x ∧ y) Kn(L − x ∨ y)
Km + n(L)K0(x) = a−1sh(ax), K1(x) = ch(ax), K2(x) = ash(ax)� ��H = u � u, u� ∈ L2(0, L),⎧u(L) − u(0) = 0
⎪
⎪
⎪
⎪
⎪
⎨ u(0) = u(L) = 0
u(0) = 0⎪⎪⎪u(L) = 0⎪⎪⎩なし(u, v)H =

� �
Ω

⎫
(P)
⎪
⎪
⎪
⎪
(m, n) = (0, 0) ⎪⎬�(m, n) = (0, 1)⎪⎪(m, n) = (1, 0) ⎪⎪⎪⎭(m, n) = (1, 1)�u�(x)v �(x) + a2u(x)v(x) dx

u(y) = (u(·), G(·, y))H =� ��u�(x)∂xG(x, y) + a2u(x)G(x, y) dx
Ω

－119－

�

�2

sup |u(y)|0≤y≤L≤ C �u�2H ≡ C

� L�
0

�
� �
�
�u (x)�2 + a2 |u(x)|2 dx

C0 = sup G(X; a; y, y) = G(X; a; y0, y0) =
0≤y≤L
⎧
1
⎪
⎪
G(P; a; y0, y0)
=
⎪
⎪
⎪
2a
th(aL/2)⎪⎪⎪th(aL/2)⎪⎪=⎨ G(0, 0; a; L/2, L/2)2ath(aL)
⎪
⎪
G(0,
1;
a;
L,
L)
=
G(1,
0;
a;
0,
0)
=
⎪
⎪
⎪
a
⎪
⎪
⎪
1
⎪
⎪
G(1,
1;
a;
0,
0)
=
G(1,
1;
a;
L,
L)
=
⎩
a th(aL)u(x) = G(X; a; x, y0)












(0 < x < L)

























f (x)


u(x)


f (x)

:

荷重密度

u(x)

:

糸のたわみ

BVP(P; 0)⎧��⎪(0 < x < L)⎪
⎨ −u = f (x)(i)
u (L) − u(i)(0) = 0(i = 0, 1)⎪⎪⎩ f (x) と u(x) は ϕ (x) = 1/√L と直交
0�

u(x) =� L

G(P; 0; x, y) f (y) dy0
G(P; 0; x, y) =��11
|x − y|1|x − y|2 − |x − y| + L = L b2
2L
2
12
L

－120－

BVP(m, n; 0)⎧��(0 < x < L)
⎨ −u = f (x)(m)(0) = u(n)(L) = 0(m, n√= 0, 1)
u⎩(1, 1)：f (x) と u(x) は ϕ0(x) = 1/ L と直交
�� L
G(m, n; 0; x, y) f (y) dy (0 < x < L)u(x) =
01(x ∧ y) (L − x ∨ y)L
G(0, 1; 0; x, y) = x ∧ yG(0, 0; 0; x, y) =G(1, 0; 0; x, y) = L − x ∨ yG(1, 1; 0; x, y) =
� L
1
1 � 2
− |x − y| +x + (L − x)2 + y 2 + (L − y)2 −
2
4L
6

� �
�
H = u�u� ∈ L2(0, L),
⎧
� L
⎪
⎪
⎪
u(L)
−
u(0)
=
0,
u(x)ϕ(x)dx = 0
⎪
⎪
⎪
0
⎪
⎪
⎪ u(0) = u(L) = 0⎪⎨
u(0) = 0⎪⎪⎪⎪u(L) = 0
⎪
⎪
� L
⎪
⎪
⎪
⎪
⎩
u(x)ϕ(x)dx = 00(u, v)A =� L0u�(x)v �(x)dx,

u(y) = (u(·), G(·, y))A =

⎛

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(0, 0) ⎪⎪⎬�
(0, 1)⎪⎪⎪(1, 0) ⎪
⎪
⎪
⎪
⎪
⎪
⎭
(1, 1) ⎪(P)

�u�2A = (u, u)A� L
0u�(x)∂xG(x, y)dx

⎞2

� L�
�2
� �
�
sup |u(y)|⎠ ≤ C �u�2
≡
C
(x)
�u
� dx
A
0
0≤y≤L

⎝

C0 =

sup G(X; 0; y, y) = G(X; 0; y0, y0) =
0≤y≤L
⎧
L
⎪
⎪
G(P; 0; y0, y0)
=
⎪
⎪
12
⎪
⎪
⎪
⎪
L
⎪
⎨ G(0, 0; 0; L/2, L/2)
=
4
⎪
⎪
G(0,
1;
0;
L,
L)
=
G(1,
0;
0;
0,
0)
=
L
⎪
⎪
⎪
⎪
⎪
⎪
L
⎪
⎩ G(1, 1; 0; 0, 0) = G(1, 1; 0; L, L) =3

u(x) = G(X; 0; x, y0)(x ∈ Ω)－121－

離散糸のたわみ問題

f(i)

-1

0

1

a

i

N-2

N-1

N

u(i)

DBVP(X; a)⎧−u(i − 1) + (2 + a)u(i) − u(i + 1) = f (i) (0 ≤ i ≤ N − 1)
⎪
⎧
⎪
⎪
⎪
⎪ u(−1) = u(N − 1), u(0) = u(N )
⎪⎪
(P)
⎪
⎨⎪
⎪
u(N ) = 0(0, 0)⎨ u(−1) = 0,u(−1) = 0,u(N − 1) − u(N ) = 0(0, 1)⎪⎪⎪⎪⎪ u(−1) − u(0) = 0, u(N ) = 0⎪(1, 0)⎪⎪⎪⎪⎩⎩(1, 1)u(−1) − u(0) = 0, u(N − 1) − u(N ) = 0
�

u(i) =

N−1�
j=0

G(X; a; i, j) f (j)(0 ≤ i ≤ N − 1)����UN −|i−j| a+2+ U|i−j| a+222��G(P; a; i, j) =2(TN a+2− 1)
2

G(m, n; a; i, j) =
�
�
�
�� �
�
�
�
��
�
− mUi∧j a+2
UN −i∨j a+2− nUN −1−i∨j a+2
Ui∧j+1 a+2
2
2
2
2
�
�
�
�
�
�
− (m + n)UN a+2+ mnUN −1 a+2
UN +1 a+2
2
2
2

－122－

DBVP(X; a)(A + aI)u = f⎛⎞
2 + a −1−1⎟
⎜ −1 2 + a −1
⎟
⎜
...
...
...
⎟
A(P) + aI = ⎜⎟⎜⎝−1 2 + a −1 ⎠−1
−1 2 + a N ×N⎛⎞
2 + a − m −1⎜⎟
−1
2 + a −1
⎜
⎟
...
...
...
⎟
A(m, n) + aI = ⎜⎜⎟⎝⎠−1 2 + a−1−1 2 + a − n N ×N�u = G(a)f

G(a) = (A + aI)−1 =��

G(X; a; i, j),

0≤i,j≤N −1����+ U|i−j| a+2
UN −|i−j| a+222��G(P; a; i, j) =− 1)
2(TN a+22
G(m, n; a; i, j) =
�
�
�
�
�� �
�
�
�
��
Ui∧j+1 a+2− mUi∧j a+2
UN −i∨j a+2− nUN −1−i∨j a+2
2
2
2
2
�
�
�
�
�
�
− (m + n)UN a+2+ mnUN −1 a+2
UN +1 a+2222
TN (cos(θ)) = cos(N θ),

UN (cos(θ)) =sin(N θ)sin(θ)(u, v) = v ∗u�u�2 = (u, u) =

N
−1
�
i=0

|u(i)|2(u, v)H = ((A + aI)u, v) = v ∗(A + aI)u�u�2H = (u, u)H =
⎧
N
−2
�
⎪
⎪
⎪
⎪
|u(i) − u(i + 1)|2 + |u(N − 1) − u(0)|2
⎪
⎪
⎪
⎪
i=0
⎪
⎪
N
−2
⎪
�
⎪
⎪
2
⎪
|u(i) − u(i + 1)|2 + |u(N − 1)|2
⎪ |u(0)| +
⎪
⎪
⎪
i=0
⎪
⎪
N
−2
⎨
�
|u(i) − u(i + 1)|2
|u(0)|2 +
⎪
⎪
i=0
⎪
⎪
⎪
N
−2
⎪
�
⎪
⎪
⎪
|u(i) − u(i + 1)|2 + |u(N − 1)|2
⎪
⎪
⎪
⎪ i=0
⎪
⎪
N
−2
⎪
�
⎪
⎪
⎪
|u(i) − u(i + 1)|2⎪⎩
i=0

u(j) = (u, G(a)δ j )H(u ∈ CN ),

⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(0, 0) ⎪
⎪
⎪
⎪
⎪
⎪
⎪
N
−1
⎬
�
+a
|u(i)|2(0, 1)
⎪
⎪
i=0
⎪
⎪
⎪
⎪
⎪
⎪
(1, 0) ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(1, 1) ⎪⎪⎭(P)δ j = t(· · · , δij , · · · )0≤i≤N −1

－123－

�

max

0≤j≤N −1

�2

| u(j) |≤ C � u �2H(u ∈ CN )C0(a) = max tδ j G(a)δ j = G(X; a; j0, j0) =0≤j≤N −1��⎧UN 2+a⎪⎪� 2+a2�
⎪ G(P; a; j0, j0) =
⎪
⎪
2(T
−
1)
⎪
N
2
⎪
⎪
⎪
� 2+a �� 2+a �
⎪
⎪
�
⎪
N +1
U
U
⎪
� 2 �
� N2+2 �
�
2
⎪
⎪
�
� 2
=
⎪ G(0, 0; a; j0, j0)�
N −1
⎪
2
+
a
j0 =� 2 �
⎪
⎪
UN +1
⎪
⎪
⎪
2�
⎪
�
⎪
⎪
2+a
⎨
UN
�
2
�
⎪
�
�
�
�
=
⎪ G(0, 1; a; j0, j0)�⎪j0 =N −1
− UN 2+a
UN +1 2+a
⎪
⎪
2
2
⎪
⎪
⎪
�
�
⎪
�
⎪
UN 2+a
⎪
�
⎪
⎪
�
� 2
�
�
⎪
⎪ G(1, 0; a; j0, j0)�j0=0 =⎪− UN 2+a
UN +1 2+a
⎪
2
2
⎪
⎪
⎪
�
�
�
�
⎪
2+a
�
⎪
UN 2 − UN −1 2+a⎪⎪ G(1, 1; a; j , j )�2⎪��
� 2+a �� 2+a �=⎩0 0 �
j0 =0,N −1
UN +1 2+a
−
2U
+
U
N
N −1
2
2
2

u = G(a)δ j0 = t(G(X; a; i, j0))0≤i≤N −1

f(i)

-1

0

1

N-2

i

N-1

u(i)

DBVP(X; 0)⎧−u(i − 1) + 2u(i) − u(i + 1) = f (i) (0 ≤ i ≤ N − 1)
⎪
⎧
⎪
⎪
⎪
(X) = (P)⎪ ⎪⎪ u(−1) = u(N − 1), u(0) = u(N )⎪⎪⎨ u(−1) = 0,u(N ) = 0
(X) = (0, 0)⎨ ⎪u(−1) = 0,u(N − 1) − u(N ) = 0(X) = (0, 1)⎪⎪⎪⎪(X) = (1, 0)⎪⎪⎩ u(−1) − u(0) = 0, u(N ) = 0⎪⎪
u(−1) − u(0) = 0, u(N −
1)
−
u(N
)
=
0
(X) = (1, 1)
⎪
⎪
√
⎩
t
N
(P) と (1, 1)：u と f は ϕ0 = (1/ N ) (1, · · · , 1) ∈ C と直交
�
⎧
N −1
⎪
⎪ �
⎪
⎪
G(X; 0; i, j) f (j)(X) = (0, 0), (0, 1), (1, 0)⎪⎨
j=0
u(i) =
N
−1
�
⎪
⎪
⎪
⎪
G∗(X; i, j) f (j)(X) = (P), (1, 1)⎪⎩
j=0

G(m, n; 0; i, j) =(i ∧ j + 1 − m(i ∧ j)) (N − i ∨ j − n(N − 1 − i ∨ j))

G∗(P; i, j) = b2(N ; |i − j|),N + 1 − (m + n)N + mn(N − 1)b2(N ; i) =12N

N2 − 11i2 − i +2
12N

G∗(1, 1; i, j) = b2(2N ; |i − j|) + b2(2N ; 1 + i + j)

－124－

N

DBVP(X; 0)�
Au = f√(P) と (1, 1)：u と f は ϕ0 = (1/ N )t(1, · · · , 1) ∈ CN と直交
⎞
⎛
2 −1
−1
⎟
⎜−1 2 −1⎟⎜... ... ...
⎟A(P) = ⎜⎟⎜⎝−1 2 −1⎠−1−1 2 N ×N⎞⎛
2 − m −1
⎟
⎜ −1
2 −1
⎟
⎜
... ... ...
⎟A(m, n) = ⎜
⎟
⎜
⎝
−1 2
−1 ⎠
−1 2 − n N ×N
�

u=

�

G(0)f
G∗f

G(0) = A−1 =(X) = (0, 0), (0, 1), (1, 0)(X) = (P), (1, 1)��G(X; 0; i, j)

G(m, n; 0; i, j) =,0≤i,j≤N −1(i ∧ j + 1 − m(i ∧ j)) (N − i ∨ j − n(N − 1 − i ∨ j))
N + 1 − (m + n)N + mn(N − 1)��G∗ = lim G(a) − a−1E 0 =
a→0

�

�

G∗(X; i, j)

E 0 = t ϕ 0 ϕ00≤i,j≤N −1

1 2 1
N2 − 1i − i+
2N
2
12N
G∗(1, 1; i, j) = b2(2N ; |i − j|) + b2(2N ; 1 + i + j)

G∗(P; i, j) = b2(N ; |i − j|),b2(N ; i) =(u, v)A = (Au, v) = v ∗Au
�u�2A = (u, u)A =
⎧
N
−2
�
⎪
⎪
⎪
⎪
|u(i) − u(i + 1)|2 + |u(N − 1) − u(0)|2
⎪
⎪
⎪
⎪
i=0
⎪
⎪
N
−2
⎪
�
⎪
⎪
⎪
|u(0)|2 +|u(i) − u(i + 1)|2 + |u(N − 1)|2
⎪
⎪
⎪
⎪
i=0
⎪
⎪
N
−2
⎨
�
|u(0)|2 +|u(i) − u(i + 1)|2
⎪
⎪
i=0
⎪
⎪
⎪
N −2
⎪
⎪ �
⎪
⎪
|u(i) − u(i + 1)|2 + |u(N − 1)|2
⎪
⎪
⎪
⎪ i=0
⎪
⎪
−2
⎪ N
�
⎪
⎪
⎪
|u(i) − u(i + 1)|2⎪⎩(P)
(0, 0)(0, 1)(1, 0)(1, 1)

i=0

u(j) =�(u ∈ CN )(u ∈ CN0 )(u, G(0)δ j )A(u, G∗δ j )AN
CN
0 = C ∩

�t(0, 0), (0, 1), (1, 0)(P), (1, 1)�

ϕ0 u = 0

－125－

�

�2

max

| u(j) |⎧⎨0≤j≤N −10≤j≤N −1

C0 =

max

t

δ j G(0)δ j⎩ max tδ j G∗δ j
0≤j≤N −1⎧⎪⎪⎪G∗(P; j0, j0) =
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
�
⎪
⎪
�
⎪
⎪
⎪ G(0, 0; 0; j0, j0)�j0=� N −1 � =
⎪
⎪
2
⎪
⎪
⎨
�
�
=
G(0, 1; 0; j0, j0)�⎪j0 =N −1
⎪
⎪
⎪
⎪
�
⎪
⎪
�
⎪
⎪
=
⎪ G(1, 0; 0; j0, j0)�
⎪
⎪
j0 =0
⎪
⎪
⎪
⎪
⎪
�
⎪
⎪
�
⎪
=
⎩ G∗(1, 1; j0, j0)�j0 =0,N −1u=�

G(0)δ j0 = t(G∗δ j0 = t(�

≤ C � u �2A(u ∈ CN )(u ∈ CN0 )(0, 0), (0, 1), (1, 0)(P), (1, 1)⎫(0, 0), (0, 1), (1, 0) ⎬(P), (1, 1)

⎭

2

N −1
12N

�

1
N +1

N +1
2

��

=

N +2
2

�

N
N
1
6N

(N − 1)(2N − 1)

G(X; 0; i, j0)
G∗(X; i, j0))0≤i≤N −1)0≤i≤N −1(0, 0), (0, 1), (1, 0)(P), (1, 1)

正多面体と離散ソボレフ不等式の最良定数

多面体
正 4 面体正 6 面体
正 8 面体正 12 面体
正 20 面体
切頂正 4 面体
切頂正 6 面体
切頂正 8 面体
切頂正 12 面体
切頂正 20 面体

�

記号 面 F 頂点 N 辺 E 炭素
R4

4

4

6

R6

6

8

12

R8

8

6

12

R12

12

20

30

R20

20

12

30

T4

8

12

18

T6

14

24

36

T8

14

24

36

T12

32

60

90

T20

32

60

90

F +N =E+2

－126－

C20

C60

R6
4

7

6

5

4

7

3

0

1

2

3

0

e = {(0, 1), (0, 3), (0, 7), (1, 2), (1, 6), (2, 3),(2, 5), (3, 4), (4, 5), (4, 7), (5, 6), (6, 7)}

R6

⎞
⎛
⎞
1 −1 000000u(0) − u(1)⎜1 0
⎜u(0) − u(3)⎟0 −1 0
0
0
0 ⎟⎛
⎞
⎟
⎜
⎟
⎜
0
0
0
0
0 −1⎟ u(0)⎜1 0
⎜u(0) − u(7)⎟⎟ ⎜u(1)⎟⎜0 1 −1 0⎜u(1) − u(2)⎟
0
0
0
0
⎟⎜
⎜
⎟
⎜
⎟
⎜0 1
⎜u(1) − u(6)⎟
u(2)⎟
0
0
0
0 −1 0 ⎟
⎟⎜
⎜
⎟
⎜
⎟
⎜
⎜0 0
⎜u(2) − u(3)⎟1 −1 000
0 ⎟⎜u(3)⎟⎟⎜⎟⎜Bu = ⎜⎜u(4)⎟⎟ = ⎜0 01
0
0 −1 00 ⎟⎟⎟⎜⎜⎜u(2) − u(5)⎟⎜0 0⎜u(3) − u(4)⎟u(5)⎟
01 −1 0
0
0 ⎟⎜
⎟
⎟⎜
⎜
⎟
⎜
0
0
1 −1 00 ⎟ ⎝u(6)⎠⎜0 0⎜u(4) − u(5)⎟⎟ u(7)⎜0 0⎜u(4) − u(7)⎟
0
0
1
0
0
−1
⎟
⎜
⎟
⎜
⎝0 0
⎝u(5) − u(6)⎠000
1 −1 0 ⎠
0 0
0
0
0
0
1 −1
u(6) − u(7)⎛�Bu�2 = (Bu)∗(Bu) = u∗B ∗Bu = u∗Au = �u�2A =|u(0) − u(1)|2 + |u(0) − u(3)|2 + |u(0) − u(7)|2 + |u(1) − u(2)|2+|u(1) − u(6)|2 + |u(2) − u(3)|2 + |u(2) − u(5)|2 + |u(3) − u(4)|2+|u(4) − u(5)|2 + |u(4) − u(7)|2 + |u(5) − u(6)|2 + |u(6) − u(7)|2

R6

⎛

3

−1

0

−1

0

0

0

−1

⎞

⎟
⎜
⎜−1 3 −1 000 −1 0 ⎟⎟⎜⎟⎜⎟⎜ 0 −1 3 −1 0 −1 00⎟⎜⎟⎜−1 0 −1 3 −1 0
0
0
⎟
⎜
∗
A=B B=⎜
⎟
⎟
⎜ 0
0
0
−1
3
−1
0
−1
⎟
⎜
⎟
⎜
0 −1 0 −1 3 −1 0 ⎟⎜ 0⎟⎜⎜ 0 −1 00
0 −1 3 −1⎟
⎠
⎝
−1 0
0
0 −1 0 −1 3

－127－

A=

�

a(i, j)a(i, j) =

d=

⎧
⎪
⎪
⎨3

⎧
⎪
⎪
⎨

�

,

0≤i,j≤N −1

d
−1

⎪
⎪
⎩ 0

4

(i = j)(i, j), (j, i) ∈ e(otherwise)(R4, R6, R12, T4, T6, T8, T12, T20)
(R8)

⎪
⎪
⎩5

(R20)

DEVP
Aϕk = λk ϕk(0 ≤ k ≤ N − 1)

0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λN −11
ϕ0 = √ t(1, · · · , 1) ∈ CN
N

DBVP
�
Au = ftϕ u = 0,0
�tϕ f = 00u = G∗fG∗ = lim

a→+0�

G(a) − a−1E 0

G(a) = (A + aI)−1,�

E 0 = ϕ 0 t ϕ0

－128－

CN
0 =





u ∈ CN | tϕ0u = 0(u, v) = v ∗u(u, v)A = (Au, v) = v ∗Auu(j) = (u, G∗δ j )A
δ j = t(· · · , δij , · · · )0≤i≤N −1



2

max
| u(j) |0≤j≤N −1

C0 =≤ C �u�2A≡C(i,j)∈e|u(i) − u(j)|2N−1
1tδ G δ = tδ G δ = 1
max
j ∗ j
j0 ∗ j0Nλ0≤j≤N −1
k=1 k

u = G∗δ j ∈ CN
0

正 4 面体0

3

C0(R4) =

1
3
16

2

= 0.187 · · ·

－129－

3

1

正 6 面体

4

7

6

5

4

7

3

0

1

2

3

0

4

5

C0(R6) =29
96= 0.302 · · ·正 8 面体

3

1

5

2

1

0

C0(R8) =1372= 0.180 · · ·正 12 面体（C20）

18

14

15

16

17

18

14

19

13

11

9

7

19

13

0

12

10

8

6

0

12

1

2

3

4

5

1

2

C0(R12) =137300= 0.456 · · ·－130－正 20 面体

7

6

8

9

5

3

11

10

1

2

8

6

0

5

3

4

C0(R20) =7
36= 0.194 · · ·

切頂正 4 面体

9

3

4

5

7

8

3

9

10

2

6

10

2

11

1

0

11

1

C0(T4) =301
720= 0.418 · · ·

切頂正 6 面体

7

16

6

15

14

17

5

18
4

19

C0(T6) =

173
288

12

11

13

22
20

9

8

10

1

21

23

0

= 0.601 · · ·

－131－

7

16

6

17

5
2

3

18
4

19

切頂正 8 面体

11

12

13

10

11

12

6

5

18

17

14

23

0

9

6

5

18

17

7

4

19

16

15

22

1

8

7

4

19

16

3

20

21

2

3

20

1019

C0(T8) == 0.505 · · ·2016

切頂正 12 面体

30

31

52

32

33
34

53

54

36

37

49

48

58

46

47

38

C0(T12) =

42

64

1

22

16
21

20

2

18

17

15

14

19

3
4

43

29
28

23
0

59

44
41

27

26
24

45

39
40

25

57

50
35

55
56

51

11

7
5

6

41

30

13
12

8

9

10

= 0.853 · · ·75

切頂正 20 面体（C60）

54

30

53

55

31

42

43

52

40

54

31

55

32

59

56

51

48

47

44

39

36

35

32

59

56

33

58

57

50

49

46

45

38

37

34

33

58

57

27

28

3

4

7

8

15

16

19

20

27

28

3

29

2

5

6

9

14

17

18

21

26

29

2

26
25

1

10

13

22

25

1

24

0

11

12

23

24

0

C0(T20) =239741376200= 0.637 · · ·

－132－

多面体
正 4 面体正 6 面体
正 8 面体正 12 面体
正 20 面体
切頂正 4 面体
切頂正 6 面体
切頂正 8 面体
切頂正 12 面体
切頂正 20 面体

記号 頂点
R4
4
R6
8
R8
6
R12 20
R20 12
T4 12
T6 24
T8 24
T12 60
T20 60オイラーの多面体定理

面
4
6
8
12
20
8
14
14
32
32

辺
最良定数
6
3/16
12
29/96
12
13/72
30
137/300
30
7/36
18
301/72036
173/28836
1019/2016
90
64/75
90 239741/376200

近似値

0.187 · · ·0.302 · · ·0.180 · · ·0.456 · · ·0.194 · · ·0.418 · · ·0.601 · · ·0.505 · · ·0.853 · · ·0.637 · · ·F +N =E+2

フラーレンと離散ソボレフ不等式の最良定数

Cn フラーレン
炭素原子 n 個

5 員環 12 個, 6 員環 0 個以上で球状に構成

C20, C24, C26, C28, C30, · · · , C60, · · ·C20= 正 12 面体,

C60 バッキーボール = 切頂正 20 面体

Cn フラーレンの異性体
5 員環と 6 員環の模様が異なる

C20 1
C24 1
C26 1
C28 2
C30 3

C32 6
C34 6
C36 15
C38 17
C40 40C42 45
C44 89
C46 116
C48 199
C50 271－133－

C52 437
C54 580
C56 924
C58 1205
C60 1812

古典力学モデル（炭素原子のたわみ問題）
炭素原子同士の距離が 1
バネ定数 1 の線形バネで結ばれる
離散ソボレフ不等式の最良定数 = 古典力学モデルの「かたさ」
値が小さいほど「かたい」

DBVP�
Au = ftϕ u = 0,0�u = G∗ f ,ϕ0 = √1 t(1, · · · , 1) ∈ CN

tϕ f = 0,0G∗ = lim

a→+0

G(a) = (A + aI)−1,N�

G(a) − a−1E 0
E 0 = ϕ 0 t ϕ0

u = t(u(0), · · · , u(N − 1)),
�
�

A=

,

a(i, j)0≤i,j≤N −1

CN
0 =

�

�

f = t(f (0), · · · , f (N − 1))⎧⎨ 3 (i = j)
−1 (i, j), (j, i) ∈ ea(i, j) =⎩ 0(otherwise)�u ∈ CN | tϕ0u = 0(u, v) = v ∗u,(u, v)A = (Au, v) = v ∗Au
�u�2A = (u, u)A = u∗Au =�(i,j)∈eu(j) = (u, G∗δ j )A
δ j = t(· · · , δij , · · · )0≤i≤N −1－134－|u(i) − u(j)|2



max

0≤j≤N −1C0 =| u(j) |max

0≤j≤N −1

2

≤ C �u�2A ≡ C(i,j)∈e|u(i) − u(j)|2tδ G δ = tδ G δj ∗ jj0 ∗ j0

u = G∗δ j0

フラーレンの安定性に関する研究成果
孤立 5 員環則→

昔の手法

離散ラプラシアンの固有値による評価

→

いくかある

離散ソボレフ不等式の最良定数で評価する先行研究 → ない

離散ソボレフ不等式の最良定数で評価する利点
離散ラプラシアンの固有値を調べる必要がない
計算が Mathematica で簡単に実行できる

C60 フラーレン
炭素原子 60 個

5 員環 12 個と 6 員環 20 個

代表はバッキーボール（切頂正 20 面体，孤立 5 員環則）
異性体は 1812 個

1970 年 大澤映二が存在を予言
1985 年 クロトー，スモーリーらが存在を確認
1996 年 ノーベル化学賞－135－カーボンナノチューブ
グラフェンを円筒にしたもの
側面が 6 員環，両端はフラーレンを 2 つに割った半球
ねじり方の違いで導体や半導体になる
アームチェア型，ジグザグ型，カイラル型
人間がつくる最も細いチューブ

1991 年 飯島澄男が発見

対称フラーレン
平面グラフの中央横軸
ジグザグラインジグザグリングL

アームチェアリング

R

A

平面グラフの中央横軸上の 1 点について点対称

A60A1（バッキーボール）

54

30

59

56

33

58

27

28

10

51

48

57

50

3

4

5

49

46

7

8

38

15

16

35

59

56

33

58

57

27

28

32
16

37

34

19

20

9
17

21

C0(A60A1)  0.63727

2

29
25

5

23

3
6

26

22
4

12

20

10
18

13
3

11

36

55

17

45

14

9
10

2

0

39

8
6

1
1

44

15

31

18

7
2

29
25

47

54
11

40

19

6

24

43

20

30
12

13

52

16

41

14

55

26

42

15

31

5

長さ 18 ジグザグライン 10 本

53

11

32

長さ 10 アームチェアリング 6 本

1
1

24

1/1812－136－

0L60A1（バッキーボール）

長さ 18 ジグザグライン 10 本

長さ 10 アームチェアリング 6 本
11

34

35

30

31

32

14
38
37
16
58

48
50

49

51

52

20

22

21

19

44

54

55

27

26

8

5

4

40

39

57

58

16

24

12
13

3

2

2

22

1

0

52

21

19
18

51

3

28

29
5

17
16

49
20

10

15

50
15

59

23
9

14

11

48

36
37

17

8

10

4

0

38

56
25

7
9

6

30
14

41
42

18

6
7

20

18

28

29
5

10

43
45

19
53

35

12

46
47

15
59

34

13

36

39

33

6
7

20

9

6

8

5

4

4

1

C0(L60A1)  0.63727

1/1812

L60A2（ツイステッドバッキーボール）

長さ 18 ジグザグライン 10 本

長さ 10 アームチェアリング 6 本
11

35

30

31

32

14
48
50

49

51

52

22

21

19

54

55
27

9

6

26

5

57

58

12

23

15

13

52

20

21

29
5

17
16

19

7

20

18

6
4

2

3

51
22

3

4

49

10

9
14

11

50
15

59

24

48

36
37

16

25

2

0

39

8

10

8

38
40

17
56

7

4

14

41
42

18

6
7

20

18

44

28

29
5

10

43
45

19
53

30

35

12

46
47

20

15
59

34

13

36
37

33

1

0

5

1

C0(L60A2)  0.657729

2/1812

A60B1（カメボール）

長さ 10 アームチェアリング 2 本
長さ 18 ジグザグライン 6 本
長さ 20 ジグザグライン 2 本
長さ 8 ジグザグリング 1 本

11
33

32

31

14

13

42

43

46

29

20

6

19

53

21

22

18

16

6

10

51

17

5

50

23

24

15

13

1

59

58

25

26

12

10

40

57

56

27

28

9

7

11
4

2

3

43

54

29

20

6

19

10

53

4

48

52

51

21

22

16

36

59

20

23

24

15

13

7

17

5

49

50

16

18

0

1

11
4

2

12/1812

－137－

12

14

1

C0(A60B1)  0.659416

25
8

3

2

1

46

6

8

35

47
45
17

55

5

30
12

44
42
18

9

14

31
13

41
39
19

8

3
0

37
20

7

2

36

49
16

52

32
14

38
48

17
54

33
15

47
45

18
55

34

35

12

44

41

11

30

3

L60B1（カメボール）

34

35

31

11

40

36
38

57

58

59

48

52

20

7

12
11

4

3

35

38

57
23

15
58

22

17

59
21

18

16
3

C0(L60B1)  0.659416

52
28

27
6

7

8

6

10
9

1
5

1

51

5

4

0

48
20

29

20
19

2

49
50

37

10

15
14

31
30

14
36

16

9

2

5

40

56
24

13

1

4

1

55

8

10
9

42
17

25

7

8

6

19

26

6

43

54

27

34

39

11

18

53

28

5

18

45
19

51

33

41

12
44

47

20

29

21
10

0

49

15

22

13
46

50

37

16

32

30

14

39

長さ 18 ジグザグライン 6 本
長さ 20 ジグザグライン 2 本
長さ 8 ジグザグリング 1 本
長さ 10 アームチェアリング 2 本

4

12/1812R60A1（カメボール）

長さ 8 ジグザグリング 1 本
長さ 18 ジグザグライン 6 本
長さ 20 ジグザグライン 2 本
長さ 10 アームチェアリング 2 本

7
30

31

8
41

32
42

43
9

38

44
59

46

17

48
47

45
20

13

16

14

12

13

6

52
51

27

16
25

28

26

3

4

33

43

23

56

22

6

7

19

45
20

18

9

10

47
13

15

17

20

46

44
59

19

5

29

16

14

12

13

6

3

11

8

5

12

58
57

18
21

24

42

9

55
17

4

2

34

54
53

15

3

11

36
10

50
49

14
29

15

37
11

41

32

35

39

40
12

2

2
1

0
1

C0(R60A1)  0.659416

12/1812

R60B1（大車輪）平行でない長さ 8 ジグザグリング 3 本

30

42

32
43
9

38

41

40

39

46

48

50

12
44

59

15

47
14

29
14

6

49
15

27

51

25

23

28

26

24

3

4

5

3
13
2

33

54

56

58

55

21

57
19

19

6

8

1

0

12
44

59
20

17

45
13

15

20

18

16

9

10

11

5
7

43
9

18

22
4

1

34

53
17

42

32

35
10

52

16

7

36
37

11

45
13

12

8

31

29
14

6

2

3
13

12

C0(R60B1)  0.666969

2

42/1812－138－

L60ZK1（ヒフミクン）

孤立長さ 18 ジグザグライン 3 本
孤立長さ 8 ジグザグリング 2 本

11
30

35

31

14

13
48

36
49

50
15

47

52

51
21

46

19

20

53

54
28

5

34

35

42

40

38

36

43

41

55

56

7

25

13

14

10

12

58

3

59

24

23

16

17

52

51
22

21

19

20

5

6

18
3

2

49

10

2
4

50
15

15

4
5

37
16

9

11

8

57

26
8

9

39
17

18

27

6
7

6

44
19

29

33

14

45

20

22

32
12

4

1

0

5

1

C0(L60ZK1)  0.67196

166/1812

AAA60A1（トリプル A）

長さ 8 アームチェアリング 3 本
長さ 20 ジグザグライン 1 本

11
30

31
12

39

42

36

42

33

32

43

41

40

38

37

35

34

32

43

41

58

45

46

49

50

53

54

57

58

45

46

13

20

19

18

17

16

15

14

13

20

59

44

47

48

51

52

55

56

59

44

17

14

29

26

25

22

21

18

17

14

10
16

11

3

4

15

28

13

2

5

6

27

24

4

5

23

8
20

7

8

6

3

12

7

9
16

10

11

9

29
3

10

19

47

15

28

13

2

12

2
1

0
1

C0(AAA60A1)  0.68392

387/1812

RR60A1（カーボンナノチューブ）32

長さ 7 ジグザグリング 3 本

37
11

38

30

33

31

34

35

36

13
39

49

51
52

14

50
20

53
24

54
23

10
20

47

18
55

29
4

22

56

5

21

9

11

0

8

7

30

57

6

40
15

58
26

7
14

39

41
42

16

27

12

3

44
17

28

10

43

45
46

48
19

38
12

59

8

53

9

10
20

13

15

17

19

5

4

3

2

54
23

24

18

50
20

14

25

16

51
52

4
22

21

9

0

8

3

2

1
1

6

1

C0(RR60A1)  0.702064

1025/1812－139－

RRR60A1（カーボンナノチューブ）

長さ 6 ジグザグリング 3 本

11
30

35

46

34

38

36
47

12

37
17

48
49

16

59

57

19

29
18
7

15
56

9

2

23

4

15
14

12

5

0

1

2

49

27

5

16

48
59

18

26

8
29

19

28
6

11

13

6

17

12

51

25

47

50

19

24

36

46

13

53

35

45

52

20

22
3

17

14

55

21

44
43

54

10

20

42
41

30

31

32

40
39

58

8

33

18
7

9

2

7

17

10

8

6

16

3

4

5

0

1

C0(RRR60A1)  0.735247

1690/1812

RR60B1（カーボンナノチューブ）37

長さ 7 ジグザグリング 2 本

32

37

11
30

38

36

35

33

34

13
51

14

40
20

53
29

59

58

9

57

21

56

19

55

17

54

15

13

59
23

10
10

11

2
0

53

9
12

40
20

29

8
14

39
52

14

28

7
16

13

50
15

27

6
18

3
8

38

51

49
48

16

26

5
20

47
46

17

25

4
22

44
18

24

10
10

45

43
42

19

23

30

12
41

39
52

31

4
22

9

21

8

0

3

1

3

5

4

6

1
7

7

2

C0(RR60B1)  0.77329

1800/1812

RRRR60A1（カーボンナノチューブ）
30

34

33

32

31

30

34

43

35

37

39

41

43

35

44
11

36
15
53

45
46

54
16

25
10

15
5
14

6

18

12

25

22

29
6
16

24
23

4
8

20
55

10

21
3

10

16

26

20
19

2

54

59

9

53

45

17

27

36
15

46

58

8

17
1

47

18

28

44
11

48

57

7
16

24

49

19

29

12

50

56

6

23

51

20

42

40
13

52

55

59

38
14

長さ 5 ジグザグリング 4 本

15
5

1
14

6

5

13

11

9

7

5

13

4

0

1

2

3

4

0

C0(RRRR60A1)  0.857813－140－

1812/1812

n
1
2
...
12
...
42
...
166
...
387
...
1025
...
1690
...
1800
...
1811
1812

C0(n)
0.6372700691
0.6577294008...
0.6594161135...
0.6669694342...
0.6719601465...
0.6839200228...
0.7020639322...
0.7352465097...
0.7732895993...
0.8325592565
0.8578134657

バッキーボール
ツイステッドバッキーボール
カメボール
大車輪
ヒフミクン
トリプルスリーAAA60A1

カーボンナノチューブ

RR60A1カーボンナノチューブ

RRR60A1

カーボンナノチューブ

RR60B1カーボンナノチューブ

RRRR60A1

C60

－141－

IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, JapanOn discrete constant principal curvature surfaces
Yuta OGATA
Kyoto Sangyo University
In this talk, we will study the discrete surface theory on a full 3-ary oriented tree andintroduce the notion of discrete principal curvatures on them. In order to investigate
the geometric meaning of discrete principal curvatures, we will also deﬁne a discrete
analog of principal directions on discrete surfaces. At the end of the talk, we also show
some examples of discrete constant principal curvature surfaces.

－142－

－143－

－144－

－145－

－146－

－147－

－148－

－149－

－150－

－151－IMI Workshop II: Geometry and Algebra in Material Science IV
September 4 - 5, 2023, West W1-D-413, Kyushu University, Fukuoka, Japan

Mathematical and computational science governing
the functions of molecules and materials in nature
Shinichiro NAKAMURA
Kumamoto University
The author has been involved in the design of molecules and materials in the ﬁelds ofindustry, government, and academia. The methods are quantum chemistry, molecular
dynamics, and ﬁrst-principles calculations. Recently also applied is a data science(DS) approach. From that experience, I have been fascinated by photosynthesis that
embodies ”natural intelligence” and the splendor of pigment molecules that exist in
Natural intelligence is beyond the current artiﬁcial intelligence. Aiming
nature.
at this understanding, the authors will present the results on the as yet unexploredmechanism by which photosynthesis generates oxygen from water and converts CO 2
into starch. In addition, I will report on research that attempted to design by learningfrom the naturally occurring pigment called porphyra-334. I would also like to touchon strategies guided by category theory.

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Article
Cite This: J. Phys. Chem. B 2019, 123, 7649−7656

pubs.acs.org/JPCB

Unique Structural Relaxations and Molecular Conformations ofPorphyra-334 at the Excited State
Makoto Hatakeyama,*,†,‡ Kenichi Koizumi,‡ Mauro Boero,¶ Katsuyuki Nobusada,§,⊥Hirokazu Hori,# Taku Misonou,# Takao Kobayashi,∥ and Shinichiro Nakamura‡,□!.<6.+1  †

Sanyo-Onoda City University, 1-1-1 Daigakudori, Sanyo-Onoda, Yamaguchi 756-0884, Japan
Cluster for Science, Technology and Innovation Hub, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan§
Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki 444-8585,
Japan
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615 8520 J‡

ABSTRACT: Quantum chemistry based simulations wereused to examine the excited state of porphyra-334, one of thefundamental mycosporine-like amino acids present in a wide
variety of aqueous organisms. Our calculations reveal threecharacteristic aspects of porphyra-334 related to either its
ground or excited state. Speciﬁcally, (i) the ground state (S0)structure consists of a planar geometry in which three units
can be identiﬁed, the central cyclohexene ring, the glycine
branch, and the threonine branch, reﬂecting the π conjugationof the system; (ii) the ﬁrst singlet excited state (S1) shows alarge oscillator strength and a typical ππ* excitation character;and (iii) upon relaxation at S1, the originally ground state
planar structure undergoes a relaxation to a nonplanar one, S1, especially at the carbon−nitrogen (CN) groups linking thecyclohexene ring to the glycine or threonine arm. The induced nonplanarity can be ascribed to the fact that the carbon atoms of
the CN groups prefer an sp3 hybridization in the S1 state. At the singlet state, these processes are unlikely to be trapped bysinglet−triplet intersystem crossing especially when these occur in the hydrophilic zwitter-ion forms of porphyra-334. These
results provide the missing information for thorough interpretation of the stability of porphyra-334 upon UV irradiation.

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Chem-VAE（Chemical Variational Auto encoder)を駆使すれば
天然色素ポルフィラ334を超える
新しい分子が設計できるか？？
〜Conventional分子設計法（類似検索）と比較〜
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Figure 1. (a) A diagram of the autoencoder used for molecular design, including the joint property prediction model. Starting from a discrete
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a continuous molecular representation. Given a point in the latent space, the decoder network produces a corresponding SMILES string. A mutilayer
perceptron network estimates the value of target properties associated with each molecule. (b) Gradient-based optimization in continuous latent
space. After training a surrogate model f(z) to predict the properties of molecules based on their latent representation z, we can optimize f(z) withrespect to z to ﬁnd new latent representations expected to have high values of desired properties. These new latent representations can then be
decoded into SMILES strings, at which point their properties can be tested empirically.

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「マス・フォア・インダストリ研究」シリーズ刊行にあたり
本シリーズは，平成 23 年 4 月に設立された九州大学マス・フォア・インダストリ研究所(IMI)が，平成 25 年 4 月に共同利用・共同研究拠点「産業数学の先進的・基礎的共同研究拠点」として，文部科学大臣より認定を受けたことにともない刊行するものである．本シ
リーズでは，主として，マス・フォア・インダストリに関する研究集会の会議録，共同研
究の成果報告等を出版する．各巻はマス・フォア・インダストリの最新の研究成果に加え，その新たな視点からのサーベイ及びレビューなども収録し，マス・フォア・インダストリの展開に資するものとする．

平成 30 年 10 月
マス・フォア・インダストリ研究所所長 佐伯

材料科学における幾何と代数 IV
マス・フォア・インダストリ研究 No.26, IMI, 九州大学
ISSN 2188-286X
発行日

2023 年 11 月 28 日

編集

松谷茂樹，緒方勇太，落合啓之，加葉田雄太朗，佐伯修，濵田裕康，松江要発行

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修シリーズ既刊
Issue

Author ／ Editor

Title

Published

マス・フォア・インダストリ
研究 No.1

穴田 啓晃
安田 貴徳
Xavier Dahan
櫻井 幸一

Functional Encryption as a Social Infrastructure and
Its Realization by Elliptic Curves and Lattices

26 February 2015

マス・フォア・インダストリ
研究 No.2

滝口
藤原

Collaboration Between Theory and Practice inInverse Problems

12 March 2015

マス・フォア・インダストリ
研究 No.3

筧

非線形数理モデルの諸相：連続，離散，超離散，その先Various aspects of nonlinear mathematical models: continuous, discrete, ultra-discrete, and beyond

24 March 2015

マス・フォア・インダストリ
研究 No.4

穴田
安田
櫻井
寺西

啓晃
貴徳
幸一
勇

Next-generation Cryptography for
Privacy Protection and Decentralized Control andMathematical Structures to Support Techniques

29 January 2016

マス・フォア・インダストリ
研究 No.5

藤原
滝口

宏志
孝志

Mathematical Backgrounds and Future Progress ofPractical Inverse Problems

1 March 2016

マス・フォア・インダストリ
研究 No.6

松谷
佐伯
中川
上坂
濵田

茂樹
修
淳一
正晃
裕康

結晶のらせん転位の数理

10 January 2017

マス・フォア・インダストリ
研究 No.7

滝口
藤原

孝志
宏志

Collaboration among mathematics, engineering andindustry on various problems in infrastructure andenvironment

1 March 2017

マス・フォア・インダストリ
研究 No.8

藤原
滝口

宏志
孝志

Practical inverse problems based on interdisciplinary
and industry-academia collaboration

20 February 2018

マス・フォア・インダストリ
研究 No.9

阿部
高島
縫田
安田

拓郎
克幸
光司
雅哉

代数的手法による数理暗号解析
Workshop on analysis of mathematical cryptography
via algebraic methods

1 March 2018

マス・フォア・インダストリ
研究 No.10

阿部
落合
高島
縫田
安田

拓郎
啓之
克幸
光司
雅哉

量子情報社会に向けた数理的アプローチ
Mathematical approach for quantum informationsociety

26 December 2018

孝志
宏志

三郎

（

）

IssueAuthor ／ Editor

Title

Published

マス・フォア・インダストリ
研究 No.11

松谷
佐伯
中川
濵田
上坂

茂樹
修
淳一
裕康
正晃

結晶転位の先進数理解析
Advanced Mathematical Investigation for Dislocations

7 January 2019

マス・フォア・インダストリ
研究 No.12

滝口孝志

Non-destructive inspection for concrete structuresand related topics

13 February 2019

マス・フォア・インダストリ
研究 No.13

宇波
長野
吉岡
田上
白井

耕一
智絵
秀和
大助
朋之

数理農学における時系列データのモデル化と解析
Modeling and Analysis of Time Series Data in MathAgro Sciences

28 February 2019

マス・フォア・インダストリ
研究 No.14

佐久間 弘文
大津 元一
小嶋 泉
福本 康秀
山本 昌宏
納谷 昌之

ドレスト光子に関する基礎的数理研究

18 March 2019

マス・フォア・インダストリ
研究 No.15

松谷
佐伯
中川
濵田
富安

結晶の界面，転位，構造の先進数理解析

2 December 2019

マス・フォア・インダストリ
研究 No.16

Takuro Abe
Yasuhiko Ikematsu
Koji Nuida
Yutaka Shikano
Katsuyuki Takashima
Masaya Yasuda

Quantum computation, post-quantum cryptography
and quantum codes

17 January 2020

マス・フォア・インダストリ
研究 No.17

河村
津曲
西澤
溝口

彰星
紀宏
弘毅
佳寛

代数・論理・幾何と情報科学―理論から実世界への展開

10 February 2020

マス・フォア・インダストリ
研究 No.18

Takashi Takiguchi

New technologies for non-destructive and noninvasive inspections and their applications

21 February 2020

マス・フォア・インダストリ
研究 No.19

Hirofumi Sakuma
Motoichi Ohtsu
Masayuki Naya
Izumi Ojima
Yasuhide Fukumoto

Basic mathematical studies on dressed photon
phenomena

19 March 2020

茂樹
修
淳一
裕康
亮子

マス・フォア・インダストリ
研究 No.20

松谷 茂樹
井上 和俊
加葉田雄太朗
佐伯
修
垂水 竜一
内藤 久資
中川 淳一
濵田 裕康

材料科学における幾何と代数 I

24 November 2020

マス・フォア・インダストリ
研究 No.21

Daisuke Sakurai
Shigeo Takahashi
Naoki Hamada
Osamu Saeki
Hamish Carr
Takahiro Yamamoto

Fiber Topology Meets Applications

10 March 2021

マス・フォア・インダストリ
研究 No.22

巴山
中丸
大垣

竜来
啓
真二

数理的生成手法による CG とデジタルファブリケー
ション

15 March 2021

松谷
井上
落合
佐伯
垂水
内藤
中川
濵田
松江

茂樹
和俊
啓之
修
竜一
久資
淳一
裕康
要

材料科学における幾何と代数 II

11 November 2021

櫻井 大督
佐伯
修
高橋 成雄
Hamish Carr
山本 卓宏
濱田 直希

Fiber Topology Meets Applications 2

15 March 2022

佐久間弘文
大津 元一
小嶋
泉
福本 康秀

解析から設計に向けたオフシェル数理科学

28 March 2022

マス・フォア・インダストリ
研究 No.23

マス・フォア・インダストリ
研究 No.24

マス・フォア・インダストリ
研究 No.25

Institute of Mathematics for Industry
Kyushu University

九州大学マス・フォア・インダストリ研究所〒819 - 0395 福岡市西区元岡744

https://www.imi.kyushu-u.ac.jp