Architect: Buckminster Fullerw3.phys.nthu.edu.tw/~mou/teach/Fullerenes.pdf · Architect:...
Transcript of Architect: Buckminster Fullerw3.phys.nthu.edu.tw/~mou/teach/Fullerenes.pdf · Architect:...
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FullerenesFullerenesFullerenesFullerenesArchitect: Buckminster Fuller
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OrbitsOrbitsbondπ
OrbitsOrbitsbondπ
...... C C
bondσ bondσ
2 2 21S 2SC rbon 2Pa 1S 2SC rbon 2Pa2 34 electrons in bonds (SP ) + bond or SPσ π4 electrons in bonds (SP ) bond or SPσ π
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Euler’s TheoremEuler’s Theorem
F=E-V+2F E V 2
F number of facesF = number of facesE = number of edgesE number of edgesV = number of verticesV = number of vertices
2
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nC n = number of carbon atomsEach carbon has three edges.
3V V3V 22 2
VE F∴ = = +2 2
SS
F F=∑
SF S= 面形之數目12
S
SS
E SF= ∑ (6 ) 12SS
S F− =∑213
S
SS
V SF= ∑S
3
3 S
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∑Molecules with 5Molecules with 5--fold symmetryfold symmetry
6(6 ) 12 is arbitrary.SS
S F F− = ⇒∑
12 t 3 5 6C V F F個606 6
5 6
20 12 pentagon 3 5 6
3 60 5 12 6C V
FF F
F= +
∴ × = × + =
∵個
6 6 203 60 5 12 6 FF∴ × × +
70 5 6 12 pentagon 3 5 6C V F F= +∵個
6 6 253 70 5 12 6 FF∴ × = × + =
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5Nature 409, 304(2001)
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奈米碳管奈米碳管
6
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Graphene: Graphite sheetNanotube
NanoribbonNanoribbon
Nanoparticle
7
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石墨的結構石墨的結構
U it llU it llUnit cellUnit cell
8
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Crystal and Orientations of CCrystal and Orientations of C6060yy 6060
250T C°>Free rotational motion
isotropic→ isotropic→
250 corner, 3 face atoms form T C°<4 penetrating Simple Cubic latticeswith 4 different orientation of 3-fold axes along 4 g(rotation about these axes smeared out 5-fold symmetry)
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2D Crystal with 52D Crystal with 5--fold symmetric moleculesfold symmetric molecules
100 100A A100 100Orientation Sudden
A A×Orientation Sudden Changed across domain
Nature 409, 304(2001)