Fullerének és nanocsövek geometriai szerkezete

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Fullerének és nanocsövek geometriai szerkezete . László István Elméleti Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem . Euler’s Theorem for closed polyhedrons. F-E +V = 2(1-g). F = # of faces E = # of edges V = # of vertices g = genus of the surface = # of handles

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Fullerének és nanocsövek geometriai szerkezete

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Fullerének és nanocsövek geometriai szerkezete

László István

Elméleti Fizika Tanszék

Budapesti Műszaki és Gazdaságtudományi Egyetem

Euler’s Theorem for closed polyhedrons

F-E+V = 2(1-g)

F = # of faces

E = # of edges

V = # of vertices

g = genus of the surface = # of handles

(0 for sphere and 1 for torus)

is the number of faces with i vertices,

If

and each vertex has 3 neighbours:

and thus

The topological coordinates

For the

if

otherwise

Fullerene

or

D. E. Manolopoulos, P. W. Fowler, JCP 96, 7603 (1992)

L. Lovász, A. Schrijver, Ann. De L`Inst. Fourier (Grenoble)

49, 1017 (1999)

r

R

Toroidal structures

I. László, A. Rassat, P.W. Fowler, A. Graovac, CPL 342, 369 (2001)

Nanotubes

I. László, A. Rassat, JCICS 43, 519 (2003)

Planarstructures

I. László, JCICS 44, 315 (2004)

Construction of polyhex nanotubes

(m, 0) zig-zag

(m, m) armchair

(m, n) chiral

Experimental observations of nanotube junctions

• -J. Li, at all Nature 402, 253 (1999),
• -P. Nagy, R. Ehlich, L.P. Bíró, J. Gyulai,
• J. Appl. Phys. A 70,481(2000)
• B. G. Satishkumar, P. John Thomas, A. Govindaraj, C. N. R. Rao
• App. Phys. Lett. 77, 2530 (2000)

- L. P. Bíró, Z. E. Horváth, G. I. Márk Z. Osváth, A. A.Koós, A. M. Benito, W. Mares,

Ph. Lambin,

Diam. And Rel. Mat. 13, 241 (2004)

Theoretical propositions for nanotube junctions

• - L. A. Chernozatonskii, Phy. Lett. A170, 37 (1992)
• G. E. Scuseria, Chem. Phys. Lett. 195, 534 (1992)
• L. Chico et al. Phys. Rev. Lett. 76, 971 (1996)
• M. Menon, D. Srivastave, Phys. Rev. Lett. 79, 4453 (1997)
• G. Treboux, P. Lapstun K. Silverbrook Chem. Phys. Lett. 306, 402 (1999)
• S. Melchor Ferrer, N. V. Khokhriakov, S. S. Savinskii, Mol. Eng. 8, 315 (1999)
• A. N. Andriotis et al. Appl. Phys. Lett. 79, 266 (2001)
• M. Terrones et al. Phys. Rev. Lett. 89, 075505 (2002)
• M. Yoon et al. Phys. Rev. Lett. 92, 075504 (2004)

Euler’s theorem and consequences

for nanotube junctions

Euler’s Theorem for closed polyhedrons

F-E+V = 2(1-g)

F = # of faces

E = # of edges

V = # of vertices

g = genus of the surface = # of handles

(0 for sphere and 1 for torus)

is the number of faces with i vertices,

If

and each vertex has 3 neighbours:

and thus

e=3

e=2

For closed ended nanotubes

For open ended nanotubes

The (u, v) coordinates of the intersection line

on the rectangle for the first cylinder

and

with

are cylindrical

Where

and

coordinates of cylinder 1 and 2

The (u, v) coordinates of the intersection line

on the rectangle for the second cylinder

with

Examples

Cylinder 1

Cylinder 2

d =1.3

(5, 2)

(10, 3)

2 - 5 - 3 =

12 - 18 =

- 6

(10, 0)

( 8, 0)

d = 0.0

= 90

(10, 0)

( 8, 0)

d = 0.0

Nanotube junction

as intersection of

Cylinders and cone

Conclusions

• By changing the positions and angles between
• nanotube axes several junctions can be generated
• between any kind of nanotubes
• Using Stone-Wales –transformations the
• unphysical polygons can be eliminated
• The method can be generalized for constructing
• junction between any kind of carbon surfaces
• As the electronic properties depend on the
• junctions, a systematic search by parameter
• variations could reach to special nanotube
• networks with new useful properties