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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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slide1

Fullerének és nanocsövek geometriai szerkezete

László István

Elméleti Fizika Tanszék

Budapesti Műszaki és Gazdaságtudományi Egyetem

slide4

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)

slide5

is the number of faces with i vertices,

If

and each vertex has 3 neighbours:

and thus

slide8

The topological coordinates

For the

adjacencymatrix

if

otherwise

slide13

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)

slide16

r

R

slide17

Toroidal structures

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

slide18

Nanotubes

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

slide19

Planarstructures

I. László, JCICS 44, 315 (2004)

slide20

Construction of polyhex nanotubes

(m, 0) zig-zag

(m, m) armchair

(m, n) chiral

slide39

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)

slide40

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)
slide41

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)

slide42

is the number of faces with i vertices,

If

and each vertex has 3 neighbours:

and thus

slide44

e=3

e=2

slide45

For closed ended nanotubes

For open ended nanotubes

slide47

The (u, v) coordinates of the intersection line

on the rectangle for the first cylinder

and

slide48

with

are cylindrical

Where

and

coordinates of cylinder 1 and 2

slide49

The (u, v) coordinates of the intersection line

on the rectangle for the second cylinder

with

slide50

Examples

Cylinder 1

Cylinder 2

d =1.3

(5, 2)

(10, 3)

slide51

2 - 5 - 3 =

12 - 18 =

- 6

slide53

(10, 0)

( 8, 0)

d = 0.0

= 90

slide54

(10, 0)

( 8, 0)

d = 0.0

slide59

Nanotube junction

as intersection of

Cylinders and cone

slide61

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