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Fullerének és nanocsövek geometriai szerkezete

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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  1. Fullerének és nanocsövek geometriai szerkezete László István Elméleti Fizika Tanszék Budapesti Műszaki és Gazdaságtudományi Egyetem

  2. 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)

  3. is the number of faces with i vertices, If and each vertex has 3 neighbours: and thus

  4. For g = 0 (fullerenes)

  5. The topological coordinates For the adjacencymatrix if otherwise

  6. 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)

  7. r R

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

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

  10. Planarstructures I. László, JCICS 44, 315 (2004)

  11. Construction of polyhex nanotubes (m, 0) zig-zag (m, m) armchair (m, n) chiral

  12. 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)

  13. 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)

  14. 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)

  15. is the number of faces with i vertices, If and each vertex has 3 neighbours: and thus

  16. For g = 0 (fullerenes)

  17. e=3 e=2

  18. For closed ended nanotubes For open ended nanotubes

  19. Nanotube junctions as intersections of cylinders d

  20. The (u, v) coordinates of the intersection line on the rectangle for the first cylinder and

  21. with are cylindrical Where and coordinates of cylinder 1 and 2

  22. The (u, v) coordinates of the intersection line on the rectangle for the second cylinder with

  23. Examples Cylinder 1 Cylinder 2 d =1.3 (5, 2) (10, 3)

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