Constraints on Small Fullerene Helices


A single heptagon together with a single pentagon can join two graphene semitubules together at a 30° angle so that all carbon atoms are three-fold coordinated and all other carbon rings are hexagons. This bend connects tubules of complementary classes. A tubule class is the set of all tubules having the same helicity. A tubule class has a uniform density of radii, which is the number of different tubules in the class per unit change in tubule radius. The classes that are joined by a heptagon and pentagon have smallest members whose radii differ by a factor of √3. Thus tubule segments joined by a heptagon and pentagon cannot have exactly equal circumferences. There are only a finite number of twist angles allowed between sequential bends along a tubule. Pentagons and heptagons should be isolated and separated as far as possible. These requirements particularly constrain the bends and twists that can occur in the smallest tubules. These considerations favor long-range order along a helix, given a driving force for curling during the formation process.

This is a preview of subscription content, access via your institution.


  1. [1]

    H.W. Kroto, J.R. Heath, S.C. O’Brien, R.F. Curl and R.E. Smalley

  2. [2]

    W. Krätschmer, L.D. Lamb, K. Fostiropoulos, and D.R. Huffman, Nature 347, 354 (1990).

    Article  Google Scholar 

  3. [3]

    H.W. Kroto, Nature 329, 529 (1987).

    CAS  Article  Google Scholar 

  4. [4]

    S. Iijima, Nature (London) 354, 56 (1991).

    CAS  Article  Google Scholar 

  5. [5]

    M.S. Dresselhaus, G. Dresselhaus, and R. Saito, Phys. Rev. B 45, 6234 (1992).

    CAS  Article  Google Scholar 

  6. [6]

    J.W. Mintmire, B.I. Dunlap, and C.T. White, Phys. Rev. Lett. 68, 631 (1992).

    CAS  Article  Google Scholar 

  7. [7]

    C.T. White, D.H. Robertson, and J.W. Mintmire, Phys. Rev. B 47, 5485 (1993).

    CAS  Article  Google Scholar 

  8. [8]

    B.I. Dunlap, Phys. Rev. B 46, 1933 (1992).

    CAS  Article  Google Scholar 

  9. [9]

    S. Iijima, RM. Ajayan, and T. Ichihashi, Phys. Rev. Lett. 69, 3100 (1992).

    CAS  Article  Google Scholar 

  10. [10]

    B.I. Dunlap, Phys. Rev. B 49, 5643 (1994).

    CAS  Article  Google Scholar 

  11. [11]

    B.I. Dunlap, Phys. Rev. B 50, 8134 (1994).

    CAS  Article  Google Scholar 

  12. [12]

    X.B. Zhang, Z.F. Zhang, D. Bernaerts, G. Van Tendeloo, S. Amelinckx, J. Van Lan-druyt, V. Ivanov, J.B. Nagy, Ph. Lambin, and A.A. Lucas, Europhys. Lett. 27, 141 (1994).

    CAS  Article  Google Scholar 

  13. [13]

    S. Amelinckx, X.B. Zhang, D. Bernaerts, X.F. Zhang, V. Ivanov, J.B. Nagy, Science 265, 635 (1994).

    CAS  Article  Google Scholar 

  14. [14]

    D.H. Robertson, D.W. Brenner, and J.W. Mintmire, Phys. Rev. B 45, 12592 (1992).

    CAS  Article  Google Scholar 

  15. [15]

    M. Fujita, R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B, 45, 12834 (1992).

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Brett. I. Dunlap.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dunlap, B.I. Constraints on Small Fullerene Helices. MRS Online Proceedings Library 359, 169–174 (1994).

Download citation