Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T13:48:04.754Z Has data issue: false hasContentIssue false

A note on the percolation probabilities of pairs of closely similar lattices

Published online by Cambridge University Press:  24 October 2008

M. F. Sykes
Affiliation:
Wheatstone Physics Laboratory, King's College, London
J. J. Rehr
Affiliation:
Wheatstone Physics Laboratory, King's College, London
Maureen Glen
Affiliation:
Wheatstone Physics Laboratory, King's College, London

Abstract

The percolation probabilities of the face-centred cubic and close-packed hexagonal lattices are found not to be identical; the identity of their critical percolation probabilities remains an open question. Other pairs of closely similar lattices are discussed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Vyssotsky, V. A., Gordon, S. B., Frisch, H. L. and Hammersley, J. M.Critical percolation probabilities (bond problem). Phys. Rev. 123 (1961), 15661567.CrossRefGoogle Scholar
(2)Frisch, H. L., Sonnenblick, E., Vyssotsxy, V. A. and Hammersley, J. M.Critical percolation probabilities (site problem). Phys. Rev. 124 (1961), 10211022.Google Scholar
(3)Frisch, H. L., Hammersley, J. M. and Welsh, D. J. A.Monte Carlo estimates of percolation probabilities for various lattices. Phys. Rev. 126 (1962), 949951.Google Scholar
(4)Dean, P.A new Monte Carlo method for percolation problems on a lattice. Proc. Cambridge Philos. Soc. 59 (1963), 397410.Google Scholar
(5)Dean, P. and Bird, N. F.Monte Carlo estimates of critical percolation probabilities. Proc. Cambridge Philos. Soc. 63 (1967), 477479.CrossRefGoogle Scholar
(6)Neal, D. G.Estimates of critical percolation probabilities for a set of two-dimensional lattices. Proc. Cambridge Philos. Soc. 71 (1972), 97106.Google Scholar
(7)Shante, V. K. S.and Kirkpatrick, Scott. An introduction to percolation theory. Advances in Physics 20 (1971), 325357.CrossRefGoogle Scholar
(8)Seitz, F.The modern theory of solids (McGraw Hill; New York, 1940).Google Scholar
(9)Dugfale, J. S. and Simon, F. E.Thermodynamic properties and melting of solid helium. Proc. R. Soc. (London), Ser. A 218 (1953), 291310.Google Scholar
(10)Harrison, W. A.Solid state theory (McGraw Hill; New York, 1970).Google Scholar
(11)Barron, T. H. K. and Domb, C.On the cubic and hexagonal close-packed lattices. Proc. Roy. Soc. (London), Ser. A 227 (1955), 447465.Google Scholar
(12)Wells, A. F.The geometrical basis of crystal chemistry. VIII. Acta Cryst. 18 (1965), 849900.CrossRefGoogle Scholar
Wells, A. F.The geometrical basis of crystal chemistry. IX. Some properties of plane nets. Acta Cryst. B 24 (1968), 5057.Google Scholar
Wells, A. F.The geometrical basis of crystal chemistry. X. Further study of three-dimensional polyhedra. Acta Cryst. B 25 (1969), 17111719.Google Scholar
Wells, A. F.The geometrical basis of crystal chemistry. XI. Further study of three-dimensional 3-connected nets. Acta Cryst. B 28 (1972), 711713.Google Scholar
(13)Domb, C. and Sykes, M. F.High temperature susceptibility expansions for the close-packed hexagonal lattice. Proc. Phys. Soc. London B 70 (1957), 896897.Google Scholar
Domb, C.On the theory of cooperative phenomena in crystals. Philos. Mag. Suppl. 9 (1960), 315316.Google Scholar
(14)Martin, J. L., Sykes, M. F. and Hioe, F. T.Probability of initial ring closure for self-avoiding walks on the face-centred cubic and triangular lattices. J. Chem. Phys. 46 (1967), 34783481.CrossRefGoogle Scholar
(15)Dienes, P.The Taylor Series (Oxford University Press, 1931).Google Scholar
(16)Sykes, M. F., Guttmann, A. J., Watts, M. G. and Roberts, P. D.The asymptotic behaviour of self-avoiding walks and returns on a lattice. J. Phys. A 5 (1972), 653660.Google Scholar
(17)Domb, C. and Sykes, M. F.High temperature susceptibility expansions for the close-packed hexagonal lattice. Proc. Phys. Soc. B 70 (1957), 896897.CrossRefGoogle Scholar
(18)Sykes, M. F., Gaunt, D. S., Roberts, P. D. and Wyles, J. A.High temperature series for the susceptibility of the Ising model I. Two-dimensional lattices. J. Phys. A 5 (1972), 624639.CrossRefGoogle Scholar
(19)Sykes, M. F., Gaunt, D. S., Roberts, P. D. and Wyles, J. A.High temperature series for the susceptibility of the Ising model II. Three-dimensional lattices. J. Phys. A 5 (1972), 640652.CrossRefGoogle Scholar
(20)Baker, G. A.Application of the Padé approximant method to the investigation of some magnetic properties of the Ising model. Phys. Rev. 124 (1961), 768774.Google Scholar
Baker, G. A. Further applications of the Padé approximant method to the Ising and Heisenberg models. Phys. Rev. 129 (1963), 99102.Google Scholar
(21)Sykes, M. F. and Essam, J. W.Exact critical percolation probabilities for site and bond problems in two dimensions. J. Math. and Phys. 5 (1964), 11171127.Google Scholar