Abstract
Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (p hcp c, S =0.199 255 5±0.000 001 0) and bond (p hcp c, B =0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of p c are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.
REFERENCES
D. Stauffer and A. Aharony, An Introduction to Percolation Theory, Revised 2nd ed. (Taylor and Francis, London, 1994).
M. Sahimi, Applications of Percolation Theory (Taylor and Francis, London, 1992).
Y. Y. Tarasevich and S. C. van der Marck, An investigation of site-bond percolation on many lattices, Int. J. Mod. Phys. C 10:1193 (1999).
S. C. van der Marck, Percolation thresholds and universal formulas, Phys. Rev. E 55, 1514 (1997).
C. D. Lorenz and R. M. Ziff, Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation, J. Phys. A 31:8147 (1998).
C. D. Lorenz and R. M. Ziff, Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices, Phys. Rev. E 57:230 (1998).
See, for example, C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley & Sons, Inc., New York, 1996).
J. L. Bocquet, Percolative diffusion of a dumbbell interstitial defect on a fcc lattice: Calculation of a percolation threshold with the use of a series method, Phys. Rev. B 50:16386 (1994).
R. M. Ziff, P. T. Cummings, and G. Stell, Generation of percolation cluster perimeters by a random-walk, J. Phys. A 17:3009 (1984).
M. E. Fisher, The theory of condensation and the critical point, Ann. Phys. (NY) 3:255 (1967).
R. M. Ziff and G. Stell, Critical behavior in three-dimensional percolation: Is the percolation threshold a Lifshitz point?, University of Michigan Report No. 88-4, (1988).
H. G. Ballesteros, L. A. Fernandez, V. Martin-Mayor, A. M. Sudupe, G. Parisi, J. J. Ruiz-Lorenzo, Scaling corrections: site percolation and Ising model in three dimensions, J. Phys. A 32:1 (1999).
S. Redner, A FORTRAN program for cluster enumeration, J. Stat. Phys. 29:309 (1982).
S. Mertens, Lattice animals: A fast enumeration algorithm and new perimeter polynomials, J. Stat. Phys. 58:1095 (1990).
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Lorenz, C.D., May, R. & Ziff, R.M. Similarity of Percolation Thresholds on the HCP and FCC Lattices. Journal of Statistical Physics 98, 961–970 (2000). https://doi.org/10.1023/A:1018648130343
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DOI: https://doi.org/10.1023/A:1018648130343