A003220 -id:A003220 - cp's OEIS frontend

A047709 Low-temperature series in u = exp(-4J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

0, 0, 1, 0, 12, 4, 129, 72, 1332, 960, 13419, 11372, 132900, 126396, 1299851, 1349784, 12592440, 14023944, 121074183, 142818336, 1157026804, 1432470300, 11001347199, 14196860272, 104161648860, 139351826712, 982653092725, 1357030991292, 9241395939636

Offset: 1

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Many sources give this sequence multiplied by 4 because the actual susceptibility per spin is this series times 4m^2/kT. (m is the magnetic moment of a single spin; the factor m^2 may be present or absent depending on the precise definition of the susceptibility.)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Y. Chan, A. J. Guttmann, B. G. Nickel, and J. H. H. Perk, The Ising Susceptibility Scaling Function, J Stat Phys 145 (2011), 549-590; arXiv:1012.5272 [cond-mat.stat-mech], 2010-2020. Gives 642 terms in the file Triangle_u642.txt (divide by 4 to get this sequence).
J. W. Essam and M. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.
M. F. Sykes, D. S. Gaunt, J. L. Martin, S. R. Mattingly, and J. W. Essam, Derivation of low-temperature expansions for Ising model. IV. Two-dimensional lattices--temperature grouping, Journal of Mathematical Physics 14 (1973), 1071.

Cf. A002919, A002920, A003488, A005399, A057383, A057387.

Cf. A002912, A002927, A002924, A002925, A002926, A003220.

Edited and extended from Chan et al by Andrey Zabolotskiy, Mar 02 2021

A003119 High-temperature series for spin-1/2 Ising magnetic susceptibility on diamond structure.

Original entry on oeis.org

1, 4, 12, 36, 108, 324, 948, 2772, 8076, 23508, 67980, 196548, 566820, 1633956, 4697412, 13501492, 38742652, 111146820, 318390684, 911904996, 2608952940, 7463042916, 21328259716

Offset: 0

N. J. A. Sloane

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
J. W. Essam and M. F. Sykes, The crystal statistics of the diamond lattice, Physica, 29 (1963), 378-388.
D. S. Gaunt and M. F. Sykes, Estimation of Critical Indices for the Three-Dimensional Ising Model, J. Phys. A 6 (1973), 1517-1526.
Peter Young, Coefficients in the series expansions.

Cf. low-temperature: A003220 (ferromagnetic), A007216 (antiferromagnetic); A002913 (cubic structure); A002923 (Heisenberg model).

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 01 2008

Name clarified by Andrey Zabolotskiy, Nov 14 2024

cp's OEIS Frontend

A047709 Low-temperature series in u = exp(-4J/kT) for ferromagnetic susceptibility for the spin-1/2 Ising model on hexagonal lattice.

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