A002924 -id:A002924 - cp's OEIS frontend

A002926 Low temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

0, 0, 1, 0, 12, -14, 135, -276, 1520, -4056, 17778, -54392, 213522, -700362, 2601674, -8836812, 31925046, -110323056, 393008712, -1369533048, 4844047090, -16947396000, 59723296431, -209328634116, 736260986208, -2582605180212, 9074182912884

Offset: 1

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. 421.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. Vohwinkel, personal communication.

Andrei Zabolotskii, Table of n, a(n) for n = 1..32
Gyan Bhanot, Michael Creutz, Ivan Horvath, Jan Lacki and John Weckel, Series expansions without diagrams, Phys. Rev. E, 49 (1994), 2445-2453; arXiv:hep-lat/9303002, 1993.
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. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
A. J. Guttmann and I. G. Enting, Series studies of the Potts model: I. The simple cubic Ising model, J. Phys. A 26 (1993) 807-821.
Elmar Siepmann, Reihenanalysen für das 3-dimensionale Ising-Modell, Westfälische Wilhelms-Universität Münster, 1993. See Eq. (A.13); divide the coefficients by 4 to get this sequence.
M. F. Sykes et al., Derivation of low-temperature expansions for Ising model VI. Three-dimensional lattices - temperature grouping, J. Phys. A 6 (1973), 1507-1516.

Cf. A002913 (high temperature series); other quantities: A002915 (antiferromagnetic susceptibility), A002891 (partition function), A002929 (magnetization); other lattices: A002927 (square), A002924 (f.c.c.), A002925 (b.c.c.).

Corrections and updates from Steven Finch

a(25)-a(27) from Bhanot et al. added by Andrei Zabolotskii, Feb 09 2022

A002925 Susceptibility series for b.c.c. lattice.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 16, -18, 0, 252, -576, 519, 3264, -12468, 20568, 26662, -215568, 528576, -164616, -3014889, 10894920, -13796840, -29909616, 190423962, -399739840, -22768752, 2803402560, -8743064909

Offset: 1

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. 421.
Claude Itzykson and Jean-Michel Drouffe, Statistical field theory, vol. 2, Cambridge University Press, 1989. See Eq. (122).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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. E. Fisher, Padé approximant studies of the lattice gas and Ising ferromagnet below the critical point, J. Chem. Phys., 38 (1963), 802-812.
M. F. Sykes et al., Derivation of low-temperature expansions for Ising model VI. Three-dimensional lattices - temperature grouping, J. Phys. A 6 (1973), 1507-1516.
Index entries for sequences related to b.c.c. lattice

Cf. A002924, A002926, A371049.

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

Original entry on oeis.org

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

A002892 Low-temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, -6, 0, 0, 8, 42, -114, 66, 24, 123, 134, -1563, 2262, -405, 846, -2532, -15182, 47961, -37992, 8044, -59694, -57117, 742394, -1233840, 597456, -798392, 1447162, 7898736, -27134598, 27649335

Offset: 0

N. J. A. Sloane

sign

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361: part 1, part 2.
C. Domb and A. J. Guttmann, Low-temperature series for the Ising model, J. Phys. C: Solid State Phys., 3 (1970), 1652-1660.
M. F. Sykes, D. S. Gaunt, J. W. Essam and C. J. Elliott, Derivation of low-temperature expansions for Ising model. VI. Three-dimensional lattices-temperature grouping, J. Phys. A: Math. Nucl. Gen., 6 (1973), 1507-1516.
Index entries for sequences related to f.c.c. lattice

Cf. A001407 (high-temperature), A002891 (cubic lattice), A002924 (susceptibility), A003196 (magnetization).

Name clarified, missing minus signs added to a(12) and a(17), and terms a(22) and beyond added by Andrey Zabolotskiy, Feb 14 2022

A003220 Low-temperature series for spin-1/2 Ising ferromagnetic susceptibility on diamond.

Original entry on oeis.org

0, 0, 1, 8, 44, 208, 984, 4584, 21314, 98292, 448850, 2038968, 9220346, 41545564, 186796388, 838623100

Offset: 0

N. J. A. Sloane, Simon Plouffe

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 421. [has a typo in a(15)]
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)
M. F. Sykes, D. S. Gaunt, J. W. Essam and C. J. Elliott, Derivation of low-temperature expansions for Ising model. VI. Three-dimensional lattices-temperature grouping, J. Phys. A: Math. Nucl. Gen., 6 (1973), 1507-1516.

Cf. A002924 (f.c.c.), A002925 (b.c.c.), A002926 (cubic); A007216 (antiferromagnetic), A002930 (magnetization), A003119 (high-temperature).

Name clarified, a(15) corrected by Andrey Zabolotskiy, Nov 14 2024

cp's OEIS Frontend

A002926 Low temperature series for spin-1/2 Ising magnetic susceptibility on 3-dimensional simple cubic lattice.

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A002925 Susceptibility series for b.c.c. lattice.

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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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A002892 Low-temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

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A003220 Low-temperature series for spin-1/2 Ising ferromagnetic susceptibility on diamond.

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