A001407 -id:A001407 - cp's OEIS frontend

A001393 High temperature series for spin-1/2 Ising free energy on 3-dimensional simple cubic lattice.

1, 0, 3, 22, 192, 2046, 24853, 329334, 4649601, 68884356, 1059830112, 16809862992, 273374177222, 4539862959852, 76744615270821, 1317316023432372, 22913901542478978, 403242080061821802, 7169757254509112094, 128654570700129670404, 2327634530912450464791, 42424918919225263486322, 778469235834728913157632, 14371906938404203811137770

Offset: 0

N. J. A. Sloane

z = exp(-f/T) = 2 * cosh(K)^3 * Sum_{n >= 0} a(n) * v^(2*n) where v = tanh(K), K = J/T, T is temperature (in the units of energy), J is the nearest-neighbor interaction, and f is the free energy per spin. See Wipf, pp. 181-182. z is the [geometric average] partition function per spin, so the original name of this entry, "Partition function for cubic lattice", is somewhat more directly related to this sequence. - Andrey Zabolotskiy, Oct 18 2021

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).
Andreas Wipf, Statistical Approach to Quantum Field Theory, LNP 864, Springer, 2013.

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; arXiv:hep-lat/9212032.
A. J. Guttmann and I. G. Enting, The high-temperature specific heat exponent of the 3-dimensional Ising model, J. Phys. A 27 (1994) 8007-8010; arXiv:cond-mat/9411002.
G. S. Rushbrooke and J. Eve, High-temperature Ising partition function and related noncrossing polygons for the simple cubic lattice, J. Math. Physics 3 (1962) 185-189. Gives correct a(0)-a(6) and incorrect a(7).
Index entries for sequences related to specific heat

Cf. A001406, A001407, A002891, A010571.

Corrections and updates from Steven Finch

a(14)-a(23) from Andrey Zabolotskiy, Oct 18 2021

A001406 High temperature series for partition function for spin-1/2 Ising model on b.c.c. lattice.

Original entry on oeis.org

1, 0, 12, 148, 2568, 53944, 1283270, 33268932, 918420570

Offset: 0

N. J. A. Sloane

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 and M. F. Sykes, The calculation of lattice constants in crystal statistics, Phil. Mag., 2 (1957), 733ff.
M. F. Sykes, D. L. Hunter, D. S. McKenzie and B. R. Heap, Specific heat of a three dimensional Ising ferromagnet above the Curie temperature. II, J. Phys. A: Gen. Phys., 5 (1972), 667-673. The power-series part of Eq. (1.2) is the logarithm of the g.f. for this sequence (even powers only).
Index entries for sequences related to b.c.c. lattice

Cf. A001393, A001407.

a(6)-a(8) added, name clarified by Andrey Zabolotskiy, Feb 14 2022

A002918 High temperature series for spin-1/2 Ising specific heat on 3-dimensional f.c.c. lattice.

Original entry on oeis.org

6, 48, 390, 3216, 26844, 229584, 2006736, 17809008, 159799812, 1446245424, 13181330772, 120849559824, 1113598633188

Offset: 0

N. J. A. Sloane

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).

G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
A. J. Guttmann and G. S. Joyce, Critical behaviour of an isotropic spin system, J. Phys. C: Solid State Phys., 6 (1973), 2691-2712. See Table 1.
Index entries for sequences related to f.c.c. lattice
Index entries for sequences related to specific heat

Cf. A002916 (cubic), A002917 (b.c.c.), A002921 (susceptibility), A001407 (partition function), A002165 (Heisenberg).

G.f.: (v^2-1) * (-q/2*f(v)^2 - (v^2-1) * f'(v)^2 + f(v) * (2*v*f'(v) + (v^2-1)*f''(v))) / f(v)^2, where f(v) is the g.f. of A001407 and q = 12 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 15 2022

Better description from Steven Finch
a(8)-a(10) from Guttmann & Joyce added by Andrey Zabolotskiy, Feb 02 2022
a(11)-a(12) from Andrey Zabolotskiy, Feb 15 2022

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

A047712 High-temperature coefficients for internal energy for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

6, 24, 132, 816, 5448, 38808, 290568, 2255232, 17981532, 146428728, 1213014960, 10192820592, 86687830596, 744919762584

Offset: 1

N. J. A. Sloane

M. E. Fisher and M. F. Sykes, Antiferromagnetic susceptibilities of the simple cubic and body-centered cubic Ising lattices, Physica, 28 (1962), 939-956.
S. McKenzie, Extended high-temperature series expansions for the spin-s Ising model, J. Phys. A: Math. Gen., 16 (1983), 2875-2880. See table 1, column c(n); divide by 2 to get a(n).
M. F. Sykes, J. L. Martin and D. L. Hunter, Specific heat of a three-dimensional Ising ferromagnet above the Curie temperature, Proc. Phys. Soc., 91 (1967), 671-677.
Index entries for sequences related to f.c.c. lattice

Cf. A010571 (cubic), A047711 (b.c.c.), A001407 (partition function), A002918 (specific heat).

G.f. 6*v + 24*v^2 + 132*v^3 + 816*v^4 ... = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = 1 + 8*v^3 + 33*v^4 + ... is the g.f. of A001407 and q = 12 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 14 2022

a(9)-a(11) from Sykes, Martin & Hunter added by Andrey Zabolotskiy, Feb 05 2022
Name clarified and a(12)-a(13) added by Andrey Zabolotskiy, Feb 14 2022
a(14) from McKenzie added by Andrey Zabolotskiy, Jan 18 2023

cp's OEIS Frontend

A001393 High temperature series for spin-1/2 Ising free energy on 3-dimensional simple cubic lattice.

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A001406 High temperature series for partition function for spin-1/2 Ising model on b.c.c. lattice.

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A002918 High temperature series for spin-1/2 Ising specific heat on 3-dimensional f.c.c. 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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A047712 High-temperature coefficients for internal energy for spin-1/2 Ising model on f.c.c. lattice.

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