A001406 -id:A001406 - 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

A001407 High temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

Original entry on oeis.org

1, 0, 0, 8, 33, 168, 962, 5928, 38907, 268056, 1918938, 14169360, 107333498, 830660688, 6546655404, 52410001448

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. (1.3) is the logarithm of the g.f. for this sequence.
Index entries for sequences related to f.c.c. lattice

Cf. A001393, A001406, A002892, A047712, A002918.

a(10)-a(14) added and name clarified by Andrey Zabolotskiy, Feb 14 2022

a(15) added using data from A047712 by Andrey Zabolotskiy, Jan 18 2023

A047711 High-temperature coefficients for internal energy for spin-1/2 Ising model on b.c.c. lattice.

Original entry on oeis.org

4, 48, 840, 19080, 501712, 14383256, 436774992, 13826204264

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.
Index entries for sequences related to b.c.c. lattice

Cf. A010571 (cubic), A047712 (f.c.c.), A001406 (partition function).

Sum_{n>=1} a(n) * v^(2*n-1) = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = Sum_{n>=0} A001406(n) * v^(2*n) and q = 8 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 14 2022

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

A002917 High temperature series for spin-1/2 Ising specific heat on 3-dimensional b.c.c. lattice.

Original entry on oeis.org

4, 140, 4056, 129360, 4381848, 153700408, 5519859080, 201714989064

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.
Index entries for sequences related to b.c.c. lattice
Index entries for sequences related to specific heat

Cf. A002916 (cubic), A002918 (f.c.c.), A001406 (partition function), A047711 (internal energy), A002914 (susceptibility), A002167 (Heisenberg model).

Sum_{n>=0} a(n) * v^(2*n) = (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) = Sum_{n>=0} A001406(n) * v^(2*n) and q = 8 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 15 2022

Better description from Steven Finch

a(5)-a(7) from Andrey Zabolotskiy, Feb 15 2022

A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 4, -4, 0, 28, -60, 44, 204, -750, 1084, 979, -8444, 18886, -7568, -82269, 280288, -348172, -576712, 3677331, -7445964, 569558, 41740944, -126624684

Offset: 1

Andrey Zabolotskiy, Mar 11 2024

The series is in the variable u = exp(-4J/kT).

The expansion of the logarithm of the g.f. of this sequence is given in Domb & Guttmann's Table 1 (with a reference to Sykes et al., 1965) and continued in Eq. (4.14) of Sykes et al., 1973.

Claude Itzykson and Jean-Michel Drouffe, Statistical field theory, vol. 2, Cambridge University Press, 1989. Eq. (120) is supposed to give the logarithm of the g.f., but its second half is erroneously switched with the second half of Eq. (121). These second halves are Eqs. (4.15) and (4.14) of Sykes et al., 1973.

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, J. W. Essam and D. S. Gaunt, Derivation of low-temperature expansions for the Ising model of a ferromagnet and an antiferromagnet, J. Math. Phys. 6 (1965), 283-298.
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 b.c.c. lattice

Cf. A002891 (simple cubic), A002892 (f.c.c.); A003193 (magnetization), A002925 (ferromagnetic susceptibility), A007218 (antiferromagnetic susceptibility); A001406 (high temperature).

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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A001407 High temperature series for partition function for spin-1/2 Ising model on f.c.c. lattice.

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A047711 High-temperature coefficients for internal energy for spin-1/2 Ising model on b.c.c. lattice.

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A002917 High temperature series for spin-1/2 Ising specific heat on 3-dimensional b.c.c. lattice.

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A371049 Low temperature series for spin-1/2 Ising partition function on body-centered cubic lattice.

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