Critical behaviour of Ising systems on extremely anisotropic lattices: II. The simple cubic lattice

Abstract

The critical behaviour of the magnetic susceptibility χ of a spin- 1 2 Ising system on a simple cubic lattice in which the coupling constants along two of the three lattice axes, J 2 and J 3, are very small in comparison with the coupling constant along the remaining lattice axis, J 1, is investigated on the basis of the series expansion of χ in the variables t r = tanh βJ r , r = 1, 2, 3. For t 2, t 3 ⪡ 1, 1 − t 1 ⪡ 1 the reduced susceptibility χ′ ≡ ( kT μ 2 )χ is found to behave as χ (t 1,t 2,t 3)≈χ′ 0(t 1,t 2,t 3)≡ 1 1−t 1 ∑ m,n=0 ∞ b mn0 t 2 1−t 1 m t 3 1−t 1 n , where the b mn 0 are constants. The three-dimentional nature of the system is contained in the double power series ∑ m,nb mn0 [ t 2 (1 − t 1 )] m [ t 3 (1 − t 1 )] n , of which the singular behaviour is investigated for the case t 2 = at 3, a = 1, 2, 4, 6, 8. Rewriting for these cases the double power series as a single power series ∑ nb nO (a) [ t 2 (1 − t 1 )] n and assuming for this series a singular behaviour of the well-known type [ 1 − b(a)t 2 (1 − t 1 )] −p(a) we find in all five cases that the power p( a) is consistent with the value γ = 5 4 of the critical exponent of χ for the isotropic simple cubic lattice. Using this value we find for b( a) the values 6.10 ± 0.02, 4.48 5 ± 0.02, 3.54 ± 0.02, 3.19 ± 0.02, 2.99 ± 0.02 for a = 1, 2, 4, 6, 8, respectively. These numerical results are used for the investigation of the surface which represents the critical equation in the t 1, t 2, t 3 space. It turns out that very close to the point (1, 0, 0) this surface behaves as a cone with its apex at (1, 0, 0); in this respect the Bethe-Peierls approximation according to which the surface behaves near (1, 0, 0) as a plane rather than as a cone, is essentially in error.