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The above proof can be extended to more than two dimensions. Griffiths ha, also shown the existence of spontaneous magnetization without imposing the special boundary condition, but we shall forego the proof. It should be noted that the existence of long-range order in two-dimension, is a special feature of the Ising model. In more realistic models with a continuou, order parameter, one can show that long-range order is not tenable in two-dimensions, because of large fluctuations (see Section 16.7).

A 4 5

FIGURE 6.29 This graph shows an options chart by itself. Note the relatively low price on the y-axis.

In the Ising model the energy of a configuration of the spin lattice depends not on the detailed distribution of spins over lattice sites but only on the tW() numbers N + and N ++. The number N +IN is said to be a measure of the N "long-range order" in the lattice, and N ++/( Y 12) is said to be a measure of the "short-range order." The reason for this terminology is as follows. Let us imagine that the distribution of spin is random, except for the restriction that it possesse, the given values of N + and N ++. If we know definitely that a given spin is up. then the number N ++/( Y 12) is the fraction of its nearest neighbors with spin N up. This number, however, imposes less and less correlation as we consider the second-nearest neighbors, third-nearest neighbors, etc. It is therefore a measure of the local correlation of spins; hence the name short-range order. On the other hand, the number N +IN requires no correlation between nearest neighbors. It does, however, require that in the entire lattice a fraction N +IN of all spins must be up. Thus, if the number N +1 N is known in the neighborhood of a given spin. we will know that no matter how far we go away from the given spin the order measured by it is the same. Hence the name long-range order. Define the parameter of long-range order L and that of short-range order a as follows: 1 N+ 1 == -(L + 1) (-I~L~+I) N 2 (14.32) N++

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==Ha+l)

(-I~a~+I)

L SiS} =

(ij)

hN(2a - 2L + 1)

(14.33)

Thus the ensemble average of the long-range order is the magnetization per particle. The energy per spin is, by (14.1), 1 NEAL, a) = - ~ y(2a - 2L + 1) - HL (14.34) The Bragg-Williams approximation* is contained in the statement that "there is no short-range order apart from that which follows from long-range order." More precisely, the approximation consists of putting N ++/( hN) z (N +/N)2 or a z HL+l)2- 1 (14.35)

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(14.36)

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This approximation clearly has heuristic value, but it is difficult to estimate the error involved. With (14.36) the partition function (14.2) becomes

QAH, T) =

{s, }

ePN(~qL2+HL)

(14.37)

The sum extends over all sets {Si}' but the summand depends only on L. Hence we want to find the number of sets {Si} that share the same L. According to (14.32), L is determined by N +. The number we seek is the number of ways to pick N + things out of N, namely N!jN + !(N - N +)1. Therefore +1 N! 1 2 " ePN(,qL +HL) (1438) Q (H T) I , = L~1 [tN(1 + L)]![~N(1 - L)]! . As N ~ 00 the logarithm of QI is equal to the logarithm of the largest term in the summand. Using Sterling's approximation for N! we find that

1 N log QI(H, T)

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