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same isotherm with T1 < T,.., whereas the point marked Tz lies on an isotherm with Tz > T,... Thus with H = 0 it is only possible to obtain two points on each isotherm for T < T,. and one point for T> T,... To obtain a complete isotherm we must consider the case B =1= O. We then obtain the isotherms represented by the dotted curves. In the Bragg-Williams approximation, the transition region is empty. This indicates that the Bragg-Williams approximation is not satisfactory. We can see, however, there is a first-order transition in the lattice gas and that T,. is the critical temperature of the transition region. It is to be noted that v = 1 is the smallest specific volume possible, because the volume of a unit cell of the lattice has been chosen to be unity. The profile of the transition region is shown in the P-T diagram of Fig. 14.11. The results of the Bragg-Williams approximation can be obtained by other methods. One way is through the Gibbs variational principle, as suggested in Problem 10.6. Another way is suggested in Problem 14.3. It is an example of a general type of approximation known as "mean-field theory," a general formulation of which will be given in Section 17.4.

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The Bethe-Peierls approximation is an improvement over the Bragg-Williams approximation, in that the former takes into account specific short-range order. In the Bragg-Williams approximation the assumption N HI hN = (N +IN)z ignores the possibility of local correlation between spins. The BethePeierls approximation replaces it by a better assumption. An outline of the method follows. We try to find a more accurate relation between N Hand N + by focusing our attention not on the entire lattice, but only on a sublattice composed of any lattice site plus its y nearest neighbors. We form the mental picture of this sublattice being "immersed" in the background provided by the rest of the lattice, just as a small-volume element in a liquid is "immersed" in the

background provided by the rest of the liquid. It is assumed that the background influences the sublattice only through a single parameter that is similar to the fugacity in a liquid. The relation between N ++ and N + will be obtained for the sublattice through heuristic arguments. It is then assumed that the same relation holds throughout the entire lattice. We consider only the case H = O. To begin, consider a sublattice consisting of any lattice site whose spin state is denoted by s, together with its y nearest neighbors, as shown in Fig. 14.12. Let P(s, n) be the probability that n of the nearest neighbors are in a spin-up state while the center site has the spin state 5. If s = + 1, then P(s, n) refers to configurations of the sublattice in which there are n pairs (+ + ) and y - n pairs (+ - ). If s = -1, then P(s, n) refers to configurations in which there are n pairs (+ - ) and n - y pairs (- - ). For a given n there are (~) ways to decide which of the y neighbors are the n spins in question. Thus we assume that

P( +1, n) = q(~)

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