Bethe-Guggenheim approximation

Equations (5.43) with (5.51) give the activities as functions of the densities for this model in the Bethe-Guggenheim approximation. If the occupancies of the cells neighbouring any one cell were indeed independent of each other, these relations would be exact. They are therefore exact for the class of lattices called trees, on which there are no closed paths between any two neighbours of one site there is no path other than via that site, so that, once the state of occupancy of the cell at that site is specified, the cells at the neighbouring sites are decoupled from, and independent of, each other. Because the Bethe-Guggenheim approximation is thus exact for one class of models, it is necessarily therm ynami-cally consistent and, indeed, it may be verified by explicit calculation that (5.43) with (5.51) satisfies (5.44). 

The one-component lattice gas of 5.3 may also be treated in the Bethe-Guggenheim approximation, which is a generalization and improvement upon, the simple mean-field theory. The latter follows from the former in Uie limit of large c and small e. The resulting mean-field theory is then necessarily thermodynamically consistent, because the Bethe-Guggenheim approximation is consistent for all c and e. In the present two-component model, in which the only interactions are infinitely strong repulsions, there is no simplification we can make beyond the Bethe-Guggenheim approximation and still retain thermodynamic consistency there is no parameter e, and, while the coordination number c is at our disposal, there is no limit to whidi we can usefully take h. 

We have from (5.43), (5.51), and (5.53) that in the Bethe-Guggenheim approximation the coexistence curve is determined by the roots (other than the trivial root q.=qb)... 

Bethe-Guggenheim approximation are given by (S.S6) as Pa-Pb (c- l)/(c -2), which are indeed small when the coordination number c is large. Thus, near the critical point, the coupled equations (5.60), with (5.62), are roughly of the form... 

Bethe-Guggenheim pair distribution approximation, 122 Bethe Salpeter s Hamiltonian operator, 240... 

In fact, a more complicated pattern of interaction is likely to give rise to different types of short-range order. The symmetries of the Bragg-Williams approximation can be readily broken if, instead of the chain molecules, the pairs of chains corresponding to the interaction described in Eq. (2.21) are regarded as statistical units. This leads to the Bethe-Peierls approximation or to the quasi-chemical approximation of Fowler and Guggenheim defect polymer crystal appears then to be a sub-... 

However, one can proceed beyond this zeroth approximation, and this was done independently by Guggenheim (1935) with his quasi-chemicaT approximation for simple mixtures and by Bethe (1935) for the order-disorder solid. These two approximations, which turned out to be identical, yield some enliancement to the probability of finding like or unlike pairs, depending on the sign of and on the coordmation number z of the lattice. (For the unphysical limit of z equal to infinity, they reduce to the mean-field results.)... 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

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