Bragg and William

The treatment of such order-disorder phenomena was initiated by Gorsky (1928) and generalized by Bragg and Williams (1934) [5], For simplicity we restrict the discussion to the synnnetrical situation where there are equal amounts of each component (x = 1/2). The lattice is divided into two superlattices a and p, like those in the figure, and a degree of order s is defined such that the mole fraction of component B on superlattice p is (1 +. s)/4 while that on superlattice a is (1 -. s)/4. Conservation conditions then yield the mole fraction of A on the two superlattices... 

This equation for the free energy is due to Gorsky and to Bragg and Williams.t The affinity, corresponding to the parameter is thus... 

Those theories of ordering that assume that the thermodynamics of ordering can be explained by Q alone are termed mean-field theories. Here two such theories are explored and compared that of Bragg and Williams (1934) and that of Landau (1937). While some attempt has been made to consider the pressure-dependence of order-disorder phenomena in minerals (Hazen and Navrotsky 1996), here I shall limit the discussion of these phenomena to their temperature-dependence alone. 

In the early 1900s Max von Laue had predicted that X rays would be diffracted by the atomic nuclei in a crystal. The father and son team of William Henry Bragg and William Lawrence Bragg developed equipment and equations, respectively, for extracting information about the structure of molecules from the X-ray diffraction pattern in a process that has been described as a three-dimensional jigsaw puzzle. 

This atatisticdl mechanical treatment, due to Bragg and Williams, of the transition in / brass led on to the modem concept of order-disorder transitions and this was further reinforced by Onsctger s work. The corresponding thermodynamic theory, as developed by Tisza and others, depends on the use of an internal parameter of the system (as well as T and p) and is closely related to the thermodynamic theory of chemical reactions, as described in Chapter 4. 

We have perfectly defined the two states of order and disorder. However, we can imagine intermediary states - e.g. a state where a certain number of atoms of A are on sites that are normally attributed to A in the perfectly ordered solution. This number would obviously be between the average and the total number of atoms of A. In order to characterize such an intermediary state, Bragg and Williams defined a degree of order or long-distance order parameter s, such that this degree is equal to 1 if the solution is perfectly ordered, and 0 in the case of a completely random distribution solution. 

The degree of order at long distance, 5, or the Bragg and Williams degree of order, is defined by the relation ... 

Thus, (X) is independent of the temperature. This is known as the zero-order approximation or the Bragg and William approximation. 

Note.- We have seen that relation [2.70] of the strictly-iegular solution model was called the Bragg and Williams zero-order approximation. Similarly, in view of relation [2.89], the quasichemical solution is called an approximation of order 1. The order at hand is, in fact, the power to which the exponential appearing in relation [2.89] is raised power zero for the Bragg and Williams model, and power 1 for the quasi-chemical model. 

In order to evaluate the functions g(5) and E s), we need to know the distribution of the atoms on the lattice for the given value of s. Two models have been developed the Gorsky, Bragg and Williams model and the quasi-chemical model. The hypotheses upon which these models are based are similar, respectively, to those used for the model of a strictly-regular solution (see section 2.3.3) and those used for Fowler and Guggenheim s quasi-chemical solution model (see section 2.3.5). 

The Bragg-WilUams description (Bragg and Williams, 1934) of long-range atomic order 5 in a binary alloy is given as follows ... 

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