Bragg-Williams

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

Nix and Shockley [6] gave a detailed review of the status of order-disorder theory and experiment up to 1938, with emphasis on analytic improvements to the original Bragg-Williams theory, some of which will be... 

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.

The simple Bragg-Williams treauiient of this behaviour assumes an unlike atom bond strengdi which is greater than the like atom bonding and calculates the entropy of mixing as a function of die disorder which counterbalances this negative heat of formation. The relationship between the Curie temperature, Tc, and the bond energies is... 

The crucial experiments that determined the structures of a number of very simple crystals, beginning with sodium chloride, were done, not by von Laue and his helpers, but by the Braggs, William (1862-1942) and Lawrence (1890-1971), father... 

We have employed the Bragg-Williams approximation (BWA) to obtain rough estimates of the ordering/segregation critical temperatures. It is well known that the BWA usually overestimates critical temperatures (approximately by 20 %) in comparison with the exact value obtained from Monte Carlo simulations, or by other highly accurate methods of statistical mechanics. This order of accuracy Is nevertheless sufficient for our present purposes. 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

Bragg, William H., 248 Branched chain alkane Saturated hydrocarbon in which not all the carbon atoms are in a single continuous chain, e.g., C—C—C, 580... 

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. 

Fig. 3. Schematic diagram for the variation with concentration of the partial molar heat of solution of the liquid noble metals into liquid tin, taken from reference 51. The numbers are the experimental and calculated AHt for the solutes, in cal/g atom, at the two concentrations of 0 and 0.02 mole fraction. Q-C labels the curves calculated by the quasichemical theory in first order B-W labels the curves calculated by the Bragg-Williams, or zeroth-order approximation, which assumes a random...

Bragg-Williams treatment of convergent ordering in solid solutions... 

Figure 9.10 Order parameter a for the Bragg-Williams model as a function of reduced temperature.

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

It should be clear that the Mayer method provides a convenient and economic framework within which to correct the major omissions inherent in the use of the crudest order-disorder results, e.g. the quasi-chemical and Bragg-Williams approximations, for crystals with short-range forces. However, detailed calculations by this method do not appear to have been attempted so far. 

If the Bragg-Williams approximation (69) is employed to describe the attractive interactions in the adsorbed layer, and the transition state is taken to be free of such interactions, the rate can be described by... 

Applying the Bragg-William approximation (assuming that the H-H interaction is given by the mean field of the H-atoms) and with the concentration Ch = Nh/ N, the free energy is... 

The nc value for palladium-hydrogen is 0.25 from magnetic susceptibility measurements (52) with Tc = 564°K (52), and from P-C-T data, de Ribaupierre and Manchester (26) estimate nc = 0.29 and Tc = 566°K. The Bragg-Williams approximation gives a reasonable WfiH value by using an average value of nc, the critical temperature, and an analytical expression for ie(n) determined... 

The simplest approximation to the total entropy of mixing is the mean-field Bragg-Williams-Gorsky configurational entropy, proportional to X > lnp [4J 3... 

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