Williams model, Bragg

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

In the Bragg-Williams model studied in 6, not only G (given by 19.54) but also 8 and F change continuously on passage through the Curie point. On the other hand the second derivatives... 

The discussion of co-operative phenomena given here is based on the simple Bragg-Williams model. The modern theories of order-disorder changes have undergone rapid development recently. The situation in 1938 is admirably reviewed by Nix and Shockley 1 more recent summaries of both theoretical and experimental developments will be found in papers by Lipson and Wannier. See also Guggenheim,Rush-brooke, and footnote p. 305. 

Figure 4. The dependence of the free energy on the degree of order, Q, given by the Bragg-Williams model, at three temperatures (above E, below E and at E)-...

This shows that the Bragg-Williams model predicts that the transition temperature is a direct function of the exchange energy for the order-disorder process. Furthermore, the equilibrium condition dddQ = 0 gives the solution... 

Figure 9. The results of the Monte Carlo simulations of ordering of A1 and Si in the cordierite framework (from Thayaparam et al. 1995) and feldspar (Meyers et al. 1998), compared wi the equilibrium order parameter predicted by the Bragg-Williams model and the pair CVM model, shown on a rescaled temperature axis.

Figure 11. r dependence of Q for members of the ilmenite-hematite solid solution, determined from neutron powder diffraction (solid symbol Harrison et al. 2000a) and quench magnetization (open symbols Brown et al. 1993). Sohd hues are fits using a modified Bragg-Williams model. 

The ordering behaviour in ilmSO, ilm90 and ilmlOO appears to be fully reversible, but the data close to Tc can only be fitted with a critical exponent for the order parameter, /3, which is of the order of 0.1, which does not correspond to any classical mean field Landau-type model. Instead, a modified Bragg-Williams model is required, that describes the free energy phenomenologically in terms of a configurational entropy alongside an enthalpy that contains terms up to... 

For a multiple-component clathrate hydrate where the number of molecules denoted by type a is highly affinitive but is fixed to n, the thermodynamic properties of the clathrate hydrate under deficiency of the molecules a are examined. The guest a, called promoter molecule, is a species that forms a clathrate hydrate at a given thermodynamic condition provided that enough amount of a is supplied. The other molecules are tentatively removed and the distribution of a among available cage types (the number of n j molecules) is determined so that Z" has the maximum, which is equivalent to Bragg-Williams model, and is written as... 

When H2 > Mii 2 > 02c and the two-dimensional gas partially condenses, when M2 < Ml 2 < 02c and a part of condensate evaporates into the two-dimensional gas. For the Bragg-Williams model the queintities 0jc, Hjc are obtained from relations... 

This maximum value suggests a definition of a critical temperature for this system. However, an another critical temperature is usually introduced to describe order-disorder transitions [24]. Its determination is strongly connected to the Bragg-Williams model. In this model, the solution of... 

LRO parameter r approaches an equilibrium curve in accordance with the Bragg-Williams model (dashed line). 

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