Abstract
Consider a lattice \({\mathcal {N}}\) of dimension \(d\), coordination number 
, \(N\) sites, periodic boundary conditions and an Ising variable \(\sigma ({\varvec{r}})=\pm 1\) at every \({\varvec{r}}\in {\mathcal {N}}\).
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Notes
- 1.
Where the subscript ‘1’ is now a reminder that \({\varvec{r}}\) and \({\varvec{r}}'\) are first-neighbour primary sites.
- 2.
In the Ising model
(Sect. 4.5.6). - 3.
These authors used a different model, but the same general principles must apply here.
- 4.
- 5.
The Lambert W function (see e.g. Corless et al. 1996) is the multi-valued inverse of the function \(z\exp (z)\).
- 6.
We use \(\sigma ({\varvec{r}},{\varvec{r}}')\) rather that \(\tau ({\varvec{r}},{\varvec{r}}')\) for the secondary site variable since it is an Ising spin variable.
- 7.
For the sake of clarity we have used \(\xi =0.235\), in Fig. 10.6, rather than \(\xi =0.21\) for the (c) curve.
- 8.
This result was derived for a somewhat different decorated solution model by Bartis and Hall (1974).
- 9.
If it were true that
, the coexistence curves would form a double cusp at their meeting point. - 10.
Meaning that the equivalent Ising model on the primary sites is ferromagnetic.
- 11.
We are using the grand-canonical distribution and scaling with respect to the number of primary–secondary–primary triplets rather than the number of primary sites.
- 12.
This decorated model can be looked on as one of the class of interstitial models (see, for instance, Perram 1971) in which the open structure is disordered by displacement of molecules from ‘framework’ onto interstitial sites.
- 13.
Curves B and C respectively, given for comparison, show the results of using formulae from first-order and mean-field approximations for
and 
. - 14.
This has been treated extensively by Mulholland and Rehr (1974).
- 15.
There are interesting similarities between this analysis and the discussion of solution of the Bethe-pair approximation for the Ising model using the function \(g(\tau )\) in Sect. 7.3.
- 16.
The diagrams in Fig. 10.11 are, unlike Fig. 10.10, qualitative rather than quantitative. Accurate calculations for the simple cubic lattice are described by Mulholland and Rehr (1974) who use them to plot diagrams for all these cases in the \(\rho \)–\(T\) plane. They also investigate the ‘antiferromagnetic’ case \(\varepsilon <0\), \(\xi =0\).
(Sect. 
, the coexistence curves would form a double cusp at their meeting point.
and 
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Lavis, D.A. (2015). Edge-Decorated Ising Models. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_10
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DOI: https://doi.org/10.1007/978-94-017-9430-5_10
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