Generalized Ising model

The transformation to the (generalized) Ising model is useful since it clearly brings out the symmetries of the problem Eq. (9) is invariant under the transformation... 

In order to express the formation energy of any Al-Zr compound based on a perfect fee lattice, we make a cluster expansion [25] of our FP-LMTO calculations to fit a generalized Ising model. This allows us to obtain the energy of any configuration of the fee lattice. 

It is clear that the form of Equation 11.24 constitutes indeed a generalized Ising model. The CE sums introduced here still suffer from multiple counting of... 

Blum, V., Hart, G.LW., Walorski, M.J., and Zunger, A. (2005) Using genetic algorithms to map first-principles results to model Hamiltonians application to the generalized Ising model for alloys. Phys. Rev. B, 72, 165113. 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

Figures 1 a, 2a to compare with djd l and mj(n — I)a , in Figures 1 b, 2b. The limit and slopes in Figure 1 b are exact but the general pattern of behavior of the other plots is sufficiently similar to give us confidence in the conclusions. (The convergence in three dimensions is more rapid since excluded volume plays a smaller part. Similarly the self-avoiding walk approximation provides a closer fit to the correct behavior of the Ising model.)...

The Ising-type Hamiltonian is usually employed for H-bonded ferroelectrics. In the more general quantum-mechanical approach (dynamic Ising model [2,3]) it has a form ... 

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

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