Ising models

The energy 7i(S) of a given configuration of N spins is made up of two parts (1) a contribution that arises solely from the inter-spin molecular forces (= Hst s), and (2) a contribution that is due to the interaction between the spins and any external magnetic fields (= Since 5, is effectively the magnetic moment 

Once HsoS and Hs h are both specified, the basic problem is to once again find a way to compute the sum-over-states appearing in the expression defining the partition function (equation 7.2)  

It is important to understand that critical behavior can only exist in the thermodynamic limit that is, only in the limit as the size of the system N — = oo. Were we to examine the analytical behavior of any observables (internal energy, specific heat, etc) for a finite system, we would generally find no evidence of any phase transitions. Since, on physical grounds, we expect the free energy to be proportional to the size of the system, we can compute the free energy per site f H, T) (compare to equation 7.3) 

From this we can calculate any observables of interest. For example, we can immediately find the internal energy per site, u H,T), specific heat per site, c H,T) and magnetization, M[H,T) —  

It is expected to depend oidy on the vector distance 7) between the sites i and j and, away from the critical temperature Tc, to decay exponentially as j f j becomes large i.e. 

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Our discussion of solids and alloys is mainly confined to the Ising model and to systems that are isomorphic to it. This model considers a periodic lattice of N sites of any given symmetry in which a spin variable. S j = 1 is associated with each site and interactions between sites are confined only to those between nearest neighbours. The total potential energy of interaction... 

An alternative fomuilation of the nearest-neighbour Ising model is to consider the number of up f T land down [i] spins, the numbers of nearest-neighbour pairs of spins IT 11- U fl- IT Hand their distribution over the lattice sites. Not all of the spin densities are independent since... 

The Ising model is isomorphic with the lattice gas and with the nearest-neighbour model for a binary alloy, enabling the solution for one to be transcribed into solutions for the others. The tlnee problems are thus essentially one and the same problem, which emphasizes the importance of the Ising model in developing our understanding not only of ferromagnets but other systems as well. 

The relationship between tlie lattice gas and the Ising model follows from the observation that the cell occupation number... 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

Our discussion shows that the Ising model, lattice gas and binary alloy are related and present one and the same statistical mechanical problem. The solution to one provides, by means of the transcription tables, the solution to the others. Flistorically, however, they were developed independently before the analogy between the models was recognized. 

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. 

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

Lee T D and Yang C N 1952 Statistical theory of equations of state and phase transitions II. Lattice gas and Ising models Phys. Rev. 87 410... 

In 1925 Ising [14] suggested (but solved only for the relatively trivial ease of one dunension) a lattiee model for magnetism m solids that has proved to have applieability to a wide variety of otiier, but similar, situations. The mathematieal solutions, or rather attempts at solution, have made the Ising model one of tlie most famous problems in elassieal statistieal meehanies. 

The standard analytic treatment of the Ising model is due to Landau (1937). Here we follow the presentation by Landau and Lifschitz [H], which casts the problem in temis of the order-disorder solid, but this is substantially the same as the magnetic problem if the vectors are replaced by scalars (as the Ising model assumes). The themiodynamic... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

The classical treatment of the Ising model makes no distinction between systems of different dimensionality, so, if it fails so badly for d= 2, one might have expected that it would also fail for [Pg.644]

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

A2.5.7 THE CURRENT STATUS OF THE ISING MODEL THEORY AND EXPERIMENT... 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

We have seen in previous sections that the two-dimensional Ising model yields a syimnetrical heat capacity curve tliat is divergent, but with no discontinuity, and that the experimental heat capacity at the k-transition of helium is finite without a discontinuity. Thus, according to the Elirenfest-Pippard criterion these transitions might be called third-order. 

Binder K 1981 Finite size scaling analysis of Ising-model block distribution-functions Z. Phys. B. Oondens. Matter. 43 119-40... 

Figure C2.10.1. Potential dependence of the scattering intensity of tire (1,0) reflection measured in situ from Ag (100)/0.05 M NaBr after a background correction (dots). The solid line represents tire fit of tire experimental data witli a two dimensional Ising model witli a critical exponent of 1/8. Model stmctures derived from tire experiments are depicted in tire insets for potentials below (left) and above (right) tire critical potential (from [15]).

The Ag (100) surface is of special scientific interest, since it reveals an order-disorder phase transition which is predicted to be second order, similar to tire two dimensional Ising model in magnetism [37]. In fact, tire steep intensity increase observed for potentials positive to - 0.76 V against Ag/AgCl for tire (1,0) reflection, which is forbidden by symmetry for tire clean Ag(lOO) surface, can be associated witli tire development of an ordered (V2 x V2)R45°-Br lattice, where tire bromine is located in tire fourfold hollow sites of tire underlying fee (100) surface tills stmcture is depicted in tlie lower right inset in figure C2.10.1 [15]. 

Suppose now that the sites are not independent, but that addition of a second (and subsequent) ligand next to a previously bound one (characterized by an equilibrium constant K ) is easier than the addition of the first ligand. In the case of a linear receptor B, the problem is fonnally equivalent to the one-dimensional Ising model of ferromagnetism, and neglecting end effects, one has [M] ... 

Docherty, R., Clydesdale, G., Roberts, K.J. and Bennema, P., 1991. Application of the Bravais-Freidel-Donnay-Harker attachment energy and Ising models to predicting and understanding the morphology of molecular crystals. Journal of Physics D Applied Physics, 24, 89-99. 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

In the light of the above questions, it is tempting to refer to the results emerging from numerous theoretical and computer simulation studies [40,41,85-88,129-131] of the random field Ising model, and we shall do so, but only after completing the present discussion. 

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