Ising lattice

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

Onsager L and Kaufman B 1949 Orystal statistics III. Short range order in a binary Ising lattice Phys. Rev. 65 1244... 

Yang 0 N 1952 The spontaneous magnetization of a two-dimensional Ising lattice Phys. Rev. 85 809 (87 404)... 

Moreover, some uncertainty was expressed about the applicability to fluids of exponents obtained for tlie Ising lattice. Here there seemed to be a serious discrepancy between tlieory and experiment, only cleared up by later and better experiments. By hindsight one should have realized that long-range fluctuations should be independent of the presence or absence of a lattice. 

Table 7.1. Ratio of the critical temperature calculated for an f.c.c. Ising lattice by different methods, assuming only constant nearest-nei bour interactions and using MC as the standard (after Kikuchi 1977)...

Figure 7.18. Comparison of experimental and predicted phase equilibria in the system CaC03-MgC03 using CVM in the tetrahedron approximation for a trigonally distorted f.c.c. Ising lattice. Semi-quantitative agreement is achieved for the calcite-dolomite segment but the Mg-rich side of the diagram indicates the need to include a more complex model (Burton and Kikuchi 1984b).

Another model which includes interaction and for which partial results are available on the decay of initial correlations is that of the one dimensional time-dependent Ising model. This model was first suggested by Glauber,18 and analyzed by him for one-dimensional Ising lattices. Let us consider a one-dimensional lattice, each of whose sites contain a spin. The spin on site,/ will be denoted by s/t) where Sj(t) can take on values + 1, and transitions are made randomly between the two states due to interactions with an external heat reservoir. The state of the system is specified by the spin vector s(t) = (..., s- f), s0(t), Ji(0>---)- A- full description of the system is provided by the probability P(s t), but of more immediate interest are the reduced probabilities... 

For a binary Ising lattice, we introduced a nonrandom factor that was observed from simulation to have a linear relation with composition. The characteristic parameter of the linear relation was found by combining a series expansion and the infinite dilution properties. On this basis, an accurate expression for the Helmholtz energy of mixing... 

For a simple cubic Ising lattice with a total Nr sites is occupied by K types of molecules, each molecule occupies one site, the constrains i Ni = Nr and 2Nu + Nij = zN) are satisfied, where N,- is the... 

The nonrandom factor /,y characterizes the degree of deviation from ideal mixing, and its numerical value can be estimated directly by simulation. It is shown that l fjy has a fairly well linear relation with mole fraction. For binary Ising lattice, the expression of/,y was obtained by combining simulation and statistical mechanics (Yan et al., 2004). For the multicomponent Ising lattice, a generalized expression has been proposed as... 

Figure 6 Coexistence curves of binary Ising lattice (Yan et al., 2004).

Figure 7 Internal energy of mixing for a ternary Ising lattice (Yang et at, 2006ca).

The residual Helmholtz energy due to the dissociation of polymer chains in pure state and the association of polymer chains in mixture state can be calculate by Equation (5). The pair correlation functions of component i in the corresponding Ising lattice system are calculated by gf = 1 / fyfij (Liu al., 2007). The residual... 

Figure 8. Polynucleotide replication as multitype branching process is compared with spin lattice models (A) One-dimensional model is based on generalized one-dimensional Ising lattice Every spin is assumed to exist in n different states corresponding to n different polynucleotide sequences. Genealogy of branching process is considered as analog of particular one-dimensional arrary of spins.

Figure 9. Critical temperture of order-disorder transition in two-dimensional Ising lattice as function of coupling constants and Jy. Note that transition temperature approaches...

Ising model this case corresponds to the limit Jx O, where the two-dimensional Ising lattice degenerates to the one-dimensional spin system. We are dealing with a degenerate phase transition ... 

Thus many investigations support an idea about ordered regions in water. Since the water-gaseous solution is characterized by the temperature of structurization Tc discussed above, we can hypothesize that the water network features an order parameter. As a first cmde approximation, in Ref. 359 an ordering of OH groups of water molecules was treated in the framework of a simplified model in which water molecules were located in knots of the Ising lattice. The model was based on the Blinc s formalism stated briefly in... 

Recently, Binder et al. [118] considered the Ising lattice of a binary atomic (N=l) mixture confined in a very thin film by antisymmetric surfaces each attracting a different component. It was shown that the segregation of each blend component to opposite surfaces may create antisymmetric (with respect to the center of the film z=D/2) profiles ( >(z) even for temperatures above critical point T>TC, where flat profiles are expected when external interfaces are neglected. Such antisymmetric profiles would not be distinguished in experiments (with limited depth resolution) from coexisting profiles described by a hyperbolic tangent. 

Fisher, M.E. (1967). Critial temperatures of anisotropic ising lattices. II. General upper bounds. Phys. Rev., 162, 480-5. 

For a theoretical discussion of these various phase transitions associated with surfaces and interfaces, the proper thermodynamic functions need to be characterized. In order to be specific, we consider an Ising lattice model (Binder, 1983 fig. 8) in which each site i of a d-dimensional cubic lattice carries an Ising spin = 1. Let us assume a thin film-geometry, where a film of thickness L — N )a (i.e., we have N atomic layers and a is... 

In fig. 60 for the case of strong substrate attraction all layering transition lines accumulate at the point T = 0, H = 0 (i.e., ix = ixC0ltx). However, this is only true due to the specific assumption of a short range force arising from the surface, which in the Ising lattice gas framework leads to a surface... 

The intermolecular potential, Eq. III.7, is not a necessary condition for Eq. III. 13. The necessary and sufficient condition is that N-1 In ZQ is a function of kTv jX only. In other words, the condition is that the ratio of the potential energy (> 0) in one configuration with the specific volume to the potential energy in another similar configuration with, the specific volume always equals (r2M) - Under this weaker condition, where the additivity of the intermolecular potential is not assumed, most relations in the present section hold. Namely, if a phase transition between crystalline and amorphous states occurs, then the (l-fs 1)th power of the transition temperature is proportional to the pressure and the latent heat is proportional to the transition temperature. In the next section we treat an Ising-Lattice model with the aforementioned similarity condition. 

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