Ising critical exponents

Gi is determined in terms of the ratio of the Ising and mean-field critical amplitudes of the susceptibility and the exponent l/(y-l). The exponent is about 4 as determined from the Ising critical exponent and Gi is identified as the reduced temperamre of Tx, according to 1 - Tc/Tx. 

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. 

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 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. 

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]).

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

Table 1. Comparison of the critical exponents of NIPA gel with some known Ising systems. The number in the parentheses indicates the error of the corresponding exponent. The row next to the last is obtained directly by using an = —0.05. The last row is the Fisher renormalized results...

As was demonstrated by Kikuchi and Brush [88], using the Ising model as an example, an increase of mo in the expansion in the form secures the monotonic approach of the calculated critical parameters to exact results, except for the critical exponents which cannot be reproduced by algebraic expressions. It is important to note here that the superposition approximation permits exact (or asymptotically exact) solutions to be obtained for models revealing the critical point but not the phase transition. This should be kept in mind when interpreting the results of the bimolecular reaction kinetics obtained using approximate methods. 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

One of the few attempts to tackle the problem of ionic criticality more quantitatively was made by Hafskjold and Stell in 1982 [36], and was later taken up by H0ye and Stell [17, 302, 303]. Based on a comparative analysis of the correlation functions for nonionic and ionic fluids, these authors asserted that the critical point of the RPM is Ising-like. To this end, they argued that the density-density correlation function hpp(r) and the associated direct correlation function cpp(r) obey essentially the same OZ equation and closure as that of a single-component, nonionic fluid. It was assumed that this analogy suffices to ensure that the critical exponents are Ising-like. 

The different values of the exponents can be obtained either by exact calculations (molecular field and 2-D Ising model) or by interpolations of series expansions. Table 1 gives the critical exponents for different models. 

Fig. 21. Log-log plot of Tc(oo)-Tc(D) versus D, for the bond-fluctuation model of a symmetric polymer mixture with NA=NB=N=32. For small D, the straight line corresponds to a shift Tc(°o)-Tc(D)oc1/d, while the second straight line for larger D shows the result Tc(°o)-Tc(D)ocD"1/v, with v=0.63 being the critical exponent of the three-dimensional Ising model correlation length [229,230]. From Rouault et al. [55]...

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

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