Mean field model order parameter, temperature dependence

The Bean-Rodbell model (Bean Rodbell, 1962) adds a phenomenological description of magneto-volume effects to the classical molecular mean-field model of Weiss. The dependence of exchange interaction on interatomic spacing is then considered, taking into account three new parameters fi, which corresponds to the dependence of ordering temperature on volume, and also the volume compressibility, K and thermal expansion ai-The formulation behind the model is as follows ... 

The three adjustable parameters are determined, A/kB = 90 K, Jo/kB = -36 K, and J /kB = 125 K, so as to reproduce the spin-crossover transition temperature Tc = 48 K, the virtual Jahn-Teller transition temperature rJT = 6 = 26 K, and the effective LS-HS gap in the LS phase Acff/kB = 340 K. (Note Aeff is approximated by A + 2Jx in this mean-field model.) This choice of model parameters gives a phase sequence from the LS to HS with increasing temperature, corresponding to the arrow path in Fig. 7. Temperature dependence of thermodynamic quantities (Fig. 8) is calculated along the path indicated by the arrow in Fig. 7, where the discontinuities arising from the first-order spin-crossover transition are recognized Ap0 = 0.99, AH = 0.64 kJ mol-1, and AS = 13.3 J K-1 mol-1 These theoretical... 

Fig. 6.22 Mean field model of the nematic phase temperature dependence of the order parameter. The two branches correspond to stable nematic phase with positive order parameter (solid line), and unstable phase with negative order parameter (dash line). Order parameter discontinuity at 5 = 0.429 indicates the first order N-I transition...

Fig. 28. Elastic and inelastic data characterizing the antiferromagnetic phase transition in U2Zn,7 at r,., = 9.7 K. The top fraine shows the development of the order parameter at (1,0,2) compared to an S = mean-field model. The inset shows a double logarithmic plot of the reduced squared staggered magnetization versus reduced temperature in the critical regime. The bottom frame shows the temperature dependence of the quasielastic magnetic neutron scattering at an energy transfer of fico = 0.75 meV. (From Bioholm 1988.)...

In the isotropic phase, ry = r = cr = 0 in the nematic phase, ry 0, r = <7 = 0 in the smectic-A phase ry 0, r 0, a 0. For perfect order all three tend to unity. Part of the task of molecular theory is, of course, to calculate the temperature dependence of these order parameters. Again we point out that although the three quantities of Eq. [11] are sufficient to parametrize simple mean field models, a good approximation to the true distribution function, /(cos, 2) requires many terms in Eq. [7]. 

Figure 27 presents values of W(T) calculated by Dietl et al. (2001a) in comparison to the experimental data of Shono et al. (2000). Furthermore, in order to establish the sensitivity of the theoretical results to the parameter values, the results calculated for a value of Xc that is 1.8 times larger are included as well. The computed value for low temperatures, W = 1.1 /xm, compares favorably with the experimental finding, W = 1.5 fim. However, the model predicts a much weaker temperature dependence of IV than observed experimentally, which Dietl et al. (2001a) link to critical fluctuations, disregarded in the mean-field approach. 

For polymers, x is usually defined on a per monomer basis or on the basis of a reference volume of order one monomer in size. However, x is usually not computed from formulas for van der Waals interactions, but is adjusted to obtain the best agreement between the Flory-Huggins theory and experimental data on the scattering or phase behavior of mixtures (Balsara 1996). In this fitting process, inaccuracies and ambiguities in the lattice model, as well as in the mean-field approximations used to obtain Eq. (2-28), are papered over, and contributions to the free energy from sources other than simple van der Waals interactions get lumped into the x parameter. The temperature dependences of x for polymeric mixtures are often fit to... 

Phase transitions are defined thermodynamically. However, to model them, we must turn to theories that describe the ordering in the system. This is often done approximately, using the average order parameter (here we assume one will suffice to describe the transition) within a so-called mean field theory. The choice of appropriate order parameter is discussed in the next section. The order parameter for a system is a function of the thermodynamic state of the system (often temperature alone is varied) and is uniform throughout the system and, at equilibrium, is not time dependent. A mean field theory is the simplest approximate model for the dependence of the order parameter on temperature within a phase, as well as for the change in order parameter and thermodynamic properties at a phase transition. Mean field theories date back to when van der Waals introduced his equation of state for the liquid-gas transition. 

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