Mean-field theory short range

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

The mean-field theory can also be applied to short range interactions. 

We note that even short-range interactions may, however, allow a mean-field scenario, if the system has a tricritical point, where three phases are in equilibrium. A well-known example is the 3He-4He system, where a line of critical points of the fluid-superfluid transition meets the coexistence curve of the 3He-4He liquid-liquid transition at its critical point [33]. In D = 3, tricriticality implies that mean-field theory is exact [11], independently from the range of interactions. Such a mechanism is quite natural in ternary systems. For one or two components it would require a further line of hidden phase transitions that meets the coexistence curve at or near its critical point. 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

In the intermediate domain of values for the parameters, an exact solution requires the specific inspection of each configuration of the system. It is obvious that such an exact theoretical analysis is impossible, and that it is necessary to dispose of credible procedures for numerical simulation as probes to test the validity of the various inevitable approximations. We summarize, in Section IV.B.l below, the mean-field theories currently used for random binary alloys, and we establish the formalism for them in order to discuss better approximations to the experimental observations. In Section IV.B.2, we apply these theories to the physical systems of our interest 2D excitons in layered crystals, with examples of triplet excitons in the well-known binary system of an isotopically mixed crystal of naphthalene, currently denoted as Nds-Nha. After discussing the drawbacks of treating short-range coulombic excitons in the mean-field scheme at all concentrations (in contrast with the retarded interactions discussed in Section IV.A, which are perfectly adapted to the mean-field treatment), we propose a theory for treating all concentrations, in the scheme of the molecular CPA (MCPA) method using a cell... 

The second-order magnetic transition is usually preceded by local (short-range) magnetic order. As mentioned in sect. 3 the mean-field theory, which does not take into account the magnetic fluctuations, predicts a jump of AC, = 12.5 J mol K at Tq for a spin 1/2. In this region there are some examples of a practically pure second-... 

Within the mean-field theory there are no spontaneous elastic deformations since any deformation increases the free energy. However, when a nematic LC is subject to interactions with the confining walls the homogeneous order can be perturbed. On the microscopic level, the molecules of the walls and of the liquid crystal attract each other via a short-range van der Waals interaction. In the macroscopic description this is modeled with a contact quadruple-quadruple interaction, known as the Rapini—Papoular model [32,33], which to the lowest order reads... 

A copolymer-homopolymer mixture provides us with a variety of domain morphology since both macro- and micro-phase separations take place simultaneously. (Equi-hbrium property has been studied extensively by a mean-field theory [1].) We consider a mixture of A-B diblock copolymer and C homopolymer assuming a short-range repulsive interaction between A and B monomers and B and C monomers. One may expect a multiple domain structure in a sense that microphase separated domains are developed in a macrophase separated domain [2, 3]. It is also expected that formation of vesicles is also possible... 

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