The molecular field approximation

In the case when k 0 the correlation functions pm, rn 2 are no longer multiplicative but have angular dependence, e.g., 52 depends on ri, r2 and the angle between them 32 = 92 ri,r2 d). However, the molecular field approximation neglects this fact and again yields (5.2.9) which is now only an approximate solution. 

The conclusion can be drawn that the traditional approach [30] corresponds to a zero-level approximation, i.e., a cut-off of the hierarchy at m = 1 thus linearising (5.2.9) for gi and neglecting all integral terms in wi (equation (5.2.7)) which describe competition of several B s for some reactant A. It is not surprising that the standiird critical exponent a = 1 obtained from equations (5.2.5) and (5.2.9) is independent of the relative diffusion coefficient, since k enters into equation (5.2.6) only for m 2. Therefore, in an approximation at the next level we have to consider the equation for the correlation function p2 at least. This level of approximation could be acceptable (at least to reproduce equation (5.2.1)) if the reaction for 7 0 remains weakly non-ideal (the g is nearly multiplicative). 

Equation (5.2.1) is derived for k = 1. There is a reason to think that this single point k = 1 only is the range of the validity of the critical exponent 

Therefore, the weakness of the fluctuation effects in the A -f B —B reaction as compared to the case of A + B - 0 (Section 5.1) does not permit us to restrict ourselves to the radial correlations between A s and B s subtle effects leading to equation (5.2.1) arise due to the angular correlations of reactants B in the vicinity of a single A. The demonstrated peculiarity of the A + B — B reaction requires an adequate specified mathematical approach. 

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In applying this approach to the equation of state of the hard-sphere fluid [57], it was found that the molecular-field approximation... 

In a magnetic compound we have to consider exchange interactions in the molecular field approximation, the Curie law takes the Curie Weiss form... 

Note that the introduction of the correlation functions gm in (5.2.4) instead of (m + l)-point densities p >m in fact enabled us to reduce the number of variables. For instance, the molecular field approximation, g2 = g (n )g (r2), corresponds to that for superposition approximation (equation (2.3.55)) for pii2 whereas, in its turn, equation (5.2.13) for <73 corresponds to the higher-order superposition approximation (equation (2.3.56)) for When substituting (5.2.13) into (5.2.12) with m = 2, we obtain an exact equation for g with... 

Early theoretical treatments of liquid crystals were not surprisingly based on the molecular field approximation. However, it is neccessary to make assumptions about the pair potential employed in the calculation and it is impossible to know whether the predictions of a particular model really arise from the pair potential employed or whether they arise, at least in part, from the deficiencies of the basic approximation employed. The general problem is so complex that a better mathematical treatment of the molecular interactions in a liquid crystal is out of the question. However, with the introduction of ever more powerful computers, it has become possible to carry out meaningful numerical simulations of model liquid crystals. 

The molecular-field approximation suggests the following relation for the field and temperature dependence of the sublattice magnetization ... 

Now we recall the discussion on the magnetic order in a ferromagnetic spin system through the molecular-field approximation as is described in the usual textbook [10], In the spin system of S = the magnetization M at the temperature T is described as... 

The NMR experiment was performed on nB in C11B2O4. The spectrum in the commensurate phase is discussed. The magnetic moment at the Cu(A) site is estimated to be 0.45 /tg, which is almost 50% of the moment derived from the neutron diffraction experiment. The magnetic moment at the Cu(B) sites is absent in the commensurate phase. The asymmetric nature of the spectral pattern is not understood so far. A phenomenological understanding of the commensurate to incommensurate transition is discussed on the basis of the molecular-field approximation. More precise discussion of the commensurate phase and also of the incommensurate phase will be presented in the near future. 

To understand the behavior above the Curie temperature, we will use the molecular field approximation. In the presence of an applied external field B, we have to add the term due to the N spins interacting with the field,... 

This Hamiltonian cannot be solved exactly because the first term is nonlinear. We can, however, use the molecular field approximation, which consists of taking the mean value (Sp over the j spins ... 

Fig. 31 Temperature behavior of the energy levels of the cluster in the molecular field approximation for / < 0, > 0, y =...

This way, the exchange interactions between adjacent units are treated in the molecular field approximation. It appears that these interactions are anti-ferromagnetic and weak compared to intercluster ones. On the other hand, local distortions which stabilize the Sz=0 component of Ni(II) ions strongly reinforce... 

The molecular field approximation assumes that the magnetoactive entity is under the influence of the external magnetic field as well as the internal... 

Aburto, S., Jim6nez, M., Marquina, M. L. Valenzuela, R. (1982). The molecular field approximation in Ni-Zn ferrites. In Ferrites Proceedings of the Third International Conference. Eds H. Watanabe, S. lida and M. Sugimoto. Center for Academic Publications, Tokyo, pp. 188-91. 

For systems containing localized magnetic moments, the thermopower has not been theoretically investigated in such detail as the resistivity. An expression for the thermopower of ferromagnetic materials with localized moments has been obtained by Kasuya (1959) in both the molecular field approximation and the spin wave approximation. In the former case, Kasuya used a molecular field approximation to obtain the energy spectrum of the conduction electrons and the localized magnetic moments. In addition he assumed that the spin-flip transition probabilities for scattering of electrons by local moments dominate the non-spin-flip transition probabilities. 

Stoner and Wohlfarth were able to derive a self-consistent equation for the spontaneous magnetization Mo in the molecular field approximation (MFA) using the model described above. Mo is then written in terms of the average occupation numbers (n ) for electrons of spin cr ... 

Sketching a comprehensive picture of the topological constraints in the molecular field approximation, we observe two characteristic features ... 

It should be emphasized that the quantitative interpretation of all these methods relies on the molecular field approximation, ie., it neglects any fluctuation effects. The same remark holds for the treatment of the ordered phase, the phase diagram obtained using exchange and quadrupolar interactions (Koetzler et al. 1979, Morin and Schmitt 1983). 

During recent decades the molecular theory of flexoelectricity in nematic liquid crystals was developed further by various authors. " In particular, explicit expressions for the flexocoefiicients were obtained using the molecular-field approximation taking into account both steric repulsion and attraction between the molecules of polar shape. The influence of dipole-dipole correlations and molecular flexibility was later considered. Recently flexoelectric coefficients have been calculated numerically using the mean-field theory based on a simple surface intermolecular interaction model. This approach allows us to take into consideration the real molecular shape and to evaluate the flexocoefiicients for mesogenic molecules of different structures including dimers with flexible spacers. 

This chapter is arranged as follows. In Section 1.2 we consider in more detail the dipolar and quadrupolar mechanisms of flexoelectricity, and in Section 1.3 we derive the general expressions for the flexocoefficients in terms of the direct pair correlation function. These results are used in Section 1.4 to obtain approximate expressions for the flexocoefficients in the molecular-field approximation taking into account both intermolecular repulsion and attraction. In that section we also consider the dependence of the flexocoefficients on the absolute value of the molecular dipole and on the orientation of the electric dipole with respect to the molecular long axes and the steric dipole. In Section 1.5 the effect of dipole-dipole correlations is analysed and in Section 1.6 we discuss the mean-field theory of flexoelectricity, which allows us to account for the real molecular shape. 

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