Molecular-field approximation

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

In the following discussion, we treat the surface effect on the basis of the Weiss field (molecular field) approximation (17-19), assuming no relaxation (fluctuation of the electron spins). In the treatment, the reduced magnetization m (magnetization at a certain temperature divided by that at OK) of the surface ferric ions at temperature T K is described by... 

The Stoner model retains only the first term, proportional to (molecular field approximation which leads to terms such as (26) in the Hamiltonian). 

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

In order to take into account the complex structure of the valence band, Dietl et al. (2000, 2001c) and Abolfath et al. (2001) have computed hole energies by diagonalizing the 6 x 6 k p Luttinger matrix together with the p-d exchange contribution taken in the virtual crystal and molecular field approximation,... 

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 Bragg-Williams molecular field approximation assumes a binomial... 

Various approximation of this model (e.g. those of molecular splitting, pseudo-chemical, chaotic and molecular field approximations) were reported and analyzed by Tovbin and Fedyanin [88, 89]. In particular, the pseudochemical approximation for chemisorption kinetics is of the form... 

Our quantum chemical approach has been applied to explain the thermodynamics of the TKHS-like materials on the level of simple mean (molecular) field approximation [1,4] which, however, is not well adapted to provide for quantitative estimations of the critical temperature of phase transitions. To examine the... 

Exact solutions do not exist for the exchange Hamiltonian of two-dimensional layers and three-dimensional networks. These situations can, however, be described using approximation methods such as molecular field approximation and high-temperature series expansion. 

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 ... 

Show that in the present molecular field approximation, the preceding derivative is expressed as... 

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. 

Recent theoretical treatments of the soft-mode behaviour include a detailed study by Onodera using classical mechanics, and a theory of hydrogen-bond mechanics, including tunnelling effects, by Stamenkovic and Novakovic. ° Onodera assumes a quartic potential function for his individual oscillators, with a bilinear interaction which reduces to c x, where x is the displacement, under the Weiss-molecular-field approximation. The model is soluble without further approximation (in series of elliptic functions), yielding the temperature variation of frequency and damping. If the quartic potential has a central hump larger than kTc,... 

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

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