Molecular mean field model

The general representation of the molecular mean-field model is then... 

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 ... 

Fig. 5. a) Isothermal magnetization versus applied magnetic field, from 200 to 400 K, at a 1 K temperature step and 100 Oe field step and b) Isomagnetic H/T versus 1 /T plot, of data from the molecular mean-field model, from M = 5 emu/g (dark blue line) to M = 75 emu/g (orange line), with a 5 emu/g step. 

Molecular mean-field models of capillary d mamics... 

Mean-field models are obviously approximations whose accuracy must be determined so scientists can know to what degree they can be "trusted". For electronic structures of atoms and molecules, they require quite substantial corrections to bring them into line with experimental fact. Electrons in atoms and molecules undergo dynamical motions in which their coulomb repulsions cause them to "avoid" one another at every instant of time, not only in the average-repulsion manner that the mean-field models embody. The inclusion of instantaneous spatial correlations among electrons is necessary to achieve a more accurate description of atomic and molecular electronic structure. 

In summary, the dynamical interactions among electrons give rise to instantaneous spatial correlations that must be handled to arrive at an accurate picture of atomic and molecular structure. The simple, single-configuration picture provided by the mean-field model is a useful starting point, but improvements are often needed. 

Gerber, R.B. and Ratner, M.A. (1988a). Mean-field models for molecular states and dynamics New developments, J. Phys. Chem. 92, 3252-3260. 

R.B. Gerber, V. Buch and M.A. Ratner, Time-dependent self-consistent field approximation for intramolecular energy transfer. I. Formulation and application to dissociation of van der Waals molecules, J. Chem. Phys., 77 (1982), 3022 M.A. Ratner and R.B. Gerber, Excited vibrational states of polyatomic molcecules the semiclassical self-consistent field approach, J. Phys. Chem., 90 (1986) 20 R.B. Gerber and M.A. Ratner, Mean-field models for molecular states and dynamics new developments, J. Phys. Chem., 92 (1988) 3252 ... 

Khelashvili, G., Pandit, S.A., Scott, H.L. Self-consistent mean-field model based on molecular dynamics Application to lipid-cholesterol bilayers. J. Chem. Phys. 2005,123, 034910. 

Fig. 2. Orientational probability distributions of the molecular axes in (a) a-nitrogen and (b) y-nitrogen. Contours of constant probability for the molecule in the origin, calculated in the mean field model, are plotted as functions of the polar angles (0, ) with respect to the crystal axes (Fig. 1). The angle 0 increases linearly with the radius of the plots from 0 (in the center) to tt72 (at the boundary) d> is the phase angle.

The influence of the molecular weight distribution of diblock copolymers on their segregating properties is considered in Sect. 4.2.4. It describes first experimental study [255] on a bimodal mixture of short symmetric and long asymmetric copolymers added to a polymeric matrix. Shorter copolymers were found to adsorb preferentially at the homopolymer interfaces in accord with brush formation observed from a solvent host matrix [274-276]. The mean field model is able [255] to predict the segregation isotherm of the bimodal mixture of copolymers, basing on single component adsorption data [254]. 

In fact the molecular field fluctuates in time and depends on the instantaneous magnetic moments fij. A mean field model abstracts from this fluctuating character and assumes a thermal average in the form... 

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