Mean-field model magnetism

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

The temperature dependence of the magnetic hyperfine splitting in spectra of interacting nanoparticles may be described by a mean field model [75-77]. In this model it is assumed that the magnetic energy of a particle, p, with volume V and magnetic anisotropy constant K, and which interacts with its neighbor particles, q, can be written... 

The results of the asymmetry parameter r as a function of nuclear charge at o 0 is shown in Fig. 1. This curve of the asymmetry parameter shown is completely analogous to curves representing the behavior of magnetization as a function of the temperature in mean field models of ferromagnetic systems [41] as shown in Fig. 2. 

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... 

For a general introduction to the field of NMR in magnetically ordered compounds with non-S state lanthanide ions, we refer the reader to Taylor (1971) or McCausland and Mackenzie (1980), where the interplay of the exchange and crystal-field interactions has been analysed. Generally, for cubic systems like RAI2 at least a three-parameter mean-field model is adopted, based on a single-ion Hamiltonian comprising a crystal field of cubic symmetry [CEF parameters W, x Barnes (1979)] and an isotropic molecular field constant (A) ... 

The above results pertain mainly to amorphous alloys rich in 3d metal. Magnetic ordering temperatures for amorphous alloys of lower 3d metal concentrations are included in fig. 39. In the mean field model the magnetic ordering temperatures are given by the expression (Heitnan et al., 1976) ... 

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. 

The mean-field model also allows the study of mixed-state transitions, by considering a proportion of phases (high and low magnetization) within the metastabUity region. Magnetization curves are shown in the inset of Fig. 19, for A3 = 2 Oe (emu/g) , corresponding to a critical field lOT. The mixed-phase temperature region is from 328 to... 

In accordance with the described mean-field model, for pure LCs a sharp, weakly first-order I-N transition is typically observed, exhibiting a narrow (only a few mK) two-phase region. Large, externally applied electric or magnetic fields are needed to impose a continuous conversion because the electric or magnetic susceptibility of LCs is low [9-13]. Nevertheless, transition smearing can be observed when strong random fields are introduced to the system, such as the mechanical stresses imposed on LCs confined in random porous media [14]. 

Fig. 84. High-field magnetization curves along the c and a axes at 1.3 K in tetragonal PrCOjSij the dashed line is calculated by using a simple incommensurate mean field model (after Fujii and Shigeoka 1990).

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.)...

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