Order-disorder theory ferromagnetism

ORDER-DISORDER THEORY AND APPLICATIONS. Phase transitions in binary liquid solutions, gas condensations, order-disorder transitions in alloys, ferromagnetism, antiferromagnetism, ferroelectncity, anti-ferroelectricity, localized absorptions, helix-coil transitions in biological polymers and the one-dimensional growth of linear colloidal aggregates are all examples of transitions between an ordered and a disordered state. 

More generally, the dynamic behavior of domain walls in random media under the influence of a periodic external field gives rise to hysteresis cycles of different shape depending on various external parameters. According to a recent theory of Nattermann et al. [54] on disordered ferroic (ferromagnetic or fe) materials, the polarization, P, is expected to display a number of different features as a function of T, frequency, / = iv/2tt, and probing ac field amplitude, E0. They are described by a series of dynamical phase transitions, whose order parameter Q = uj/2h) Pdt reflects the shape of the P vs. E loop. When increasing the ac... 

With w 7 0 the problem is not simple. It can in fact be shown to be essentially equivalent to certain problems in the theory of the order-disorder transition in alloys, regular solutions, ferromagnetism, etc. 

Nearly all experimental coexistence curves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetic materials, are far from parabolic, and more nearly cubic, even far below the critical temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Verschaffelt (1900), from a careful analysis of data (pressure-volume and densities) on isopentane, concluded that the best fit was with p = 0.34 and 5 = 4.26, far from the classical values. Van Laar apparently rejected this conclusion, believing that, at least very close to the critical temperature, the coexistence curve must become parabolic. Even earlier, van der Waals, who had derived a classical theory of capillarity with a surface-tension exponent of 3/2, found (1893)... 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

Landau developed a theory for second-order phase transition [16], such as from diamagnetic phase to ferromagnetic phase, in which the order parameter increases continuously from zero as the temperature is decreased across the transition temperature Tc from the high temperature disordered phase to the low temperature ordered phase. For a temperature near the order is very small. The free energy of the system can be expanded in terms of the order parameter. 

Most modem theories of phase transitions are based on Landau theory. Landau introduced the concept of an order parameter, a thermodynamic quantity that vanishes in one phase (the disordered phase) and is nonzero and generally nonunique in the other phase (the ordered phase). For the ferromagnetic... 

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