Ferromagnet classic

The transition from a ferromagnetic to a paramagnetic state is normally considered to be a classic second-order phase transition that is, there are no discontinuous changes in volume V or entropy S, but there are discontinuous changes in the volumetric thermal expansion compressibility k, and specific heat Cp. The relation among the variables changing at the transition is given by the Ehrenfest relations. 

We will present the equation of motion for a classical spin (the magnetic moment of a ferromagnetic single-domain particle) in the context of the theory of stochastic processes. The basic Langevin equation is the stochastic Landau-Lifshitz(-Gilbert) equation [5,45]. More details on this subject and various techniques to solve this equation can be found in the reviews by Coffey et al. [46] and Garcia-Palacios [8]. 

A plot of tlie reciprocal of the measured susceptibility 1 /Zmoi vs. r is a straight line with slope 1/C, and which crosses the abscissa at T = 0 (Fig. 19.6). For 0 = 0 the equation is simplified to the classic Curie law = C/T. Generally, values of 0 0 are found when cooperative effects arise at low temperatures (ferro-, ferri- or anti ferromagnetism). The straight line then has to be extrapolated from high to low temperatures (dashed lines in Fig. 19.6). 

It is well known that ferromagnetism favors reentrant behaviour (see subsection 1.3). The formation of such secondary phases is supported by nonstoichiometry. Therefore the chemical characterization of the sample is of prime importance. However, due to the presence of the two light elements B and C the various classical characterization techniques as chemical analysis, intensity analysis of x-ray or neutron diffraction, transition electron... 

There exist systems, namely systems which undergo processes involving hysteresis (plastic deformation or ferromagnetism, lor example) for which no equation of slate can he indicated. Although Ihe laws of thermodynamics may apply to such systems, the rigorous results of classical thermodynamics arc not applicable because the science of thermodynamics is developed on the assumption of the existence of the single-valued function. 

Much of our understanding of critical phenomena is based on the Landau-Ginzburg model of a ferromagnet. This model concentrates on the local magnetization, represented by an m-component vector field Sa (r), a = 1,..., m. often called a classical spin field . The interaction of the spin field is described by the Landau-Ginzburg Hamiltonian... 

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