Neel relaxation ferrofluids

The magnetization dynamics of ferrofluids is characterized by the distinction between Brownian and Neel relaxations. Brownian relaxation refers to the mechanical rotation... 

Here the ratio Xri/xb represents the coupling between the magnetic and mechanical motions arising from the nonseparable namre of the Langevin equations, Lqs. (121) and (122). Thus the correction to the solid-state result imposed by the fluid is once again of the order 10 Hence we may conclude, despite the iionseparability of the equations of motion, that the Neel relaxation time of the ferrofluid particle should still be accurately represented in the IHD and VLD limits by the solid-state relaxation time formulae, Eqs. (87) and (90). Furthermore, Eq. (122) should be closely approximated by the solid-state relaxation equation... 

We may summarize the contents of this chapter in more detail as follows. In Section I we demonstrate how the explicit form of Gilbert s equation describing Neel relaxation may be written down from the gyromagnetic equation and how, in the limit of low damping, this becomes the Landau-Lifshitz equation. Next the application of this equation to ferrofluid relaxation is discussed together with the analogy to dielectric relaxation. 

In a ferrofluid particle where the Neel relaxation mechanism is blocked, orientational changes in M will be due to the rotational motion of the particle only. Consequently the magnetization behaves like that of a bar magnet when placed in a magnetic field. The gyromagnetic or precessional term is not present in its equation of motion. Consequently that equation is Eq. (5.2) with the precessional terms set equal to zero corresponding to rjyM 1 so that now (see Section 1 of [16])... 

These equations govern the Neel relaxation of a single domain ferromagnetic particle. They bear a resemblance to equations (5.26)-(5.28) for the Debye relaxation of ferrofluid particles (with the Neel mechanism blocked) subjected to a weak AC field superimposed on a strong DC magnetic field H. They differ from the ferrofluid equations, however, insofar as they contain processional terms g and... 

We have mentioned that the question posed above was answered in part by Shliomis and Stepanov [9]. They showed that for uniaxial particles, for weak applied magnetic fields, and in the noninertial limit, the equations of motion of the ferrofluid particle incorporating both the internal and the Brownian relaxation processes decouple from each other. Thus the reciprocal of the greatest relaxation time is the sum of the reciprocals of the Neel and Brownian relaxation times of both processes considered independently that is, those of a frozen Neel and a frozen Brownian mechanism In this instance the joint probability of the orientations of the magnetic moment and the particle in the fluid (i.e., the crystallographic axes) is the product of the individual probability distributions of the orientations of the axes and the particle so that the underlying Fokker Planck equation for the joint probability distribution also... 

The Shliomis Stepanov approach [9] to the ferrofluid relaxation problem, which is based on the Fokker Planck equation, has come to be known in the literature on magnetism as the egg model. Yet another treatment has recently been given by Scherer and Matuttis [42] using a generalized Lagrangian formalism however, in the discussion of the applications of their method, they limited themselves to a frozen Neel and a frozen Brownian mechanism, respectively. 

In order to illustrate how precession-aided relaxation effects may manifest themselves in a ferrofluid, it will be useful to briefly summarize the differences in the relaxation behavior for axially symmetric and nonaxially symmetric potentials of the magnetocrystalline anisotropy and apphed held, when the Brownian relaxation mode is frozen. Thus only the solid-state (Neel) mechanism is operative that is, the magnetic moment of the single-domain particle may reorientate only with respect to the crystalline axes. 

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