Neel relaxation field

Another process responsible for a fluctuation of the local magnetic field is Neel relaxation. It corresponds to the flip of the crystal magnetization vector from one easy direction of anisotropy to another. The correlation time of this... 

Fig. 3. Illustration of the origin of proton nuclear magnetic relaxation induced by a super-paramagnetic crystal. The water molecule (symbolized by a bee) experiences a magnetic field which fluctuates because of the translational diffusion and because of Neel relaxation. The bottom curve represents a typical time evolution of this field.

Another effect of increasing the anisotropy is the lengthening of Neel relaxation time, which generally dominates xq. Since the zero-field relaxation rate is proportional to xq, this lengthening will increase the low field relax-ivity, a consequence opposed to the slackening due to the hidden transitions. 

In Section II, Gilbert s equation describing the Neel relaxation is augmented by a random field term, representing thermal fluctuations. The underlying Fokker-Planck equation is then constructed from this augmented equation. The time constant in this equation is the Neel relaxation time... 

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... 

Here we note that this result differs from that of Raikher and Shliomis [17] (Eq. (42) of their paper) where the relaxation time used in the denominator is simply the Neel relaxation time Tjv(o-). The relaxation time used here is Tj (ct), the (perpendicular field-dependent) relaxation time in the presence of uniaxial anisotropy which from Eq. (7.58) to 0(cr) from the binomial expansion is (see Eq. 6.78)... 

Fig. 7 Schematic representation of the evolution of the magnetic properties of magnetic nanoparticles as a function of their volume and of the models suitable to describe them. The label (1) illustrates that the maximum magnetic field for which the linear response theory (Neel relaxation model) is valid decreases with increasing volume. The label (2) is the domain where incoherent reversal modes occur so Stoner-Wohlfarth model based theories are not valid anymore. The label (3) shows a plateau in the volume dependence of the coercive field. Reprinted with permission from Ref 41. Copyright 2011, American Institute of Physics.

The a-term comes from the series development of the Langevin function, the b-tsrm is the anisotropy effect, with K the anisotropy constant. /I is a constant that depends only on the material. The fit of the experimental curve with Eq. (13.74) leads to the determination of the fitting parameters and b. On the other hand, the Neel relaxation time for a superparamagnetic particle to respond to the magnetic field is [70] ... 

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 field-dependent expressions obtained in the previous sections for relaxation in an external magnetic field are applicable to both Neel and Debye relaxation and their specific application merely requires the use of the relevant time constant or t. ... 

Equations (6.68) and (6.69) coincide precisely as far as their field dependencies are concerned with the formulae obtained in Section V.C. Note that in our formulae the relaxation time is the Neel time rather... 

The inconsistencies were explained by Dezsi and Fodor [43], who found that three out of five deposits of natural a-FeOOH and a synthetic sample all gave a unique hyperfine pattern, while the other two did show clear evidence of two fields. The latter result was found to be true of samples containing excess water deposits of stoichiometric a-FeOOH do not show evidence for more than two magnetic sublattices. The Neel temperature was found to be only 367 K, and relaxation occurred at up to 30° below this. It would appear that the defect structure and particle size are both important in determining the properties of a specific sample. 

Fig. 23. Left Zero field spectra of polycrystalline DyAg at two temperatures within the paramagnetic regime. The Neel temperature is 60 K. The two spectra are shifted vertically for clarity. Right Temperature dependence of the relaxation rate A derived from fits of an exponential decay of polarization to spectra of the type shown in the left hand panel. Typical is the sharp rise on approach to the magnetic transition temperature. The line is a guide to the eye, but fits to a critical power law are often possible. After Kalvius et al. (1986).

Fig. 82. Left Temperature dependence of spontaneous muon spin precession frequency and of relaxation rate of the associated non-oscillating signal in magnetically ordered PrCojSij. Right Field dependence of the paramagnetic relaxation rate near the Neel temperature (top) and fit to a critical law (bottom) for ZF and LF measurements as indicated. The measuring geometry is always c. After Gubbens et al. (1997a, 1998).

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