Root-exponential relaxation

Whether this is more than a convenient parameterization is debatable. The stretched exponential in eq. (35) has no direct relation to stretched exponential relaxation of bulk magnetization (see Campbell et al. 1994). As will be discussed in sect. 8 the only clear-cut case is the highly dilute spin glass. It was shown by Uemura et al. (1984) that above the glass transition temperature root-exponential relaxation occurs, that is p = 0.5. That... 

ZF and LF-ftSR has now been reported by Dunsiger et al. (2000). The ZF relaxation function is root-exponential at all temperatures down to 0.025 K, indicative of a dilute spin system with substantial dynamics. This supports the idea of isolated islands nucleated around defects, but indicates only slowed fluctuations, not full freezing. The apparent spin fluctuation rate drops starting near 1 K (where bulk probes see effects they attribute to short-range magnetic order, Schiffer et al. 1994), but does not extrapolate to zero, and shows no effect around 0.14K. Thus p,SR sees no spin-glass transition. All of this is generally consistent with the neutron diffraction results. In LF at 0.1 K, the relaxation... 

For a given type of x-values distribution, the size of the x-values array (number of blocks ra, ) plays approximately the same role as the number of scans N. Theoretically, the relative precision of any relaxation rate estimate is proportional to the square root of both and N. This, of course, presumes that Ub is anyway large enough to carry out the analysis. For example, values as small as 4 may be sufficient in mono-exponential cases, while continuous distributions spreading over several orders of magnitude require a logarithmic distribution of x-values and Ub values of over 100. 

A typical shape of the cross relaxation kinetics is an exponential function of the square-root of tj. A particular distribution function... 

Below the crossover temperature, the square root term becomes dominant and the relaxation of the magnetization is no longer exponential. In contrast with the magnetic susceptibility, the mono-disperse and poly-disperse models predict qualitatively different dynamic behaviors. However, the dominant characteristic time r is still given by Eqs. 37 or 38. 

In other experiments, Joos et al. [95,148] established that sometimes the dynamics of adsorption from micellar solutions exhibits a completely different kinetic pattern the interfacial relaxation is exponential, rather than inverse square root, as it should be for diffusion-limited kinetics. 

The computer modeling [150] shows that exhibits two exponential (kinetic) regimes, AB and CD, and two inverse-square-root (diffusion) regimes, BC and DE, see Figure 4.8. In particular, the point C corresponds to the moment Xq = (Dilhj)tc where is the characteristic time of the slow micellar process see Ref. [149]. x also serves as a characteristic relaxation... 

As emphasized in the introduction to relaxation kinetics, the methods described in this section can, in principle, be extended to derive equations for mechanisms with any number of relaxation times. Qearly these become progressively more complex as the number of roots increases. Assumptions have to be made in terms of limiting conditions, to extract useful information from them. The practical difficulties of resolving multiple exponentials from noisy experimental records have been alluded to before and helpful hints on this topic are presented in section 2.3. The discussion of examples of investigations by temperature and pressure jump techniques in... 

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