Fast fluctuation limit

According to eq. (21), transverse field measurements allow, in principle, separation of the static field width (oc o) from the fluctuation rate (1/r) in an intermediate case. In practice, this is rather difficult and the combination of zero and longitudinal field measurements are more powerful in this respect, as will be shown further below. In the static limit (r —> oo) we can extract the second moment of the field distribution (see eq. 20). In the fast fluctuation limit only the product a r appears (eq. 24) and independent information on one of the two quantities is needed. 

Recall (above) that in ZF, fast field fluctuations in a Gaussian distribution result in muon spin relaxation at a rate determined solely by AVv, so that ZF (in that limit) cannot separate the distribution width from the fluctuation rate, but LF in principle can. For a Lorentzian distribution, the ZF fast-fluctuation limit relaxation rate A(0) 4a/3 depends... 

The first task is to formulate a theoretical signal fimction, which we denote as representing muon polarization as a fimction of time. For example, a transverse field measurement in the fast fluctuation limit should have the form... 

One basic problem (which has already been pointed out in sect. 3.2.2) of the fast fluctuation limit is that a separation of ( B ) and tc is not possible. Furthermore, since the expression (58) for the different directional components i contain diflerent correlation times T, it is incorrect to express the total relaxation simply as ( B Tc in a strongly anisotropic case. Therefore, from a measurement of A and Ax it is not trivial (among other considerations, one must know the muon interstitial site) to determine how much of the anisotropy comes from an anisotropic field distribution and how much relates to anisotropic fluctuation rates. One may hope that not too close to the magnetic transition (i.e., well inside the paramagnetic regime) ( fiu ) shows little anisotropy and that one senses the anisotropy of tc in ZF. But an exact treatment of a p.SR measurement on these terms has not been given to our knowledge. 

An overview of the temperature dependence of depolarization rates in ZF and TF (0.02 T) for both the Nd and the Eu compounds is presented in fig. 169. The Nd material itself shows some usual behavior the relaxation rate in it begins to rise at 200 K due to the approach to a magnetic phase transition (although depopulation of CEF levels may play a role). The magnetic state was not explored. The fact that the depolarization rate in TF is virtually the same proves that the local fields are in the fast-fluctuation limit. 

Even without using numerical methods, one can analyse some physically sound limiting cases of the exact solution for the case of strong collisions (7.64). First of all, (7.64) evidently reduces to the results of Robert and Galatry in the quasi-static case, i.e. when tc — oo. An opposite limiting case of fast fluctuations... 

The Landau-Zener formula in the limit J2/ v - oo gives PLZ = 1 whereas Eq. (183) for the case of fast fluctuations in this limit gives P — At small values of J2/ u, both formulas give the same result, P = 2itJ2/ v. ... 

In the fast modulation limit [Equation (8)], the loss of information is fundamental. The lifetime of the frequency perturbations is short compared to their magnitude, and the uncertainty principle precludes a full characterization of w(t) by any experimental technique. However, in the slow modulation limit and in the intermediate regimes, the loss of information in the FID is not fundamental. The next section shows that the Raman echo contains additional information about the rate of the frequency fluctuations that is not present in the FID. By using a combination of Raman echo and... 

Figure 15 (a) The 4.2 K Mossbauer spectra of [(Fe(IV)=0)(TMC)(NCCH3)](0Tf)2 in acetonitrile recorded in (A) zero field and (B)a parallel field of 6.5 T. The solid line represents a spin Hamilton simulation with the parameters described in the text, (b) Mossbauer spectra of [Fe(lV)=(0)(TMCS)] recorded at temperatures and applied fields that are indicated. The solid lines represent spin Hamiltonian simulations with parameters described in the text. The spectra were simulated in the slow (at 4.2 K) and fast (at 30 K) spin fluctuation limit. The applied field was directed parallel to the observed y radiation. The doublet drawn above the topmost experimental spectrum (0 T, 4 K) represents a 7% Fe(ll) contribution from the starting complex. (From J. U. Rohde et al. (2003) Science 299 1037-1039. Reprinted with permission from AAAS)... 

We have chosen to encompass our methodology in the necessarily limited framework of rotational FPK operators for describing the solute molecule and the solvent cages (slow fluctuating solvent structures) with translational FPK operators for describing stochastic fields (fast fluctuating solvent structures). We are aware that a truly complete description... 

In spite of the absence of any quasistatic moment formation, CeCee must be considered not far from magnetic ordering. Neutron data (Regnault et al. 1987) found dynamic short-range AFM correlations below 10K. Since no effect is seen on the pSR relaxation rate, the fluctuations of the moments must remain in the fast dynamical limit. Similarly magneto-resistance (Qnuki et al. 1985) and thermoelectric power (Amato et al. 1987) show anomalies at low temperature that point towards incipient magnetism. The possibility of magnetic order below 5mK has been pointed out (Schuberth et al. 1995, Pollack et al, 1995). 

The fundamental requirement for longitudinal relaxation of a proton nucleus is a time dependent magnetic field fluctuating at the Larmor frequency of the nuclear spin. In the fast motion limit, the frequency distribution of the fluctuating magnetic fields associated with the molecular motion, i.e. the spectral density function J(a)), has, in the simplest case, a Lorentzian shape, as described by Eq. (1), and is characterized by the correlation time xc ... 

If the limit of fast fluctuations is not accurate, we can expect dynamic features influencing the lineshape. The situation will be more complicated than for the zero field case discussed in Section F.6.4.2.d, because of the polarization effect on the relative intensities of the absorption lines. 

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