Effect of spin fluctuations [#!KuboJPSJ54!#,#!HayanoPRB79!#]

In the framework of the Kubo-Toyabe theory, the effects of field fluctuations have been taken into account with the `strong collision model' [36,6]. This model assumes that (1) fluctuations occur suddenly, and that (2) every time the local field fluctuates, the muon forgets the previous local field information. Hereafter, the field fluctuation rate ($\nu$) is defined as the Markoffian fluctuation rate, namely, the exponential decay rate of the autocorrelation function of the local fields:

The strong collision model generally calculates the dynamical muon spin relaxation $G^{\rm D}(t;\nu)$ from the original static relaxation function G(t) as follows:

The terms of this series account for the muons which experienced $0,1,2,\cdot\cdot\cdot$field fluctuations in the time interval of $0\sim t$.

Fig.18 shows the dynamical Gaussian Kubo-Toyabe function in zero-field for various fluctuation rates ($\nu$). In the slow fluctuation regime ($\nu/\Delta\stackrel{<}{\sim}0.1$), the fluctuation induces slow relaxation of the 1/3-component. The asymptotic behavior of this relaxation has been obtained as $\sim 1/3\exp(-2\nu t/3)$ [6]. In the intermediate fluctuation regime ($0.1\stackrel{<}{\sim}\nu/\Delta \stackrel{<}{\sim}2$), the relaxation has a Gaussian behavior in the beginning, but loses the 1/3-component. Hence, the existence/absence of the 1/3-component is a clue which distinguishes static/dynamic relaxation.

  

Figure 18: The zero-field muon spin relaxation in the fluctuating Gaussian local field (dynamical Gaussian Kubo-Toyabe function $G^{\rm DGKT}(t;\Delta,H_{\rm LF},\nu)$).

In the fast fluctuation regime ($\nu/\Delta\stackrel{\gt}{\sim}10$), the relaxation is approximated by an exponential function [7,37]:

where the relaxation rate is:

In this fast fluctuation regime, the relaxation rate ($\lambda$)decreases with faster fluctuation rates. This phenomenon is known as the `motional narrowing' of the T₁ relaxation rate. The longitudinal field dependence of the relaxation rate ($\lambda$)is consistent with that of the T₁ relaxation theory [34], which has been developed for nuclear magnetic resonance (NMR).