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## 3.3 A minor correction to the Lorentzian theory

In the previous section, the Lorentzian theory was introduced as a muon spin relaxation theory for dilute spin systems. This theory is based on the Lorentzian field distribution at the muon sites. Still, a truly Lorentzian distribution is unphysical, because some fraction of the muons must locate at an infinitesimally small distance from a magnetic ion, in order to realize the diverging second moment of the Lorentzian distribution . To restore the physicality of the local field distribution, it will be natural to introduce a large cut-off field () to the Lorentzian distribution. This idea is easily formulated in the Gaussian decomposed picture of the Lorentzian distribution (eq.29,30), which has been introduced to obtain the dynamical Lorentzian relaxation function .

In this picture, a weighting function was introduced to sum up the contributions from every muon sites (see eq.29). In real spin systems, the upper bound of the site-sum integral should be replaced by a cut-off field :

where is normalized for the new upper bound :

The physical meaning of the cut-off field is the largest possible Gaussian field width for the dilute spin system considered. This quantity may be of the same order of magnitude as the hypothetical Gaussian width (), which is expected when all the lattice points are filled up with moments. Since the relation between and the Lorentzian width (a) has been obtained (eq.26), is estimated to be:

where c is the dilute spin concentration and a is the Lorentzian width.

The most significant correction because of the cut-off field appears in the fast fluctuation regime. In the traditional Lorentzian theory, the relaxation function in this regime is a square-root exponential function (eq.32). When the cut-off field is introduced, the relaxation at large fluctuation rates loses the fast front-end of the square-root exponential behavior, because the fast front-end originates from the T1 relaxation of muons which locate at sites with large local fields. In Fig.22, the -corrected dynamical Lorentzian functions [] are compared with the square-root exponential functions of the ideal case.

The -corrected dynamical Lorentzian function is well approximated by a `stretched' exponential function, ; in Fig.23, the stretching power () is shown as a function of the normalized fluctuation rate (). At small fluctuation rates, the power converges to 1/2, as expected for the square-root exponential behavior, and in the large fluctuation limit, approaches 1.

Experimentally, muon spin relaxation in paramagnetic dilute spin systems often exhibits a stretched exponential behavior, with its power approaching 1 at high temperatures [40]. The above mentioned cut-off field effect may explain at least part of the phenomena.

Next: 3.4 Summary of the Up: 3 Spin relaxation theories Previous: 3.2 Lorentzian theory [#!UemuraPRB85!#]