The Lorentzian distribution width (*a*) is a calculable quantity, if
one knows the concentration of the dilute moments (*c*) and the
hypothetical Gaussian width for the *c*=1 dense spin system
(). In the low concentration regime (), the Lorentzian width is expressed [7] as:

The static muon spin relaxation for the Lorentzian field distribution has
been obtained [39] as:

In Fig.20, the static Lorentzian relaxation function
is shown. In zero-field, the relaxation
converges to 1/3 of the full amplitude, which is, again, the
signature of static relaxation functions. The `dip' at is shallower
and broader than that of the Gaussian Kubo-Toyabe function (Fig.17),
reflecting the broadness of the Lorentzian distribution. The relaxation at early times
shows an exponential decay, as the result of Fourier transform of the Lorentzian
distribution.

In the presence of field fluctuations, the Lorentzian relaxation
function is modulated in a similar manner as the Gaussian Kubo-Toyabe
function. Still, one must notice that the Lorentzian distribution
results from many *inequivalent* muon sites. A particular muon,
which resides at site A (Fig.19),
never experiences the local field at site B during the field
fluctuation processes. If one is not aware of this point, and applies
the strong collision series (eq.21) to the
static Lorentzian relaxation function ,one obtains an unphysical result: the absence of motional narrowing in the
fast fluctuation regime.

The proper treatment to dynamisize the Lorentzian relaxation function is as follows [7]:

- (1)
- Decompose the Lorentzian field distribution to the sum of many Gaussian distributions, each of which represents the local field distribution at an individual muon site.
- (2)
- Obtain the dynamical
*Gaussian*Kubo-Toyabe function for each muon site. This treatment reflects the inequivalence of each muon site for the Lorentzian distribution. - (3)
- Add each contribution, to restore the Lorentzian field distribution.

This procedure has been formulated [7], using a weighting function
, which is the probability of finding
a muon site (Gaussian field width: ) in a dilute spin system
environment (Lorentzian field width: *a*). The dynamical Lorentzian relaxation function is
obtained:

In Fig.21, the dynamical Lorentzian relaxation
function in zero-field is shown. The
effect of the field fluctuations is similar to that of the Gaussian
case: in the slow fluctuation regime, the 1/3-component suffers a slow
relaxation as [7], and in the fast
fluctuation regime, motional narrowing is exhibited. For the Lorentzian
distribution, the relaxation in the fast fluctuation regime is
approximated by a square-root exponential function [7]:

Experimentally, the Lorentzian relaxation function, as well as the square-root exponential behavior in the fast fluctuation regime, have been observed in dilute spin glass alloys [7], and the theory has been quite successful in dilute spin systems.