3.2 Lorentzian theory [#!UemuraPRB85!#]

 The Gaussian Kubo-Toyabe theory introduced above is based on Gaussian local field distribution, which is often realized in dense spin systems. In dilute spin systems, such as dilute spin glass alloys, it is known that the dipolar fields from the local moments take a more Lorentzian distribution [38]:

where a is the width of the Lorentzian field distribution. The origin of the Lorentzian distribution is the large variety of the muon sites relative to the local moments (see Fig.19). Since some muons locate relatively far from the local moments (site A of Fig.19), and some close (site B of Fig.19), the local field distribution has a sharper peak around zero (from site A's) and a broader tail (from site B's) than the Gaussian field distribution.

  

Figure 19: A dilute spin system. The local field takes an isotropic Lorentzian distribution.

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 ($\Delta_{100\%}$). In the low concentration regime ($c\stackrel{<}{\sim} 5\%$), the Lorentzian width is expressed [7] as:

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

for zero-field, and in the presence of a longitudinal field ($H_{\rm LF}$) [7]:

where the j_i(x) are spherical Bessel functions.

In Fig.20, the static Lorentzian relaxation function $G^{\rm L}(t;a,H_{\rm LF})$ 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 $at\sim 2$ 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.

  

Figure 20: The muon spin relaxation in the frozen dilute spin system: static Lorentzian Kubo-Toyabe function $G^{\rm L}(t;a,H_{\rm LF})$.

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 $G^{\rm L}(t;a,H_{\rm LF})$,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 $\rho_a(\Delta)$, which is the probability of finding a muon site (Gaussian field width: $\Delta$) in a dilute spin system environment (Lorentzian field width: a). The dynamical Lorentzian relaxation function is obtained:

where,

This weighting function, by definition, converts a Gaussian distribution to a Lorentzian distribution:

  

Figure 21: The zero-field muon spin relaxation in a fluctuating dilute spin system: dynamical Lorentzian relaxation function $G^{\rm DL}(t;a,H_{\rm LF},\nu)$.

In Fig.21, the dynamical Lorentzian relaxation function in zero-field $G^{\rm DL}(t;a,H_{\rm LF}=0,\nu)$ 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 $\sim 1/3\exp(-2\nu t/3)$ [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]:

with the relaxation rate:

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.