Lorentzian from Gaussian

In publications on dilute spin glass moment dynamics probed by µSR, Uemura[1] modeled the Lorentzian field distribution at the muon site due to the dilute spins

$$P_{L}(B_{i})=\frac{\gamma _{\mu }a}{\pi (a^{2}+\gamma _{\mu }^{2}B_{i}^{2})}$$

as a Gaussian distribution of fields at each "class'' of muon site

$$P_{G}(B_{i})=\frac{\gamma _{\mu }}{\sqrt{2\pi }\Delta }\exp \ . . . . . . ( -\frac{% \gamma _{\mu }^{2}B_{i}^{2}}{2\Delta ^{2}}\right)$$

convoluted with a strange distribution of such classes of site (see Fig.1)

$$\rho (\Delta )=\sqrt{\frac{2}{\pi }}\frac{a}{\Delta ^{2}}\exp \left( -\frac{% a^{2}}{2\Delta ^{2}}\right) ,$$

$$P_{L}(B_{i})=\int_{0}^{\infty }\rho (\Delta )P_{G}(B_{i})d\Delta .$$

This convolution also works to generate the distribution of magnitudes of field:

$$P_{L}(\vert{\bf B}\vert)=\int_{0}^{\infty }\rho (\Delta )P_{G}(\vert{\bf B}\vert)d\Delta ,$$

where

$$P_{L}(\vert{\bf B}\vert)=\frac{4wB^{2}}{\pi \left( w^{2}+B^{2}\right) ^{2}},\textrm{% \qquad }w=\frac{a}{\gamma _{\mu }}$$

$$P_{G}(\vert{\bf B}\vert)=\sqrt{\frac{2}{\pi }}\frac{B^{2}}{\s . . . . . . t) ,\textrm{\qquad }\sigma =\frac{\Delta }{% \gamma _{\mu }},$$

and to generate the static ZF Lorentzian Kubo-Toyabe relaxation function:

$$G_{L}(t)=\int_{0}^{\infty }\rho (\Delta )G_{G}(t)d\Delta ,$$

where

$$\begin{array}{c} G_{L}(t)=\frac{1}{3}+\frac{2}{3}\left( 1-at . . . . . . exp \left( -\frac{\Delta ^{2}t^{2}}{2}\right) . \end{array}$$

Figure 1: Uemura's trick distribution

In spite of the apparent strangeness $\rho (\Delta )$, Uemura was able to use this to model moment dynamics (for a stationary muon) to produce dynamic muon spin relaxation functions G(t) that decoupled properly in the fast fluctuation limit (direct "dynamicization'' of the Lorentzian field distribution, equivalent muon diffusion in a the spin glass, does not result in fluctuation decoupling), and were consistent with observed µSR. So, this convolution does seem to bear some relation to reality.