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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

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

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

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

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

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


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

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

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

where

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


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

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

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

where

\begin{displaymath}\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}
\end{displaymath}

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.
next up previous
Next: Between Gaussian and Lorentzian Up: Uemura's Site-Distribution Trick Previous: Uemura's Site-Distribution Trick
Jess H. Brewer
2002-09-24