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\section*{Phenomenological model for homogeneous disorder --- Noakes, Kalvius}

\noindent This is a short summary of the formula\"{e} derived by Noakes and
Kalvius \cite{noakes97,noakes99}. However, I will stick to the notation as used
in \cite{yaouanc11}.

\subsection*{Noakes-Kalvius, ZF, static}

\noindent For the static ZF depolarization function, $P_{Z}^{\rm stat}$ on finds [see \cite{yaouanc11}, p.135, Eq.(6.101)]:

\begin{eqnarray*}
 P_{Z}^{\rm stat}(t, R_b, \Delta_{\rm eff}) &=& \frac{1}{3} + \frac{2}{3} \left(\frac{1+R_b^2}{1+R_b^2+R_b^2 \gamma_\mu^2 \Delta_{\rm eff}^2 t^2}\right)^{3/2}
  \left(1-\frac{\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}{1+R_b^2+R_b^2\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}\right) \times \\
 && \times \exp\left[-\frac{1}{2}\left(\frac{\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}{1+R_b^2+R_b^2\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}\right)\right]
\end{eqnarray*}

\noindent with $R_b = \Delta_{\rm GbG} / \Delta_0$ and $\Delta_{\rm eff}^2 = \Delta_0^2 + \Delta_{\rm GbG}^2$. However, this can be substantially simplified by 
using the fact that $\Delta_{\rm eff}^2 = (1+R_b^2) \Delta_0^2$. Using this, the above formula reduces to

\begin{equation}
 P_{Z}^{\rm stat}(t, R_b, \Delta_0) = \frac{1}{3} + \frac{2}{3} \left(\frac{1}{1+R_b^2 \gamma_\mu^2 \Delta_0^2 t^2}\right)^{3/2}
  \left(1 - \frac{\gamma_\mu^2 \Delta_0^2 t^2}{1+R_b^2 \gamma_\mu^2 \Delta_0^2 t^2}\right) 
  \exp\left[-\frac{1}{2}\left(\frac{\gamma_\mu^2 \Delta_0^2 t^2}{1+R_b^2\gamma_\mu^2 \Delta_0^2 t^2}\right)\right]
\end{equation}

\noindent where $\Delta_{\rm eff}$ could be removed alltogether.

\subsection*{Noakes-Kalvius, TF, static}

\noindent For the static TF depolarization function, $P_{X}^{\rm stat}$ on finds [see \cite{yaouanc11}, p.135, Eq.(6.100)]:

\begin{eqnarray*}
 P_{X}^{\rm stat}(t, R_b, \Delta_{\rm eff}) &=& \sqrt{\frac{1+R_b^2}{1+R_b^2+R_b^2 \gamma_\mu^2 \Delta_{\rm eff}^2 t^2}} 
   \exp\left[-\frac{1}{2}\left(\frac{\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}{1+R_b^2+R_b^2\gamma_\mu^2 \Delta_{\rm eff}^2 t^2}\right)\right]
   \cos(\gamma_\mu B_{\rm ext} t + \phi),
\end{eqnarray*}

\noindent which reduces to 

\begin{equation}
  P_{X}^{\rm stat}(t, R_b, \Delta_0) = \sqrt{\frac{1}{1+R_b^2 \gamma_\mu^2 \Delta_0^2 t^2}} 
    \exp\left[-\frac{1}{2} \left(\frac{\gamma_\mu^2 \Delta_0^2 t^2}{1+R_b^2\gamma_\mu^2 \Delta_0^2 t^2}\right)\right] \cos(\gamma_\mu B_{\rm ext} t + \phi)
\end{equation}

\subsection*{Noakes-Kalvius, ZF, dynamic}

\noindent With
$$ \Theta(t) = \frac{\exp(-\nu_c t - 1 + \nu_c t)}{\nu_c^2} $$

the dynamic polarization function $P_{Z}^{\rm dyn}$ reads [see \cite{yaouanc11}, p.137, Eq.(6.106)]:

$$ P_{Z}^{\rm dyn}(t, R_b, \Delta_{\rm eff}, \nu_c) = \sqrt{\frac{1+R_b^2}{1+R_b^2+4 R_b^2 \gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}}
  \exp\left[-\frac{2\gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}{1+R_b^2+4 R_b^2\gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}\right],
$$

\noindent which reduces to 

\begin{equation}
 P_{Z}^{\rm dyn}(t, R_b, \Delta_0, \nu_c) = \sqrt{\frac{1}{1+4 R_b^2 \gamma_\mu^2 \Delta_0^2 \Theta(t)}}
  \exp\left[-\frac{2 \gamma_\mu^2 \Delta_0^2 \Theta(t)}{1+4 R_b^2 \gamma_\mu^2 \Delta_0^2 \Theta(t)}\right]
\end{equation}

\subsection*{Noakes-Kalvius, TF, dynamic}

the dynamic polarization function $P_{X}^{\rm dyn}$ reads [see \cite{yaouanc11}, p.136, Eq.(6.105)]:

$$ P_{X}^{\rm dyn}(t, R_b, \Delta_{\rm eff}, \nu_c) = \sqrt{\frac{1+R_b^2}{1+R_b^2+2 R_b^2 \gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}}
  \exp\left[-\frac{\gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}{1+R_b^2+2 R_b^2\gamma_\mu^2 \Delta_{\rm eff}^2 \Theta(t)}\right]
  \cos(\gamma_\mu B_{\rm ext} t + \phi),
$$

\noindent which reduces to 

\begin{equation}
 P_{X}^{\rm dyn}(t, R_b, \Delta_0, \nu_c) = \sqrt{\frac{1}{1+2 R_b^2 \gamma_\mu^2 \Delta_0^2 \Theta(t)}}
  \exp\left[-\frac{\gamma_\mu^2 \Delta_0^2 \Theta(t)}{1+2 R_b^2 \gamma_\mu^2 \Delta_0^2 \Theta(t)}\right]
  \cos(\gamma_\mu B_{\rm ext} t + \phi).
\end{equation}

\begin{thebibliography}{99}
 \bibitem{noakes97} D.R. Noakes, and G.M. Kalvius, Phys. Rev. B, \textbf{56}, 2352, (1997).
 \bibitem{noakes99} D.R. Noakes, J. Phys.: Condens. Matter \textbf{11} 1589, (1999). 
 \bibitem{yaouanc11} A. Yaouanc, and P. Dalmas de R\'{e}otier, ``Muon Spin Rotation, Relaxation, and Resonance'', Oxford University Press 2011. 
\end{thebibliography}

\end{document}
