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\title{Notes on dirty clean limit}

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\section{Cleanliness dependence of LSW power}

The power dissipated via \textforcesc{LSW} from Ford and Weber\cite{Ford1984} is at local, quasistatic limit.
The dirty limit (on the left) matches up with the Mattis-Bardeen case as expected\cite{Mattis}, cf. \cref{fig:namvsmb}.
This is good because it gives us a bit of confidence in the Nam expressions as we move to the cleaner case.
However, the clean case looks off in \cref{fig:pvstau}.
The increase in dissipation as the material gets cleaner doesn't make physical sense.
This isn't an issue with the full $T_1$ calculations without approximations, as graphed in \cref{fig:t1calc}.

There are a few differences that could be responsible:
\begin{itemize}
	\item The \textforcesc{LSW} power calculations are done in the local limit without a cutoff
	\item Local limit vs nonlocal.
	\item Quasistatic issues
\end{itemize}

The first doesn't make a ton of sense because the dielectric functions are in the local limit, with $k\rightarrow 0$.
The cutoff shouldn't actually matter for that.
The second is on its own has the same issue, seemingly.
If it were local vs nonlocal, might be able to compensate anyway by just increasing $z$.

The quasistatic limit seems like the right place to look.
I don't think there's any reason to assume it would be valid in the superconducting case at this point.
There may be a more appropriate way of taking the quasistatic limit, somehow with the local limit as well.

Useful step might be to look at quasistatic limit on its own, without the local assumption.
And vice-versa.

\begin{figure}[htp]
	\centering
	\includegraphics[width=12cm]{dirtynamvsmb}
	\caption{The dirty limit of Nam vs Mattis-Bardeen} \label{fig:namvsmb}
\end{figure}

\begin{figure}[htp]
	\centering
	\includegraphics[width=12cm]{lswPower-tau}
	\caption{$P_{lsw}$ vs $\tau$} \label{fig:pvstau}
\end{figure}

\begin{figure}[htp]
	\centering
	\includegraphics[width=12cm]{fullt1calc}
	\caption{Full $T_1$ calculation} \label{fig:t1calc}
\end{figure}

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