The X-ray analysis is used to determine the number of stopped muons in the
AlCap target by counting the number of
muonic X-rays produced. In addition to the X-rays, there are gamma rays
that can be observed and are relevant to Mu2e/COMET as well as AlCap.

The entrance beam scintillator
counters can be used to count the muons
as they enter the vacuum chamber, however
collimation and accounting for
those muons that pass through the thin target,
the stopping efficiency in the target is significantly
less than 100\%. 

The factor of interest for normalisation is the rate
of muon nuclear capture inside the target. The branching ratio of
capture is known for the targets of interest \cite{MeasdayAl} and is
proportional to the number of muon stops.

With the active silicon target
we have a reliable method of detecting a stop, namely, an energy
deposited in the silicon corresponding to the incoming muon beam energy, and
a time coincidence between signals from the beam counters and the silicon
detector. This is not the case with
the passive silicon and aluminium targets, for which we use instead the muonic
X-rays for normalisation.

When a muon is captured by an atom, it gives off
characteristic X-rays as it falls to the 1s state that can be counted
to determine the stopping rate.
For aluminium and silicon, the energies and intensities of the X-rays
from the $2p\to1s$ transitions 
are well known and listed in Table~\ref{tab:xray_ref}.
The stopping rate
can then be inferred from the number of these X-rays measured,
after accounting for
geometric and photo-efficiencies. The rate from this method was cross-checked with that determined directly from stops in the active
silicon target, and the numbers are within each other's uncertainties.

A peak from the natural background ${}^{214}$Pb (351.9 keV) exists near
the ${}^{27}$Al \atrn{2p}{1s} ($K_\alpha$) X-ray.
To suppress this neighbouring peak and the
background, we required an entering muon in time coincidence with the
germanium pulse. However, we noticed that an unexpected
$\gamma$ line prompt with muon nuclear capture on lead appears at a slightly
lower energy than the ${}^{214}$Pb line (Figure \ref{fig:xrayanalysis:tl207}).
This is consistent with an intermediate excited
state of $^{207}$Tl, produced by muon capture on $^{208}$Pb.
Though not spoiling our measurement, the classification of
this peak is important so that it can be confirmed in the next run.

\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{figs/tl207.png}
  \caption{To reduce the nearby pollution of the Al$_{K\alpha}$ by natural $^{214}$Pb,
    only germanium signals within 300 ns of an entering muon were examined. When
    the background peak persisted, we realised it was a prompt $\gamma$ from
    muon capture on lead going via an intermediate excited $^{207}$Tl$^*$ state. This was
    confirmed by the time structure of photons in that peak, which matches
    the muonic lead lifetime.}
  \label{fig:xrayanalysis:tl207}
\end{figure}

Though muonic X-rays are the primary method of normalisation in AlCap,
there are others that can be used and since both Mu2e and COMET are interested
in alternative normalisation schemes, it is important to examine the
viability of other peaks as
indicators of stopped muons. One is the $\gamma$ from the reaction
$^{27}_{13}\mbox{Al}+\mu^-\to\nu_{\mu}+n+\gamma+^{26}_{12}\mbox{Mg}$, with an intensity of
about 50\% per stopped
muon and an energy of 1809 keV \cite{MeasdayAl}.
What is appealing about this peak
is that there are few nearby peaks to worry about, and the signal-to-noise
ratio is
favourable, see the data in Figure \ref{fig:xrayanalysis:mg26_1809}.
The ratio of the number of observed
counts in the 1809 keV gamma ray line relative to
the $2p\to 1s$ muonic X-ray line is in good agreement with the value in the
literature.

\begin{figure}
  \centering
  \includegraphics[width=0.6\textwidth]{figs/mg26_1809keV}
  \caption{The $\gamma$ produced in
           $\mu^-+^{27}_{13}\mbox{Al}\to\nu_{\mu}+n+^{26}_{12}\mbox{Mg}^*$
           followed by $^{26}_{12}\mbox{Mg}^*\to ^{26}_{12}\mbox{Mg}+\gamma$,
           occurs at 1809 keV with an intensity of 0.51 per $\mu$-capture \cite{MeasdayAl}.
           Because this line occurs in such a clean region of the photon spectrum and is so intense,
           it could possibly be used for monitoring the number of stopped muons
           in Mu2e and COMET.}
  \label{fig:xrayanalysis:mg26_1809}
\end{figure}

A second gamma line at 844 keV results from the beta decay of
the relatively long-lived (9.5 minutes) $^{27}$Mg isotope
produced in the reaction
$^{27}_{13}\mbox{Al}+\mu^-\to\nu_{\mu}+^{27}_{12}\mbox{Mg}$.
Though counting this peak agreed within error with that expected from published
branching ratios, the uncertainty is large due to poor statistics.
By improving the statistics
under this peak in the proposed run we will
determine a more precise branching ratio, making it useful as a potential
normalisation to the number of muon stops in Mu2e and COMET.
The current data for this peak, as well as nearby peaks,
is illustrated in figure \ref{fig:xrayanalysis:mg27_844kev}.

\begin{figure}
  \centering
  \includegraphics[width=0.5\textwidth]{figs/mg27_844keV}
  \caption{A peak at 844 keV from the decay of $^{27}_{12}$Mg is
           correlated with the number of stopped muons in the target.
           Unfortunately the yield is low and additionally polluted
           by a nearby iron peak. With the statistics from our first run,
           we could not determine to a sufficient precision the number
           of these $\gamma$s we expect per captured muon, however we
           will be able to achieve this in the proposed next run when the statistics will be much improved.}
  \label {fig:xrayanalysis:mg27_844kev}
\end{figure}
