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\title{Alignment of the ATLAS Muon Spectrometer with Muon Tracks}

\author{B.~Bittner$^1$, S.~Kaiser$^1$, O.~Kortner$^1$, H.~Kroha$^1$, S.~Kotov$^1$, I.~Potrap$^1$\\
R.Harrington$^2$, J.~Love$^2$, N.~Nation$^2$}

\date{}
\maketitle

\begin{center}
$^1$ Max-Planck-Institut f\"ur Physik, Munich, Germany

$^2$ Boston University, Boston, USA
\end{center}


The muon spectrometer~\cite{muon_tdr} of the ATLAS experiment 
is equipped with three layers of muon detectors in a toroidal
magnetic field of $3-6\,$Tm bending power generated by a superconducting
air-core magnet system. 
The spectrometer is designed to provide muon momentum resolution of better
than $10\%$ for transverse momenta up to 1~TeV over a pseudo-rapidity
range of $|\eta|\le 2.7$. This requires very accurate track
sagitta measurement with three layers of drift-tube chambers which have to be aligned
relative to each other with an accuracy of about 30~$\mu$m in the bending direction.
The relative positions of the chambers
are continuously measured with optical alignment monitoring systems~\cite{muon_tdr}\cite{mdt_align1}.

\begin{figure}[!h]
\centering
\includegraphics[width=0.7\textwidth]{ATLAS_MDT_alignment_sag_cor}
\caption{Accuracy $\sigma$ in the determination of the muon track sagitta corrections
due to chamber misalignment as a function of pseudo-rapidity $\eta = -\ln\tan {(\theta /2)}$
(with the polar angle $\theta$ of the track defined with respect to the beam axis) in the barrel part of the muon spectrometer
using in each azimuthal sector 100000 simulated straight muon tracks of 20 GeV momentum originating at the
interaction point (full circles) and 400000 straight cosmic muon tracks from the detector 
commissioning data (open circles).
\label{fig:ATLAS_MDT_alignment_straighttracks}}
\end{figure}

\clearpage
In the barrel part of the spectrometer, only the large chamber sectors mounted in between the
magnet coils can be fully aligned with optical sensor measurements.
The small chamber sectors mounted on the coils 
have to be aligned with respect to the large sectors using muon tracks passing through the
overlap regions between the small and the large sectors. 
The alignment corrections will be calculated in intervals of a few hours
at dedicated computing centers for the calibration and alignment of the muon chambers \cite{mdt_align1}.

Straight muon tracks from proton-proton collisions, measured while the toroid magnets
are turned off, are needed for the precise determination the initial chamber positions
after installation as a starting point for the monitoring of further chamber movements 
by optical sensors mounted on the chambers. An efficient algorithm  
has been developed for this purpose~\cite{mdt_align2}
and successfully tested with cosmic ray data taken with the installed ATLAS detector 
(see Fig.~\ref{fig:ATLAS_MDT_alignment_straighttracks}).
The chamber displacements from nominal positions determined this way
agree well with mechanical and optical measurements.
Tests with simulated data show that the relative chamber
positions in each barrel sector can be determined with the required accuracy
on the track sagitta corrections~\cite{mdt_align2}
(see Fig.~\ref{fig:ATLAS_MDT_alignment_straighttracks}).

The method has being extended to the alignment of the muon chambers with
curved muon tracks in the magnetic field during normal operation of the experiment
in order to verify the optical alignment corrections. This requires an
independent measurement of the muon momentum which is insensitive to
misalignment of the MDT chambers along the muon track. 
The MDT chambers with their two triple or quadruple layers of precisely positioned drift tubes
measure not only track coordinates with high accuracy but also the local
track direction. This feature can be utilized for independent momentum determination from
measurement of the track deflection angles between the inner and the outer chamber layer
and between the two multilayers of the chambers in the middle layer located inside the magnetic field.
Studies with simulated data show that the required alignment accuracy
can be achieved within two days of data taking at the nominal LHC luminosity 
of $10^{33}\,$cm$^{-2}$s$^{-1}$.


\begin{thebibliography}{1}

\bibitem{muon_tdr}
ATLAS collaboration, \emph{Technical Design Report for the ATLAS
Muon Spectrometer}, CERN/LHCC 97-22, May 1997.

\bibitem{mdt_align1}
Ch.~Amelung et al.,
\textsl{The ATLAS Muon Alignment System},
proceedings of the First LHC Alignment Workshop, CERN, Geneva, Switzerland, 4--6 September 2006,
CERN report, CERN-2007-004, 2007; \\
J.C.~Barriere et al.,
\textsl{The Alignment System of the Barrel Part of the ATLAS Muon Spectrometer},
ATLAS Note, ATL-MUON-PUB-2008-007, April 2008.

\bibitem{mdt_align2}
I.~Potrap, O.~Kortner, S.~Kotov, H.~Kroha,
\textsl{Alignment of the ATLAS Muon Spectrometer with Straight Tracks Using MILLEPEDE Method},
ATLAS Note, ATL-COM-MUON-2007-017, December 2007.

\end{thebibliography}
\end{document}
