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\begin{titlepage}

\begin{large}
\hspace*{14cm}
\parbox[t]{50mm}{
20 September 1999\\
}
\end{large}
\vspace{1.5cm}

\begin{center}
{\LARGE\bf Monitoring of the Assembly of the}

\vspace{0.4cm}
{\LARGE\bf BOS~98 Prototype MDT Chamber}
\vspace{1cm}

{\Large H.~Kroha, A.~Manz, A.~Ostapchuk and S.~Schael}
\vspace{0.5cm}

{\Large Max-Planck-Institut f\"ur Physik, Munich}

\end{center}


\vspace{3.5cm}
\begin{abstract}
A detailed description of the monitoring
procedures and results during the construction of the BOS~98 prototype MDT chamber is given.
The analysis of the optical monitoring data taken during the construction of the
BOS~98 prototype chamber shows the importance of this information for the control
of the assembly process. The data have been used to optimize the assembly procedure.
The influence of the thermal environment in the assembly room 
on the chamber and on the optical monitors (laser beams) has been studied
and understood in detail.
The results of the analysis give confidence in the capability of the assembly method to
achieve the required mechanical accuracy of the chambers in series production.
\end{abstract}
\clearpage

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\tableofcontents
\clearpage

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\section{Introduction}

The purpose of this note is to provide the details of the monitoring
procedures during the construction of the BOS~98 prototype MDT chamber 
as a supplement to note \cite{BOS1} where the assembly method and
measurements of the prototype chamber are described.

The description of the monitoring procedures
starts after the installation and measurement of the tube jigs and the sphere
support towers on the granite table and of the optical monitors on the spacer structure (see \cite{BOS1}).
The environmental conditions in the climatized assembly room, the
preparations of the sphere support towers and of the spacer are explained as well as
the zero-readings of the optical monitoring devices and how they were used during chamber
assembly.

The principle layout of the assembly table is shown in
Figure~\ref{fig:frame} with the naming scheme for the spacer parts, cross plates CP
on the high voltage (HV) and readout (RO) side and in the middle (MI) of the chamber,
long beams LB-L and LB-R and spheres supported on the sphere towers T1--T6 on the table,
and the different optical monitors. The following optical monitoring systems have been used
extensively during assembly of the prototype:

\begin{enumerate}

\item The four RASNIK in-plane alignment systems IP1--IP4.

\item Three RASNIK cross plate sag monitors S-HV, S-MI and S-RO 
mounted on the three cross plates with the lens in the center and mask
and CCDs at the ends at a distance of 1~m from the lens.

\item Two multi-point sphere monitors MP-L and MP-R with two laser beams directed parallel
to the long beams on either side of the chamber and passing through several transparent
MPA-ALMY sensors, reference sensors MP-LR1,2 and MP-RR1,2 on the assembly table
at start and end of the laser beams and sensors MP-L1--MP-L3 and MP-R1--MP-R3 mounted
on all six sphere holders at the ends of the cross plates (see \cite{ALMY}).

\end{enumerate}


\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=frame-almy.eps,width=0.75\textwidth}}
\caption{\small
   Layout of the chamber assembly table and chamber coordinate system with the spacer support structure
   on the table in the initial (INI) orientation.
   The naming of the spacer components and optical monitoring devices is explained in the text.}
\label{fig:frame}
\end{large}
\end{center}
\end{figure}
%

The first test of the assembly procedure was the reproducibility of the optical monitor readings
when the spacer was repeatedly released on the sphere towers. The test
lead to the definition of a minimum of mechanical constraints
of the sphere towers on the spacer needed for the correct relative orientation
of the two multilayers.

After good reproducibility of the spacer positioning on the sphere towers was achieved,
the optical monitor readings in the initial and the inverted orientation of the spacer 
on the towers were compared. The spacer is inverted by rotations of $180^\circ$
around the $x$- and the $y$-axis (see Figure~\ref{fig:inv}). 
The comparison provides an information about
the relative height of the six sphere supports.

Finally, the monitor readings in the two orientations of the spacer on the
granite table were compared with the readings in four orientations of the
spacer supported on the crane used for assembly: the two vertical and the two 
horizontal orientations (see Figure~\ref{fig:crane}). This study
leads to a procedure to adjust the long-beams to the center of the middle cross plate
in order to avoid internal stresses and to minimize up-down asymmetries in the gravitational sag
of the cross plates and left-right asymmetries in the sag of the long beams.

After completion of the above steps, the zero-readings of the RASNIK monitors on the spacer could be
taken to be available as references during the assembly of the chamber. During chamber construction,
the stability of the monitor readings was studied which is essential for the
performance of the assembly procedure and for the prediction of the mechanical
properties of the chamber.


\section{Preparations Before Assembly}
%

\subsection{Temperature Control in the Assembly Room}
%
The BOS prototype chamber was assembled in a climatized clean room
with the following specifications in accordance with \cite{TDR}:
humidity $(45\pm 5)\%$ and air temperature above the assembly table 
$(20\pm 0.5)^\circ$C. Temperature and humidity were monitored during the
chamber construction with an analog plotter device. The continuous
monitoring revealed a cycle of the air conditioning system with a period
of 30 minutes and amplitudes of temperature and humidity variations
within the tolerances.

\begin{figure}[thb]
\begin{center}
\vspace{-8mm}
\mbox{\epsfig{figure=temp_air_table_comb_2.eps,width=1.0\textwidth}}

\vspace{-10mm}
\caption{\small
Temperature measured in the climatized assembly room over a period of 24 hours
in the air and at the top (combs) and bottom of the granite table (see text)}
\label{fig:2_regimes}
\end{center}
\end{figure}

\begin{figure}[thb]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=temp_night_2.eps,width=0.9\textwidth}}

\vspace{-13mm}
\caption{\small
As Figure~\ref{fig:2_regimes} with the night period enlarged}
\label{fig:night}
\end{center}
\end{figure}

\begin{figure}[thb]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=temp_day_2.eps,width=0.9\textwidth}}

\vspace{-13mm}
\caption{\small
As Figure~\ref{fig:2_regimes} with the day period enlarged}
\label{fig:day}
\end{center}
\end{figure}

\begin{figure}[thb]
\begin{center}
\vspace{-5mm}
\mbox{\epsfig{figure=temp_diff.eps,width=0.7\textwidth}}

\vspace{-30mm}
\caption{\small
Temperature differences with respect to the air of the top and of the bottom of
the granite table in the clean room during the day}
\label{fig:diff}
\end{center}
\end{figure}


After the prototype construction, the clean room was equipped with a set
of 12 high-precision temperature sensors, 4 sensors at the corners of the bottom side of
the granite table, 4 sensors attached to the ends of the two end combs on top of the assembly table
and 4 sensors distributed in the room to measure the air temperature.
One of them was delivered with absolute calibration
by the manufacturor. The others were calibrated with respect to the reference sensor.
Both the measurement resolution and the calibration accuracy are better than $\pm 0.05^\circ$C.
The sensors can be read out continuously with a LabVIEW program on a PC.

The purpose of the measurements was to determine with high accuracy

\begin{enumerate}
\item the stability of the air temperature in the clean room,
\item the uniformity of the air temperature and
\item the stability and uniformity of the temperature of granite table and combs.
\end{enumerate}

The first observation was that the air conditioning system operates in two regimes 
with different characteristics: {\sl cooling} when the incoming air from outside the clean
room is warmer than the air inside, and {\sl heating} when the input air temperature is
lower than the clean room temperature.
In the first case, the air temperature in the clean room oscillates with an amplitude 
of $\pm 0.1^\circ$C and a period of about 5~minutes.
In the second regime, the amplitude of the air temperature oscillations in the clean room 
is $\pm 1^\circ$C with a period of approximately 30~minutes.
The measured temperature variations over a 24 hour period are shown in 
Figures~\ref{fig:2_regimes}, \ref{fig:night} and \ref{fig:day}.
The two regimes can be clearly distinguished: the cooling regime during the day and
heating during the night.

The BOS 98 prototype was assembled in February 1998 when the outside air
temperature was always below the specified clean room temperature. 
Therefore the air conditioning system worked in the heating regime
with the larger air temperature variations of $\pm 1^\circ$C
at a period of 30~minutes which also was observed during the chamber construction.

To study temperature gradients in the air, four sensors were placed in different
locations in the clean room: one on the wall in a
corner of the room close to the floor, one at the same height in an opposite corner,
one at 3 meters height above the first sensor and the fourth sensor
at a distance of 5~cm above the granite table.
No temperature differences larger than $\pm 0.1^\circ$C have been detected between the
four sensors in both air conditioning regimes.

The four temperature measurements at the bottom of the granite table
agree with each other with an accuracy of $\pm 0.1^\circ$C. The same is true for the
four measurements on top of the granite table at the combs.
The averages of the temperature measurements in the four corners at the top (on the combs)
and at the bottom of the granite table are shown in
Figures~\ref{fig:2_regimes}, \ref{fig:night} and \ref{fig:day}.
The temperature variations of granite table and combs are much smaller than the
periodic temperature variations of the air in both air conditioning regimes.

There is, however, a small temperature gradient between the granite table surface and the air
(see Figure~\ref{fig:diff}).
The average temperature difference of the bottom of the granite table with respect to the air
is $+0.22^\circ$C during the night and $+0.34^\circ$C during the day. The combs on top of
the table show an average temperature difference relative to the air of $+0.22^\circ$C during the night
and of $+0.50^\circ$C during the day. Therefore there is an
average temperature difference of maximum $+0.16^\circ$C
between top (combs) and bottom of the granite table only during the day
which is within the range recommended by manufacturers for a long lifetime of the flatness 
of the granite table (see \cite{QAQC}).

The temperature difference between granite table and air can be explained 
by the illumination of the assembly room heating the granite table.
The electric lights were switched on continuously 
during the measurements shown. After switching off the lights in the clean room,
the temperature of the granite table approaches the average air temperature during the night.
During the day there is an additional source, most likely the day-light, heating mostly the 
top surface of the granite table.

The temperatures of the end combs define the
$z$-pitch of the tube layer placed on them. The $z$-pitch of the tube layers
already glued to the spacer is also sensitive to the air temperature.
The maximum difference between comb pitch and $z$-pitch of the tube layers already 
glued to the spacer which can be caused by the observed temperature difference of $0.4^\circ$C between
combs and air is

\[ \Delta_{z} = 0.4^\circ {\mathrm C}\cdot 27~\frac{\mu {\mathrm m}}{{\mathrm m}\; {}^\circ {\mathrm C}}\cdot \
\frac{2.160~{\mathrm m}}{72} \simeq 0.3~\mu{\mathrm m}. \]

If this difference in pitch between successive tube layers were completely frozen in the chamber
it would be at the limit of what can be tolerated for the precision of the chamber.

During the prototype assembly, the lights in the clean room were
switched off during the night when the glue between successive tube layers
was curing. The $z$-pitch of the 6 tube layers of the prototype chamber is very uniform
with a maximum variation of $0.4~\mu$m on the high-voltage and the readout side
as verified by the X-ray tomograph measurements (see \cite{xtomo}).


\subsection{Constraints of the Sphere Towers}
%
Originally \cite{TDR} the sphere support towers provided the following
mechanical constraints on the spacer orientation on the assembly table: the sphere
support of tower T1 was fixed both in $x$- and in $z$-direction, the sphere
supports of towers T2 and T3 were only free in $x$-direction and fixed in $z$
and the spheres on towers T4, T5 and T6 were free both in $x$- and in
$z$-direction.

The results of the reproducibility test with this setup are shown
in Table~\ref{tab:repr1}. The spacer was set on the sphere towers, lifted
with the crane and released again several times. Every time the spacer was released on
the sphere towers the crane was disconnected and the crane supports on the spacer
were dismounted.

\begin{table}[htb]
\caption{\small 
Reproducibility of the (longitudinal) in-plane monitor readings before
releasing the $z$-constraint of sphere tower T2}
\begin{center}
\begin{tabular}{|c|c|c|c||} \hline
{Attempt}&{$y_{\mathrm IP-1}$ $[\mu$m$]$}&{$y_{\mathrm IP-4}$ $[\mu$m$]$}&{$z_{\mathrm IP-1,4}$ $[\mu$m$]$}\\ \hline
{1}&{4}&{2}&{1-2}\\
{2}&{20}&{25}&{20-30}\\
{3}&{2}&{6}&{5-7}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:repr1}
\end{table}

The in-plane monitor readings varied by about $20-30~\mu$m in these tests and were also unstable.
It was found that the sphere supports in their original configuration over-constrained
the spacer structure when it was inverted, namely the $z$-position of the middle
cross plate. This led to irreproducible positioning of the spheres on tower T2 in $z$ and $y$
measured by the in-plane monitors.
After releasing the $z$-constraint on sphere tower T2,
the reproducibility of the monitor readings improved to the expected level of a few micron.

\subsection{Test of Sphere Tower Planarity}
%
The planarity of the system of six sphere towers can be verified by comparing
the in-plane monitor readings in the initial and the inverted orientation of the
spacer. As explained in Figure~\ref{fig:inv}, half the difference in
the $y$-readings of in-plane monitor IP-1 in the two spacer orientations determines the
deviation of the heights on the sphere towers T4, T5 and T6 from a straight line. 
The same applies for towers T1--T3 and monitor IP-4. Equal height of the towers on the
right- and the left-hand side cannot be tested with this method.

\begin{figure}[thb]
\begin{center}
\mbox{\epsfig{figure=inv.eps,width=0.9\textwidth}}
\caption{\small
   Effect of unequal height of the sphere towers on the $y$-readings of the in-plane monitors}
\label{fig:inv}
\end{center}
\end{figure}

The $y$-readings of monitor IP-1 showed an initial deviation of $-25~\mu$m of the
height of tower T5 from the line connecting towers T4 and T6 which was confirmed
by direct measurement and corrected. After the correction, the test showed
deviations of both tower lines, T1--T3 and T4--T6, from a straight line of 
not more than $\pm$ a few micron.


\subsection{Preparation of the Spacer}
%
The in-plane and cross plate RASNIK systems were continuously operational 
during the chamber construction including when the spacer
was supported on the crane and not on the assembly table. Comparison of the
readings of the RASNIK monitors in the four orientations of the spacer on the crane, 
two vertical (V1, V2) and two horizontal (H1, H2) orientations (see Figure~\ref{fig:crane}), 
and in the two orientations, initial (INI) and inverted (INV), of the spacer on the
sphere towers allows to detect internal stresses of the assembled spacer.
Using this information, the long-beam fixations on the middle cross plate were adjusted 
to the center of the cross plate in order to minimize internal stresses caused
by asymmetric fixation. Such built-in stresses lead to unpredictable up-down asymmetric
sag of the cross plates and left-right asymmetric sag of the chamber along the long beams.

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=crane.eps,width=0.7\textwidth}}
\caption{\small
The four orientations of the chamber when supported on the crane}
\label{fig:crane}
\end{large}
\end{center}
\end{figure}

\subsubsection{Adjustment of the Long-Beam Fixations}
%
Initially, the long-beams were fixed to the middle cross-plates
in preliminary positions close to their median plane.
The $y$-readings of the longitudinal in-plane monitors IP-1 and IP-4 in four
successive states of the spacer, INI, V1, V2, INV, before further adjustments 
are shown in Table~\ref{tab:adj1}.

The readings in the states INI and INV are equal for each
monitor since they are defined by the height of the sphere towers.
Without internal stresses in the spacer, the monitor readings
in the vertical orientations V1 and V2 on the crane are expected 
to be the same as for the spacer on the sphere towers since there are
no forces to deform the spacer. 
The data in Table~\ref{tab:adj1} indicate a torsional deformation of the spacer
in the vertical orientations on the crane which can be explained
by internal stresses in the spacer due to imperfect centering of the
long beam fixations on the middle cross plate.

\begin{table}[thb]
\caption{\small 
The $y$-readings $[$mm$]$ of the longitudinal in-plane monitors IP-1 and IP-4 and of the
sag monitors S-RO, S-MI and S-HV on the three cross plates in the
initial (INI) and inverted (INV) orientations of the spacer on the sphere towers
and in the vertical positions V1 and V2 on the crane and the average of the latter 
{\bf before} adjustment of the long beams.
The gravitational sags $[\mu$m$]$ of the cross plates in the INI and INV positions
calculated from these readings are also given.}

\begin{center}
\begin{tabular}{|l|c|c||c|c|c||c|c|c||} \hline
{Orientation}&{$y_{\mathrm IP-1}$}&{$y_{\mathrm IP-4}$} &
$y_{\mathrm S-RO}$ & $y_{\mathrm S-MI}$ & $y_{\mathrm S-HV}$ & 
${\mathrm Sag}_{\mathrm CP-RO}$ & ${\mathrm Sag}_{\mathrm CP-MI}$ & ${\mathrm Sag}_{\mathrm CP-HV}$ \\
& $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ \\
\hline\hline
{INI}&{16.841}&{17.039}&{10.377}&{13.834}&{10.403}& 55 & $-7$ & 56 \\
\hline
{V1}&{17.203}&{17.422}&{10.263}&{13.846}&{10.289}& & & \\
{V2}&{17.129}&{17.472}&{10.271}&{13.845}&{10.292}& & & \\
$<V1,V2>$ &{17.166}&{17.457}&{10.267}&{13.846}&{10.291}& & & \\
\hline
{INV}&{16.866}&{17.043}&{10.252}&{13.675}&{10.269}& \hphantom{5}8 & 86 & 11 \\
\hline\hline
\end{tabular}
\end{center}
\label{tab:adj1}
\end{table}

The difference between the in-plane monitor readings in the two vertical orientations can
be explained by measurement errors in the $y$- and $z$-coordinates
due to the unknown relative angular orientations of masks and CCDs of the RASNIK monitors.
The relative vertical displacement of the middle cross plate 
between the states V1 and V2 (sag in $z$-direction in the chamber coordinate frame)
is as large as $\sim 5~mm$. If masks and CCDs are not perfectly aligned which was not 
attempted, the large displacement leads to sizable differences also in the horizontal 
($y$-) readings of the in-plane monitors in orientations V1 and V2.

The internal stresses of the spacer have an additional consequence.
In the absence of stresses, the cross-plates should show no
deformations in the vertical orientations on the crane and
the gravitational sags of the cross plates in the states INI and INV 
should be of equal size with respect to vanishing
sag in the vertical states V1 and V2. 

The gravitational sags of the cross-plates in the INI and INV states
can be calculated from the corresponding $y$-readings
of the cross plate RASNIK monitors in Table~\ref{tab:adj1} and the
average of the $y$-readings in the two vertical orientations as reference according to the
following relations (the factor 1/2 arises from the conversion of RASNIK readings, i.e.\
displacement of the mask image on the CCD, to lens displacement):

\begin{eqnarray*}
{\mathrm Sag}_{CP}^{INI} &=& \frac{y_{\mathrm S-CP}^{INI} - y_{\mathrm S-CP}^{V}}{2}, \\
{\mathrm Sag}_{CP}^{INV} &=& \frac{y_{\mathrm S-CP}^{V}   - y_{\mathrm S-CP}^{INV}}{2}.
\end{eqnarray*}

The sags of the three cross plates in the four states
INI, V1, V2 and INV of the spacer
are given in Table~\ref{tab:adj1} and Figures~\ref{fig:cpro} and \ref{fig:cphv}.
Before further adjustments of the long beams, the measured cross plate sags show a considerable asymmetry
between the INI and INV orientation providing ambiguous input for the cross plate sag compensation
during assembly of the tube layers.

\begin{figure}[p]
\begin{center}
\vspace{-12.5mm}
\mbox{\epsfig{figure=cpro.eps,width=0.67\textwidth}}
\vspace{-7mm}

\caption{\small $y$-deformations (with respect to the INI orientation) of the
readout (RO)-side cross plate measured with the cross plate RASNIK monitor
before and after long beam adjustment
(see text) in the initial (INI) and inverted (INV) orientation of the spacer
on the sphere towers and in the four orientations (V1, V2, H1, H2) of the spacer on the crane.
After long beam adjustment, the cross plate sags in the INI and INV states are symmetric
with respect to the bending in the vertical
orientations and can be determined as the average of the INI and INV
readings.}
\label{fig:cpro}

\mbox{\epsfig{figure=cphv.eps,width=0.67\textwidth}}
\vspace{-7mm}

\caption{\small Same as Figure~\ref{fig:cpro} for the high voltage (HV)-side cross plate}
\label{fig:cphv}
\end{center}
\end{figure}

\begin{figure}[thb]
\begin{center}
\mbox{\epsfig{figure=longbeam_1.eps,width=0.6\textwidth}}
\caption{\small
 Adjustment of the long beams on the middle cross plate}
\label{fig:longbeam}
\end{center}
\end{figure}

The following model was employed for the final adjustment of the long beams to 
remove the internal stresses of the spacer.

The relative $y$-displacements of the middle cross-plate in the chamber coordinate system are considered
in two states of the spacer, on the sphere-holders (INI) and in vertical orientation on the crane, 
as illustrated in Figure~\ref{fig:longbeam}. 
The $y$-displacements $\Delta y_{\mathrm IP-1,4}$ of the lenses of the 
two longitudinal in-plane RASNIK monitors IP-1 and IP-4
in the vertical orientation (average for V1 and V2) relative to the reference position on the sphere towers (INI)
determine the required adjustments $\Delta y_{\mathrm LB-L}$ and $\Delta y_{\mathrm LB-R}$ of the two long beams on the
middle cross plate using a linear interpolation between the $z$-positions $z_{\mathrm IP-1,4}$ of the lenses:

\begin{equation}
\Delta y_{{\mathrm LB-}i} = -\Delta y_{\mathrm IP-1} -
   \frac{\Delta y_{\mathrm IP-4} - \Delta y_{\mathrm IP-1}}{z_{\mathrm IP-4} - z_{\mathrm IP-1}}\cdot
    (z_{{\mathrm LB-}i} - z_{\mathrm IP-1}),\;\;  i = L,\, R,
\label{eq:inter}
\end{equation}

where $z_{{\mathrm LB-}i}$ are the $z$-coordinates of two long-beams on the middle cross plate.

From the data in Table~\ref{tab:adj1} one derives

\begin{eqnarray*}
\Delta y_{\mathrm IP-1} &=& \frac{<y_{\mathrm IP-1}^{V}> - y_{\mathrm IP-1}^{INI}}{2} = 166~\mu{\mathrm m}, \\
\Delta y_{\mathrm IP-4} &=& \frac{<y_{\mathrm IP-4}^{V}> - y_{\mathrm IP-4}^{INI}}{2} = 206~\mu{\mathrm m}.
\end{eqnarray*}

According to equation (\ref{eq:inter}) the long beam fixations on the middle
cross-plate have been shifted by

\begin{eqnarray*}
\Delta y_{\mathrm LB-L} &=& -y_{\mathrm IP-1} -
\frac{\Delta y_{\mathrm IP-4}-\Delta y_{\mathrm IP-1}}{z_{\mathrm IP-4}-z_{\mathrm IP-1}}\cdot
(z_{\mathrm LB-L}- z_{\mathrm IP-1}) \\
&=& -166~\mu m +    \frac{206~\mu m - 166~\mu m}{1500~mm}\cdot
    300~mm \simeq -150~\mu{\mathrm m}
\end{eqnarray*}

and

\begin{eqnarray*}
\Delta y_{\mathrm LB-R} &=& -y_{\mathrm IP-1} -
\frac{\Delta y_{\mathrm IP-4}-\Delta y_{\mathrm IP-1}}{z_{\mathrm IP-4}-z_{\mathrm IP-1}}\cdot
(z_{\mathrm LB-R}- z_{\mathrm IP-1}) \\
&=& -166~\mu m -    \frac{206~\mu m - 166~\mu m}{1500~mm}\cdot
    1200~mm \simeq -200~\mu{\mathrm m} 
\end{eqnarray*}

in the chamber coordinate frame in Figure~\ref{fig:frame} (initial orientation of the spacer).

The $y$-readings of the in-plane and
cross-plate monitors and the corresponding cross plate sags after the adjustement are given in
Table~\ref{tab:adj2} and Figures~\ref{fig:cpro} and \ref{fig:cphv}.
The cross plate sags in INI and INV orientation are now very symmetric.
One concludes that internal stresses of the spacer have effectively been removed.
An accuracy of $50~\mu$m in the adjustment of the long beam fixations was sufficient
to achieve this.

The residual stresses in the spacer after the adjustment still cause an asymmetry in the longitudinal
gravitational sag of the chamber measured on the left- and right-hand side with
monitors IP-1 and IP-4 as well as between the sag in initial and inverted orientation  
while the average sag of both sides is quite equal for the two chamber orientations (see Table~\ref{tab:lrsag}).
With the in-plane monitor  measurements, the sag can be compensated individually for each long beam
with the integrated sag compensation systems in the actual chamber orientation in the experiment.
The asymmetry in the chamber sag
which cannot be predicted by calculations can in principle be further reduced by adjusting the
long beams with higher accuracy than $50~\mu$m.

\begin{table}[bht]
\caption{\small 
The $y$-readings of the longitudinal in-plane monitors IP-1 and IP-4 and of the
sag monitors S-RO, S-MI and S-HV on the three cross plates in the
initial (INI) and inverted (INV) orientations of the spacer on the sphere towers
and in the vertical positions V1 and V2 on the crane and the average of the latter 
{\bf after} final adjustment of the long beams.
The gravitational sags of the cross plates in the INI and INV positions
calculated from these readings are also given.}

\begin{center}
\begin{tabular}{|l|c|c||c|c|c||c|c|c||} \hline
{Orientation}&{$y_{\mathrm IP-1}$}&{$y_{\mathrm IP-4}$} &
$y_{\mathrm S-RO}$ & $y_{\mathrm S-MI}$ & $y_{\mathrm S-HV}$ & 
${\mathrm Sag}_{\mathrm CP-RO}$ & ${\mathrm Sag}_{\mathrm CP-MI}$ & ${\mathrm Sag}_{\mathrm CP-HV}$ \\
& $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[$mm$]$ & $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ \\
\hline\hline
{INI}&{16.759}&{16.985}&{10.316}&{13.933}&{10.353}& 24 & 37 & 23  \\
\hline
{V1}&{16.735}&{17.029}&{10.265}&{13.858}&{10.285}& & & \\
{V2}&{16.661}&{17.086}&{10.271}&{13.862}&{10.292}& & & \\
{$<V1,V2>$}&{16.648}&{17.057}&{10.268}&{13.860}&{10.289}& & & \\
\hline
{INV}&{16.792}&{16.996}&{10.205}&{13.789}&{10.235}& 32 & 36 & 27 \\
\hline\hline
\end{tabular}
\end{center}
\label{tab:adj2}
\end{table}

\begin{table}[hbt]
\caption{\small 
Longitudinal sag (in negative $y$-direction)
on the left- and right-hand side of the chamber supported on the crane
measured with the in-plane RASNIK monitors IP-1 and IP-4 relative to their zero-readings (no sag) during
assembly with increasing number of tube layers glued and after dismounting of the spheres from the completed
chamber. The measurements on the pseudo-kinematic supports in the X-ray tomograph are also given.}

\begin{center}
\begin{tabular}{|l|c|c|c||c|c|c||} \hline
Orientation & \multicolumn{3}{c||}{Initial} & \multicolumn{3}{c||}{Inverted} \\
\hline
Monitor     & Sag${}_{IP-1}$ & Sag${}_{IP-4}$ & Average sag & Sag${}_{IP-1}$ & Sag${}_{IP-4}$ & Average sag \\
& $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ & $[\mu$m$]$ \\
\hline\hline
Zero-reading & 0 & 0 &  & 0 & 0 & \\
\hline
Spacer       & 444 & 366 & 405 & 367 & 447 & 407 \\
\hline
Layer 1 (ML~1)     & 430 & 356 & 393 &     &     &     \\
Layer 2 (ML~1)     & 471 & 394 & 433 &     &     &     \\
Layer 3 (ML~1)     & 485 & 416 & 451 & 424 & 484 & 454 \\
\hline
Layer 4 (ML~2)     &     &     &     & 429 & 471 & 450 \\
Layer 5 (ML~2)     &     &     &     & 453 & 502 & 478 \\
Layer 6 (ML~2)     & 504 & 470 & 487 & 490 & 522 & 506 \\
\hline
Without spheres    & 433 & 392 & 413 & 386 & 425 & 406 \\
\hline
Tomograph support  & 423 & 381 &     &     &     &     \\
\hline\hline
\end{tabular}
\end{center}
\label{tab:lrsag}
\end{table}

\subsubsection{Thermal Properties of the Spacer}
%
The purpose of the RASNIK cross-plate sag monitors is to measure
the gravitational sag of the cross plates in $y$-direction during chamber assembly. 
But they are also able to measure cross-plate deformations in the second coordinate,
in $x$-direction parallel to the long beams. From the continuous monitoring
of the spacer stability on the sphere towers, periodic variations
of bending of the cross plates in $x$-direction have been observed 
(see Figure~\ref{fig:long_breath})
which show a clear correlation with the air temperature in the temperature
controlled assembly room which varied with a period of about 20 minutes 
due to the control cycle of the air-conditioning system.

\begin{figure}[thb]
\begin{large}
\mbox{\epsfig{figure=long_breath.eps,width=1.1\textwidth}}
\caption{\small
Variation with time of the bending of the three cross plates in $x$-direction 
before glueing of the tube layers}
\label{fig:long_breath}
\end{large}
\end{figure}

\begin{figure}[thb]
\mbox{\epsfig{figure=bos_cp_sags.eps,width=1.1\textwidth}}

\caption{\small Sag of the cross plates on the high voltage (HV) and
readout (RO) side and of the middle (MI) cross plate
before and after sag compensation during glueing of the tube layers 1--6
as measured with the cross plate RASNIK monitors (the $y$-axis is always
pointing upwards)}
\label{fig:cp_sag_comp}
\end{figure}

\begin{figure}[thb]
\begin{center}
\mbox{\epsfig{figure=bos_y_0123.eps,width=1.1\textwidth}}

\caption{\small
Stability of the $y$-readings of the in-plane RASNIK monitors IP-1...IP-4 during assembly
of the chamber (glueing of six tube layers) relative to the zero-readings taken with the spacer alone.
The corresponding relative displacements of the middle cross plate are a factor 1/2 smaller than
the readings shown (in the chamber coordinate system for the initial (INI) orientation).}
\label{fig:ip_z_stable}
\end{center}
\end{figure}

\begin{figure}[thb]
\begin{center}
\mbox{\epsfig{figure=bos_z_0123.eps,width=1.1\textwidth}}

\caption{\small
Stability of the $z$-readings of the in-plane RASNIK monitors IP-1...IP-4 during assembly
of the chamber (glueing of six tube layers) relative to the zero-readings taken with the spacer alone.
The corresponding relative displacements of the middle cross plate are a factor 1/2 smaller than
the readings shown (in the chamber coordinate system for the initial (INI) orientation).}
\label{fig:ip_y_stable}
\end{center}
\end{figure}


\begin{figure}[p]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=click-clack.eps,width=0.73\textwidth}}
\vspace{-7mm}

\caption{\small
   Variation with time of the $y$-sag of the two outer cross-plates after glueing of the $4^{\mathrm th}$ tube layer
   while the chamber is still on the jigs and the vacuum of the combs is on}
\label{fig:click-clack}

\vspace{-5mm}
\mbox{\epsfig{figure=click-clack_crane.eps,width=0.73\textwidth}}
\vspace{-7mm}

\caption{\small
   Variation with time of the $y$-sag of the two outer cross-plates after glueing of the $5^{\mathrm th}$ tube layer
   while the chamber is supported horizontally on the crane}
\label{fig:click-clack_crane}
\end{center}
\end{figure}

\begin{figure}[p]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=almy_bos98.eps,width=1.10\textwidth}}
\vspace{-62mm}

\caption{\small
Variations of the laser beam position measured on a reference sensor of the sphere monitoring system
on the assembly table at a distance of about 4~m from the laser collimator
as a function of time $[$hours$]$: $z$-coordinate $[$mm$]$ (top) and $y$-coordinate $[$mm$]$ (bottom) with
arbitrary offsets}
\label{fig:almy}
\end{center}
\end{figure}


\begin{figure}[p]
\begin{center}

\mbox{\epsfig{figure=plastic_nofoil.eps,width=1.10\textwidth}}
\vspace{-7mm}

\caption{\small
Correlation between the variations of the air temperature in the climatized assembly room
and of the vertical temperature difference between top and bottom of the plastic laser beam shielding
tube (2~cm diameter)}
\label{fig:grad1_y}
\end{center}
\end{figure}


\begin{figure}[p]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=plastic_foil.eps,width=0.73\textwidth}}
\vspace{-7mm}

\caption{\small
Correlation between the variations of the air temperature 
and of the vertical temperature difference between top and bottom of the plastic 
tube wrapped in aluminum foil}
\label{fig:grad2_y}

\mbox{\epsfig{figure=plastic_foil_horiz.eps,width=0.73\textwidth}}
\vspace{-7mm}

\caption{\small
Correlation between the variations of the air temperature 
and of the horizontal temperature difference between the walls of the plastic
tube wrapped in aluminum foil}
\label{fig:grad2_z}
\end{center}
\end{figure}


\begin{figure}[p]
\begin{center}
\vspace{-15mm}
\mbox{\epsfig{figure=clean_room_laser_test_02.eps,width=1.1\textwidth}}
\vspace{-7mm}

\caption{\small
Variations of the laser spot position with time at a distance of about 4~m from the laser collimator
on the granite table in the assembly room with different beam shielding tubes (MDT tubes have a wall
thickness of $400~\mu$m). The periodic variations follow the cycle of the air conditioning system in the
heating regime.}
\label{fig:laser}
\end{center}
\end{figure}


The measured sagittae of the two outer cross plates CP-RO and CP-HV
have opposite signs and equal amplitude (about $\pm 5~\mu$m 
while the effect for the middle cross plate is much smaller.
The effect can be explained by the thermal expansion and contraction
of the aluminum long-beams (by $\pm 10~\mu$m corresponding to a temperature
variation of the long beams of $\pm 0.1^\circ$C)
which makes the outer cross plates bend while there is little effect
on the middle cross plate.

The cross plates were supported on the sphere towers at both ends during the
measurements. All sphere supports except one are allowed to move in $x$-direction
with a certain friction. Uniform expansion of the long beams and movements
of the spheres in $x$ are not measured by the assembly monitoring systems.
Therefore the temperature variations and the associated length variations of the long beams
could still be larger than the above estimate if the spheres at the
outer cross plates moved in $x$-direction in addition to the observed bending
of the cross plates.

In any case, the observed small deformations of the spacer in $x$-direction will 
have no impact on the chamber assembly procedure.

\subsection{Zero-Readings of the RASNIK Monitors}
%
An important step of the assembly procedure is the calibration
of the RASNIK monitors. All RASNIK monitors were mounted on the spacer
with a positioning accuracy of a few millimeters such that they provided
sharp and analyzable images of the center area of the mask.
In order to monitor deformations of the chamber during chamber assembly
and afterwards, zero-readings of the RASNIK
in-plane alignment and cross plate sag monitors in the non-deformed state
on the assembly table have to be recorded as a reference.

The first zero-readings of the RASNIK systems at the start of the chamber construction
were taken with the assembled spacer alone: as the average of the
RASNIK readings in two spacer positions on the sphere towers, in the initial
(INI) and in the inverted (INV) orientation. Subsequently, the stablility
of the in-plane monitor readings was monitored over the whole construction.
The cross plate sag monitor data were used for the compensation of the cross plate sag
with the force compensation system (see \cite{BOS1}) before the glueing
of each tube layer and to measure the cross plate sag on the supports during the
X-ray tomograph measurements.

\subsubsection{In-Plane Alignment System}

As described above, comparison of the $y$-readings of the two longitudinal
in-plane monitors IP-1 and IP-4 in the initial and
inverted positions of the spacer on the assembly table
allowed to test the planarity the sphere tower heights. After a small
correction to one sphere tower, deviations from planarity and differences
in the readings of the monitors for the two orientations were
virtually negligible (see Table~\ref{tab:adj2}). 
The average values of the readings for both orientations
were taken as zero-readings for monitors IP-1 and IP-4 during chamber
construction.

The $y$-readings of the two diagonal in-plane monitors IP-2 and IP-3 are
affected by the gravitational sag of the middle cross-plate. When the
long-beam fixations on the middle cross-plate were re-adjusted to minimize
internal stresses of the spacer, the gravitational sags of all cross-plates
became equal in size in the initial and the inverted spacer orientation
(see Table~\ref{tab:adj2}). This means that the average values of the 
$y$-readings of IP-2 and IP-3 in the INI and INV positions must correspond 
to the situation when these monitors are aligned on straight lines and therefore
can be taken as the zero-readings.

The readings of the four in-plane monitors in the INI and INV positions
of the spacer and their averages (the zero-readings) are given in Table~\ref{tab:ip-calib}.
During chamber assembly, monitoring of the stability of the $y$-measurements 
of the in-plane monitors is essential to control the planarity of the chamber (see section 3.1,
Table~\ref{tab:ip_y_stable} and Figure~\ref{fig:ip_y_stable}).

\begin{table}[htb]
\caption{\small Zero-readings $y_0$ and $z_0$ of the in-plane RASNIK monitors at the start of chamber assembly
defined as the averages of the monitor readings in the initial (INI) and
the inverted (INV) spacer position on the sphere towers.}

\begin{center}
\begin{tabular}{|c||c|c|c|c||c|c||} \hline
{Monitor}&{$y_{INI}$ $[$mm$]$}&{$z_{INI}$ $[$mm$]$}&{$y_{INV}$ $[$mm$]$}&{$z_{INV}$ $[$mm$]$}&$y_0$ $[$mm$]$&$z_0$ $[$mm$]$\\ \hline
{IP-1}&{16.752}&{14.580}&{16.800}&{14.585}&{16.778}&{14.583}\\
{IP-2}&{17.344}&{14.343}&{17.476}&{14.346}&{17.407}&{14.345}\\
{IP-3}&{16.118}&{\hphantom{1}7.499}&{16.256}&{\hphantom{1}7.499}&{16.178}&{\hphantom{1}7.499}\\
{IP-4}&{16.961}&{10.236}&{17.005}&{10.238}&{16.983}&{10.237}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:ip-calib}
\end{table}

After removal of the over-constraint in the sphere supports on the assembly table,
differences between the $z$-readings of the in-plane monitors in the initial and the
inverted spacer position are very small and only reflect the reproducibility of the
monitor measurements themselves.
After the glueing of the first tube layer to the cross plates, movements of the
middle cross plate are prevented by the stiffness of the tube layers.
The averages of the $z$-readings in INI and INV orientation were defined as the
zero-readings (see Table~\ref{tab:ip-calib}) which were very stable during the
whole construction with relative displacements of the middle cross plate less than $8~\mu$m
(see Table~\ref{tab:ip_z_stable} and Figure~\ref{fig:ip_z_stable}). This shows that
no significant temperature gradients occurred between the two chamber ends during the
construction.

\subsubsection{Cross-Plate Sag Monitors}
%
For the cross plate sag monitors the same procedure to determine the zero-readings
can be applied as for the in-plane alignment system.
After the re-adjustment of the long-beams the cross-plate
sags became symmetrical in the INI and INV positions such that the average
values of the monitor readings for both orientations corresponded to the non-deformed
state of the cross-plates.
The readings of the cross-plate monitors in the two orientations and the 
zero-readings derived from them are given in Table~\ref{tab:cp-calib}.

\begin{table}[htb]
\caption{\small Zero-readings $x_0$ and $y_0$ of the cross plate RASNIK monitors at the start of chamber assembly
defined as the averages of the monitor readings in the initial (INI) and
the inverted (INV) spacer position on the sphere towers.}

\begin{center}
\begin{tabular}{|c||c|c|c|c||c|c||} \hline
{Monitor}&{$x_{INI}$ $[$mm$]$}&{$y_{INI}$ $[$mm$]$}&{$x_{INV}$ $[$mm$]$}&{$y_{INV}$ $[$mm$]$}&$x_0$ $[$mm$]$&$y_0$ $[$mm$]$\\ \hline
{CP-HV}&{14.441}&{10.316}&{14.446}&{10.205}&{14.444}&{10.271}\\
{CP-MI}&{12.846}&{13.933}&{12.843}&{13.789}&{12.845}&{13.865}\\
{CP-RO}&{12.088}&{10.353}&{12.076}&{10.235}&{12.082}&{10.291}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:cp-calib}
\end{table}

\section{Monitoring of the Assembly Process}
%

\subsection{In-plane Alignment System During Assembly}
%
The role of the in-plane monitors during chamber assembly was the following:

\begin{enumerate}

\item Verification of the planarity of the sphere supports before glueing
of each tube layer.

\item Verification of the planarity of the chamber after glueing
of each tube layer. 

The readings of the in-plane monitors should be
stable during the assembly process and show no significant deviations from the zero-readings.

\item Input for the reconstruction of the sag of the middle cross-plate 
after removal of the sag monitor on this cross plate for the assembly of the
second multilayer.

\item Measurement of the longitudinal gravitational sag of the chamber for 
adjustment to the wire sag (separate for each long beam; see Table~\ref{tab:lrsag}).

\end{enumerate}

The deviations of the $y$-measurements of the in-plane monitors from their zero-readings
after glueing of each tube layer are shown in Table~\ref{tab:ip_y_stable} and Figure~\ref{fig:ip_y_stable}.
Changes of the longitudinal sags of the chamber on the assembly table 
as measured by the in-plane monitors IP-1 and IP-4 relative to the zero-readings
are below $15~\mu$m. 

During assembly of the first
multilayer (tube layers 1--3), the $y$-readings of all monitors were very stable.
Larger deviations ocurred during the assembly of the second multilayer (tube
layers 4--6) which can be explained by the larger sag of the middle cross plate
increasing with the number of tube layers glued
(see \cite{BOS1} and Figure~\ref{fig:cp_sag_comp}).
The sag of the middle cross plate has no effect on the wire positioning accuracy
and has not been fully compensated during the assembly of the second multilayer.
The diagonal monitors IP-2 and IP-3 show about twice as large deviations
from the zero-readings than the longitudinal monitors  
(maximum: $30~\mu$m) since they are more affected by the sag of the middle cross plate.

The $z$-readings of all four in-plane monitors were very stable during the
whole assembly process (see Table~\ref{tab:ip_z_stable} and Figure~\ref{fig:ip_z_stable}).

%The stability of the in-plane monitor readings during curing of the glue over night for the
%assembly of a tube layer is shown in Figure???

\begin{table}[thb]
\caption{\small  Stability of the $y$-measurements of the in-plane monitors during chamber assembly:
movements of the lenses (middle cross plate) in $y$ $[\mu$m$]$ with respect to the zero-readings
after glueing of a tube layer ($0.5\times$ the values of the $y$-readings shown in Figure~\ref{fig:ip_y_stable}).}

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c||} \hline
{Monitor}&{Layer 1}&{Layer 2}&{Layer 3}&{Layer 4}&{Layer 5}&{Layer 6}\\
\hline\hline
{IP-1}&{$-3$}&{\hphantom{$-$}3}&{$-\hphantom{1}1$}&{\hphantom{1}8}&{\hphantom{1}2}&{\hphantom{1}9}\\
\hline
{IP-2}&{$-6$}&{$-2$}&{$-15$}&{15}&{24}&{13}\\
{IP-3}&{$-2$}&{\hphantom{$-$}6}&{\hphantom{-1}0}&{18}&{22}&{30}\\
\hline
{IP-4}&{$-1$}&{\hphantom{$-$}1}&{$-\hphantom{1}9$}&{\hphantom{1}5}&{16}&{13}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:ip_y_stable}
\end{table}

\begin{table}[htb]
\caption{\small  Stability of the $z$-measurements of the in-plane monitors during chamber assembly:
movements of the lenses (middle cross plate) in $z$ $[\mu$m$]$ with respect to the zero-readings
after glueing of a tube layer ($0.5\times$ the values of the $z$-readings shown in Figure~\ref{fig:ip_z_stable}).}

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c||} \hline
{Monitor}&{Layer 1}&{Layer 2}&{Layer 3}&{Layer 4}&{Layer 5}&{Layer 6}\\
\hline\hline
{IP-1}&{4}&{4}&{2}&{1}&{2}&{8}\\
\hline
{IP-2}&{4}&{5}&{4}&{1}&{0}&{4}\\
{IP-3}&{4}&{2}&{4}&{1}&{4}&{9}\\
\hline
{IP-4}&{3}&{2}&{4}&{3}&{3}&{7}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:ip_z_stable}
\end{table}

\subsection{Cross-Plate Monitors During Assembly}
%
The cross plate monitors were used to measure the gravitational sag of the
cross plates supported on the sphere towers with increasing number of tube layers glued to the spacer
in order compensate it with the force compensation system of the assembly setup (see \cite{BOS1}).
The cross plate sags measured with the cross plate monitors before and after
sag compensation (see \cite{BOS1}) are shown in Figure~\ref{fig:cp_sag_comp}.

For the RASNIK monitor on the
middle cross-plate space was only available during the assembly of the first
multilayer. For the assembly of the second multilayer, the sag of the middle
cross plate was reconstructed from the data of the two outer cross plate monitors
and of the four in-plane monitors. This method was tested with the spacer before
the glueing of the tube layers and found to be very accurate with only $1-2~\mu$m
error (see \cite{BOS1}).

During continuous monitoring of the cross plate sags while the chamber was supported on the sphere towers 
or on the crane small periodic instabilities of the measurements in close
correlation with the temperature cycle in the climatized room were observed which are
shown in Figures~\ref{fig:click-clack} and \ref{fig:click-clack_crane}, respectively.


\subsection{Sphere Monitors}
%
For the monitoring of the $y$- and $z$-positions of the six spheres on the sphere towers
two multi-point monitors were foreseen, each with a laser beam pointing along one side of the chamber
and passing through transparent MPA-ALMY sensors mounted on the assembly table as reference and on the
sphere holders at the ends of the cross plates (see \cite{ALMY} and Figure~\ref{fig:frame}).
The laser beam paths were protected with plastic tubes of 2~cm~$\times$~2~cm cross section.

When the laser beam sphere monitors were operated in the climatized assembly room
with the plastic laser beam shielding tubes, it turned out that their performance
was considerably deteriorated by unexpectedly large instabilities in the pointing
of the two laser beams.

The laser spot position in vertical direction ($y$-coordinate)
on the reference sensor MP-RR2 the furthest away
from the laser source (see Figure~\ref{fig:frame}) is shown in Figure~\ref{fig:almy}
as a function of time. Oscillations with
amplitudes as large as $80~\mu$m were observed with a period of 30~minutes
perfectly correlated with the cycle of the air conditioning system and the air temperature.
In horizontal ($z$-) direction, the
oscillations of the laser beam position on the same sensor have the same period but
show a smaller amplitude of $20~\mu$m  (see Figure~\ref{fig:almy}).

The effect can be explained by a gradient of the air
temperature and therefore of the air density and refractive index $n(y)$ transverse to the laser beam
inside the shielding tubes.
To produce a deflection of the light rays of $\Delta y =80~\mu m$ over a $L=4$~m long path a
temperature gradient of $dT/dy = 0.1^\circ$C/cm is required according to the relation

\begin{eqnarray*}
\Delta y &\simeq& {d n\over d y}\cdot {L^2\over 2} \\
&=& a\cdot (n_0 -1) {d T\over dy}\cdot {L^2\over 2} \simeq {10^{-6}\over {}^\circ\, {\mathrm C}}\cdot {dT\over dy}
\cdot {L^2\over 2}
\end{eqnarray*}

where the temperature dependence of the refractive index of air is parametrized by

\[ n(T) = 1 + {n_0 -1\over 1 + a\cdot T [{}^\circ {\mathrm C}]} \]

with $n_0 = n(T=0^\circ {\mathrm C}) = 1+ 3\cdot 10^{-4}$ and $a=3.6\cdot 10^{-3}/{}^\circ {\mathrm C}$.

Measurements of the temperature of the plastic tube walls confirmed that such large
temperature gradients are indeed present inside the plastic shielding tubes
in the climatized room. Figure~\ref{fig:grad1_y} shows the
oscillations of the measured temperature difference between top and bottom of the plastic tube
with an amplitude of $0.2^\circ$C and with the period of 30~minutes of the
air conditioning system. The temperature gradient inside the
2~cm~$\times$~2~cm tubes therefore varies by $\pm 0.1^\circ$C/cm in good agreement with 
the expected value. In addition, the average temperature difference between top and bottom
differs from zero by about $+0.2^\circ$C. A constant vertical temperature gradient on the shielding tube
can be caused by the continuous illumination of the tube from above during the measurements (in a similar
way as observed for the granite table; see section 1).

The following solutions of the problem have been investigated:

\begin{enumerate}

\item Attempts to sufficiently insulate the shielding tubes thermally from the environment
did not succeed.

\item Wrapping of the plastic tubes in aluminum foil.

Measurements of the vertical and horizontal temperature difference between the walls of
of a plastic tube with and without wrapping in the assembly room (in the heating regime of the
air conditioning system) are shown in Figures~\ref{fig:grad2_y} and \ref{fig:grad2_z}.
The amplitude of the oscillations of the temperature gradient with the cycle of 
the air conditioning system as well as the average temperature gradient
are both reduced by an order of magnitude compared to the unwrapped plastic tube
due to reflection of the ambient light and the better thermal conductivity of the tube wall. 
In horizontal direction, oscillations of the temperature gradient of the same magnitude as in vertical
direction are measured but the average temperature gradient is zero in contrast to the vertical case.

\item Heat conducting metal shielding tubes.

Such tubes can relatively easily be installed. The decoupling of the inside
of the tubes from temperature gradients developing on the outer surface improves
with increasing wall thickness (see Figure~\ref{fig:laser}).

With a round aluminum tube with 30~mm outer diameter and a wall thickness of 1~mm
the temperature gradients inside the tube can be reduced such that the position variations of the laser
spot at a distance of about 4~m from the source are below $2~\mu$m in the clean room
in the heating regime. Figure~\ref{fig:laser} shows the measured variations with the cycle of the
air conditioning system. The laser beam stability achieved with the ``thick'' aluminum tube
is sufficient for the laser sphere monitoring system.

The effect of a temperature gradient constant in time, e.g.\ due to
illumination of the shielding tube from above was not measured.
If it exists for the ``thick'' aluminum tube, it can be reduced with better reflectivity of the tube outer wall.

Without shielding tube, the laser beam deflections in vertical direction are significant.
Figure~\ref{fig:laser} also shows that plastic shielding tubes stronly enhance the effect
of tenperature gradients in the air on the laser beam.

\item Increased local air flow in the room transverse to the
shielding tubes (using fans) in order to smoothen temperature gradients on the surface of the tubes.

\item Longitudinal air flow through the shielding tubes to smoothen temperature gradients inside.

The effect on the laser beam stability is rather sensitive to the magnitude of the air flow.
If it is too low, there is no improvement. If it is too high, turbulance, especially at the tube ends,
can even increase deflections of the laser beam. Therefore and because of the extra effort needed, this method
will not used.

\end{enumerate}
\clearpage

\section{Conclusions}
%
The detailed analysis of the optical monitoring data 
taken during the construction of the BOS~98 prototype chamber shows their 
importance for the control of the assembly process. 
The data have been used to optimize the assembly procedure.
The influence of the thermal environment in the assembly room 
on the chamber and on the optical monitors (laser beams) has been studied
and understood in detail.
The results of the analysis give confidence in the capability of the assembly method to
achieve the required mechanical accuracy of the chambers in series production.


\begin{thebibliography}{99}

\bibitem{BOS1} V.~Bartheld et al., {\em Construction of the Full-Scale
Prototype of a BOS MDT Chamber}, ATL-MUON-98-256, October 1998.

\bibitem{ALMY} H.~Kroha and S.~Schael, {\em Optical Monitoring System for
the MDT Assembly Stations}, ATL-MUON-97-156, April 1997.

\bibitem{TDR} {\em ATLAS Muon Spectrometer--Technical Design Report},
CERN/LHCC/97-22, ATLAS TDR 10, 31 May 1997.

\bibitem{xtomo} H.~Kroha, A.~Manz and A.~Ostapchuk, {\em Analysis of the X-Ray Tomograph
Data of the BOS MDT Prototype Chamber}, ATL-COM-MUON-99-025, August 1999.

\bibitem{QAQC} H.~Kroha, {\em Quality Assurance and Control Reference Document
for ATLAS MDT Chamber Construction}, ATL-COM-MUON-99-026, May 1999.

\end{thebibliography}

\end{document}
