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{\Large \bf TRIUMF Experiment E614 \\}
{\Large \bf Technical Note \\}

\vspace{0.4cm}

{\Large \bf Optimization of the Number of Sense Wires per Plane in the Dense
Chamber Array \\}

\vspace{0.4cm}

\rm{\bf D.H. Wright, TRIUMF \\}

\vspace{0.4cm}

\rm{\bf 9 February 1998 \\}

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\begin{abstract}
The number of sense wires per chamber in the dense array of 10 chambers at
either end of the E614 spectrometer was optimized and found to be 48.  This
was the smallest number of wires required to add extra track points for small
angle events and events in which the helix half-wavelength matches the spacing
between chamber pairs in the sparse chamber array.  This result assumes
that the spacing between chamber pairs in the sparse array is 4.8 cm.  If this
spacing were increased to 6.8 cm, 60 wires per plane in the dense array would
be required. 
\end{abstract}

The 10 closely spaced drift chambers (dense array) at either end of the E614
spectrometer were included to provide a profile monitor of the incoming muon
beam and to add extra track points to small-angle positron tracks.  It is
clear that the full 80 wires per plane used in the widely spaced drift chamber
pairs (sparse array) are not necessary in this case.  An optimum number was
found when the positron helix wavelength was compared to the spacing between
chamber pairs in the sparse array where the momentum and angle of the track are
determined.

In the present design the spacing between pairs in the sparse array is 4.8 cm.
Hence some positron tracks with wavelengths in the neighborhood of 9.6 cm will
generate track points with nearly identical $x$ (or $y$) coordinates
throughout the sparse array, leading to a poorly determined momentum and angle.
The helix wavelengths for which this is a problem occur roughly within the band
\begin{equation}
   \frac{1.25\pi}{4.8cm} > k > \frac{0.75\pi}{4.8cm} .
\end{equation}
As shown in Fig. 1 this band crosses a corner of the stated E614 acceptance,
$10^\circ < \theta < 60^\circ$, $ 0.4 < x < 1.0 $.

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\begin{figure}
\begin{center}
\epsffile{tn8fig1.ps}
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\caption{Acceptance coverage for a chamber-pair spacing of 4.8 cm.  Solid
lines: contours of helix wavenumber $k$ as a function of positron momentum and
angle.  Dashed lines: contours of helix radius as a function of momentum and
angle, with the number of wires per plane required to contain the helix
diameter.  Bold lines: acceptance region of E614 spectrometer.}
\end{figure}

\epsfysize=8cm
\begin{figure}
\begin{center}
\epsffile{tn8fig2.ps}
\end{center}
\caption{Acceptance coverage for a chamber-pair spacing of 6.8 cm.  Solid
lines: contours of helix wavenumber $k$.  Dashed lines: contours of helix
radius, with the number of wires per plane required to contain the helix
diameter.  Bold lines: acceptance region of E614 spectrometer.}
\end{figure}

This defect can be corrected by adding extra track points from the dense array,
but only if there are enough wires to contain the possible helix diameters when
the wavelength is small ($k > 0.75\pi/4.8cm$).  The number of wires per plane
$N_{wires}$ sufficient to contain a helix of diameter $2R_{max}$ is given by
\begin{equation}
   N_{wires} = 2(2R_{max}+2.0cm)/0.4cm
\end{equation}
where 0.4 cm is the sense wire spacing and 2.0 cm is the likely maximum offset
of the positron track origin from the $z$-axis due to the muon beam spot size
at the target.

In Fig. 1, the area beneath the dashed line represents all the momenta and
angles for which the dense array adds a full 10 points (5 $x$ and 5 $y$) to a
track.  The dashed line representing 80 wires per plane covers the entire
spectrometer acceptance indicating that many of the wires in that case are
redundant.  The dashed line representing 48 wires per plane covers only part
of the desired acceptance but enough to remedy the potential poor fits in the
short wavelength corner.  Coincidentally, considerations of wiring and packing
of front-end electronics determined that more than 48 wires per plane in the
dense array would be difficult \cite{Ama97}.  Hence 48 wires per plane in the
dense array was chosen for the final design.

This design decision has an impact on another question under consideration.
It was shown that increasing the spacing between chamber pairs in the sparse
array from 4.8 cm to 6.8 cm improved the angle and energy resolution
significantly \cite{TN2}.  This would involve substituting 6.0 cm spacers for
the 4.0 cm spacers which go between chamber pairs.  The effect on the above
wavenumber analysis is shown in Fig. 2.


Here the angle-energy region where poor fits would be likely has moved deeper
into the E614 acceptance region.   Assuming 48 wires per plane, part of the
acceptance around 33 MeV/c and 55$^\circ$ is not covered by the dense chamber
array.  60 wires per plane would be required.  If that number is not possible
then the increased spacing between chamber pairs in the sparse array should be
avoided.  It would however be possible to alternate 4 cm spacers with 6 cm
spacers in the sparse array, which would entirely remove the problem of
wavelength-spacing coincidence.

\begin{thebibliography}{10}

\bibitem{Ama97} P. Amaudruz and G. Sheffer, private communication 1997.
\bibitem{TN2} D.H. Wright, E614 Technical Note \# 2, 1997.

\end{thebibliography}

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