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\begin{titlepage}
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%%Nor%%\bf
%%Nor%%KEK Preprint 98-?? \\
%%Nor%%DPNU 98-42 \\
%%Nor%%KUDP 98-01 \\
%%Nor%%TUAT-HEP 98-02 \\
%%Nor%%KINDAI-HEP 98-01 \\
%%Nor%%November 1998 \\
%%Nor%%H \\
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\begin{center}
\large \bf
Designing of Stereo-Wire Geometry \\
for a Long Cylindrical Drift Chamber with \\
Small Jet Cells
\end{center}

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\begin{center}
\large 
	$N.Khalatyan^a$, $K.Fujii^a$, $H.Okuno^a$,
%%Nor%%	$T.Abe^c\footnote{Now at SLAC, California, USA}$, 
	$T.Abe^{c,1}$, 
	$K.Hoshina^b$, $Y.Kato^e$, $Y.Kurihara^a$, 
	$H.Kuroiwa^b$, $T.Matsui^a$, 
        $O.Nitoh^b$,
	$A.Sugiyama^c$, $K.Takahashi^b$, $T.Watanabe^d$,
	$T.Yoshida^a$,
\end{center}

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\begin{center}
$^a$ High Energy Accelerator Research Organization(KEK),
 Tsukuba, 305-0801, Japan\\ 
$^b$Tokyo University of Agriculture and Technology,
 Tokyo 184-8588, Japan\\
$^c$Department of Physics, Nagoya University, Nagoya 464-8602, Japan\\
$^d$Kogakuin University, Tokyo 163-8677, Japan \\
$^e$Kinki University, Osaka, 577-8502, Japan
\end{center}

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\begin{abstract}
Common problems in designing stereo-wire geometry
for a long cylindrical drift chamber with small jet cells are
described and possible ways to approach them are discussed in detail.
Among these problems, special emphasis is put on
the surface field variation along an anode wire due to 
the geometrical deformation of the cell structure.
Also estimated is the  surface field variation due to electrostatic 
and gravitational sags, based on the sags
measured for a 4.6~m-long test chamber.
These field variations are translated into gain
variation along the stereo wires and used to
constrain the stereo geometry.
\end{abstract}
\vfil

\footnotesize ${^1}$ \small\rm \underline {Now at SLAC, California, USA}

\end{titlepage}

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%  ------------
%  Introduction
%  ------------

\section{Introduction}

One of the most striking features of experiments
at a future linear $e^+e^-$ collider such as JLC\cite{Ref:JLC-I}
is the novel possibility of recognizing all physics processes
in terms of fundamental particles (leptons, quarks, and gauge bosons).
This involves explicit reconstructions of heavy unstable particles
such as $W$, $Z$, and $t$ through jet invariant-mass measurements.
In order to achieve the best attainable jet invariant-mass resolution,
it is crucial to make maximum use of information from a
central tracking device as well as that from calorimetry. 
Essential is thus good track-cluster matching capability
to avoid double counting, not to mention a high
momentum resolution with a wide angular acceptance.
A large cylindrical drift chamber with small jet cells is
a candidate central tracking device to
fulfill these requirements.
From the track-cluster matching point of view,
what we are concerned most is
track extrapolation errors, 
which depend on both $r$-$\phi$ and $z$ resolutions.
Improvement of the latter necessitates
introduction of stereo wires, since the $z$ resolution
in charge division or time difference readout
is typically $1~\%$ of the wire length or worse,
which is a few cm in the linear collider use.

In this paper we describe
common problems in designing stereo-wire geometry
for a long cylindrical drift chamber with small jet cells
and discuss possible ways to approach them in detail.
The surface field variation along an anode wire 
is one of the most important among these problems,
and deserves special attention.
Possible sources of such a field variation include
(1) the geometrical deformation of the cell structure
as a function of $z$ (the coordinate along the chamber axis)
due to stereo geometry, 
and (2) that due to electrostatic and gravitational sags.
These field variations are estimated through
explicit electric field calculations using
the chamber simulation program GARFIELD\cite{Ref:garfield}.
As for the second source, we use
the electrostatic and gravitational sags directly measured
for a 4.6~m-long test chamber\cite{Ref:sag}.
The estimated field variations are translated into gain
variation along the anode wires using the gas gain data
taken for this purpose.
We then use this gain variation as a key parameter to
constrain the stereo geometry.

The paper is organized as follows.
We begin with a general description of geometrical parameters
that determine the configuration of a cylindrical drift
chamber with axial and stereo layers of small jet cells,
and then outline the decision process for these parameters
in section \ref{Sec:outline}.
Taking a candidate central tracker for JLC as an example,
we will demonstrate the procedure in the following sections.
Finally section \ref{Sec:conclusions} summarizes our results
and concludes this paper.

%  ------------
%  Sec:outline
%  ------------
\section{Outline of Geometrical Parameters Decision Process}
\label{Sec:outline}

The parameters that determine the geometrical configuration 
of a small-jet-cell cylindrical drift chamber can be classified
into two groups:
(1) global parameters that specify the size and layout of
axial and stereo super-layers, and
(2) local parameters that describe the geometry of a
jet cell in each super-layer.
The first class includes
the length,
the inner and outer radii, and
the stereo angle\footnote{
Strictly speaking, stereo angle,
which is defined as the wire angle measured from
the axial direction, 
slightly varies even within a single jet cell: 
it increases linearly with the radial position of a wire.
As discussed later, the stereo angle of the innermost
wire is the most important in deciding the stereo
geometry.
}
of each super-layer.
If there is a layer of shielding wires between
adjacent super-layers, its radius and stereo angle also
belong to this class.
The parameters in the second class are, essentially, 
the relative locations of wires that comprise a
single jet cell, and determine the local properties
of the chamber.
These parameters are optimized to achieve performance goals
such as required spatial resolution, two-hit separation
capability, etc, together with other non-geometrical
parameters including wire high voltages, gas mixture,
magnetic field, and so on.
Detailed discussions on decision process for these parameters
are beyond the scope of this paper and will not
be covered unless necessary in the context of
stereo geometry determination.

\subsection{Basic Chamber Parameters in Terms of Physics Requirements}

Now let us outline a typical parameters decision process.
The chamber performance goals required by physics of interest
are often expressed in terms of transverse momentum resolution,
dip angle resolution, two-track separation capability, 
and polar angle coverage.
The transverse momentum resolution $\sigma_{p_T}$ is approximately
given by the following formula\cite{Ref:sigma}:
\begin{equation}
   \frac{\sigma_{p_T}}{p_T} 
	\simeq \sqrt{ \left( \frac{\alpha' \sigma_{r\phi}}{B {l}^2} \right)^2
		 \left( \frac{720}{n+4} \right) {p_T}^2
	       + \left( \frac{\alpha' C}{B {l}} \right)^2
	         \left( \frac{10}{7}\left(\frac{X}{X_0}\right) \right)
	       },
\end{equation}
where 
\begin{eqnarray}
  \left\{
  \begin{array}{lll}
    \sigma_{r\phi} & = & \mbox{spatial resolution in the $r$-$\phi$ plane per wire} \cr
    n & = & \mbox{the number of sampling points} \cr
    \alpha' & =  & 333.56 ~({\rm cm}\cdot{\rm T}\cdot{\rm GeV}^{-1}) \cr
    C & = & 0.0141 ~({\rm GeV}) \cr
    (X/X_0) & = & \mbox{thickness measured in radiation length units } \cr
    {l} & = & \mbox{lever arm length (cm)} \cr
    B & = & \mbox{magnetic field (T)}  
  \end{array}
  \right. .
\end{eqnarray}  
The number of sampling points is practically determined by the lever arm
length and the minimum possible wire-to-wire distance.
The best result is thus expected for the largest possible
$B$ field and lever arm length with the smallest possible
spatial resolution.
Since the required polar angle coverage determines the wire length
when the lever arm length and the radial position of the innermost
wires are given, the lever arm length is limited by the maximum
possible wire length for stable chamber operation\footnote{
There are some external constraints on the radial positions of
the inner- and outermost wires:
the radial position of the innermost wires is constrained,
in the case of the present JLC design, by the size of the support
tube for the final focusing quadripole magnets
that reside in the detector system.
The radial position of the outermost wires has to be
consistent with the surrounding barrel calorimeter.
}.
The $B$ field essentially determines the Lorentz angle
given a chamber gas mixture\cite{Ref:Lorentz} 
and, if a tilted cell
structure is adopted, it imposes a limit on
the two-hit separation capability.
%It also sets a momentum threshold to muon identification.
There will also be a technical limit to the maximum
possible $B$ field
attainable with a superconducting solenoid 
from the mechanical strength
and the stored energy to be released upon quench.

On the other hand, the dip angle resolution $\sigma_{\tan\lambda}$ 
can be written in the form\cite{Ref:sigma}:
\begin{equation}
   \sigma_{\tan\lambda} 
	\simeq \sqrt{ \left( \frac{\sigma_z}{{l}} \right)^2
	              \left( \frac{12}{n} \right)
	       + \left( 1 + \tan^2\lambda \right)
	         \left( \frac{C}{p_T} \right)^2
	         \left( \frac{13}{35} \left( \frac{X}{X_0} \right) \right)
	       },
\end{equation}
where $\sigma_z$ is the spatial resolution per wire in the axial direction
which is given by
\begin{equation}
   \sigma_z = \sigma_{r\phi}/\tan\alpha
\end{equation}
in the case of stereo wires with a stereo angle $\alpha$.
Provided that ${l}$ and $B$ are predetermined by the requirement
on the transverse momentum resolution, the only free parameter
left here is the stereo angle.
Apparently the larger the stereo angle becomes the better the
dip angle resolution.
There are, however, various restrictions to the stereo angle
which are the main topics of this paper.

The two-hit separation capability is controlled by the width of
a sense wire signal which is determined by the arrival time distribution of 
track electrons, space charge effects in the avalanche formation,
and the subsequent readout electronics.
Apart from the optimization of the readout electronics,
appropriate choices of the drift line configuration and chamber gas
are thus important.
In any case, the relation of the two-hit separation capability issue
to the stereo geometry is at most indirect and can be treated
separately as a problem concerning 
local properties of the chamber\cite{Ref:two-track}.




\subsection{Global Stereo-Geometrical Parameters}

At this point we have a rough idea about basic parameters
describing the chamber: the inner and outer radii, the length,
the number of sampling points, and the $r\phi$ spatial resolution 
per wire.
In order to decide the stereo geometry, we further need to know
the super-layer layout of the chamber and problems introduced
by the use of finite stereo angles.
A stereo layer can be formed by rotating the end point
of each wire belonging to this layer by a small angle ($\Delta\phi$)
with the other end point fixed.
As illustrated in Fig.\ref{Fig:3d-stereo-wire},
the stereo angle $\alpha$ is related to $\Delta\phi$ 
and the wire length ($L$) through
\begin{equation}
   \label{Eq:alpha(r)}
   { \everymath{\displaystyle}
   \begin{array}{lll}
   	\alpha & = & \tan^{-1}\left( \frac{2r(z=\pm L/2)}{L} 
   		\sin\left(\frac{\Delta\phi}{2}\right) \right) \cr
   		\rule{0in}{5.0ex}
   	     ~ & \simeq & \frac{r(z=\pm L/2)}{L} \cdot \Delta\phi  ,
   \end{array}
   }
\end{equation}
which suggests $\alpha$ being $r$-dependent even within a single
stereo jet cell as mentioned earlier.

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig1.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:3d-stereo-wire]{\small \label{Fig:3d-stereo-wire}
%%Fig-NIM%%		3-dimensional view of a single stereo wire.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\hfill
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig2.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:hyperboloid]{\small \label{Fig:hyperboloid}
%%Fig-NIM%%		Formation of a hyperboloidal surface due to
%%Fig-NIM%%		stereo wires.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\end{figure}

The otherwise cylindrical surface of this wire layer turns
into a hyperboloidal surface as shown in Fig.\ref{Fig:hyperboloid}.
As a result, both the azimuthal angle and the radial position
of the stereo wire becomes $z$-dependent:
\begin{equation}
\label{Eq:phi(z)}
{ \everymath{\displaystyle}
\begin{array}{lll}
   \phi(z) & = & \phi(z=0) 
           + \tan^{-1} \left[ \left(\frac{2z}{L}\right)
                               \tan\left(\frac{\Delta\phi}{2}\right)
                       \right] \cr
   \rule{0in}{5.0ex}
%   \rule{0in}{3.0ex}
%      ~ & \simeq & \phi(z=0) + \left( \frac{z}{L} \right) \Delta\phi
\label{Eq:r(z)}
   r(z) & = & \sqrt{ \left( r(z=0) \right)^2 + \left( z\tan\alpha \right)^2 } \cr
   \rule{0in}{4.0ex}
   ~ & = & \sqrt{ \left( r(z=\pm L/2) \right)^2 
   	+ \left( z^2 - (L/2)^2 \right) \tan^2\alpha },
\end{array}
} 
\end{equation}
where $z$ is measured from the middle of the chamber along the
chamber axis
\footnote{
The azimuthal wire position becomes a linear function,
and the radial wire position a quadratic function of $z$ measured
from the middle of the chamber:
\begin{equation}
  \left\{
  \begin{array}{lll}
   \phi(z) & \simeq & \phi(z=0) + \left( \frac{\alpha z}{r(z=0)} \right) \cr
   \rule{0in}{3.0ex}
   r(z)    & \simeq & r(z=0) + \frac{1}{2}\left(\frac{(\alpha z)^2}{r(z=0)}\right)
  \end{array}
  \right.
\end{equation}
to the lowest order of $\alpha z$.
}.

This inhomogeneity distorts the electric field to some extent
and limits the maximum possible stereo angle, thereby
restricting the $z$ resolution $\sigma_z$.
Fig.\ref{Fig:stereo-cell} demonstrates how a single stereo cell
structure deforms as one moves from one endplate to the other.
This $z$-dependent geometrical deformation of the stereo cell
is parametrized by a so called shrink factor and
imposes another important restriction on the
possible range of stereo angles.

%\begin{figure}[htb]
%\begin{minipage}[htb]{7cm}
%\centerline{
%\epsfxsize=6cm 
%\epsfbox{\figdir/fig3.eps}
%}
%\end{minipage}
%\hfill
%\begin{minipage}[htb]{6cm}
%\caption[Fig:stereo-cell]{\small \label{Fig:stereo-cell}
%Exaggerated illustration of the Stereo-cell formation through
%		rotation by an angle $\Delta\phi$.
%}
%\end{minipage}
%\end{figure}

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\begin{minipage}[htb]{12cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig3.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:stereo-cell]{\small \label{Fig:stereo-cell}
%%Fig-NIM%%Exaggerated illustration of the stereo-cell formation through
%%Fig-NIM%%rotation by an angle $\Delta\phi$ and its deformation along
%%Fig-NIM%%the wires.
%%Fig-NIM%%$ABCD$ and $A_1B_1C_1D_1$ are the cross sections of a single stereo cell
%%Fig-NIM%%at the end plates $(z=\pm L/2)$.
%%Fig-NIM%%$A_2B_2C_2D_2$ is that of the same stereo cell inbetween
%%Fig-NIM%%$(-L/2~<~Z<~+L/2)$.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%}
%%Fig-NIM%%\end{figure}

The shrink factor $f(z)$ is defined to be
the ratio $A_2B_2(z)/AB(z=\pm L/2)$ in Fig.\ref{Fig:stereo-cell}
and is readily derived from
Eqs.\ref{Eq:alpha(r)} and \ref{Eq:r(z)}:
\begin{equation}
\label{Eq:f(z)}
{ \everymath{\displaystyle}
\begin{array}{lll}
	f(z) & = & \frac{r_B(z)-r_A(z)}{r_B(z=\pm L/2)-r_A(z=\pm L/2)} \cr
   \rule{0in}{7.0ex}
	& = &  \sqrt{1 + \left(\frac{z^2 - (L/2)^2}{L^2}\right)4\sin^2\left(\frac{\Delta \phi}{2}\right)} \cr
   \rule{0in}{7.0ex}
	& = & \frac{\sqrt{R^2+\left(z^2 - (L/2)^2\right)\tan^2\alpha}}{R},
%	f(z) & = & \left(r_B(z)-r_A(z)\right)/\left(r_B(z=\pm L/2)-r_A(z=\pm L/2)\right) \cr
%	& = &  \sqrt{1 + \left(z^2 - (L/2)^2\right)4\sin^2\left(\frac{\Delta \phi}{2}\right)/L^2} \cr
%	& = & \sqrt{R^2+\left(z^2 - (L/2)^2\right)\tan^2\alpha}/R,
\end{array}
}
\end{equation}
where $R=r(z=\pm L/2)$ and $f(z=\pm L/2) = 1$ by definition.


It is a common practice to use pairs of stereo super-layers ($V$ and $U$)
with alternating stereo angles ($\pm \alpha$) interleaved with 
axial super-layers ($A$)\footnote{
The axial super-layers greatly facilitate track finding in
the $r$-$\phi$ projection, since stereo super-layers alone
cannot provide any absolute coordinate before its $z$ coordinate
is determined through tracking.
There is, however, a logical possibility to do away with
axial layers.
}.
In this case, the inhomogeneity is more serious for the gap between
a stereo super-layer ($V$) and 
the adjacent axial super-layer ($A$) just inside.
The minimum gap attained at the middle point of the chamber
has to be large enough for stable operations of the chamber.

In the following sections, we will illustrate in detail
the parameters decision process for a stereo geometry outlined above, 
taking an example of a possible JLC central tracking system.


%  ---------------
%  Sec:jlc-example
%  ---------------
\section{Illustration of Stereo Geometry Determination}
\label{Sec:jlc-example}

\subsection{A Possible Central Tracker for JLC}

Basic parameters of a possible JLC central tracker read
$B=2~T$, $r_{in}=45~{\rm cm}$, 
$r_{out}=230~{\rm cm}$, $L=460~{\rm cm}$, 
$n~=~80$, 
and $\sigma_{r\phi}=100~\mu{\rm m}$,
whose feasibility is the key issue of our
on-going R\&D\cite{Ref:sag}.
The tracker consists of 6 axial and 10 stereo super-layers,
each having 5 anode wires, 
arranged with increasing $r$ in the following order:
$AVUAVUAVUAVUA$,
where $A$, $V$, and $U$ again denote
an axial super-layer, stereo super-layers with
stereo angles of $+\alpha$ and $-\alpha$, respectively.
The gap between adjacent super-layers is
correlated with the stereo angle and is equal to 
either $4.0$ or $5.0~{\rm cm}$ at the endplates, 
depending on its radial location:
the $U$-to-$A$ and the $V$-to-$U$ gaps are equal to $4.0~{\rm cm}$,
while the $A$-to-$V$ is set at $5.0~{\rm cm}$
(see Fig.\ref{Fig:jlc-cdc})\footnote{
We can further optimize the inter-super-layer distances.
}.

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\begin{minipage}[htb]{9cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig4.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:jlc-cdc]{\small \label{Fig:jlc-cdc}
%%Fig-NIM%%		A possible super-layer layout for the central tracking 
%%Fig-NIM%%		system of the JLC.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%}
%%Fig-NIM%%\end{figure}

In the current design, there is a layer of shielding wires
in each gap between adjacent super-layers,
in order to electrostatically isolate different super-layers.
These shielding wires in a $V$-to-$U$ gap
will have the same stereo angle ($\alpha$) with the $V$ wires.
On the other hand, the shielding wires in
$A$-to-$V$ or $U$-to-$A$ gaps will have half the stereo
angle of the adjacent stereo super-layers ($\pm \alpha/2$).
%
% The shielding wires in an $A$-to-$V$ gap are
% $3.0$ away from the $A$
% and $2.0~{\rm cm}$ away from the $V$ layers
% at the endplates,
% respectively. 
%
Note also that the stereo angle varies even within a
single jet cell, according to Eq.\ref{Eq:alpha(r)}.




\subsection{Geometrical Limitation to the Stereo Angle}

It is already clear that there is a definite geometrical limit
to the maximum possible stereo angle:
the stereo angle should be small enough to avoid
mechanical contact of wires belonging to different layers.
The decisive parameter is therefore
the minimum distance between shielding wires and the stereo wires.
This is more important for the $A$-to-$V$ gaps,
since an $A$ super-layer stays as a cylinder, while
the $V$ super-layer surrounding it shrinks in the middle:
it changes its shape to a hyperboloid.
In the above mentioned JLC example, the shielding 
wires are $2.5~{\rm cm}$ away 
from the nearest stereo super-layer ($V$)
at the both wire ends.
Along the length of the chamber 
the distance between the innermost stereo layer belonging to
a $V$ super-layer and the adjacent layer of the shielding
wires decreases as a quadratic function of $z$
and attains its minimum at $z = 0$.

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig5.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:rphi-proj]{\small \label{Fig:rphi-proj}
%%Fig-NIM%%		$r$-$\phi$ projection of a stereo wire ($A_0A_1$):
%%Fig-NIM%%                $MN$ is the maximum displacement of the stereo wire.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\hfill
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig6.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:d(alpha)]{\small \label{Fig:d(alpha)}
%%Fig-NIM%%		Maximum radial displacement of a stereo wire
%%Fig-NIM%%		as a function of the stereo angle for different gaps.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\end{figure}



Fig.\ref{Fig:rphi-proj} illustrates the $r\phi$ projection
($A_0A_1$) of a single stereo wire.
Then we have 
\begin{equation}
   A_0A_1 = L \cdot \tan\alpha,
\end{equation}
where the chamber length is $L=4.6~{\rm m}$
in the JLC example (see Fig.\ref{Fig:3d-stereo-wire}).
$MN$ is the maximum displacement of the stereo wire 
with respect to its 
original position on the endplates and corresponds to the middle point
($z=0$) along the wire:
\begin{equation}
{ \everymath{\displaystyle}
\begin{array}{lll}
        MN \equiv  d  & = & r(z=\pm L/2) - r(z=0) \cr
   \rule{0in}{5.2ex}
        ~  & = & R - \sqrt{R^2 - \frac{L^{2} \cdot \tan^2 \alpha}{4}} \cr
   \rule{0in}{5.2ex}
        ~  & = & R\left(1-\cos\left(\frac{\Delta\phi}{2}\right)\right),
\end{array}
}
\end{equation}
where the nominal radius
$R = r(z=\pm L/2)$ is the radial position of the stereo wire
at the endplates (see Eq.\ref{Eq:r(z)}).

In order to prevent the stereo wire from touching
any shielding wire,
% which is assumed to be axial here though
% it can have some finite stereo angle,
this maximum displacement $MN$ should be 
less than the nominal distance between 
the shielding and stereo wires measured at the endplates.
The maximum displacement $MN$, however, strongly depends on
the stereo angle $\alpha$ and the nominal radius $R$. 
Given our example central tracker consisting of
super-layers having a cell size of $70~{\rm mm} \times 100~{\rm mm}$
(the maximum drift length is thus $5~{\rm cm}$)
and being arranged in the $AVUAVUAVUAVUA$ configuration from
$r_{in}=45$ to $r_{out}=230~{\rm cm}$,
we can easily calculate
the radial position of each axial or stereo super-layer 
as well as the gap between any two neighboring super-layers.
Fig.\ref{Fig:d(alpha)} plots this maximum displacement $MN$
for each super-layer as a function of stereo angle $\alpha$.
The dashed line is an example of
the geometrical upper bound on the maximum
displacement ($2.0~{\rm cm}$)\footnote{
The actual maximum allowable displacement here should be decided
in terms of electrostatic stability when high voltages are
applied.
Our experience with a $4.6~{\rm m}$-long test chamber tells us
that a distance equal to or greater than
$1~{\rm cm}$ is enough for wire stability in the case of
the present cell design.
}.

%\begin{figure}[htb]
%\centerline{
%\epsfxsize=6cm 
%\epsfbox{\figdir/fig6.eps}
%}
%\caption[Fig:d(alpha)]{\small \label{Fig:d(alpha)}
%		Maximum radial displacement of a stereo wire
%		as a function of the stereo angle for different gaps.
%}
%\end{figure}


%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig7.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:alpha(r)]{\small \label{Fig:alpha(r)}
%%Fig-NIM%%		Upper limit of the stereo angle as a function of 
%%Fig-NIM%%		the radius for all of the 10 stereo super-layers.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\hfill
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/tab.eps}
%%Fig-NIM%%}
%%Fig-NIM%%{\small
%%Fig-NIM%%	Table 1: The radial position ($R$), stereo
%%Fig-NIM%% angle ($\alpha$) and rotation angle ($\Delta\phi$) for each
%%Fig-NIM%% stereo super-layer.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\end{figure}

Fig.\ref{Fig:alpha(r)}, on the other hand, shows the upper limit of
the stereo angle as a function of the radial 
position of each super-layer.
From these  we can read off
the geometrical upper limit on the stereo angle for
each stereo super-layer. 
As pointed out above, it should be noted here again that
the stereo angle varies
even within a single cell according to
Eq.\ref{Eq:alpha(r)}: it grows linearly with the
radial position.
On the other hand, $\Delta\phi$ is constant within
a single super-layer.





\subsection{$\protect \mathbold{Z}$-dependent Geometrical Deformation of Stereo Cells}

As shown earlier, the wires belonging to a stereo layer form 
a hyperboloidal surface.
Consequently, any stereo cell shrinks as one moves
from an endplate to the middle of the wires.
As derived in the previous section, Eq.\ref{Eq:f(z)}
tells how the stereo cell shrinks as a function of $z$:
\begin{equation}
\nonumber
	f(z) = \frac{\sqrt{R^2+\tan^2\alpha\left(z^2 - (L/2)^2)\right)}}{R}.
\end{equation}
In this formula the shrink factor $f$ is a function of $z$,
$R$, and $\alpha$.
In order to calculate $f$ as a function of $z$,
we use $R$ and $\alpha$ tabulated in 
Table~1
% Table~\ref{Tab:z(r-alpha)} 
for each
stereo super-layer.

%\begin{table}[htb]
%\caption{The radial position ($R$), stereo
% angle ($\alpha$) and rotation angle ($\Delta\phi$) for each
% stereo super-layer.}
%\label{Tab:z(r-alpha)}
%\vspace{0.4cm}
%\begin{center}
%\footnotesize 
%\begin{tabular}{|c|c|c|c|}
%\hline
%&  &  & \\
%Super-layer &  R (cm)  &  $\alpha$ (mrad) &  $\Delta\phi$ (mrad) \\
%&  &  &  \\
%\hline
%V &  61  &  67  &  512 \cr
%U &  72  &  75  &  483 \cr
%V &  95  &  84  &  411 \cr
%U & 106  &  90  &  395 \cr
%V & 129  &  99  &  356 \cr
%U & 140  & 104  &  346 \cr
%V & 163  & 111  &  316 \cr
%U & 174  & 116  &  309 \cr
%V & 197  & 122  &  287 \cr
%U & 208  & 126  &  282 \cr
%\hline
%\end{tabular}
%\end{center}
%\vspace {1cm}
%\end{table}



The resultant shrink factor is shown in Fig.\ref{Fig:f(z)}
for all of the 10 ($V$,$U$) stereo super-layers.

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig8.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:f(z)]{\small \label{Fig:f(z)}
%%Fig-NIM%%		Shrink factor ($f$) as a function of $z$ for 
%%Fig-NIM%%		all of the 10 super-layers:
%%Fig-NIM%%		from the bottom to the top, the lines correspond to
%%Fig-NIM%%		super-layers ordered in the radial direction.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\hfill
%%Fig-NIM%%\begin{minipage}[htb]{7cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig9.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:jlc-cell]{\small \label{Fig:jlc-cell}
%%Fig-NIM%%		A typical geometry of a single cell:
%%Fig-NIM%%                the circles ($\circ$) and  ($\bullet$) correspond to sense
%%Fig-NIM%%                and    potential wires, respectively.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%\end{figure}

The variation of the
shrink factor $\Delta f/f$ is at most $3~\%$ 
observed for the innermost stereo super-layer which has
$r = 68~{\rm cm}$ and $\alpha = 75~{\rm mrad}$.
The $z$-dependent geometrical deformation of the stereo cell
induces variation of the surface field on the sense wires
in it, thereby introducing gain nonuniformity along the wires.
We calculated the surface field variation on the sense wires,
using the drift chamber simulation program, GARFIELD.
The cell structure used in the calculation is shown
in Fig.\ref{Fig:jlc-cell} whose size depends on the shrink factor $f$ and
varies also with the radial position
or the super-layer that the cell belongs to.

%\begin{figure}[htb]
%\centerline{
%\epsfxsize=6cm 
%\epsfbox{\figdir/fig9.eps}
%}
%\caption[Fig:jlc-cell]{\small \label{Fig:jlc-cell}
%		A typical geometry of a single cell.
%}
%\end{figure}

Fig.\ref{Fig:E_0(z)} plots the calculated surface fields against
the $z$ coordinate.
The maximum variation of the
surface field is $\Delta E/E \simeq 0.43~\%$.

%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\begin{minipage}[htb]{10cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=6cm 
%%Fig-NIM%%\epsfbox{\figdir/fig10.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:E_0(z)]{\small \label{Fig:E_0(z)}
%%Fig-NIM%%		Surface field as a function of $z$ for a sense wire
%%Fig-NIM%%		in the innermost stereo super-layer at $r = 68~{\rm cm}$.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%}
%%Fig-NIM%%\end{figure}

\subsection{Surface Field Variation Due to Electrostatic and Gravitational Sags}

In addition to the stereo geometry, there is another
source of geometrical deformation of the cell 
structure that is wire sags due to gravitational and electrostatic
forces on the wire.
These gravitational and electrostatic sags also contribute
to the surface field variation and are common to
both axial and stereo wires.
It suffices to investigate the effects
of these wire sags for axial wires only.
We, therefore, calculate the
surface field variation with GARFIELD, 
using as input the electrostatic and 
gravitational sags actually measured 
as the difference of the wire coordinates at
$z=\pm L/2$ (wire ends) and $z=0$ (mid point)
for a $4.6~{\rm m}$-long test chamber\cite{Ref:sag}.
The maximum wire-sag-induced variation of the surface field
is $\Delta E_0/E_0 \simeq 1.4~\%$.
It should be emphasized that the fact that
the field variation is comparable with those from
the stereo geometry for our example
is largely due to the use of Aluminum field wires.
We can in principle reduce the field variation
from the gravitational and electrostatic sags,
by optimizing the wire materials and by
stringing these wires with higher tensions.
On the other hand, the limitation coming from
the stereo geometry is inherent in any stereo chamber
and is unavoidable.


\section{Gas Gain Variation along Stereo Wires}

Now that we know the surface field variation along any
wire, we can study its $z$-dependent gas gain variation
and discuss the operation feasibility of the chamber,
provided that the relation of the surface field
to the gas gain is known.
Conversely, the stereo angle should be so determined to make 
the gain variation acceptable.

We have measured the gas gain for the 
${\rm CO}_2(90\%)$-Isobutane $(10\%)$ mixture,
by using a single wire proportional counter\cite{Ref:gas-gain}.
This gas mixture is a possible candidate for the JLC central 
tracker because of its very small diffusion coefficients.
The counter is made of an aluminum cathode tube
with a $10~{\rm mm}$ inner diameter and a gold-plated
tungsten anode wire with a $30~\mu{\rm m}$ diameter.
For the $^{55}$Fe $X$-rays ($5.9~{\rm keV}$)
and $^{90}$Sr $\beta$-rays,
the charges collected on the anode within 
$1~\mu{\rm sec}$ were measured as a function of the
applied voltage.

The results of the measurements are
depicted in Fig.\ref{Fig:Q(E_0)},
where the collected charge $Q$ is plotted as a function
of the anode surface field $E_0$.
%%Fig-NIM%%\begin{figure}[htb]
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\begin{minipage}[htb]{10cm}
%%Fig-NIM%%\centerline{
%%Fig-NIM%%\epsfxsize=8cm 
%%Fig-NIM%%\epsfbox{\figdir/fig11.eps}
%%Fig-NIM%%}
%%Fig-NIM%%\caption[Fig:Q(E_0)]{\small \label{Fig:Q(E_0)}
%%Fig-NIM%%		Signal charge as a function of the surface field on a sense wire.
%%Fig-NIM%%}
%%Fig-NIM%%\end{minipage}
%%Fig-NIM%%}
%%Fig-NIM%%\end{figure}

These data are fitted to the following empirical formula:
\begin{equation}
	\ln Q = a + b E_0,
\end{equation}
where $a$ and $b$ are the parameters to be determined.
From these data, we obtain the following relation:
\begin{equation}
   \frac{\Delta Q}{Q} = (15 \sim 20) \left( \frac{d E_0}{E_0} \right)
\end{equation}
around $E_0 \simeq 300~{\rm kV/cm}$, where
the gas gain is about $10^4$.

By combining these results with the surface field variations
calculated in the previous subsections,
we estimate the gain variation of the stereo wire
due to the stereo geometry
to be at most $8~\%$ which is acceptable.

The surface field variation due to the wire displacements
induced by gravitational and electrostatic forces for
our $4.6~{\rm m}$ test chamber could give rise to some
additional gain variation as large as $25~\%$ at worst.
The wire displacements can, however, be reduced by appropriate
choices of wire materials enduring higher tensions,
a higher gain gas mixture allowing
lower high voltages on sense wires, etc.
On the other hand, the gas gain variation due to the
stereo geometry is irreducible without geometry
redesigning.

\section{Discussions}

In the above, we have been trying to demonstrate a general
designing procedure of stereo-wire geometry.
In order to make the procedure more transparent and easier
to understand, we took a concrete example of a possible
central tracker for a future JLC experiment.
However, the parameters of the tracker we took as an example are
by no means final ones, and need refinements and 
optimizations: 
our example of a possible JLC central tracker has
$r_{in}=45~{\rm cm}$, $r_{out}=230~{\rm cm}$,
and a cell size of $70~{\rm mm} \times 100~{\rm mm}$
as required by detector performance goals and
external constraints from other detector components.
The super-layer arrangement ($AVUAVUAVUAVUAVUA$) 
and inter-super-layer gap sizes 
($4.0$ or $5.0~{\rm cm}$ at the endplates)
are less well motivated.
We discuss below some ideas for further possible improvements.
 
 \begin{itemize} 
 \item The super-layer structure
 
 As mentioned before,
 the axial super-layers greatly facilitate track finding in
 the $r$-$\phi$ projection, since stereo super-layers alone
 cannot provide any absolute coordinate before its $z$ coordinate
 is determined through tracking.
 Because of this, we assumed throughout this paper the
 use of axial super-layers together with stereo ones.
 There is, however, a logical possibility to do away with
 axial layers.
 We need more feasibility studies of this point.

 \item The gaps between adjacent super-layers and the
 location of shielding wires
 
 The gaps between adjacent super-layers 
 can be rearranged, for instance,
 to equalize the minimum distances between any
 adjacent super-layers, regardless of whether
 they are $A$, $V$, or $U$-types.
 We can also allow a larger
 stereo angle for inner super-layers by
 giving larger gaps to them.

 The inter-super-layer gaps have single shielding layers.
 It is now clear from Fig.\ref{Fig:d(alpha)} 
 that, by increasing 
 the distance between a stereo super-layer and its
 adjacent layer of shielding wires,
 we can increase the stereo angle.
 This procedure can be particularly important
 for inner super-layers
 where stereo angles are less than $100~{\rm mrad}$. 


 \item The shielding layers consisting of stereo wires
 
 In order to allow a larger displacement of stereo wires
 in the middle, we proposed to use, for shielding wires,
 half a stereo angle of the adjacent stereo wires.
 This also brings more symmetry to gaps between shielding
 layers and adjacent super-layers.

\end{itemize} 
  
 
 
\section{Conclusions}
\label{Sec:conclusions}

We have outlined a general design procedure
for a long cylindrical drift chamber with small jet cells
with an emphasis put on the decision process for
stereo wires geometry.
The procedure was illustrated, taking a concrete example of
a possible central tracker for JLC.
The surface field variation along an anode wire 
is one of the decisive factors to determine
stereo geometry.
We have examined the surface field variation originating from
(1) the geometrical deformation of the cell structure
as a function of $z$ (the coordinate along the chamber axis)
due to stereo geometry, 
and (2) that due to electrostatic and gravitational sags.
These field variations were estimated through
explicit electric field calculations using
the chamber simulation program GARFIELD\cite{Ref:garfield}.
The estimated field variations were translated into gain
variation along the anode wires using the gas gain data
taken for this purpose and used to constrain the stereo
geometry.



\section*{Acknowledgments}
The authors would like to thank all the members of 
the JLC physics working group.
In particular, they are very grateful to
Y.Nakamura for his great contribution in gas gain measurements,
N.Ishihara and T.Ohama for useful discussions,
and S.Iwata for continuous encouragements and supports.
This work was supported in part by the JSPS Japanese-German
Cooperation Program.

%%\section*{References}
\begin{thebibliography}{99}
\bibitem{Ref:JLC-I} JLC group, KEK Report 92-16, December,1992.
\bibitem{Ref:sigma} R.L.~Gluckstern, Nucl. Inst. and Meth. {\bf 24} (1963) 381,
         JLC physics group, http://www-jlc.kek.jp/subg/offl/lib/docs/helix\_manip.ps.gz.
\bibitem{Ref:garfield} Veen~Hove, 
	A Drift Chamber simulation program GARFIELD, 1994.
\bibitem{Ref:sag} S~Sudou, N.~Khalatyan, Y.~Kurihara, T.~Abe, K.~Fujii,
	A.~Sugiyama, K.~Takahashi, T.~Watanabe,
	Nucl. Inst. and Meth. {\bf A383} (1996) 391.
\bibitem{Ref:Lorentz} JLC CDC group, a paper on Lorentz angle measurement, in preparation.
\bibitem{Ref:two-track} JLC CDC group, a paper on two-track separation paper, in preparation.
\bibitem{Ref:gas-gain} JLC CDC group, a paper on Gas gain measurement, in preparation.

\end{thebibliography}
%%%%\end{document}

\newpage
\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig1.eps}
}
\caption[Fig:3d-stereo-wire]{\label{Fig:3d-stereo-wire}
		3-dimensional view of a single stereo wire.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig2.eps}
}
\caption[Fig:hyperboloid]{ \label{Fig:hyperboloid}
		Formation of a hyperboloidal surface due to
		stereo wires.
}
\end{figure}








\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig3.eps}
}
\caption[Fig:stereo-cell]{ \label{Fig:stereo-cell}
Exaggerated illustration of the stereo-cell formation through
rotation by an angle $\Delta\phi$ and its deformation along
the wires.
$ABCD$ and $A_1B_1C_1D_1$ are the cross sections of a single stereo cell
at the end plates $(z=\pm L/2)$.
$A_2B_2C_2D_2$ is that of the same stereo cell inbetween
$(-L/2~<~Z<~+L/2)$.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig4.eps}
}
\caption[Fig:jlc-cdc]{ \label{Fig:jlc-cdc}
		A possible super-layer layout for the central tracking 
		system of the JLC.
}
\end{figure}



\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig5.eps}
}
\caption[Fig:rphi-proj]{ \label{Fig:rphi-proj}
		$r$-$\phi$ projection of a stereo wire ($A_0A_1$):
                $MN$ is the maximum displacement of the stereo wire.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig6.eps}
}
\caption[Fig:d(alpha)]{ \label{Fig:d(alpha)}
		Maximum radial displacement of a stereo wire
		as a function of the stereo angle for different gaps.
}
\end{figure}



\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig7.eps}
}
\caption[Fig:alpha(r)]{ \label{Fig:alpha(r)}
		Upper limit of the stereo angle as a function of 
		the radius for all of the 10 stereo super-layers.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/tab.eps}
}
{\small
	Table 1: The radial position ($R$), stereo
 angle ($\alpha$) and rotation angle ($\Delta\phi$) for each
 stereo super-layer.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig8.eps}
}
\caption[Fig:f(z)]{ \label{Fig:f(z)}
		Shrink factor ($f$) as a function of $z$ for 
		all of the 10 super-layers:
		from the bottom to the top, the lines correspond to
		super-layers ordered in the radial direction.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig9.eps}
}
\caption[Fig:jlc-cell]{ \label{Fig:jlc-cell}
		A typical geometry of a single cell:
                the circles ($\circ$) and  ($\bullet$) correspond to sense
                and    potential wires, respectively.
}
\end{figure}





\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig10.eps}
}
\caption[Fig:E_0(z)]{ \label{Fig:E_0(z)}
		Surface field as a function of $z$ for a sense wire
		in the innermost stereo super-layer at $r = 68~{\rm cm}$.
}
\end{figure}


\begin{figure}[htb]
\centerline{
\epsfxsize=12cm 
\epsfbox{\figdir/fig11.eps}
}
\caption[Fig:Q(E_0)]{ \label{Fig:Q(E_0)}
		Signal charge as a function of the surface field on a sense wire.
}
\end{figure}

\end{document}



%\section*{List of Tables}
%\begin{itemize}
%\item {\bf Table 1.} Geometrical parameters of the possible central tracking
%	system of the JLC 
%\end{itemize}

%\end{document}
