\section {Optics Commissioning of the High Resolution Spectrometers}
\subsection {Introduction}
The HRSs have focusing properties that are
point-to-point in the dispersive direction. The optics matrix
elements allow the reconstruction of the interaction vertex
from the coordinates of the detected particles at the
focal plane. Data obtained with a set of foil targets (which define a
set of well-defined interaction points along the beam) and a
sieve-slit collimator were used to determine the optical matrix
elements. The Hall A event analyzer ESPACE
(section~\ref{sec:espace}) and a
newly written C++ optimization routine~\cite{optim}
were used to
derive optical matrix elements from these calibration data.
\subsection {Acceptance}
Due to the combination of a 10 cm extended target capability and a
10\% momentum
acceptance, the angular acceptance of the HRS devices is a
complex function of momentum and target position. The acceptance of
the
spectrometers has been modeled by means of Monte Carlo simulations.
In that modeling the raytracing program SNAKE~\cite{vernin} was used
to trace trajectories,
spanning the full acceptance and more, through the magnetic fields in
each of
the spectrometers. The descriptions of the magnetic fields are based
on a
combination of the original design parameters, survey information
regarding
the achieved physical locations of the magnets, and the results of
the magnetic
field mapping~\cite{vernin2}. The location and direction of
these trajectories are recorded at each of the critical apertures
along the
spectrometers. The critical apertures are those that have been
identified as defining the acceptance. They are the exit of the Q$_1$
quadrupole,
the entrance of the dipole, the exit of the dipole, and both the
entrance and
exit of the Q$_3$ quadrupole. Transfer functions from the target to
each of these
apertures of up to fifth order are determined using a polynomial
fitting program.
These transfer functions along with the knowledge of the physical
size of each
of the apertures are used to provide a test of whether or not any
given
trajectory in a Monte Carlo simulation will make it through the
spectrometer.
Except at extreme values of $y_{0}$ and $\delta$ comparisons of
actual data with
simulations have been found to agree at the 3\% level or better.
Updated optics models have been generated which improve the
performance of the acceptance model and
develop schemes for defining an acceptance through software cuts on
trajectories that make optimal use of the full acceptance.
\subsubsection {Approach}
\label{sec:approach}
{ For each event, two angular coordinates (\( \theta_{det} \) and
\( \phi _{det} \)) and two spatial coordinates (\( x_{det} \) and
\( y _{det} \)) are measured at the focal plane detectors. The
position of the
particle and the tangent of the angle made by its trajectory along
the dispersive direction are given by \( x_{det} \) and \( \theta
_{det} \), while \( y_{det} \) and \( \phi _{det} \) give the
position and tangent of the angle perpendicular to the dispersive
direction. These focal plane variables are corrected for any
detector offsets from the ideal central ray of the spectrometer to
obtain the focal plane coordinates $x_{fp}$, \( \theta_{fp} \),
$y_{fp}$, and \( \phi_{fp} \). A detailed description of the Hall
A optics coordinate systems is given in Ref.~\cite{nilanga1}. These
observables are used to calculate $\theta_{tg}, y_{tg}, \phi_{tg}$
and $\delta$ for the particle at the target by matrix inversion of
Eq.~2 which links the focal-plane coordinates to the target
coordinates (in a first-order approximation). }
In practice, the expansion of the focal plane coordinates is
performed up to fifth order. A set of tensors $Y_{jkl}, T_{jkl},
P_{jkl}$ and $D_{jkl}$ links the focal-plane coordinates to target
coordinates according to\footnote{The superscripts denote
the power of each focal plane variable.}
\begin{align}
y_{tg} &=\sum_{j,k,l} Y_{jkl}\theta ^{j}_{fp}y^{k}_{fp}\phi
^{l}_{fp}, \label{eq:2.1} \\
\theta_{tg} &=\sum_{j,k,l} T_{jkl}\theta ^{j}_{fp}y^{k}_{fp}\phi
^{l}_{fp},\label{eq:2.2} \\
\phi_{tg} &=\sum_{j,k,l} P_{jkl}\theta ^{j}_{fp}y^{k}_{fp}\phi
^{l}_{fp}, \,\,\,\rm{and}\label{eq:2.3} \\
\delta &=\sum_{j,k,l} D_{jkl}\theta ^{j}_{fp}y^{k}_{fp}\phi
^{l}_{fp}, \label{eq:2.4}
\end{align}
where the tensors $Y_{jkl}, T_{jkl}, P_{jkl}$ and
$D_{jkl}$ are polynomials in x$_{fp}$. For example,
\begin{equation}
Y_{jkl}=\sum_{i=0}^{m} C_{i}x^{i}_{fp}.\label{eq:3.1}
\end{equation}
Mid-plane symmetry of the spectrometer requires that
\it{(k+l)} \rm{is odd for \( Y_{jkl} \) and \( P_{jkl} \),
while \it{(k+l)} \rm{ is even for} \( D_{jkl} \) and \( T_{jkl} \).}
In practice, the basic variables $y_{tg}, \theta_{tg}$ and
$\phi_{tg}$ do
not form a good set of variables for the optics calibration
procedure. For a foil target not located at the origin of the target
coordinate system $y_{tg}$ varies with $\phi_{tg}$. In the case of a
multi-foil target, $\phi_{tg}$ calculated for a given sieve-slit
hole depends on $y_{tg}$; furthermore, all three variables depend on
the
horizontal and vertical beam positions ($x_{beam}$ and
$y_{beam}$\footnote{Beam variables are measured in the
hall coordinate system, centered at the center of the hall with
$\hat{x}$ towards the left of the beam, $\hat{z}$ along the beam
direction
and $\hat{y}$ vertically up.}).
On the other hand, the interaction position along the beam,
$z_{react}$, and vertical and horizontal positions at the sieve
plane, $x_{sieve}$ and $y_{sieve}$, are uniquely determined for a set
of foil targets and a sieve-slit collimator. These three variables
are calculated by combining the ``basic'' variables defined above
using the equations (see Fig.~\ref{tg_coord}):
\begin{align}
z_{react} &= -(y_{tg}+D)\frac{\cos(\arctan \phi_{tg})}{\sin(\theta_0+
\arctan \phi_{tg})}+x_{beam}{\rm cot}(\theta_0+\arctan \phi_{tg})\\
y_{sieve}&=y_{tg}+L\phi_{tg}\\
x_{sieve}&=x_{tg}+L\theta_{tg}
\end{align}
L is the distance from Hall center to the sieve plane, while D
is the horizontal displacement of the spectrometer axis from its
ideal
position. The spectrometer central angle is denoted by $\theta_0$.
The vertical
coordinate $x_{tg}$ is calculated using $\theta_{tg}$, $z_{react}$
and the vertical displacement of the spectrometer from its ideal
position.
The optics tensor coefficients are determined from a $\chi^2$
minimization procedure in which the events are reconstructed as close
as possible to the known position of the corresponding foil target
(in the case of $z_{react}$) or the sieve-slit hole (in the case
of $y_{sieve}$ and $x_{sieve}$). The quality of the track
reconstruction procedure is illustrated in Figs.~\ref{slit} and \ref{zreact}.
\subsubsection {Optics Commissioning results} \label{results}
The following results were obtained from elastic scattering data
taken at $E_0=845$ MeV on a thin $^{12}$C target. An example of a
momentum spectrum used in this analysis is shown in Fig.~\ref{dpkin}.
All quantities have been measured at the target.
\begin{itemize}
\item Relative angular-reconstruction accuracy
\begin{itemize}
\item in-plane (transverse):\hspace*{5.6cm} $\pm$0.2 mrad
\item out-of-plane (dispersive):\hspace*{5.1cm}$\pm$0.6 mrad
\end{itemize}
\item Angular Resolution (FWHM)
\begin{itemize}
\item in-plane (transverse):\hspace*{5.6cm} 2.0 (2.6) mrad
\item out-of-plane (dispersive):\hspace*{4.9cm} 6.0 (4.0) mrad
\end{itemize}
\item Momentum Resolution (FWHM)
\hspace*{3.7cm} $2.5~(2.6) \times10^{-4}$
\item Relative transverse position reconstruction accuracy
\hspace*{0.3cm}$\pm$0.3 mm
\item Transverse position resolution (FWHM)
\hspace*{2.6cm}4.0 (3.1) mm
\end{itemize}
%
Values in parentheses are an estimate of the resolution values based
on the design optics and multiple scattering in the various windows
and detectors. The momentum resolution quoted above is for the center
of
the focal plane. Away from the center, the resolution increases
(see
Fig.~\ref{res_ch}). This is because the actual focal plane of the
spectrometer does not coincide with the ideal focal plane used for
measurements. As a result, the first-order $\theta$-dependent term
becomes increasingly large as $\delta$ moves away from 0, making the
momentum determination increasingly sensitive to the measurement of
the trajectory direction and, hence, more sensitive to multiple
scattering.
\subsection {Scattering angle determination}
\label{sec:angledet}
The electron scattering angle, $\theta_{scat}$, is calculated by
combining
$\phi_{tg}$ and $\theta_{tg}$
(measured relative to the central ray of the spectrometer) and the
spectrometer central angle $\theta_0$ between the beamline and the
spectrometer nominal central ray:
\begin{equation}
\theta_{scat} = \arccos(\frac{\cos(\theta_0) - (\phi_{tg}) \sin(\theta_0)}
{\sqrt{1 + (\theta_{tg})^2 + (\phi_{tg})^2}})
\end{equation}
Thus, the accuracy of the scattering angle determination depends on
the accuracy of $\phi_{tg}$ and $\theta_{tg}$ relative to the
central ray of the spectrometer. The accuracies quoted above
are the relative accuracies of $\phi_{tg}$ and $\theta_{tg}$ with
respect to the central sieve-slit hole. This makes the determination
of $\phi_{tg}$ and $\theta_{tg}$ angles for the central sieve-slit
hole crucial for the determination of the scattering angle.
The values of $\phi_{tg}$ and $\theta_{tg}$ for the central sieve-slit hole are
calibrated by using electron scattering from a thin $^{12}$C target
located close to the Hall A center. The two angles for the electrons
passing through the central hole are calculated from the following
survey information:
\begin{itemize}
\item the target position
\item the spectrometer central angle
\item the displacement of the spectrometer nominal central ray from
the hall center
\item the position of the sieve-slit center with respect to the
nominal central ray
\item the position of the beam position monitors with respect to the
ideal beam line.
\end{itemize}
Each of these measurements has approximately a 0.5 mm systematic
uncertainty. With the distance from the target to the sieve
collimator being about 1.18 m, the total contribution of these measurements
(added in quadrature with relevant weights) to the scattering-angle
accuracy is about 0.6 mrad. However, once the central
sieve-slit hole angles are calibrated relative to the nominal central
ray, this uncertainty will remain a constant systematic offset to
the scattering angle.
An alternate method utilizes the elastic
$^1$H($e,e^\prime p$) reaction. After correcting for energy losses
in the target along with internal bremsstrahlung for the
incident electron, one can relate the beam energy to the electron
scattering angle and the proton emission angle relative to the
beam. By making several such measurements over a range of
spectrometer angles at a fixed and independently measured beam
energy, one arrives at a set of equations, each equation involving
only the particle angles. Under the assumption that the angles are correctly
measured except for a possible overall offset for each
spectrometer (common to all kinematics within the set), one
can determine these absolute offsets through a maximum likelihood
analysis. The details of the method of extracting the
spectrometer angular offsets can be found in Ref.~\cite{tn_epscan}.
In April of 2000 the first of such a series of $^1$H($e,e^\prime p$)
measurements was made. Several additional scans were
made subsequently in order to assess the stability of the
spectrometer angular offsets over time. To date, the first three
scans have been analyzed~\cite{tn_epscan,tn_epscan2}, though the
third lacked a direct, independent measurement of the beam
energy. Based on the average of these three scans, the angular
offsets were found to be $-0.91\pm 0.08$ mrad and $-0.40\pm 0.07$
mrad for the HRS-L and HRS-R arm, respectively. These averages
were obtained by fitting only the angular offsets for the first two
scans, constraining the beam energy to the values measured via
the Arc method, and fitting both the angular offsets and the beam
energy for the third scan. The error was estimated from the spread
in survey results.
\subsection {Absolute momentum determination}
The relative momentum $\delta$ for an event measured by the
spectrometer is used to calculate the absolute momentum $p$ for that
event:
\begin{equation}
p=p_0 (1+\delta)
\end{equation}
where $p_0$ is the central momentum of the spectrometer,
related to the dipole magnetic field $B_0$ by,
%\begin{equation}
%p_0=\Gamma B_0
%\end{equation}
\begin{equation}
p_0=\Sigma_{i=1}^{3}{\Gamma_i B_0^i}
\end{equation}
where $\Gamma_i$ are the spectrometer constants.
%High-precision beam energy measurements in Hall A make it possible
%to determine the spectrometer constant, and hence the absolute
%momentum for each particle, to better than $4\times 10^{-4}$.
The extraction of the spectrometer constants for the HRS
pair is described in detail in Ref.~\cite{nilanga2}.
At beam energies below 1.5 GeV, elastically scattered
electrons from a thin $^{12}$C target were used to calculate the
spectrometer constant for each spectrometer. For the ground state of
$^{12}$C, the scattered electron momentum $p$ is related to the beam
energy $E_i$ by
\begin{equation}
p=E_f=\frac{E_i-E_{loss1}}{1+\frac{2E_i{\rm sin}^2(\theta
/2)}{M_t}}-E_{loss2}
\end{equation}
where $E_f$ denotes the energy of the scattered electrons,
$M_t$ the mass of the target nucleus, and
$E_{loss1}$ and $E_{loss2}$ the energy loss before and after
scattering.
At higher beam energies the cross section for elastic scattering
from $^{12}$C vanishes, making it impossible to use the above method
to calculate spectrometer constants. Instead, missing-energy
measurements of the 1p$_{1/2}$ state in the $^{12}$C($e,e^{\prime}p$)
reaction were used
to calculate spectrometer constants for momenta above 1.5 GeV. For
this method the kinematics were chosen such that one spectrometer
momentum is set at the lower region where the constant has already
been
determined using the previous method, while the second
spectrometer momentum is at the higher value where the constant needs
to be determined.
Due to the relatively heavy mass of the $^{12}$C nucleus, the
spectrometer constant calculated using the above methods depends
only
weakly on the spectrometer angle. Furthermore, for a thin foil target
the
energy loss before and after scattering is relatively small and can
be calculated accurately.
Spectrometer constants were thus determined with an accuracy
of $4 \times 10^{-4}$ over the full momentum range
of both HRS. The beam
energy for these measurements was measured using the Arc and eP
methods to a few times $10^{-4}$.
In the 3~GeV region (where the Arc and eP results disagree) the
spectrometer constants are fully consistent with the Arc results.
Table~\ref{gamma} lists the spectrometer constant coefficients for the
HRS pair.