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\documentclass{chowto}
%%
\usepackage{graphicx} 
\usepackage{epsfig}
\usepackage{subfigure}
%%
\title{HKS Optical Calibration}
\howtotype{expert} % ``expert'', ``user'', ``reference''
\experiment{HKS} % Optional
\author{Lulin Yuan and Liguang Tang}
\category{hks}
%\maintainer{Name of person maintaining document} % Optional
\date{May 10, 2005}

\begin{document}

\begin{abstract}
This document describes the method and procedure of optical calibration
for HKS spectrometer system. 
\end{abstract}
 
\section {Overview}
The purpose of HKS optical calibration is to optimize target angle
($xptar$, $yptar$) and
momentum ($\delta p$) reconstruction matrices to get the best
resolutions.
% HKS experiment aims at obtaining high resolution spectroscopy of hypernuclei.
%. To calculate hypernuclear excitation energies, we need
%to determine scattered Kaon and electron momenta and angles. Actually,
%the Kaon momentum is the dominant factor for the final missing mass
%resolution. In order to ensure the proposed missing mass resolution
%(e.g., $\sim 400$ KeV FWHM for $^{12}_\Lambda$B), it is important to
%optimize the spectrometer optics to get the required momentum and
%angle reconstruction resolution.

%\begin{table}[!hbt]
%\caption{required momentum and angle reconstruction resolution}
%\begin{center}
%\begin{tabular}{ll} \hline
%K+ momentum   & 110 KeV/c (rms):1$\times10^{-4}$ \\ 
%E' momentum      & 120 KeV/c(rms):2$\times10^{-4}$  \\
%K+ angle    & 2.9 mrad(rms)\\\hline
%\end{tabular}
%\end{center}
%\end{table}

The optical calibration procedure will use the data from Enge and HKS
sieve slit runs for angle reconstruction calibration and CH$_2$ target
runs for momentum calibration. Before the experiment, the procedure
has been tested by simulation.

\section {General Procedure}

The spectrometer optical calibration procedure is specially designed
for HKS spectrometer system. It is somewhat different from other Hall
C basic equipment experiment using only HMS and SOS, because of the
existence of splitter magnet right after the target. This complicate
both the angle and momentum calibrations. 

%For angle reconstruction calibration, because target is inside the field of
%Splitter. The sieve slit plate is mounted on the Enge magnet
%entrance. The particle trajectories are already bended by the splitter
%field before they pass through the sieve slit holes. Thus the
%particles passing through one hole no longer have a single, fixed
%angle. 

%For momentum calibration, because the Enge, HKS two arms and the beam dump line
%are all coupled through splitter field, thus suitable kinematics
%setting for typical single arm momentum calibration (e.g., by
%production of carbon excited states) is difficult to implement. 

The required calibration runs for HKS optical calibration are:

\begin{itemize}
\item Separate Enge and HKS sieve slit(SS) runs, using Carbon target
without raster. Check the Enge and HKS focal plane correlations:
$Yfp$ vs. $Xfp$, $Xpfp$ vs. $Xfp$, $Ypfp$ vs. $xdp$. Statistics is enough when all the bands can be
clearly seen (see fig.\ref{fig:engefpss1} and fig.\ref{fig:engefpss2} for Enge, fig.\ref{fig:hksfpss} for HKS).

\item Two arm coincidence runs in production kinematics, on CH$_2$
target to obtain $\Lambda$ and $\Sigma^0$ spectra. We also need data
on Carbon target for the $^{12}_\Lambda$B ground state spectrum. The
yield of $\Sigma^0$ and $^{12}_\Lambda$B gs should reach at least
1500 events and 1500 events, respectively. We should also look
at the S/A ratio under the $\Lambda$ peak to keep it more than
5 to 1 after raster correction.

\end{itemize}

The general procedure for the calibration is:
\begin{enumerate}
\item Separate Enge and HKS sieve slit calibration to obtain angle
reconstruction matrices using original momentum reconstruction
matrix. 

\item Raster correction for CH$_2$ target data using existing raster
correction matrix.

\item Kinematics calibration to find deviation of beam energy, Enge
and HKS central momenta from their nominal values. The
central angles of the spectrometers are not used in the missing mass
calculation and they are not needed for fixed-angle spectrometer.

\item Two arm coupled momentum calibration to obtain momentum
reconstruction matrix. 

\item After getting the calibrated momentum matrix, go back to first step 
again to do SS calibration using the new momentum matrix. Iterate until best
resolution is reached.

\end{enumerate}

\section {Sieve Slit Calibration}
\subsection {Procedure}
In HKS experiment, although the Splitter field smears the one to one
correspondence between target angles and sieve slit hole position and introduces additional momentum dependence.
particle trajectories scattered from a point target with given momenta
still have a fixed target angle when they pass a given hole. 
The angle in this case is also a function of particle momentum, and
depend on splitter optics. Based on this observation, the Sieve slit
calibration procedure is:

\begin {enumerate}
\item Fit function $F_{s2t}$: target angles as a function of SS
positions and momentum based on Splitter optics.

\item Separate events of the SS calibration run data hole by hole,
thus determine corresponding SS hole center
position for each event.

\item Using function $F_{s2t}$, calculate corrected target angles
($xptar$, $yptar$) for each event from SS center position and
momentum.

\item Fit transfer matrix from focal plane to target: ($xfp$, $xpfp$, $yfp$,
$ypfp$)$\longrightarrow$($xptar$, $yptar$).
\end {enumerate}

A simulation of the SS calibration procedure with initial inaccurate
Splitter, Enge optics reaches final resolution for e' (Enge) target
angles ($\sigma$): $xptar$: 1.8 mr, $yptar$: 1.1 mr, for Kaon (HKS)
angles: $xptar$: 0.4 mr, $yptar$: 2.4 mr.  

\subsection{Code}

The codes are written in Perl script, which calls a series of Fortran
programs. Currently, they work on simulated data. All the
codes start from the
directory:\verb+jlabs1:/home/yuan/HKS/optics/+

\begin{itemize}

\item \verb+.enge/fsv2tarang+: Perl script to fit target angles as function
of sieve slit positions ($X_{sv}$, $Y_{sv}$) and momentum by 6-order
polynomials.

\item \verb+.enge/opticscalib+: Perl script to fit target angles as 6-order
		       polynomials of  focal plane quantities.
\item Nominal electron arm Raytrace data
card:\verb+.enge/hksearm_tilt.dat+
\item nominal electron arm reconstruction matrix:\verb+.enge/enge_recon_coeff.dat+ 

\end{itemize}

\section{Momentum Calibration}

The method of HKS momentum calibration is making use of the known
masses of $\Lambda$,$\Sigma^0$ hyperons and the narrow width of
$^{12}_\Lambda$B hypernuclear ground state. The momentum
reconstruction matrices for Enge and HKS arms are optimized
simultaneously by minimizing the Chisquare defined as the sum of
squared mass differences between the calculated mass and the PDB
values. A Nonlinear Least Square method is used to optimize the
matrices. The procedure uses the existing target angle reconstruction
matrices.

Although there is emulsion measurement of $^{12}_\Lambda$B gs binding
energy at $11.37\pm0.06$ MeV, the associate statistical error and the composite nature
($2^-$/$1^-$) of the state may cause the peak center to deviate from this
value. So in the calibration process, the peak position is a
adjustable parameter and the calibraion is carried out in a iterative
way, the gs peak center is adjusted after each iteration according to
the actual fitted position of calculated  $^{12}_\Lambda$B gs
distribution, until an minimum Chisquare is reached.    

\subsection {procedure}
The procedure is:

\begin{enumerate}
\item Calculate missing mass using the existing angle matrix and the initial
(un-calibrated) momentum matrix. From CH$_2$ target runs, select
$\Lambda$,$\Sigma^0$ events, from Carbon target runs, select
$^{12}_\Lambda$B gs events from a window located around the center
of the missing mass peaks.

Inside the 5mm thick CH$_2$ target, the energy loss and
radiative processes are significant. It shifts the missing
mass distributions and forms a tail.  The width of the event
window has to be selected appropriately, because a too wide
window will dilute the sample with too many radiated events while a too
narrow window will cause a insufficient statistics in the fitting.

\item Define  Chisquare as the sum of squared mass differences $\Delta
m_i^2$ between the
calculated mass and the PDB values or the initial assumed
$^{12}_\Lambda$B gs binding energy:
$$ \chi^2=\sum {w_i\Delta m_i^2}$$
where $w_i$ is the relative weight of $\Lambda$,$\Sigma$ and
$^{12}_\Lambda$B gs events. 

\item $\chi^2$ is a function of missing mass $emiss$,  so a function of e' and
Kaon momenta $\delta p_e$ and $\delta p_k$:
$$\chi^2=f(emiss(\delta p_e,\delta p_k)$$
Minimizing $\chi^2$ by by a Nonlinear Least 
Square method to optimize the Enge and HKS momentum
reconstruction matrices .

\item fit the actual peak center of the calculated  $^{12}_\Lambda$B gs
missing mass distribution. Using this value as the new
$^{12}_\Lambda$B gs binding energy.  Go back to first step. Iteration
until an minimum Chisquare is reached.   

\end{enumerate} 

\subsection{Code}

All files have a path name in the front: \verb+jlabs1:/home/yuan/HKS/optics/+:

\begin{itemize}
\item \verb+.dpcal_eloss/emiss_cal.pl+: Perl script to reconstruct
missing mass from focal plane quantities based on existing momentum and angle matrices.
\item \verb+.ml_dpcal/dp_cal_nls.pl+: Perl script to select events and
optimize the momentum matrices.
\item \verb+.dpcal_eloss/steer.dat+: input data card to specify the
parameters for the
momentum calibration, such as the number of events used in the
calibration, relative weights, the centers of missing mass peaks for
event selection,etc. 
 
\end{itemize}

%\section{Kinematics calibration}
%The purpose of kinematics calibration is to find the offsets of beam
%energy, Enge and HKS central momentum from nominal values. The method
%is also to use the known masses of $\Lambda$,$\Sigma^0$ and
%$^{12}_\Lambda$B ground state, same as momentum calibration.  The
%$\chi^2$ is defined the same way as in momentum calibration. But for
%kinematics calibration, the $\chi^2$ is a function of beam energy offset $\Delta
%E_{e0}, e' central momentum offset $\Delta p_{e'} and Kaon momentum
%offset $\Delta p_k$, corresponding to the zero order terms in the
%momentum matrices. At correct offsets,
%this $\chi^2$ will be minimized.
%
%IF we define sum of kinematics offsets as:
%
%$$\Delta p_{kin}=\Delta E_{e0} - \Delta p_{e'}-\Delta p_k$$.
%
%The $\chi^2$ will dominantly depend on $\Delta p_{kin}$. Correct
%offsets can be found by scan on $\Delta p_{kin}$ over all possible
%values  to locate the minimum Chisquare.   
%
\section{Raster Correction}
\subsection{Procedure}
To avoid burning the CH$_2$ target, beam will be rastered
5mm$\times$5mm on the target. A direct reconstruction with matrices
obtained for point target gives much worse resolution, for example,
the $\delta p$ resolution for HKS is $1.16\times10^{-4}$(rms) without
raster, it becomes $6.64\times10^{-4}$ with 5mm$\times$5mm raster. 

To do raster correction, the beam positions on target need to be
determined by raster magnets X and Y current event by event. For
rastered beam, The reconstructed target quantities can be expressed as
functions of focal plane quantities and beam positions at target. It
can be written as, for example:

$$\delta p = f(xfp, xpfp, yfp, ypfp)+g(xtar, ytar)$$

The raster correction function $g$ depends only on beam position.  It
can be fitted from nominal Raytrace data card. Even if the actual
optics of the spectrometer system may deviate from the nominal data
card, its effect on raster correction function $g$ is small so we can
still use the nominal function (fig.\ref {fig:engeres_ras} and
fig.\ref {fig:hksdp_ras}).

\begin{figure}
\begin{center}
  \subfigure
{\includegraphics[width=7cm]{optics-calib-engeangle_ras.eps}}
  \subfigure
{\includegraphics[width=7cm]{optics-calib-engedp_ras.eps}}
\end{center}
\caption{Comparison of Enge momentum(up) and angle(down) reconstruction resolution with
rastered beam,with and without raster correction. Both resolutions are
shown when the correction function $g$ fitted as a 3-order and 4-order
polynomials. The rightmost point is the resolution after raster
correction with wrong Enge optics.}
\label{fig:engeres_ras}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=7cm]{optics-calib-hksdp_ras.eps}
\end{center}
\caption{Comparison of HKS momentum reconstruction resolution with
rastered beam,with and without raster correction.}
\label{fig:hksdp_ras}
\end{figure}

\subsection{Code}
Under directory jlabs1:/home/yuan/HKS/optics/:
\begin{itemize}

\item \verb+.matrixfit/matfit6_backward.f+: Fortran program to fit
target quantities as a function of focal plane quantities and beam
position in the form of the above equation. It fits function $g$ as a
3-order polynomial of beam position. Need Raytrace
simulation output as input data.

\item \verb+.matrixfit/expon_rastercor.dat+: input data file specifying
the power of polynomial terms in functions $f$ and $g$. 

\end{itemize}

\section{Simulation of the Calibration Procedure}
Simulated events are used to test the calibration method. The
simulation take account of target physics processes, spectrometer
optics and detector resolution.

The simulation of target processes is  based on SIMC, a physics
simulation program for Hall C basic equipment experiment, adapted
for HKS experiment. The ionization energy loss, Bremsstrahlung,
multiple scattering and beam spread effects are included in the
simulation. The simulation uses 5mm thick CH$_2$ target and 0.1 g/cm$^2$
Carbon target.

Simulated events from target are send through The
spectrometers. RAYTRACE is used to simulate events in the electron
arm. Since Raytrace data card is not available for the Kaon arm, the
events are transfered from target to HKS focal plane by a 5-variable
($xtar$, $xptar$, $ytar$,$yptar$,$\delta p$), 6-order forward
matrix. Only optical properties (no physical processes) are considered in
this step. 

At focal plane of the spectrometers, the positions and angles are
smeared according to detector resolution and multiple
scattering. For Enge, the resolutions ($\sigma$) are: 86 $\mu$m (x),
0.7 mr(xp), 210 $\mu$m(y), 2.8 mr(yp). For HKS, they are: 160 $\mu$m
(x and y), 0.33 mr (xp and yp). 

The simulated focal plane events are then reconstructed back to the
target using backward reconstruction matrix. The beam energy and
particle momentum are corrected by the average energy loss in the
target. The corrections are 0.4923 MeV (beam energy), 0.6015 MeV (e'
momentum), 0.4890 MeV (Kaon momentum) for CH$_2$ target, 0.0820 MeV
(beam energy),
0.1026 MeV (e' momentum), 0.0866 MeV (Kaon Momentum) for Carbon
target. 

The simulated $\Lambda$, $\Sigma^0$ and $^{12}_\Lambda$B missing mass
spectra are shown in fig.\ref{fig:mmco}. The predicted $1^-$ state of
$^{12}_\Lambda$B  at
2.73 MeV is also included in the simulation for comparison. The quasi-free carbon background on CH$_2$ target is simulated
with $\Lambda$ and $\Sigma^0$ events by a assumed S/B ratio of 6:1
under $\Lambda$ peak. The $^{12}_\Lambda$B GS has a missing mass
resolution of 397 KeV (FWHM) with correct optics.   

The simulated Enge focal plane sieve slit correlation patterns are
shown in fig.\ref{fig:engefpss1}. Fig.\ref{fig:engefpss2} shows the
yfp vs. xfp correlations for each X-column of sieve slit holes.

The simulated HKS focal plane sieve slit patterns are
shown in fig.\ref{fig:hksfpss}.

%To test the calibration method by simulation, We intentionally use wrong
%spectrometer optics in reconstruction. The splitter field is changed
%from nominal value of 1.546 Tesla to 1.550 Tesla in forward simulation
%for e' events, but still using the nominal 
%reconstruction matrix in reconstruction. In addition, the beam energy,
%electron and Kaon arm central momenta have offsets from nominal
%values, they are 1.2 MeV, -0.7 MeV and -0.9 MeV respectively. The
%missing mass distributions using the before the calibration is shown
%in fig. \ref{fig:mmwonc}. The missing mass distributions after the
%calibration is in fig. \ref{fig:mmwoac}. 

\begin{figure}
\begin{center}
\includegraphics [width=15cm]{optics-calib-mmco.eps}
\end{center}
\caption{Simulated $\Lambda$, $\Sigma^0$ and $^{12}_\Lambda$B missing mass
spectra with Carbon QF background from CH$_2$ target}
\label{fig:mmco}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=15cm] {optics-calib-engefpss1.eps}
\end{center}
\caption{Simulated Enge focal plane sieve slit correlation and hole patterns.}
\label{fig:engefpss1}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=15cm] {optics-calib-engefpss2.eps}
\end{center}
\caption{Simulated Enge focal plane $yfp$ vs. $xfp$ correlations for each X-column of sieve slit holes.}
\label{fig:engefpss2}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=15cm] {optics-calib-hksfpss.eps}
\end{center}
\caption{Simulated HKS focal plane sieve slit correlation and hole patterns.}
\label{fig:hksfpss}
\end{figure}
 
\end{document}

% Revision history:
% $Log: optics-calib.tex,v $
% Revision 1.1.2.1  2005/05/13 13:42:07  saw
% Initial version
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