\begin{quotation}
{\it Objective:} The Period and Angle analysis is used to determine
the dispersion directions (angles) and a measure of the grating periods
based on the dispersion distance of known narrow lines.
\end{quotation}

\subsection{Analysis Method}

At XRCF Encircled Energy measurements of the diffracted orders were
performed with HXDS Beam Centering in order to precisely locate the
diffracted images.  The available EE data sets are listed in
Table~\ref{tab:beam_cen_sets}.

The beam centers obtained from these measurements are put into data
files, {\tt beam\_cen\_yymmdd.rdb}.  The centers are reported in HXDS
coordinates ({\tt ypos, zpos} as in the .sum files) in microns,
Figure~\ref{fig:beam_cen_example}.  Each of these data files is
analyzed with {\tt xrcf\_beam\_cen\_werr.pro} to get a best-fit period
and angle (with errors) for each grating,
Figure~\ref{fig:beam_cen_anal_output}.  The results are put, along
with sub-assembly predicted values, into a results file, {\tt
beam\_cen\_results.rdb}, Figure~\ref{fig:beam_cen_results}.

For these analyses the following parameter values were assumed:
Rowland distance = 8788.04 mm, hc = 12.3985, Al-K is at 1.4867 keV,
and TOGA rowland distance = 5366.55 mm.

The Period and Angle contents of this results file are plotted using
{\tt beam\_cen\_plots.pro} and shown in Figures~\ref{fig:disp_periods} and
\ref{fig:disp_angles}.  The weighted-average period for each grating based
{\it only} on the XRCF Al-K data sets is shown in the plot titles.  The HETG
opening angle and the HETG mean angle (MEG HEG bisector) are also
calculated and plotted.

\begin{table}[hb]
\begin{center}
\begin{tabular}{cccll}
\hline
Phase & Date code & Line & Orders & TRW IDs (runids) \\
\hline
\hline
R &  960827  & Al-K & MEG -1,0,+1 & 102931, 102926, 102929 \\
 &    &  & HEG -1,0,+1 & 102938, 102933, 102935\\
\hline
1C & 961223 & Al-K & MEG 0,+1,-1& C-HXF-EE-3.003,4,10\\
 &  &  & HEG 0,+1,-1,0& C-HXF-EE-3.005,6,13,5a\\
\hline
1D & 970104 & Al-K & MEG 0,+1,-1,-3,+3 & D-HXF-EE-3.004,5,6,12,13\\
 &  &  & HEG 0,+1,-1,-2,+2 & D-HXF-EE-3.007,8,9,14,15\\
\hline
1D & 970116 & Mg-K & MEG 0,+1,-1 & D-HXF-EE-3.021,22,23\\
 &  &  & HEG 0,+1,-1 & D-HXF-EE-3.024,25,26 \\
\hline
1E & 970208 & Al-K & HEG 0,-2,+2 & E-HXF-EE-3.007,14,15 \\
 &  &  & MEG -3,+3 & E-HXF-EE-3.007,14,15 \\
\hline
\end{tabular}
\end{center}
\caption[Beam Center Measurements of Diffracted Order Locations]
{Beam Center Measurements of Diffracted Order Locations}
\label{tab:beam_cen_sets}
\end{table}

\subsection{Results}

Both the MEG and HEG periods agree much better with the Rowland spacing value
derived from design plus shim of 8782.8 mm.  Yet the LETG period agrees
well with the 8788.0 mm value.  The working hypothesis is that there
is an HETG-LETG spacing difference that is not yet uncovered.

\begin{quotation}
{\it To-do:} \\
\indent$\bullet$ Improve analysis of EE data \\
  \indent\indent$\bullet$ Add mst\_date and run\_id values to 961223 and 970104 data files.\\
  \indent\indent$\bullet$ Re-do full analysis: .pha files to .sum files \\
  \indent\indent$\bullet$ Plot .sum files, compare with expected 1D scans (simulations); defocus? \\
  \indent\indent$\bullet$ Better "center" location and errors using 1D simulated shapes \\

\indent$\bullet$ ( Explain the LETG / HETG difference in spacing ) \\
\indent$\bullet$ Is the TOGA MEG period difference due to old LR version used for TOGA? \\
\indent$\bullet$ Why the HEG Al/Mg difference? ( by 1 part in 4000 ) \\
\indent$\bullet$ Use HSI observations for period determination too. \\
\indent$\bullet$ Use HRC and ACIS-S data sets to determine period/angle as well. \\
\end{quotation}

\begin{figure}
\begin{center}
\epsfig{file=beam_cen_example_plot.eps,height=9cm}
\caption[Beam center example plot]
{Beam center example plot}
\label{fig:beam_cen_example}
\end{center}
\end{figure}

\begin{figure}
{\tt\small
\begin{tabbing}
\input{beam_cen.txt}
\end{tabbing}
}
\caption[Period and angle analysis output]
{Period and angle analysis output}
\label{fig:beam_cen_anal_output}
\end{figure}

\begin{figure}
{\tt\small
\begin{tabbing}
\input{beam_cen_results.txt}
\end{tabbing}
}
\caption[Beam center results file]
{Beam center results file}
\label{fig:beam_cen_results}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=beam_cen_periods.ps,height=20cm}
\caption[Measured Grating Periods]
{Measured Grating Periods.  Parameter values assumed:
$X_{\rm RS}$ = 8788.04 mm, $hc$ = 12.3985, Al-K is at 1.4867 keV,
and for TOGA $X_{\rm RS}$ = 5366.55 mm.  Note that the XRCF derived
periods for both MEG and HEG are larger by $\approx$500 ppm than the
LR-NIST values, Section~\ref{sec:lr_nist}.
}
\label{fig:disp_periods}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=beam_cen_angles.ps,height=20cm}
\caption[Measured Dispersion Angles]
{Measured Dispersion Angles.  The LETG and HETG (mean) dispersion angles
(w.r.t. HXDS axes) track very well through the XRCF Phase I subphases.
The HETG opening angle is very stable as would be expected.}
\label{fig:disp_angles}
\end{center}
\end{figure}