\chapter{Calibration Products}
\label{chap:cal_prods}

\section{Overview of Products}
\label{sec:prod_overview}

This section provides a general overview to the types of calibration
products, the next sections provide the HETG and HETGS products themselves.

{\it Calibration Data} are all the raw data acquired that has information
to constrain the instrument models; these are the basis of a calibration.

{\it Calibration Measurement Products} are the specific results from
specific manipulations of the Calibration Data and cover many levels
of complexity.  They represent the results when the raw data have been
``analyzed'' to create something, {\it e.g.}, ``counts per second in the
Al-K BND bump'', ``ACIS-I 5 arc min Mo-L off-axis encircled-energy
curve'', or ``witness flat reflectivity at 8.04 keV''.  There are lots of
these!  And their meaning and organization and synthesis can require a
lot of human knowledge.  Many results may be ammenable to database
storage, others may require memos to describe and define them. 
Error estimates are produced for each measurement result as well.

{\it Fundamental Calibration Products} are the most detailed
``encoding'' of the calibration activity into detailed (complex)
models, {\it e.g.}, SAOSAC as a model of the HRMA.  This requires a
synthesizing by calibrationists of the various Calibration Measurement
Products and model expectations.

{\it Calibration Interface Products} are specific, well-defined,
useful parameters, tables, models which describe the essence of the
instrument to sufficient accuracy to be useful yet in a more
accessible form than the Fundamental Calibration Products.  (The
distinction is blurred and need not be too strict.  For example the
ACIS pixel size is about as Fundamental as can be yet it is also a
very useful CIP value.)  These CIPs are the main interface between the
calibration activity and the rest of the system.

{\it Analysis Reference Data} is used to explicitly identify those
data that are needed and used by, {\it e.g.}, the ASC Data System for
simulation and data analysis.  The formats are dictated by the
analysis algorithms and software architecture; often these data may be
simply reformatted versions of the CIPs.  The \mx parameters,
Section~\ref{sec:marx_prods}, are a good example of Analysis
Reference Data.

The relationship of these products is schematically shown in
Figure~\ref{fig:cal_prods}.

\begin{figure}
\begin{center}
\epsfig{file=cip_figure.eps,height=12cm}
\caption[Schematic Diagram of the Types of Calibration Products]
{Schematic Diagram of the Types of Calibration Products}
\label{fig:cal_prods}
\end{center}
\end{figure}

\clearpage
\section{Fundamental Calibration Products for HETG}
\label{sec:fcps}

As we have seen, the LRF functions depend not only on shell-level
quantities but on grating-level data, {\it e.g.}, grating
mis-alignment angles\cite{marshall97}.  For this reason, {\it the
fundamental HETG calibration or model is at the grating facet level}.

For each grating facet, $f = 1$ to 336, we have a diffraction
efficiency $g_f(E,m)$, period map in facet coordinates $p_f(x,y)$, and
alignment distribution $\Gamma_f(\phi _{\rm roll})$ as well as mechanical
parameters.  These fundamental data can be combined with other
information to produce, {\it e.g.}, $\nu_s$, $G_s(E,m)$,
$SEA(E,m,\dots )$, and $LRF(E,m,\dots )$.

\clearpage
\section{HETG Calibration Interface Products}
\label{sec:cips}

A discussion of and values for the HETG CIPs are presented here; data
files, discussions, and updates are available at {\tt
http://space.mit.edu/HETG/xrcf.html} under ``Calibration Products''.

\subsection{Basic Parameters}
\label{sec:cip_basic}

\begin{figure}[ht]
\begin{center}
\epsfig{file=facet_figure.eps,height=9cm}
\caption[Basic Geometry for HETG Facet Location]
{Basic Geometry for HETG Facet Location}
\label{fig:basic_facet}
\end{center}
\end{figure}

Basic parameters of the HETG design are tabulated in the file {\tt
HETGbasic.rdb} for computer input and reproduced here in
Table~\ref{tab:cip_basic}.  The HETG radii given here are really a
method to record the angle of the central ray through the shell's
facets, ``theta\_shell'' in Figure~\ref{fig:basic_facet}:

\begin{equation}
\tan \theta_s ~=~ {\rm HETGrad}_s ~/~ {\rm HETGrd}
\end{equation}

\begin{quotation}
{\it To-do:} \\
Produce and include reference to 3-D facet location information.
\end{quotation}

\begin{table}[hb]
\begin{center}
\begin{tabular}{llll}
name & value & error & unit \\
\hline
\hline
HETGrd & 8633.69 & 0.1 & mm \\
HETGrad1 & 521.66 & 0.1 & mm \\
HETGrad3 & 419.90 & 0.1 & mm \\
HETGrad4 & 370.66 & 0.1 & mm \\
HETGrad6 & 275.44 & 0.1 & mm \\
HETGvign1 & 0.937 & 0.01 & none \\
HETGvign3 & 0.940 & 0.01 & none \\
HETGvign4 & 0.931 & 0.01 & none \\
HETGvign6 & 0.936 & 0.01 & none \\
\hline
\end{tabular}
\caption[Basic HETG Design Parameters]
{Basic HETG Design Parameters, from {\tt HETGbasic.rdb}.}
\label{tab:cip_basic}
\end{center}
\end{table}

\clearpage
\subsection{Periods, Angles, and Spacing}
\label{sec:cip_periods_angles}

{\bf Periods}

Lab measurements were made on the flight gratings for HETG as
well as on the TOGA gratings used in XRCF rehearsal.  These laboratory
measurements were corrected based on NIST-measured reference samples
(Section~\ref{sec:lr_nist}) resulting in our best estimate ``LR-NIST'' periods.

The current CIP adopted periods (in \AA ) are the sub-assembly values
because XRCF measurements are not accurate enough to require a change.
In particular a spacing discrepancy still exists,
Section~\ref{sec:periods_angles}.  The period errors are all taken as
the sub-assembly values, again because (with an HETG displacement
assumption) XRCF measurements are in agreement and do not provide any
smaller error bounds.

{\bf Angles}

The mean dispersion angles have been measured at XRCF and the HETG
opening angle should be the same for XRCF and flight.  The flight
clocking and grating angles have been modified slightly from the
XRCF values by the as-installed
+46 arc second HETG roll.

{\bf Spacing}

The HEG/MEG period difference seen between sub-assembly (HeCd,HeNe)
and XRCF-AL-K (assuming a spacing of 8788.04 mm) of about 500 ppm is
assumed to be due to an inaccurate value for the HETG Rowland spacing
at XRCF, Section~\ref{sec:hetg_location_at_xrcf}.  Thus, for now
the HETG position at XRCF is set to be 8782.8 mm to have reasonable
agreement between XRCF and sub-assembly periods (and hence calculated
line energies).  The cause for the (alleged) displacement has not been
determined.

For flight spacing, Scott Texter reports that EK measures (mechanical
means) the as-installed HETG to be 0.009 inches closer to the HRMA
than designed, so expected flight spacing is 8633.69+0.009*25.4 =
8633.92 mm.  The error on this is probably 0.2 mm or less
(HRMA-grating spacing).  However the HRMA-focus spacing may still have
error of order 0.5 mm.

\begin{table}[hb]
\begin{center}
\begin{tabular}{llll}
name & value & error & unit \\
\hline
\hline
HEGp & 2000.81 & 0.05 & Angstrom \\
MEGp & 4001.41 & 0.10 & Angstrom \\
HETGrsX & 8782.8 & 0.50 & mm \\
HETGrsF & 8633.92 & 0.50 & mm \\
HETGopenX & 9.934 & 0.008 & degrees \\
HETGopenF & 9.934 & 0.008 & degrees \\
HETGclockX & -0.225 & 0.05 & degrees \\
HETGclockF & -0.215 & 0.05 & degrees \\
HEGangleX & -5.19 & 0.05 & degrees \\
MEGangleX & 4.74 & 0.05 & degrees \\
HEGangleF & -5.18 & 0.05 & degrees \\
MEGangleF & 4.75 & 0.05 & degrees \\
\hline
\end{tabular}
\caption[HETG Period, Angle, and Spacing Parameters]
{HETG Period, Angle, and Spacing Parameters, from {\tt HETGperiod.rdb}.}
\label{tab:cip_period}
\end{center}
\end{table}

\clearpage
\subsection{Period and Roll Variation Parameters}
\label{sec:cip_core}

A coarse characterization of the gratings LRF effect is with the
period and roll variations, listed as $dp/p$ and $\gamma$
in the error budget of Table~\ref{sec:error_budget}.
These values are measured in the laboratory by the LR and Alignment
setups.  Confirmation of these laboratory values at XRCF
with the XRCF LRF Core analysis has only begun, Section~\ref{sec:lrf_core}.

The MEG mis-aligned gratings have pointed out a mechanism whereby the
alignment measurement differs from the actual grating bar
orientations.  It is likely that the same effect operating at a lower
and randomized level will lead to an increase in the actual $\gamma$
value over the alignment system determined one.  A simple analysis of
one XRCF data set suggests that $\gamma$ may be between 1 or 2 arc
minutes rms, hence a value of 1.5 arc minutes (with a large 0.5 arc
minute error) is being adopted preliminarily.

As for the period variations, the LR predictions of 127 ppm and 106
ppm, Figures~\ref{fig:meg_period_hist} and~\ref{fig:heg_period_hist},
are a lower limit to these variations.  Again the limited XRCF
analysis has not produced a precise value but suggests $dp/p < 300$~ppm.
Unlike the alignment effect, there is currently no expectation that
the $dp/p$ measured by the LR is in error, so the proposed baseline
values are the LR values increased by rss'ing them with 100 ppm to
account for any mounting induced distortions.


\begin{table}[hb]
\begin{center}
\begin{tabular}{llll}
name & value & error & unit \\
\hline
\hline
HEGdpop & 146. & 50. & ppm \\
MEGdpop & 162. & 50. & ppm \\
HEGroll & 1.5 & 0.5 & arc minutes \\
MEGroll & 1.5 & 0.5 & arc minutes \\
\hline
\end{tabular}
\caption[Parameters Effecting the HETG LRF Core]
{Parameters Effecting the HETG LRF Core, from {\tt HETGcore.rdb}.}
\label{tab:cip_core}
\end{center}
\end{table}

\clearpage
\subsection{Efficiency Products}
\label{sec:cip_effic}

Table~\ref{tab:cip_effics} lists the available files that contain the
current best estimates of the HETG diffraction efficiency, $G_s(E,m)$,
for $m=-11,\dots ,+11$; note that error files are available as well.
These files have equal plus and minus orders and include the shell vignetting
$\nu_s$ factor.  Samples of this data are plotted in
Figures~\ref{fig:pairs_cip_ee} and~\ref{fig:hetg_cip_ee}.

\begin{table}[hb]
\begin{center}
\begin{tabular}{llll}
Shell & Data File & Error File \\
\hline
\hline
1 & HETG\_shell1\_effic.rdb.gz & HETG\_shell1\_effic\_err.rdb.gz \\
3 & HETG\_shell3\_effic.rdb.gz & HETG\_shell1\_effic\_err.rdb.gz \\
4 & HETG\_shell4\_effic.rdb.gz & HETG\_shell1\_effic\_err.rdb.gz \\
6 & HETG\_shell6\_effic.rdb.gz & HETG\_shell1\_effic\_err.rdb.gz \\
MEG & MEG\_effic.rdb.gz & MEG\_effic\_err.rdb.gz \\
HEG & HEG\_effic.rdb.gz & HEG\_effic\_err.rdb.gz \\
HETG & HETG\_effic.rdb.gz & HETG\_effic\_err.rdb.gz \\
\hline
\end{tabular}
\caption[CIP Efficiency Files for the HETG]
{CIP Efficiency Files for the HETG, available from \\
 {\tt http://space.mit.edu/HETG/cal\_prods.html/\#HETG\_CIP}}
\label{tab:cip_effics}
\end{center}
\end{table}

%
% These figures produced by ~dd/idl/xrcf/cip_plots.pro
%

\begin{figure}
\begin{center}
\epsfig{file=pairs_cip_ee.eps,height=18cm}
\caption[Effective Efficiency for First-order MEG and HEG]
{Effective Efficiency for First-order MEG and HEG.  These efficiencies
are the mirror weighted average of the shell-by-shell efficiencies
for the appropriate pair of shells.
Plus and minus orders have been combined.
The dotted lines are the ``2-sigma'' errors of the
shell efficiencies with additional error due to uncertainties in the
relative area (weighting) of the HRMA shells.}
\label{fig:pairs_cip_ee}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{file=hetg_cip_ee.eps,height=18cm}
\caption[Effective Efficiency for HETG 1st and 0 orders]
{Effective Efficiency for HETG 1st and 0 orders.  These efficiencies
are the mirror weighted average of the shell-by-shell efficiencies.
The first-order efficiency is for combined plus and minus orders for
HEG and MEG.  The dotted lines are the ``2-sigma'' errors of the
shell efficiencies with additional error due to uncertainties in the
relative area (weighting) of the HRMA shells.}
\label{fig:hetg_cip_ee}
\end{center}
\end{figure}

\clearpage
\subsection{\mx Parameters}
\label{sec:marx_prods}
\input{pre_marx_prods}

\clearpage
\section{HETGS Calibration Interface Products}

\subsection{Resolving Power}

Using the resolving power error budget described
in Section~\ref{sec:error_budget}, estimates of the
flight HETGS resolving power have been made, Figure~\ref{fig:res_power}.
The ``optimistic'' and ``conservative'' $E/dE$ curves
are available in rdb format in the files: \\

\begin{center}
{\tt 
MEG\_res\_opt.rdb \\
HEG\_res\_opt.rdb \\
MEG\_res\_con.rdb \\
HEG\_res\_con.rdb \\
}
\end{center}

\subsection{Effective Area}

In order to compute the HETGS effective area, the HRMA effective area,
$A_s(E)$, and the ACIS-S quantum efficiencies,
$QE_{\rm ACIS}(E,{\rm grade~set})$, are needed and should be available
soon.


\clearpage
\subsection{XSPEC Response Matrices}
\label{sec:xspec}

Please see {\tt http://space.mit.edu/HETG/xspec/xspec.html} for a
discussion of HETG and XSPEC.