

\section{Expected rates}


\subsection{Overview}

We estimate trigger and background rates in \hd\ using measurements of the
hadronic cross section
combined with the CLAS experience. The hadronic rate between any two photon
energies $E_1$ and $E_2$ can be written as
$$R=\int_{E_1}^{E_2}n\sigma(E)\frac{dN}{dE} dE$$
where $n$ is the number of target protons per unit area, $\sigma(E)$ is
the hadronic cross section as a function of energy, and $dN/dE$ is the
photon energy spectrum. The photon flux is composed of a coherent and
incoherent
sum as detailed in Chapter 4.
Background rates are dominated by
the broad-band incoherent flux. The signal rates result from the photon flux
in the coherent peak, which will depend on the radiator crystal structure and
its orientation. The coherent peak will be optimized to the specific
physics program. For our rate estimates, we use the typical case for
the flux computed on a
diamond radiator with the coherent peak at E$_\gamma$ = 9~\eunit.
%table is from Richard, should be included in chapter 4, was available from
%''meeting notes /Hadronic Rate Estimates for \hd'' June 9, 1998.

Both coherent and incoherent fluxes are proportional to the electron beam
current and radiator thickness.  Multiplying the number of electrons
per second by the radiator thickness in radiation lengths gives the
product $N_0$ which we will use in the following calculations.
For conditions which we will refer to as ``low intensity''  (300 nA
beam on a $10^{-4}$ radiator), $N_0$ = 1.9$\times$ 10$^8$/s.
For the coherent peak at 9 \eunit\, the tagged photon flux between 8.4 and
9.0 \eunit\ is $R_{tag}$ = 0.14 $N_0$.
The average tagging efficiency over this interval is 0.375, so the tagged
photon flux on target is 1.0 $\times$ 10$^7$/s.
``High intensity'' running, where the tagger becomes ineffective as part of
the level 1 trigger, nominally corresponds to
$N_0$ = 1.9$\times$ 10$^9$/s and yields $10^8$/sec tagged photons on target.

The total hadronic $\gamma p$ cross section\footnote{We use  measured cross
sections \cite{Armstrong,Caldwell} with
 actual data obtained from
the Durham Data Base \cite{durhamDB}} is plotted
in Fig.~\ref{fig:rates_sig}.
\begin{figure}
  \centerline{\epsfig{file=ch8_gpsigADR.eps,height=3.0in,width=5.25in}}
  \caption{Total cross section for $\gamma p\to$~{\em hadrons} as a function
    of photon energy.}
  \label{fig:rates_sig}
\end{figure}
For the experimental conditions defined above and
a 30\cmlunit\ liquid hydrogen target,
the total hadronic rate in the detector is
\begin{equation} 
  R_0 = 2\times 10^{-4}N_0 \label{eqn:rate_r0} 
\end{equation}
and a tagged hadronic rate
\begin{equation} 
  R_T = 7.4\times 10^{-6} N_0 
  \label{eqn:rate_rt}
\end{equation}
For low intensity, the expected total hadronic
rate is 37\khz\ and the tagged hadronic rate is 1.4\khz.

\subsection{Trigger elements}

We make some rudimentary assumptions about the trigger elements in order
to estimate various rates. These assumptions are discussed further in the
Trigger section of this document. Initial commissioning of the detector
at low rates will use a level 1 trigger to select events of interest.
At higher rates a sophisticated level 3 software trigger
\footnote{We are reserving level 2 for a possible intermediate level
hardware trigger}
 is required. We
concentrate here on discussion of rates at lower photon beam fluxes.

The trigger consists of coincidences between several counter
elements of the detector. It must select the tagged hadronic rate in the
presence of accidentally coincident backgrounds.
The first trigger element is the  photon tagger, essentially
a segmented scintillation counter.
The rate in this counter is determined by $N_0$, which is
controllable (within limits) by the experiment.

The second trigger element is the  start counter.  A start counter
similar in design to that used in CLAS would suffice for
low intensity running, although modifications will likely be
needed to handle high intensity conditions.
One possible design might use several layers of scintillating fibers.
Such a counter would also provide position information with sufficient
resolution for track reconstruction.  In comparing the demands of
the \hd\ start counter to the CLAS experience, it is useful to note that
the \hd\ target is inside a solenoidal magnetic field which will protect
the start counter from the flux of low-energy Compton scattered electrons
emerging from the target. The CLAS start counter does not enjoy this
protection.

The tagger and start counters are small, and are therefore the best
candidates for determining the precise event timing. For this discussion,
we will assume that coincidences between them can be identified within a
time window $\Delta T_1=15$\nstunit.

Interesting events will have particles in the final state other than the
one that satisfied the start counter requirement.
These particles may be energetic,
forward-going charged particles, forward or large angle photons, and/or charged
particles with sufficient transverse momentum to reach
the bore of the solenoid.
Any particles of this type will be registered in other elements of the
detector
and these signals can be used as further requirements in the trigger. This
refines the loose interaction definition given above. We refer to this
collection of signals as the global level 1 trigger. As its elements
are counters of extended size, we take a coincidence time window
$\Delta T_2=100$\nstunit\ when the global level 1 trigger is required.

\subsection{Accidental rates}

The rate of interesting events given by Eq.~\ref{eqn:rate_rt}
is 1.4\khz\ ($N_0$ = 1.9$\times$ 10$^8$/s) and 
14\khz\ ($N_0$ = 1.9$\times$ 10$^9$/s) for low and high intensity beams respectively. 
However, various other processes will form accidental coincidences at the different trigger
stages, and we need to recognize these. It is most important that these do
not form the bottleneck for the data acquisition system, regardless of our
ability to reject them offline. 

% 
% Furthermore, these accidental rates are
% at least quadratic in the photon flux, so the accidental-to-true ratio grows
% as we increase the beam intensity.

We consider two sources of accidental background. They are not
entirely mutually exclusive, but we consider them separately for
ease of explanation. The first ($A_1$) of these comes from purely random
time coincidences between the trigger elements, in which case we compute
the time overlap based on the various counter singles rates. The second
($A_2$) is more ``physical'', considering hadronic photoproduction that is outside
the tagging range, but in accidental coincidence with the tagging system.

First consider purely random coincidence events.
A coincidence between the tagger and start counter loosely defines an
interaction in the target. The rate $A_0$ of this coincidence is given by
\begin{equation} A_0=S R_{tag}\Delta T_1 \label{eqn:rate_a0} \end{equation}
where $S$ is the total rate in the start counter. Based on the
experience in CLAS, we take $S=0.03 N_0$, scaled using appropriate factors for
collimation and beam intensity. This is most certainly
an upper limit because of the solenoidal shielding effect.
%(In CLAS, $\approx7$\% of this rate is generated by tagged photons.)
For $R_{tag}=2.7\times10^7$/s we find $A_0=2.3\times10^6$/s,
considerably larger than the tagged hadronic rate $R_T=1400$/s.
Further refinements are achieved by the global level 1 trigger.

The rate of the global level 1 trigger, $f_{L1}\times R_0$, is taken to
be the total hadronic rate\footnote{The cosmic-ray rate is small and has been
neglected.} reduced by the rate for single pion production for
E$_\gamma~ \leq$ 0.5 \eunit\ ($f_{L1}=0.5$). A loose trigger which uses a
charged particle track count in the start counter
and requires neutral energy in the barrel and/or
forward calorimeter (see Section \ref{sec:lev1trigger} below)
%Section 7.2.2 in current version
should easily be able to eliminate these low energy events.
The accidental
rate using both the interaction and global level 1 triggers is
\begin{eqnarray}
A_1 &=& A_0 f_{L1} R_0 \Delta T_2 = 
    S f_{L1} (0.28\times 10^{-4}) N_0^2 \Delta T_1 \Delta T_2
\label{eqn:rate_a1}
\end{eqnarray}
where we have substituted from Eqs.~\ref{eqn:rate_r0} and~\ref{eqn:rate_a0}
and used $R_{tag} = 0.14 \times N_0$.
The second accidental background comes from true hadronic events,
and therefore would pass the global level 1 trigger. They are out of time
with the precise RF signal, but that is much smaller than the online resolving
time $\Delta T_1$ of the interaction coincidence.
Ignoring the ``true'' events that are part of this rate, one calculates
\begin{eqnarray}
A_2=f_{L1} R_0 R_{tag}\Delta T_1=f_{L1} 
 (0.28\times 10^{-4})N_0^2\Delta T_1
\label{eqn:rate_a2}
\end{eqnarray}

In order to evaluate the total accidental contribution numerically, 
correlations must be taken into account. This reduces the sum of
the above estimates. For $N_0$ = 1.9$\times$ 10$^8$/s,
the accidental contribution to the trigger is 7.3\khz, and the 
tagged hadronic rate is $R_T$ = 1.4\khz. We note that as the
photon flux increases, the start counter and
tagger lose their effectiveness in
reducing trigger rates, so the trigger rate asymptotically becomes
proportional to the hadronic rate. At higher currents, a 
DAQ system with a software level 3 trigger is
required. A summary of the rates is shown in 
Fig.~\ref{fig:rates_acc} as a function of electron beam current.
\begin{figure}
\centerline{\epsfig{file=ch8_rates.eps,height=5.0in,width=5.25in}}
\caption[Estimated rates as a function of electron beam current.]
{Estimated rates as a function of electron beam current.
Plotted is the total hadronic rate and the estimated trigger rate, 
which is the sum of accidental coincidences and the tagged
hadronic signal.}
\label{fig:rates_acc}
\end{figure}

\subsection{Rates in tracking chambers}

At the high photon flux anticipated for \hd, one concern is that the
occupancy rates in the drift chambers may be too high
to allow reconstruction.
In order to estimate these occupancies, a test of high intensity 
running with photons 
was performed in the CLAS detector.\footnote{This information is taken from
CLAS-NOTE-2000-004 High-Rate-Test by Elton Smith.}
Measurements were taken at 10, 80, 250 and 320 nA with 
a 10$^{-4}$ radiator,
and rates were measured in the forward TOF
scintillators (7.5-12.5 deg), the electromagnetic calorimeter (8-45 deg), 
and the drift chambers. The drift chamber occupancies at the highest
current (320 nA) are given in  Table \ref{table:occupancies}.


%Occupancies for each superlayer:
%(logbook entry \#7031), Run number 21998.

\begin{table}[bth]
 \caption{\label{table:occupancies} Drift chamber occupancies for each
superlayer (in percent) for run
21998  at the maximum beam current of 320 nA (logbook entry \#7031).}
\begin{center}
  \begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Beam & Region 1 & Region 1 & Region 2  & Region 2 & Region 3 & Region 3   \\
 ~    & S1     & S2       & S3        &   S4     & S5       &     S6   \\
\hline
 Photon & 2.3\% &   2.3\%  &   0.3\% &  0.4\%  & 0.7\%  & 0.9\% \\
% Electron & ?\% &   ?\%  &   ?\% &  ?\%  & ?\%  & ?\% \\
\hline
  \end{tabular}
\end{center}
\end{table}

Although the conditions of the test did not duplicate precisely the 
conditions expected in \hd, reasonable estimates
can be made by appropriate scaling. In Table 
\ref{table:comparison} we compare the differences in target, collimation
and beam energy. As the majority of background results from lower energy
photons, we assume the energy dependence of the measured rates is small.
The rates are scaled by a factor of 1.7 (ratio of target lengths) and the
beam current is scaled up by a factor of 5.33, which is the expected
collimation ratio.  The drift chambers
in region 1 are completely unshielded by any magnetic field in CLAS,
whereas the drift chambers in region 2 are shielded by the field of the CLAS
torus. The 2.2 T solenoidal field for \hd\
is expected to be at least as effective as a shield as the CLAS torus.
Therefore, we expect the oocupancies in the \hd\ drift chambers to be as low or
lower than those in CLAS for comparable granularity. 

\begin{table}[bth]
 \caption{\label{table:comparison} Comparison between conditions in Hall B
during high rate test and anticipated running parameters for \hd.
A current of 3 $\mu$ A in \hd\ corresponds to 10$^8$ photons/s in the 
coherent peak.}
\begin{center}
  \begin{tabular}{|l|c|c|}
\hline
~  & Hall B & Hall D  \\
 ~    & Test &   \\
\hline
Beam Current  & 80 nA $\rightarrow$ 320 nA & 300 nA $\rightarrow$ 3 $\mu$ A \\
Radiator & 10$^{-4}$ &  10$^{-4}$  \\
Collimation keeps & 80\%  & 15\% \\
Target Length &  18 cm	 &   30 cm \\
Beam Energy	 &   2.4 GeV	&  12 GeV \\
Trigger	 &   Restricted	&  Open \\
% DAQ Rate & 3 KHz &  15 KHz \\
\hline
  \end{tabular}
\end{center}
\end{table}


Extrapolating measured occupancies in region 2 to a 
current of 3 $\mu$A (\hd\ with 10$^8$ photons/s in the coherent peak), 
we expect an 
occupancy of 0.6\%. The rates are plotted versus electron current scaled
to \hd\ in Figure \ref{fig:rates}.
This is well below the typical operational limits of 2.3\%
imposed for the region 1 drift
chambers in CLAS during electron beam running, a rate at which tracks 
can still be reconstructed
with reasonable efficiency. We note that the extrapolated rates in region 1
for a beam current of 3 $\mu$A is approximately 5\%, exceeding usual operational
limits by a factor of 2, but this figure is for a configuration 
which is completely unshielded by any magnetic field
whatsoever and thereby represents an absolute maximum to the expected rates.
We note that
the operation of a polarized target in Hall B (which replaces the mini-toroid
with a solenoidal field) allows running at twice the normal luminosity. Thus
we expect that for comparable segmentation, raw rates in the \hd\ detector 
at the maximum design current will be similar to the current experience with
CLAS.  The conclusion is that the \hd\ detector should be able to handle
rates up to 10$^8$ photons/s.


\begin{figure}[p]
\begin{center}
\mbox{\epsfig{file=ch8_luminosity.eps,width=0.9\textwidth,
height=0.9\textwidth}}
\end{center}
\label{fig:rates}
\caption{Drift chamber occupancies (scaled by target
thickness = 1.7) plotted versus 
beam current (scaled by collimation factors = 5.33) expected for
\hd\ operation. The drift chambers
in region 1 (squares) are completely unshielded by any magnetic field
in CLAS, whereas the drift chambers in region 2 (triangles) are shielded 
from backgrounds by the main torus
field. The nominal low current operation in \hd\ (10$^7$ photons/s in
the coherent peak) corresponds to 300 nA. The 2.2 T solenoidal field for \hd\
is expected to be at least as effective as a shield as the CLAS torus.}
\end{figure}


