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\title{Simulation of Tagger Backgrounds \\
coming from the 12~GeV Electron Beam Halo}
\author{Richard Jones, University of Connecticut}
\maketitle

\begin{abstract}
Detailed ray-tracing simulations of the 12 GeV electron beam at the
position of the Hall D photon radiator have recently become available.
These results make it possible for the first time to estimate the
backgrounds in the tagger focal plane that arise from beam particles
that are outside the nominal beam envelope.  This note describes how a
beam halo model was extracted from the accelerator simulation data,
and estimates the impact that the expected halo fractions will have on
the observed rates in the tagging counters.
\end{abstract}

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In the original report on the electron beam requirements for
GlueX~\cite{reqnote}, a preliminary estimate was made of how much
background would be produced in the tagging counters for a given
fraction of electron beam particles that fall outside the nominal
beam envelope.  An upper limit of $10^{-5}$ for the halo fraction
was estimated, based on the criterion that no more than 1\% of the
rates in the tagging counters should be produced by background coming
from halo, where halo was defined to be any particles crossing the
plane of the radiator outside a $5\sigma$ ellipse around the beam centroid.
This estimate was based on the very conservative assumption that halo
particles are uniformly distributed over a disk completely filling the
1.5~in.\ diameter electron beam pipe.

The Jefferson Lab CASA group has recently produced a report describing
the detailed shape and momentum profile of the electron beam at the Hall D
radiator~\cite{halonote}.  The report describes a detailed Monte Carlo
simulation of the accelerator and beam delivery complex that is capable
of tracking individual electrons through the accelerator and transport line
up to the Hall D tagger.  Electrons were injected into the simulation with
a Gaussian phase-space distribution, but as they were tracked through the 
accelerator, some particles diffused far from the core of the beam and
eventually formed an extended tail that appeared superimposed on top of the 
central normal distribution.  The reason the halo formed is that, once a
particle wandered outside the core, perhaps because of energy loss from
synchrotron radiation or small-angle scattering from residual gas in the beam
pipe, it began to sample magnetic fields in the lattice elements outside their
linear region.  This caused halo particles to continually accummulate 
deviations from the nominal orbit, and eventually to be lost by collisions
with the walls of the vacuum chamber.  As soon as a particle reached the
inner radius of the beam pipe in the simulation, it was considered lost
and removed from the simulation.  It is the competition between processes
that generate new halo particles and their eventual loss that determines
the amount of halo seen at any point along the beam line.

Besides synchrotron radiation and beam-gas scattering, other processes
feeding the halo include displaced or mis-powered magnets, mismatched
optics, and intrabunch scattering.  All of these effects are taken into
account together in the simulation by introducing random errors in the
alignment and field parameters of magnetic elements in the lattice, which
result in displacements of the main beam from the nominal orbit within 
certain predefined envelopes.  These perturbations are not large enough
to degrade the quality of the central part of the beam, but they contribute
to the growth of the halo in the way described above.  Ref.~\cite{halonote}
reports halo distributions for simulations with a 3~mm orbit perturbation
envelope, a 6~mm envelope, and a 10~mm envelope.  The recommendation of
the authors was to consider the 3~mm envelope as a realistic design goal
for the 12~GeV beam.

\section{\label{model}halo model extraction}

In Ref.~\cite{halonote}, the beam halo is defined as the fractional integral
of the beam current that lies outside the central peak in the $x$ or $y$
projection of the transverse beam intensity profile.  The region of the
central peak is defined by $\pm 5.10$~mm in $x$ and $\pm 3.15$~mm in
$y$ of the beam centroid.  Each of the three simulation data sets presented
contain $10^8$ beam particles, of which over 95\% reach the Hall D radiator.
The $x$ and $y$ projections of the impact points on the transverse plane at
the radiator position are plotted in Fig.~6 and fitted to the sum of a
central Gaussian and a second-order polynomial halo parameterization.  The
parameters resulting from the fits are given in Tables~2-5.

One difficulty with extracting a halo model from these plots is that there
is no unique way to convert a pair of $x$ and $y$ projections into a 
two-dimensional halo intensity distribution $h(x,y)$.  Upon request, the
authors were able to provide some two-dimensional histograms that showed
the $x,y$, $x,\theta_x$, $y,\theta_y$, and $\theta_x,\theta_y$ distributions
for the 3~mm perturbation envelope simulation.  These are shown in the
first panels of Figs.~\ref{fig:xy}-\ref{fig:ythy}. In order to make the
halo events visible in these plots, it was necessary to mask off the
areas of the plot inside the central $5\sigma$ region of the beam.  In
spite of the limited statistics of the halo events, after applying these
masks it was possible to discern the main features of the halo distribution
and extract a plausible analytical approximation to the phase-space
density.  The second panels in Figs.~\ref{fig:xy}-\ref{fig:ythy} show
1000 points each, generated according to the analytical model using Monte
Carlo techniques.

\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{datahalo_xy.eps}
\includegraphics[width=2.5in]{modelhalo_xy.eps}
\caption{\label{fig:xy}
Caption goes here.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{datahalo_xthx.eps}
\includegraphics[width=2.5in]{modelhalo_xthx.eps}
\caption{\label{fig:xthx}
Caption goes here.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{datahalo_ythy.eps}
\includegraphics[width=2.5in]{modelhalo_ythy.eps}
\caption{\label{fig:ythy}
Caption goes here.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=2.5in]{dmhalo_prox.eps}
\includegraphics[width=2.5in]{dmhalo_proy.eps}
\caption{\label{fig:proxy}
Caption goes here.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{micro_rate.eps}
\caption{\label{fig:micro_rate}
Caption goes here.}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{hodo_rate.eps}
\caption{\label{fig:hodo_rate}
Caption goes here.}
\end{center}
\end{figure}

\subsection{\label{sec:parameters:halo}beam tails}

Transverse emittance reflects the r.m.s.\ widths of the transverse
position and angle distributions of electrons in the beam.  These 
distributions follow a Gaussian profile, typically over several orders
of magnitude.  Beyond some radius, however, the distribution begins to
fall more slowly than the Gaussian profile or even becomes relatively
flat.  These tails (or halo) contain relatively few particles relative
to the central core of the beam, but can be important because they can
interact with the dense materials surrounding the beam line and target
and produce background.  In the case of GlueX, the distance of order
100~m from the electron beam to the target (with the photon collimator
in between) prevents such off-axis electrons from producing background
in the GlueX detector.  The same is not true of the tagging counters,
however. 

Photons produced by halo electrons have essentially zero chance of 
getting through the photon collimator.  The reason for this is that
they are produced by bremsstrahlung in materials that are orders of
magnitude thicker than the crystal radiator, otherwise their photon
yield would be negligible by reason of the low intensity of halo
electrons relative to the core of the beam.  Electrons passing through
such a thick radiator undergo so much multiple scattering that the
photon spot that they produce projected out to the distance of the
collimator plane is orders of magnitude larger than the collimator
aperture.  By contrast, the probabilities are
somewhat higher that the degraded halo electron might find its way into
one of the tagging counters.  Beam particles which create hits in the
tagging counters but no corresponding photon in the photon beam inflate
the tagging efficiency.  To make a quantitative estimate of this effect
a model is needed for the distribution of materials in the vicinity of
the beam axis upstream of the tagger and a spatial and momentum
distribution for the halo particles.

A zeroth-order estimate for a safe upper limit for beam halo intensity
is obtained as follows. Fit the transverse beam position distribution to
a Gaussian function and subtract this distribution from the full beam
population.  The remaining particles, described as the halo population,
are spread out over a relatively large spot compared to the original
Gaussian radius. Assume that all of the halo beam particles end up
striking some vacuum, support or magnetic element on their way through
the tagger and that they all either scatter into a tagging counter
themselves or produce secondaries that do.  Under these assumptions, one
may ask what fraction of the original beam may belong to the halo
population without degrading the tagging efficiency by more than 1\%.
Under these somewhat perverse assumptions, the upper limit on the halo
integral is $10^{-6}$. 

If this halo level were easy to guarantee coming from CEBAF at 12~GeV
then our work would be done.  However we have been assured that it is not.
To make progress in refining this estimate, more information regarding the
distribution of materials around the beam and the halo phase space
distribution is needed.  Experience with CEBAF at 6~GeV has shown that
the latter can be difficult to predict and sometimes to control.  On the
other hand, we can configure the beamline elements to minimize the
amount of material in the region upstream of the tagger which may cause
the electron beam tails to scrape and produce background in the tagging
counters.
The following preliminary description of the region surrounding the beam in
the region of the radiator has been studied using a GlueX tagger Monte
Carlo simulation.
\begin{itemize}
\item A square aluminum frame with an inner cutout region of dimensions
$1.5\times 1.5$~cm$^2$, outer dimensions $3\times 3$~cm$^2$ and
thickness 3~mm.
\item A stainless steel beam pipe leading through the tagger quadrupole
from the radiator housing to the tagger vacuum box of outer diameter
3.8~cm and thickness 1.5~mm.
\item A pressure of $10^{-4}$ Torr in the tagger vacuum.
\item A 3~cm gap between the poles of the tagger dipoles.
\end{itemize}
In order to estimate the background in the tagging spectrometer,
particles were generated uniformly in a disk of radius 2.5~cm at
the position of the Hall D radiator.  This is a very pessimistic model
for what might emerge from a 1.5 inch beam pipe coming from the tunnel,
so the results can be taken as a conservative estimate for the expected
halo rates in the tagger.

Using this model, $10^6$ halo events were tracked through the tagger
geometry with magnetic field simulation and full shower generation.
Fig.~\ref{halo_image} shows the transverse profile at the position of
the radiator of all electrons in this sample which produced hits in
the tagging counters.  The second panel in the figure is provided to
help with interpreting the hit pattern.  The central red square in
the second panel is the diamond radiator.  The crystal mounting frame
is shown by a light blue rectangular outline around the crystal.
Next in order of increasing radius is the entrance flange to the
tagging spectrometer vacuum box (purple ring) whose inner diameter
is equal to the gap between the poles of 3~cm, somewhat smaller than
the 3.5~cm ID pipe leading into it (solid blue ring).  The green
horizontal structures at the top and bottom of the figure are the
pole shoes of the first spectrometer dipole.  Surrounding the beam
pipe with a gap of a few mm is the spectrometer quadrupole (solid
yellow) which is represented in the model as a large block of iron
with a circular hole cut out of the middle.  Clearly visible in the
hit pattern are the outlines of the radiator crystal holder and the
vacuum chamber entrance flange, while the beam pipe material and the
quadrupole and dipole magnets appear as partial voids because they
act as absorbers rather than sources of background events.

Fig.~\ref{halo_image} shows that the entrance flange to the vacuum
box is the largest single source of tagger backgrounds in the
present simulation geometry.  This conclusion may change, however,
when the halo distribution is improved beyond the crude model of a
uniform disk extending out to 2.5~cm in radius.  For the purposes
of an initial estimate, the current model is sufficient.
of 
expected to reduce the background rates by a factor of 3-5 in 

\begin{figure}[t]
\begin{center}
\includegraphics[width=4in]{halo_image.eps}
\includegraphics[width=4in]{hi.eps}
\caption{\label{halo_image}
Transverse profile (top frame) at the radiator of particles in the
beam halo that went on to create hits in the tagging counters.
Each point is the intersection of an electron track with the plane
containing the radiator.  The hit pattern reflects the material
distribution (bottom frame) as seen by the incoming electron beam.
The two panels have matching dimensions.  See text for details of
the material map.}
\end{center}
\end{figure}

In this sample a total of 9480 hits were observed leaving energy
more than 200~keV in the tagging microscope counters.
One million beam particles represents 50~ns of
real time under full-intensity running conditions of $10^{8}$ tagged
$\gamma/s$ on the GlueX target.  Hence if $10^{-5}$ of the total electron
beam population were in the halo then the halo rate in the tagging
counters would be 10k per 5~ms or 2~MHz, which corresponds to
a inflation factor of 1\% in the tagging efficiency.  From this it
follows that an upper bound of $10^{-5}$ on the halo fraction is
sufficient to insure that its effect on the tagging efficiency is
negligible.  The only part of the halo that contributes significantly
to the background in the tagging counters is what lies between the radii
of 1.0 and 1.5~cm.

In addition to the tagger microscope counters, there is also an second
array of tagging counters which covers a much broader range in photon
beam energy is more coarsely segmented.  This broad-band array is not
used for tagging during polarized photon running, but it is extremely
important for monitoring the quality and stability of the photon beam.
It is also needed to align the crystal at the beginning of each run
period.  The simulation showed that a $10^{-5}$ halo creates a hit
rate below 1\% of the tagged photon rate across most of the counters,
with the exception of the very low-energy electron end of the spectrum.
The broad-band array extends up to 95\% of the bremsstrahlung end-point,
which means that it sees electrons of only 600~MeV energy at the
extreme end.  This end of the array is also the closest to the radiator,
which means that the solid angle of these counters relative to sources in the
radiator region is the greatest in the same place where the bremsstrahlung
spectral intensity is at its minimum.  Fig.~\ref{halo_frac} shows the
fraction of the total hits in each of the counters in the broad-band array
that would be generated by a halo with a beam fraction of $10^{-5}$.
Maintaining a 1\% upper bound on the halo contribution to the count rate
in the tagger at the high-energy photon end of the broad-band array would
push the requirement on the total halo fraction down to $10^{-6}$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=5in]{halo_frac.eps}
\caption{\label{halo_frac}
Percentage of hits in the broad-band tagging counters that 
are generated from electron beam halo particles, assuming a beam halo
fraction of $10^{-5}$.  The electron beam energy scale is roughly
linear in the $z$ coordinate, varying from 600~MeV at the left end
of the plot to 9~GeV at the right.  The shaded region indicates the
coverage of the microscope.}
\end{center}
\end{figure}

\begin{thebibliography}{123}
\bibitem{reqnote}
R.T.~Jones, ``GlueX Requirements for 12~GeV Electron Beam Properties'',
GlueX-doc-645 (2006).
\bibitem{halonote}
Y.~Roblin and A.~Freyberger, ``Studies of Beam Halo Formation in the 
12~GeV CEBAF Design'', JLAB-TN-06-048 (2006).
\end{thebibliography}

\end{document}
