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\begin{center}

{\Large \bf TRIUMF Experiment E614 \\}

{\Large \bf Technical Note \\}

{\Large \bf 3-D Magnetic Field Calculations of the E614 Magnet - Part II\\}

\vspace{0.4cm}

\rm{\bf D.H.Wright TRIUMF \\}

\vspace{0.4cm}

\rm{\bf 13 March 2000 \\}

\end{center}

\begin{abstract}

  Additional questions about the effect of design changes on the E614
magnet are addressed by magnetic field calculations using OPERA-3D.  It was
found that there is little difference between 1010 steel and 1020 steel.
Body forces on the coils were calculated and found to be within the 
manufacturer's limits provided the coils can be positioned to $\pm$ 0.2 cm.
However the coils must be positioned to $\pm$ 0.05 cm to keep field 
asymmetries small.  Cutting a hole in the bottom of the yoke to balance the 
cryogenic port at the top balances the $y$ force on the coils and increases 
field symmetry.  Reinforcing rods in the floor of M13 affect the field in the 
region of the beam holes but not in the region of the drift chambers.  
Reducing the radius of the beam entry hole from 20 to 10 cm reduces the width 
of the radial field distribution in that region by half.  Optical positioning 
holes in the yoke doors do not affect either the central field or the radial 
feild around the beam entry hole.

\end{abstract}

\section{Introduction}

Calculations of the magnetic field in the E614 detector have been performed
in three dimensions using OPERA-3D \cite{VF00}.  Details of the magnet/yoke
design and the resulting fields are given in Technical Note 40 \cite{TN40}.
Additional design questions are addressed here and include:

- is there a difference between 1010 and 1020 steel? \\

- what are the forces on the coils ? \\

- should another hole be cut in the yoke to balance the coil forces ? \\

- do reinforcing rods in the floor affect the field ? \\

- will a smaller beam entry hole narrow the radial field distribution ? \\

- do optical positioning holes in the doors of the yoke affect the field ? \\

\section{1010 Steel vs. 1020 Steel}

  Previous field calculations were performed with a B-H curve for 1020 steel, 
however the yoke will be now be constructed of 1010 steel.  Using the same 
finite element mesh, field calculations were performed with both B-H curves 
with the result that no significant differences, either in field uniformity or
magnitude, were observed between the two.  All subsequent calculations in this
report were carried out assuming 1010 steel.   

\section{Forces on Coils}

The cryogenics opening at the top of the yoke introduces an asymmetry in the
field which in turn produces a net force on the coil assembly.  According to 
the manufacturer \cite{Ox00} net forces in any direction should not exceed
one ton.  In order to check this the net forces on the 6 coil-pairs in the 
E614 magnet were calculated and are shown in Table 1.

As expected the net sideways force ($F_x$) is zero.  The net downward force 
($F_y$) is 4940 N or 1113 lbs. which is well within the one-ton limit.  The
net force along the beam axis ($F_z$) is zero, although the compressive
forces on four of the six coil-pairs are very large.  As a check the coil
forces were also calculated without a yoke.  In this case the compressive 
forces were unchanged and the downward forces disappeared. 

While coil forces in a slightly asymmetric field may be acceptable if the
coils are exactly centered in the yoke, a small displacement may cause such
forces to increase.  The forces were re-calculated for coil displacements of
-0.5, 0, +0.1, and 0.5 cm in $y$.  The net downward force on the coils is 
shown for each case in Table 2.  The force is linear in the $y$ displacement 
and exceeds the one-ton limit for a downward displacement of 0.5 cm.  Hence
the coils must be placed to within $\pm$ 0.2 cm to safely avoid high forces. 

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline

  Coil pair  & $z$ (cm) & $F_y$ (N) & $F_z$ (N) \\ \hline

    1        & -76.695  &  -1704    &  2009789  \\
    2        & -33.230  &   -436    &   351222  \\
    3        & -10.315  &   -330    &      968  \\
    4        &  10.315  &   -330    &     -968  \\
    5        &  33.230  &   -436    &  -351222  \\
    6        &  76.695  &  -1704    & -2009789  \\ \hline
 Net force   &          &  -4940    &        0  \\ \hline

\end{tabular}
\caption{Force in Newtons on the six coil pairs situated in the E614
yoke with a cryogenic services opening in the top.  $F_x$ is 0 (within 
$\pm$ 1 N) in all cases.  $z$ values give the location of the coil centers
relative to the center of the yoke.}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline

  Coil displacement ($y$) (cm)  & $F_y$ (N) \\ \hline

           -0.5                 & -9443     \\
            0.0                 & -4941     \\
            0.1                 & -4046     \\
            0.5                 &  -438     \\ \hline

\end{tabular}
\caption{Net downward force in Newtons on full coil assembly as a function of 
coil displacement in $y$.}
\end{center}
\end{table}

The results of similar calculations, performed for displacements in $x$ and 
$z$, are shown in Tables 3 and 4, respectively.  The forces $F_x$ are linear 
in the $x$ displacement of the coils but because there is no field asymmetry 
in the $x$ direction, the forces do not approach the one-ton limit for any of 
the calculated displacements.  Similarly, the forces $F_z$ are linear in the 
$z$ displacement of the coils and, while larger for a given displacement, do 
not approach the one-ton limit.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline

  Coil displacement ($x$) (cm)  & $F_x$ (N) \\ \hline

           -0.1                 & -908     \\
            0.0                 &    0     \\
            0.5                 & 4578     \\ \hline

\end{tabular}
\caption{Net sideways force in Newtons on full coil assembly as a function of 
coil displacement in $x$.}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline

  Coil displacement ($z$) (cm)  & $F_z$ (N) \\ \hline

           -0.5                 & -7529    \\
            0.0                 &     0    \\
            0.1                 &  1505    \\ \hline

\end{tabular}
\caption{Net force along $z$ in Newtons on full coil assembly as a function of 
coil displacement in $z$.}
\end{center}
\end{table}

Another consideration is the effect of coil displacements on the field 
quality.  For all of the above displacements in $x$ and $y$ the field 
uniformity in the region of the drift chambers is unaffected to any significant
degree.  This is not true for the radial field in the beam entry/exit holes.
Fig. 1 shows the radial field 1 cm from the $z$ axis when the coil assembly
has been displaced 0.5 cm upward.  The effect is large, especially in the 
region $70 < z < 130$ cm.  For a 1 mm displacement Fig. 2 shows a smaller,
but still significant, effect.  In order to reduce the size of this asymmetry
to 5 gauss or less, it will be necessary to position the coils to $\pm$ 
0.5 mm.    

When the coils are displaced in $z$ the radial field in the region of either
beam hole is unaffected.  When the coils are displaced 1 mm there is a small 
upstream-downstream asymmetry in the $B_z$ field in the drift chamber region 
as shown in Fig. 3.  The maximum size of the asymmetry is 1 gauss which is 
at or below the precision of a possible field measurement.  The error in
coil position in the $z$ direction should therefore be kept to 1 mm or less.
 
\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig1.ps}
\end{center}
\caption{Effect of coil assembly shifted +0.5 cm in $y$.  $B_r$ is calculated
on a line parallel to and 1 cm from the $z$ axis.  Top curve: $y = -1, x = 0$ 
cm.  Middle curve: $y = 0, x = +1$ cm.  Bottom curve: $y = +1, x = 0$ cm.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig2.ps}
\end{center}
\caption{Effect of coil assembly shifted +0.1 cm in $y$.  $B_r$ is calculated 
on a line parallel to and 1 cm from the $z$ axis.  Top curve: $y = -1, x = 0$ 
cm.  Middle curve: $y = 0, x = +1$ cm.  Bottom curve: $y = +1, x = 0$ cm.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig3.ps}
\end{center}
\caption{Effect on $B_z$ of -1 mm shift in $z$ of coil assembly.  Top curve:
$B_z$ along $z$ axis for $-48 < z < 0$ cm.  Bottom curve: $B_z$ along
$z$ axis for $0 < z < 48$ cm.}
\end{figure}

\section{Effect of a Hole in the Bottom of the Yoke}

As indicated in the previous section, the coil forces do not exceed the
manufacturer's limit, provided the coils are well positioned.  However the
net downward force can be eliminated if a hole identical to the cryogenics 
port is cut in the bottom of the yoke.  This would increase the safety 
margin with respect to unknown position shifts of the coil.
  
In order to study the effect on the field, the model was changed by placing
a hole in the bottom of the yoke at the same position as the one in the top.
The extra steel bars, placed around the top hole to take up flux, were 
removed.  The resulting field in the detector region was very symmetric and 
was sampled over a circular path of radius 16 cm in the $x-y$ plane, centered 
on the $z$ axis at $z$ = 15 cm.  The $B_z$ field varied by only a fraction of 
a gauss over the entire path.  There is no up-down asymmetry in $B_r$ in the 
region of the beam holes as long as the coils are well-positioned.  When the
coils are shifted by 0.1 cm in $y$ an asymmetry appears as shown in Fig. 4.  
Comparing Figs. 2 and 4, it is clear that the asymmetry for a given coil
displacement is smaller when there is a balancing hole in the bottom of the
yoke.   

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig4.ps}
\end{center}
\caption{$B_r$ field from yoke with cryogenic port at the top and a mirror
image of the port at the bottom.  The coil assembly has been shifted 
+0.1 cm in $y$.  $B_r$ is calculated on a line parallel to and 1 cm from the 
$z$ axis.  Top curve: $y = -1, x = 0$ cm.  Middle curve: $y = 0, x = +1$ cm.  
Bottom curve: $y = +1, x = 0$ cm.}
\end{figure}

\section{Steel Reinforcing Rod in M13 Floor}

Blueprints of the floor in the M13 area indicate a large amount of steel in the
form of reinforcing rods beginning 1.9 cm beneath the surface of the concrete
and extending to a depth of 27.4 cm.  The rods are 1.79 cm and 1.61 cm in 
radius and are spaced at 6 inch (15.24 cm) and 9 inch (22.86 cm) intervals.
This amount of steel may be sufficient to affect the field in and around the 
magnet.  The effect of buried steel was studied by adding reinforcing rods
in approximately the correct amount and distribution to the model, but only 
down to a depth of 15.5 cm.  Limitations in the Opera software make it very 
difficult to model the rods exactly, and the small cross section of the rods
compared to the size of all other facets in the model reduce the precision of
the calculated field.  Hence it is best to view the resulting field outside the
magnet yoke as an underestimate.  

The distribution of reinforcing rods in the model is shown in Fig. 5.  The 
resulting field in the region of the drift chambers shows no effect whatsoever
due to the rods.  This is true whether or not an additional hole is cut in the
bottom of the yoke to balance the cryogenic port.  However the radial field in
the beam hole region is affected.  Fig. 6 shows an up-down asymmetry of 
about 20 gauss outside the yoke.  In the beam hole itself and further inside
the yoke this asymmetry vanishes.  The magnitude of the asymmetry at any point
along the beamline outside the yoke is not large but it is nearly constant out
to at least 4 meters.  It may therefore pose a significant problem for the 
incoming muon beam and its polarization.  A good prediction of the field at 
the 20 gauss level will require that all other steel masses in the vicinity 
be modeled.  However, irregular shapes and orientations will make modeling 
even more difficult.  In this case the best approach may be to measure the 
fields directly and correct them by adding steel in various places. 

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig5.ps}
\end{center}
\caption{Steel distribution in the model used to calculate the effect of 
reinforcing rods in the M13 floor.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig6.ps}
\end{center}
\caption{$B_r$ field due to yoke and reinforcing rod in M13 floor.  $B_r$ was
calculated on lines parallel to and 1 cm from the $z$ axis.  Bottom curve: 
$y = 1, x = 0$ cm.  Top curve: $y = -1, x = 0$ cm.}
\end{figure}


\section{Effect of 10 cm radius Beam Entry Hole}
 
The standard yoke design of Ref. \cite{TN40} had a beam entry hole of radius
20 cm.  Reducing this radius may reduce the extent of the fringe field which
in turn could reduce the muon depolarization.  The new field was calculated
with a hole radius of 10 cm and compared to the 20 cm case.  The radial field
$B_r$ is shown for both cases in Fig. 7.  With a radius of 10 cm the field in 
the beam hole (large peak) is twice as intense and roughly half the width of 
the field obtained with a 20 cm radius.  The lower and broader peak, centered 
at $z = 100$ cm is due to radial field lines which pass between the end of
the last coil and the inner face of the yoke.  The size and shape of this peak
is the same for both radii.  The field uniformity in the detector region is 
unaffected by this change.  Fig. 8 shows that $B_z$ in the region 
$0 < z < 50$ cm is nearly identical for both radii except that the field from 
a 10 cm radius hole is 5 gauss larger at all points.  

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig7.ps}
\end{center}
\caption{$B_r$ on a line 1 cm above and parallel to the $z$-axis.  The 
long-dashed curve is the field due to a 10 cm radius beam hole.  The 
short-dashed curve is the field due to a 20 cm radius beam hole.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn43fig8.ps}
\end{center}
\caption{$B_z$ calculated along the $z$ axis.  Top curve: 10 cm beam hole 
radius.  Bottom curve: 20 cm beam hole radius.}
\end{figure}

\section{Effect of Optical Positioning Holes}

  The laser-pinhole-CCD system \cite{TN27} used to position the detector 
package within the magnet yoke requires six holes of radius 2.54 cm in both 
the upstream and downstream faces of the yoke box.  These holes will be placed 
at 60 degree intervals with their centers located between 40 and 50 cm from
the beam axis, depending on the final detector cradle design.  In order to
simulate the effect of the holes on the field, the magnet model referred to
in Ref. \cite{TN40} was modified by replacing six facets in the steel endplate
with air.  The facets were not circular but nearly rectangular with cross
sections of 5 cm $\times$ 6.2 cm, or about 53\% larger than the desired 
circular holes.  The air facets were placed at 60 degree intervals at a radius
of 47.5 cm from the beam axis.

    The resulting field calculations were compared to those made with no holes
in the endplate (other than the beam hole) with primary attention given to the 
field in the drift chamber volume and in the region surrounding the beam hole.
In the drift chamber region the azimuthal dependences in $B_z$ were indentical
showing no effect from the extra holes.  The holes did however reduce the
magnitude of $B_z$ by about 4 gauss out of 22014.  In the region of the 
beam hole there was no difference between the two models in the azimuthal 
dependence of $B_z$ or $B_r$.  The extra holes caused a 0.7\% increase in
$B_z$ for $(r,z) = (5,147)$ cm and a 1.4\% decrease in $B_r$ at the same point.
At points further from the beam hole the effect was smaller and in some cases 
the sign of the effect was reversed.

\section {Conclusions}

  Additional Opera-3D calculations showed that the magnet coils must be 
positioned as a unit to within $\pm$ 0.5 mm in $x$ and $y$ and to $\pm$ 1 mm 
in $z$ to maintain field symmetry.  Within these limits forces on the coils
are small.  A hole should be cut in the bottom of the yoke which is a mirror
image of the cryogenics port at the top of the yoke.  This improves field 
uniformity and more importantly reduces downward forces on the coils. 
Reinforcing rods in the M13 floor have no effect on the field inside the
yoke but the radial field outside the yoke and along the beam line shows a
significant asymmetry which is difficult to calculate.  Ultimately this
field should be measured directly and corrected by adding steel at various 
places along the beam line.  The radius of the beam hole should be decreased
from 20 cm to 10 cm.  This narrows the radial field distribution in the region
of the beam hole and will decrease beam depolarization.  Finally it was
found that placing six optical positioning holes in the upstream and downstream
faces of the yoke will not affect the field either in the drift chamber 
region or in the beam hole region.


\begin{thebibliography}{10}

\bibitem{VF00}{OPERA-3D, V7.1, Vector Fields Co., 2000.}
\bibitem{TN40}{D.H. Wright, E614 Technical Note 40, 2000.}
\bibitem{Ox00}{F. Davies, Oxford Magnet Technology Ltd., private communication
to D. Gill, 2000.}
\bibitem{TN27}{P. Kitching, E614 Technical Note 27, 1999.}
\end{thebibliography}

\end{document}