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{\Large \bf TRIUMF Experiment E614 \\}

{\Large \bf Technical Note \\}

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

\vspace{0.4cm}

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

\vspace{0.4cm}

\rm{\bf 18 January 2000 \\}

\end{center}

\begin{abstract}

  The three dimensional magnetic field for the downstream half of the E614 
magnet coils and yoke was calculated using OPERA-3D.  Earlier results 
indicating that a square cross section yoke produced a cylindrically symmetric
field near the drift chambers were confirmed.  Construction details such as
access ports, hinges and braces were added to the model and their effects 
studied.  The addition of a steel shim at the beam entrance to the yoke was
found to narrow the radial field distribution in that region at the expense of
greater non-uniformity in the drift chamber region.  Recommendations are made 
for the placement of ports and extra steel which will remove some asymmetries 
in the field.   

\end{abstract}

\section{Introduction}

Calculations of the magnetic field in the E614 detector have already been 
performed in two and three dimensions.  A summary of these studies and the 
design decisions derived from them are given in E614 Technical Note 37 
\cite{TN37}.  In the 2-D case the return yoke was assumed to be cylindrical 
and in the 3-D case only 1/8 of the total volume was solved.  Because access 
ports or steel attachments to the yoke may potentially disrupt the symmetry of
the field and because it is not always be wise to cut holes or add steel 
solely for the sake of symmetry, 3-D calculations are required for the entire
downstream half of the detector.  Questions which must be answered before 
construction begins are: \\

- does a square cross section yoke introduce azimuthal asymmetries in 
  the beam entry hole and the chamber region of the detector which would be 
  absent in a cylindrical yoke? \\

- do the cryogenic port and electronics cable access ports in the top of 
  the yoke introduce up-down asymmetries, and if so, can they be reduced 
  or removed ? \\

- do the magnet door hinges or blocks of iron near the cryogenic services
  port have an effect? \\

- can the radial field distribution at the beam entry hole be made narrower
  by the addition of a steel shim around the hole? \\

These questions were answered using the OPERA-3D finite element code 
\cite{VF00}. 
  
\section{Details of the Model}

   The downstream half of the magnet was modeled, assuming symmetry about 
the plane $z = 0$.  The details included in the finite element mesh are
listed here.  The magnet yoke consists of 1020 steel and has the following 
outside dimensions: half-length(along $z$ axis) = 148.0 cm, width = height = 
261.0 cm.  The walls (top, bottom ($\pm y$) and sides ($\pm x$) are 20 cm 
thick with solid triangular steel braces running along each of the four 
corners.  The magnet door is 8 cm thick with a 20 cm radius beam hole 
centered on the $z$ axis.  The door occupies the space between $z = 140$ and
$z = 148$ cm.  In the center of the top face of the yoke is a cryogenic 
services port with a half-length (along $z$) of 30.48 cm and a width of 61 cm.
On either side of this port is a solid block of steel which is intended to take
up some of the return flux which was expelled by the hole.  Immediately
upstream of the magnet door, in the top face of the yoke is an access hole
for electronics cables.  The hole extends from 130 to 140 cm in $z$ and from 
-30.5 to 30.5 cm in $x$.  Near each of the four corners of the magnet door is 
placed a steel hinge.  The hinges on the right-hand side of the door are 
functional while those on the left are merely pieces of steel added to 
maintain field symmetry.  More details of the steel yoke can be found in TN37
 \cite{TN37} .

Inside the yoke is a set of six current-carrying coils, three upstream and 
three downstream.  The current density has been adjusted to give approximately
2.2T at the center of the detector.  The yoke and coils are displayed in 
Fig. 1 along with the external facets of the finite element mesh.

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig1.ps}
\end{center}
\caption{View of the downstream half of the E614 steel yoke and coils.  Lines 
on the surface of the steel show exterior facets of the finite element mesh.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig2.ps}
\end{center}
\caption{Contours of $B_z$ in the plane $z=0$.  Contour 19 represents the
largest flux coming out of the page (along $+z$) and contour 2 represents
the largest flux going into the page (along $-z$).  Contour 9 represents 
a $B_z$ flux of nearly zero.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig3.ps}
\end{center}
\caption{Contours of $B_z$ in the $y-z$ plane in the region of the drift 
chambers. Each contour represents an increment of 10 gauss.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig4.ps}
\end{center}
\caption{$B_z$ on a circle (0 to 360 degrees) in the $x-y$ plane of radius 
16 cm, centered on $x = y = 0, z = 15$ cm.}
\end{figure}

Not shown in the figure is the volume of air that must also be meshed and 
assigned boundary conditions in order to obtain a solution for the fields.  
It extends from $z = 0$ to $z= 550$cm, $x = -250$ to $x = 250$ cm, and
$y = -250$ to $y = 250$ cm.  

\section{Field Symmetry}

Using the above model the first of the above questions was addressed.
A contour plot of $B_z$ in the plane $z=0$ (Fig.2) shows a positive flux along
the $z$-axis near the center of the magnet and a negative flux returning 
through the yoke.  The flux avoids the hole of the cryogenic services port, as
expected, but it does not enter the steel bars on either side of the port.
This may indicate that the bars do not improve field symmetry.
Fig. 3 shows a contour plot of $B_z$ in the $y-z$ plane in the region of
the drift chambers.  The field in this region is very uniform and symmetric 
despite the obvious asymmetry observed near the cryogenics port in the
yoke.  A more complete view of the field asymmetry is given in Fig. 4 where 
$B_z$ is plotted along a circle in the $x-y$ plane with radius 16 cm, centered
at $x = 0, y = 0, z = 15$ cm.  The maximum up-down asymmetry in the chamber 
region is seen to be about 1.3 gauss or 0.006\%.  This is compared to an
expected error on the field mapping measurements of 1 gauss.

The radial field in the beam entrance hole is also of interest because it 
imparts an off-axis kick to the momentum and spin of the incoming muon.
Asymmetries here can seriously distort the spin distribution at the stopping 
target.  The radial field for the above model is shown in Fig. 5 along four
lines, each parallel to and 1 cm away from the $z$-axis and located at  
($x=0$,$y=1$), ($x=0$,$y=-1$), ($x=1$,$y=0$) and ($x=-1$,$y=0$).  While there 
is exact symmetry along the ($\pm x$) lines, the asymmetry along the $\pm y$
lines is well outside the expected measurement error.  The difference is due 
to the electronics cable access port which is almost directly above the beam 
hole.  Another model calculation shows that when the cable port is removed the
radial fields along all four lines are identical within errors.  Not shown in 
Fig. 5 is the radial field on the $z$-axis.  It is non-zero with a maximum 
value of 7.5 gauss.  As the field is sampled further and further from the 
$z$ axis, the \% asymmetry decreases.  Removal of the cable port causes the 
field along the $z$ axis to go to zero (within errors), as expected for a 
solenoid. 

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig5.ps}
\end{center}
\caption{$B_r$ on lines 1 cm from and parallel to $z$-axis.  Top curve: 
$x=0$, $y=-1$.  Bottom curve: $x=0$, $y=1$.  Middle curves (overlapping):
$x=1$, $y=0$ and $x=-1$, $y=0$.}
\end{figure}

\section{Repairing Field Asymmetries}

The largest asymmetry in the above model to which the experiment is sensitive 
is in the radial field in and around the beam access hole.  As stated above 
this is due to the 10 cm x 61 cm cable access hole in the top plate of the 
yoke.  With respect to electronics cabling it would be easier to split the 
hole into two 10 cm x 30.5 cm holes and move the two halves as far as possible
away from each other in $x$.  The model yoke was modified accordingly and the 
fields recalculated.  The results are shown in Fig. 6.  Here the radial field
is calculated over the same four lines as in Fig. 5 but now they all nearly
overlap.  In fact they are identical within the estimated error of the 
calculation.  A comparison of the fields in the drift chamber region shows that
splitting the cable port does not significantly disturb the cylindrical 
symmetry.  Because the radial field due to the split cable port is so nearly 
symmetrical in both the beam entrance region and the drift chamber region, 
these ports need only be placed on the downstream end of the yoke and not on 
the upstream end where the beam enters.  
 
\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig6.ps}
\end{center}
\caption{Cable access port split in two. $B_r$ on lines 1 cm from and parallel
 to $z$-axis.  Top curve: $x=0$, $y=-1$.  Bottom curve: $x=0$, $y=1$.  Middle 
curves (overlapping): $x=1$, $y=0$ and $x=-1$, $y=0$.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig7.ps}
\end{center}
\caption{View of the downstream half of the E614 steel yoke and coils with a 
20 cm thick steel cover placed over the cryogenic services port.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig8.ps}
\end{center}
\caption{Cover over cryogenics port.  Contours of $B_z$ in the plane $z=0$.  
Contour 19 represents the largest flux coming out of the page (along $+z$) and
 contour 2 represents the largest flux going into the page (along $-z$).  
Contour 9 represents a $B_z$ flux of nearly zero.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig9.ps}
\end{center}
\caption{Left-right asymmetry in radial field when fake hinges are removed 
from downstream yoke door.  Upper curve ($x=1, y=0$ cm), lower curve 
($x=-1, y=0$ cm).}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig10.ps}
\end{center}
\caption{Contour plot of $B_z$ in $x-z$ plane when fake hinges are removed 
from downstream yoke door.}
\end{figure}

It has been suggested \cite{Jam99} that a thick steel cover over the cryogenics
port would take up enough of the return flux to increase the azimuthal symmetry
in the drift chamber region.  Fig. 7 shows such a cover added to the model of
the yoke.  Fig. 8 shows that indeed some flux is taken up by the steel cover,
however it is not enough to symmetrize the central flux region.  In fact the
flux distribution is very little different from the original model as shown in
Fig. 2.  Another look at Fig. 2 shows that no flux enters the steel bars 
placed on either side of the cryogenics port.  This suggests that the bars are
not necessary for improving symmetry and may be removed. 
 
As stated above two of the steel hinges are used to hold the downstream yoke
door and the other two are used for symmetry (see Fig. 7).  When the two 
non-functional hinges were removed, a 10 - 15\% left-right asymmetry in the 
radial field was observed for $z > 180$ cm, as shown in Fig. 9.  However, no 
left-right asymmetry was observed in the drift chamber region (see Fig. 10).  
As long as there is no interest in the beam profile as it leaves the yoke it 
is unnecessary to add fake hinges to the downstream door.  Because the 
asymmetry at the beam exit does not propagate to the chamber region fake 
hinges on the upstream face of the yoke are also unnecesary.

\section{Modifying the Radial Field with Shims}

  It has been suggested that the addition of a steel shim around the beam
entrance to the yoke could narrow the radial field distribution, thereby 
reducing the overall transverse kick imparted to the incoming muon
\cite{Lob97}.  The shim would have the same effect as a coil placed at
the beam entrance.  To test this proposal, a cylinder of steel 10 cm thick
with inner and outer radii of 20 and 50 cm, respectively, was centered on the
$z$ axis and placed immediately downstream of the beam entrance to the yoke.
The resulting radial field is shown in Fig. 11 where it is compared to the 
field obtained without the shim.  The radial field with the shim is about
10\% narrower than without it, while the integral varies by less than 1\%.
However the addition of the steel at the beam entrance disturbs the field
uniformity at the center of the magnet.  Fig. 12 shows that $B_z$ along the
$z$ axis now increases by about 70 gauss over 40 cm, compared to 20 gauss
over 40 cm for the model without the shim (see Fig. 3).
  
\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig11.ps}
\end{center}
\caption{Radial field on a line parallel to the $z$ axis with $x=5$ cm and 
$y=0$.  Long-dashed curve: radial field with shim at beam entrance.  
Short-dashed curve: no shim.}
\end{figure}

\begin{figure}
\begin {center}
\includegraphics[angle=0,height=20.0cm]{tn40fig12.ps}
\end{center}
\caption{Contour plot of $B_z$ in the $x-z$ plane with the shim at the beam
entrance.  Each contour increment represents 8.6 gauss.}
\end{figure}

\section {Conclusions}

  A square cross section yoke is sufficient to produce a magnetic field which 
is cylindrically symmetric to within 2-3 gauss in the drift chamber region of 
the detector ($r < 20$ cm).  This confirms the calculation by Don Lobb of
a 1/8 model magnet \cite{TN37}.  The addition of solid triangular braces along
the corners of the yoke, access holes in the top for cryogenics and cables, and
hinges on the magnet door did not affect the symmetry in this region. 

A single electronics cable hole in the top face of the yoke at its downstream 
end and centered over the beam exit hole causes an up-down asymmetry in the 
radial field in the region of the beam exit.  This asymmetry can be almost
totally removed by splitting the hole in two.  The holes cut in the yoke will 
extend from +130 to +140 cm in $z$, with one between +60.5 and +90.18 cm in $x$
and the other between -60.5 and -90.18 in $x$.  These need only be placed at 
the downstream end of the yoke. 

  The opening in the yoke for the cryogenic services creates an up-down
asymmetry near the yoke which propagates only slightly into the drift chamber
region.  Placing a thick steel cover over the cryogenics port does not draw 
sufficient flux to reduce the asymmetry in the field near the yoke, nor does 
the addition of steel bars on either side of the opening.  However the
asymmetry in the drift chamber region is at most 2-3 gauss without any 
additional steel, so it may safely be omitted.

  Adding two fake hinges to the downstream yoke door helps to symmetrize the
radial field where the beam leaves the yoke, but the beam in this region is
probably of no interest so the extra hinges may be omitted.

  Placing a cylindrical shim at the beam entrance to the yoke decreases the 
width of the radial field distribution by about 10\% which may be useful in
reducing muon depolarization at the target.  However it also causes greater
non-uniformity (by a factor of 3.5) near the center of the magnet where the 
drift chambers are located.

\begin{thebibliography}{10}

\bibitem{TN37}{J.A. Macdonald, E614 Technical Note 37, 1999.}
\bibitem{VF00}{OPERA-3D, V7.1, Vector Fields Co., 2000.}
\bibitem{Jam99}{J.A. Macdonald, E614 Meeting, 1999.}
\bibitem{Lob97}{D. Lobb, private communication, 1997.}
\end{thebibliography}

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