\section{Pion Capture Magnet}
\label{TGT:hollow-mag}
%Robert Weggel$^*$, Charles Pearson$^*$; Joe Minervini$^\#$, Joel Schultz$^\#$, 
%Peter Titus$^\#$, Richard Camille$^\#$; Greg Naumovich$^+$, Peter Hwang$^+$
%${}^*$BNL, Upton, NY
%${}^\#$M.I.T., Cambridge, MA
%${}^+$Everson Electric Co., Bethlehem, PA
An efficient Neutrino Factory should capture nearly all the pions that the 
high-energy proton beam generates when it bombards the target. To do so, we employ a solenoidal magnetic 
field to bend the pion trajectories into helices bound to the surface of 
%flux-tube 
cylinders that enclose an invariant amount of flux. A solenoid captures 
those pions with trajectories small enough to fit inside its bore. Pions of 
high transverse momentum require a solenoid of large bore and intense field. 
For example, capture of transverse momenta up to 225~MeV/$c$, the baseline for 
Feasibility Study-II, requires a product of field and bore of 3~T$\cdot$m. 
Study-II employs a capture field of 20~T, about the maximum that is feasible; the corresponding bore is 0.15~m.

The least costly magnet of this transverse-momentum reach has a large bore 
but only modest
field. However, such a magnet would require that the magnets and other
downstream components all be 
inconveniently large. Minimum overall system cost dictates a 
modest bore but high field. 
%An additional benefit of capturing pions 
%at a higher field is the lesser angular momentum associated with any given 
%transverse momentum; the pions are ``cooler." This angular momentum remains 
%invariant as the pions travel adiabatically toward a region of lower field. 
%{\sl Not sure I believe this.}

The desired field profile of the pion capture magnet is uniform over the target,
followed by a gradual transition to the much lower field of subsequent 
components of the Neutrino Factory, as shown in Fig.~\ref{captureB}.
For minimal particle loss the optimum field profile is 
\begin{equation}
B(z) = \frac{B_0}{1 + k z / L}\, ,
\label{Tgt.m1}
\end{equation}
where $B_0$ is the field at 
$z = 0$, the
downstream end of the target, and $(k + 1)$ is the ratio of $B_0$ 
to the field at $z = L$, the downstream end of the transition region. 
For Study-II, $B_0 = 19$ T, $k = 14.2$ and $L = 18$ m. 
Within the target region itself, $-l < z < 0$, where $l = 0.6$~m,
the field need be 
only approximately uniform. Near the upstream end ($z = -l$) 
the drop in field should be at most 5\%, in order to limit shearing of the 
incoming jet of mercury by the field gradient.  Near the 
downstream end ($z = 0$) the field drops a similar amount in order to blend 
smoothly, satisfying $\nabla \cdot {\bf B} = 0$, 
with the rapid decrease with $z$ of the field at the upstream end of the 
transition region.

To generate this field we employ magnets of three types: superconducting (SC), 
resistive, and iron. SC magnets generate the entire field everywhere except in 
the vicinity of the target. There, the intense field and high density of 
energy deposition from radiation make it more economical to supplement the 
SC magnet 
with a resistive one.  Contributing to the field at the very upstream end of 
the target region is a stepped cylinder of ferromagnetic material. A cobalt-iron alloy such as Permendur 
could contribute  nearly 1.2~T, but cobalt may be undesirable from the 
standpoint of activation. Pure iron would contribute slightly more than 1~T. More valuable than the modest and highly localized 
field contribution is the favorable field gradient, which corrects much of the 
field inhomogeneity of the other coils that would otherwise cause excessive
shear of the jet of mercury entering the target region. 
\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=3in]{../template/report/ps-and-eps/field_-06_12.eps}
\caption[On-axis field of pion capture magnet near the target region]
{On-axis field of the pion capture magnet near the target region, 
$-0.6 < z < 1.2$~m.
At $z = -0.3$~m, the superconducting magnet generates about 14~T and the 
resistive insert 6~T. The iron improves the entry of the mercury
jet into the region by reducing the field inhomogeneity by a factor of two.
} 
\label{TgtM:fg1}
\end{center}
\end{figure}

Figure~\ref{TgtM:fg1} shows the on-axis field profile 
of the proposed pion capture magnet with parameters $B_{\rm max} = 20$~T, 
$B_0 = B(-l) = 19.0$~T, $k = 14.2$, and $B(L) = 1.25$~T. 
Figure~\ref{TgtM:fg2} sketches the magnets and cryostat for the region
$-l < z < 6$~m.  Tables~\ref{TgtM:tb1a} and \ref{TgtM:tb1b} list the most 
important parameters of the hollow-conductor and first eight superconducting 
coils of the pion capture magnet.

\begin{figure}[htb!]
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/main_coil-cryostat.eps}
\caption[Cryostat and coils of the pion capture magnet]
{Cryostat and coils of the pion 
capture magnet in the region $-1 < z < 6$ ~m. 
The proton beam enters from the right of the section view in the lower right of
the figure.
Shown: iron plug (of stepped, T cross section), hollow-conductor (H-C) insert 
magnet, tungsten shielding outside H-C insert, cryostat, and first five 
superconducting (SC) coils. The bores of the SC coils range from 1.27 to 
1.55~m.
The first SC coil generates 14~T; the field at the downstream end of the fifth
coil is 3.3~T. Not shown: conical beam tube and shielding between it and 
cryostat. Further downstream are additional SC coils to extend the field tail 
to 1.25 T at $z = 18$~m.} 
\label{TgtM:fg2}
\end{center}
\end{figure}

\begin{table}[htb!]
\begin{center}
\caption[Parameters of the hollow-conductor magnets]
{Parameters of the hollow-conductor magnets.}
\label{TgtM:tb1a}
\begin{tabular}{|lccc|}
\hline
&
H-C 1&
H-C 2&
H-C 3\\
\hline
Avg. current density (A/mm$^2$) & 244 & 191 & 149 \\
Winding inner radius (cm) & 17.8 & 23.2 & 35.3 \\
Winding outer radius (cm) & 23.2 & 35.3 & 49.0 \\
Radial build of windings (cm) & 5.4 & 12.2 & 13.7 \\
Upstream end, $z_1$ (cm) & $-71.2$ & $-71.2$ & $-71.2$ \\
Downstream end, $z_2$ (cm) & 3.7 & 16.5 & 36.1 \\
Coil length, $z_2 - z_1$ (cm) & 74.9 & 87.7 & 107.3 \\
Volume of windings (m$^3$) & 0.052 & 0.196 & 0.389 \\
Approx.\ peak field (T) & 20.0 & 18.6 & 16.1 \\
Avg.\ hoop tension (MPa) & 118 & 124 & 115 \\
Conductor fraction (\%) & 33.2 & 32.9 & 33.4 \\
Copper fraction (\%) & 48.9 & 48.3 & 49.2 \\
Structural fraction (\%) & 11.2 & 12.1 & 10.7 \\
Copper mass (tons) & 0.243 & 0.893 & 1.77 \\
Stainless steel mass (tons) & 0.048 & 0.194 & 0.334 \\
\hline
\end{tabular}
\end{center}
\end{table}

Note that, Table~\ref{TgtM:tb1a} incorporates  minor modifications to many of 
the coil parameters, that have not been taken into account in 
Table~\ref{capturecoils}. For example, coils downstream of 6~m are shorter and 
more numerous; this is a consequence, of trying to maintain the field quality, 
in spite of the larger intercoil gaps, introduced to facilitate cryostat 
construction and installation.   
%\begin{sidewaystable}
\begin{table}[htb!]
\begin{center}
\caption[ Parameters of the upstream SC solenoids ]{
Parameters of the upstream eight superconducting 
solenoids of the pion capture system.}
\label{TgtM:tb1b}
\begin{tabular}{|lcccccccc|}
\hline
&
SC~1 & SC~2 & SC~3 & SC~4& SC~5 & SC~6 & SC~7 & SC~8 \\
\hline
Avg.\ current density (A/mm$^2$) &
234 & 255 & 297 & 383 & 484 & 679 & 705 & 705 \\
Winding inner radius (cm) &
63.6 & 68.6 & 77.6 & 77.6 & 77.6 & 42.4 & 42.2 & 42.2 \\
Winding outer radius (cm)&
127.8 & 101.1 & 98.8 & 88.3 & 84.1 & 45.1 & 45.9 & 45.9 \\
Radial build of windings (cm)&
64.2 & 32.5 & 21.2 & 10.7 & 6.56 & 2.69 & 3.69 & 3.69 \\
Upstream end, $z_1$ (cm)&
$-125.3$ & 62.8 & 145.7 & 255.6 & 420.6 & 600.8 & 657.7 & 720.7 \\
Downstream end, $z_2$ (cm)&
52.8 & 135.7 & 245.6 & 410.6 & 606.5 & 643.7 & 707.3 & 770.3 \\
Coil length, $z_2 - z_1$ (cm)&
178.1 & 72.9 & 99.9 & 155.0 & 185.9 & 42.9 & 49.6 & 49.6 \\
Volume of windings (m$^3$)&
6.88 & 1.26 & 1.17 & 0.866 & 0.619 & 0.032 & 0.051 & 0.051 \\
Approx.\ peak field (T)&
14.0 & 11.8 & 8.74 & 6.21 & 4.33 & 3.33 & 3.03 & 3.03 \\
Avg. hoop tension (MPa) &
209 & 206 & 201 & 184 & 163 & 96 & 90 & 90 \\
Conductor fraction (\%) &
7.8 & 6.2 & 5.3 & 5.5 & 6.2 & 8.2 & 8.3 & 8.3 \\
Copper fraction (\%)&
10.4 & 10.9 & 12.1 & 16.4 & 21.8 & 38.5 & 39.9 & 39.9 \\
Structural fraction (\%)&
31.8 & 32.9 & 32.6 & 28 & 22 & 3.4 & 1.8 & 1.8 \\
Vol.\ of superconductor (liters)&
538 & 79 & 62 & 48 & 38 & 3 & 4 & 4 \\
Copper mass (tons)&
6.42 & 1.24 & 1.28 & 1.27 & 1.21 & 0.11 & 0.18 & 0.18 \\
Stainless steel mass (tons)&
17.1 & 3.24 & 2.98 & 1.89 & 1.06 & 0.01 & 0.01 & 0.01 \\
\hline
\end{tabular}
\end{center}
\end{table}
%\end{sidewaystable}
\subsection{Hollow-Conductor Resistive Coils}
In the baseline design of this Study, the resistive insert that 
surrounds the target region employs hollow conductors
rather than a Bitter  magnet, as was used in Feasibility Study-I~\cite{feas00}. 
The penalty in performance is significant, (see Section~\ref{APP-Bitter}), 
but this technology should survive much better in the harsh radiation 
environment around the target. 
%but one avoids the uncertainty that a Bitter magnet, with its 
%wet insulation, might suffer immediate electrical breakdown in the high 
%radiation environment. In any case, Bitter magnets require development of
%plate-to-plate insulator that is orders of magnitude 
%better than the present organic films at coping with the intense radiation. 
%Even were such an insulator to become available, radiation damage to the 
%conductor, combined with deterioration of electrical contacts 
%(corrosion cracking and conductor erosion by high-velocity water) likely will 
%limit Bitter-magnet lifetime to only a few months %-quite possibly less-
%at full 
%power. The several-thousand-hour lifetimes cited for the best Bitter magnets 
%may be deceptive: Bitter magnets typically spend little time at full power, 
%and have not had to cope with intense radiation.  

The hollow-conductor magnet also presents formidable engineering challenges. 
Radiation doses and neutron flux densities are very high. According to 
calculations using the MARS code \cite{muc0194}, 
each operational year (taken for radiation estimates to be $2 \times 10^7$ s) 
adds a dosage 
of $\approx 10^9$ J/kg ($10^9$ grays, or $10^{11}$ rads) 
and a neutron flux of $\approx 2 \times 10^{19}$/cm$^2$, 
despite $\approx 10$ cm of shielding by water-cooled tungsten carbide 
that attenuates the 
neutron flux by more than an order of magnitude and the gamma dose by a 
factor of about 40. The intense ambient field, combined with the fairly large 
bore and fairly high current density, induces hoop stresses that are high 
compared with the low strength of typical hollow conductors, whose copper is in 
the annealed state for ease of processing. The neutron flux will strengthen 
the conductor to values associated with considerable cold work, but will also  
embrittle the conductor \cite{Fabritsiev}
so that the conductor must be supported as if it were 
glass. The alternative is to operate the conductor at 150$^\circ$C or more 
(barely acceptable because of the penalty in conductor resistivity) or 
periodically to heat the conductor to that temperature, so as to anneal out 
much of the embrittlement before it becomes too severe.


The Study-II baseline design employs mineral insulated conductor (MIC) such as 
developed \cite{Tanaka94} %by K. Tanaka et al.[2] 
for the Japan Hadron Facility. The insulation 
is a layer of MgO sandwiched between the conductor and its copper sheath. 
The conductor, shown in Fig.~\ref{TgtM:fg4}, is 18~mm square, 
with a cooling hole that is 10~mm square, 
surrounded by insulation 1.8 mm thick and a copper sheath 
1.1~mm thick, for an overall size of 23.8~mm. As employed by the JHF, 
in lengths of 60~m and with only modest water pressure, the conductor can 
carry 3~kA. By limiting each hydraulic length to 15~m and using a high 
water pressure differential, 30~atm, as used at the National High Magnetic Field Laboratory (NHMFL), 
such a conductor will carry the required current, 15.5~kA, 
with a bulk temperature rise limited to 60$^\circ$C. With an inlet water 
temperature of 10$^\circ$C, as at the NHMFL, the peak conductor temperature is 
80$^\circ$C.
%{\bf (add ref here)} for NHMHL 
\begin{figure}[htb!]
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/jhf_conductor.eps}
\caption[Mineral-insulated hollow conductor ]
{Mineral-insulated hollow conductor
developed for Japan Hadron Facility. The end-on view shows the white layer of 
powdered MgO insulation sandwiched between the copper hollow conductor and 
its sheath, also of copper. 
%The conductor is 18 mm square with a 10~mm square 
%cooling hole. The MgO is 1.8~mm thick, and the outer copper sheath, 1.1 mm, 
%for an overall dimension of 23.8 mm. 
Of the cross section, 17\% is cooling 
passage, 37\% conductor, 28\% insulation and 18\% sheath. The side 
view shows a conductor termination, brazed of several parts that confine the 
MgO and hold the glossy white ceramic ring that keeps the sheath isolated from
the current-carrying conductor.} 
\label{TgtM:fg4}
\end{center}
\end{figure}
%\begin{figure}
%\begin{center}
%\includegraphics*[width=1in]{section.eps}
%\caption[Cross section of coil form and windings ]{Cross section of coil form 
%and windings and of hollow-conductor insert magnet. The conductor is 
%mineral-insulated conductor (MIC) of the sort developed for the Japan Hadron 
%Facility. The thick-walled cylinders reinforce the conductor against the 
%radial Lorentz hoop stresses engendered by the combination of high field (20 T)
% and large bore (0.36 m).} 
%\label{TgtM:fg3}
%\end{center}
%\end{figure}

Figure \ref{TgtM:fg5} shows the cross section and a longitudinal section
of the resistive insert magnet, built
from three grades of such hollow conductor.  The magnet consists of three 
nested coils, the innermost of two layers and the outer two coils of four 
layers each.  Surrounding each coil is a reinforcing cylinder of Inconel 718, 
maraging steel, or other high-strength material. These cylinders hold the 
downstream flange against the downstream load of $\approx 0.6$ MN 
(60 metric tons) 
from the other magnets in the system.  Simultaneously, the cylinders contain 
the conductor against the high Lorentz forces. To restrict all terminations 
to the upstream end, the conductor spirals to the downstream end in 
odd-numbered layers and back in even-numbered layers. To achieve water flow 
adequate to limit the bulk temperature rise to 60$^\circ$C with conductors 
within the capacity of the JHF drawing bench, all layers have six conductors 
hydraulically in parallel, \textit{i.e.}, a six-in-hand winding. That is, all conductors are electrically in series and hydraulically in parallel.
\begin{figure}
\begin{center}
\includegraphics*[width=4in,clip]{../template/report/ps-and-eps/solenoid-pearson.eps}
\caption[End view and section of resistive insert magnet ]
{End view (left) and vertical section (right) of the resistive insert 
 of the hollow-conductor magnet. Every layer employs six hydraulic 
 paths in parallel to achieve the short hydraulic path length necessary for 
 adequate water flow.  The conductor is 
mineral-insulated conductor (MIC) of the sort developed for the Japan Hadron 
Facility, shown in Fig.~\ref{TgtM:fg4}.
The thick-walled cylinders reinforce the conductor against the 
radial Lorentz hoop stresses engendered by the combination of high field (20~T)
 and large bore (0.36~m).
} 
\label{TgtM:fg5}
\end{center}
\end{figure}

The inner coil uses conductor exactly as shown in Fig.~\ref{TgtM:fg4}. 
The other coils 
use conductor of the same proportions, to permit fabrication from billets of 
the same dimensions as that for the inner coil.  These outer coils, being 
longer as well as larger in diameter, have longer passages that require bigger 
conductor, 26.8~mm for the coil of intermediate size and 30.6~mm for the outer 
coil. Each conductor in the outermost double layer is 35~m long, with a mass 
equivalent to 57~m of 23.8~mm square conductor. 
This is within 5\% of the maximum so far produced on the JHF drawing bench, 
and thus sets the limit on conductor size throughout the magnet.
%The resistive insert magnet could be more efficient if one could replace the 
%copper sheath with 
%one of stronger material, thereby eliminating the need for reinforcement 
%against hoop stresses.  Higher efficiency could come also from a better cooling 
%geometry, with cooling passages much shorter and therefore smaller.  This is 
%possible, while still keeping the insulator dry, if one were to use solid 
%conductor, such as the alternative developed by Tanaka \cite{Tanaka00}, 
%cooled externally 
%by water flowing axially through narrow radial gaps between consecutive layers.
%However, only if the insulation layer were much thinner (or the thermal 
%conductivity much better than the $\approx 1.5$ W/m-C for the MgO in JHF 
%conductor) would
%the temperature rise across the insulation be acceptable, instead of hundreds 
%of degrees. Thinner insulation alone, even without a change in the hydraulic 
%geometry, could make the magnet more efficient by improving the percentage of 
%conductor, which is only 37\% in the JHF conductor.  Worth pursuing is the 
%development of conductor insulated by an anodized thin sheath of 
%aluminum \cite{Leonhardt}, or by a ceramic coating \cite{Celik},
% or wrap \cite{Rice}.  
 
%If the ceramic insulation 
%could withstand water as well as radiation, then one could reconsider a 
%Bitter magnet, cooled either axially, as at the NHMFL, or radially, as 
%formerly at MIT's Francis Bitter National Magnet Laboratory, where a hybrid 
%magnet with a 12-T SC coil and 9-MW insert generated 22.5~T in a 6$``$ bore
%\cite{Iwasa}. 
%With radial cooling the cooling passages are shorter, and one need not 
%the insulators to permit water to flow through the turns of the magnet.
\subsection{Superconducting Coils}
One of the superconducting coils of the pion capture magnet is also a 
formidable 
engineering challenge: SC~1, with its 14-T field and 1.3-m bore. Fortunately 
there are two precedents for this coil. One, is a collaboration of MIT and the 
NHMFL for its 45-T hybrid magnet~\cite{45thybrid}. It can generate 15~T when 
operated alone, 
and therefore not restricted to 14~T by the need for current margin to survive 
the current surge from a tripout or burnout of the insert coil of the hybrid 
system. However, the bore of this magnet is only half that of pion capture 
magnet SC~1. More relevant is the central solenoid model coil (CSMC),
shown in Fig.~\ref{TgtM:iter_magnet}, for 
ITER, the International Thermonuclear Experimental Reactor \cite{Jayakumar}. 
The coil has 
generated 13~T in a bore 26\% bigger than necessary for SC~1. The CSMC weighs 
140 tons and stores 600~MJ, the same as the entire pion capture magnet, 
including the coils in its 18-m-long transition region. 

\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=3in]{..//template/report/ps-and-eps/ITER_CS_magnet.eps}
\caption[Superconducting magnet of same scale as pion capture magnet  ]
{Superconducting magnet of the same scale as the pion capture magnet. 
The CSMC for ITER weighs 100~tons, generates 13~T in a 1.6~m bore, and 
stores 600~MJ.} 
\label{TgtM:iter_magnet}
\end{center}
\end{figure}

The pion capture magnet has the additional complexity of energy deposition 
from radiation, up to 1~kW/m$^3$, despite shielding about 30~cm thick.  However, 
it does not have to cope with energy deposition from the high sweep rate that 
the CSMC must survive.  It also does not have to cope with so high a discharge 
voltage, 15~kV for the CSMC. Therefore, its insulation need not be so 
thick, nor its current density quite so low.
\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=4in]{../template/report/ps-and-eps/ITER_CS_conductor.eps}
\caption[Concept of high-current cable-in-conduit conductor ]
{Concept of high-current cable-in-conduit conductor needed by 
the intense-field coils of the pion capture magnet. Liquid helium in the 
central tube flows through the spiral gap in its wall to cool the strands of 
superconducting cable that parallel the central tube. The outer jacket, 
typically of stainless steel or Inconel, provides most of the mechanical 
strength.} 
\label{TgtM:iter_conductor}
\end{center}
\end{figure}
  
Cable-in-conduit conductor (CICC) will be used
in the highest field superconducting coils of the pion capture magnet, 
coils SC~1--5.  Figures~\ref{TgtM:iter_conductor} and \ref{TgtM:cmsc_conductor}
show the CSMC conductor, 
which is about 50~mm square and can carry 46~kA in a field 
of over 13~T.  Liquid helium in the central tube flows through the spiral gap 
in its wall to cool the strands of superconducting cable that parallel the 
tube.  The fine strands have a high ratio of surface to volume, to keep each 
strand at nearly the same temperature as the helium.  The outer jacket, 
typically of stainless steel or Inconel 908, protects the delicate 
strands within and provides almost all of the mechanical strength to resist 
huge Lorentz forces in large magnets that generate intense fields.  
Cable-in-conduit conductor is 
appropriate for large magnets operating at 10~kA or more. For the 
downstream coils of the pion capture magnet, which experience lower hoop 
and much lower axial loads, solid conductors or Rutherford cables are 
simpler and more economical.
\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=4in,clip]{../template/report/ps-and-eps/cmsc_conductor.eps}
\caption[The cable-in-conduit conductor for the ITER central solenoid ]
{The cable-in-conduit conductor for the ITER central solenoid.} 
\label{TgtM:cmsc_conductor}
\end{center}
\end{figure}

\subsection{Magnetic Forces}
The axial loads on the upstream, high-field coils of the pion capture magnet 
are immense. Figure~\ref{TgtM:force} shows that the peak cumulative axial load 
(which is at the downstream end of SC~1) to be over 100~MN, or 10,000 metric
tons. All of the biggest loads involve only the first five SC coils. 
 To manage this load, we support the coils with a structure that is 
cold at both ends to minimize heat leaks into the cryostat. 
The obvious way to do this is to house them all in the same cryostat.
This is the only feasible way given that the separation between consecutive 
coils is 10~cm, which is not enough for two sets of coil flanges and cryostat 
walls. 
The loads on all the low-field coils (beyond $z = 6$~m) sum to only 2.5~MN. 
We group these coils in sets, as in the case of phase rotation 
coils,  with each cryostat of convenient length.
%{\bf (give pointer)} right efter coils above
\begin{figure}[!bht]
\begin{center}
\includegraphics*[width=3in,clip]{../template/report/ps-and-eps/coil_force.eps}
\caption[Cumulative axial force on components of the pion capture magnet ]
{Cumulative axial force on components of the pion capture magnet. Upper 
curve: the peak force is over 100~MN, or about 10,000~metric tons. The forces 
between the first five superconducting coils, with their high field and large 
size, dictate that they share a common cryostat. The forces on coils beyond 
$z = 6$~m are much less, and allow individual cryostats for each coil or 
convenient group of coils. Lower curve: the force on the resistive insert 
magnet and iron is only 1.2~MN (note the semi-logarithmic scale).} 
\label{TgtM:force}
\end{center}
\end{figure}
\subsection{Field Quality}
The gaps between consecutive coils can introduce considerable field ripple, 
especially beyond 
$z = 6$~m, the downstream end of the proton beam absorber, where the coils 
are of smaller bore. Fig.~\ref{TgtM:fieldonaxis} shows the field ripple for 
coils with 14~cm gaps as indicated in Table~\ref{TgtM:tb1b}. Whereas 
Table~\ref{capturecoils} had 
only 7 coils downstream of $z = 6$~m, Table~\ref{TgtM:tb1b} has 19~coils, each of only 
$\approx 50$~cm in length. This geometric distribution maintains the field 
ripple within 5.3\% peak-to-peak and 10\% rms.   
%The ripple is larger than that 
%further upstream because the gaps between consecutive coils are larger 
%($\approx 14$~cm instead of 10~cm) and the coils are of smaller bore 
%(0.84~m instead of 
%0.27-1.55 m).  
An on-axis field ripple of this size does not affect the transmission of pions to the phase rotation region.
%{\bf (Give pointer)}.
\begin{figure}[bht!]
\begin{center}
\includegraphics*[width=3in,clip]{../template/report/ps-and-eps/field_3_18.eps}
\caption[On-axis field of pion capture magnet for $3 < z < 18$ m    ]
{On-axis field of the pion capture magnet from 3 to 18 m downstream of 
the target region, where the solenoids have 0.42~m inner radius and 0.50~m length 
separated by axial gaps 0.14~m. The desired field (dashed line), to maximize 
the adiabatic retention of captured pions, declines from 19~T at $z = 0$ to 
1.25~T at 18~m according to Eq.~\ref{Tgt.m1}. 
The actual field (solid line) differs from the desired value by 5.3\% 
peak-to-peak (from $- 2.5 \%$ to +2.8\%), with an rms deviation of 1.0\% 
(Note the semi-logarithmic scale).} 
\label{TgtM:fieldonaxis}
\end{center}
\end{figure}
%The field ripple shown in Fig.~\ref{TgtM:fieldonaxis}
%is not the complete story, however. 
%Off axis the ripple becomes much worse, increasing above its on-axis value 
%approximately as the square of the radial distance, as shown in
%Fig.~\ref{TgtM:ripple}.  At the beam pipe wall the 
%ripple is about four times worse than on axis. This field ripple, with its 
%sizable radial component of field, may be a significant contributor to the loss
%of pions from the end of the target to the downstream end of the transition. 
%If so, it may pay to use coils of larger diameter, or go to greater 
%lengths to reduce the spacing between consecutive coils.

%\begin{figure}[!bht]
%\begin{center}
%\includegraphics*[width=3in]{../template/report/ps-and-eps/field_ripple.eps}
%\caption[On-axis and off-axis field ripple ]
%as function of axial gap between successive coils ]
%{On-axis and off-axis field ripple as function of axial gap between 
%successive coils. The ripple is approximately proportional to the gap width 
%between successive coils. Off-axis the ripple increases approximately as the 
%square of the radial distance; at the beam pipe wall it is about four times 
%worse than on axis.
%} 
%\label{TgtM:ripple}
%\end{center}
%\end{figure}




