#  -Decay, -Capture and Bunching Section

## Introduction

Following the target, each proton bunch from the four bunches extracted
from the proton driver produces a cloud of pions that is transported
downstream within a focusing channel, where the pions decay to muons
following + . The initial pions, and the product muons, are produced
with a large energy spread. Due to this energy spread the length of the
beam bunch increases in this transport and a longitudinal
position-energy correlation develops with the lower-energy end of the
beam trailing behind the higher-energy muons. This energy spread is much
larger than the acceptance of the following cooling system. The plan is
to reduce this energy spread by an acceleration system that decelerates
the high-energy head of the bunch while accelerating the low-energy
tail, obtaining at the end of this "phase-rotation" system a beam with
minimal energy spread and a relatively long bunch length.

In the previously developed ^+^-^-^ collider scenarios \[1\], the
phase-rotation system was constrained by the need to keep the entire
muon beam from an initial proton bunch within a single bunch throughout
the system. The bunch length had to be short, in order to fit into the
longitudinal acceptance of the cooling and bunching system which was
based on \~30MHz rf. For a neutrino source, the long bunch can be split
into a string of higher-frequency bunches (with that frequency matched
to the following acceleration systems) \[1\].

In the -source, a phase-rotation system is used to obtain a very long
bunch and that greater length coupled with a long acceleration pulse
(obtained from an induction linac) obtains a much-reduced energy spread.
The long bunch is then inserted into an RF system, which forms the beam
into 201.25 MHz bunches matched to the following cooling system. A
design goal of the present scenario is to obtain an energy spread small
enough so that the following cooling system can capture and cool a
maximal number of muons without any additional energy cooling.

## Baseline Scenario Description

A schematic overview of the precooling decay, RF capture and bunching
section is displayed \[2\] in Figure 1. It shows a 50m decay channel
following the target and capture solenoid. This channel is matched into
an induction linac phase-energy rotation system (100m long), which is
followed by a "minicool" energy absorber (2.45m of liquid hydrogen), and
a bunching section (16.4m long) matching the beam into a 201.25 MHz
cooling system. Simulation results are displayed in Table 1.

A critical parameter for this channel is the total voltage difference
that can be provided by the induction linac, since that also determines
the energy range which can be accepted by the capture system. Our
baseline scenario contains a voltage difference of 200 MV (+150 to --50
MV). With that range, we can accept the particles with energies centered
about the maximal production energy of \~130 MeV kinetic energy,
obtaining an acceptance range of about 30 to 230 MeV. The transverse
focusing magnets and the apertures are selected to accept most of the
particles produced within that energy range. The acceptance of the
overall system is matched to the precooling design goal, which is to
provide \~0.1 per initial proton from the target to the cooling system,
and our simulations show that performance at this level should be
possible.

![](media/image1.png){width="6.249305555555556in"
height="1.6166666666666667in"}

In the following subsections we describe the various subcomponents of
this "precooling" system in greater detail: the decay channel, the
induction linac, the minicooling section and the buncher, and discuss
matching into the following cooling section. Results of tracking
simulations will be presented to estimate the performance of the various
sections \[2\]\[3\]. The simulation results are summarized in Table 1
and these results are discussed in the remainder of the chapter.

###  The Decay Channel

As discussed in the previous section, in the baseline scenario 16 GeV
proton beam bunches collide with a 0.8m long graphite (C) target, that
is immersed in a 20 T solenoidal focusing field. 's produced in the
target are captured in the solenoidal field and directed into a decay
channel for + decay. The focusing solenoidal field is gradually reduced
to 1.25 T over a \~3m transition region, while the transport region
radius is increased to 30cm. At the end of transition region \~0.242 +
per initial proton are both within the energy acceptance cut (E \< 500
MeV) and captured within the transport. The rms transverse emittance
matched into the decay channel is \~15 mm, corresponding to a beam size
of ~x~ = ~y~ 9 cm, and ~Pt~ 30 MeV/c. The decay channel immediately
follows this transition region.

The decay channel consists simply of a long large-diameter beam pipe
with a 30cm radius and with a solenoidal focusing field of 1.25 T. In
the present baseline, the total length of this decay channel is 50 m,
including a final 3 m subsection for matching into the following
induction linac.

By the end of the 50m decay channel section, \~95% of the 's in the
initial beam have decayed to 's. Simulations show that very few (\<
\~10%) of the initially captured + 's (compared to 3 m after the target)
are lost in this section. The transverse momentum introduced by decay is
smaller than the intrinsic beam momentum spread and adds little to the
beam emittance. The transverse emittance increase is only \~10% to
~t,rms~ 16.5 mm.

Figure 2 shows projections of the beam phase space distribution at the
end of the decay channel. The initial energy spread of \~300 MeV is
stretched over a \~50 m length. The transverse phase space projections
(x-P~x~ and x-P~y~) are also displayed.

![](media/image2.png){width="5.75in" height="4.545833333333333in"}

At the end of the decay channel the focusing magnetic field is gradually
increased from 1.25 T to 3 T over a 3m length, while the beam transport
radius is reduced from 30 cm to 20 cm. This change matches the beam
dimensions and focusing system to that used in the following induction
linac capture system. Figure 3 shows a cross section of this matching
section. The rms beam size is reduced from ~x~ = ~y~ = 9 cm in the decay
channel to \~5.7 cm at the end of the matching section. This fits easily
within the 20 cm radius aperture of the induction linac.

###  The Induction Linac![](media/image3.png){width="3.55in" height="2.341666666666667in"}

The decay channel is followed by an induction linac. The channel
consists of 100 1m modules of the form displayed schematically in Figure
4. Each module has an accelerating gap of 10cm, which provides an
accelerating voltage of between --0.5 and 1.5 MV to the particles. The
focusing, longitudinal magnetic field is provided by an array of
superconducting solenoids placed in separate cryostats inside the
induction modules. The gaps between the cryostats are shaped to serve as
accelerating gaps

Figure 4 shows cylindrical coils with 22.2 cm inner radius and 1.46 cm
thickness separated by the induction gap of 14 cm length and spacers of
length a. Cylindrical magnetic induction cores surround these coils. The
coils provide a mean focusing of 3 T on axis, with a periodic
fluctuation of \~10% due to the coil position. The 1 m, \~3 T periodic
focusing system has betatron oscillation resonance\'s for \~200 MeV/c
particles, which would lead to particle losses. However, these
resonances are eliminated if the inter-coil spacing length is chosen to
be a=14cm, the same as the induction gap coil spacing. This changes the
transverse focusing period to 0.5m and removes the medium energy
resonance condition \[2\]. A preliminary engineering layout of an
integrated superconducting coil will be discussed later on (see also
Figure 12).

The cores are excited by a waveform matched to the beam energy-phase
distribution, with a voltage range from 0.5 MV/gap for the high-energy
head of the bunch to \~1.5 MV for the low energy tail. This asymmetric
form, which accelerates the beam centroid by \~0.5 MV/gap, was chosen as
a preferred mode for the pulse forming network, since it facilitates a
fast reset of the cores, which enables sequential very fast excitation
of the 4 primary bunches \[3\]\[4\]. The pulse lengthens from 150 ns to
200 ns along the linac, matching the bunch lengthening. Figure 5 shows
the desired module waveforms along the linac (from gap 1 to 100).

![](media/image5.png){width="4.55625in" height="3.6680555555555556in"}

Energy-phase distributions of the beam along the induction linac are
displayed in Figure 6. The bunch lengthening with energy spread
flattening of the captured beam is displayed. A core of a beam with a
full energy width of \~300 MeV and a bunch length of \~50 m (with a
curved position-energy correlation) is phase rotated into a \~80 m long
bunch centered about \~280 MeV beam energy (\~175MeV kinetic energy)
with a full-width of \~100 MeV. The width is correlated with position
along the bunch, but the mean beam energy is not. About 10% of the
initial beam is lost in the transport. The beam transverse emittance and
beam size are approximately unchanged in the RF rotation. The rms
transverse emittance ~T~ remains at \~16 mm (rms, normalized) while the
beam radius remains ~x~ 6
cm.![](media/image6.png){width="5.668055555555555in"
height="2.9784722222222224in"}

In the baseline design a string of 4 proton bunches, spaced by \~500 ns,
is sent onto the target. The induction cells must provide approximately
identical waveforms for the muons from each proton bunch coming from the
proton driver. Thus, the induction cells must be driven by a 4-pulse
waveform, with each pulse \~150---200 ns long, spaced by 500 ns. Note
that the four-pulse design at these parameters was selected because it
is estimated that this was within demonstrated induction linac
capabilities \[6\]. The induction linac itself will be discussed in
greater detail later on in this chapter.

### "Minicool" Absorber Section

At the end of the induction linac the mean beam kinetic energy is T \~
175 MeV (total energy E 280 MeV), while the following bunching and
cooling section is designed for muons with a kinetic energy of
\~100 MeV. In order to match these central energies the beam passes
through an energy-loss absorber. This absorber also provides some
transverse cooling.

Following the induction linac a 3 m transition region is placed with
coils matching the beam optics into a solenoidal magnetic field of 5 T
at the absorber. The physical aperture is increased to a radius of 30cm
to minimize possible losses. The absorber is a 2.45 m long liquid H~2~
bottle, which lowers the mean beam kinetic energy from 175 MeV to 100
MeV. Simulations show that this insert decreases the rms transverse
emittance ~t,rms~ from 16.4 mm to 12.4 mm. However, the rms energy
spread is somewhat increased by the absorber. Figure 7 shows the
longitudinal (E-cphase space distributions of the beam before and after
the minicool insert; the energy spread increase is not large compared to
the large intrinsic beam energy spread.

![](media/image7.png){width="5.5in" height="2.875in"}

In our baseline scenario the "minicool" energy loss is used primarily to
reduce the central energy of the phase-rotated beam to that of the
cooling channel, but it also provides some initial transverse cooling.
If a longer energy loss section is used, more transverse cooling is
obtained and the beam energy could be lowered to the level where an
adiabatic capture and acceleration with a 201.25 MHz section can match
the beam energy to the cooling rf. However, at present parameters the
energy spread of the beam is initially large and energy loss at low
energies would increase that energy spread far beyond the acceptance of
the cooling channel. This limitation could be removed if energy cooling
were incorporated into the scenario.

### Buncher 

After the mean energy is reduced by the "minicool" absorber to the
cooling kinetic energy of T 120 MeV (P 200 MeV/c), the bunches from the
induction linac must be formed into 200 MHz bunches. Several scenarios
for transforming a long-bunch beam from the induction linac into a
string of short bunches have been proposed and these options have been
described in \[8\]. The optimal scenario depends on the beam properties
at the exit of the induction linac as well as the beam cooling scenario
requirements. Two styles of matching have been considered:

-   An adiabatic match obtained by gradual bunching of the beam with 200
    MHz RF. This can be done at a fixed energy corresponding to the mean
    capture and cooling energy, or by lowering the energy to a level
    where the beam longitudinal motion is enhanced, and then bunching
    and accelerating the beam to cooling energy. An adiabatic match
    could minimize phase space dilution, and maintain a small
    longitudinal phase space.

-   A short bunching RF section followed by a drift in which bunch
    length is minimized to match the cooling acceptance. This drift
    incorporates transverse matching. This option can be modified to
    include multiple steps (2 or 3).

Since in the baseline scenario (as currently implemented) the beam
energy spread (and longitudinal emittance) is large and not well matched
to adiabatic scenario parameters, we have decided to minimize length and
cost of the buncher by choosing a short buncher scenario as our
baseline.

The buncher used in the baseline scenario is displayed schematically in
Figure 8. It has a total length of 16.4 m and contains three 1.944m
201.25 MHz RF modules. The longitudinal beam distributions at the end of
the buncher are displayed in Figure 9. It shows the longitudinal
distribution of the entire beam; the 201 MHz structure can be seen. The
energy width of the beam has been increased, but the degree of increase
depends upon position along the bunch. The RF system has partially
formed the \~80 m long beam distribution into a string of about fifty
201 MHz bunches. Figure 9 also shows the distribution folded over the
201 MHz period. A 201MHz RF bucket corresponding to the estimated
acceptance of a E 200 MeV, f=201MHz cooling system is displayed
overlapping the distribution. About 2/3 of the 's are within that
acceptance. ![](media/image8.png){width="5.75625in"
height="1.3604166666666666in"}

At the end of the buncher the beam is matched into a 5 T focusing
system, and has a transverse emittance of ~t,rms~ 12 mm. Longitudinally
the beam overflows the expected acceptances of the cooling channel.
However, \~0.123 per initial proton are within that acceptance,
indicating that the precooling section is able to meet the goal of \~0.1
/p.

This buncher has not yet been fully optimized. Ref. \[8\] discusses some
of the possible variations, and these will be studied in more detail.
Some significant improvements may be expected from future parameter
optimization studies, which will include more complete integration with
the upstream phase rotation and downstream cooling systems.

### Integration with Cooling Section

The optimal beam capture and bunching parameters depend critically on
the acceptance and parameters of the downstream cooling and acceleration
sections. The cooling channel and some of its design options are
described in more detail in the following cooling chapter.

If there is no energy cooling in the scenario, the most critical
parameter in downstream acceptance is the longitudinal emittance for the
beam entering the cooling channel. A cooling channel has a longitudinal
acceptance corresponding to an rf bucket determined by the average
accelerating gradient eVand stable phase ~0~ set such that the energy
gain (eV sin~0~) cancels the mean energy loss rate from absorbers.
Figure 9 shows a cooling bucket acceptance superimposed on a simulated
-distribution. The smaller size of the bucket implies a significant
design mismatch.

![](media/image10.png){width="2.582638888888889in"
height="2.1868055555555554in"}

In the present example we have centered the final beam energy of the
precooling section at E 200 MeV, since that is the expected energy for
the cooling channel. However it is not known whether that is an optimum
cooling energy. A larger longitudinal acceptance with less longitudinal
heating would be obtained by a higher energy cooling system; however, it
would also have a reduced cooling rate. Variation of the cooling energy
within the context of global scenario optimization would be an important
goal of future research. Other parameters of the precooling (and
cooling) systems may also be varied to improve the match.

### Discussion: Variations and Alternatives

A number of variations and improvements on the baseline scenario could
be considered and studied. Many of these could improve -acceptance. In
addition to incremental improvements, greatly different -capture
scenarios (using different technologies) could be considered. We first
consider some incremental improvements.

The present precooling scenario has been shown by simulations to provide
\~0.12 per initial 16 GeV proton into the acceptance of the cooling
channel from an initial Carbon target. Optimization of this scenario
should continue with the attempt to increase the yield as much as
possible without increasing technical complexity of the system
significantly.

One critical parameter (discussed above) is the target material; a
high-Z target could provide up to twice as many 's from the same primary
beam. However, because of the much more demanding technology that would
be required to build and operate high-Z targets, they are not considered
for this study but do provide an obvious upgrade path.

Acceptance could be somewhat improved by increasing the voltage
bandwidth of the RF capture system, although the current bandwidth does
cover the production peak. Increasing the induction linac voltage range
from 200 MV to 300 MV could increase the number of accepted 's by \~25%
but would lengthen the induction linac by 50%.

The energy spread at the end of the precooling region can be reduced by
lengthening the decay channel + induction linac system (including
lengthening the induction linac pulse beyond 200 ns). In principle the
energy spread can be reduced inversely proportional to the final
macrobunch width. (For example, A. Van Ginneken has presented a
simulation that obtains an energy spread a factor of two smaller, when
using a 200 m decay drift together with a 400 ns induction linac pulse.
\[9\])

The energy spread (and longitudinal emittance) at the end of the
induction linac, and the subsequent energy spread after the 200MHz
buncher, is much larger than desired for the cooling system, and is
significantly larger than the longitudinal acceptance of the cooling
channel. Thus large particle losses occur at the beginning of the
cooling channel. The difficulty is exacerbated by the fact that the
cooling system increases the energy spread (and longitudinal emittance),
which means increased beam losses in the cooling system. Modification of
the capture section to reduce energy spreads (with further scenario
optimization and/or more acceleration modules) and modification of the
cooling system to include some energy cooling would both be very
desirable.

An alternative collection system is described in the PJK scenario \[10\]
and more recently in an updated version of \[10\]. This scenario begins
with a low frequency rf capture section, obtained by 42m of 30---90MHz
RF at (\~5 MV/m), immediately after the target. This initial capture
section obtains a \~1.5m long (rms) -bunch with \~15% rms momentum
spread. This is followed by an energy loss "minicool" section, a drift
section (\~100m), and an induction linac (\~100m), to obtain a long
bunch with much smaller energy spread. Since this system has more total
rf capture voltage, and has an initial rf capture section reducing the
energy spread closer to the target, the capture efficiency will be
larger. Recent calculation indicate \[11\] that total efficiency could
be up to 0.15/proton. The final energy spread will be somewhat smaller
than our baseline scenario. Also, this RF enables better sorting of the
beam polarization, obtaining higher initial -beam polarization (2545% or
so). However, the PJK scenario adds significant complexity and power
requirement to the capture scenario, requiring development and
construction of 30---80 MHz RF systems as well as induction linacs,
which will increase risk and cost.

The present scenario uses induction linac systems, which have the great
advantage that the acceleration waveform can in principle be tailored to
match the phase-energy distribution of the beam, and thereby provide an
ideal phase-rotation system. However it is possible that the actual
performance of an induction linac system may be deficient in
acceleration waveform quality, gradient, reliability, and/or lifetime or
simply be too expensive. Alternative acceleration systems such as the
\~30 MHz RF system used in the baseline ^+^-^-^ Collider scenarios could
be used, without an additional induction linac. Recently scenarios using
\~5MHz RF systems (at \~1MV/m) have also been suggested, and are
investigated \[12\]. These (higher-frequency) alternatives may not
develop energy spreads as small as those that can be developed from very
long induction linac pulses; they may require additional energy cooling
in a complete scenario.

## The Technical Design of the Induction Linac

Since the induction linac modules are the primary component of the
precooling (phase-energy rotation) system, we include a more detailed
discussion of the properties of these modules. Several designs for such
an induction linac \[4\] have been presented and different designs for
pulse power production have been studied\[5\]\[6\]. Figure 10 shows a
detailed view of an induction linac module from ref. \[4\], including
acceleration and focusing components. Some parameters are included in
Table 2. The 1 m long induction module consists of a set of toroidal
magnetic cores surrounding an induction gap. A pulser sends high-current
pulses to the module that change the core magnetic fields, and the
changing magnetic field generates an accelerating voltage across the
induction gap. As compared to the design discussed in the first part of
this chapter technical detail, like for example the exact cell length
etc may vary. This is a result of the ongoing optimization with the
technical design going on simultaneously.

The induction module includes a 3 T solenoidal field for transverse beam
focusing. To minimize the leakage of magnetic field into cores (to \<
\~0.1 T), the superconducting solenoidal coils are placed inside the
cores. It is essential to investigate early on what the maximum
tolerable field is going to be at the cores. Separate cryostats and
current leads are required for each magnet. A \~20 cm annular gap
between the coils and the induction cores is sufficient for cryogenic
and magnetic separation.

A number of materials can be considered for the magnetic cores of the
induction modules, such as Finemet, Metglas and nanocrystalline
compounds. In the baseline design of ref. \[4\] Finemet and Metglas 2605
are preferred materials.

### Overall Topology

Each Induction Linac cell is 1m long operated at 2.00 MeV (-0.50 MeV,
+1.50 MeV \_ head to tail). This allows the pulsed power pulse train to
be delivered to the 10 induction cores per cell at 150 kV, a reasonable
voltage level for cables. The cathode of the accelerating gap is located
upstream to provide the proper polarity to the beam, but the center
conductor of the cables are connected to the cores in the sense that
allows the pulse generator to operate in negative polarity, which
minimizes the size of its components. The configuration of the major
cell components was dictated by the need to maintain a high acceleration
gradient at conservative electric field stress levels on the vacuum
insulator and cathode surfaces in vacuum, and to minimize the
perturbation of the axial magnetic field of the superconducting magnets
caused by the acceleration gap. A 1 m long inter-cell module is
positioned every 10 cells to provide vacuum pumping for the beam tube
and beam position/net current monitors. A constant solenoid gap pitch is
maintained by choosing the inter-cell module length to be the same as
that of the acceleration cells.

A cross section view of a pair of cells is shown in Figure 10. The bore
radius of 20 cm corresponds to an axial magnetic field of 3.0 T. The
induction cell's acceleration gap is 10 cm and its solenoid gap is 14
cm, which allows 2 cm per side for thermal isolation between the end of
the superconducting coil and the vacuum feed conductor. The
superconducting solenoid is 1.46 cm thick with an inside radius of 22.2
cm. The Mycalex insulator stack is 82 cm long by 5 cm thick with an
inside radius of 36.4 cm. The induction cores are located outboard of
the insulator with an inside radius of 45 cm and an outside radius of 77
cm.

### Electric Field Stresses

An equipotential plot of the electric field distribution in the
induction cell is shown in Figure 11. The field in the oil dielectric,
radial feeds between individual cores is 150 kV/cm or 42 % of the
calculated breakdown strength of 345 kV/cm. The peak axial field along
the Mycalex insulator is 23 kV/cm for an effective stress time of 50 ns
(89% of peak). This value compares to a measured breakdown field of \>
33 kV/cm for a 2.2 s pulse on the Mycalex insulator of the DARHT 2^nd^
axis injector, which scales to \>55 kV/cm at 50 ns (33 \* (1000 ns/50
ns)^1/6^). On the DAHRT and AIRIX 1^st^ axis injectors vacuum cathode
surfaces have been operated in a non-emitting mode at fields of 250
kV/cm, pulse lengths of 60 ns, and areas of 1000 cm^2^. With some
preliminary cathode conductor shaping the peak field in the accelerating
gap is only 183 kV/cm, and in the region where emitted electrons could
impact the insulator it is further reduced to 103 kV/cm.

### Induction Cell Cores

Each 2.0 MV cell contains 10 individual 200 kV cores. Diametrically
opposed cables drive alternate cores, and these core pairs are then
clocked azimuthally to provide clearance for the pulse shaping network
enclosures that couple the cables to the cell.

The induction cores provide the dominant load to the pulse train
generator (2.5 kA of core current versus \~0.1 kA of beam current).
Minimizing the core losses reduces pulsed power system costs, but lower
core losses require the use of either more expensive core material or
larger core volumes. The results of a scan which optimizes system costs
are summarized in Table 3. Two types of Metglas manufactured by Allied
Signal (USA) were considered, 2605SC and 2714A. 2605SC is the least
expensive with the largest losses and 2714A is the most expensive with
the lowest losses. A compromise between these extremes, which minimized
system costs, was obtained with Finemet FT-1H, a nano-crystalline alloy
manufactured by Hitachi (Japan) at a flux density swing of 1.0 T. This
optimum gave a mean core loss current of 2.5 kA, which defined a core
load impedance of 80 used for the pulsed power design.

![](media/image12.wmf){width="5.977952755905512in"
height="2.689763779527559in"}

### Superconducting Magnet for the Induction Cell

The requirements for the phase rotation linac solenoids are as
follows: 1) the solenoid outside diameter should minimized. This means
that the magnetic induction in the channel should be maximized. 2) The
radial thickness of the solenoid cryostat should also be minimized. This
allows the induction linac acceleration structure to be brought closer
to the axis of the machine. As the magnetic field increases, the
thickness of the superconducting magnet that creates the field must also
increase. 3) The space between the induction linac cells must be
minimized. This means that the space used for the cold mass support
system, the electrical leads and the cryogen feed system must fit in
this minimum space. To further reduce the space occupied by the
cryostat, the solenoid was designed to be cooled indirectly using
flowing two-phase helium in a cooling tube attached to the support
structure. The 40 K helium used to cool the shield is carried in tubes
attached to the
shields.![](media/image14.wmf){width="6.302777777777778in"
height="3.654861111111111in"}

Figure 12 below shows a cross-section of a superconducting solenoid that
is designed to generate an average induction of 3 T on the axis of the
phase rotation linac. The inner bore of the solenoid cryostat is 400 mm.
This allows a 200 MeV muon beam with a nominal diameter of 384 mm (at 3
T) to pass through the solenoid without loss (except from muon decay).
The distance from the end of the superconducting coil to the outside end
of the cryostat can be reduced to 20 mm. If an additional support clip
is needed at the end of the coil, it can be accommodated in the space
shown.

The proposed conductor for the coil shown in Figure 12 above is a
standard MRI magnet conductor that is 1 part NbTi and 4 parts RRR=70
copper. This conductor has fifty-five 85-µm filaments twisted with a
twist pitch of 12.7 mm. The bare matrix dimensions of the conductor are
0.955 mm by 1.65 mm. The insulation on the conductor is 0.025 mm thick.
At an average design induction of 3.0 T on axis, the coil design current
is about 521.3 A. As a result, the coil would be a 10 layer coil that is
14.6 mm thick (including 2 mm of ground plane insulation).

It is proposed that the coils be wound and cast on a form that is
removed after the coil is cured. After curing the coils are removed from
the mold and machined at the ends and on the outer radial surface. After
the coils are machined they can be shrunk fit into a 6061 aluminum
support structure that has been machined so that the coils closely fit
with in it. Table 4 on the next page shows the design parameters for the
induction linac solenoids.

The 6061-aluminum support structure on the outside of coils serves the
following functions: 1) It limits the coil strain by carrying some of
the magnet hoop forces, and 2) it serves as a shorted secondary to
protect the magnet during a quench. A single magnet is entirely
self-protected through quench back from the support structure. One can
use quench back to protect a string of these magnet as well. When a
quench is detected in one magnet, the current in the string can be
discharged through a varistor resistor, causing all coils to go normal
through quench back from the support structure.

The space available longitudinally for leads, cryogenic services and
cold mass supports is about 85 mm wide at the center of the magnet. The
cold mass of phase-rotation solenoid (including the 40 K shield and
lower lead assembly) is estimated to be about 250 kg. The primary forces
that will be seen between the cold mass and room temperature will be
forces do to shipping and forces introduced due to unbalanced magnetic
fields. The magnet cold mass supports are designed for a force of 10000
N in any direction. It is proposed that a pair of 60 mm diameter
oriented carbon fiber tubes (with a wall thickness of about 3 mm) be
used to carry forces from the cold mass to room temperature.

Since there is a solenoid magnet every meter down the phase rotation
channels and the drift spaces between the phase rotation linac sections,
leads must be brought out of each of these magnets. One has two
choices: 1) All of the magnets or 20 to 25 meter long subsets can be
hooked in series and run off of a common power supply. Inter-connects
between the solenoids can be either superconducting or conventional
copper cable. 2) Each magnet can have its own set of leads to room
temperature and its own power supply, so individual cells can be tuned.
It is proposed that the leads between 4 K and 40 K be made from high
temperature superconductor (HTS). The leads from room temperature to the
top of the HTS leads at 40 K should be gas-cooled. Gas from the
refrigerator that is used to cool the magnet shields and cold mass
support intercepts can be used to cool the gas-cooled leads. This gas
must be returned to the refrigerator compressor intake at room
temperature. Figure 13 shows a schematic representation of the cold mass
support system, the helium supply system, and the current leads.

The induction linac superconducting solenoids are cooled by conduction
from the 6061-aluminum support structure. The aluminum support structure
will be cooled by two-phase helium flowing in tubes attached to the
support structure. Two-phase helium cooling is commonly used to cool
detector magnets and magnets that are used in physics detectors. The
advantages of two-phase tubular cooling are as follows: 1) There is very
little helium inventory within the magnet. 2) The two-phase helium tubes
have a high-pressure rating. This means that the magnet cryostat is not
a pressure vessel. 3) Two-phase helium cooling does not require a cold
compressor or a helium pump to circulate the helium through the magnet
cooling system. 4) The helium in a two-phase helium cooling circuit
decreases as along the flow circuit. 5) The pressure drop along a
two-phase helium flow circuit is lower than for a supercritical helium
flow circuit. A large number of superconducting solenoid magnets have
been cooled using force flow two phase helium in tubes.

It is assumed that the magnet shown is one of twenty to twenty-five
magnets that are cooled in series from the two-phase helium refrigerator
and control cryostat. If twenty to twenty-five magnets are cooled, the
mass flow rate through the flow circuit should be about 2.5 grams per
second. The two-phase helium tube would be attached to the
superconducting coil support structure, the base of the HTS leads and
the attachment points of the cold mass supports. The heat that is added
to the two-phase helium flow stream in each of the solenoid is expected
to be about 0.5 W.

### 4 Pulse per Booster Cycle System Architecture

The pulser system is certainly the most ambitious piece of hardware for
the induction linac layout. While single pulse a comparatively easy to
provide, multiple pulses at the required voltage rating are difficult to
produce. Although two approaches have been considered, the first one
presented here is our present baseline. Figure 15 represents a layout of
the pulsed power system showing the major components to scale. They
consist of cell cables, a water dielectric delay line, four series pulse
forming lines, PFLs, magnetic compressor pulse charge modules, MCPCs,
and a thyristor switched prime power unit. One of these units is
required per induction module, overall 100 of them.

Figure 16 is a block diagram of the system that documents key parameters
of the components. The generation of four bipolar, 100 ns pulses spaced
400 ns apart is made possible by the use of rapid cell core reset that
returns them to the same state between the pulses. 8.3 kA of cell reset
current is supplied via a 17 H inductor housed in a cable adapter at the
end of the water delay line. The inductor energy is supplied in 220 s by
a simple Thyristor switched capacitor bank (72 F at 4.0 kV). The reset
current resets the cell cores towards -- saturation, blocking leakage
current from the four pulse generator, and allowing voltage to develop
across the accelerating gap. A + 1.5 MV pulse from the 4 pulse generator
drives the cores towards + saturation. When the pulse disappears, the
reset current produces --0.5 MV across the accelerating gap while
providing rapid seset of the cores in preparation for the next pulse.
Without rapid reset the core area would have to be a factor of four
larger, a significant cost impact, and the pulse generator would have to
produce the bipolar pulse. It is desirable that the volt-second product
of both the acceleration pulse and the reset pulse be equal so that the
cell cores are in the same state for each pulse to ensure pulse to pulse
reproducibility. Even more importantly the reset pulse must not drive
the cell cores completely into saturation before the arrival of the next
acceleration pulse or the head of the beam will see 0.0 MV instead of
--0.5 MV. The reset pulse width must also be at least 3 times the
acceleration pulse width to allow switching transients from the pulse
generator to be damped out to an acceptable level. These criteria lead
to the asymmetric bipolar acceleration pulse and a net acceleration of
0.5 MV per cell. They also lead to the addition, of a pre-pulse
generator not shown in Figure 15 and Figure 16 that will be attached in
parallel with LS1 at the entrance to the delay line. Without it, the
cell cores would be at -- saturation when the first pulse arrived,
because of the long time scale of the reset circuit. The pre-pulse
generator will supply a \> 8.3 kA pulse to the cell cores to overcome
the reset current, and drive the cores towards + saturation past their
desired operating point. When this +80 kV, 100 ns long pre-pulse ends
100 ns before the beginning of the 1^st^ pulse, the reset current will
develop -- 50 kV across each cell core and return the cores to the
desired operating point for the start of the 1^st^ pulse. The pre-pulse
PFL energy is only 42% of the individual main pulse
PFLs.![](media/image17.jpeg){width="4.901968503937008in"
height="4.493700787401575in"}

The 1^st^ pulse of the 4 pulse train is generated by the closure of the
saturable reactor LS1, which discharges PFL1, launching a 184 kV, 100 ns
square wave pulse with a 20 ns risetime into the water dielectric delay
line. At the end of the delay line the pulse is routed via ten 67 cables
to the ten 80 cores of the cell. The cable to core mismatch developes a
change in core voltage of 200 kV superimposed on the reset voltage of
--50 kV. The pulse shaping networks at the cable/cell interface convert
the fast risetime square wave to the required hyperbolic shape. For this
study a simple inductor was used to approximate the required shape. The
delay line prevents first pulse reflections from this mismatch from
returning to the cell for the pulse train length of 1.3 s. Once
discharged, PFL1 acts as a matched output line for the PFL2 when LS2
closes launching the 2^nd^ pulse into the delay line 400 ns later. The
sequence repeats until the 4^th^ pulse of the train is generated by
closure of LS4. The risetime of the 4^th^ pulse is slowed to \~40 ns
because it has to trnsverse the saturated inductance of 4 switches, but
this is largely compensated for by the pulseshaping network at the cell.

Each series PFLs is pulse charged to 368 kV by a MCPC. For this study
the MCPC design was based a MAG1D currently in use at 1 pule per second
at LLNL so that realistic costs could be developed. A modified version
of a MAG1D was run at 100 Hz at PSI. As part of the R&D effort the MCPC
design would be optimized for this application. An intemediate energy
store, IES, magnetic compression stage was added to the back end of the
MAG1D to replace its thyratron switched feed. Extension of the charge
time from 4.0 s to 20 s allows the use of commercially available
thyristors for high reliability switching.

### Preliminary Circuit Simulations

A simple circuit simulation of the four series PFLs with realistic
saturated switch inductances, the delay line, cables, simple inductor
waveshape network, and a resistive load for the core was done to check
the feasibility of producing the required pulse waveshape with this
approach. Figure 17 shows a four pulse train of the square-wave injected
into the delay at the top and the cell voltage on the bottom. Although
the 4^th^ pulse of the top pulse train shows significant distortion on
the top of the waveform and a slower risetime, much of this is
integrated out on the core voltage waveform. Figure 17 shows an overlay
of the 1^st^ and 4^th^ pulse of the core voltage pulse train with the
pulses time shifted for comparison. Based on these results it seems
reasonable that a combination of rear PFL impedance tailoring and
special pulse train generators along the Induction Linac will allow
matching of the four pulses in the
train.![](media/image19.wmf){width="6.3in" height="4.993055555555555in"}

### Alternative Approach for the Pulsing System

A second design for the pulser system was initiated in order to
investigate alternative methods for generating the power pulse driving
the induction cores. Figure 19 shows a simplified electronic layout for
the pulse power generator\[4\]. The inductor voltage is formed by the
common discharge of the forming capacitor C0 and pulse forming line
(PFL) into the inductor. A detailed scheme description is provided in
\[7\], and the resulting pulse shape is shown in Figure 20. For this
power system approach, LIA accelerating gradient changes from -1.5 MV/m
in the beginning of the cycle to 0.5 MV/m in the end, which is similar
to what was found optimal in \[5\]. The time duration of the negative
and the positive parts of the voltage pulse are chosen so that the
volt-second areas are equal. As a result, magnetic reset of the
induction cell core is reached automatically, and after one voltage
pulse ends, the system is ready to accept the next one. This power
system approach allows generation of several pulses by adding fast
charging circuits in parallel at points A and B in Figure 19.
![](media/image21.png){width="5.1in" height="3.189583333333333in"}

The impedance of the circuit in Figure 19 can not be made arbitrarily
small. Nevertheless, it appears possible to use one generator to feed
ten inductors (0.5-m of total length) in parallel. Core power loss is
the major factor that dictates the size and the cost of a pulsed power
system. It was shown in \[6\] that it is important not to underestimate
core power loss because it scales nonlinearly with applied voltage. In
order to calculate the average power loss, careful calculation of the
instant power loss has to be made followed by integration over the pulse
length. Calculations made according to this scheme resulted in a minimal
induction core weight of about 100 Tons and an average power required
for the total system of about 5 MW. The next step of the development
will include thorough simulation of pulsed power circuits that take the
nonlinear induction cell impedance into account, as well as stray
inductance, and stray capacitance of each section element.

## Summary

The reduction of the energy spread of the muon beam is the primary goal
of this phase rotation section. The central component that provides the
necessary time dependent voltage, is the induction linac. All the other
technical components (solenoids in the decay channel, mini cooling
absorber, buncher cavities) are simple compared to the induction linac.
The superconducting coils inside the induction cores are an additional
technical complication. The assessment of all these difficulties and the
initial design approach being presented here indicate that the technical
problems are solvable with a reasonable amount of R&D.

The phase rotation design presented here performance of the phase
rotation on the other hand, as being presented here, meets our intensity
performance goal, however the energy spread is still too large in order
to guarantee good performance through the buncher section and the
cooling channel. More work is needed.

REFERENCES

1.  C. Ankenbrandt et al., \"Status of Muon Collider Research and
    Development and Future Plans\", Phys. Rev. ST Accel. Beams 2, 081001
    (1999)

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1.  S. Geer, C. Johnstone, and D. Neuffer, "Muon Storage Ring Neutrino
    Source: The Path to a Muon Collider?\", FN-2073, March 1999.

2.  V. Balbekov, "Pre-cooling Stage of Neutrino Factory", MUCOOL note
    96, March 2000.

3.  V. Balbekov, "Optimization of Phase Rotation Using an Induction
    Linac", MUCOOL note 63, October 1999

4.  J. Fockler et al., "Induction Linac Technology for Muon Beam Phase
    Rotation", Neutrino Factory Feasibility Study Workshop, February
    2000.

5.  Terechkine, "Neutrino Factory Phase Rotation LIA: Pulsed Power
    System Optimization", January 2000
    http://www.fnal.gov/projects/muon_collider/nu-factory/subsys/ss-phrot/Optimization.pdf.

6.  S. Yu, LBNL, private communication, Jan. 2000.

7.  V. Kazacha, A. Sidorov, I. Terechkine \"Pulsed Power System of
    Linear Induction Accelerator for FNAL Neutrino
    Factory\".http://www.fnal.gov/projects/muon_collider/nu-factory/subsys/ss-phrot/proposal_inductl.pdf.

8.  D. Neuffer, "RF Capture Variations for the Muon Collider Neutrino
    Source", MUCOOL note 52, October, 1999.

9.  A. Van Ginneken, unpublished, January 2000.

10. R. B. Palmer, C. Johnson, and E. Keil, "A Cost-Effective Design for
    a Neutrino Factory", MUCOOL-67, November 1999, Proc. NuFact99, Lyon,
    1999.

11. Bob Palmer, The draft of the current results of this paper are now
    at \<http://pubweb.bnl.gov/people/palmer/nu/pjksim/pjksim.ps\>, and
    will appear shortly as a MUC note.

12. W. Chou and A. Van Ginneken, "Phase Rotation Using Low-frequency
    RF", MUCOOL note 90, February 2000.