%18. HIGH-POWER RF SYSTEMS: 201.25 AND 402.5 MHZ
%R. Rimmer*, J. Corlett*, N. Hartman*, A. Ladran*, D. Li*, R. Macgill,* A. 
%Moretti\%, J. Reid\%, H. Padamsee\&, G. Rongli\&, V. Shemelin\&
%*LBNL, \% FNAL, \& Cornell
\section{Introduction to rf Systems}
        The rf systems for the buncher and the cooler are required to
        match the muon beam into the longitudinal acceptance of the
        cooling channel and to replenish the beam energy lost during
        ionization cooling. Since they must operate inside the strong
        solenoid fields they must be normal conducting. These systems
        require a large number of rf cavities operating at
        high gradient, and a large amount of pulsed rf power. They are
        technically challenging and expensive, and have therefore been
        the focus of continued development during Study-II. The
        cooling channel layout has continued to evolve since Study-I,
        with emphasis on integration of realistic components into the
        available space along with optimization of the channel
        performance. The buncher and cooling channel systems must
        accommodate liquid-hydrogen absorbers, high-gradient rf
        cavities, windows, tuners, superconducting solenoids,
        diagnostics, pumping, harmonic cavities and other
        equipment. The system must be designed in such a way as to
        allow assembly and access for maintenance. The buncher and cooling channel comprises a large number
        of modules (cells). The cell layouts are described in Section~\ref{RF:sec18.4}.  Each module contains two or four 201.25~MHz
        closed-cell cavities and is powered by one or two high-power
        multibeam klystrons. The density of equipment in the building is
        therefore high and the systems must be carefully laid out to
        allow access for installation and maintenance. Following the
        cooling channel is a matching section containing rf and
        solenoids, but no absorbers.

The proposed buncher, cooling channel, and matching section is
        approximately 183~m long and requires 184~cavities and 84
        klystrons at 201.25~MHz and an additional 6~cavities and 3~klystrons at 402.5~MHz. The total installed power is approximately 780~MW
        ($\approx 1.56$~MW average), and the installed voltage is 1080
        MV.

The cooling channel is followed by an acceleration section
        employing 299 two-cell superconducting rf cavities operating at 201.25
        MHz. These structures are also challenging because of the
        high gradient and large physical size. Peak power requirements
        for the acceleration section are not as high as for the normal conducting rf sections, but the pulse length is much longer. Many superconducting cavities
        can be powered from a single klystron station. Several
        multi-cell rf cavities may share a common cryostat. The final
        energy at the end of the accelerating section is 20 GeV,
        compared with the 50 GeV of Study-I. This reduces the size and
        cost of the acceleration section significantly.


\section[NCRF Specifications for Buncher and Cooling] {NCRF Specifications for Buncher and Cooling Channels }
        Table~\ref{RF:tb18.A} summarizes the inventory of normal conducting rf cavities (NCRF). The cooling channel simulations were based upon ideal pillbox cavities with lengths 
        determined by the space available in the chosen lattices (and
        zero space between cavities). The gradients and phases of
        these cavities were adjusted to optimize the cooling
        channel performance while keeping the gradients and rf power
        requirements within feasible limits. Table~\ref{RF:tb18.B} shows the peak
        cavity power and klystron output power to meet these
        requirements, and the total power for each cavity type. Both
        tables also show how the required voltages are obtained
        using realistic re-entrant or ``omega'' shaped cavities with
        closed-off irises of finite thickness. The loss of active
        length in the realistic case is compensated by the greater efficiency
        of the rounded design. To be conservative, the iris diameter
        used for the omega cell was sufficient to accommodate any
        reasonable beryllium foil. In practice, the foils may be
        smaller and in any case will decrease in size towards the end
        of the cooling channel. Ideally, the cavity shape would be
        optimized for each foil size. This would maximize the
        efficiency and minimize the power cost. Note that the rf power
        requirements are dominated by the cooling sections (1,1)-(1,3) and
        (2,1)-(2,3), which have the largest number of cavities and the
        highest gradients.

\begin{table}[!tbh]
\begin{center}
\caption[Parameters for the ideal and realistic  NCRF cavities]{Parameters for the ideal (pillbox) and practical (omega) NCRF cavities. ${}^* \textrm{Note: Kilpatrick number is about 15 MV/m at 201.25 MHz}.$}
\label{RF:tb18.A}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\multicolumn{7}{|c|}{Ideal pillbox dimensions used in the simulations (see Chapter~\ref{BandPR:chap})}\\
\hline
Section&Radius&Length&Freq.&No. of &$E_{pk}^*$&$V_{eff}$\\
&(m)&(m)&(MHz)&cavities &(MV/m)&(MV)\\
\hline
b1&0.570&0.373&201.25&4&6.40&2.07\\
b2&0.570&0.373&201.25&8&6.00&1.94\\
b3&0.570&0.373&201.25&8&8.00&2.59\\
(1,1)-(1,3)&0.570&0.466&201.25&68&15.48&5.76\\
(2,1)-(2,3)&0.570&0.559&201.25&74&16.72&6.71\\
match &0.570&0.559&201.25&22&16.72&6.71\\
b1 402.5 MHz&0.285&0.186&402.5&2&6.40&1.03\\
b2 402.5 MHz&0.285&0.186&402.5&4&8.00&1.29\\
\hline
\multicolumn{7}{|c|}{Omega cavities}\\
\hline
Section&Radius&Length&Freq.&No. of &$E_{pk}^*$&$V_{eff}$\\
&(m)&(m)&(MHz)&cavities &(MV/m)&(MV)\\
\hline
b1&0.607&0.405&201.25&4&7.41&2.07\\
b2&0.607&0.405&201.25&8&6.95&1.94\\
b3&0.607&0.405&201.25&8&9.27&2.59\\
(1,1)-(1,3)&0.607&0.405&201.25&68&20.62&5.76\\
(2,1)-(2,3)&0.615&0.483&201.25&74&23.06&6.71\\
match &0.615&0.483&201.25&22&23.06&6.71\\
b1 402.5 MHz&0.308&0.288&402.5&2&6.57&1.03\\
b2 402.5 MHz&0.308&0.288&402.5&4&8.21&1.29\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[!tbh]
\begin{center}
\caption[Voltage and power requirements for the NCRF cavities]{Voltage and power requirements for the NCRF cavities. \\${}^{\dag}$~$Rs,$ calculated, = $V^2/P$, ${}^*$ Real cavity, $Q_0$ assumed 85\% of theoretical; ${}^{**}$ Klystron forward power for $3\tau$ filling.}
\label{RF:tb18.B}
\begin{tabular}{|lccccccc|}
\hline
\multicolumn{8}{|c|}{Ideal pillbox}\\
\hline
Section&$Veff$&$Rs {}^{\dag}$ &$P_c {}^*$&$P_{kly}{}^{**}$&No. of&$P_{tot}$&Sum\\
&(MV)&$(M\Omega)$&(MW)&(MW)&cavities&(MW)&(MW)\\
\hline
b1&2.07&8.899&0.567&0.628&4&2.51&\\
b2&1.94&8.899&0.499&0.552&8&4.42&\\
b3&2.59&8.899&0.886&0.982&8&7.85&\\
(1,1)-(1,3)&5.76&10.701&3.646&4.038&68&274.60&\\
(2,1)-(2,3)&6.71&11.428&4.635&5.134&74&379.91&\\
match &6.71&11.428&4.635&5.134&22&112.95&782.0\\
b1 402.5 MHz&1.03&6.275&0.200&0.222&2&0.444&\\
b2 402.5 MHz&1.29&6.275&0.313&0.347&4&1.387&1.8\\
\hline
\multicolumn{8}{|c|}{Omega cavities}\\
\hline
b1&2.07&10.220&0.494&0.547&4&2.19&\\
b2&1.94&10.220&0.434&0.481&8&3.85&\\
b3&2.59&10.220&0.77&0.855&8&6.84&\\
(1,1)-(1,3)&5.76&10.220&3.818&4.228&68&287.54&\\
(2,1)-(2,3)&6.71&11.794&4.491&4.974&74&368.09&\\
match &6.71&11.794&4.491&4.974&22&109.43&778.0\\
b1 402.5 MHz&1.03&8.368&0.150&0.166&2&0.333&\\
b2 402.5 MHz&1.29&8.368&0.235&0.260&4&1.040&1.4\\
\hline
\end{tabular}
\end{center}
\end{table}

\section{RF Station Description}
\label{RF:sec18.3}
        Each rf station consists of a modulator for two klystrons,
        a distribution system, and low-level rf and controls driving two or
        more cavities. The modulator must provide a flat top DC pulse
        of up to $125~\mu s$ with a recharge time of less than 20~ms. (This
        is equivalent to a repetition rate of 50 Hz, but not every
        50~Hz pulse is required. The output from the AGS appears as 6
        pulses spaced at 20~ms followed by a 300~ms gap, 
with  a repetition rate of  2.5~Hz.) The \textit{average} duty factor is $\approx 1.9 \times 10^{-3}.$

        Each rf station must provide approximately 10 MW of peak
        power to drive two cavities. The power source is a
        multi-beam klystron, which should give good reliability and a
        long operational lifetime.

        Power distribution will be via high-power coaxial lines, with
        the power split between two or more cavities using appropriate
        delays to maintain the proper phase . The cavities will use
        coaxial feedthroughs and loop-type couplers. The high peak power
        requirements require careful design of the components,
        although the average power, of about 10~kW per coupler, is
        quite modest. Provision is made for adjusting the phase
        of individual cavities and for handling the reflected power
        during the initial part of the cavity fill time.

        Each station includes a water distribution system and a rack of 
       low-level rf hardware and controls.

\subsection{Power Source and Equipment}
\label{RF:sec18.3.1}
The bunching, ionization cooling channel, and the match to the acceleration system requires high peak rf power sources at
201.25~MHz and 402.5~MHz to efficiently bunch, cool the muon
beam and prepare it for acceleration. Table~\ref{RF:tb18.B} lists the peak rf power requirements for each section. There are 184 201.25-MHz cavities in the
channel that require 782~MW of rf power for a pulse length of $125~\mu$s
at 15~Hz (average) and six 402.5~MHz cavities that require 1.8~MW at 15~Hz.

 An examination of the requirements shows that an rf source of about 6
 or 12~MW would be ideal for the 201.25 MHz~cavities and a source of
 500 to 750 kW for the 402.5 MHz cavities. The rf power for the 201.25 MHz
 cavities could be supplied by existing gridded tubes at about the 5~MW 
level. However, the low gain and lifetime of gridded tubes make
 the R\&D effort to develop an alternative most attractive. Preliminary
 calculations at SLAC~\cite{RF:ref18.4} have shown that a 201.25~MHz klystron
 could be built with a reasonable amount of R\&D. The gain, efficiency,
 and lifetime are all higher than a gridded tube at 50~dB, 50-70\% and
 50,000 hours, respectively. SLAC has examined two designs, a
 single-gun diode design, and a multibeam klystron (MBK). The multibeam
 klystron is the more attractive in that it reduces the overall length
 of the tube from 7.5~m to about 3.5--4.0~m. The length reduction
 factor of the multibeam klystron, and its potential for higher
 efficiency, make it the optimum candidate for the Neutrino
 Factory. Moreover, the length of the multibeam klystron is consistent
 with the manufacturing capabilities of current tube
 manufacturers, whereas  the manufacture of a 7.5~m diode tube would be
 a big step and would require new and costly facility upgrades. Figure~\ref{RF:fg18.AA} shows a 7~beam MBK developed by Thompson for TESLA.
\begin{figure}[htb!]
\begin{center}
%\vskip 2cm
\includegraphics[width=3in]{../template/report/ps-and-eps/fig_18.AA.eps}
\caption{Thomson TH 1801 multi-beam klystron.}
\label{RF:fg18.AA}
\end{center}
\end{figure}
 Two such
 tubes have been built and tested, and have demonstrated efficiencies
 of $63 - 66$\%~\cite{RF:ref18.1A}. To provide rf power overhead for dynamic
 regulation of the rf phase and amplitude, a 12~MW multibeam klystron
 has been selected as the high-power rf source for the Neutrino
 Factory. This provides an rf power overhead margin of about 20\% for
 regulation. The design will be a fully integrated horizontal
 package incorporating the tube, solenoid, and high-voltage terminal, 
 as pioneered at CERN for LEP. This facilitates the replacement
 and installation of tubes in the facility. Another advantage of the horizontal design, besides the ease
 of handling, is the reduced cost of the rf building because of the
 lower building height requirement. With a mean-time-between-failures 
(MTBF) of 50,000 hours and 84~tubes, one tube will need to be replaced about every 30 days (after the initial
 break-in period). Since there is a comparable number of klystrons for the acceleration system, the rate of klystron failures for the facility as whole will be about two per month, on average. Many of these failures towards the end of life are
 gradual, and replacement can be scheduled for routine maintenance
 periods.  Because of the large size and
 high cost of waveguide, the transmission lines from the tubes to the
 cavities will be large coaxial lines of 0.31 to 0.36~m diameter,
 pressurized to 1.75 atmospheres of dry air.  Power splitters 
 divide the rf power from each tube to supply the cavities. Sections b1 and b2 of the buncher (see Chap.~\ref{bunchcool} and, in particular, Table~\ref{BandC:tbbunchz}) will require a
 12-way splitting of the power; section b3 an 8-way split, and
 sections (1,1) to (2,3) of the cooling  a 2-way split. Splitters with proper built-in
 phase delays further divide the power to each cell or cavity
 section of the cooling channel.
        
\begin{figure}[htb!]
\begin{center}
%\vskip 2cm
\includegraphics[width=5in]{../template/report/ps-and-eps/rf_fg1.eps}
\caption{Cross section of cooling channel equipment gallery.}
\label{RF:fg18.A}
\end{center}
\end{figure}

\begin{figure}[htb!]
\begin{center}
%\vskip 2cm
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg2.eps}
\caption{Cooling channel equipment gallery, plan view.}
\label{RF:fg18.B}
\end{center}
\end{figure}
The 402.5 MHz system can use currently existing 900 kW diode rf
klystron amplifiers. Because of their long length, it would be
advisable to fund a small R\&D effort aimed to  development of an integrated horizontal
package for the tube. As for the multibeam klystron, this would
improve the efficiency of tube handling and provide cost savings
because of reduced building height requirement. Coaxial
transmission lines with splitters would be used to provide the rf
power to the cavities. Only 3 klystron tube amplifiers are required to
supply the requirements of the 402.5 MHz buncher rf.

Figures~\ref{RF:fg18.A} and ~\ref{RF:fg18.B} show a cross section and plan view of a portion of
the rf building gallery along a 201.25~MHz section. The rf building is
approximately 190 m long and 30 m wide. With horizontal packaging
of the 201.25 and 402.5 MHz klystrons, the height of the building roof
line need only be 5.5 m. Because of the large footprint of the
equipment, the klystrons are arranged side by side and on both sides
of the gallery. Not shown in the figures are the transmission line
splitters, required to supply the rf power to the cavities, and 
the utilities. The 402.5 MHz klystron system footprint will be much
the same, but about half the size, and these klystrons will be located
in sections b1 and b2 interspersed between the 201.25 MHz equipment.

\subsection{Station Controls and Low-Level rf}
\label{RF:sec18.3.2}
The low-level rf (LLRF) and control system provides the drive power
for the final klystron amplifier, contains feedback loops for phase,
amplitude and cavity frequency control, and circuitry for  personnel safety and equipment
protection. A frequency reference line running the length of the
complex provides an rf phasing reference to which each cavity is
locked. A microprocessor in each rf station processes error
information to control the amplitude and phase and thus keep the cavity
tuned to the reference frequency. The microprocessor communicates
with, and accepts directions from, the central control room. The
system is similar to systems currently in use at Fermilab or planned
for the SNS project.  The LLRF system will include fast circuits to
detect sparks and malfunctions and immediately inhibit the rf to
protect the equipment and cavities. Other, hard-wired, fast circuits
will monitor for high rf leakage from equipment or contact with
high-voltage and current, and then activate interlocks for personnel
protection. The equipment would be housed in five standard racks next
to the klystron and associated equipment, Fig.~\ref{RF:fg18.B}. 
\subsection{High-Voltage Modulator and Power Supply}
\label{RF:sec18.3.3}
The high-voltage modulator and power supply for the 201.25 MHz rf system will
use the latest solid-state design. Currently available Insulated Gate
Bipolar Transistor (IGBT) modulator technology will be built by
industry to provide the pulsed power requirements of the klystron, see
Fig.~\ref{RF:fg18.AAA}. The Neutrino Factory will use IGBT modulators similar to
designs currently being built for the SNS project. They are very
reliable, efficient, and cost effective. A 19-beam klystron, the basis
of this design, has a calculated efficiency of 70\% and a klystron tube 
perveance of $0.5\times 10^{-6}.$ The specifications for the modulator and power
supply are given in Table~\ref{RF:tb18.C}. The overall efficiency of the
modulator and power supply from the AC mains is about 95\%.
\begin{figure}[htb!]
\begin{center}
%\vskip 2cm
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fig_18.AAA.eps}
\caption[Compact modulator]{Compact modulator from Diversified Technologies, Inc., Medford, MA (Capacitor bank and regulator not shown in this picture).}
\label{RF:fg18.AAA}
\end{center}
\end{figure}

\begin{table}[tbh!]
\begin{center}
\caption{High-voltage modulator parameters.}
\label{RF:tb18.C}
\begin{tabular}{|lcc|}
\hline
Klystron frequency (MHz)&
201.25&
402.5\\
High-voltage (kV)&
80&
60\\
Current (A)&
215&
31\\
Duty factor (\%)&
0.19&
0.0525\\
Average power (kW)&
33&
1.0\\
Voltage droop (\%)&
0.1&
0.1\\
\hline
\end{tabular}
\end{center}
\end{table}


\subsection{NCRF AC Electrical Power and  Water System}
\label{RF:sec18.3.4}
The AC power for the normal conducting rf must support 84 tubes
with 33 kW average power and three tubes with 1.0 kW average power,
solid-state amplifiers and solenoid power supplies, cooling water
systems and miscellaneous other loads. These all require a 480-V three-phase 
supply. In addition to this, AC power is required at 120 V and
208 V for racks and other miscellaneous equipment. This gives a total
 AC power requirement of 6.8~MW. Table~\ref{RF:tb18.D} shows a summary of the AC power
requirements.
\begin{table}[tbh!]
\begin{center}
\caption{NCRF systems AC power requirements.}
\label{RF:tb18.D}
\begin{tabular}{|lc|}
\hline
Item&
Power\\
& (MW)\\
\hline
Klystron modulators (95\% efficiency)&
2.9\\
Amplifiers \& supplies&
1.0\\
Cooling + miscellaneous loads&
2.3\\
Racks etc.&
0.6\\
Total&
6.8\\
\hline
\end{tabular}
\end{center}
\end{table}

The cooling water system will be sized to accommodate the average
power of 6.8 MW with a proper temperature rise for safe and efficient
operation of the equipment. Each klystron station requires 75 gpm of
low-conductivity water (LCW) for cooling the klystron and associated
equipment, and 20 gpm LCW to cool and provide temperature control of
the cavity. This gives a total water requirement of 7,980 gpm. This
could be divided up between room-temperature (and above) water and chilled-water 
systems for cavity control at 20 gpm per station. For all these
systems, we assume a supply header pressure of 100 psi and return
pressure of 40 psi.
\section{Specification of NCRF Cavities for Cooling}
\label{RF:sec18.4}
        The 201.25 MHz normal-conducting cavities in the cooling
        sections must operate at very high accelerating
        gradients. This would be impractical with conventional
        open-iris structures, given the large size of the beam iris
        required. A great improvement can be made in the shunt
        impedance of the cavity by closing the iris with a thin
        conducting barrier. This barrier must use the smallest amount
        of material to minimize scattering of the muon beam. In the design we   close the irises with thin beryllium foils. Alternative 
        methods of closure, such as grids of thin-walled tubes will be
        evaluated in the future. The foils must be thick enough to
        conduct away the heat from the rf currents and keep the
        temperature below a predetermined level. The foils are 
        pre-stressed in tension during manufacture in order to keep
        them flat. This method ha2 been tested experimentally and
        works well up to the point where the thermal expansion exceeds
        the pre-stress and the foils begin to move. Foil thicknesses
        have been chosen for Study-II that will keep the temperatures
        below this critical level. The use of tapered foils, or foils
        with stepped thickness, can reduce the amount of material
        intercepted by the core of the beam, reducing the amount of
        scattering significantly. Table~\ref{RF:tb18.D2} shows the thickness of
        the foils used in the simulations of the various types of
        cavities in the buncher and cooling channel.

\begin{table}[tbh!]
\begin{center}
\caption[Beryllium foil thicknesses]{Beryllium foil thicknesses for various cells in the buncher and cooling channel. ${}^*$ dual values imply a stepped-thickness foil.} 
%\textit{note 119/239 might need to be 128/256RR}}  
\label{RF:tb18.D2}
\begin{tabular}{|lcccccc|}
\hline
Type&Section&Frequency&Length&Gradient&Thickness*&Radius*\\
&&(MHz)&(m)&(MV/m)&($\mu$m)&(cm)\\
\hline
end&b1&402.5&0.186&6.4&75&18\\
end&b2&402.5&0.186&6&75&18\\
end&b1&201.25&0.3728&6.4&10&21\\
middle&b1&201.25&0.3728&6.4&120/240&14/21\\
end&b2&201.25&0.3748&6&100&21\\
middle&b2&201.25&0.3748&6&105/210&14/21\\
end&b3&201.25&0.3748&8&180&21\\
middle&b3&201.25&0.3748&8&187/374&14/21\\
end&(1.1)&201.25&0.466&15.48&200/400&12/18\\
middle&(1,1)&201.25&0.466&15.48&700/1400&14/21\\
end&(1,3)-(2,1)&201.25&0.5592&16.72&248/495&12/18\\
middle&(1,3)-(2,1)&201.25&0.5592&16.72&917/1834&14/21\\
end&(2,1)&201.25&0.5592&16.72&128/256&10/15\\
middle&(2,1)&201.25&0.5592&16.72&495/990&12/18\\
\hline
\end{tabular}
\end{center}
\end{table}

        The normal-conducting cavities in the buncher are the
        same design as those in the first cooling section, though they
        are  operated at lower gradient, allowing the use
        of thinner foils. The buncher
        section also contains harmonic cavities
        operating at 402.5~MHz. These fit into the spaces that are
        occupied by the liquid-hydrogen absorbers in the cooling cells
        farther downstream. For these cavities, the foils occupy most
        of the diameter of the end walls, but the gradients are sufficiently 
        low that the losses in the foils are manageable.

        The normal-conducting cells must have some cooling to remove
        the average power losses in the walls and to stabilize the
        frequency. The Study-II design is based upon room
        temperature operation, although the option of operating at
        reduced temperature (\textit{e.g.}, liquid-nitrogen temperature) has been kept
        open. This option would lower the wall resistance and reduce
        the peak power requirements, at the expense of adding an
        additional refrigeration system.
\subsection{201.25~MHz Closed-Cell Description}
        The cooling channel simulations were based upon simple pillbox
        cavities that have continuous, flat, conducting end walls from
        the center all the way to the outer radius. The cavity lengths
        assumed for the simulations are just the available space
        divided by the appropriate number of cells. In practice the
        cavities must be closed by assemblies of foils or grids that
        are demountable to permit assembly or
        repair. This requires a non-zero thickness for each iris,
        reducing the length available for rf and lowering the
        effective shunt impedance. We have mitigated the losses by rounding
        the outer walls of the cavity to improve the quality factor
        and restore the shunt impedance. Any practicable assembly of
        foils (or grids), requires some space for flanges and
        access. We used a minimum spacing of 50 mm between
        cavities, as shown in Fig.~\ref{RF:fg18.C}. The dimensions of the
        cavities were adjusted to fit the remaining available
        space. Note that the resulting cavity lengths are
        significantly shorter than the optimum for a particle of this
        velocity $(\beta = 0.87).$ A cavity length that
        is more optimal could be achieved by adjusting the total cell
        length appropriately and this will be done as part of the overall optimization process later. The cavity shape is slightly reentrant
        in order to maximize the inductance, minimize the capacitance,
        and, hence, get the highest shunt impedance~\cite{RF:ref18.1}. 
Figures~\ref{RF:fg18.C} and \ref{RF:fg18.D} show the cavities separated by a pair of
        foils. This allows variable thickness foils to be used
        where the stepped side is not exposed to rf. Figure~\ref{RF:fg18.E} shows
        a MAFIA simulation of the electric field in two half-cells
        separated by a pair of foils. Some field enhancement can be
        seen on the noses. Alternatively, a single foil of twice the
        thickness could be used in the center of the iris, heated from
        both sides (except for the end cells). Another advantage of
        the closed-cells is that there is no rf coupling through the
        iris, so the cavities can be individually phased for optimum
        performance of the cooling channel. One penalty of the omega
        shape is some field enhancement on the nose, see Figs.~\ref{RF:fg18.Fa} and ~\ref{RF:fg18.Fb}. Although the nose is made with as large a radius as
        practical, it still has an enhancement factor of about 1.7 over 
the field on axis. However, the highest surface
        field in Table~\ref{RF:tb18.B} is only about 1.5 times the Kilpatrick 
        number for this frequency. Furthermore, a positive aspect of this field
        concentration is that it is not on the foil but on the solid
        copper. A breakdown to this point may be less of an
        issue. Figure~\ref{RF:fg18.Fb} shows the azimuthal magnetic field. The
        distribution on the foil, and therefore the rf heating, is
        similar to the pillbox model, although there is some shielding
        due to the noses.  Figure~\ref{RF:fg18.G} shows the profile of the cavity
        from the downstream part of the cooling channel where only two
        cavities are used per cooling cell (Lattice 2). The cavities are longer
        and closer to the optimum for this particle velocity (though
        there is still room for some improvement).  Figures~\ref{RF:fg18.Ha} and ~\ref{RF:fg18.Hb} show the 2D electric and magnetic field profiles for this
        case.


\begin{figure}[hbt!]
\begin{center}
%\vskip 2cm
\includegraphics[width=6in]{../template/report/ps-and-eps/rf_fg3.eps}
\caption{Profile of cavities for buncher and Lattice 1 cooling section.}
\label{RF:fg18.C}
\end{center}
\end{figure}
%
\begin{figure}[hbt!]
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg4.eps}
\caption{Section of one cavity.} \label{RF:fg18.D}
\end{minipage}%
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg5.eps}
\caption{MAFIA model with two foils.} \label{RF:fg18.E}
\end{minipage}
\end{figure}
%
\begin{figure}[hbt!]
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=1.5in]{../template/report/ps-and-eps/rf_fg6.eps}
\caption{URMEL 2D E-field.}  \label{RF:fg18.Fa}
\end{minipage}%
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=1.5in]{../template/report/ps-and-eps/rf_fg7.eps}
\caption{URMEL 2D azimuthal H-field.} \label{RF:fg18.Fb}
\end{minipage}
\end{figure}

\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/RF_fg9.eps}
\caption{Profile of cavities for Lattice 2 cooling section.}
\label{RF:fg18.G}
\end{center}
\end{figure}
\subsection{Foil Requirements}
\label{RF:sec18.4.2}
\begin{figure}[hbt!]
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=1.5in]{../template/report/ps-and-eps/RF_fg10.eps}
\caption{Lattice 2 cooling cavity. URMEL 2D E-field.}
\label{RF:fg18.Ha}
\end{minipage}%
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=1.5in]{../template/report/ps-and-eps/RF_fg11.eps}
\caption[URMEL 2D azimuthal H-field] {Latice 2 cooling cavity. URMEL 2D azimuthal H-field.}
\label{RF:fg18.Hb}
\end{minipage}%
\end{figure}
        The closed-cell cavity design described above assumes that
        beryllium foils will be used to seal off the beam
        irises. Other methods, including grids of thin-walled tubes,
        have been discussed, and show promise, but are not as far
        advanced in understanding or testing as the foils. Hence
        pre-stressed foils have been chosen as the baseline design for
        Study-II. The foils are made of thin high-purity beryllium
        sheet bonded to a thicker ring of slightly lower grade
        material, see Fig.~\ref{RF:fg18.I}. The exact details of this process
        are proprietary but the combination of materials used results
        in a small but significant difference in the thermal expansion
        of the foil relative to the ring assembly. This produces a
        tensile pre-stress on cool down from the joining operation,
        which helps to keep the foil flat.

        When the foils are heated by rf, and only cooled by conduction
        to the edges, they assume an approximately parabolic
        temperature profile, see Fig.~\ref{RF:fg18.J}. The calculated
        rf-induced profile is slightly flatter than parabolic and can
        be used in ANSYS as a load set for the stress
        calculations. Figure~\ref{RF:fg18.K} shows an example of the temperature
        distribution in a thin foil from such an analysis.

        The foils remain flat until the thermal expansion exceeds the
        tensile pre-stress. At this point compressive stress is
        generated in the foil, and it starts to deflect by buckling
        into a gently bowed shape, see Fig.~\ref{RF:fg18.L}. The maximum
        allowed temperature difference is about $35~{}^{\circ}C$ and is
        approximately independent of the radius and thickness. Of
        course a thicker foil can take more power before reaching the
        buckling temperature, as shown in Fig.~\ref{RF:fg18.M}. A set of foils
        (Table~\ref{RF:tb18.D2}), has been specified for the set of cavities used
        in Table~\ref{RF:tb18.A}, that keeps the temperatures below the critical
        point. For the larger irises, the foils become quite thick and
        the scattering of the muon beam becomes significant. One way
        to reduce this is to make the windows thinner in the middle,
        where the core of the beam passes, and thicker towards the
        outside, where there are fewer particles, see Fig.~\ref{RF:fg18.N}. It
        is thus possible to reduce the scattering while maintaining
        the same temperature rise in the foil. Figure~\ref{RF:fg18.O} shows the
        temperature profile for a thin window of uniform thickness and
        for windows with thicker profiles starting at different radii.
        As can be seen from the figure, adding material at large
        radius has a significant effect on the temperature profile up
        to about one third of the way in. Beyond this point, there is
        diminishing return and much past halfway there is little to be
        gained by adding more material. Simulations have shown that
        such a stepped window reduces the multiple scattering
        significantly compared with a uniform foil for the same
        temperature. Going to multiple steps in thickness, or to a
        continuous taper, should yield further small improvements in
        scattering but the simulations do not show a significant
        improvement in transmission through the cooling channel.

\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg11.eps}
\caption{Layout of beryllium test window (all dimensions in mm).}
\label{RF:fg18.I}
\end{center}
\end{figure}

\begin{figure}[htb!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg12.eps}
\caption[ Actual temperature profile for rf heating ]{Actual temperature profile for rf heating and parabolic approximation from halogen lamp tests.}
\label{RF:fg18.J}
\end{center}
\end{figure}

The pre-stressed foil properties have been investigated experimentally
in a low-power test cavity at 805 MHz using a halogen lamp as a heat
source~\cite{RF:ref18.2}. These experiments used small (160 mm diameter)
foils and the results have been extrapolated to larger foils. We have
assumed that the same pre-stress can be achieved in the larger foils,
but this must be validated experimentally as part of the future R\&D
program. It should be straightforward to obtain the desired pre-stress
by adjusting the combination of materials in the outer ring but some
experimentation may be required to find the optimum combination.

\begin{figure}[bth!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg13.eps}
\caption[ANSYS calculated temperature profile for thin window ]{ANSYS calculated temperature profile for thin window with 60~W loading.}
%\caption[~ ANSYS calculated displacement \textit{vs.} power]{ANSYS calculated displacement \textit{vs.} power for larger windows}
\label{RF:fg18.K}
\end{center}
\end{figure}

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg14.eps}
\caption[ANSYS model showing buckling displacement ]{ANSYS model showing example of buckling displacement (dimensions in m).}
\label{RF:fg18.L}
\end{center}
\end{figure}

\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg15.eps}
\caption[ANSYS calculated displacement \textit{vs.} power]{ANSYS calculated displacement \textit{vs.} power for larger windows. The three curves represent three window thicknesses, $127\mu$m, $190.5\mu$m, and $254\mu$m.}
\label{RF:fg18.M}
\end{center}
\end{figure}
        One issue with the closed-iris structures is the possibility
        of multipactoring due to the high secondary yield of beryllium
        or aluminum (foils or tubes). This could cause outgassing and
        possibly breakdown in the cavity, which might damage the
        delicate structures. Persistent multipactor discharge may also
        heat the surfaces involved. Unlike copper, the secondary yield
        of aluminum does not reduce with rf conditioning because of a 
stable surface oxide layer. It is expected that beryllium may
        behave similarly, although the handbook values for beryllium
        oxide are lower than those for aluminum oxide. It is proposed
        to suppress this problem by the application of low secondary
        emission coatings, such as titanium nitride (TiN). This issue
        will be investigated experimentally in a high-power cavity as
        part of the ongoing Muon Collaboration 805 MHz R\&D
        program. The cavity is designed to use demountable foils or
        copper blank-off plates and can be conditioned to very
        high-gradient using the high-power klystron test stand in the
        Lab G facility at FNAL. The foils will be coated on one side
        with TiN and conditioning tests can thus be run with all-copper 
surfaces, uncoated beryllium windows, coated beryllium
        windows or combinations of these. Windows of various thickness
        and with stepped profiles will be tested and the conditioning
        can be attempted with a wide range of magnetic fields in an
        available 5~T superconducting solenoid.
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=3in]{../template/report/ps-and-eps/rf_fg16.eps}
\caption{Stepped window concept.}
\label{RF:fg18.N}
\end{center}
\end{figure}

 \begin{figure}[hbt!]
\begin{center}
\includegraphics[width=3in]{../template/report/ps-and-eps/rf_fg17.eps}
\caption[Temperature profile of uniform thin window ]{Temperature profile of uniform thin window and windows with steps to thicker outer region starting at various radii.}
\label{RF:fg18.O}
\end{center}
\end{figure}
\afterpage{\clearpage}
\subsection{2.75~m Lattice Implementation (Lattice 1)}
        The cooling channel lattice is a tightly packed assembly of
        equipment including liquid-hydrogen absorbers, superconducting
        solenoids, high-gradient rf cavities, instrumentation, vacuum
        equipment, etc. Our studies show that it is possible to
        integrate all these components into the available cell length.
        Several iterations have been performed on this layout to try
        to make the most efficient use of the space. Constraints
        include the size of the rf cavities, which is dictated by the
        frequency, the size of the absorbers, which is determined by
        the beam size, and the cell length, which has been fixed for
        this Study at 2.75~m for the buncher and Lattice 1 cooling section
        and 1.65~m for the Lattice 2 cooling section. The two lattice cell dimensions will be re-evaluated as part of
        the overall system optimization. 

The sizes of the coil packs
        and cryostats have been chosen to allow practical current
        densities and the coil diameters have been kept small to
        minimize the amount, and therefore
        the cost, of superconductor required. The largest coil is the central one (``coupling coil'') that surrounds
        the rf cavities. The inner diameter of this coil is left
        large enough to allow the cavity structures to pass through
        during assembly. The rf feeds must come out through the wall
        of the cryostat, and may be angled to give clearance to other
        hardware. Pumping ports will be short and wide to give good
        conductance and may also penetrate the cryostat. Clearance is
        required at the end of each cooling cell for
        installation or removal of one absorber/rf module from the
        channel. This is achieved by using a collapsible flange in the
        outer cryostat wall, which is reinforced after it is made up
        in order to handle the possible magnetic forces. RF shields
        will be used to keep beam-induced signals from escaping into
        the outer cryostat and vacuum system. 

Figure~\ref{RF:fg18.P} shows the
        proposed cooling channel layout for the first cooling section
        lattice, including all major components except the beam
        instrumentation package, which will occupy the clearance
        opening at the end of each cell or, possibly, the space
        between the rf cavities and the hydrogen absorber. The space
        in the cryostat outside of the cavities will be evacuated to
        minimize the load on the rf structures. This approach would 
        provide insulation for the cavities if they were operated
        below room temperature. It also obviates the need
        for UHV connections between each cavity and between the
        cavities and the hydrogen absorbers. The flanges are required
        only to provide rf continuity (for screening) and to separate
        the UHV of the rf system from the guard vacuum of the
        cryostat.
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg18.eps}
\caption{Cooling channel Lattice 1, four cavities per cell.}
%\caption{402.5 MHz buncher harmonic cavity}
\label{RF:fg18.P}
\end{center}
\end{figure}
\subsection{1.65~m Lattice Implementation (Lattice 2)}
\label{RF:18.4.3}
        The 1.65 m lattice for the downstream part of the cooling
        channel will use a layout similar to the upstream part, but
        with smaller hydrogen absorbers and only two rf cavities per
        cell. The density of equipment is similarly high. In this case
        the cavity lengths are closer to the optimum for this particle
        velocity, but could still be improved if the cell
        length were increased slightly. Figure~\ref{RF:fg18.Q} shows the proposed
        cooling channel layout for lattice 2, including
        all major components except the instrumentation package.
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg19.eps}
\caption{Cooling channel Lattice 2, two cavities per cell.}
\label{RF:fg18.Q}
\end{center}
\end{figure}
\subsection{402.5 MHz Buncher Cavity}
        The buncher harmonic cavities, Fig.~\ref{RF:fg18.U}, are smaller,
        simpler versions of the 201.25 MHz cavities. They are rounded
        pillboxes and are closed by similar foils  that are
        smaller and thinner than those used for the large cavities. There is adequate
        space for the cavities to be the optimal length for this
        particle velocity. Though the power requirements are modest,
        cooling water is used to stabilize the frequency and
        remove the small amount of average power dissipated in the
        walls. Harmonic cavities are installed in some of the
        buncher cells in the location corresponding to that where the hydrogen absorbers are
        placed in the normal cooling sections, \textit{i.e.}, inside the bore
        of the focusing solenoid coils. 
%If required due to radial space
%        constraints, the diameter of the cavity could be reduced
%        slightly at the cost of a little more power and somewhat
%        thicker foils.
 \begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg20.eps}
\caption{402.5~MHz buncher harmonic cavity.}
\label{RF:fg18.U}
\end{center}
\end{figure}
\subsection{Tuning Requirements}
        Since there is negligible beam loading, the tuning
        requirement for the cavities is simply to compensate for
        temperature variations due to water supply changes and rf heating. If
        we assume bulk water temperature fluctuations are of the order
        of $1{}^{\circ}$C or less, and a thermal expansion coefficient of copper
        of approximately 17~ppm/${}^{\circ}$C, then the frequency variation would
        be about 3.4 kHz. Since the average power is modest, it should
        be easy to limit the temperature rise due to rf heating to
        10${}^{\circ}$C or less. A worst-case cold start with the cavities around
        0${}^{\circ}$C and a normal operating temperature of 40${}^{\circ}$C would produce a
        frequency detuning of about 136 kHz. Simple 2D calculations
        show that if the length of the cavity is varied from the
        nominal value, the frequency sensitivity is about 236 kHz/mm,
        so a small range of motion would be adequate to achieve the
        required tuning range. A tuning scheme similar to that used
        for superconducting cavities, where the cavity is
        mechanically stretched or compressed within elastic limits,
        will easily achieve this range of motion. Alternatively, a
        moving plunger tuner can be used to tune the cavity
        inductively but this would require an additional aperture in
        the cavity and would be harder to package within the confines
        of the cryostat.

        It would also be possible to tune the cavities over a limited
        range by controlling the water temperature, but the water
        stability would have to be a fraction of a degree to keep the
        frequency stable to within the bandwidth of the cavity (3.3~kHz 
unloaded, 6.6~kHz critically coupled). Moreover, each
        cavity would then require an independent water circuit and
        controller, which would be inconvenient.

        Depending on the elastic range of motion of the cavities, it
        may be desirable to have some kind of \textsl{fixed} tuning after
        assembly to account for manufacturing tolerances (analogous to
        the ``dimpling" of linac cavities). This could be a specific
        part of the cavity which is designated to be deformed or the
        cavity as a whole could be designed such that it can be
        stretched or compressed beyond the elastic limit to achieve a
        permanent tuning. If detailed analysis shows that the cavity
        has a sufficiently large elastic tuning range, it may even be
        possible to relax the requirement of keeping the foils flat
        and allow some movement to take place. (Pre-bowing of the
        foils would ensure that this happens in a predictable manner.) 
        This would allow thinner foils to be used with a concomitant
        reduction in scattering.

        In the event that vibrations of the foils or other parts of
        the system should produce troublesome fluctuations in the rf
        fields, the deforming type tuner could be augmented with a
        fast piezo-electric actuator allowing feedback at audio
        frequencies. This has been demonstrated to reduce the effect
        of microphonics in superconducting cavities.

\subsection{Vacuum Requirements}
        The operating vacuum in the high-gradient cavities should be
        in the 10~nTorr range or better. Operating much above this
        range is likely to produce more frequent arcing and would
        require significantly longer time to condition the cavities
        initially and after any vent. The reliability of the rf window
        is also strongly influenced by the vacuum level. The frequency
        of window arcs and the lifetime of anti-multipactor coatings
        on the ceramic are both degraded by operating at pressures
        above about 100~nTorr. These conditions will require strong
        pumping and good conductance to the rf cavities. Because of
        the presence of strong solenoid magnetic fields, ion pumps may
        not be used in close proximity to the cavity during operation,
        though they may be useful during initial conditioning with
        solenoids off. Cryopumps or titanium sublimation pumps may be
        useful close to the cavities with magnetic fields on. It would
        be advantageous to pump the cavities through the rf coupler if
        there proves to be sufficient conductance, since this will
        ensure the best possible vacuum at the rf input window. A
        large diameter coaxial feed with a short distance to the pump
        may have sufficient conductance by itself. If not, it can be
        supplemented by an additional pumping port on the cavity
        body. A thorough bakeout to above 150${}^{\circ}$C after assembly would
        be advantageous but may be incompatible with the
        superconducting components. In that case, the individual
        components will be baked separately before final assembly into
        the cryostat.

\section{SCRF Cavities for Acceleration}
Based on the high-real estate gradient desired to minimize muon loss,
superconducting cavities are selected for the acceleration section of the Neutrino Factory to provide an active gradient of
15-17 MV/m, and a real-estate gradient of 7.4 MV/m.  At such
high-gradients, the peak rf power demand for copper cavities that
provide 7.5 GV would become prohibitively expensive.  By virtue of low
losses, SC cavities can be filled slowly, reducing the
peak power demand to roughly 0.5~MW per cell for a 3~ms rise time.

As a result of experience at LEP, CEBAF, TTF, Cornell, KEK and
CEA-Saclay, the science and technology of superconducting cavities and
associated components are highly developed~\cite{RF:ref18.H1}.  In all, SCRF
systems totaling 1~km in active length have been installed in a
variety of accelerators and routinely operated to provide a total of 5
GV.  The largest installation was at LEP-II, where 500 m of niobium-coated copper cavities provided more than 3 GV of acceleration~\cite{RF:ref18.H2}. The Neutrino Factory calls for nearly 500 m to provide 7.4 GV.

Although the sheet-metal Nb cavites used for TESLA are capable of
providing gradients of the order of 20 MV/m and higher~\cite{RF:ref18.H3}, we have
chosen Nb/Cu technology, developed at CERN~\cite{RF:ref18.H4} for LEP-II, for
several reasons:
\begin{itemize}
\item Because of the low rf frequency (201.25 MHz), and the
accompanying thicker wall (\textit{e.g.}, 6~mm), the cost of raw sheet niobium
would be  prohibitive for the roughly 600
cells needed.  
\item High thermal conductivity copper provides better
stability against quenching of superconducting cavities than does sheet Nb.
This is especially beneficial at 201.25 MHz because of the high stored
energy per cell (roughly 1~kJ per cell at design gradient).  
\item
The wall thickness of 201.25~MHz cavities may need to be
even greater than 6 mm for mechanical stability against atmospheric load, 
for reducing Lorentz force detuning, and for avoiding microphonics from external vibrations.  
\item A coated copper cavity allows the use of pipe
cooling instead of the more usual bath cooling.  Pipe cooling saves
liquid-helium inventory (estimated at 100,000~L for standard bath
cooling of 600 cells).  It also opens additional avenues for improving
the mechanical stability for large scale cavities.  
\end{itemize}
Recent results from CERN~\cite{RF:ref18.H5} on 400 MHz Nb/Cu cavities (Fig.~\ref{RF:fg18.H1}) demostrate accelerating gradients of 15~MV/m at 2.5 K at a $Q$ of $2
\times 10^9.$  Because of the lower frequency used here, we can expect the $Q$
to be four times higher.  We have chosen an operating temperature of
2.5~K and a $Q$ value of $6\times 10^9.$  Extrapolating LEP results at 4.5 K would imply a much
lower $Q < 2\times 10^9$ at the design gradient for the Neutrino Factory, even when scaled
for the lower frequency. Moreover, LEP cavities never reached the Neutrino Factory design gradients at 4.5 K.
 
Modeling the $Q$ \textit{vs.} $E$ (Fig.~\ref{RF:fg18.H2}) obtained for LHC 400~MHz cavities
and incorporating the $Q$ increase for 201.25~MHz, ANSYS studies
conclude that it will not be possible to reach $E_{acc} = $15--17~MV/m at a $Q$
 of $6\times 10^9,$ unless the operating temperature is reduced to 2.5~K. 
Figure~\ref{RF:fg18.H2} shows the peak magnetic field expected for 17~MV/m in
a 2-cell cavity with 300~mm beam aperture.  It corresponds to $E_{acc} =
13$~MV/m for the LHC cavity geometry because of the relatively smaller
beam pipe and optimized cavity. (The Neutrino Factory cavity geometry is 
discussed below.)

 \begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg21.eps}
\caption{Q$_0$ \textit{vs.} gradient for Nb/Cu CERN 400 MHz, LHC cavity.}
\label{RF:fg18.H1}
\end{center}
\end{figure}

\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg22.eps}
\caption[Q$_0$ \textit{vs.} gradient expected for 201.25~MHz cavity]{Qo \textit{vs.} gradient expected for 201.25 MHz cavity. Although the Neutrino Factory design is Q$_0 = 6\times 10^9$ at $E_{acc} = 17$~MV/m, for a
2-cell, 300~mm aperture cavity, it corresponds to only 13~MV/m (marked X) for the LHC cavity geometry due to the smaller
aperture of this cavity and the optimized geometry.}
\label{RF:fg18.H2}
\end{center}
\end{figure}

        Accelerator physics studies show that an aperture of 300 mm
        (diameter) is acceptable for the Neutrino Factory, except for
        the first 1000 MeV of the pre-accelerator linac, where an
        aperture of 460 mm has been chosen.  Because of the higher
        peak fields arising from the larger aperture, the gradient for
        the first section of the pre-accelerator has been reduced to
        15~MV/m.  The corresponding surface magnetic field is still
        12\% less than the peak magnetic field for the LHC cavity at 15
        MV/m.

In selecting the rf pulse length $(T_{rf}),$ a trade-off must be made
between peak rf power on the one hand, and refrigerator load,
tolerance to microphonics and  Lorentz force (LF) detuning on the
other hand.  Increasing $T_{rf}$ will lower the peak power, but increase
the average rf power and the refrigeration load.  Increasing $T_{rf}$ will
also drive $Q_L$ toward higher values, decreasing the cavity bandwidth
and thereby increasing its sensitivity to LF detuning and
microphonics.  The peak rf power $(P_{pk})$ needed to establish the fields
depends on the stored energy (U), the cavity time constant 
$(\tau=\frac{Q_L}{\omega})$ and the
amount of detuning $\delta \omega$ expected from Lorentz force and 
microphonics, as follows~\cite{RF:ref18.H1}
\begin{equation}
P_{pk}=\frac{U(\frac{\omega}{Q_L})\left\{(Q_L\frac{\delta \omega} {\omega})^2 
+\frac{1}{4}\right\}} {\left\{(1-\exp{-\frac{T_{rf}}{2\tau}})\right\}^2}
\end{equation}
Once the fill time and detuning tolerance are selected, the loaded $Q$
of the cavity can be found to minimize the peak power required.  A
conservative estimate for detuning tolerance in these large 201.25 MHz
structures is 40 Hz.  Cavities at TTF and CEBAF show microphonic
excitation of $< 10$~Hz~\cite{RF:ref18.H6}.  For a fill time of 3 ms, the optimum $Q_L$
is $1\times 10^6$ (bandwidth = 200 Hz) and the required peak power is about 500
kW per cell.  Coaxial couplers developed for KEKB~\cite{RF:ref18.H7}
 have delivered 380 kW CW to a 1~A beam.  In pulsed mode, higher
power performance is expected.  For a wall
thickness of 8 mm, the calculated Lorentz force detuning at 17 MV/m is
128 Hz.  Most of this can be handled with feed-forward techniques
developed at TTF for TESLA~\cite{RF:ref18.H8}.
\subsection{SCRF Structures at 201.25~MHz}
To improve the real-estate gradient, it is important to have a large
filling factor of cavities in the cryomodule.  This pushes structures
towards multi-cell cavities.  On the other hand, because of the low
frequency and high-gradient, the coupler power and stored energy per
structure increase with the number of cells.  Also, the mechanical
resonance frequency of multi-cell cavities drops, demanding stiffening
schemes. Compromising between these factors, 2-cell units are
chosen.  In the first 1000~MeV of the preaccelerator linac, where
apertures of 460~mm are needed, gradients are lowered to 15~MV/m to
keep the peak surface fields comparable to those of the 300-mm bore cavity at 17~MV/m; input coupler power is
kept at the 500~kW level by providing one coupler at each end.
\begin{figure}[hbt!]
\begin{minipage}[b]{.5\linewidth}
\centering\includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg23.eps}

\end{minipage}%
\begin{minipage}[b]{.5\linewidth}
\centering \includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg24.eps}
%\caption{Two-cell geometry, small aperture} \label{RF:fg18.H4}
\end{minipage}
\caption{Two-cell geometry: (left) small aperture; (right) large aperture.} \label{RF:fg18.H3}
\end{figure}
The performance of a superconducting cavity depends on the peak
surface fields.  Minimizing $E_{pk}$ is important to avoid field emission
that lowers the cavity $Q$ and increases heat load.  Minimizing $H_{pk}$ is
also important, since the $Q$ of these cavities falls with surface
magnetic field, one of the characteristic features of Nb/Cu cavities
(Fig.~\ref{RF:fg18.H1}).  In the 400 MHz LHC cavity, which reached $E_{acc} = 15$~MV/m, the corresponding peak surface fields were $E_{pk} = 33$ MV/m and $H_{pk}
= 750$ Oersted.  The LHC cavity has a beam pipe diameter of 300 mm.
Keeping the same beam pipe diameter for 201.25 MHz, 2-cell cavities,
it is possible to improve the Neutrino Factory cavity geometry (see
Fig.\ref{RF:fg18.H3}) to reduce the peak fields to 14\% below LHC-cavity
values.  Relative to CERN cavity performance, there is adequate safety
margin for both improved structure choices.  Tables~\ref{RF:tb18.H1} and 
\ref{RF:tb18.H2} list the
properties of the 2-cell 300~mm aperture unit and the 2-cell large aperture 
unit, respectively.  Figure~\ref{RF:fg18.H3} (left panel) shows the 2-cell geometry with 300
mm aperture and (right panel) shows the 2-cell geometry with 460 mm
aperture. Figure~\ref{RF:fg18.H8} shows the deformation (exaggerated) due to
Lorentz force detuning for the 2-cell, 300~mm diameter cavity.
 \begin{figure}[hbt!]
\begin{center}
\includegraphics[width=3in]{../template/report/ps-and-eps/rf_fg27.eps}
\caption{Lorentz force detuning for 8~mm wall thickness cavity.}
\label{RF:fg18.H8}
\end{center}
\end{figure}

\begin{table}[hbt!]
\begin{center}
\caption{2-cell, 300~mm-diameter cavity parameters.} \label{RF:tb18.H1}
\begin{tabular}{|lc|}
\hline
RF freq (MHz) & 201.25\\
No. of cells per cavity & 2\\
Active cavity length (m)&  1.5\\
No. of cavities &  268\\
Linac & 76\\
RLA & 192\\
Aperture diameter (mm)&
300\\
$E_{acc}$ (MV/m)&
17\\
Energy gain per cavity (MV)&
25.5\\
Stored energy per cavity (J)&
2008\\
$R/Q$ ($\Omega$/cavity)&
258\\
$E_p/E_{acc}$&
1.43\\
$H_p/E_{acc}$ (Oe/MV/m)&
38\\
$E_{pk}$ at 17 MV/m (MV/m)&
24.3\\
$H_{pk}$ at 17 MV/m (Oe)\label{RF:18.4.2}&
646\\
$Q_0$&
$6\times 10^9$\\
Bandwidth (Hz)&
200\\
Input power per cavity (kW)&
1016\\
RF on-time (ms)&
3\\
RF duty factor (\%)&
4.5\\
Dynamic heat load per cavity (W)&
18.9\\
Operating temperature (K)&
2.5\\
$Q_L$&
$10^6$\\
Microphonics detuning tolerable (Hz)&
40\\
Wall thickness (mm)&
8\\
Lorentz force detuning at 15 MV/m (Hz)&
128\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{2-cell, 460~mm-aperture cavity parameters.} \label{RF:tb18.H2}
\begin{tabular}{|lc|}
\hline
RF freq (MHz)&
201.25\\
No. of cells per cavity&
2\\
Active cavity length (m)&
1.5\\
No. of cavities&
43\\
Aperture diameter (mm)&
460\\
$E_{acc}$ (MV/m)&
15\\
Energy gain per cavity (MV)&
22.5\\
Stored energy per cavity (J)&
1932\\
$R/Q$ ($\Omega$/cavity)&
208\\
$E_p/E_{acc}$&
1.54\\
$H_p/E_{acc}$ (Oe/MV/m)&
44\\
$E_{pk}$ at 15 MV/m (MV/m)&
23.1\\
$H_{pk}$ at 15 MV/m (Oe)&
660\\
$Q_0$&
$6\times 10^9$\\
Bandwidth (Hz)&
200\\
Input power per cavity (kW)&
980\\
RF on-time (ms)&
3\\
RF duty factor (\%)&
4.5\\
Dynamic heat load per cavity (W)&
18.3\\
Operating temperature (K)&
2.5\\
$Q_L$&
$10^6$\\
Microphonics detuning tolerable (Hz)&
40\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Input Power Coupler}
The antenna type coaxial design was chosen based on the successful
experiences of CERN (LEP-II), DESY (HERA and TTF) and especially the
success of the input coupler for KEKB~\cite{RF:ref18.H7}. Figure~\ref{RF:fg18.H6} shows the dimensions of the KEK, 508 MHz coupler, which will be scaled
proportionately to 201.25 MHz.  The lengths of the various sections
will be adjusted to fit the final cavity and cryostat designs adopted.
The waveguide-to-coaxial transition is of the door-knob variety.  As
in all high-power applications, the main window will be at room
temperature and remote from the cavity.  At KEKB, it is a disk
shaped, water cooled, 95\% pure alumina ceramic with a central hole for
the inner conductor. A teflon coaxial centering disk between the
window and doorknob serves to limit the flow of air to the cavity in
the unlikely event of a ceramic window break.  The inner conductor is
made of OFHC copper pipe and is water cooled. The outer conductor is
made of copper plated (30~mm) stainless steel and has fins cooled by a
4.5 K stream from the refrigerator.  This reduces both the dynamic and
static  heat leaks associated with the coupler.

Benefiting from simulation codes recently available for calculating
and avoiding multipacting~\cite{RF:ref18.H9}, dimensions of the inner and
outer conductors are chosen so that multipacting will not be a
serious problem.  The coaxial design also permits application of a DC
bias voltage between the two conductors to curtail any possible
multipacting that may develop near the window or other sensitive
regions.
 
The coupler will be equipped with standard diagnostics for vacuum, gas
species, temperature and light monitoring.  Vacuum and light levels
can be used to trip the rf power source in case of arcs.

The $Q_{ext}$ value of the input coupler is fixed after initial
adjustment of the position of the inner conductor by the use of
appropriate spacing washers during final assembly.  From experience at
KEK, we expect that the $Q_{ext}$ for the non-accelerating modes of the
fundamental pass band will be of the same order as the $Q_{ext}$ for the
accelerating mode, \textit{i.e.}, a few $\times 10^5.$

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=2in]{../template/report/ps-and-eps/rf_fg25.eps}
\caption{KEKB 508-MHz coupler.}
\label{RF:fg18.H6}
\end{center}
\end{figure} 
\subsection{Higher-Order Mode (HOM) Couplers}
The function of the HOM couplers is to damp the higher-order modes to
$Q_{ext}$ values of $10^4 - 10^5$ to prevent resonant build up of beam-induced
fields that may make the beam unstable or increase the HOM power.  The
HOM couplers extract beam-induced HOM power from the cavity and
deposit it in room-temperature loads.  In view of the large muon bunch
length, we do not expect HOMs to be a serious issue.
  
Two couplers are needed, with a relative azimuthal angle of about $90^{\circ}$
to ensure damping of both polarizations of dipole modes.  One
coupler is attached to each end of the cavity.  The HOM couplers must
reject the accelerating mode by means of a narrow-band filter built
into the coupler.

Detailed calculations will be carried out during the prototyping stage
for the HOM spectra, possible trapped modes, and expected HOM power.
Codes exist and procedures have been well established for electron
applications. Our baseline device is a loop type coupler (Fig.~\ref{RF:fg18.H7})
because it is demountable, compact, has relaxed mechanical tolerances,
and provides demonstrated performance in mode damping~\cite{RF:ref18.H10}.  
The plane of
the loop is orthogonal to the beam axis.  The loop couples mainly to
the magnetic field of dipole modes and mainly to the electric field of
longitudinal modes. The rejection filter is formed by the inductance
of the loop and the capacitance between the loop end and the outer
conductor.  The loop is capacitively coupled to the external load
via a type-N connector, and is cooled by conduction through an
upper stub.  Final tuning of the filter can be carried out outside the
clean room once the coupler is attached and the cavity sealed.  $Q_{ext}$
values are typically $10^3$ to $10^5$ for high impedance modes in a 9-cell
TESLA cavity.  These $Q$ value will be even lower for the 2-cell cavities.

Power tests carried out for TESLA cavities under CW operating conditions showed good
thermal behavior up to an accelerating field of 21~MV/m.
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=3.5in]{../template/report/ps-and-eps/rf_fg26.eps}
\caption{TESLA-type HOM coupler.}
\label{RF:fg18.H7}
\end{center}
\end{figure}
\subsection{Tuner}
The function of the tuner is to match the cavity resonance frequency
to the desired accelerator operating frequency.  If the cavity is
not being used for acceleration, the tuner must detune the cavity
frequency a few bandwidths away from resonance, so that the beam will
not excite the fundamental mode. During accelerator operation, the
tuner must correct for slow changes in the cavity frequency due to
changes in the liquid-helium bath-pressure, or in the lengths of the
cavity and He vessel support system. Tuning is achieved by varying the
total length of the cavity within its elastic limit, so that the field
flatness is preserved.  The tuning coefficient of a 2-cell cavity is of
the order of 50~Hz/mm. (Plunger tuners are not advisable in
superconducting cavities because of moving parts and the danger of
dust.)

With a mechanical tuner, the length of the cavity is
controlled by an electromechanical system acting differentially with
respect to the cavity body.  If each cavity is enclosed in its own
helium vessel, the latter must have some flexibility built in.

A mechanical tuning system is generally composed of a stepping motor,
a gear box, a screw and nut assembly, and a lever arm with a flex
mechanism attached. A fast piezoelectric element can be added for
fine tuning, compensating Lorentz force detuning, Fig.~\ref{RF:fg18.H8}, as
well as microphonics.  Figure~\ref{RF:fg18.H9} shows the lowest frequency
vibrational mode of the two-cell cavity that could be excited by
vibrations. The stiffness will be increased to raise the frequencies
of this and other mechanical modes.

\begin{figure}[!hbt]
\begin{center}
\includegraphics[width=4in]{../template/report/ps-and-eps/rf_fg28.eps}
\caption{Lowest vibrational mode of 2-cell cavity.}
\label{RF:fg18.H9}
\end{center}
\end{figure}
Alternatively a thermal tuner could be considered, modeled after the
LEP system~\cite{RF:ref18.H11}.  This uses three Ni tubes as tuner bars located
in the cryomodule insulation vacuum.  The tuner rib cage can also help
increase the mechanical resonant frequency of cavity longitudinal modes.  For
slow tuning in one direction (constriction) the temperature of the
tubes is lowered by flowing cold helium gas.  For tuning in the
opposite direction the temperature is raised by centrally located
electric heaters.  The typical tuning speed is 10 Hz/sec.  Heat losses
are minimized by counter-flowing cold He gas through the tuner tubes.
For fast tuning, coils can be wound around the Ni tubes to produce a
magnetic field that changes the length of the tubes by the
magnetostrictive effect.  Rapid (ms) tuning ranges of kHz are
possible.
    
Tuners are an active part of the complete rf low-level control system, which
stabilizes the frequency, amplitude and phase variations induced by
sources such as the rf drive, beam current variations, Lorentz force
detuning, and microphonics.  
\subsection{Cryogenics for SCRF}
Figure~\ref{RF:fgH10} shows a 3D CAD model of the long cryomodule with four
2-cell units and a focusing magnet.  Each cavity has two input couplers,
one on each end, and two HOM couplers, also one on each end.  Mature
cryomodule designs (see Fig.~\ref{RF:fgH11}), available at CERN for LEP-II
and LHC will be adapted to the Neutrino Factory needs.  Based on scaling from 
 LEP 12.5-m long cryomodules, 4.5 K static
heat leaks of 100~W per cryomodule are expected. Thin beryllium
windows will be placed on the beam line at each end of the cryomodule
to protect the cavity vacuum and to keep the cavity surfaces clean
during installation into the beam line. Tables~\ref{RF:tbH3}, ~\ref{RF:tbH4} and ~\ref{RF:tbH5} give cryomodule parameters. Table~\ref{RF:tbH6} gives a summary for the total
SCRF requirements. The hardware implementation of the refrigerator is
described in Chapter~\ref{CHAP:cryo}.

%Figure 18.H10. Long cryomodule
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=\textwidth]{../template/report/ps-and-eps/rf_fg29.eps}
\caption{Long cryomodule.}
\label{RF:fgH10}
\end{center}
\end{figure}


%Figure 18.H11. LEP cryomodule
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=3in]{../template/report/ps-and-eps/rf_fg30.eps}
\caption{LEP cryomodule.}
\label{RF:fgH11}
\end{center}
\end{figure}

%Table18H3. Short Cryomodule Parameters
\begin{table}[hbt!]
\begin{center}
\caption{Short cryomodule parameters.}
\label{RF:tbH3}
\begin{tabular}{|lc|}
\hline
No. of cryomodules (in linac)&     11\\
No. of 2-cell cavities in one cryomodule &    1\\
No. of input couplers&  2\\
Overall length (m)& 5\\
Active length (m)&  1.5\\
Cavity dynamic heat load at 2.5~K (W)&        18 \\
Couper dynamic heat load at 2.5~K (W)&        1 \\
Coupler static heat load at 2.5~K, 5--8~K, 40--80~K (W)& 2, 4, 40\\
Cryomodule static heat load at  2.5~K, 5--8~K, 40--80~K (W)& 6, 60,600 \\
\hline  
Total 11 cryomodule heat load @ 2.5~K, 5--8~K, 40--80~K (W) &  300, 700, 7000 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[htb!]
\begin{center}
\caption{Intermediate cryomodule parameters.}
\label{RF:tbH4}
\begin{tabular}{|lc|}
\hline
No. of cryomodules (in linac)&     16\\
No. of 2-cell cavities in one cryomodule &     2\\
No. of input couplers&  4\\
Overall length& 8 m\\
Active length&  3 m\\
Cavity dynamic heat load at 2.5~K (W)&        36\\
Couper dynamic heat load at 2.5~K (W)&        2 \\
Coupler static heat load at 2.5~K, 5--8~K, 40--80~K (W)& 4, 8, 80 \\
Cryomodule static heat load at  2.5~K, 5--8~K, 40--80~K (W)& 7, 70, 700 \\
\hline
Total 16 cryomodule heat load at 2.5, 5--8~K, 40--80~K & 790, 1250, 12,500 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[hbt!]
\begin{center}
\caption{Long cryomodule parameters.}
\label{RF:tbH5}
\begin{tabular}{|lc|}
\hline
No. of cryomodules (19 linac + 48 RLA)&     67\\
No. of 2-cell cavities in one cryomodule &  4\\
No. of input couplers&  8\\
Overall length (m)& 13 \\
Active length (m)&  6 \\
Cavity dynamic heat load at 2.5~K (W)& $4\times 19 = 76$ \\
Couper dynamic heat load at 2.5~K (W)& $8 \times 0.5 = 4$ \\
Coupler static heat load at 2.5~K, 5--8~K, 40--80~K (W)&  8, 16, 160 \\
Cryomodule static heat load at 2.5~K, 5--8~K, 40--80~K (W)&   10, 100, 1000  \\
\hline
Total 67--cryomodule heat load at 2.5~K, 5--8~K, 40--80~K (kW)&6.6,  7.8, 78\\
\hline
\end{tabular}
\end{center}
\end{table}


\begin{table}[htb!]
\begin{center}
\caption{SCRF overall parameters for a Neutrino Factory.}
\label{RF:tbH6}
\begin{tabular}{|lc|}
\hline
No. of cryomodules  &   94\\
No. of 2-cell cavities  &       311\\
No. of input couplers & 622\\
Overall length (m)&        1054\\
Active length (m)& 467 \\
Filling factor  &       0.44\\
Total voltage (GV) & 7.5\\
Average real estate gradient (MV/m) &  7.8 \\
Total heat load at 2.5~K, 5--8~K, 40--80~K (kW)&  7.7, 9.7, 94 \\
Cryo load (with $\times 1.5$ safety factor) 2.5~K, 5--8~K, 40--80~K (kW) & 11.6, 14.6, 141\\
  Assuming efficiency multipliers of 600, 225 , 20 &\\ 
AC power for refrigeration (MW)&    13\\
Total peak rf power (with 20\% margin for control/losses)(MW) &362\\
Average rf power (MW)  &     16.3\\
AC power for rf (efficiency multiplier = 2) (MW) &   35.6\\
\hline
Total AC power (MW)&        49 \\
\hline
\end{tabular}
\end{center}
\end{table}

\subsection{Power Source for SCRF}
The superconducting linac and recirculating linear accelerator (RLA)
designs employ a total of 311 cavities. The linac contains 119
cavities running at a gradient of up to 17~MV/m. (The early part of the linac
operates at a gradient of 15 MV/m). The rf pulse length is 3~ms and the average
repetition rate is 15 Hz, although the recovery time between pulses is
only 20~ms. Each cavity is driven by two 500~kW couplers. With a 20\% rf
power overhead, this works out to 1.2~MW per cavity and a total rf
requirement of 375~MW.

Examination of the average power requirement demonstrates that a
very high efficiency source is required. The best candidate for the
required source is again a multi-beam klystron (MBK). This could be
the same basic design as that used for the NCRF system but with increased
thermal capacity to handle the increased duty factor. Thompson has 
developed a 7-beam MBK with an efficiency near 70\%, see
Section~\ref{RF:sec18.3.1} and reference~\cite{RF:ref18.1A}. Scaling this design to 201.25 MHz
would produce a 5 MW, 19 beam MBK with each beam having a perveance of
$5\times 10^{-5}.$ The tube, with gun and collector, would be about 5.7 m in length
and could be manufactured by industry after some initial R\&D. A 37-beam 
MBK has also been developed by other groups~\cite{RF:ref18.3}.

Each tube will drive four cavities through 8-way power splitters. The
specifications for the modulator, an IGBT-type like that described in Section~\ref{RF:sec18.3.3} for the NCRF, are 50~kV at 142~A, with an average power of 
320~kW. To save costs, the modulator
will be designed to operate two 5~MW tubes, requiring twice the current
and average power rating. A total 0f 78~tubes and 39~modulators are
required to supply the rf power requirements. The multi-beam klystron
may be designed with a vapor-cooled collector to save on the cooling
water requirement.
 Such a system is at least ten times more efficient than conventional
 water cooling. With vapor cooling, each tube will require 120~gpm of
 near-room-temperature cooling water with a total installed capacity of
 9,360~gpm. Assuming an efficiency of 95\%, each modulator station will
 require 694 kW of installed AC power for a total of 27~MW.


\section{Conclusions}   
The normal conducting and superconducting systems have
        continued to evolve since Study-I. Both cavity designs have
        been studied in some detail and feasible solutions have been
        developed for the required cooling channel and acceleration
        parameters. Ongoing R\&D programs are addressing the practical
        aspects of cavity fabrication and conditioning. The cooling channel
        layout, though densely packed, has shown the feasibility of
        assembling all of the vital components. There is room for
        further optimization of the cooling channel, notably by
        adjusting the total cell length, to reduce the rf power
        requirements and reduce the superconducting magnet
        costs. The superconducting rf accelerating section has been
        developed using design choices that are consistent with the
        state of the art at various laboratories around the world.



\begin{thebibliography}{9}
\bibitem{RF:ref18.4}Daryl W. Sprehn, \textsl{MBK Calculations}, Feb 26, 2001; N.~Holtkamp and D.~Finley, eds.,
{\sl A Feasibility Study of a Neutrino Source Based on   a Muon Storage Ring}, 
   Fermilab-Pub-00/108-E (2000), Chapter 10, p. 10-17, and 19.
 \hfill\break
http://www.fnal.gov/projects/muon\_collider/nu-factory/nu-factory.html

\bibitem{RF:ref18.1A}  A. Beunas, G. Faillon, THOMSON TTE, France, S. Choroba, A. Gamp, DESY, Germany \textsl{A High Efficiency Long Pulse Multi Beam Klystron For The Tesla Linear Collider}, Submitted to PAC 2001.

\bibitem{RF:ref18.1} R.A. Rimmer, N. Hartman, D. Li, Al. Moretti, T. Jurgens \textsl{Closed-Cell 201.25 MHz rf Structures for a Muon Cooling Channel},  submitted to PAC 2001.

\bibitem{RF:ref18.2} N. Hartman, D. Li, J. Corlett, \textsl{Thin Beryllium Windows - Analysis and Design Status},\\
http://www-mucool.fnal.gov/mcnotes/muc0180.pdf.

\bibitem{RF:ref18.3}A. Larionov, International Workshop on Pulsed RF Sources For Linear Colliders (RF93), July 5-9, 1993, Dubna-Protvino, Russia.

\bibitem{RF:ref18.H1}H.~Padamsee, J.~Knobloch and T.~Hays, \textsl{RF Superconductivity for Accelerators}, John Wiley and Sons, 1998.

\bibitem{RF:ref18.H2}P.~Brown et al., \textsl{Proceedings of the 9th Workshop on rf Superconductivity} (1999) ed. by B.~Rusnak, p. 1.

\bibitem{RF:ref18.H3}D.~Proch, \textsl{Proceedings of the 9th Workshop on rf Superconductivity} (1999) ed. by B.~Rusnak, p. 19 and D.~Trines.,\textsl{ibid.}, p. 605.

\bibitem{RF:ref18.H4}E.~Chiaveri, \textsl{Proceedings of the 1996 European Particle Accelerator Conference}, ed., S.Myers et al., Barcelona, Spain, IOPP Publishing, Bristol, p. 200 (1996).

\bibitem{RF:ref18.H5}E. Chiaveri, \textsl{Proceedings of the 9th Workshop on rf
Superconductivity} (1999) ed. by B.~Rusnak, p. 352.
and S.~Bauer et al., \textsl{ibid.} p. 437.

\bibitem{RF:ref18.H6}C. Reece, \textsl{Proceedings of the 8th Workshop on rf Superconductivity}, ed. by V.~Palmieri and A.~Lombardi (1998) p. 138.

\bibitem{RF:ref18.H7}T.~Furuya et al., \textsl{Proceedings of the 9th Workshop on rf Superconductivity} (1999) ed. by B.~Rusnak, p. 31.T; and
S. Mitsunobu, \textsl{ibid.} p. 505.

\bibitem{RF:ref18.H8}S.~Simrock et al., \textsl{Proceedings of the 9th Workshop on rf Superconductivity} (1999) ed. by B.~Rusnak, p. 92.

\bibitem{RF:ref18.H9}E.~Somersalo et al., \textsl{Proceedings of the 1995 Particle Accelerator Conference}(1996), Dallas Texas, p. 1623.

\bibitem{RF:ref18.H10}J. Sekutowicz, TESLA Note, 94-07 (1994).

\bibitem{RF:ref18.H11}G.~Cavallari et al., \textsl{Proceedings of the 3rd Workshop on rf Superconductivity} (1988) ed. by K.~Shepard, p. 625.
\end{thebibliography}




