Rapid Acceleration in an FFAG using High-Frequency RF
C. Johnstone * and S. Koscielniakt
* Fermi National Accelerator Laboratory, P.O. Box 500, MS 220, Batavia, IL, 60510
^TRIUMF, Vancouver, B.C., V6T2A3 Canada
Abstract. When large transverse and longitudinal emittances are to be transported through a circular machine,
extremely rapid acceleration holds the advantage that the beam becomes immune to nonlinear resonances because
there is insufficient time for amplitudes to build up. Uncooled muon beams exhibit large emittances and require
fast acceleration to avoid decay losses and would benefit from this style of acceleration. The approach here
employs a fixed-field alternating gradient or FFAG magnet structure and a fixed frequency acceleration system.
Acceptance is enhanced by the use only of linear lattice elements, and fixed-frequency rf enables the use of
cavities with large shunt resistance and quality factor.
INTRODUCTION
FFAG LATTICES
A circular accelerator system can be designed with magnetic fields that remain constant during acceleration by
adopting an alternating gradient focussing lattice. The
arcs of such machines, composed of large aperture magnets, can be designed to accommodate the large energy
range in acceleration. The beam centroid orbit is not
fixed as in a ramped machine, but rather moves across the
magnet aperture during acceleration. Lattices have been
developed which can contain up to an energy change of
afactoroffour[6][7][8].
In the nonscaling FFAG, not only do the central orbits move across the aperture, but also the optics functions vary with the central momentum. When acceleration occurs so rapidly that the beam experiences only
a few turns in the machine one does not have to avoid
resonances or control lattice parameters as a function of
momentum. Instead, one has the freedom to choose parameters optimal for the application such as minimizing
circumference and supporting a large transverse dynamic
aperture[2].
The example, which provides the focus of this paper,
is a 6-20 GeV non-scaling FFAG cell optimized for ultrarapid, stable acceleration, as is required for intense muon
sources. The entire lattice is comprised solely from 314
simple arc FODO cells. (The periodicity is, therefore,
314). The rf system is assumed distributed over most of
the ring with 3 cavities filling the 3 m straight in each
half cell. (To keep field gradients to conventional values,
« 3MV/m, the rf needs to occupy about 300 of the cells,
given a few number of acceleration turns, although much
higher-gradient cavities are currently under construction
and testing.) The remaining empty cells are filled with
the kickers necessary for injection and extraction, processes which require several cells to complete. The simplicity of lattice design; i.e. its unbroken periodic structure, is responsible for its stability over a large range
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
Acceleration of large emittance beams, particularly those
with simultaneously large transverse and longitudinal
emittances, present a challenging new direction in accelerator design. Conventional accelerators such as synchrotrons or linacs cannot support acceleration of ultralarge emittances for either case. Scaling FFAG accelerators, such as the radial or spiral sector[l], display an
almost unlimited momentum acceptance, but transverse
acceptance is sensitive to design and is generally restricted. The approach described here is development of
a nonscaling FFAG[2] wherein the ideal optics demonstrate strong linearity; that is, the acceptance is not sensitive to energy or amplitude. The discerning feature of
the nonscaling version, in keeping with the condition of
linearity, is that the optics are not held constant, as in the
scaling machine, but change slowly with momentum.
However, a signature of fixed field acceleration is that
orbit length unavoidably changes with energy; it can
be substantial and can result in a significant phase-slip
relative to the accelerating waveform[2] [3]. For relativistic beams and rapid acceleration, this poses a nonstandard problem which must be addressed by the rf
system. Recent workshops have focussed on the phasing problems of FFAGs and a number of solutions
are being advanced[4]. These include: (i) momentumdependent chicanes[5] to correct pathlength differences,
(ii) broad-band RF that can be phased quickly, or (iii)
a frequency low enough (25 MHz, for example), to
make the phase errors ineffectual. Outside of the obvious solutions of broadband rf, or very low frequency rf,
this paper outlines alternative approaches using high-Q,
high-frequency rf systems. This paper reports significant
progress on both a lattice and rf acceleration system for
a high-energy FFAG in the context of rapid acceleration.
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
207
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208
(hundreds of microseconds) can be achieved in either a
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When acceleration is to be completed in a few turns,
the large energy gain per turn forces one to consider
on-crest acceleration as in a cyclotron. In this case, the
rf is used almost entirely to provide acceleration. One
may envisage use of low-Q (< 103) cavities and on-crest
operation with phase shifting of accelerating stations
to make up for non-isochronous behaviour of orbits in
the FFAG. Even when the phases of the cavities are
readjusted, this regime can be sustained only for a few
tens of turns, due to the near absence of phase focusing.
A more affordable approach for rapid acceleration employs high-Q (> 106) and high frequency (> 100 MHz).
Because high-Q cavities cannot be rephased on a rapid
acceleration timescale (microsecond), the phase relation
of the beam to the acceleration waveform is completely
determined by the changes in the orbital pathlength during acceleration, and these changes are substantial at
high cavity frequency. (For the example in this paper,
the total pathlength changes by w 50 cm which is 1/3
the wavelength at 200 MHz). Thus acceleration does not
remain fixed at the crest of the rf waveform, but rather
crosses over it one or more times. The nonscaling FFAG
demonstrates at least two operational modes for rapid
acceleration, termed cross-crest and near-crest acceleration, with the difference being a common initial phase
versus independently-optimized starting phases for individual cavities.
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FIGURE 5. Circumference change as a function of momenturn
length as a function of energy. The pathlength dependence is clearly linear with momentum for radiallystaggered, parallel orbits as in the radial-sector FFAG,
but it is parabolic in nonscaling FFAGs. (The following
discussion of scaling FFAGs refers to the radial-sector
FFAG explicitly.)
Pathlength and therefore traversal time change with
the reference energy. In a regime of slow acceleration
this variation is distributed over many turns, but for rapid
acceleration non-negligible changes occur from cell to
cell. Of course, the cell traversal times must be synchronized with the waveforms in the RF cavities responsible
for acceleration. This consideration leads to the following discussion of acceleration modes and the rf systems
that are compatible with them.
FFAG RF system
The present work investigates the simplest approach:
the application and optimization of a single highfrequency, high-Q rf system to operate in the near-crest
regime', since this allows lower voltage and may furnish
a less distorted phase space than cross-crest regime
when an equal number of turns are employed. Modest
elaborations, such as second harmonic or a mixing of
several fundamental frequencies to produce a waveform
that better matches the beam traversal times, are also
considered.
There are CERN[9,10] designs available for 200 MHz
normal T conducting (NC) and super conducting (SC)
cavities, and for 400 MHz SC cavities that provide
a starting point for extrapolation. A design with R =
14 Mohm, Q = 1 x 104 and 2 MV gap voltage is within
reach of present NC technology and the peak rf power is
some 250 MW distributed between 1800 cavities. However, the filling time of 350 jus is too long to allow
rephasing. The modulation could be achieved by vec-
FFAG ACCELERATION
In a circular machine, particles make repeated passages
through the same cavities; and so on every revolution of
the machine the frequency and phasing of each cavity
must be readjusted. To make this possible, the demanded
phase change per cavity filling time should be small; and
this leads to the condition 2n(AC/C) < l/Q where Q
is the quality factor and AC/C is the relative fractional
change in circumference per turn. If no attempt is made
to adjust the cavity frequency or phase, then errors accumulate linearly with time and inversely as harmonic
number. A fixed-field machine can be operated in several
modes, classifiable in terms of the timescale for acceleration; for example, conventional and rapid, which range
over tens of milliseconds and tens of microseconds, respectively. Between these extremes, ies moderate acceleration: theoretically, a few hundred acceleration turns
209
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Unfortunately,
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NC
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NC
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NC
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benefit
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77 to 10999. In ei7
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There
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There
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There
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bunch
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The
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demonstrated
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demonstrated
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demonstrated
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initial
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different,
initial
cavity
phases,
then
at
least
5-turns
of
different,
initial
cavity
phases,
then
at least
5-turns
different,
initial
cavity
phases,
then
at least
5-turns
of
acceleration
can
besustained
sustained
with
modest
over-voltage.
acceleration
can
be
sustained
with
modest
over-voltage.
acceleration
can
be
sustained
with
a modest
over-voltage.
acceleration
can
be
with
aaamodest
over-voltage.
(The
amount
of
over-voltage
represents
the
increase
inrf
(The
amount
of
over-voltage
represents
the
increase
in
rf rf
amount
of over-voltage
represents
increase
in
(The(The
amount
of over-voltage
represents
thethe
increase
in
voltage
required
relative
to
pure
on-crest
acceleration.)
voltage
required
relative
to
pure
on-crest
acceleration.)
A
voltage
required
relative
to pure
on-crest
acceleration.)
voltage
required
relative
to pure
on-crest
acceleration.)
AAA
25%
over-voltage
yields
a
1.5
eV.s
admittance
(Fig.
7).
If
25%
over-voltage
yields
a
1.5
eV.s
admittance
(Fig.
7).
over-voltage
yields
a 1.5
admittance
(Fig.
25%25%
over-voltage
yields
a 1.5
eV.seV.s
admittance
(Fig.
7).7).
IfIf If
one
adds
2nd
harmonic
(Fig.
8),
then
one
needs
(4/3)
x
one
adds
2nd
harmonic
(Fig.
8),
then
one
needs
(4/3)
×
harmonic
(Fig.
then
needs
(4/3)
oneone
addsadds
2nd2nd
harmonic
(Fig.
8), 8),
then
oneone
needs
(4/3)
××
1.25
of
the
nominal
voltage,
and
the
admittance
rises
1.25
of
the
nominal
voltage,
and
the
admittance
rises
to
of the
nominal
voltage,
admittance
rises
1.251.25
of the
nominal
voltage,
andand
thethe
admittance
rises
to to
2.28
eV.s.
The
input
admittances
areare
similar
inin
shape
2.28
eV.s.
The
input
admittances
are
similar
in
shape
to to
2.28
eV.s.
The
input
admittances
similar
shape
2.28
eV.s.
The
input
admittances
are
similar
in
shape
to
that
inin
figure
6 except
except
that
their
time-width
is ishalved.
halved.
that
in
figure
6
that
their
time-width
is
that
figure
6
except
that
their
time-width
halved.
that in figure 6 except that their time-width is halved.
Again,
when
considering
the
entire
region,
thethe
transport
Again,
when
considering
the
entire
region,
the
transport
Again,
when
considering
the
entire
region,
transport
Again,
when
considering
the
entire
region,
the
transport
is
nonlinear,
but
the
emittance
which
corresponds
toto
thethe
is
nonlinear,
but
the
emittance
which
corresponds
to
the
is
nonlinear,
but
the
emittance
which
corresponds
is central
nonlinear,
butisthe
emittancewell
which
corresponds to
the
region
reasonably
conserved.
central
region
reasonably
well
conserved.
central
region
is reasonably
well
conserved.
central
region
isisreasonably
well
conserved.
8. Output
emittance,
dual
harmonic.
FIGURE
Output
emittance,
dual
harmonic.
FIGURE
Output
emittance,
dual
harmonic.
FIGURE
8. 8.Output
emittance,
dual
harmonic.
ofof"best
initial
using
“best
phases”which
are
using
acombination
combinationof
“bestinitial
initialphases"which
phases”whichare
are
using
aa combination
“best
phases”which
are
cavity
basis
and
an
overoptimized
on
an
individual
cavity
basis
and
an
overoptimized
on
an
individual
cavity
basis
and
an
overoptimized on an individual cavity basis and an overvoltage.
When
second
harmonic
is employed,
the
useful
employed,
the
useful
voltage.
When
second
harmonic
employed,
the
useful
voltage.
When
second
harmonic
is is
employed,
the
useful
acceptance
is
greater
when
the
"best
phases"
scenario
isisis
“best
phases”
scenario
acceptance
is
greater
when
the
“best
phases”
scenario
acceptance is greater when the “best phases” scenario is
adopted,
and
is
typically
doubled.
In
all
cases,
the
overInIn
allallcases,
cases,
the
overadopted,
typically
doubled.
cases,the
theoveroveradopted,
andand
is is
typically
doubled.
In
all
allalltransport
is isnon-linear
with
thetheuseful,
or
preserved,
useful,
or
preserved,
transport
non-linear
with
useful,
or
preserved,
all transport is non-linear with the useful, or preserved,
phase
space
area
comprising
roughly
one
half
of
the
full
phase
space
area
comprising
roughly
one
half
ofof
the
full
phase
space
area
comprising
roughly
one
half
the
full
phase
space
area
comprising
roughly
one
half
of
the
full
admittance.
However,
since
most
beams
do
not
approach
admittance.
However,
since
most
beams
do
not
approach
admittance.
However,
since
most
beams
do
not
approach
admittance.
However, since
most beams do not transport
approach
such
large
longitudinal
emittances,
such
large
longitudinal
emittances,
nonlinear
transport
such
large
longitudinal
emittances,nonlinear
nonlineartransport
transport
such
large
longitudinal
emittances,
nonlinear
issues
do
not
pose
serious
concerns
with
the
operation
issues
do
not
pose
serious
concerns
with
the
operation
issues
do
not
pose
serious
concernswith
withthetheoperation
operation
issues
do
not
pose
serious
concerns
ofof
these
machines
in
a rapid
acceleration
mode.
of
these
machines
inin
rapid
acceleration
mode.
these
machines
a rapid
acceleration
mode.
of
these
machines
in
aa rapid
acceleration
mode.
Acceleration
with
100
MHz
Acceleration
with
100
MHz
RF
Accelerationwith
with100
100MHz
MHzRF
RF
Acceleration
RF
i"
&..
Since
thethe
phase
slips
accumulate
half
as
quickly
when
Since
the
phase
slips
accumulate
half
asas
quickly
when
Since
phase
slips
accumulate
half
quickly
when
the
phase
slips
accumulate
half
asMHz
quickly
when
theSince
RF
is
halved,
one
anticipates
that
100
accelerathe
RF
is
halved,
one
anticipates
that
100
MHz
accelerathe
RF
is
halved,
one
anticipates
that
100
MHz
accelerathe
RF
is halved,
one anticipates
that
100a MHz
acceleration
will
bebe
less
compromised
by
using
number
tion
will
be
less
compromised
byby
using
a larger
number
tion
will
less
compromised
using
alarger
larger
number
tion
will
be
less
compromised
bythe
using
aMHz
larger
number
of
turns.
Based
on
the
results
of
200
case
studofof
turns.
Based
onon
thetheresults
studturns.
Based
resultsofofthe
the200
200MHz
MHzcase
case
studof
turns.
Based
on the results
of the
200
MHz frequency
case studies,
we
have
considered
only
the
use
of
a
"best
ies,
wewe
have
considered
only
thethe
use
ies,
have
considered
only
useofofa “best
a “bestfrequency
frequency
ies,
we
have considered
only
the
use ofgeneral
a “besttrends
frequency
and
phases"
forfor100
100
MHz
rf.
Certain
can
1.1
and
phases”
for
rf.rf.
Certain
general
can
and
phases”
100MHz
MHz
Certain
generaltrends
trends
can
and
phases”
for
100admittance
MHz rf. Certain
general
trendsfrom
can
be
noted.
The
input
rises
almost
linearly
bebe
noted.
The
input
admittance
rises
noted.
The
input
admittance
risesalmost
almostlinearly
linearlyfrom
from
be
noted.
The
input
admittance
rises
almost
linearly from
from
0.20.2
toto4.1
4.14.1
eV.s
asasthe
the
FIGURE
±10%
Band
from
input
acceptance
0.2
to
eV.s
number
ofofturns
turns
is isreduced
reduced
eV.sas
thenumber
numberof
turnsis
reducedfrom
from
FIGURE
6.6. 6.±10%
Band
from
input
acceptance
FIGURE
±10%
Band
from
input
acceptance
0.2
to 4.1toeV.s
as
thethenumber
of turns isisreduced
from
fourteen
six
and
voltage-per-turn
raised
from
FIGURE 6. ±10% Band from input acceptance
fourteen
to
six
and
the
voltage-per-turn
is
raised
from
fourteen
to
six
and
the
voltage-per-turn
is
raised
from
Conclusion:
For
5-turn
acceleration,
there
little
difto six
and
the
voltage-per-turn
is raised from
1.25
toto
2.92
GV.
For
example,
10-turn
acceleration
with
Conclusion:
For
5-turn
acceleration,
there
little
dif- fourteen
Conclusion:
For
5-turn
acceleration,
there
isisis
little
dif1.25
2.92
GV.
For
example,
10-turn
acceleration
with
1.25
to
2.92
GV.
For
example,
10-turn
acceleration
with
Conclusion:
For
5-turnemittance
acceleration,
there conventional
is little difference
the
output
between
to 2.92
GV.over-voltage
For example,
10-turn acceleration
with
a modest
modest
25%
super-posed
on
the
nominal
ference
the
output
emittancebetween
betweenconventional
conventional 1.25
ference
ininin
the
output
emittance
a
modest
25%
over-voltage
super-posed
on
the
nominal
a
25%
over-voltage
super-posed
on
the
nominal
ference
in the where
outputcavity
emittance
between
conventional
acceleration
phases
are
adjusted,
versus
modestof25%
over-voltage
super-posed
on the nominal
voltage
1.4
GV/turn,
yields
eV.s
acceptance;
see
acceleration
where
cavityphases
phasesare
areadjusted,
adjusted,versus
versus avoltage
acceleration
where
cavity
voltage
1.4
GV/turn,
yieldsaa 2.3
a 2.3
eV.s
acceptance;
see
ofof
1.4
GV/turn,
yields
2.3
eV.s
acceptance;
see
acceleration where cavity phases are adjusted, versus
voltage of 1.4 GV/turn, yields a 2.3 eV.s acceptance; see
210
figures 9 and 10. If one adds second harmonic, the adfigures
one
second
the
adfigures9rises
9and
andto10.
10.
one adds
adds
second
harmonic,
admittance
4.2IfIfeV.s.
Varying
the harmonic,
number ofthe
rf stamittance
rises
to
4.2
eV.s.
Varying
the
number
of
rf
stamittance
4.2 eV.s.
the influence;
number of admitrf stations
from rises
100 to 600
againVarying
has little
tions
100
has
admittionsfrom
fromfrom
100 to
to 600
600
again
has little
little influence;
influence; admittances
vary
2.22
toagain
2.33 eV.s.
tances
tancesvary
varyfrom
from2.22
2.22to
to2.33
2.33 eV.s.
eV.s.
FIGURE
FIGURE9.9.9. ±10%
±10%Band
Bandfrom
frominput
input acceptance.
acceptance.
FIGURE
±10%
Band
from
input
acceptance.
HO?
FIGURE10.
10. Maps
Maps to
to the
the output
output emittance
emittance
FIGURE
FIGURE
10. Maps
to the
output emittance
Conclusion:For
Foracceleration
acceleration with
with 100
100 MHz
Conclusion:
MHz RF
RF using
using
Conclusion:
For acceleration
with
100 MHz
RF
using
a
single
frequency
and
fixed
“best
phases”,
the
optimum
singlefrequency
frequency and
andfixed
fixed “best
"best phases”,
phases", the
optimum
a asingle
the
optimum
admittance of the machine is achieved when the numadmittance
the machine
machine isis achieved
achieved when the
admittance
ofofappears
the
the numnumber of turns
to be eight or ninewhen
combined
with
ber
of
turns
appears
to
be
eight
or
nine
combined
with
bermodest
of turns
appears
to
be
eight
or
nine
combined
with
overvoltages. However, admittances comparable
modest
overvoltages.
However,
admittances
comparable
modest
overvoltages.
However,
admittances
comparable
to the 200 MHz case can be achieved in 10,11 or even 12
the200
200MHz
MHzcase
case can
can be
be achieved
achieved in
or
tototurns
the
in 10,11
10,11decrease
oreven
even12
12
with a corresponding
and advantageous
in
turns with a corresponding and advantageous decrease in
turns
with a corresponding
anduse
advantageous
in
rf gradient.
For 100 MHz rf,
of the seconddecrease
harmonic
rf gradient. For 100 MHz rf, use of the second harmonic
combination
withMHz
an overvoltage
andsecond
best phasing
prorf in
gradient.
For 100
rf, use of the
harmonic
in combination with an overvoltage and best phasing proa tremendous
in a FFAG
in duces
combination
with anlongitudinal
overvoltageacceptance
and best phasing
produces
a tremendous longitudinal
acceptance in a FFAG
for aarapid-acceleration
application.
duces
tremendous
longitudinal
acceptance
in
a
FFAG
for a rapid-acceleration application.
for a rapid-acceleration application.
SUMMARY
SUMMARY
SUMMARY
Nonscaling FFAGs have a strong advantage in appliNonscaling
FFAGsrapid
haveacceleration
a strong advantage
in applications requiring
of large-emittance
Nonscaling
FFAGsrapid
haveacceleration
a strong advantage
in applications
of large-emittance
beams requiring
by providing
a transverse admittance
beyond
cations
requiring
rapida acceleration
of large-emittance
beams
by
providing
transverse
admittance
conventional scaling FFAGs. The magnet beyond
layout
beams
by providing
admittance
beyond
conventional
scaling a transverse
FFAGs.
The
layout
(horizontally-focussing
quadrupole
andmagnet
horizontallyconventional
FFAGs. magnet)
Theandmagnet
layout
(horizontally-focussing
quadrupole
horizontallydefocussing scaling
combined-function
with appro(horizontally-focussing
quadrupole
and
horizontallydefocussing
combined-function
magnet)
with
appropriate optimization represents an innovation in FFAG
defocussing
combined-function
with
appropriate
represents
anmagnet)
innovation
in
FFAG
latticeoptimization
design,
and has
been described
in detail
in earlier
priate
represents
an innovation
FFAG
latticeoptimization
design, and has
been described
in detail in earlier
lattice design, and has been described in detail in earlier
211
publications. This design approach produces the most
publications.
This
designlinear
approach
produces
thein
most
publications.
design
approach
produces
the
most
efficient andThis
compact
machine
design
terms
efficient
and
compact
linear
machine
design
in
terms
efficient
and compact
machine
design
in terms
of circumference
andlinear
magnet
aperture.
(Circumference
of
circumference
and magnet
magnet
aperture.
(Circumference
of
circumference
and
aperture.
(Circumference
reduction
approaches
a factor
of two
from scaling
reduction
approaches
a
factor
of
two
from
scaling
reduction
approaches
a factor
from
scaling
FFAG designs
developed
for ofthetwosame
application.)
FFAG
designs
developed
for
the
same
application.)
FFAG
designs
developed
for
the
same
application.)
The slowly-changing optics which characterize this
slowly-changing
optics
which
characterize
this
The
slowly-changing
optics
which
characterize
this
nonscaling
FFAG design
have
an additional
advantage
FFAG
design
have
an
additional
advantage
nonscaling
FFAG
design
have
an
additional
advantage
in the suppression of nonlinear resonances for rapid,
suppression of
of nonlinear
nonlinear resonances
resonances for
for rapid,
inor the
suppression
moderately-rapid
acceleration
applications. rapid,
At least
moderately-rapid acceleration
acceleration applications.
applications.At
Atleast
least
or
moderately-rapid
initially, it was felt that the circumference change,
or
initially, itit was
was felt
felt that
that the
the circumference
circumference change,
change,oror
initially,
phase-slip, posed a serious problem, but since then nuphase-slip, posed
posed aa serious
serious problem,
problem,but
butsince
sincethen
thennunuphase-slip,
merous solutions have been proposed. However, most of
solutions have
havebeen
beenproposed.
proposed.However,
However,most
mostofof
merous solutions
theprevious
previoussolutions
solutionshad
had
the
disadvantage
of
requiring
the
the
disadvantage
ofofrequiring
previous
solutions
had
the
disadvantage
requiring
impractical
rf
power
by
applying
very
low-frequency
impractical
impractical rf
rf power
power by
by applying
applyingvery
verylow-frequency
low-frequencyrf,rf, rf,
or
broadband
rf
to
accomodate
the
phase-slip.
This
work
or broadband
work
broadband rfrf to
to accomodate
accomodatethe
thephase-slip.
phase-slip.This
This
work
represents
the
first
successful
study
of
the
application
represents
represents the
the first
first successful
successful study
studyofofthe
theapplication
applicationofof of
reasonablyhigh-frequency
high-frequency
and
high-Q
cavities
to
rapid
reasonably
and
high-Q
cavities
high-frequency and high-Q cavitiestotorapid
rapid
acceleration
in
a
fixed-field
accelerator,
thereby
dramatiacceleration
in
a
fixed-field
accelerator,
thereby
dramatiacceleration in a fixed-field accelerator, thereby dramaticallyreducing
reducingthe
therfrfrf
power
required
in
previous
solutions
cally
power
required
ininprevious
solutions
reducing
the
power
required
previous
solutions
(thatis,
is,when
whenaaabunch
bunch
train
being
accelerated).
(that
train
isisis
being
accelerated).
is,
when
bunch
train
being
accelerated).
Rapid
adjustments
possible
inin in
Rapidacceleration
accelerationwith
withno
adjustments
possible
acceleration
with
nono
adjustments
possible
the
show
strict
operational
conthecavity
cavityphasing,
phasing,however,
however,
show
strict
operational
conthe
cavity
phasing,
however,
show
strict
operational
constraints
stablility
requirements
ininthe
outstraintsin
orderto
meet
stablility
requirements
in
outstraints
ininorder
order
totomeet
meet
stablility
requirements
thethe
output
distribution.
For
near-crest
putphase
phasespace
spaceand
andenergy
energy
distribution.
near-crest
put
phase
space
and
energy
distribution.
ForFor
near-crest
operation,
across
operation,which
whichdisplays
displaysstable
stableperformance
performance
across
operation,
which
displays
stable
performance
across
aaa large
longitudinal
machine
acceptance
three
factors
large
large longitudinal
longitudinalmachine
machineacceptance
acceptancethree
threefactors
factors
were
interrelated:
optimal
cavwere
found
totobe
bebecritical
critical
and
interrelated:
optimal
cavwerefound
foundto
criticaland
and
interrelated:
optimal
cavity
frequency
and
starting
phases,
overvoltages,
and
the
ity
and
thethe
ityfrequency
frequencyand
andstarting
startingphases,
phases,overvoltages,
overvoltages,
and
timescale
(the
total
number
ofofturns
inin in
timescale
for
acceleration
(the
total
number
turns
timescalefor
foracceleration
acceleration
(the
total
number
of
turns
the
machine).
Only
a
few
turns
are
supportable
in
the
the
in in
thethe
the machine).
machine).Only
Onlya afew
fewturns
turnsarearesupportable
supportable
context
of
aa large
and
practically
stable
machine
admitcontext
of
large
and
practically
stable
machine
admitcontext of a large and practically stable machine admittance
and, even
then,
itit isis only
achieved by
applying
tance
even
then,
only
byby
applying
tance and,
and,
even
then,on
itrfisstations
onlyachieved
achieved
applying
different
initial
phases
and
allowing
moddifferent
initial
phases
on
rf
stations
and
allowing
moddifferent
initial
phases
on
rf
stations
and
allowing
modest
overvoltages
(typically
20-40%).
Even
though
little
est
overvoltages
(typically
20-40%).
Even
though
little
est overvoltages
(typically
though
phase-space
increase
is evident20-40%).
in the finalEven
emittance,
thelittle
phase-space
increase
isisevident
the final
emittance,
the
phase-space
increase
evidentinin
final
emittance,
transport,
which
tends to
increase
thethe
momentum
spread the
transport,
which
tends
to
increase
the
momentum
spread
transport,
which
tendslength,
to increase
momentum spread
and
decrease
the bunch
is stillthe
nonconventional,
and
decrease
bunch
length,
nonconventional,
and
decreasethe
the
bunch
length,isisstill
Specifically,
in the
application
ofstill
200nonconventional,
and 100 MHz
Specifically,
in the
ofof200
and 100
MHz
Specifically,
theapplication
application
200
100
MHz
rf, the
number ofinturns
increases from
5
to and
10, respecrf,
the
number
of
turns
increases
from
5
to
10,
respecrf, theindicating
number the
of turns
increases
5 to 10,
tively,
expected
generalfrom
dependence
onrespecthe
tively, indicating the expected general dependence on the
inverse
the frequency.
In this approach,
a hard limit octively, of
indicating
the expected
general dependence
on the
inverse of the frequency. In this approach,
hard limit oc◦ from acrest
curs
when
exceeds
at which
inverse
ofthe
thephase-slip
frequency.
In this 90
approach,
a hard
limit occurs when
thefalloff
phase-slip
exceeds
90° from
crest at which
point
sharp
of stable
and useful
curs awhen
the phase-slip
exceeds
90◦ transmission
from crest atocwhich
point
a
sharp
falloff
of
stable
and
useful
transmission
occurs.
(One
can falloff
force an
increase
in the
number
of turns,
point
a
sharp
of
stable
and
useful
transmission
occurs.
(One
can
force
an
increase
in
the
number
of
turns,
but
at great
expense
in an
the increase
overvoltage
andnumber
at the cost
of
curs.
(One
can
force
in
the
of
turns,
but atgreater
great expense
in theofovervoltage
and at the
cost of
even
the phase space.)
Applying
but atgreater
great convolution
expense
in the
and at
the cost of
convolution
of overvoltage
the phase
space.)
Applying
aeven
second
harmonic
has
a
dramatic
effect
on
the
transmitgreater convolution
of the phase
space.)
Applying
aeven
second
dramatic
on
the transmitted
phase harmonic
space
area,has
buta had
only aeffect
minor
impact
on
the
a
second
harmonic
has
a
dramatic
effect
on
the
transmitted phase
number
ofspace
turns.area, but had only a minor impact on the
ted phase
area, but had only a minor impact on the
number
of space
turns.
In practical
terms, when 200 MHz RF is utilized, usenumber
of
turns.
practical terms,
whencan
200beMHz
RF iswith
utilized,
usefulIn
admittances
(≥ 1 eV.s)
achieved
accelerIn
practical terms,
when
200
MHz
RF is
utilized,
useful
admittances
(>
1
eV.s)
can
be
achieved
with
acceleration in 6 or less turns using either either a phase-agile
ful
admittances
(≥
1
eV.s)
can
be
achieved
with
acceleration
in 6 or fixed-frequency
less turns using RF
either
eitherand
a phase-agile
or
a high-Q
system
a modest
ation
in 6 orin
less
turns using
either
either
a aphase-agile
or
a high-Q
fixed-frequency
RF
system
andappears
modest
over-voltage
a FFAG.
Surprisingly,
there
to
or little
a high-Q
RF system
and
a cavmodest
over-voltage
infixed-frequency
a FFAG.
there
appears
to
be
advantage
in usingSurprisingly,
a phase-agile
or low-Q
over-voltage
in a in
FFAG.
appears
be
little advantage
using Surprisingly,
a phase-agile there
or low-Q
cav- to
be little advantage in using a phase-agile or low-Q cav-
ity to the overall performance when operating in the onor near-crest regime. For the case of 100 MHz RF, study
of a fixed frequency system, shows that a useful output
emittance of (> 2 eV.s) is realized for acceleration in 10
or less turns. In all cases, addition of second harmonic
roughly doubles the phase space conserved within the
defined cuts.
In conclusion, the nonscaling FFAG coupled with the
rf approach developed here presents not only a solution to rapid acceleration, but also a new acceleration
technique. Ultra-large emittances are successfully transported in a conventional machine with minimal complexity in components (when compared with the more
design-intense magnets required for a scaling FFAG).
The use of only linear fields form the basis for the
tremendous transverse and longitudinal acceptance, one
that is large in comparison with conventional acceleration, including traditional FFAG machines. With appropriate rf technique and design, this large acceptance was
found to be preserved despite the problems introduced by
phase-slippage. Specifically, this approach provides the
necessary transverse and longitudinal acceptance match
to high-energy muon beams with little or minimal cooling in place. Based on this work, it looks promising to
build a chain of muon accelerators from FFAGs and replace the costly and somewhat restrictive RLAs, which
so far have been the baseline accelerator for the feasibility studies of a Neutrino Factory in the U.S [11, 12].
KEK, Tsukuba, Japan, Oct. 11-13, 2000.
A. Garren, presented at the CERN FFAG Workshop,
Geneva, Switzerland, July3-4, 2000.
6. C. Johnstone, W. Wan, and A. Garren, 'Fixed Field
Circular Acceleration Design", Proceedings of the 1999
Particle Accelerator Conference, New York, NY, Mar.
29-Apr. 2, 1999, pp. 3068.
7. S. Machida, et al, 'Beam Optics Design of an FFAG
Synchrotron", (MOP1B20), submitted to EPAC 2000,
and Y. Sato, et al, 'Development of a FFAG Proton
Synchrotron", (MOP1B21), submitted to EPAC 2000.
8. K. Symon, MURA-KRS-6, (MURA-43) (1954); K.
Symon, et al, Phys. ReV. 103 p. 1837, (1956)..
9. D. Boussard et al: 'Design Considerations for the LHC
200 MHz RF System", LHC Project Report 368, January
2000.
10. E. Chiaveri & R. Losito: private communication.
11. N. Holtkamp and D. Finley, eds., A Feasibility Study
of a Neutrino Source Based on a Muon Storage Ring,
Fermilab-Pub-OO/108-E (2000).
12. S. Ozaki, R. Palmer, M. Zisman, and J. Gallardo,
eds., 'Feasibility Study-II of a Muon-based Neutrino
Source", BNL-52623, June, 2001, available at
http://www.cap.bnl.gov/mumu/studii/FS2-report.html.
5.
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Johnstone, W. Wan, and A. Garren, 'Fixed Field
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