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Muon Acceleration and FFAG
Shinji Machida
KEK
NuFact05 Summer Institute
June 12-20, 2005
Content
1.
2.
3.
4.
Acceleration of muons
Evolution of FFAG
FFAG as a muon accelerator
Design example of muon acceleration
•
Reference (among others):
– BNL-72369-2004, FNAL-TM-2259, LBNL-55478
– NuFactJ Design study report
1
Acceleration of muons
Requirement (1)
• Acceleration: as quick as possible
– Life time of muon is ~2.2 us.
– Example
• At momentum of 0.3 GeV/c
• Lorentz factor g~3, Velocity b~0.94.
• Flight path length ~2000 m
– That is even true on the lower momentum side.
Requirement (2)
• Acceptance: as large as possible
– Muons are produced as secondary particles of protons
– Cooling before acceleration if necessary
• Longitudinal emittance
– dp/p ~ +-100%
– dt or dx can be controlled by the width of primary proton:
~1 ns or 300 mm
– dp/p * dx * bg = 1000 mm at 0.3 GeV/c
• Transverse emittance
– 10 ~ 100 mm
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Machine candidate (1)
• Everyone knows modern high energy accelerator is
synchrotron. Why not for muons?
• VRCS (very rapid cycling synchrotron)
– Rapid (or fast) cycling means time required for acceleration
from injection to extraction is short.
– The most rapid cycling machine at the present is ISIS at RAL,
which has 50 Hz repetition rate. It still takes 10 ms to
complete a whole cycle.
ISIS
Rep. rate 50 Hz
J-PARC
booster
KEK-PS
booster
Fermilab
booster
AGS
booster
CPS
booster
25 Hz
20 Hz
15 Hz
~7.5 Hz
1 Hz
VRCS (continued)
• In order to accelerate muons, rep. rate must be much
faster.
• 4600 Hz design exists. (D.J.Summers, et.al.)
Power supply and Eddy current are issues.
dI/dt is too much.
Machine candidate (2)
• If we cannot use AC (ramping) magnet, the alternative
is to use only RF cavities. This is a linear accelerator.
• Linac (linear accelerator)
– To accelerate muons to 20 GeV, the length becomes 4000 m
with 5 MV/m accelerating cavity.
Linac (continued)
• Linear collider assumes 35~45 MV/m, why not for
muons?
– Muon emittance is much larger than electron emittance in
linar collider.
– To make acceptance larger, RF frequency must be relatively
lower (200 MH instead of 1.5 GHz) and field gradient is
lower as well.
• Rule of thumb is that field gradient is proportional to
square root of frequency.
– Cost is another issue.
Machine candidate (3)
• Synchrotron radiation is not a problem unlike electron.
We can use bending arcs and reuse linac several
time.
• RLA (recirculating linear accelerator)
– Use 400 m linac with energy gain of 2 GeV 10 times, we can
accelerate muons to 20 GeV.
– Need 10 arcs to bend 10 different momentum separately
because we give up ramping magnet. This machine looks
like JLAB machine.
RLA (continued)
• This was a baseline for muon acceleration until a few
years ago.
• Switchyard becomes complex with more number of
arcs and large muon emittance.
Machine candidate (4)
• Suppose if we can make orbit in bending arc less
sensitive to momentum, the same arc can be used
for different momentum.
• FFAG (fixed field alternating gradient)
– Large field index in radial direction makes orbit shift as a
function of momentum small. In accelerator terminology,
dispersion function is small.
– How small it should be? Beam size is something we can
compare with.
– Such an optics can be realized with high periodicity lattice.
There is no clear separation of straight for acceleration and
bending arc.
FFAG (continued)
• Easy to understand with alternative bending.
• Alternative bending with finite field gradient gives
alternative focusing.
RF
RF
FFAG compared with others
• Cost effective. Use RF cavity several times.
• Large acceptance.
• Machine is simple.
– Fixed field magnet
– No switchyard
• Accelerating gradient is relatively low or must be low.
Acceleration of muons
Summary
• Muons have to be accelerated as quick as possible
against muon life time.
• Muon accelerator has to have large acceptance
because a muon beam is produced as a secondary
particle and emittance is huge.
• Several schemes are considered: VRCS, Linac, RLA,
and FFAG. At the moment, FFAG seems most
feasible and cost effective.
• Requirement for muon collider is different. Although
machine is similar, muon collider has to assume
small emittance to increase luminosity.
2
Evolution of FFAG
Invention
• AG principle was invented in 1950s.
– By Courant, Synder, Christofilos
– Combination of convex (focusing) and concave (defocusing)
elements makes net focusing.
horizontal
vertical
• FFAG principle was invented a few years later
– By Ohkawa, Symon, Kolomenski
FFAG vs. ordinary AG
• Fixed field (DC field) makes a machine simpler.
– Cost of power supply for magnet is less.
• No synchronization between magnet and RF
frequency.
– Repetition rate is only determined by RF frequency change.
– Repetition rate of oAG is determined by ramping speed of
magnet.
• Large momentum acceptance.
– +-100% vs. +-1%
• Magnet size tends to be large.
– Even it is small, orbit moves in horizontal direction.
Field profile
• Sharp rise of field makes orbit shift small.
 r 
Br,   B0  F  
r0 
k
Bz(r)

k >>1

r
Transverse focusing
• Alternating gradient can be realized by two ways.
 r 
Br,   B0  F  
r0 
k
• F() has alternating sign.

radial sector
Bz(r)
Bz(r)

r
+
r
• Add edge focusing.
spiral sector

r 
F    F   h ln 
r0 


Radial and spiral sector
Radial sector consists of
normal and reverse bends.
Spiral sector use edge
as vertical focusing.

machine center
machine center
MURA days
(Midwest University Research Associate)
• In US, electron model was constructed at MURA.
– Radial sector (400 keV)
– Spiral sector (180 keV)
– Two beam accelerator (collider)
• In Russia and Japan
– Magnet design and fabrication.
“Two beam accelerator”
Particles with the same charge can rotate in both
directions.
– Sign of neighboring magnets is opposite.
– Outer radius has more bending strength.
Colliding point
Extinction
• People at that time aimed at high energy frontier.
• Because orbit moves, magnet tends to be bigger.
– Magnet of AG focusing machine has to be small compared
with ZGS.
– Magnet pole face has a bit complicated shape.
 r 
Br,   B0  F  
r0 
k
• To accelerate protons, broadband RF cavity with high
 gradient has to be developed.
Revival
• The right machine in the right place.
• Large magnet can be made with 3D modeling code.
• RF cavity with new material.
Three factors above are combined together in 2000.
The right machine in the right place
• From 1980s’, high intensity machine is demanded,
not only high energy.
• Ordinary AG machine needs large aperture magnet
to accommodate large emittance beam.
Large magnet can be made with 3D
modeling code
With an accuracy of 1%, 3D design of magnet with
complex shape becomes possible.
Gradient magnet with gap shape
• A magnet with field index k=7.6
RF cavity with new material (MA)
Magnetic Alloy has
• Large permeability
~2000 at 5 MHz
• High curie temperature
~570 deg.
• Thin tape
~18 mm
• Q is small
~0.6
Q can be increased with cutting core if necessary.
mQf (shunt impedance)
• A mQF remains constant at high RF magnetic RF
(Brf) more than 2 kG
• Ferrite has larger value at low field, but drops rapidly.
– RF field gradient is saturated.
Proton FFAG at KEK
• With all those new technology, proton FFAG (proof of
principle) was constructed and a beam is accelerated
in June 2000.
Evolution of FFAG
summary
• FFAG is an old idea back to 1950s.
• FFAG concept was not fully appreciated because
people want accelerator for energy frontier.
• Technology was not ready yet.
• RF cavity with new material and 3D calculation tool
make it possible to realize proton FFAG.
• Proof of principle machine demonstrates that FFAG
machine works as it designed.
3
FFAG as a muon accelerator
Scaling FFAG
Originally, FFAG design satisfied scaling law,
– Geometrical similarity
  r 
0
 
p r0   const.
r0 : average curvature
r : local curvature

 : generalized azimuth
– Constancy of k at corresponding orbit points
k
r B 
 0 k   
p   const.
B r 
k : index of the magnetic field
 r 
Br,
  B0  F  
r0 
k
The field

satisfies the scaling law.
Tune is constant independent of momentum: scaling FFAG
Resonance in accelerator
• Why we need to keep constant tune during
acceleration?
• Because
there are many resonances
near operating tune. Once
a particle hits one of them,
it will be lost.
In reality, however,
operating tune moves
due to imperfection
of magnet (red zigzag line).
ny
nx
Non scaling FFAG
• Muons circulate only a few turns in FFAG.
• Is resonance really harmful to a beam?
• Forget scaling law ! Let us operate ordinary AG
synchrotron without ramping magnet.
• Orbit shifts as momentum is increased.
• Focusing force decreases as momentum increases.
1 B
k1  
f Br

Orbit for different momentum
• Orbit shifts more at larger dispersion section.
Tune variation in a cycle
• Tune decreases as a beam is accelerated.
Resonance crossing simulation
• Animation
• If the acceleration is fast, resonance is not a problem.
Acceleration (1)
• Acceleration is so quick that RF frequency cannot be
synchronized with revolution frequency of muons.
• Revolution frequency changes because orbit shifts
and path length changes although speed of mouns is
already a speed of light.
• If you look at orbits carefully,
path length at the central
frequency is shortest.
Acceleration (2)
• In a first half of a cycle, path length becomes shorter
and revolution frequency becomes higher.
• In a second half of a cycle, path length becomes
longer and revolution frequency becomes lower.
Acceleration (3)
• Suppose we choose RF frequency that is
synchronized with revolution frequency at the center.
• In the first half of a cycle, a particle lags behind the
RF.
• At the center, a particle is synchronized with RF.
• In the second half, a particle lags again.
voltage
low
center
high
time
Acceleration (4)
• This is called
“Gutter acceleration”.
dp/p (normalized)
• In the longitudinal phase space, a particle follows the
path with constant color.
• If there is enough RF voltage, a particle can be
accelerated to the top
energy.
Phase (1/2 pi)
FFAG as a muon accelerator
summary
• FFAG used to satisfy scaling law, that assures
geometrical similarity of orbit and tune independent of
momentum.
• If resonance crossing is not harmful, scaling law is
not necessary.
• Just ordinary synchrotron without ramping magnet
makes a new concept of FFAG, namely non-scaling
FFAG.
• Acceleration is o fast that RF frequency cannot be
synchronized with revolution frequency.
• “Gutter acceleration” is one possible way.
4
Design example of muon
accelerator
Japanese scheme
• Scaling FFAG
• Acceleration with a bucket of low frequency RF,
5~20 MHz
Acceleration
• No time to modulate RF frequency.
• 1 MV/m (ave.) RF voltage gives large
longitudinal acceptance.
• From 10 to 20 GeV/c within 12 turns.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Accelerator chain
• Before acceleration
– Target and drift
– No cooling section
• Four scaling FFAGs,
–
–
–
–
0.3 - 1.0 GeV
1.0 - 3.0 GeV
3.0 - 10.0 GeV
10. - 20. Gev
• If physics demands, another FFAG
– 20. - 50. GeV
Longitudinal emittance vs acceptance
(after target and drift)
Acceptance of US scheme is 0.167 eV.sec (150 mm).
Difference comes from frequency of RF (5 vs. 201 MHz).
Transverse emittance
~100 mm (100,000 pi mm-mrad)
Hardware R&D (1)
Low frequency RF (ferrite loaded)
Shunt impedance
Ferrite core
Hardware R&D (2)
Low frequency RF (air core)
Hardware R&D (3)
Superconducting magnet
US scheme (Europe’s similar)
• Combination of RLA (LA) and Non scaling FFAG
• High frequency RF, 201 MHz
Accelerator chain
• Before acceleration
– Target, drift, buncher, rf rotator, and cooling
• Linac
– 0.220 GeV - 1.5 GeV
• RLA
– 1.5 - 5. GeV
• Two non-scaling FFAGs
– 5. - 10. GeV
– 10. - 20. GeV
• If physics demands, another non-scaling FFAG
– 20. - 50. GeV
Before acceleration
• Three more stages compared to Japanese scheme.
A way to make small emittance fit into 201
MHz RF
There is some stage to make longitudinal emittance
smaller so that 201 MHz RF can be used.
Emittance evolution before FFAG injection
• Cooling is also necessary to fit into the acceptance.
longitudinal
Emittance [mm]
transverse
Path length [m]
Acceleration system requirements
From Reference 1.
Initial momentum
0.3
GeV/c
Final momentum
20
GeV/c
Normalized transverse acceptance
30
mm
Normalized longitudinal acceptance
150
mm
Bunching frequency
201.25
MHz
Maximum muons per bunch
1.1 x 1011
Muons per bunch train per sign
3.0 x 1012
Bunches in train
89
Average repetition rate
15
Hz
Minimum time between pulses
20
ms
Scaling vs. non-scaling
• Scaling machine principle is proven.
• Large acceptance so that cooling is not needed.
• Magnet tends to be larger. Cost more.
• Non-scaling machine can be more compact. Cost
less.
• Need cooling to fit a beam into the acceptance.
• Principle have to be proven.
– Resonance crossing
– Gutter acceleration
– Demonstration by electron model is scheduled in UK.
Design example of muon acceleration
summary
• Japanese scheme assumes low frequency (~5 MHz)
RF and no cooling is necessary. It uses scaling FFAG.
• US and Europe scheme assumes high frequency
(~200 MHz) RF. It uses non-scaling FFAG.
• Hardware R&D is going on.
• Proof of principle model for non-scaling FFAG is
scheduled in UK.
Appendix
FFAG as a proton driver
Requirement of proton driver (1)
• Beam power
= energy x current
= energy x (particles per bunch) x (repetition rate)
• Energy
– MW using a few GeV or more energetic protons.
• Particles per bunch and Repetition rate
– From accelerator point of view, low ppb is preferable.
– Probably rep. rate does not matter as long as the beam power
above is obtained.
Requirement of proton driver (2)
• Beam quality
– Short bunch is preferable for smaller longitudinal emittance.
– Momentum spread of protons is not important because that
of muons can not be small.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
– Beam size (transverse emittance) is not important either.
Machine candidate (1)
• Slow cycling synchrotron (0.1 ~ 1 Hz)
• J-PARC is one of examples
– Maximum energy is 50 GeV.
– Particles per bunch is high, 3e14 to obtain 0.75 MW
– Should be more to upgrade to a few MW facility
– Space charge and beam instability are problems.
Machine candidate (2)
• Rapid cycling synchrotron (10 ~ 50 Hz)
• ISIS upgrade is one of examples
– Maximum energy is 50 GeV.
– Particles per bunch can be reduced,
– Design of 30 GeV with 50 Hz is feasible.
Machine candidate (3)
• Rapid cycling linac (10 ~ 50 Hz)
• SPL is one of example
– Maximum energy is limited to a few GeV.
– More particle per bunch is needed compared with RCS
– Space charge and beam instability problem are less
because acceleration is quicker.
Machine candidate (4)
• FFAG (100 ~ 1000 Hz)
– Maximum energy can be as high as synchrotron.
– Particles per bunch can be much less.
– Space charge and beam instability problem are less
because acceleration is quicker.
SCS
RCS
RCL
FFAG
energy
~50 GeV
~50 GeV
~3 GeV
~20 GeV
rep. rate
0.1~1
10~50
50
100~1000
ppb
high
low
low
much low
Space
charge
etc.
serious
moderate
less
No
problem
Exercise (1)
• Life time of a muon is 2.2 ms. However, it becomes longer when
it is accelerated and Lorentz boosted. Calculate analytically or
numerically what percentage of muons does survive when it is
accelerated from 0.3 GeV/c to 20 GeV/c assuming two cases of
average energy gain. One is 1 MeV/m and the other is 5 MeV/m.
• This exercise can be extended to more complex system. For
example, assume there are two FFAGs, one from 0.3 GeV to 3
GeV, and the other from 3 to 20 GeV. Also assume the number
of RF cavity is 5 times more in the bigger ring and RF cost is
proportional to square of average energy gain. To make the cost
of muons minimum, how we can choose the average energy
gain in the first and the second ring?
Exercise (2)
• Consider periodic beam transport line consisting of focusing and
defocusing quadrupole with the same absolute strength k (sign
is opposite). There is drift space in between and separation is L.
– Using thin lens approximation, show phase advance as a
function of k and L.
– Assume that non-scaling FFAG consists of the simple FODO
cell. If phase advance per cell is limited between 30 degrees
and 150 degrees, what is the maximum momentum ratio
from injection to extraction?
– Show Courant-Synder parameters a,b,g and phase
advance m at the entrance of focusing and defocusing
quadrupole at injection, extraction and at the center
momentum.
Exercise (3)
•
Scaling FFAG has magnetic field shape as
 r 
Br,   B0  F  
r0 
k
– Momentum compaction factor ac is defined as

dR
dp
 ac
R
p
– Show momentum compaction factor of scaling FFAG.
– RF bucket (half) height is

2  eV  E
E  b
 h h
where E is total energy, h is harmonic number, h is slippage factor
defined as

h  ac 
1
g2
how much RF voltage is required to accelerate 10 to 20 GeV when
h=20 and k=280.
Exercise
• Any questions, you can send to
[email protected]
Subjects to be studied
• Electron model of non scaling FFAG
– New scheme of acceleration
– Resonance crossing
• High intensity operation
• Optimization of scaling magnet
• Make the magnet superconducting