Muon Acceleration with FFAG

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Muon Acceleration with FFAG

• Shinji Machida
• RAL/ASTeC
• 10 November, 2005

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Content

• 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

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1

• Acceleration of muons

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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 g3, Velocity b0.94.
• Flight path length 2000 m
• That is even true on the lower momentum side.

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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

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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 J-PARC booster KEK-PS booster Fermilab booster AGS booster CPS booster
Rep. rate 50 Hz 25 Hz 20 Hz 15 Hz 7.5 Hz 1 Hz
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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.
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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.

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Linac (continued)

• Linear collider assumes 3545 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.

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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.

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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.

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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.

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FFAG (continued)

• Easy to understand with alternative bending.
• Alternative bending with finite field gradient
  gives alternative focusing.

RF
RF
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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.

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Acceleration of muonsSummary

• 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.

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2

• Evolution of FFAG

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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

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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.

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Field profile

• Sharp rise of field makes orbit shift small.
• k gtgt1

Bz(r)
r
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Transverse focusing

• Alternating gradient can be realized by two ways.
• F(q) has alternating sign.
• radial sector
• Add edge focusing.
• spiral sector

Bz(r)
Bz(r)
r

r
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Radial and spiral sector
Radial sector consists of normal and reverse
bends.
Spiral sector use edge as vertical focusing.
machine center
machine center
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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.

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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
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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.
• To accelerate protons, broadband RF cavity with
  high gradient has to be developed.

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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.

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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.

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Large magnet can be made with 3D modeling code

• With an accuracy of 1, 3D design of magnet
  with complex shape becomes possible.

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Gradient magnet with gap shape

• A magnet with field index k7.6

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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.

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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.

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Proton FFAG at KEK

• With all those new technology, proton FFAG (proof
  of principle) was constructed and a beam is
  accelerated in June 2000.

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Evolution of FFAGsummary

• 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.

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3

• FFAG as a muon accelerator

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Scaling FFAG

• Originally, FFAG design satisfied scaling law,
• Geometrical similarity
• r0 average curvature
• r local curvature
• q generalized azimuth
• Constancy of k at corresponding orbit points
• k index of the magnetic field

The field
satisfies the scaling law. Tune is constant
independent of momentum scaling FFAG
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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
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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.

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Orbit for different momentum

• Orbit shifts more at larger dispersion section.

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Tune variation in a cycle

• Tune decreases as a beam is accelerated.

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Resonance crossing simulation

• Animation
• If the acceleration is fast, resonance is not a
  problem.

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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.

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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.

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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.

low center high
voltage
time
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Acceleration (4)

• 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.
• This is called
• Gutter acceleration.

dp/p (normalized)
Phase (1/2 pi)
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FFAG as a muon acceleratorsummary

• 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.

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4

• Design example of muon accelerator

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Japanese scheme

• Scaling FFAG
• Acceleration with a bucket of low frequency RF,
• 520 MHz

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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.

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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

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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).
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Transverse emittance
100 mm (100,000 pi mm-mrad)
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Hardware RD (1)
Low frequency RF (ferrite loaded)
Shunt impedance
Ferrite core
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Hardware RD (2)
Low frequency RF (air core)
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Hardware RD (3)
Superconducting magnet
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US scheme (Europes similar)

• Combination of RLA (LA) and Non scaling FFAG
• High frequency RF, 201 MHz

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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

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Before acceleration

• Three more stages compared to Japanese scheme.

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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.
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Emittance evolution before FFAG injection

• Cooling is also necessary to fit into the
  acceptance.

transverse
longitudinal
Emittance mm
Path length m
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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
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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.

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Design example of muon accelerationsummary

• 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 RD is going on.
• Proof of principle model for non-scaling FFAG is
  scheduled in UK.

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Appendix

• FFAG as a proton driver

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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.

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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.
• Beam size (transverse emittance) is not important
  either.

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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.

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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.

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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.

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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 0.11 1050 50 1001000
ppb high low low much low
Space charge etc. serious moderate less No problem
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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?

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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.

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Exercise (3)

• Scaling FFAG has magnetic field shape as
• Momentum compaction factor ac is defined as
• Show momentum compaction factor of scaling FFAG.
• RF bucket (half) height is
• where E is total energy, h is harmonic
  number, h is slippage factor defined as
• how much RF voltage is required to
  accelerate 10 to 20 GeV when h20 and k280.

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Exercise

• Any questions, you can send to
• shinji.machida_at_kek.jp

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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