Muon Acceleration and FFAG II

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Muon Acceleration and FFAG II

• Shinji Machida
• CCLRC/RAL/ASTeC
• NuFact06 Summer School
• August 20-21, 2006

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Content

• Acceleration of muons
• Evolution of FFAG
• FFAG as a muon accelerator
• Scaling FFAG
• Non-scaling FFAG
• Design example of muon acceleration
• Japanese scenario
• US and Europe scenario
• Appendix Proton driver
• Reference (among others)
• BNL-72369-2004, FNAL-TM-2259, LBNL-55478
• NuFactJ Design study report

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3

• FFAG as a muon accelerator

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Betatron oscillation and Tune

• In transverse plane, a particle in an accelerator
  oscillates around the centeral orbit.
• It is called betatron oscillation and its
  oscillation frequency is tune. Precisely
  speaking tune is

x
s
FFAG as a muon accelerator
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Cardinal conditions

• 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
FFAG as a muon accelerator
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Beyond scaling FFAG

• If we can break scaling law, FFAG will be much
  simpler and magnet will be smaller.
• Why do we keep scaling (constant tune) during
  acceleration?
• Because of resonance in
  accelerator.

No gentle slope at low momentum. - Orbit
excursion is shorter. Constant gradient. -
Linear magnet.
FFAG as a muon accelerator
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Resonances in accelerators

• There are many resonances near operating tune.
  Once a particle hits one of them, probably it
  will be lost.
• In reality, however,
• operating tune moves
• due to imperfection
• of magnet (red zigzag line).
• Particles can survive after
• crossing resonances
• if resonance is weak and
• crossing is fast.

Tune diagram of 150 MeV FFAG
FFAG as a muon accelerator
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Non-scaling FFAG

• Muons circulate only a few (15) turns in FFAG.
• Is resonance really harmful to a beam? Maybe not.
• Forget scaling law !
• Let us operate ordinary AG synchrotron without
  ramping magnet.
• Orbit moves as momentum increases.
• Large ap makes the orbit shift small.
• Focusing force decreases as momentum increases.

FFAG as a muon accelerator
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Orbit for different momentum

• Orbit shifts more at larger dispersion section.
• No similar shape unlike scaling FFAG.

high p
low p
non-scaling
FFAG as a muon accelerator
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Tune variation in a cycle

• Tune decreases as a beam is accelerated.
• dn(tune)/dT(turn)1 for muon rings.

low p
high p
FFAG as a muon accelerator
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Beam dynamics issues

• Acceleration out of RF bucket.
• Crossing of many resonances during acceleration.
• Structure resonance has some effects.
• With alignment errors, integer resonances have to
  be considered.
• Huge acceptance (30,000 p mm-mrad) for muons.
• Dynamic aperture without acceleration at
  injection energy.

FFAG as a muon accelerator
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Synchrotron oscillation and RF bucket (1)

voltage
time
momentum
lt lt
lt lt
voltage
time
lt lt
lt lt
FFAG as a muon accelerator
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Synchrotron oscillation and RF bucket (2)

momentum
time
If momentum is too large, it cannot be trapped in
sinusoidal RF voltage. -gt RF bucket.
FFAG as a muon accelerator
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Acceleration in non-scaling FFAG (1)

• 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
• momentum is shortest.

FFAG as a muon accelerator
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Acceleration in non-scaling FFAG (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.

1/revolution freq.
momentum
FFAG as a muon accelerator
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Acceleration in non-scaling FFAG (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
FFAG as a muon accelerator
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Acceleration in non-scaling FFAG (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.

momentum
time
FFAG as a muon accelerator
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Transverse amplitude effects
Longitudinal phase space (phi, momentum)
5 to 10 GeV ring

• without transverse amplitude

with finite transverse amplitude
Horizontal is 5,000 pi mm mrad Vertical is zero
FFAG as a muon accelerator
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Source of longitudinal emittance growth

• A particle with large transverse amplitude has
  longer path length.
• This effects become visible because muon
  emittance (amplitude) is huge.

x
s
FFAG as a muon accelerator
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Matching between two FFAG rings
20GeV
10GeV
5GeV

• Second harmonics and 10 increase of RF voltage
  partially cure the problem.

FFAG as a muon accelerator
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RLA

• Synchrotron oscillation helps to mix momentum
  spread.

FFAG as a muon accelerator
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Resonance crossing without errors

• Vertical is 5,000 p mm-mrad, normalized, zero
  horizontal emittance.
• Shows the coupling due to nx-2ny0 (structure)
  resonance.
• If we start finite horizontal and zero vertical
  emittance, no exchange of emittance.

vertical emittance
5 GeV
horizontal emittance
10 GeV
FFAG as a muon accelerator
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Resonance crossing without errors, amplitude
dependence
5,000 pi mm-mrad
500 pi mm-mrad
0.5 pi mm-mrad
FFAG as a muon accelerator
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Resonance crossingwith alignment errors

• Beam has to face many integer
  tunes.

tune per cell
tune per ring
FFAG as a muon accelerator
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Resonance crossing with alignment errors,
envelope

• Horizontal is 10,000 p mm-mrad, normalized, zero
  vertical emittance.
• Errors of 0, 0.05, 0.10, 0.20 mm (rms).

Horizontal phase space (x, xp)
0. mm
0.05 mm
0.10 mm
0.20 mm
FFAG as a muon accelerator
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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.

FFAG as a muon accelerator
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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 so fast that RF frequency cannot
  be synchronized with revolution frequency.
• Gutter acceleration is one possible way.
• Transverse amplitude makes longitudinal growth.

FFAG as a muon accelerator
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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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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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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

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Exercise (4)

• 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 (5)

• Scaling FFAG has magnetic field shape as
• where B0 and r0 are magnet strength and average
  radius at injection.
• Ignore the azimuthal dependence and calculate
  orbit shift between 10 to 20 GeV when k is 280
  and r0 is 120m.
• The present design assumes the muon emiitance of
  30 pi mm (or 30,000 pi mm-mrad) normalized.
  Calculate beam size at 10 GeV and 20 GeV and
  compare them with the orbit shift.
• Use the relation
  where R is average radius of an
• accelerator (120 m) and n is tune (20).