Muon Acceleration and FFAG

Muon Acceleration and FFAG

Shinji Machida

KEK

NuFact05 Summer Institute

June 12-20, 2005

Content

1. Acceleration of muons

2. Evolution of FFAG

3. FFAG as a muon accelerator

4. 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 ~3, Velocity ~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 * = 1000 mm at 0.3 GeV/c

• Transverse emittance– 10 ~ 100 mm

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Machine candidate (1)• Everyone knows modern high energy accelerator is s

ynchrotron. Why not for muons?

• VRCS (very rapid cycling synchrotron)– Rapid (or fast) cycling means time required for acceleration f

rom injection to extraction is short.– The most rapid cycling machine at the present is ISIS at RA

L, 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

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 mu

ons?– Muon emittance is much larger than electron emittance in lin

ar 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 electro

n. 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 bec

ause 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 sen

sitive 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 fun

ction 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 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.

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 m

agnet.

• 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.

k >>1

€ Bz(r)

r

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B r,θ( ) = B0

r

r0

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⎝ ⎜

⎞

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k

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Transverse focusing• Alternating gradient can be realized by two ways.

• F() has alternating sign.

radial sector

• Add edge focusing.

spiral sector

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B r,θ( ) = B0

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r0

⎛

⎝ ⎜

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Bz(r) r

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F ϑ( ) = F θ − h lnr

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Radial and spiral sector

machine center

€

machine center

Radial sector consists ofnormal and reverse bends.

Spiral sector use edgeas vertical focusing.

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.

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

B r,θ( ) = B0

r

r0

⎛

⎝ ⎜

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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.

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 m

• Q is small~0.6Q can be increased with cutting core if necessary.

Qf (shunt impedance)• A QF remains constant at high RF magnetic RF (Br

f) 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 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.

3

FFAG as a muon accelerator

Scaling FFAG

Originally, FFAG design satisfied scaling law,– Geometrical similarity

: average curvature

: local curvature

: generalized azimuth– Constancy of k at corresponding orbit points

k : index of the magnetic field

€

∂∂p

ρ

ρ 0

⎛

⎝ ⎜

⎞

⎠ ⎟ϑ = const.

= 0

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∂k

∂pϑ = const.

= 0

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k =r

B

∂B

∂r

⎛

⎝ ⎜

⎞

⎠ ⎟

The field satisfies the scaling law.

Tune is constant independent of momentum: scaling FFAG

€

B r,θ( ) = B0

r

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

k

F ϑ( )

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). x

y

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.

€

k1 =1

f=

′ B

Bρ

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.

low center high

timevolt

age

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 (n

orm

aliz

ed)

Phase (1/2 pi)

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.

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.



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 emittancesmaller so that 201 MHz RF can be used.

Emittance evolution before FFAG injection

• Cooling is also necessary to fit into the acceptance.

transverse longitudinal

Path length [m]

Emitt

ance

[mm

]

Acceleration system requirements

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

From Reference 1.

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 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 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 abo

ve 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.

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




• 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.


• 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.


• 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 becaus

e 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 becaus

e 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 s. 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 fu

nction 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 and phase advance 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

– Momentum compaction factor c is defined as

– Show momentum compaction factor of scaling FFAG.– RF bucket (half) height is

where E is total energy, h is harmonic number, is slippage factor defined as

how much RF voltage is required to accelerate 10 to 20 GeV when h=20 and k=280.

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r

r0

⎛

⎝ ⎜

⎞

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k

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dR

R= α c

dp

p

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ΔE = β2 ⋅eV ⋅E

π ⋅h ⋅ η

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= c −1

γ 2

Exercise

• Any questions, you can send to

[email protected]

mailto:[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

Documents

Muon Acceleration and FFAG