Part I

Optics

FFAG is “Fixed Field Alternating Gradient”.

• Ordinary synchrotron needs ramping magnets to keep the orbit radius constant.

• FFAG has Alternating Gradient focusing with DC magnets. Orbit moves depending on momentum like cyclotron.

• Although orbit moves, focusing (or tune) is the same for all momentum.– zero chromaticity

Storage rings such as LANL-PSR, SNS are FFAG?

• They were not.

– They are Fixed Field and Alternating Gradient.

– However, do not satisfy zero-chromaticity within a wide momentum range, say a factor of 3.

– They are ordinary synchrotrons. Since there is no acceleration or ramping of magnet, DC magnet can be used.

• Nowadays they are, however, called FFAG.

– New concept of “non-scaling” FFAG.

– Non-scaling means no zero-chromaticity condition satisfied.

– If the orbit excursion due to acceleration is small (namely, small dispersion), acceleration without ramping magnet is possible.

– Since chromaticity is finite, tune moves in a wide range. Tune may cross even integer resonance several times.

Non-scaling FFAG

• Essentially only bends and quads, no nonlinear elements

• As small dispersion as possible to make orbit excursion small

• Large swing of phase advance, say 150 deg. at low momentum and 30 deg. at high momentum.

• Nonlinear longitudinal dynamics.

Non-scaling FFAG example by Trbojevic at BNL

• Orbits corresponding to dp/p=-33% to 33%.• Integer part of tune moves for about 2 units.

Cardinal conditions of scaling FFAG

• Geometrical similarity

: average curvature : local curvature : generalized azimuth

• Constancy of k at corresponding orbit points

k : index of the magnetic field

[figures]

€

€

∂∂p

ρ

ρ 0

⎛

⎝ ⎜

⎞

⎠ ⎟ϑ = const.

= 0

€

∂k

∂pϑ = const.

= 0

€

k =r

B

∂B

∂r

⎛

⎝ ⎜

⎞

⎠ ⎟€

ϑ

Solutions

Magnetic field profile should be

radial dependence€

F ϑ( ) = F θ − h lnr

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

B r,θ( ) = B0

r

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

k

F ϑ( )

€Bz(r)

r

Two kinds of azimuthal dependence (1)

“radial sector type” satisfies

€

F ϑ( ) = F θ( )

machine center

€

B r,θ( ) = B0

r

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

k

F ϑ( )

Two kinds of azimuthal dependence (2)

“spiral sector type” satisfies

since

€

B r,θ( ) = B0

r

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

k

F ϑ( )

€

€

F ϑ( ) = F θ − tanζ ⋅lnr

r0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

rdθ

dr= tanζ

€

θ −θ0 = tanζ ⋅lnr

r0

machine center

Radial and Spiral

From K.R.Symon, Physical Review, Vol.103, No.6, p.1837, 1956.

Days of invention

• In 1950s, the FFAG principle was invented independently by– Ohkawa, Japan– Symon, US– Kolomensky, Russia

• FFAG development at MURA (Midwestern University Research Associate)– Radial sector electron FFAG of 400 keV– Spiral sector electron FFAG of 180 keV

• Both has betatron acceleration unit, not RF.• There was a proposal of 30 GeV proton FFAG.• Even collider was proposed called “two beam accelerator”.

– Same magnet (lattice) will give counter rotating orbit for the same charge.

Two beam accelerator

• The same charged particle can rotate in both directions.– Sign of neighboring magnets is opposite.

– Outer radius has more bending strength.

Colliding point

Comparison with cyclotron

Cyclotron FFAGMagnetic field static (small field index) static (large field index)Orbit radius move in wide range move in small rangeTrans. focusing weak (n<1) strongLong. focusing no yesDuty factor 100% 10-50%RF frequency fixed variedExtraction energy fixed variable

Pros: - Small orbit excursion assures small magnet.- Strong focusing in transverse and synchrotron oscillation

s keep bunch tight.

- Extraction energy is variable.Cons: - Field with large index may be more involved.

- Duty factor is not 100%.- RF frequency must be varied.

Comparison with synchrotronsynchrotron FFAG

Magnetic field time varying staticOrbit radius non move in small rangeTrans. focusing strong strongLong. focusing yes yesDuty factor 1% 10-50%RF frequency varied and synchronized varied

with bending fieldParticles per bunch large small

Pros: - Much rapid acceleration without synchronization of magnet and RF.

- Higher duty factor.- Intensity effects are not critical.

Cons: - Orbit excursion need bigger aperture magnet.

Prospects of FFAG

• Repetition rate can be 1 kHz or even more.– Only RF pattern determines a machine cycle because magnetic field is DC

and no need of synchronization between RF and magnets.

• High beam current can be obtained with modest number of particles per bunch.– Space charge and other collective effects are below threshold because of

small number of particles per bunch.

• Transverse acceptance is huge.

Design procedure

• Rough design with approximated methods.– Elements by elements (LEGO-like) or matrix formalism– Smooth approximation

• 3D design of magnets with TOSCA• Particle tracking

– Runge-Kutta integration– More systematic way

If necessary, back to the previous phase.

Combination of gradient of body and angle at edge

• Focusing of gradient magnet

• Focusing of Edge

• Type

– Radial sector

• Singlet (FODO)

• Doublet

• Triplet (DFD, FDF)

– Spiral sector

Elements by elements

• In a body, focal length is proportional to r.

• Length of drift space is proportional to r.

• At an edge, focal length is proportional to r.

€

1

f=

′ B L

Bρ=

kB

rrθ( )

Bρ∝

1

r

€

L = rθ

€

1

f=

tanε

ρ∝

1

r

Orbit (assumption)

Assume orbit consist of

• arc of a circle

• straight line

Example of triplet radial

Sector.

Model of singlet

From the center of F to the center of D.

€

F

r0

=tanβ F

sinθF + 1− cosθF( )tanβ F

€

D

r2

=sinβ D

sinθD

€

r2

r0

= 1−ρ F

r0

1−1

sin π −θF( )

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

×tan π −θF( )

cosπ

N− β D

⎛

⎝ ⎜

⎞

⎠ ⎟ tan

π

N− β D

⎛

⎝ ⎜

⎞

⎠ ⎟+ tan π −θF( )

⎡

⎣ ⎢

⎤

⎦ ⎥

€

r1 =ρ F sinθF

sinβ F

€

εF =θF − β F

2θF

εD =θD + β D

2θD

Example of singlet

8 cells

Collider (two beam accelerator)

Additional conditions to singlet (approximation)

F and D has the same strength, only the sign is opposite.

Bending angle is scaled with radius.€

βF = β D

€

θF

θD

=r0

r3

⎛

⎝ ⎜

⎞

⎠ ⎟

k +1

Example of two beam accelerator

16 FODO cells

Model of DFD triplet

From the center of F to the center of drift

Edge focusing

€

F

r0

=tanβ F

sinθF + 1− cosθF( )tanβ F

€

r1 =ρ F sinθF

sinβ F

€

D

ρ F

=sinθF

sinβ F

×sin

π

N− β F

⎛

⎝ ⎜

⎞

⎠ ⎟− cos

π

N− βF

⎛

⎝ ⎜

⎞

⎠ ⎟tan

π

N− β F − βD

⎛

⎝ ⎜

⎞

⎠ ⎟

sin θF −π

N

⎛

⎝ ⎜

⎞

⎠ ⎟− 1− cos θF −

π

N

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥tan

π

N− β F − β D

⎛

⎝ ⎜

⎞

⎠ ⎟

€

εF =θF − β F

2θF

εD,F =θF − β F

θD

εD,O = −

π

N− β F − β D

θD

€

εF =θF − β F

2θF

€

εD,F =θF − β F

θD

€

εD,O = −

π

N− β F − β D

θD

Example of DFD triplet

8 cells, similar to POP FFAG at KEK.

Model of FDF triplet

From the center of D to the center of drift.

Edge focusing terms.

€

D

r0

=tanβ D

sinθD − tanβD 1− cosθD( )

€

r1 =ρ DsinθD

sinβ D

€

F

r1=

cosπ

N− β D

⎛

⎝ ⎜

⎞

⎠ ⎟tan

π

N− β F − β D

⎛

⎝ ⎜

⎞

⎠ ⎟+ sin

π

N− β D

⎛

⎝ ⎜

⎞

⎠ ⎟

sinθF + 1− cosθF( )tanπ

N− β F − β D

⎛

⎝ ⎜

⎞

⎠ ⎟

€

εD =θD + β D

2θD

€

εF ,D =θD + β D

θF

€

εF ,O =

π

N− β D − β F

θF

DFD vs. FDF

• If k is the same, phase advance in horizontal is smaller in FDF.

• Injection and extraction is easier in FDF.

Model of spiral

Vertical focusing mainly comes from edge, while horizontal focusing is in the mail body.

€

ε1 =ζ +

π

N−

βF

2θF

€

ε2 =−ζ +

π

N−

β F

2θF

Example of spiral

• 16 cells

Model of doublet

• Need iteration

Example of doublet

• 8 cells

Edge of FFAG

• Edge angle of radial sector FFAG is determined once opening angle is fixed.

• Stronger vertical focusing can be realized with more edge angle.

Model of fringe in synchrotrons

• Steffen (CERN handbook): 　 linear fringe– 1/f = -1/rho [Tan[e]+b / (6 rho Cos[e])]

• e, face angle• b, fringe field region• rho, bending radius

• Enge and Brown: 　 Enge function– 1/f = -1/rho Tan[e-psi]

• psi = (g/rho) F[e]• F[e] = F1/(6 g) (1+Sin[e]^2) / Cos[e] [1-F1 / (6 g) k2 (g/rho) Tan[e]]• F1 = 6 Int[Bz/B0 - (Bz/B0)^2, {s, -Inf, Inf}]

– If linear slope, F1=b. and when psi<<1 、 it becomes the same as Steffen.

• SAD: 　 expansion of Hamiltonian to 4th order.– 1/f(fringe part only) ~ -1/rho [F1/(6 rho) - 2/3 z^2/(F1 rho)] /p^2

Model of three fringe functions

• It is not clear which is correct.

Smooth approximation(results only)

For radial sector

For spiral sector

€

ν x2 = k +1+

k +1( )2

f 2

N 2g1

2

AV

€

ν y2 = −k +

f 2

2+

k −1( )2

f 2

N 2g1

2

AV

€

ν x2 = k +1

€

ν y2 = −k +

f 2

2+ 2

1

η

∂η

∂Θ

⎛

⎝ ⎜

⎞

⎠ ⎟

2

AV

Particle tracking

• Runge-Kutta• Thin-lens kick• Symplectic map

Comparison

• Runge-kutta and map based tracking.