part i optics. ffag is “fixed field alternating gradient”. ordinary synchrotron needs ramping...
TRANSCRIPT
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Part I
Optics
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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
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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.
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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.
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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.
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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
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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
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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( )
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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
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Radial and Spiral
From K.R.Symon, Physical Review, Vol.103, No.6, p.1837, 1956.
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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.
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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
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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.
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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.
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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.
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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.
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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
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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
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Orbit (assumption)
Assume orbit consist of
• arc of a circle
• straight line
Example of triplet radial
Sector.
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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
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Example of singlet
8 cells
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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
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Example of two beam accelerator
16 FODO cells
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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
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Example of DFD triplet
8 cells, similar to POP FFAG at KEK.
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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
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DFD vs. FDF
• If k is the same, phase advance in horizontal is smaller in FDF.
• Injection and extraction is easier in FDF.
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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
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Example of spiral
• 16 cells
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Model of doublet
• Need iteration
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Example of doublet
• 8 cells
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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.
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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
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Model of three fringe functions
• It is not clear which is correct.
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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
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Particle tracking
• Runge-Kutta• Thin-lens kick• Symplectic map
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Comparison
• Runge-kutta and map based tracking.