ffag modelling #2pill-box . existing building : ions . superconduc ting : ma normal conducing :...
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FFAG Modelling #2
“How to model an FFAG in 45-minutes”
12/9/2011 1
Suzie Sheehy, ASTeC Intense Beams Group, RAL
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Outline • Requirements • Basic parameter choices • Optimisation • Detailed studies • Odds and ends
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Requirements • Beam requirements
• Beam energy • Particle species • Beam current/power • Pulse structure
• Technology limitations • Superconducting/Normal conducting magnets? • Particular RF expertise? • Adding to existing facility? • Financial constraint
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Requirements Particle species
Beam energy
Beam current
Pulse structure
Rep. rate Magnet type
RF type Space available
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Requirements Particle species
Beam energy
Beam current
Pulse structure
Rep. rate Magnet type
RF type Space available
e keV nA CW Constant Combined function
NC Hospital basement
μ± MeV μA Short pulse Hz Separated function
SC Cargo port
H- GeV mA Long pulse 10-50Hz Linear Ferrite Green field
p TeV 10 mA? Pulse train kHz Non-linear Pill-box Existing building
ions Superconducting
MA
Normal conducing
Induction
Permanent
I could combine these in innumerable ways! (though not all are sensible…)
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Where to start!?
5 to 100 MeV => 97 MeV/c to 444.6 MeV/c Magnetic rigidity: 0.3235 to 1.483 Tm
KJP: “could we make a mini-PAMELA? Something like protons from 5~100
MeV?” “This would also give ions up to around 30 MeV/u, which might be useful
(if the current is high enough) for some nuclear physics or isotope production, or (much lower current) some useful radiobiology”
Back to basics: relativity
pc = T γ +1γ −1
1/ 2
γ =EE0
E = T + E0
E0(proton)=938.272 MeV/c2
Chao & Tigner, Accelerator physics handbook
Get a good grip on these quantities before modelling begins
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Basic parameters Work with the basic geometry & constraints: Ring size, packing factor, cells, type of FFAG, lattice type?
Ring radius [m]
Packing factor
6-cell lattice 5-cell lattice
There is a fair bit of guess-work involved, but geometry is (often) a good place to start!
k=40??
sinθ2
=lB
2(Bρ)
This holds for a circular trajectory – with approximate scaling law this can be assumed. Constrained by peak B field, long drift
requirement and bore diameter
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Will it be stable? (is my design crazy?)
• Hill’s equation:
• Looks like SHM, solutions:
• For AG focusing with a doublet (FD):
• For stability we require:
x' '−Kx = 0
+Κ−+
+Κ=
)cosh(
)cos()( 0
φ
φ
zayazza
zy
−=
)0(')0(
cossin
sin1cos)(')(
yy
zKzKK
zKK
zKzyzy
K>0
K=0 K<0
y' '+Ky = 0
y(z)y '(z)
=
1 z0 1
y(0)y '(0)
−=
)0(')0(
coshsinh
sinh1cosh)(')(
yy
zKzKK
zKK
zKzyzy
trace(M) < 2
M = MD MF
It’s worth thinking about stability at a single energy first, pretend it’s linear...
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Stability regions FD focusing FDF focusing
α = 0.5
α = 0.8
Horizontal Vertical Both
f0 is the “average” focusing strength and f1 is the alternating gradient focusing strength
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Basic parameters in SCode
• Use a tool like SCode or MAD-X to explore parameter space
• Step 1: calculate input parameters (script)
• Step 2: create input file
• Step 3: trial and
error…
See http://www-bd.fnal.gov/icfabd/Newsletter43.pdf, pp.54 or contact S. Machida
Scode & PTC separate particle orbits from lattice geometry
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SCode stability diagram • Perl script from S. Machida
– Run SCode with range of k values and D/F ratios to find stability regions.
Can use SCode to calculate beta functions, tunes and to do some tracking. But… let’s move onto another code!
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ZGOUBI
• ZGOUBI – Ray tracing code – Use an interface! i.e. Pyzgoubi
• Choice of analytic magnet elements – DIPOLE-S
• Excellent and flexible but lots of parameters
– FFAG • Uses a scaling field only
– MULTIPOL • Has FF issues far from magnet centre
– TOSCA • Could use field maps if you have them – lack of flexibility?
First step: how to describe the lattice?
ZGOUBI manual & source: http://sourceforge.net/projects/zgoubi
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Using DIPOLE-S for rectangular magnets?
Centre of machine
Want field lines to be parallel with magnet faces, NOT curved around machine centre!
• Make R (reference radius) very large • Magnet faces are effectively parallel • Magnetic field lines OK
TIP: Make sure the magnetic fields look as you expect
1 1polyb b=
Where is the dipole field and
1
1 ( )n
y x n nr
ZB iB B b iaR
−
+ = +
∑
Z x iy= +
2 20 1 2( , ) ( , )(1 ( ) / ( ) / ...)z zB R B F R b R RM RM b R RM RMθ θ= + − + − +
DIPOLE-S description in ZGOUBI manual:
Multipole description:
2 2polyb scale b= ⋅ ( )2
3 3polyb scale b= ⋅
Etc… easy in linear case!
1B
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PyZGOUBI tools • Closed orbit finder (basic):
x0,guess =xmax + xmin
2
′ x 0,guess =′ x max + ′ x min
2
(xmax − xmin )( ′ x max − ′ x min ) < thresholdStop iterating when:
Or if difference between successive guesses is less than some threshold.
[x,x’,y,y’]=find_closed_orbit(lattice, [guess co-ords], turns, threshold)
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ZGOUBI tools • Tune calculation (internal to ZGOUBI) • 3 main methods:
1. Fourier transform of particle position in tracking 2. Fourier transform of positions of 2 particles (one
on closed orbit, one offset) 3. Construct (1st or 2nd order) transfer matrix from
trajectories of a number of tracked particles (11 in ZGOUBI) & identify with Twiss form
3. Has the best accuracy in my experience – check what your code is using!!
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Optimisation • Minimising tune variation
Set range of parameters (k, multip, E range/step)
Construct lattice
Calculate closed orbits +
Calculate tunes/tof
Check against criterion
Save data
Educated guesswork + physics
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Optimisation
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Optimisation • Fringe fields
– Crucial in almost every type of FFAG! – Need to go past the ‘hard edge’ model – Usually use ‘Enge’ model
– What if fringe field varies with radius? – In ZGOUBI can use kappa parameter
Fringe field extent [cm]
Cell tune
F(R,θ) =1
1+ expP(d)
P(d) = C0 + C1(dg0
) + C2( dg0
)2 + C3( dg0
)3 + C4 ( dg0
)4 + C5( dg0
)5
κ = +1
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Detailed studies • Dynamic aperture (single particle)
Set range of parameters (k, multip, E range/step)
Construct lattice
Calculate closed orbit at injection
Lost? Finer range Not Lost? Continue
Track particle with amplitude
over many turns
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Detailed studies • Dynamic aperture (single particle)
Acceleration rate
Number of turns
Dynamic aperture (π mm mrad normalised)
None (@250 MeV)
1000 >420
1MV/turn 750 374
2MV/turn 375 411
4+MV/turn 188 450+
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Detailed studies • Error analysis
Set range/type of error
Construct lattice
Calculate closed orbit at injection (or use previous)
Track particle/s over n turns With/out acceleration Save data & loop
Post-tracking analysis of emittance, closed orbit distortion etc…
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Detailed studies • Error analysis
– Crucial for resonance crossing studies with realistic alignment errors
– Often requires significant computation time (for statistics) – Many degrees of freedom & sources of error! (Alignment, stray
field, BPM offsets, etc…)
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Detailed studies • Space charge
– Which codes can handle space charge in an FFAG?
• Start of development in ZGOUBI (S. Tygier) • SCode? • Development in COSY Infinity? • GPT • OPAL-CYCL
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Code comparison chart
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OPAL-CYCL • Object Oriented Parallel Accelerator Library • “a tool for charged-particle optic calculations in large accelerator
structures and beam lines including 3D space charge” • Massively parallel – from 1 to 8000 processors! (C++ based)
– various beam line element descriptions – methods for single particle optics – maps up to arbitrary order – symplectic integration schemes – (RK4 & 2nd order leapfrog) – time integration
Multibunch effects •Takes into account neighbouring bunches •Usually: Lorentz transform into local frame – calculate SC – transform back •BUT energy is different turn-by-turn and even within a bunch •SO particles are binned according to energy
This code is new to us, but has been tested with PSI 590 MeV ring! So, watch this space…
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Summary • No one-code-fits-all approach! • I have tried to introduce some codes which have
been commonly used within the UK for EMMA & PAMELA
• I have missed out LOTS of codes (COSY, GPT, MAD-X, PTC) due to personal experience
• New & old codes are constantly being developed & tested against experiment
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Resources Scode: http://www-bd.fnal.gov/icfabd/Newsletter43.pdf ZGOUBI code: http://sourceforge.net/projects/zgoubi/ OPAL code: http://amas.web.psi.ch/ Any scripts/macros I’ve presented: [email protected]
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