FFAG Modelling #2

“How to model an FFAG in 45-minutes”

12/9/2011 1

Suzie Sheehy, ASTeC Intense Beams Group, RAL

Outline • Requirements • Basic parameter choices • Optimisation • Detailed studies • Odds and ends

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

Requirements Particle species

Beam energy

Beam current

Pulse structure

Rep. rate Magnet type

RF type Space available

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…)

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

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

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...

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

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

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!

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

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

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)

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!!

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

Optimisation

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

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

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+

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…

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…)

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

Code comparison chart

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…

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

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: suzie.sheehy@stfc.ac.uk

27 12/9/2011