FFAG's wonderful world

1
Optimization of Non-scaling FFAGs LatticesFor
Muon Acceleration
Shane Koscielniak, TRIUMF, January 2004
With thanks to S.Berg, M.Blaskiewicz,
M.Craddock, E.Courant, A.Garren, C.Johnstone, E.
Keil, A.Machida, F. Mills, Y.Mori, R.Palmer,
A.Sessler, D.Trbojevic, etc.

• How did we get here? - History
• Where are we now? Optimization and Status of ?
  lattices
• Where are we going? - Conclusions and outlook

2
Milestones

• Physics potential of ?-? Colliders, Dec. 1997
  Johnstone Mills brainstorm a F0D0 FFAG with
  large momentum compaction followed by a 4-16
  GeV nonscaling FFAG design presented at PAC March
  1999.
• ICFA/ECFA ?-Factory, July 1999 Mori suggests
  scaling FFAG for proton Driver, Johnstone a
  F0D0 FFAG decay ring.
• HEMC, Montauk, Sept. 1999 Garren and others
  suggest scaling FFAG accelerators and Trbojevic
  emphasizes importance of minimum normalized
  dispersion lattice design has 6 elements/cell.
• FFAGs for ? Acceleration, CERN, July 2000 Mori
  announces construction of PoP proton FFAG
  Johnstone acknowledges problem of quadratic
  pathlength in nonscaling FFAG.
• Snowmass 2001, July Berg starts analytical
  treatment of longitudinal motion in nonscaling
  FFAG Koscielniak starts tracking results
  presented at LBNL workshop Oct. 2001 and KEK Nov.
  2001 .
• NFMC Collaboration, May 2002 Trbojevic starts
  simplifying his lattice (removes sextupoles,
  starts removing quads reducing tune) --
  impressively small pathlength variation.
• EPAC 2002, June Berg and Koscielniak present
  tracking results and start to understand the
  longitudinal phase-space topolgy.

3
Milestones continued

• EPAC 2002, June Johstone continues to optimize
  the F0D0 cell (smaller aperture, more symmetric
  ?T). Trbojevic starts code testing on sector
  cyclotron.
• FFAG Racetrack designs abandoned cannot match
  arcs to straights.
• LBNL FFAG Workshop, Nov. 2002 Trbojevic
  Blaskiewicz present longitudinal tracking results
  from their lattice. Carol proposes triplet.
• PAC 2003, Koscielniak shows behaviour of
  quadratic pendula, etc, may be understood in
  terms of libration versus rotation manifolds and
  gives criteria for opening/closure of gutter
  paths. (Berg Keil also at work.)
• KEK FFAG Workshop, July. 2033 Minor
  improvements to F0D0 and tripet lattices
  Blaskiewicz adds beamloading effect to tracking.
• BNL FFAG Workshop, October 2003 almost
  universal understanding of how to design
  nonscaling FFAG lattices and Berg, Johnstone,
  Keil and Trbojevic begin optimization in earnest.
  Craddock and Koscielniak present thin lens models
  for displacements and pathlength variation in
  doublet, F0D0 and triplet lattices as basis of
  further optimizations.
• Sessler visions that nonlinear acceleration in
  non-scaling FFAGs is viable and proposes the
  electron demonstration model.

4
TPPG009
Conditions for connection of fixed points by
libration paths may be obtained from the
hamiltonian typically critical values of system
parameters must be exceeded.
5
Hamiltonian H(x,y,a)y3/3 y -a sin(x)
For each value of x, there are 3 values of y
y1gty2gty3
We may write values as y(z(x)) where
2sin(z)3(ba Sinx) y12cos(z-?/2)/3, y2-2sin(
z/3), y3-2cos(z?/2)/3.
The 3 libration manifolds are sandwiched between
the rotation manifolds (or vice versa) and become
connected when a?2/3. Thus energy range and
acceptance change abruptly at the critical value.
6
Quadratic Pendulum Oscillator
Phase space of the equations x'(1-y2) and
y'a.Cos(x)
a1/6
Animation evolution of phase space as strength
a varies.
Condition for connection of libration paths a ?
2/3
7
Design starts from the longitudinal dynamics
requirement
?E energy increment, ?E acceleration range, ?T
spread in transit times, ? angular frequency of
rf. W depends on the longitudinal emittance and
allowed dilution. Formula ? ?T
W may be obtained from tracking or estimated from
formulae.

• Thin lens model consists of thin D F
  quadrupoles with, possibly, thin dipoles
  superimposed at D and/or F. Drift spaces are
  D?F l1 F?D l2 in doublet F?F 2 l1
  F?D l2 in triplet where l1l22l0 and l1l2l0
  in F0D0. Let ??l1?l2/l0. Let pc be reference
  momentum and ? the bend angle (for ½ cell).
• Geometric considerations for cells of equal
  length and equal integrated quadrupole strength
  ?, pathlength variation is smallest for F0D0,
  then doublet, then triplet. For example, the
  inscribed triangle (F0D0) within a trapezium
  (triplet) has a smaller perimeter.
• The pathlength increment is

F0D0 ll0 Doublet l??l0 Triplet ll2??
8

• Range of transit times is minimized when
• leading to

• Betatron tune considerations

in thin-lens limit, for cells of equal length L0
and equal phase advance per cell ?, the
quadrupole strength is given by
F0D0 ll0, Doublet l??l0, Triplet ll2??

• Thus, remarkably, F0D0, doublet and triplet cells
  with equal L0 and equal ? have equal path length
  performance (in thin-lens limit).

• For an optimized nonscaling FFAG lattice,
  independent of how the bend angle is shared
  between D F, the spread in cell transit times
• the design must satisfy the longitudinal
  criterion

?pimpulse/cell, ?0L0/c reference transit time.
Note dependencies cubic on momentum range,
quadratic on bend angle, and linear on cell
length.
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• Design Goal Find the minimum momentum compaction
  lattice ?(?L/L)/ (?p/p) with the fewest number
  of cells consistent with longitudinal requirement
• Previous formulae give a bare bones starting
  point. Need constraints optimization criteria
  for complete design.
• Constraints
• Drift length(s) to accommodate rf cavity, i.e. ?
  2m
• Intermagnet spacing to accommodate auxil
  hardware, i.e. ? 0.5m
• Peak pole-tip fields ? 7T.
• Betatron phase advance/cell ? 0.7? at injection
  momentum.
• Desirable (but less essential)
• Small magnet apertures coupled to issue of peak
  fields
• Symmetric pathlength parabola for longitudinal
  dynamics
• A principle of optimization the minimum of two
  (like) quantities combined quadratically and with
  an additive constraint (xyc) occurs when they
  are equal (and possibly opposite).

10

• The range of transit times is minimized when
• The peak fields are minimized when they are equal
  and opposite.
• The cell phase advances ?x, ?y (at injection) are
  minimized when they are equal.
• These are sufficient conditions to choose
  reference momentum, the ratio of magnetic element
  lengths ldlf, and the quadrupole gradients if
  one has formulae (or numerical values) for the
  beam displacements and sizes, the pathlength
  variation, and the tunes.
• How the dipole bend is apportioned between D and
  F no influence on path length if pc is allowed to
  float but there are aperture implications
• Though they come to similar designs, there are
  individual preferences. E.g. Some designers have
  small split of peak fields.
• E.g. Berg constructs the parabola about its
  minimum at mean momentum.
• E.g. Trbojevic chooses strengths based on
  minimizing the normalized dispersion function.

11

• Optimal designs have maximal horizontal focusing
  giving the lowest disperion at the horizontal
  lowest-beta waist and placing the dipole at this
  location maximizes the momentum compaction.
• Remarkably, in thick lens models with cells of
  equal length and phase advance, etc., triplet
  lattices have superior momentum compaction
  compared with F0D0 probably because of the
  greater distortion of orbits compared with the
  thin-lens model.
• One other thick element effect for sector
  magnets additional horizontal focusing in ve
  bending D and -ve bending F.

Electron Model FDF Triplet
cells Cell length B1d (T/m) B1f (T/m) ld (m) lf (m) Bmax (T) ?T (ps)
Berg 24 0.4 2.8 4.18 .14 .05 0.20 8
Johnstone 28 0.4 2.3 6.8 .18 .03 0.20 6
Koscielniak 24 0.4 5.4 9.9 .08 .06 0.25 7.6
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Comparison of some recent (circa November 2003)
10-20 Gev F0D0 lattices (nonscaling FFAG)
Quantity Berg Johnstone Koscielniak
cells 82 108 91
Cell length 5.7 5.7 5.7
B1d (T/m) 23.7 18. 21
B1f (T/m) 30.6 60. 40.6
ld (m) 1.0 1.3 1.1
lf (m) 0.37 0.39 0.48
Bmax ?7 ?7 4.6
?T (ps) 6.4 7.8 11
December 2003
13
Comparison of some recent (circa Nov. 2003)
10-20GeV FDF triplet lattices (nonscaling FFAG)
Quantity Berg Johnstone Koscielniak Trbojevic
cells 93 101 82 72
Cell length 5.2 5.5 4.7 4.56
B1d (T/m) 30.4 20 24.6 56.6
B1f (T/m) 50.1 60 34.9 62.4
ld (m) 1.28 1.9 1.0 1.0
lf (m) 0.45x2 0.31x2 0.36x2 0.58x2
Bmax (T) ?7 ?7 6.5 ?7
?T (ps) 6.3 5.1 8.7 9.7
20cm magnet separation
14
Examples of closed orbits and pathlength from
C.Johnstone
F0D0
Triplet
15
Conclusions

• We have the understanding to design nonscaling
  FFAG lattices with ?T/?0 a few parts in 103
• Circumference has fallen from 2 to 0.5km
• These are ? 10 turn machines majority of
  acceln over 8 turns
• FDF Triplet shortest ??T, shorter cells, beta
  function less variable, BUT cavity at largest
  dispersion, slightly harder to symmetrize (higher
  peak field?)
• F0D0 adequate ??T performance, fewer magnets,
  extra slots for high harmonic cavities
• Do not forget the doublet option! it could be
  cheaper

Questions

• Should we be designing triplets for minimum
  UN-normalized dispersion?
• How essential is it to symmetrize the pathlength
  parabola?

16
Outlook it is time to consider 2nd order effects

• Beamloading when do libration paths close off?
• Coherent dipole and quadrupole oscillations?
• Is our particle tracking mature?
• Fringe fields multipole errors encountered
  systematically
• How to tune these lattices for the pathlength
  properties?
• Injection and Extraction?
• Cavity transverse HOMs?