TRANSITION-FREE LATTICES

1
TRANSITION-FREE LATTICES

• C Johnstone and B. Eredelyi
• Fermilab
• NuFact02
• Imperial College, London
• July1-6, 2002
• special thanks to A. Thiessan

WG1 July 4 NuFact02 Imperial College, London July
1-6, 2002
2
TOPICS

• 1. What is transition in an accelerator -
  definition and reality
• 2. Transitionless Lattices types
• 3. Performance Resonance and Dynamic Aperture
  (DA) studies of Transitionless Lattices
• 4. Recent results on Proton Driver Lattices
  comparison with standard FODO
• 5. Preliminary Conclusions

3
Transition

• Transition is defined as the point during
  acceleration where there is no deviation in the
  revolution period as a function of momentum
• DT/T0 (1/gt2 1/g2) Dp/p0 0
• where g is the Lorentz relativistic factor for
  the synchronous on-momentum particle and gt is a
  property of the lattice optics
• 1/gt2 1/C0?D(s)/r(s) ds
• where D is the momentum-dispersion function, s is
  the longitudinal coordinate, and r is the radius
  of curvature (in dipoles only)
• So, transition in a lattice is changed by
  controlling the
• dispersion function
• location of dipoles

4

• In general, transition decreases with cell
  length, keeping the phase advance constant I.e.
  dispersion function decreases.
• BUT
• A standard FODO cell lattice for the Proton
  Driver would require ultra short cells with
  inter-quad spacing of only 4m.
• Clearly this is not an option, If only from beam
  injection/extraction standpoint

5
Types of Transitionless Lattices

• Missing-Dipole FODO
• Based on the standard FODO module--but dipoles
  are removed from high dispersion regions
• Transition increases both from the missing
  dipoles, but also from the decrease in the
  dispersion function which occurs.
• Strong-focussing
• Control the dispersion function by increasing the
  horizontal focussing strength over and above the
  FODO through a horizontal low-beta insertion
• Regions of negative dispersion can created,
  often driving transition imaginary (historically
  referred to as Flexible Momentum Compaction or
  FMC module).
• a more recent doublet lattice by G. Reese is
  not studied in this work

6
Comparative Features Missing
Dipole and Strong Focussing Lattices

• Missing Dipole FODO Strong Focussing
  (FMC)
• Simplest Structure Low Beta Insertion
• Lowered Dipole packing Standard Dipole
  packing
• requires spaces in arcs comparable to
  standard FODO
• Limited range in ?t Large range in ?t
  (real-imaginary)
• Dispersion suppression Dispersion suppression
• standard or phase-induced generally
  efficient
• Shortened utility straights More generous
  utility straights

7
Example FODO-based Missing Dipole Arc Module
8
Example Strong Focussing Arc module, low bx
insert (original FMC)
9
Example Strong Focussing Arc module with added
low by insert

• for increased ?t

10
Example Strong Focussing Arc for 8GeV Ring

• ?t ranges from 11i to 14i for this design

11
Arrangements of Sextupoles in the three
configurationsstandard FODO vs. the PD
Horz. Sextupole locations
Vert. Sextupole locations
12
Performance FODO-based arc moduleTotal arc
module phase advance of 270

• First, look at 3 x standard FODO cells with
  standard chromatic correction
• with ?x , ?y 0.750000 (x 2?)
• DA is almost nonexistent due to 4th and
  other HO
• order resonances
• For the Proton Driver module tune 0.75 the map is
  not as clear, but later tracking results showed
  an unacceptable sensitivity to any changes in its
  nonlinear composition.
• there is an enhancement in the DA at phase
  advances which are odd multiples of 90, 0.25,
  and 0.75, for example

13
Arc module tune

• How far from 0.75?
  Answer gt 0.03
• 0.75 coefficients order exp.
  New coefficients
• 19 -15.37743243630632 4 4 0 0 0
  -4.685819636401535
• 20 875.6921990923597 4 3 1 0 0
  4846.570946996038
• 21 -152602.0766201559 4 2 2 0 0
  4915.864249415151
• 22 -1551378.006500757 4 1 3 0 0
  -340048.9047066236
• 23 -2949880.021197357 4 0 4 0 0
  -19231593.78153443
• 24 -203.7740240642712 4 2 0 2 0
  -64.21627192997873
• 25 -11429.99602178208 4 1 1 2 0
  -7178.460145068667
• 26 -216025.9940977090 4 0 2 2 0
  -199617.5814816107
• 27 -3322.702237255702 4 2 0 1 1
  -1227.186498329944
• 28 -55510.77100820898 4 1 1 1 1
  11255.74739937733
• 29 -1407927.049676028 4 0 2 1 1
  -320886.3452778102
• 30 -8596.828047477644 4 2 0 0 2
  -3998.749005011314
• 31 10069.93733752568 4 1 1 0 2
  126330.6797013642
• 32 -2069862.948423631 4 0 2 0 2
  -570823.2944196387
• 33 -607.0883626214262 4 0 0 4 0
  -377.6830629790347
• 34 -3767.383725082195 4 0 0 3 1
  -1831.664889008636
• 35 -35265.17142457649 4 0 0 2 2
  -5245.639853101801

14
Tracking PerformanceImpact of Tune Change from
?x , ?y 0.75 to 0.72

• Standard FODO displays about half the dynamic
  aperture of the Proton Driver module
• Proton Driver module very slightly enhanced at
  new module tune
• The primary nonlinear components, the chromatic
  correction sextupoles comprise two orthogonal
  (90) families in both lattices, so
• Why the dramatic improvement in DA of the proton
  driver module over the standard FODO?

15

• Hypothesis If this enhancement is due solely to
  the chromatic correction sextupoles, then of
  their location and definition dictate the
  performance this lattice.
• Test
• Relocation of the sextupoles in a standard 3-cell
  FODO should reproduce this effect.

16
PerformanceImpact of Chromatic Correction
Sextupole Placement

• To study the role of the chromatic correction
  sextupoles in the PD lattice, three modules were
  studied and compared
• FODO 3-cell module with standard sextupole
  familes (two per plane)
• PD arc module, which,again, has two sextupole
  families per plane, but are fewer in number and
  have different relative phasing between planes
• Rearranged FODO 3 FODO cells with sextupoles
  placed in the same relative location (phase
  advance) as in the PD arc module.

17
Arrangements of Sextupoles in the three
configurationsstandard FODO, PD, and rearranged
FODO
Horz. Sextupole locations
Vert. Sextupole locations
18
PerformanceRearranged FODO

• Rearranged FODO shows identical DA to the PD arc
  module it is increased by a factor of 2 over
  the normal arrangement.
• Confirmation that the large DA evidenced in the
  PD lattice is an artifact of the exact sextupole
  arrangement used.
• How will such a lattice perform to a change in
  the nonlinear optics?

19
Nonlinear PerformanceSensitivity to the
chromatic correction sextupoles

• Turn off the horizontal sextupoles
• DA aperture in FODO increases in both planes, as
  expected, from the removal of a nonlinearity.
• In the PD module, for ?x , ?y 0.75, the
  vertical DA drops by an order of magnitude (from
  15 cm to 1.5 cm) I.e. removal of a strong
  nonlinearity causes a tremendous decrease in the
  acceptance of the machine.
• In the PD module, for ?x , ?y 0.72, the
  decrease is still unacceptable, but is now 30.
• The Rearranged FODO verifies the unexpected
  decrease in DA
• WHAT IS GOING ON?

20
Nonlinear Performance and DA FODO

• Standard sextupole placement in a 90 FODO
  lattice is relatively insensitive to sextupole
  cross-correlations
• The two planes are strongly independent of each
  other
• Removal of nonlinearities results in the expected
  enhancement of performance.

21
Nonlinear Performance and D A PD

• PD module tune ?x , ?y 0.750000
• singular solution in a nonlinear optics regime
• DA relies entirely on a delicate cancellation of
  tuneshift contributions between sextupole
  families in different planes. This solution has
  no inherent stability.
• In this particular module, the vertical DA shows
  an extreme dependence on the horizontal
  sextupoles, but the reverse is not true, the
  horizontal DA is not sensitive to the vertical
  sextupoles (the horizontal sextupole placement
  is more near the standard).

22
Nonlinear Performance and DA PD

• PD module tune ?x , ?y 0.72
• more robust, but performance still dictated
  strongly by the nonlinear rather than the linear
  optics
• DA still relies on cancellation of nonlinear
  terms between sextupole families in different
  planes.
• These conclusions hold for the rearranged FODO.

23
How will this delicate balance of nonlinear
behavior withstand the introduction of
nonlinearitiessuch as unavoidable magnet field
errors?

• IT DOESNT--
• Dramatic decrease in DA, up to an order of
  magnitude in PD module performance.
• No significant tune dependence of DA in any
  implementation of the arc module. (0.75, or 0.25
  is no longer a magic tune)
• No significant performance difference between the
  arc modules.
• magnet field errors are taken from the MI design
  report

24
Preliminary Conclusions

• Lattices which rely on the delicate cancellation
  of nonlinear terms do not survive.
• The DA enhancement of such lattices is
  artificial.
• All of the modules show identical performance
  after introducing MI magnet errors