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

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

Transition:

Transition is defined as the point during acceleration where there is no deviation in the revolution period as a function of momentum:

T/T0 = (1/t2 –1/2) p/p0 = 0

where is the Lorentz relativistic factor for the synchronous on-momentum particle and t is a property of the lattice optics:

1/t2 = 1/C0D(s)/(s) ds

where D is the momentum-dispersion function, s is the longitudinal coordinate, and is the radius of curvature (in dipoles only)

So, transition in a lattice is changed by controlling the dispersion function location of dipoles

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

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

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

Example: FODO-based Missing Dipole Arc Module

Example: Strong Focussing Arc module, low x insert (original FMC)

Example: Strong Focussing Arc module with added low y insert

for increased t

Example: Strong Focussing Arc for 8GeV Ring

t ranges from 11i to 14i for this design

Arrangements of Sextupoles in the three configurations:standard FODO vs. the PD

Horz. Sextupole locations

Vert. Sextupole locations

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

Arc module tune How far from 0.75? Answer: > ±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 36 -60166.60709072684 4 0 0 1 3 42101.58355180608 37 -2342.472689445537 4 0 0 0 4 19035.25442672397

Tracking Performance:Impact 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?

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.

Performance:Impact 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.

Arrangements of Sextupoles in the three configurations:standard FODO, PD, and rearranged FODO

Horz. Sextupole locations

Vert. Sextupole locations

Performance:Rearranged 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?

Nonlinear Performance:Sensitivity 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?

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.

Nonlinear Performance and D A: PD

PD module tune: x , y = 0.750000singular 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).

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.

How will this delicate balance of nonlinear behavior withstand the introduction of nonlinearities:such as unavoidable magnet field errors*?

IT DOESN’T-- 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

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