Optimization of Non-scaling FFAG’s LatticesFor Muon Acceleration

1. Optimization of Non-scaling FFAG’s 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 . • “NF&MC 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. Rotation manifold Libration manifold y1 y2 y3 Hamiltonian: H(x,y,a)=y3/3 –y -a sin(x) For each value of x, there are 3 values of y: y1>y2>y3 We may write values as y(z(x)) where 2sin(z)=3(b+a Sinx) y1=+2cos[(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. a=1 a=1/2 a=2 Quadratic Pendulum Oscillator Phase space of the equations x'=(1-y2) and y'=a.Cos(x) a=1/6 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 l1+l2=2l0; and l1=l2=l0 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: l=l0 Doublet: l=l0 Triplet: l=l2

8. Range of transit times is minimized when leading to • Betatron tune considerations: in thin-lens limit, for cells of equal length L0and equal phase advance per cell , the quadrupole strength is given by: F0D0: l=l0, Doublet: l=l0, Triplet: l=l2 • 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 p=impulse/cell, 0=L0/c reference transit time. Note dependencies: cubic on momentum range, quadratic on bend angle, and linear on cell length.

9. 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 (x+y=c) 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 ld:lf, 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

12. Comparison of some recent (circa November 2003) 10-20 Gev F0D0 lattices (nonscaling FFAG) * December 2003

13. Comparison of some recent (circa Nov. 2003) 10-20GeV FDF triplet lattices (nonscaling FFAG) * 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 accel’n 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. Electron demonstration model Proceeding with designs for an “electron demonstration model” will be less challenging than muons – chosen 3 GHz, not 40 GHz. However, do not forget “small ring effects”. 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?