1. Novel Constant-Frequency Acceleration Technique for Nonscaling Muon FFAGs Shane Koscielniak, TRIUMF, October 2004 Classical scaling FFAGs (MURA) have geometrically self-similar orbits that lead to constant betatron tune vs momentum. New nonscaling FFAGs break this tradition. In particular, the variable-tune linear-field FFAG offers very high momentum compaction. For several GeV muons, and s.c. magnets, the range of spiral orbits with  p/p up to  50% is contained in an aperture of a few cms. Fixed-Field Alternating Gradient (FFAG) accelerators were originally developed by the MURA group in late 1950s. In the few turns timescale intended for muon acceleration, the magnet field and the radio-frequency cannot be other than fixed. This leads to a machine with novel features: crossing of transverse resonances and asynchronous acceleration.

2. 36 Cell F0D0 lattice; electrons Variable-tune linear-field FFAGs Orbits vs Momentum 96 Cell F0D0 lattice for muons

3. Per-cell path length variation for 10-20 GeV F0D0 lattice for muons Per-cell path length variation for 10-20 MeV F0D0 lattice for electrons Lattice cell is F quadrupole & combined function D

4. Kick model of FFAG consists of thin D & F quads and thin dipoles superimposed at D and/or F. Equal integrated quad strengths . Let p c be reference momentum and  the bend angle (for ½ cell). Drift spaces in F0D0 are equal to l From Pythagoras, the pathlength increment is: Displacements at centre of D & F: In thin-lens limit, for cells of equal length L 0 and equal phase advance per cell , the quadrupole strength is given by: Range of transit times is minimized when leading to Thus, for an optimized nonscaling FFAG lattice, the spread in cell transit times:

5. Longitudinal Equations of motion from cell to cell: E n+1 =E n +eV cos(  T n ) - energy gain T n+1 =T n +  T(E n+1 ) - arrival time  s =reference cell-transit time,  s =2  h/  T n =t n -n  s is relative time coordinate Conventional case:  =  (E),  T is linear, yields synchronous acceleration: the location of the reference particle is locked to the waveform, or moves adiabatically. Other particles perform oscillations about the reference particle. Non-scaling FFAG case:  fixed,  T is parabolic, yields asynchronous acceleration: the reference particle performs nonlinear oscillation about the crest of the waveform; and other particles move convectively about the reference.

6. Phase space of the equations x'=y and y'=a.Cos(x) Linear Pendulum Oscillator For simple pendulum, rotation paths cannot become connected. Manifold: set of phase-space paths delimited by a separatrix Libration: bounded periodic orbits Rotation: unbounded, possibly semi- periodic, orbits

7. Quadratic Pendulum Oscillator Phase space of the equations x'=(1-y 2 ) and y'=a.Cos(x) a=2/3

8. Phase space of the equations x'=(1-y 2 ) and y'=a.Cos(x) a=1/6 a=1 a=1/2 a=2 Condition for connection of rotation paths: a  2/3 Quadratic Pendulum Oscillator

9. Hamiltonian: H(x,y,a)=y 3 /3 –y -a sin(x) For each value of x, there are 3 values of y: y 1 >y 2 >y 3 We may write values as y[z(x)] where 2sin(z)=3(b+a Sinx) y 1 =+2cos[(z-  /2)/3], y 2 =-2sin(z/3), y 3 =-2cos[(z+  /2)/3]. Libration manifold Rotation manifold The 3 rotation manifolds are sandwiched between the libration manifolds (& vice versa) and become connected when a  2/3. Thus energy range and acceptance change abruptly at the critical value. y1y1 y2y2 y3y3

10. Phase portraits for 3 to 12 turn acceleration Acceptance and energy range versus voltage for acceleration in 4 to 12 turns Small range of over-voltages

11. The fixed points z i =(T,E) i are solutions of: T n+1 =T n (instantaneous synchronism) and E n+1 =E n (no energy change) c.f. fixed “point” of the transverse motion is the closed orbit x n+1 =x n and x ’ n+1 =x ’ n The direction of phase slip reverses at each fixed point, so the criterion is simply that voltage be large enough that another fixed point be encountered before a  phase slip has accumulated. Essentially the scheme operates by allowing the beam to slide from one condition of synchronism to another; but a threshold voltage is required to achieve this. General principle for acceleration over a range spanning multiple fixed points: The condition is simply that hamiltonian be equal at the unstable fixed points: H(z 1 )=H(z 2 )=H(z 3 ), etc The rf voltage must exceed the critical value to link the unstable fixed points in a zig-zag ladder of straight line segments.

12. TPPG009 Conditions for connection of unstable fixed points by rotation paths may be obtained from the hamiltonian; typically critical values of system parameters must be exceeded.

13. Quartic Pendulum Oscillator Phase space of the equations x'=y 2 (1-b 2 y 2 )-1 and y'=a.Cos(x) b held fixed at b=1/3 a=1/4 a=1 a=3 a=2  3/5 Critical value to link fixed points

14. Isochronous AVF Cyclotron (TRIUMF) Cyclotron is much more isochronous than muon FFAG. So do not need GeVs or MeVs per turn, 100 keV enough Longitudinal trajectory as measured by time-of-flight Craddock et al, 1977 PAC Longitudinal trajectory as computed, 2004, by Rao & Baartman

15. Conclusion The asynchronous acceleration principle devised to explain and predict the properties of the variable-tune linear-field FFAG is seen to be perfectly at home in the world of the (imperfectly) isochronous cyclotron. Animations showing the evolution of phase space, as parameters are varied, for the quadratic, cubic and quartic pendulum may be viewed @W3 *.avi movie files are located at http://www.triumf.ca/people/koscielniak/FFAGws/ Files in GIF format are located at http://www.triumf.ca/people/koscielniak/nonlinearffag.htm