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Dynamics of Nonlinear Acceleration in FFAG Systems - Shane Koscielniak, TRIUMF, Vancouver

Detailing the principles and equations governing acceleration in nonscaling FFAG systems, with illustrations of muon and electron machines. Explore phase space properties, manifold dynamics, nonlinear system behavior, and operational modes.

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Dynamics of Nonlinear Acceleration in FFAG Systems - Shane Koscielniak, TRIUMF, Vancouver

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  1. Bucketless Acceleration in nonscaling FFAG’s & Sketches of muon and electron machines Shane Koscielniak, TRIUMF, Vancouver, BC, Canada April 2004 • Phase space properties of pendula • Manifold: set of phase-space paths delimited by a separatrix • Rotation: bounded periodic orbits • Libration: unbounded, possibly semi-periodic, orbits • When dependence of ‘speed’ on momentum is nonlinear system is characterized by discontinuous behaviour w.r.t. parameters Remember to close media player before proceeding to next slide

  2. Equations of motion from cell to cell of accelerator: 0=reference cell-transit duration, s=2h/ Tn=tn-ns is relative time coordinate En+1=En+eV cos(Tn) Tn+1=Tn+T(En+1)+(0-s) Conventional case: =(E), T is linear, 0=s, yields synchronous acceleration: the location of the reference particle is locked to the waveform, or moves adiabatically. Other particles perform (usually nonlinear) oscillations about the reference particle. Non-scaling FFAG case:  fixed, T is nonlinear, yields asynchronous acceleration: the reference particle performs a nonlinear oscillation about the crest of the waveform; and other particle move convectively about the reference.Two possible operation modes are normal0=s and slip0s (see later).

  3. Linear Pendulum Oscillator For simple pendulum, libration paths cannot become connected. Click on video clip to run animationRemember to close media player before proceeding to next slide Phase space of the equations x'=yand y'=a.Cos(x)

  4. a=1 a=1/6 a=1/2 a=2 Quadratic Pendulum Oscillator Condition for connection of libration paths: a  2/3 Phase space of the equationsx'=(1-y2) and y'=a.Cos(x)

  5. Quadratic Pendulum Oscillator Animation: evolution of phase space as strength `a’ varies. Click on video clip to run animationRemember to close media player before proceeding to next slide Phase space of the equations x'=(1-y2) and y'=a.Cos(x)

  6. Bi-parabolic Oscillator Topology discontinuous at a =1 Phase space of the equations x'=(1-y2)and y'=a(x2-1) • For a < 1 there is a sideways serpentine path • For a > 1 there is a upwards serpentine path • For a 1 there is a trapping of two counter-rotating eddies within a background flow. a=1/10 a=1/2 Click on video clip to run animationRemember to close media player before proceeding to next slide a=1 a=2 Condition for connection of libration paths: a  1 Animation: evolution of phase space as strength `a’ varies.

  7. 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.

  8. Small range of over-voltages Phase portraits for 3 through 12 turn acceleration; normal rf Acceptance and energy range versus voltage for acceleration completed in 4 through 12 turns Small range of over-voltages

  9. 6-turn 6-turn fundamental fundamental input input output Acceptance, =.49 eV.s Acceptance, =.52 eV.s Phase spaces for normal rf and slip rf

  10. Muon Lattice and Magnet specifications 10-20 GeV/c muons; cell tunes x=y=0.35, Bmax7 Tesla, W1/12=.0833, 11 MV/cell @ 200 MHz Radiation hard, DC magnets, vacuum chamber not included

  11. Muon Lattice and Magnet specifications 10-20 GeV/c muons; cell tunes x=y=0.35, Bmax 4.5 Tesla, W  1/12=.0833, 11 MV/cell @ 200 MHz Radiation hard, DC magnets, vacuum chamber not included

  12. 90 Cell F0D0 lattice for muons 24 Cell F0D0 lattice; electrons

  13. Path length variation for F0D0 muon lattice Path length variation for F0D0 electron lattice

  14. Electron Lattice and Magnet specifications 10-20 MeV/c electrons; cell tunes x=y=0.35, Bmax 0.20 Tesla, W  1/4=.250, 0.5 MV/cell @ 2.86 GHz Radiation soft, DC magnets, vacuum chamber not included

  15. Electron Lattice and Magnet specifications 10-20 MeV/c electrons; cell tunes x=y=0.35, Bmax 0.20 Tesla, W  1/4=.250, 0.25 MV/cell @ 2.86 GHz Radiation soft, DC magnets, vacuum chamber not included

  16. Cross-section of Combined Function Magnet - has dipole and quadrupole field components

  17. 100 cell F0D0 lattice(Johnstone) Pathlength variation for F0D0 Pathlength variation for Triplet 100 cell Triplet lattice(Johnstone)

  18. Phase space of the equations x'=y2(1-b2y2)-1 and y'=a.Cos(x) Quartic Pendulum Oscillator Animations: evolution of phase space as strengths `a,b’ vary. Click on video clip to run animation Click on video clip to run animation Parameter `a` is varied from 0.1 to 2.9 while `b` held fixed at b=1/3. Parameter `b` is varied from to 0.1 to 0.5 while `a` held fixed at a=3/4. Remember to close media player before proceeding to next slide/animation

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