Novel Constant-Frequency Acceleration Technique for Nonscaling Muon FFAGs

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Novel Constant-Frequency Acceleration Technique
for Nonscaling Muon FFAGs
Shane Koscielniak, TRIUMF, October 2004
Fixed-Field Alternating Gradient (FFAG)
accelerators were originally developed by the
MURA group in late 1950s.
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 .
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.
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96 Cell F0D0 lattice for muons
Orbits vs Momentum
36 Cell F0D0 lattice electrons
Variable-tune linear-field FFAGs
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Per-cell path length variation for 10-20 GeV F0D0
lattice for muons
Lattice cell is F quadrupole combined function D
Per-cell path length variation for 10-20 MeV F0D0
lattice for electrons
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Kick model of FFAG consists of thin D F quads
and thin dipoles superimposed at D and/or F.
Equal integrated quad strengths ?. Let pc be
reference momentum and ? the bend angle (for ½
cell). Drift spaces in F0D0 are equal to l
Displacements at centre of D F
From Pythagoras, the pathlength increment is
Range of transit times is minimized when leading
to
In thin-lens limit, for cells of equal length L0
and equal phase advance per cell ?, the
quadrupole strength is given by
Thus, for an optimized nonscaling FFAG lattice,
the spread in cell transit times
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Longitudinal Equations of motion from cell to
cell
?sreference cell-transit time,
?s2?h/? Tntn-n?s is relative time coordinate
En1EneV cos(?Tn) - energy gain Tn1Tn?T(En
1) - arrival time
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.
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Linear Pendulum Oscillator
Phase space of the equations x'y and y'a.Cos(x)
Manifold set of phase-space paths delimited by a
separatrix
Libration bounded periodic orbits
Rotation unbounded, possibly semi-periodic,
orbits
For simple pendulum, rotation paths cannot become
connected.
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Quadratic Pendulum Oscillator
Phase space of the equations x'(1-y2) and
y'a.Cos(x) a2/3
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Quadratic Pendulum Oscillator
Condition for connection of rotation paths a ?
2/3
Phase space of the equations x'(1-y2) and
y'a.Cos(x)
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Hamiltonian H(x,y,a)y3/3 y -a sin(x)
For each value of x, there are 3 values of y
y1gty2gty3
We may write values as yz(x) where
2sin(z)3(ba Sinx) y12cos(z-?/2)/3, y2-2sin
(z/3), y3-2cos(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.
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Small range of over-voltages
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
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General principle for acceleration over a range
spanning multiple fixed points
The rf voltage must exceed the critical value to
link the unstable fixed points in a zig-zag
ladder of straight line segments.
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.
The fixed points zi(T,E)i are solutions
of Tn1Tn (instantaneous synchronism) and
En1En (no energy change) c.f. fixed point
of the transverse motion is the closed orbit
xn1xn and xn1xn
The condition is simply that hamiltonian be equal
at the unstable fixed points H(z1)H(z2)H(z3),
etc
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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.
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Quartic Pendulum Oscillator
Critical value to link fixed points
b held fixed at b1/3
a1/4
a2?3/5
a3
a1
Phase space of the equations x'y2(1-b2y2)-1 and
y'a.Cos(x)
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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
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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 _at_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/nonlinearf
fag.htm