bucketless acceleration in nonscaling ffag’s & sketches of muon and electron machines shane...
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Bucketless Acceleration in nonscaling FFAG’s & Sketches of
muon and electron machinesShane 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
pendulumgold.avi
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Equations of motion from cell to cell of accelerator:
En+1=En+eV cos(Tn)
Tn+1=Tn+T(En+1)+(0-s)
0=reference cell-transit duration, s=2h/
Tn=tn-ns is relative time coordinate
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 0s (see later).
Phase space of the equations
x'=y and y'=a.Cos(x)
Linear Pendulum Oscillator
For simple pendulum, libration paths cannot become connected.Click on video clip to run Click on video clip to run
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Phase space of the equations x'=(1-y2) and y'=a.Cos(x)
a=1/6
a=1
a=1/2
a=2
Condition for connection of libration paths: a 2/3
Quadratic Pendulum Oscillator
Quadratic Pendulum Oscillator
Animation: evolution of phase space as strength `a’ varies.
Phase space of the equationsx'=(1-y2) and y'=a.Cos(x)
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Bi-parabolic Oscillator
Phase space of the equations
x'=(1-y2) and y'=a(x2-1)
Animation: evolution of phase space as strength
`a’ varies.
Condition for connection of libration paths: a 1
Topology discontinuous at a =1For a < 1 there is a sideways serpentine
pathFor 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=2a=1
a=1/2a=1/10
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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: 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].
Rotation manifold
Libration manifold
The 3 libration manifolds are sandwiched between the rotation manifolds (or vice versa) and become connected when a2/3. Thus energy range and acceptance change abruptly at the critical value.
y1
y2
y3
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
Small range of over-voltages
Phase spaces for normal rf and slip rf 6-
turnfundamental
input
outputAcceptance, =.49
eV.s
6-turnfundamental
input
Acceptance, =.52 eV.s
Muon Lattice and Magnet specifications
LatticeLattice ElementElement LengthLength
(metre)(metre)
DipoleDipole
(T)(T)
GradientGradient
(T/m)(T/m)
RadiusRadius
(m)(m)
AngleAngle
(rad)(rad)
B-peakB-peak
TeslaTesla
ApertureAperture
(cm)(cm)
F0D0F0D0 DD 0.7430.743 5.7225.722 34.8934.89 10.6410.64 .03491.03491 6.9066.906 6.06.0
F0D0F0D0 FF 0.3570.357 00 69.3569.35 ---- 00 6.9066.906 14.214.2
DoubletDoublet DD 0.9880.988 4.8374.837 39.9439.94 13.0413.04 .03785.03785 6.9906.990 7.257.25
DoubletDoublet FF 0.5120.512 00 72.772.7 ---- 00 6.9906.990 12.0612.06
TripletTriplet DD 1.201.20 4.8854.885 34.6334.63 13.3813.38 .04488.04488 6.7526.752 6.06.0
TripletTriplet FF 0.3500.350 00 68.8968.89 ---- 00 6.7526.752 10.510.5
LatticeLattice # cells# cells Cell lengthCell length
(metre)(metre)
Circumference Circumference (metre)(metre)
Drift1Drift1 Drift2Drift2 WW PcPc
GeV/cGeV/c
F0D0F0D0 9090 5.15.1 459459 22 22 .0846.0846 18.25818.258
DoubletDoublet 8383 4.04.0 332332 22 0.50.5 .0878.0878 18.91618.916
TripletTriplet 7070 4.94.9 343343 22 0.50.5 .0908.0908 19.59519.595
10-20 GeV/c muons; cell tunes x=y=0.35, Bmax7 Tesla, W1/12=.0833, 11 MV/cell @ 200 MHz
Radiation hard, DC magnets, vacuum chamber not included
Muon Lattice and Magnet specifications
LatticeLattice ElementElement LengthLength
(metre)(metre)
DipoleDipole
(T)(T)
GradientGradient
(T/m)(T/m)
RadiusRadius
(m)(m)
AngleAngle
(rad)(rad)
B-peakB-peak
TeslaTesla
ApertureAperture
(cm)(cm)
F0D0F0D0 DD 1.1391.139 3.5333.533 21.1221.12 17.40717.407 .03272.03272 4.3944.394 6.756.75
F0D0F0D0 FF 0.5610.561 00 40.6640.66 ---- 00 4.3944.394 15.015.0
DoubletDoublet DD 1.3301.330 3.1733.173 22.3122.31 19.68619.686 .03378.03378 4.3204.320 7.477.47
DoubletDoublet FF 0.6700.670 00 41.5541.55 ---- 00 4.3204.320 13.6513.65
TripletTriplet DD 1.7011.701 3.1103.110 21.0221.02 21.11421.114 .04028.04028 4.4464.446 6.946.94
TripletTriplet FF 0.49960.4996 00 40.3240.32 ---- 00 4.4464.446 11.711.7
LatticeLattice # cells# cells Cell lengthCell length
(metre)(metre)
Circumference Circumference (metre)(metre)
Drift1Drift1
(metre)(metre)
Drift2Drift2
(metre)(metre)
WW PcPc
GeV/cGeV/c
F0D0F0D0 9696 5.75.7 547.2547.2 22 22 .0843.0843 18.43818.438
DoubletDoublet 9393 5.05.0 465465 22 11 .0876.0876 18.72318.723
TripletTriplet 7878 5.75.7 444.6444.6 22 0.50.5 .0899.0899 19.68419.684
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
90 Cell F0D0 lattice for muons
24 Cell F0D0 lattice; electrons
Path length variation for F0D0 muon lattice
Path length variation for F0D0 electron lattice
Electron Lattice and Magnet specifications
LatticeLattice ElementElement LengthLength
(cm)(cm)
DipoleDipole
(T)(T)
GradientGradient
(T/m)(T/m)
RadiusRadius
(m)(m)
AngleAngle
(rad)(rad)
B-peakB-peak
TeslaTesla
ApertureAperture
(cm)(cm)
F0D0F0D0 DD 9.1499.149 0.15190.1519 3.6113.611 0.39320.3932 0.11640.1164 0.19290.1929 1.951.95
F0D0F0D0 FF 4.8514.851 00 6.1476.147 ---- 00 0.19290.1929 4.414.41
DoubletDoublet DD 11.7111.71 0.12770.1277 3.6663.666 0.48470.4847 0.12080.1208 0.19280.1928 2.462.46
DoubletDoublet FF 6.296.29 00 6.1946.194 ---- 00 0.19280.1928 4.04.0
TripletTriplet DD 13.013.0 0.13370.1337 3.3433.343 0.47620.4762 0.13660.1366 0.19240.1924 2.082.08
TripletTriplet FF 4.04.0 00 5.9035.903 ---- 00 0.19240.1924 3.693.69
LatticeLattice # cells# cells Cell lengthCell length
(cm)(cm)
Circumference Circumference (metre)(metre)
Drift1Drift1
(cm)(cm)
Drift2Drift2
(cm)(cm)
WW PcPc
MeV/cMeV/c
F0D0F0D0 2727 4040 10.810.8 1515 1515 .2586.2586 17.90817.908
DoubletDoublet 2626 3737 9.629.62 1515 55 .2628.2628 18.56018.560
TripletTriplet 2323 4646 10.5810.58 1515 55 .2676.2676 19.09519.095
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
Electron Lattice and Magnet specifications
LatticeLattice ElementElement LengthLength
(cm)(cm)
DipoleDipole
(T)(T)
GradientGradient
(T/m)(T/m)
RadiusRadius
(m)(m)
AngleAngle
(rad)(rad)
B-peakB-peak
TeslaTesla
ApertureAperture
(cm)(cm)
F0D0F0D0 DD 6.636.63 0.15780.1578 5.2335.233 0.37990.3799 .08727.08727 0.19420.1942 1.251.25
F0D0F0D0 FF 3.373.37 00 9.4909.490 ---- 00 0.19420.1942 2.932.93
DoubletDoublet DD 9.809.80 0.13870.1387 5.5995.599 0.44670.4467 .09520.09520 0.19910.1991 1.541.54
DoubletDoublet FF 5.205.20 00 9.7549.754 ---- 00 0.19910.1991 2.662.66
TripletTriplet DD 10.010.0 0.13800.1380 4.7754.775 0.46290.4629 0.10830.1083 0.19020.1902 1.311.31
TripletTriplet FF 3.03.0 00 8.9738.973 ---- 00 0.19020.1902 2.402.40
LatticeLattice # cells# cells Cell lengthCell length
(cm)(cm)
Circumference Circumference (metre)(metre)
Drift1Drift1
(cm)(cm)
Drift2Drift2
(cm)(cm)
WW PcPc
MeV/cMeV/c
F0D0F0D0 3636 4040 14.414.4 1515 1515 .2599.2599 17.97317.973
DoubletDoublet 3333 3333 10.8910.89 1515 55 .2521.2521 18.58018.580
TripletTriplet 2929 4141 11.8911.89 1515 55 .2595.2595 19.14719.147
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
Cross-section of Combined Function Magnet - has dipole and quadrupole field components
100 cell F0D0 lattice (Johnstone)
100 cell Triplet lattice (Johnstone)
Pathlength variation for F0D0
Pathlength variation for Triplet
Quartic Pendulum Oscillator
Phase space of the equations
x'=y2(1-b2y2)-1 and y'=a.Cos(x)Animations: evolution of phase space as strengths `a,b’
vary.
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.
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