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

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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

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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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