Resonances
field imperfections and normalized field errors
smooth approximation for motion in accelerators
perturbation treatment
chaotic particle motion
introduction: driven oscillators and resonance condition
sextupole perturbation & slow extraction
Poincare section
stabilization via amplitude dependent tune changes
Introduction: Damped Harmonic Oscillator
equation of motion for a damped harmonic oscillator:
0 is the Eigenfrequency of the HO
0)()()( 20
102
2 twtwQtw dtd
dtd
Q is the damping coefficient
(amplitude decreases with time)
example: weight on a spring (Q = )
0)()(2
2 twktwdtd )sin()( 0 tkatw
k
Introduction: Driven Oscillators
general solution:
an external driving force can ‘pump’ energy into the system:
)](cos[)()( tWtwst
)()()( twtwtw sttr
stationary solution:
where ‘’ is the driving angular frequency!and W() can become large for certain frequencies!
)cos()()()( 20
102
2
tm
FtwtwQtw dt
ddtd
Introduction: Driven Oscillators
stationary solution
)](cos[)()( tWtwst
stationary solution follows the frequency of the driving force:
W()
oscillation amplitude can become large for weak damping
Q>1/2
Q<1/2
Q>1/2
Q<1/2
Introduction: Pulsed Driven Resonances Example
higher harmonics:
example of a bridge:
2nd harmonic: 3nd harmonic: 5nd harmonic:
peak amplitude depends on the excitation frequency and damping
[Bob Barrett; Messiah College]
Introduction: Instabilities
weak damping:
resonance catastrophe without damping:
excitation by strong wind on the Eigenfrequencies
2)(2])(1[
1
00
2)0()(
Q
WW
Tacoma Narrow bridge1940
resonance condition: 0
Smooth Approximation: Resonances in Accelerators
revolution frequency:
betatron oscillations:
periodic kick
F rev = 2 frev)
F
driven oscillator
weak or no damping!(synchrotron radiation damping (single particle) or Landau damping distributions)
excitation with frev
Eigenfrequency:0 = 2 f
Q = 0 / rev
Smooth Approximation: Free Parameter
co-moving coordinate system:
equations of motion:
choose the longitudinalcoordinate as the freeparameter for the equationsof motion
y
x
s
dsd
dtds
dtd
2
222
2
dsdv
dtd
with: vdtds
Smooth Approximation: Equation of Motion I
Smooth approximation for Hills equation:
perturbation of Hills equation:
in the following the force term will be the Lorenz force of a charged particle in a magnetic field:
)/()),(()()( 202
2pvsswFswsw
dsd
0)()()(2
2 swsKswdsd
)cos()( 00 sAsw
(constant -function and phase advance along the storage ring)
0)()( 202
2 swswdsd
K(s) = const
LQ /2 00 (Q is the number of oscillations during one revolution)
BvqF
Field Imperfections: Origins for Perturbations
linear magnet imperfections: derivation from the design dipole and quadrupole fields due to powering and alignment errors
time varying fields: feedback systems (damper) and wake fields due to collective effects (wall currents)
non-linear magnets: sextupole magnets for chromaticity correction and octupole magnets for Landau damping
beam-beam interactions: strongly non-linear field!
non-linear magnetic field imperfections: particularly difficult to control for super conducting magnets where the field quality is entirely determined by the coil winding accuracy
Field Imperfections: Localized Perturbation
periodic delta function:
Fourier expansion of the periodic delta function:
infinite number of driving frequencies
0
1{)( 0 ssL
)/(),()/2cos()()( 202
2pvswFLsrswsw
rLl
dsd
equation of motion for a single perturbation in the storage ring:
)/(),()()()( 02
02
2pvswFlssswsw Lds
d
for ‘s’ = s0
otherwiseand 1)( 0 dsssL
Field Imperfections: Constant Dipole
rQLr LQ 0
/20
00/2
normalized field error:
resonance condition:
r
Lsrswsw Llk
dsd )/2cos()()( 02
02
2
avoid integer tunes!
remember the example of a single dipole imperfection from the ‘Linear Imperfection’ lecture yesterday!
pBqkpv
Bvq
pv
F Bv /0
equation of motion for single kick:
Field Imperfections: Constant Quadrupole
)()()( 12
2
2sxksxsx xds
d equations of motion:
0)( sy
0)()()( 12
2
2 sxksx xdsd
with:x
B
p
qk y
1
change of tune but no amplitude growth due to resonanceexcitations!
Field Imperfections: Single Quadrupole Perturbation
assume y = 0 and Bx = 0:
)()/2cos()()( 120,2
2sxLsrsxsx
rx L
lk
dsd
xklsspvsF L 10 )()/()(
exact solution: variation of constants or MAP approach see the lecture yesterday
resonance condition: 2//2 0/2
0,0,00 nQLr LQ
xx
avoid half integer tunes plus resonance width from tune modulation!
Field Imperfections: Time Varying Dipole Perturbation
time varying perturbation:
)/()/]/[2cos()()( 2
202
20 pvLsrswsw
rrevkickL
lF
dsd
resonance condition:
)(/)/(2 0/2
000 rQffLr revkick
LQrevkick
avoid excitation on the betatron frequency!
)/()/2cos()cos()( 00 pvLsFtFtFrev
kickstkick
(the integer multiple of the revolution frequency corresponds to the modes of the bridge in the introduction example)
Field Imperfections: Several Bunches:);cos()( revkickkick tBtF
FF
machine circumference
:2);cos()( revkickkick tBtF
FF
higher modes analogous to bridge illustration
Field Imperfections: Dipole Magnets
dipole magnet designs:
conventional magnet designrelying on pole face accuracyof a Ferromagnetic Yoke
LEP dipole magnet:
air coil magnet design relyingon precise current distribution
LHC dipole magnet:
Field Imperfections: Super Conducting Magnetstime varying field errors in super conducting magnets
Luca Bottura CERN, AT-MAS
t
I
192.1 A
11743A
769.1A
Field Imperfections: Multipole ExpansionTaylor expansion of the magnetic field:
with:n
nn
xy iyxfiBB n )(0
!1
1
1
n
yn
n x
Bf
multipole
dipole
quadrupole
sextupole
octupole
order
0
1
2
3
Bx
yf 1
)3( 3236
1 yyxf
0
yxf 2
By
xf 1
)3( 2336
1 xyxf
0B
)( 2222
1 yxf
normalized multipole gradients:
nn fp
qk
1
1][
nn mk
]/[
]/[3.0
cGeVp
mTfk
nn
n )(
)()/()( pvBvqpvsF
Field Imperfections: Multipole Illustrationupright and skew field errors
upright: skew:
n=0
1
2
123
4n=1
6 2
345
1n=2
Field Imperfections: Multipole Illustrationsquadrupole and sextupole magnets
LEP Sextupole
ISR quadrupole
Perturbation Treatment: Resonance Condition
perturbation treatment:
with:
equations of motion:
)/2cos()()(,
,,2
02
2Lsryxaswsw ml
rnml
rmndsd
)(....22
10nOxxxx
with: yxw ,
)/2cos()( 0,0,00 xx LsQxsx [same for ‘y(s)’]
)( ]~~[cos 0,0,
~,~
,~,~12
012
2 2srQmQlaww yx
mmllrmn
Ldsd
yxQL ,0
2
(nth order Polynomial in x and y for nth order multipole)
Perturbation Treatment: Tune Diagram I
tune diagram:
avoid rational tune values!
resonance condition: yxyx QrQmQlLL
,
22)~~(
there are resonanceseverywhere!(the rational numberslie dens within thereal number)
Qy
Qx
up to 11 order (p+l <12)
rQmQl yx
Perturbation Treatment: Tune Diagram II
coupling resonance:
avoid low order resonances!
regions with few resonances:
rQmQl yx
regions without loworder resonancesare relatively small!
Qy
Qx
< 12th order for aproton beamwithout damping
< 3rd 5th order forelectron beams withdamping
7th11th4th & 8th9th
Perturbation Treatment: Single Sextupole Perturbation
perturbed equations of motion:
)cos()( 0,00 sAsx x
r
x LsrsxLQsx ALlk
dsd )/2cos()()/2()( 21
12
0,12
2
2
)/2(cos1
2
1)()( 2
0212
012
2Lsr
Lxlksxsx
rdsd
with: LQxx /2 0,,0 and
r
x LsQrALlk
)/]2[2cos( 0,21
8
220 )()/()( 2
1 xlksspvsF L
Perturbation Treatment: Sextupole Perturbation
rQrQ xox 0,, )(22
resonance conditions:
avoid integer and r/3 tunes!
contrary to the previous examples no exact solution exist!this is a consequence of the non-linear perturbation
(remember the 3 body problem?)
rQ
rQQrQ
xQr
xQr
xox
x
x
0,2
0,2
0,,
0,
0, 3/)2(22
perturbation treatment:
graphic tools for analyzing the particle motion
Poincare Section: Linear Motion
)cos()( Rsx
unperturbed solution:
the motion lies on an ellipse
)sin(0 Rxxds
d
phase space portrait:
with 0 ds
d
x
0/x
R
linear motion is described by a simple rotation
Poincare Section: Definition
Poincare Section:
resonance in the Poincare section:
record the particlecoordinates at onelocation in thestorage ring
y
x
s
0/ x
x
x
0/x
Qturn 2 turn
3
12fixed point condition: Q = n/r
points are mapped onto themselves after ‘r’ turns
Poincare Section: Non-Linear Motion
momentum change due to perturbation:ds
pv
sFx
)(
phase space portrait with single sextupole:
x
0/x
R+R
02 Q
sextupole kick changes the amplitude and the phase advance per turn!
single n-pole kick: nn xlk
nx
!
1
222
1xlkx
2xQturn
Poincare Section: Stability?
instability can be fixed by ‘detuning’:
sextupole kick:
overall stability depends on the balance between amplitudeincrease per turn and tune change per turn:
)(xQturn motion moves eventually off resonance
)(xRturn motion becomes unstable
amplitude can increase faster then the tune can change
overall instability!
Poincare Section: Illustration of Topology
Poincare section:
fixed points and seperatrix
x
0/x
3
2
1
Rfp
Q < r/3
regular motion small amplitudes:
border between atable and unstablemotion chaotic motion
large amplitudes: instability & particle loss
220 )()/()( 2
1 xlksspvsF L
Poincare Section: Simulatiosn for a Sextupole Perturbation
Poincare Section right afterthe sextupole kick
x’
x
for small amplitudes theintersections still lie on closedcurves regular motion!
for large amplitudes and near the separatrix the intersectionsfill areas in the Poincare Section chaotic motion; no analytical solution exist!
separatrix location depends onthe tune distance from the exactresonance condition (Q < n/3)
Slow Extraction With Sextupoles
Septum magnet:
x
0/x
adjust tune closer to the resonance condition during extraction
2
02/1 ]3
[16lk
Qr
R fp
the region of stable motion shrinks and more particles reach the septum
F
Stabilization of Resonances
octupole perturbation:
instability can be fixed by stronger ‘detuning’:
perturbation treatment:
)]2cos(1[2
)cos(2
200
AxAx
if the phase advance per turn changes uniformly with increasing R the motion moves off resonance and stabilizes
336
1)/()( xlkpvsF
...)()()( 10 sxsxsx
12031
20,12
2
6
1)()/2()( xxlksxLQsx xds
d
13
2
13
22
0,12
2)2cos(
62)(
62)/2()( ][ x
lkAsx
lkALQsx xds
d
Stabilization of Resonances
resonance stability for octupole:x’
x
an octupole perturbation generates phase independent detuning and amplitude growth of the same order
amplitude growth and detuningare balanced and theoverall motion is stable!
this is not generally true in case of several resonance drivingterms and coupling between the horizontal and vertical motion!
Chaotic Motion
octupole + sextupole perturbation: x’
x
the interference of the octupoleand sextupole perturbationsgenerate additional resonancesadditional island chains in
the Poincare Section!
intersections near the resonances lie no longer on closed curves local chaotic motion around the separatrix & instabilities slow amplitude growth (Arnold diffusion)
neighboring resonance islands start to ‘overlap’ for large amplitudes global chaos & fast instabilities
Chaotic Motion‘Russian Doll’ effect: x’
x
magnifying sections of the Poincare Section reveals always the same pattern on a finer scale renormalization theory!
x’
x
Summary
field imperfections drive resonances
Poincare Section as a graphical tool for analyzing the stability
island chains as signature for non-linear resonances
(three body problem of Sun, Earth and Jupiter)
higher order than quadrupole field imperfections generate non-linear equations of motion (no closed analytical solution)
solutions only via perturbation treatment
slow extraction as example of resonance application in accelerator
island overlap as indicator for globally chaotic & unstable motion