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