Beam observation and

Introduction to Collective Beam Instabilities

Observation of collective beam instability

Collective modes

Wake fields and coupling impedances

Head-tail instability

Microwave instability

Beyond

T. Toyama KEK

Observation of collective beam instability

Example: KEK-PS 12 GeV Main Ring

At 500 MeV injection plat bottom

a beam loss occurs (red curve).

Amount and timing of the loss => random.

Proton number NB

(Feedback CT)

Magnetic field

Example: KEK-PS 12 GeV Main Ring

At phase transition energy~5.4 GeV (in kinetic energy)

a large beam loss occurs (red curve).

Amount of the loss is at random.

Proton number NB

(Feedback CT)

Magnetic field

Observation

NB

Multi-trace of horizontal betatron oscillation

NB

Amplitude of betatron oscillationMagnetic field

Observation horizontal betatron tune during acceleration

t

f

frev 2frev

frev-ffrev-f

2frev-f 2frevf

Without external kick, coherent oscillation emerged

Measurement by a wall current monitor

Real signals may be attenuatedby the loss in the cable > 100 mand limited band width of the WCM.

Beam loss: collective instabilities --- at random, a kind of

positive feedback starting from a random seed direct space charge effects --- regular

some mistake in parameterrs --- regular(B, fRF, tune, …)

Collective modes

Coasting beam / longitudinal

n=3

Beam

Coasting beam / transverse

Collective modes

n=3

Beam

betattonoscillation

x or y

Bunched beam / logitudinal

Collective modes

l=1 l=2 l=3dipole quadrupole sextupole

zz z

charge density

Phase space

…..

no momopole mode

Bunched beam / transverse

Collective modes

dipole mode density zx

zzz

l=0 l=1 l=2monopole dipole quadrupole …..

superimposed

Wake fields and coupling impedance

Electromagnetic fields is produced by the beam passed by.

+ + +++++++ + +

− − − − − − −

− − − − − − −

IB

Wall Current −IB

−

−

v

Wall Current

++e qF//

s

F⊥

€

Wake functions (W//, W⊥) : "Green function"

Force acting on a test particle (charge e)

produced by the delta-function beam (charge q, dipole moment qy)

Longitudinal component:

F // = F//ds−L /2

L /2

∫ =−eqW//(s)

Transverse component:

F ⊥ = F⊥ds−L /2

L /2

∫ =−eqyW⊥(s)

[W//]=[V /C]

[W⊥]=[V /Cm]

Wake fields and coupling impedance

€

Wake functions (W//, W⊥) of a resistive wall

Force acting on a test particle (charge e)

Wake fields and coupling impedance

€

Longitudinal impedance Z// :

Sinusoidal current J 0(s,t)= ˆ J 0ei(ks−ωt) produces

longitudinal wake potential across the section:

V(s,t) =−1v

d ′ s J 0(s,t−′ s −sc

) s∞∫ W//(s− ′ s )

=−J 0(s,t) Z//(ω)

longitudinal impedance:

Z// =dzc

e−iωz/c −∞∞∫ W//(s)

[Z//]=[Ω]

Wake fields and coupling impedance

€

Transverse impedance Z⊥ :

Sinusoidal dipole moment J1(s,t) =

Current J 0(s,t)× dipole displacement y(s,t)

J1(s,t)=y(s,t) J 0(s,t) = ˆ J 1ei(ks−ωt) produces

transverse wake potential across the section:

V⊥ =i J1(s,t) Z⊥(ω)

transverse impedance:

Z⊥ =iβ

dzc

e−iωz/c −∞∞∫ W⊥(s)

[Z⊥]=[Ω/m]

Wake fields and coupling impedance

Wake fields due to a Gaussian beam in a resistive pipe

Longitudinal wake potential Transverse wake potential

Acc

eler

atio

nD

ecce

lera

tion

Dam

pen

defl

ecti

onF

urth

er

defl

ecti

on

Wake fields and coupling impedance

Impedance of a resistive pipe

Wake fields and coupling impedance

Wake fields by cavities

Q=1 Q=10

Wake fields and coupling impedance

Impedance of cavities

Q=1 Q=10

Head-Tail InstabilityTransverse bunched beam instability

Time domain picture

Head-Tail InstabilityChromaticity = 0

Red full line: (z)x(z)Red dushed line: (z)x’(z)Blue: kick due to resistive wall

x

x’

Growth

Damp No effect

~Totally no effect

(1)

(1)

(2)

(2)

(3)

(3)

(4)

(4)

headtail

Head-Tail Instability

Head-tail phase

z

p/p

€

ˆ z

€

−̂ z 0

€

χ =ξωβˆ z

cη

€

Δνβ

νβ=ξ

Δpp

phase of betatron oscillation

phase space of synchrotron oscillation

Head-Tail Instability

x

x’

Damp

~ 1

Red full line: (z)x(z)Red dushed line: (z)x’(z)Blue: kick due to resistive wall

(1)

(2)

(1)

(2)

~Totally damping

headtail

Head-Tail Instability

x

x’

Growth

~Totally growing

(1)

(2)

~

Red full line: (z)x(z)Red dushed line: (z)x’(z)Blue: kick due to resistive wall

(1)

(2)

headtail

Head-Tail Instability

Summary of Growth rate vs. Chromaticity

Head-tail phase

Gro

wth

rat

e

Chao’s text book

mode = 0

mode = 1mode =2

mode =3

€

χ =ξωβˆ z

cη

Stab

le

Un

stable

Head-Tail InstabilityKEK-PS 12 GeV Main Ring

T. Toyama et al., PAC97, APAC98, PAC99

mode=0

mode=1 mode=2

NB

amplitude of dipole oscillation

Head-Tail InstabilityCERN PS higher order head-tail mode

R. Cappi, NIM

Head-Tail Instability

KEK-PS12GeV MR

Frequencydomainanalysis

growth rate∝Re[Z()] F()

Re[

ZT]

For

m f

a ct o

r F

(f

req.

spe

ctru

m

o f t h

e b e

a m)

m=0

m=1

m=2

€

ωξ =ξωβ

η

Head-Tail Instability

ObservationGrowth ratemode=0

Head-Tail Instability

Cure

Chromaticity control

Landau damping by octupole magnets …

Beam response and Landau dampingCoasting beamTransverse motion

€

Single particle oscillating at ω.

External driving force is on at t=0.

˙ ̇ x +ωx=AcosΩt

x(t >0) =−A

Ω2−ω2(cosΩt−cosωt), ω≠Ω

x(t >0) =−AtΩ

(cosΩt−cosωt), ω=Ω

Beam response and Landau damping

€

Magenta: x(t)=−A

Ω2 −ω2(cosΩt−cosωt),

Ω =1.1ω

Driving force

Response

€

Blue: x(t)=−AtΩ

SinΩt,

Ω =ω

€

Red: f(t)=AcosΩt

Driving force

Response of the beam

Absorbed power by the beam

The beam: ensemble of the particlesFrequency distribution:

The beam motionapproaches steady oscillation.

Velocity d<x>/dt: in phase with the forceWork is done on the beam

Absorbed power by the beam: constantStored energy in the beam:

Macroscopic aspect: a beam driven by a force approaches steady oscillation.

Microscopic aspect: Small amount of resonant particles grows infinitely large.

€

∝ tResponse of particles

Longitudinal instabilityMicrowave Instability uniform distribution

Wake: V=Z (z)The seed of density modulation is produced

V1= Z (z), slippage,

Landau damping by the spread of rev =p/p phase slippage factor = 1/t

21/2

t phase transition energy

p/p

p/p

p/p

p/p

Density modulation reduced! Larger p/p more stable

Microwave Instability

Observation & simulation

K. Takayama et al., Phys. Rev. Lett. 78 (1997) 871

Microwave Instability

Sources: Narrow-band resonances

res ~ 1GHz

Cures

Reducing Impedance

Landau dampingReducing local beam chaege line density

Artificial increasing momentum spread

p/p > rev

MethodsHigher harmonic rf cavity

Voltage modulation of foundamental rf cavity

…

Cures

Reducing Impedance Exchange ~ 2/3 BPMs new ESM BPM

~2/3 Pump port new one with slits

Growth rate reduction

Reducing local beam chaege line density

Increasing momentum spread > rev

Voltage modulation of foundamental rf cavity

T. Toyama, NIM A447 (2000) 317

BeyondImpedance calculationImpedance measurements

Beam transfer function

Vlasov equationCoupled bunch instability

Mode-coupling instability

Electron-cloud instabilityfeedback system

feedback in RF control system

feedback damper = pick-up & kicker

“… every increase in machine performance has accompanied by the discovery of new types of instabilities.” - J. Gareyte (CERN)

References

Schools:CAS, USPAS, and OHO (Japanese)

Conferences proceedings:APAC, EPAC, and PAC

Textbook etc.:• A. W. Chao, PHYSICS OF COLLECTIVE BEAM

INSTABILITIES IN HIGH ENERGY ACCELERATORS• Editors: A. Chao and M. Tigner, Handbook OF

ACCELERATOR PHISICS AND ENGINEERING

Good Luck!