Collective beam Instability in Accelerators

A circulating charged particle beam resembles an electric circuit, where the impedance plays an important role in determining the circulating current. Likewise, the impedance of an accelerator is related to the voltage drop with respect to the motion of the charged particle beams. The impedance is more generally defined as the Fourier transform of the wakefield, which is the electromagnetic waves induced by a passing charged particle beam. The induced electromagnetic field can, in turn, impart a force on the motion of each individual particle. Thus, single-particle motion is governed by the external focusing force and the wakefield generated by the beams, and the beam distribution is determined by the motion of each particle. A self-consistent distribution function may be obtained by solving the Poisson-Vlasov equation. The impedance that a charged-particle beam experiences inside a vacuum chamber resembles the impedance in a transmission wire. For beams, there are transverse and longitudinal impedances.

For beams, there are transverse and longitudinal impedances.

1. The longitudinal impedance has the dimension Ω, and by definition is equal to the energy loss per revolution in a unit beam current.

2. The transverse impedance is related to the integrated transverse force divided by the dipole current component on betatron motion, and has the dimension Ω/m. The transverse impedance arises from accelerator components such as the resistive wall of vacuum chamber, space charge, image charge on the vacuum chamber, broad-band impedance due to bellows, vacuum ports, and BPMs, and narrow-band impedance due to high-Q resonance modes in rf cavities, septum and kicker tanks, etc.

)(0||per turn )( ntj

n edeIZE

Impedances:

20sc||

2gZj

nZ

222

0sc

112 baRZjZ

deWjCdsBE

IjdsF

yIejZ j)()()(

1

dteWZ

ZIE

tj

)()(

)()(

||||

||

)//(Q12

rr

sh2RLC

jR

bcZ

skin30

rw ))sgn(1( bRZjZ

rw||2rw

2 ZbcZ

)//(Q1 rr

shRLC||

j

RZ

1,skin2/10rw||

2))sgn(1( nbnZj

nZ

Wake Field• The static electric field of a charged particle uniformly points in all directions.

When the velocity of charge particle approaches the speed of light v→c, the electromagnetic fields become pancake-like, i.e. the electric field is radial and magnetic field azimuthal (the Liénard-Wichert fields).

• Inside a perfectly conducting beam pipe, the pancake of fields travels along the beam pipe. A ring of opposite charges (image current) travel along the beam pipe. If the beam pipe is not perfectly conducting or contains discontinuities, the image charges is slowed down, and electromagnetic (wake) fields is left behind.

a. Rigid beam approximation: The beam motion is not affected by the wake fields it generated. This implies that the distance z of the test particle behind the source particle does not change.

b. Impulse approximation: At high energies , we do not need to know the instantaneous E or B separately, We only care the integrated impulse. Both E or B and F are difficult to calculate. It is easier to calculate the impulse, which obey the Panofsky-Wenzel theorem.

• In accelerator, the wakefields are proportional to the beam intensity.

• The wake fields are calculated at a distance z behind the source particle with two approximations:

Panofsky-Wenzel Theorem Maxwell equations:

Lorentz force on the test particle

Impulse (z<0):

rw||

2rw2 ZbcZ

,0 p

ps

ps

xz pz

px

A supplement of the Panofsky-Wenzel Theorem with

),,(),,( zyxWezyxp

Since

withWe find a W-function that describes the impulse on a particle

Cylindrically Symmetric Pipe

,0 pConsider a beam experience a kick force with cos mθ multipole:

Thus the transverse impulse produced by Im=eam of the m-th multipole mode is

m Moment of beam distribution

Longitudinal wake impulse

Transverse wake impulse

0 q −qI0 W0′(z) 01 q<x> −qI0<x> W0′(z) qI0<x> W1(z)

q<y> −qI0<y> W0′(z) qI0<y> W1(z)

∆E=

where Wm(s) is the transverse wake function. The longitudinal impulse becomes

Right behind a source particle, the test particle should receive a retarding force. For small z, we have

With causality, the wakefields behave as follows:

For the monopole m=0, we find ∆p=0, and ∆ps independent of r and θ, but a function of z. Particles in a thin transverse slice of the beam will receive the same impulse in the s-direction with function of W0′ on z. This impulse can lead to beam self- bunching.

Consider a beam current: , /The wake m=0 at position z is a superposition of the wakes produces by all charges of previous times, 1 , , ||where || Since the induced voltage is V=−I0Z0||, the energy gain and loss is qV. Similarly, the transverse impulse v∆p in one revolution is the integrated transverse force on a particle, i.e. ∆∆ , ,

Resistive wall impedanceBecause the resistivity of the vacuum chamber wall is finite, part of the wakefield can penetrate the vacuum chamber and cause energy loss to the beam. Penetration of electromagnetic wave into the vacuum chamber can be described by Maxwell’s equations

where the sign function, sgn(ω) = +1 if ω > 0 and −1 if ω < 0, is added so that theimpedance satisfies the symmetry property.

rw||

2rw2 ZbcZ

Space-charge impedance

where I0 = eβcλ0 and I1 = eβcλ1. The perturbation generates an electric field on the beam. Using Faraday’s law

where the factor 1/γ2 arises from cancellation between the electric and magnetic fields.

For most accelerators, the vacuum chamber wall is inductive at low and medium frequency range. Let L/2πR be the inductance per unit length, then the induced wall electric field is

The total voltage drop in one revolution is

where βc = ω0R is the speed of the orbiting particles, and Z0 = 1/ε0c = 377 ohms is the vacuum impedance. Using R(∂λ/∂s) = (∂λ/∂θ), and eβcλ1 = I1, we obtain the impedance, defined as the voltage drop per unit current, as

Space-charge impedance (transverse)Let a be the radius of a uniformly distributed beam in a circular cylinder. Let x0be an infinitesimal displacement from the center of the cylindrical vacuum chamber. The resulting beam current density is

Narrowband and Broadband impedances

)//(Q1 rr

shRLC

j

RZ

)//(Q12

rr

sh2RLC

jR

bcZ

The impedance of RLC resonator circuit has two poles located at

The contour integral of the impedance in the complex ω’-plane. Because the impedance is analytic in the lower complex ω’-plane, the Cauchy integral formula can be used to obtain the dispersion relation.

Properties of the transverse impedance Properties of the Longitudinal impedance

The wake function is real, thus

The property of Z(ω)/ω is similar to that of Z (ω).

Impedances

Frequency spectrum of beamsWe consider a coasting beam, where particles continuously fill the accelerator, the transverse coordinate at any instant of time is

Here θ is the orbital angle, n is the mode number. At a fixed azimuth θ0, the transverse motion is described by the betatron motion with betatron angular frequency Qω0:

The nth mode of the transverse oscillation is

The angular phase velocity of the beam motion is

Effect of wakefield on particle motion:

If the beam encounter collective betatron motion of mode n with coherent frequency ω, the betatron motion for each particle is

The parameters ξ represent any variables that ωnw of the beam particle depends on. The betatron tune may depend on betatron amplitude due to sextupole, octupole magnetic fields, and the fractional off-momentum parameter δ=Δp/p0.

Example 1: A beam with zero momentum (frequency) spread ρ(ξ)=δ(ξ-ξ0)

The imaginary part of the transverse impedance produce real frequency shift, while the real part of the impedance produce imaginary part of the coherent frequency. If the imaginary part is negative, the mode grows exponentially with time. For fast wave, ωnw>0, and the real part of the impedance is larger than 0, and thus there is no growth. A beam with zero frequency spread may suffer collective instability for slow-wave modes.

Example1: Instability due to broadband impedance. We consider the Hill’s equation:

ymR

IZejmtFyQy kkk

2

)()( 20

kkkktkntj

kk ynyynjyyeYy 200

)( )( ,)()( ,

YmR

IZejYnQ kk

2])()[( 2

02

0

dmQRIZej

n

)()(

41

w,0

)( ,4

][ 0w,0

w,

QnYmQRIZejY nkn

Here ξ represents sets of beam parameters, Y=∫ρ(ξ)Y(ξ)dξ. The dispersion relation can be used to solve the eigen-frequency ω. If the imaginary part of the eigen-frequency is negative, the wave grows exponentially. The transverse motion is unstable.

The set of parameters ξ represents any variables that ωn,w and the beam distribution function depend on. Since betatron tunes depend on betatron amplitudes due to space-charge force, sextupoles, and other higher-order magnetic multipoles, the betatron amplitude can serve as a ξ parameter. Since Q and ω0 depend on the off-momentum parameter δ = p/p, δ can also be chosen as a possible ξ parameter.

dmQRIZej

n

)()(

41

w,0

where δ = p/p0 is the fractional off-momentum coordinate, Here, we have assigned the fractional off-momentum coordinate as a ξ parameter. The wave frequency spread vanishes at mode number n0 = Cy/η. Thus if Cy/η <0, the mode number n0corresponds to a slow wave. The beam may become unstable against transverse collective instability.

A. Beam with zero frequency spread

For a beam with zero frequency spread, i.e. ρ(ξ) = δ(ξ − ξ0), we obtain

1. The imaginary part of the impedance gives rise to a frequency shift. 2. The resistive part generates an imaginary coherent frequency ω. If the

imaginary part of the coherent frequency is negative, the betatron amplitude grows exponentially with time, and the beam encounters collective instability.

3. For fast and backward waves, ωn,w0 is positive. Where the real part of the impedance Z (ω) is positive, and the imaginary part of the coherent frequency is positive, and there is no growth of collective instability.

4. The collective frequency ωn,w0 of a slow wave is negative, where the real part of the transverse impedance is negative. Since the imaginary part of the collective frequency is negative, a beam with zero frequency spread can suffer slow wave collective instability.

Defining U and V parameters as

where V is related to growth rate, and U is related to collective frequency shift, we obtain Re [ωcoll] = ωn,w0 − U, Im[ωcoll] = V.

B. Beam with finite frequency spreadWith parameters U and V , the dispersion relation for coherent dipole mode frequency ω becomes

1. The solution of the dispersion relation corresponds to a coherent eigenmode of collective motion. If the imaginary part of the coherent frequency is negative, the amplitude of the coherent motion grows with time.

2. If the imaginary part of each eigenmodes is positive, coherent oscillation is damped. The threshold of collective instability can be obtained by finding the solution with ω = ω −j|0+|, where 0+ is an infinitesimal positive number. The remarkable thing is that there are solutions of real ω even when −U + jV is complex.

A model of collective motion

where W = −U +jV for a broad-band impedance. If ωn,w(k)=ωn,w0 is independent of k, i.e. no frequency spread, the collective frequency is ωcoll=ωn,w0 +W. The corresponding eigenvector for collective mode is Yk,coll=ρkYk. Thus any amount of a negative real part of the impedance can produce a negative imaginary collective frequency and lead to collective instability. All other incoherent solutions have random phase with eigenvalue ωn,w0. In fact, the coupling between external force and beam particles is completely absorbed by the collective mode.

If the frequency spread ωn,w among beam particles is larger than the coherent frequency shift parameter W, the collective mode disappears, and there is no coherent motion. The disappearance of the collective mode due to tune spread is called Landau damping. The requirement of a large frequency spread for Landau damping is a necessary condition but not a sufficient one. We consider the frequency spread model ωn,w(k)Yk = ωn,w0Yk + ΔΩ(Yk − <Y>), where ΔΩ is a constant that determines the frequency spread of the beam. This model of tune spread resembles space-charge tune shift. The resulting collective mode frequency is ωcoll=ωn,w0 +W.

With frequency spread, the threshold of the collective instability is obtained by solving the dispersion integralWe consider a Gaussian distribution function:

Im(ω)=0Im(ω)=−0.5σω

B. Solutions of dispersion integral with Gaussian distribution

Laudau damping and frequency spread

The collective beam instability of a beam can be represented by the equation of motion:

where ωβ=Qω0 is the betatron angular frequency, ω is the angular frequency of the collective motion. The solution of the equation of motion is

The graph shows y(t) as a function of time for the initial value (y0=0,y0’=0), and F=0.01 mm, ωβ=0.99, ω=1. We note that the particle is driven by the external perturbation with increasing amplitude for a time

(y0=0,y0’=0), and F=0.01 mm, ωβ=0.9, ω=1.

ttttFty

sin)sin(cos)cos(1

2

ˆ)(

(y0=0,y0’=0), and F=0.01 mm, ωβ=0.8, ω=1.

T

dttyFyytyFyydtd

0

22212

2122

212

21 sinˆ)( ,sinˆ)(

ttttFty

cos)sin(sin)cos(1

2

ˆ)(

The average power that the external force acts on the particle is

Here ζ=(1/2)(ω-ωβ)T. As time T increases, the coherent driving force becomes less effective in exciting transferring external power to the particle. The coherent time is T~1/(ω-ωβ), i.e. fewer oscillators are in phase with respect to the external force.

If all particles are driving by an external force (impedance or wakefield) , and there is no frequency spread in beam particles, the collective motion will drive the beam into ever larger amplitude oscillation.

However, if the beam particles have frequency spread. As time increases, fewer and fewer particle becomes coherent with the external force, and the collective motion is stabilized. This self-stabilization effect is called Landau Damping.

Figure 2.58: The upper plot shows the coherent function (sin2 ζ)/ζ2. Note that the function becomes smaller as the ζ variable increases. This means that the external force can not coherently act on a particle if (ωβ −ω)T becomes large. The lower plot shows the response of three particles vs time t to an external sinusoidal driving force F(t) = F sin ωt. Here the units of ω and t are related: if ω is in rad/s, t is in s, and if ω is in 106 rad/s, then t is in μs. The frequency difference of these three particles is ω = 0.01, 0.05, and 0.1 shown respectively as solid line, dashes, and dots. Note that the particle with a large frequency difference will fall out of coherence with the external force.

As time T increases, the coherent frequency window decreases, i.e.|ω − ωβ| ∼ 1/T, fewer and fewer oscillators will be affected by the external force. When the external force can not pump power into a beam with a finite frequency spread, collective instability essentially disappears, i.e. the system is Landau damped.

Example 2: Longitudinal broadband instability of coasting beam

0

tdtd

)(0 )( ntj

ne

Vlasov equation:

dnE

nZeInjd

nEnZeIn

j

20

2||00

20

2||00

)(2)/(/

2)/(

1

)(0||per turn )( ntj

n edeIZE

)(||02

0 )(2

ntj

n edZeIE

)(2

)( 02

||00

d

EZeI

nj nnVlasov equation →

Note that ∂ω/∂δ=η. For a delta function distribution function, the dispersion function has the analytic solution:

EnZeI

jn 2

||02

0 2)/(

1

Beam with Gaussian distribution function:

Integrating the integral of the dispersion relation

0~ n

w(z) is the complex error function

02

~

ny

In terms of the U and V parameters, the dispersion integral becomes

,1)( G21 JjVUj ,1)(44.0 G JVjUj

Keil-Schnell condition:

2 2

2ˆ 0B IeNI

t

Effect of the longitudinal Space charge forceWe consider a cylindrical beam in a cylindrical beam-pipe. The electromagnetic field due to the beam is

Here λ is the particle line density, e is the charge, βc is the speed, ε0 and μ0 are permittivity and permeability of the vacuum. We consider a small perturbation in the line density and current. )(

10)(

10 , ntjntj eIIIe I0=eβcλ0, I1=eβcλ1. The perturbation generates electric field on the beam. Using the Faraday’s law:

Es and Ew are electric fields at the center of the beam and the surface of vacuum chamber, g0=1+2ln(b/a) is the geometric factor. Note also: ∂λ/∂t=βc∂λ/∂s.

The electric field on the beam is

The total voltage drop in one revolution is

22 2/

21

ss

s

e

22 2/32

ss

s

ess

12/

233300

10

20

1

1

22

22cos

2

ns

s

Bn

snn

nnn

semcZcRgeNs

REheV

Rss

s

0242

cos2323

002

02

0202

2

s

mcZgecRN

EeVh

dtsd

s

Bs

The space charge force is defocusing that pushes particle away from the center. The envelope equation for the longitudinal bunch becomes

0242

cos3

2||

232300

2

0202

02

2

s

ss

Bss

mcZgecRN

EeVh

dtd

• Contains 30 ferrite cores (Ni-Zn Toshiba M4C21A) with Resistive Paste

–Each core is 8 in. outer diameter, 5 in. inner diameter, 1 in. thick

Ni-Zn ferrite core with Resistive Paste

Impedance Calculation of Ferrite cavity

100 200 300 400 500 600 700−2

−1

0

1

2

3

Time in ns

Arb

itrar

y U

nits

C. Beltran et al., Microwave instability induced by the Ferrite inserts at PSR. See Ph.D. thesis (Indiana Univ., 2003). 20 40 60 80 100 120 140 160 180

−20

−10

0

10

20

30

Frequency f (MHz)

Impe

danc

e Z || (k

Ω)

Z||real for 25 °C

Z||im for 25 °C

Z||real for 125 °C

Z||im for 125 °C

Spectrum measurements can beCompared with an impedance model.PSR data: Beltran et al.

C. Beltran, draft calculation

Beam breakup instability

W(t’-t) is the transverse function. To understand the effect of transverse wakefield, we divide the beam into two macro-particles with charges Ne/2. Let ℓ be the distance between these two macro-particles, W(ℓ) be the wake-function at the location of the second macro-particle. The equation of motion becomes

If, for some reason, the head particle begins to perform betatron oscillation, the tail particle can be resonantly excited.

k2→k1 ∆k=k2−k1

By setting the growth term is eliminated.

This is called the BNS (Balakin, Novokhatsky, Smirov) damping.