Theory, observations and mitigation of dancing bunches

in the Tevatron

Alexey Burov

AD talk, May 19 20111

Dancing bunches at Tevatron

R. Moore et al, PAC’03 C. Y. Tan measurements

At injection, the synchrotron frequency shift is estimated as low as .

How may any instability happen at that low intensity (impedance)?

0 0| | / (1 2)%

p coalesced injection

p coalesced top energy

Instabilities: General Ideas

• Coherent instabilities happen because of collective interaction of the beam particles. Interaction is described by wakes / impedances:

• Instabilities normally have a certain intensity threshold. At low intensity, instabilities are weak enough to be suppressed by the beam immune system – Landau damping.

3

rf wake( ) ( ) ( ) ; , 1,...,i i i i jj

z F z F z W z z i j N

( ) ( ) exp( )2

dqW z Z q iqz

Landau Damping

• Landau damping is a transfer of energy of coherent mode into incoherent motion.

• Only particles resonant with the mode can take its energy.

• If the mode frequency stays outside the incoherent spectrum, there are no resonant particles, no Landau damping.

• If so, any potentially unstable interaction leads to an actual instability, howsoever weak it is.

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Roles of short- and long-range wakes

• For longitudinal bunch motion, the coherent and incoherent frequencies are normally determined by the RF and intra-bunch interaction (potential well distortion), while the instability is driven by couple-bunch or multi-turn wakes.

• Using the biological analogy, the short-range wakes are responsible for a crash of the immune system (Landau damping), while the long-range wakes work as bacteria.

• Thus, the short- and long-range wakes play very different roles for coherent stability.

• Loss of Landau damping (LLD) is determined by the short-range wakes, while the long-range ones are responsible for the growth rate.

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Linear focusing (parabolic RF potential)

• Multi-particle equations:

• The solution can be presented as a sum of a steady-state part and a small perturbation:

• Since for the steady state ,

• is always a solution – for any wake and distribution.

(D’yachkov & Baartman, 1993)

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( ) ; , 1,...,i i i jj

z z W z z i j N

ˆi i iz z z

ˆ ˆ ˆ ˆ( )i i i jj

z z W z z

const cosiz t

Linear focusing (cont)

• The rigid bunch mode is always a solution for the linear focusing.

• Its frequency = 1 = bare synchrotron frequency.

• Incoherent frequencies are normally either all suppressed or all elevated from 1, thus, there are no resonant particles here.

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LLD threshold = 0 for the linear focusing.

const cosiz t

0.2 0.4 0.6 0.8 1.0J

0.970

0.975

0.980

0.985

0.990

0.995

1.000

incoherent

coherent mode

LLD scaling, normal RF

For bunches, LLD was first considered by A. Hofmann & F. Pedersen (1979)

Potential well distortion incoherent tune shift scales as

LLD happens when this shift exceeds the synchrotron tune spread :

From here, the threshold

Very strong dependence on the bunch length l : for the space charge case

What is the numerical factor for this scaling law?

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1 2Im ( ) /N Z l l

| |

1 4th | Im ( ) |N Z l l

5thN l

2l

Hofmann & Pedersen LLD threshold

• Hofmann & Pedersen (arbitrarily) assumed for conventional RF the coherent mode to be the same as for the linear focusing, i.e. the rigid bunch motion. For the space charge impedance below transition and their special distribution

got the threshold

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2 5rf lim

rf

215

bI h Zk

V n

lim( ) 1 /F J J J

lim lim2J

0/bI Ne T - average bunch current

- bunch length in units of RF phase.

rf rf,h V - RF harmonic number and voltage amplitude

2lim

30 lim

| |0.75

20

k

Overestimation for the rigid-bunch model?

Dependence on the bunch shape?

Boltzmann-Jeans-Vlasov (BJV) equation

• Boltzmann-Jeans-Vlasov (BJV) equation describes free oscillations of plasma or beam in given external fields, while Coulomb fields of the plasma or wake fields of the beam have to be taken into account.

• For longitudinal motion of a bunch, BJV writes as:

• Here I, φ and Ω(I) are the canonical action, phase, and incoherent frequency in the distorted potential well;

• F(I) and f (I,φ,t ) are the steady state and perturbation phase space density; W(z) is the wake function.

• What is an eigen-system of BJV ? 10

( ) ( ) 0f f V

I F It

( , ) ( , , ) ( )V z t f I t W z z dI d

van Kampen theory

• For infinite classical plasma, van Kampen found all the eigen-modes of BJV.

• There is always a continuous spectrum of them. These modes are singular, their frequencies are identical to incoherent frequencies (kv for the plasma case).

• Landau damping can be treated as phase mixing of the continuous van Kampen modes

• There can be discrete modes as well.

• , so some of the discrete modes do not decay,

this is the LLD case11

discrete

growth rate 0

Oide-Yokoya: reduction to linear algebra

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1

( , , ) ( ) cos ( )sini tm m

m

f I t e f I m g I m

2 2 2 2

1

( ) ( ) 2 ( ) ( ) ( , ) ( );m mn nn

m I f I m I F I dI V I I f I

0 0

2( , ) cos( ) cos( ) ( ( , ) ( , ))mnV I I d d m n W z I z I

( )( ) exp( )

2

dq Z qW z i iqz

q

( ) ( ) 0 .f f V

I F It

BJV equation

Phase expansion, O&Y, 1990

*

0 0

( )( , ) 2 Im ( , ) ( , ); ( , ) cos( )exp ( , ) .mn m n m

Z q dV I I dq G q I G q I G q I m iqz I

q

Matrix BJV

Matrix elements

Impedance

How the mode may look like

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0.5 1.0 1.5 2.0J

0.5

0.6

0.7

0.8

0.9

1.0

bare bucket

Spectrum for H-P distribution, inductance above transition

lim1.0, 1.5k J

mode

0.2 0.4 0.6 0.8 1.0 1.2 1.4J

0.05

0.10

0.15

M ode

Mode should look like the blue line, since it is coupled with the low-amplitude particles. The magenta line is the rigid-bunch motion, if it would happen.

Thus, the rigid-bunch mode is a very poor approximation…

Mode general properties

• For the linear focusing, the mode frequency does not depend on the intensity at all, while the incoherent frequencies can be strongly suppressed or elevated due to the well distortion.

• It is reasonable to expect, that for the conventional RF, the mode frequency does not move as much with the intensity as the incoherent frequencies.

• Thus, if the incoherent spectrum is suppressed, the mode frequency appears above the incoherent band.

• Since the incoherent frequency decreases with the action, the mode is mostly associated with the low-amplitude particles.

• Thus, the effective bunch length is determined by the distribution steepness at low amplitudes, so the LLD threshold strongly depends on that steepness.

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5thN l

LLD thresholds, inductance above transition, m=n=1

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2.01.5

0.010

0.050

0.020

0.030

0.015

0.070th

0

| |

limI

lim( )F I I I

2lim

lim

( ) ( ) 1 cos 8I

F I I II

2lim( ) ( )F I I I

For the H-P distribution the threshold is 3 times below rigid-bunch mode result.

For the coalesced bunch and full bucket the threshold is as low as

Note how strong is dependence on the distribution function!

lim( )F I I I

lim/I I

( )F I

Bucket acceptance

max 8 / 2.5I

th 0| | / 1%

0 .2 0 .4 0 .6 0 .8 1 .0

0 .2

0 .4

0 .6

0 .8

LLD thresholds, Tevatron

• protons, injection

• protons, top energy

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2.01.5

0.005

0.010

0.050

0.100

0.500

1.000thk

limI

pp

pp

protons, top energy

pbars, top energy

protons, injection

pbars, injection

3||0 rf

20 0

Im2.

ZNrk

c nZ

2lim

lim

( ) ( ) 1 cos 8I

F I I II

2lim( ) ( )F I I I

lim( )F I I I

5/2th limk I

For Gaussian, , is ≈ twice lower then for . lim( ) exp 4 /F I I I thk 2

lim( ) ( )F I I I

Stabilization: an idea

• Since the LLD threshold is so strong function of the low-action steepness, the oscillations can be stabilized by flattening the distribution at small arguments.

• This can be done by resonance modulation of RF phase at the low-amplitude synchrotron frequency. Simulations were performed:

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( )F I

1

1 1 0

1

sin sin( )n n n

n n n n

n n

z z t p

p p t z t

t t t

The experiment (C. Y. Tan, 2011)

• Two coalesced bunches are accelerated to 980 GeV. The experiment is done at flattop energy because the bucket area is much larger than beam longitudinal beam size and so any emittance growth can be observed without beam loss.

• Beam is shaken by phase modulating the Tevatron RF at the synchrotron frequency 34 Hz. The amplitude of the shake is 3° peak. To prevent excitation at the tails, the amplitude is ramped adiabatically:

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Shake after shake

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Before versus after

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Some more diagnostics

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Bunch rms length

Beam current

Centroid offset

Dependence on bunch details (1 shake@150 GeV)

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Comparison with dampers

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Damper stops shaking, resulting with more triangular distribution

For more intensity, there can be more than 1 discrete mode, and that centroid-seeing damper could be insufficient.

RF Shaking at KEK Photon Factory

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before after

In the 2.5-GeV Photon Factory storage ring at KEK, modulating the RF phase at a frequency of 2 times the synchrotron oscillation frequency with a peak-to-peak amplitude of 3.2°.

The beam lifetime was improved from 22 to 36 h under a beam current of about 360 mA (Touschek).

At the same time, the longitudinal coupled-bunch instability was suppressed.

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS, VOLUME 3, 050701 (2000)

Conclusions

• The calculated LLD thresholds are in agreement with proton and antiproton dancing bunch observations and impedance model.

• Hofmann-Pedersen LLD power low is confirmed in BJV solution.

• Strong dependence on the distribution at small amplitudes suggests that bunch flattening has to be helpful.

• This flattening can be done with resonant shaking of RF phase at zero-amplitude synchrotron frequency. Experiments confirm that.

• Similar stabilization was seen at KEK photon factory.

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Acknowledgement

• Many thanks to Valeri Lebedev and Slava Danilov (ORNL) for multiple discussions.

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Thank you all for your attention!