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Theory of Beam Transverse Oscillations

A. Burov

AD Talk, Aug 5, 2010

Content

1. Coasting beams: Möhl–Schönauer Equation (MSE)

1. Discussion of the MSE itself

2. Solution of MSE for strong space charge

2. Bunched beams

1. No space charge

1. Jeans-Vlasov equation2. Sacherer modes

2. Strong space charge

1. MSE for bunched beams2. Blaskiewicz square-well model3. General solution of MSE for a bunch4. Modes and Landau damping5. TMCI

3. General stability area

3. Summary 2

Möhl – Schönauer Equation

• In 1974, D. Möhl and H. Schönauer suggested to describe coasting beam oscillations by the following equation:

3

.02 02

022

2

2

xxQxQQxQdt

xdiscciii

i

lattice wake space charge

.4

)0();(rms

320

rN

QQQ sciscsc J

action

Ohm3774

; 00

0 c

ZZ

Z

C

rNiQ xx

c

Coherent tune shift, in a frequency domain:

Space charge term

• The only non-trivial term in this equation is the space charge force:

4

.02 02

022

2

2

xxQxQQxQdt

xdiscciii

i

Space charge term

• The only non-trivial term in this equation is the space charge force:

5

.02 02

022

2

2

xxQxQQxQdt

xdiscciii

i

xxQ isc

Space charge term

• The only non-trivial term in this equation is the space charge force:

• Why non-linear space charge forces are described by a linear term? Does it mean that it is valid only for K-V (constant density profile) distribution?

• No, it is valid for any beam profile. The reason is following:

6

.02 02

022

2

2

xxQxQQxQdt

xdiscciii

i

xxQ isc

• The single-particle motion consists of 2 parts: – free oscillations (typically with beam size amplitude, or )

– driven by the coherent offset oscillations (much smaller than the beam size).

• The equation describes the driven oscillations only, so it results from linearization of the original non-linear space charge term over infinitesimally small coherent motion, and averaging over the betatron phases of the free incoherent oscillations.

Comments to Möhl-Schönauer Equation (MSE)

7

xxQ isc )(JSpace charge term:

rmsJ

MSE as a unique choice

• This equation is a unique possibility for– A linear equation (driven oscillations are small!)

– With time-independent coefficients (? More to explain…)

– With the given tunes and incoherent SC tune shifts

– And coherent SC tune shift = 0 (3rd Newton’s Law!)

8

( )scQ J

• The assumption for the rigid beam approximation is that a core of any beam slice moves as a whole, so there is no inner motion in it.

• This is a good approximation, when the space charge is strong enough (Burov & Lebedev, 2008):

• Otherwise, the beam shape oscillates as well, and MSE is not valid.

MSE is based on a rigid-beam approximation

9

.02 02

022

2

2

xxQxQQxQdt

xdiscciii

i

lattice wake space charge

.|| isc QQ

coasting beam

Coasting beam: Landau damping (Burov, Lebedev, 2008):

10

.

,

ImIm

,ReRe

;,),(Re

;/,)Re(

2

2)2(

)1(

)2()1(

1

sep

xxlsep

sep

xxlsep

cc

cc

sep

xxsepyxsccsep

xxcsclxxsepsep

Q

dJfQQQ

Q

dJfQQQ

Q

QQQ

Q

dJfQJJQQQ

JffdQQJfQQ

Landau Damping

• Landau damping is proportional to the phase space density of the resonance particle:

• Thus, with strong space charge, it is determined by the distribution tails.

• Typically, the far tails are neither predictable nor measurable, and not even reproducible in many cases. That is why, at strong space charge, the stability thresholds are predictable and reproducible with poor accuracy, like factor of 1.5-2.

11

dQQJfQQ csclxxsepsep )Re(

Coasting Beam Thresholds (Burov, Lebedev, 2008)

• Thresholds are determined by . For a round Gaussian beam:

12

cQ Im

Chromatic threshold

rms

||

p

pnp

Octupole threshold

ppl ppnQ /ˆˆ

yx

yxoi

JJ

JJQ ;02/)(

Direct application to bunched beams

• Coasting beam results sometimes can be applied to bunched beams.

– For short-wavelength and fast oscillations (much shorter than the bunch length, much faster than the synchrotron frequency) – the so-called microwave instability;

– For multi-bunched beam, at zero chromaticity and long wavelength (much longer than the bunch spacing)

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Bunched Beams (no SC – Sacherer theory)

• Jeans-Vlasov Equation:

14

dxzzWzdzxzzWzF

pF

t

x

xx

ss

xb

)()()()()(

;0

Bunched Beams (no SC – Sacherer theory)

• Jeans-Vlasov Equation:

15

dxzzWzdzxzzWzF

pF

t

x

xx

ss

xb

)()()()()(

;0

xixxxx

tissxxsx

eJfJDJf

eJgJfJgJf

)(2),(~

),(~),(~

)()(

0

00

Dipole motion

Linearization

Bunched Beams (no SC – Sacherer theory)

• Jeans-Vlasov Equation:

16

dxzzWzdzxzzWzF

pF

t

x

xx

ss

xb

)()()()()(

;0

xixxxx

tissxxsx

eJfJDJf

eJgJfJgJf

)(2),(~

),(~),(~

)()(

0

00

),,(~)(),(~ssnm

imsnss JgeJRJg s Thus, the only function to be found is

describing the longitudinal phase space variation of the transverse offset.

Typical solution for the offset modulation

17

),(~, ssmn Jg

3D and Contour plots for (example) ),(~4,1 ssJg

Condition for Landau damping: 5.02.0~;|| CQCQm cs

0

20 )()(

p

JZC

NriQ

b

bmp

xc

coherent tune shift (air-bag model)

)sign(

s

p

Q

• Wake yields its coherent tune shift for every mode. If it is small

(weak head-tail), it can be calculated as a diagonal matrix element (similar to QM):

• The result depends on the chromaticity through the head-tail phase

Wake+ : weak head-tail

180 1 2 3 40.4

0.3

0.2

0.1

0

0.1

0.2

b

w

NW

Q

0

Im

|||| 1 kkcQ

kWkQcˆ

)sign(

s

ps

QR

Growth rates for the constant wake for the first 5 modes.

All of them are odd functions of the chromaticity.

0Im ck

Q

Wake+ : strong head-tail

• In the opposite case , the nearest modes are coupled by the wake, leading to instability even at . This is called strong head-tail or transverse mode-coupling instability, TMCI.

• For electron machines (no space charge), typically modes k=0 and k=-1 are coupled sooner than other, since is maximal, and the modes are shifted down.

• After coupling of modes k and k+1:

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|||| 1 kkcQ 0

skk Q 1ImIm

0ˆ0WQc

B. Salvant et al, HB2008, TMCI simulations, no space charge

Möhl-Schönauer Equation (MSE) for bunched beams

Using slow betatron amplitudes ,

the MSE writes as

Here and are time and distance along the bunch, both in angle units.

20

)()exp()( ibi xiQX

)(ix

xixixxiQx iiiiii W)()(v))(,()())(()(

,/ ddxx ii ,/ ,22 t ,)/(/ ppddQb )()(v ii

wakechromaticityspace charge

MSE for bunched beam – cont.

• After a substitution with a new variable , the chromatic term disappears from the equation, going instead into the wake term:

• Thus, for no-wake case, the bunched beam modes do not depend on the chromaticity, except the head-tail modulation . Neither Landau damping, nor eigenfrequencies depend on the chromaticity for W=0 .

21

;4 2

0

bQ

Rr

))(exp()()( iii iyx y

yiyyiQy iiii W))(,()())(()(

.)()())(exp()(ˆ dssyssisWyτ

W

)exp( i

Bunched Beam: Square Well Model

• For a square potential well and KV transverse distribution, the head-tail modes with space charge were described by Mike Blaskiewicz (1998).

• For the air-bag distribution, there are two particle fluxes in the synchrotron phase space:

• Since , MSE is easier to solve in this case.

22

zp

zforward

backward

constscQ

0 2 4 6 8 1010

5

0

5

Coherent tunes, no wake

23

...2,1,0;42

222

kQkQQ

sscsc

bk

s

bk

Q

s

sc

Q

Q

general result:

sc

sscbk

sc

sbk

Q

QkQ

kQ

Qk

22

22

...2,1,0;

...2,1;

2 k

QkQ sc

sbk

small space charge:

high space charge:

incoherent tune shift

At high space charge, the + and - fluxes oscillate in phase for the positive modes and out of phase for the negative modes.

Only positive modes matter for high space charge.

Landau damping

• The wake-driven growth rate can be compensated by Landau damping . Stability requires

• For the square-well model, the Landau damping is identical to the coasting beam case, except the chromaticity is dropped for the bunched case (Blaskiewicz, 2003)

• As a result, the condition for the resonant particles to exist (LD condition) for the square well case is

• Thus, for the strong space charge case, , there is no Landau damping for square potential well.

• What about Landau damping for other bucket shapes? 24

cQIm

L

cL QIm

nn

5.02.0~;|)|,max( CQQCQk cscs

ssc QkQ

General solution of MSE

• After a substitution with a new variable , the chromatic term disappears from the equation, going instead into the wake term:

From here, a 2nd order ordinary IDE follows (A. Burov, 2009):

25

;4 2

0

bQ

Rr

))(exp()()( iii iyx y

yiyyiQy iiii W))(,()())(()(

.)()())(exp()(ˆ dssyssisWyτ

W

.

v),v(v

v),v(

0)(;ˆ

2

2

2

df

df

u

yyWQyQd

ydu

d

dscsck

)(scQ is cross-section averaged.

No-wake (space charge only) modes, Gaussian bunch

26

2 0 23

2

1

0

1

2

3ky

0

1

23

4

These modes do not look too different from no-space charge case. The only significant difference is that for strong space charge, modes are counted by a single integer, while conventional zero-space-charge modes require 2 integers: for azimuthal and radial numbers (due to their possible variations along synchrotron phase and action).

Landau damping for strong space charge (SC)

• Landau damping results from energy transfer from coherent modes to incoherent motion of resonant particles.

• For strong space charge , particles and modes are tune-separated by , so their resonance may seem to be impossible.

• However, this is not generally correct, since the SC tune shift goes to 0 at the bunch tails, where this resonance may happen.

• The higher is SC, the further to the tails it happens. Thus, space charge strongly suppresses Landau Damping.

27

ssc QQ

scQ

Landau damping – results

• According to (Burov, 2009), for Gaussian bunch LD is estimated as:

(the numerical factor ~ 0.1 – best fit of V. Kornilov, GSI, to his numerical simulations, 2010)

• Note: mode k=0 is not damped at all. The damper is needed for it and may be for a few higher modes.

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3

40.1 sk s

sc

Qk Q

Q

Vanishing TMCI

• While the conventional head-tail modes are numbered by integers,

the space charge modes are numbered by natural numbers:

• This is a structural difference, leading to significant increase of the transverse mode coupling instability: the most affected lowest mode has no neighbor from below.

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,...2,1,0, kkQsk

.2kk

Coherent tunes of the Gaussian bunch for zero chromaticity and constant wake versus the wake amplitude.

Note high value of the TMCI threshold.

EwT0

0 50 10040

20

0

20

40

k

2eff0 )0(

s

b

Q

QNW

A schematic behavior of the TMCI threshold for the coherent tune shift versus the space charge tune shift.

0.1 1 10 100 1 103

0.1

1

10

s

w

Q

QRe

sQQ /)0(eff

stable

unstable unstable

TMCI threshold for arbitrary space charge

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Summary

• Theory of transverse coherent oscillations of beams is well-established now – both for coasting and bunched cases, with any space charge.

• For strong space charge, stability threshold is determined by far tails of distribution – hardly visible and not always reproducible. This limits the prediction accuracy – both for coasting and bunched cases.

• For bunched beam with strong space charge, recent simulations (O. Boine-Frankenheim and V. Kornilov, GSI) so far confirm the theory. More simulation details are needed and expected. No observations to verify the theory are yet available.

• For TMCI, moderate SC improves stability, but too high SC makes the bunch more unstable.

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