a study of collective single bunch phenomena in the alba ringrumolo/papers/collect1.pdf · a study...

Accelerator Division

ALBA project Document No: EDMS Document No. Created: September 9, 2005 page:1 of 10AAD-SR-BD-AN-0050 3.1415 Modified: Rev. No.: 1

A study of collective single bunch phenomena in theALBA ring

Abstract

Multi- and single-bunch instabilities determine the ultimate performances ofaccelerator and storage rings. Intensity limits are usually dictated by the onset ofthese mechanisms, therefore the study of thresholds and tolerances is fundamentalat the stage of ring design [1].The longitudinal and transverse intra-bunch effect of a broad-band impedance onthe ALBA nominal bunch, including a parameter study on the resonator parameterswithin plausible ranges, is the subject of the present note. Thresholds are roughlyestimated from formulae well-known from literature. The study is subsequentlyrefined via macro-particle simulation, which can include more realistic situations.The HEADTAIL code [2] has been adapted and used for this purpose.

Prepared by:G. RumoloM. Munoz

Checked by:D. Einfeld

Approved by:

Distribution listAll

AAD-SR-BD-AN-0050 Rev. No.: 1

Contents

1 Introduction 3

2 Longitudinal plane 4

3 Transverse plane 8

4 Conclusions and outlook 8


1 Introduction

Collective instabilities usually limit the performances of accelerator rings. The maximum current thatcan be injected and stored in an accelerator or storage ring depends on the ring impedance. In thedesign stage of an accelerator, it is necessary to budget a total impedance that is consistent with thebeam intensity requirements. This total budget is then carefully allocated to individual vacuum chambercomponents by means of off-line calculation (electromagnetic codes [3]) and/or measurement (wire ordielectric probe methods) prior to installation.The impedance of an accelerator is usually thought of as made of three main contributions: resistivewall, several narrow band resonators modeling cavity-like objects, and one broad-band resonator thatmodels the rest of the ring. Both resistive wall and narrow band resonators usually produce severe multi-bunch effects, since they are associated to slowly decaying wake fields causing mainly the motion ofone bunch to affect subsequent bunches. The broad-band resonator, which models the global effect ofall discontinuities of the beam pipe and several non-resonating objects (like pick-ups, kickers, etc.), isshort range and mostly affects the particle dynamics within one single bunch.The broad-band resonator model gives an accurate description of the impedance at frequencies ∼ ωr =c/b, where b is the beam pipe half-height. In the higher frequency range, the more detailed diffractionmodel gives better results, but it needs to be applied only on a scale |z| b (where z is a longitudinalcoordinate along the bunch). In the lower frequency range, the broad band model ignores the possi-bility of resonant responses to sharply defined frequencies (cavity modes). These cavity modes occurat frequencies below the cut-off frequency c/b (since they have to be trapped) and give rise to wakefunctions which ring for long periods of time. Neglecting these long-range contributions, we certainlydo not get accurate results from the model in the range |z| b. In the case of ALBA, the full bunchlength approximately equals the pipe vertical size, therefore applying the broad band model to predictthe single bunch collective behaviour seems perfectly legitimate (if we do not apply a too fine slicing inthe macroparticle simulation). It can be easily proven [1] that, assuming an accelerator ring a fractionf of which is filled by objects contributing to the broad band impedance, the total impedance can bewritten as:

Z||0 (ωr)

n≈ f

2Z0 , (1)

where Z0 = 377 Ω is the vacuum impedance. In this equation n is the defined as the ratio betweenthe resonant and the beam revolution frequencies, n = ωr/ω0. If we assume that ∼ 8% of the ringcontributes to the broad band impedance we obtain Z

||0 (ωr)/n ≈ 15 Ω. Nevertheless, a sufficiently

smooth vacuum chamber design, as is customary to achieve in modern accelerators, can lower this valueto about 1 Ω.From the longitudinal impedance, we can also give an estimation of the expected transverse impedanceby using the following formula (exact for the resistive wall impedance, but roughly applicable also tothe broad band impedance):

Z⊥1 (ω) =

C

πb2

Z||0 (ω)

n, (2)

where C is the ring circumference. Assuming the longitudinal impedance to be between 1 and 10 Ω, wethen easily get to a transverse impedance spanning the range 0.5 − 5 Ω/m for the ALBA ring.The question of single bunch stability under the effect of the broad-band impedance is addressed inthis report. Both longitudinal and transverse dynamics are taken into consideration and analyzed inthe following two sections. After giving some simple analytical estimations about plausible values forthresholds (in terms of tolerable impedance, as we consider the bunch current fixed), the results ofdetailed simulations carried out with the HEADTAIL code are reported. HEADTAIL is a macro-particlecode, which uses slicing and slice-to-slice wake fields to simulate the bunch dynamics under the effect


of a broad-band impedance. With particle tracking more realistic situations can be handled, like a bunchwith its synchrotron motion inside the sinusoidal bucket and a combination of dipole and quadrupolewake fields due to a flat chamber (emittance damping yet to be added to the model). Therefore thetheoretical predictions can be cross-checked and refined. All narrow-band impedances, such as thoseassociated to HOM’s of cavities, which can have non-negligible effects on the beam dynamics andstability [4], are not included in the present study.The ALBA parameters used throughout this note are summarized in Table 1.

Table 1: ALBA parameters used in simulationsParameter Symbol NumberMachine circumference C = 2πR0 268.8 mEnergy E 3 GeVNumber of electrons/bunch Nb 5.6 × 109

Bunch length σz(=cσt) 4.6 mm (15.4 ps)Momentum spread σδ 1.05 × 10−3

Cavity voltage V 3.6 MVHarmonic number nrf 448Synchrotron tune Qs 8.45 × 10−3

Mom. compaction α 8.72 × 10−4

R.m.s. emittances εx,y 4.5/0.04 nmTunes Qx,y 18.286/8.4752Average beta functions < βx,y > 6.26/9.05Chromaticities ξx,y 0,0Pipe half height b 14 mm

Given the general broad band resonator expressions

Z||0 (ω)

n=

ω

ωr

Rs

1 + iQ(

ωr

ω− ω

ωr

) ,

Z⊥1 (ω) =

C

πb2

ω

ωr

Rs

1 + iQ(

ωr

ω− ω

ωr

) =RT

1 + iQ(

ωr

ω− ω

ωr

) , (3)

a parameter study on the variables Rs/n, RT and ωr (quality factor Q usually set to unity) has beencarried out. As the c/b value for the ALBA beam pipe sits around 3.4 GHz, values of fr = ωr/2π in therange of few to about 10 GHz are taken into consideration. In accordance with what was previously saidabout the expected values of the impedance, reasonable values for our study will also be in the range offew tenths to few Ω for the longitudinal part (Rs) and few to about ten Ω/m for the transverse part (RT ).

2 Longitudinal plane

We first study the longitudinal stability of a bunch under the effect of the broad band impedance. Abroad band impedance is expected to cause instability above a certain threshold, which we can roughlyevaluate using the Boussard criterion (coasting beam criterion extended to bunched beams, but expectedto be more correct for long bunches). However, even below threshold, there could be enough potentialwell distortion as to cause bunch stretching, which can be associated to bunch shape deformation orcoherent undamped dipole motion of the bunch within the bucket.


The Boussard criterion for longitudinal microwave instability (or turbulent bunch lengthening) yields[1]: ∣∣∣∣∣∣

Z||0

n

∣∣∣∣∣∣ < 1.7 ln(2)Z0|η|γNbr0

σ2δσz (4)

Plugging the ALBA values into the above formula, we obtain the expected threshold of microwaveinstability to sit around 0.75 Ω. We recall here the formula that gives the potential well distortion for aGaussian bunch when a wake field is acting on it (known as Haissinski equation [1]):

λ(z) = λ0 exp

[− z2

2σ2z

− e2

meγCσ2δc

2η

∫ z

0dz′′

∫ ∞

−∞λ(z′)W ′

0(z′′ − z′)dz′

](5)

3.9

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

0 0.002 0.004 0.006 0.008 0.01 0.012

σz

(mm

)

t (ms)

ωr=2π 3.4 GHz

Z||/n=0.98ΩZ||/n=1.96ΩZ||/n=3.26Ω

Z||/n=9.8Ω

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

0 0.002 0.004 0.006 0.008 0.01 0.012

σz

(mm

)

t (ms)

ωr=2π 8.86 GHz

Z||/n=2.75ΩZ||/n=1.5Ω

Z||/n=0.75ΩZ||/n=0.15Ω

Figure 1: Bunch length (r.m.s.) evolution of an ALBA bunch subject to the action of different impedances(different Rs as labelled and ωr = 2π × 3.4 GHz (left) or 8.86 GHz (right).

The sign of the second order term in the Taylor expansion of the second term in the exponential ofEq. (5) determines whether we have bunch lengthening or shortening. We can conveniently write thisterm in the following way:[

d2

dz2

∫ z

0dz′′

∫ ∞

−∞λ(z′)W ′

0(z′′ − z′)dz′

]z=0

=∫ ∞

−∞λ(z′)W ′′

0 (−z′)dz′ =

= −∫ ∞

−∞λ(ω)ω[Z

||0 (ω)]dω (6)

Therefore the sign of the integral of the bunch spectrum multiplied by ω[Z||0 (ω)] is a quantity of

interest. It can be easily calculated that using the impedance (3), with the resonant frequency ωr leftas a free parameter, and the spectrum of the ALBA bunch, the transition between the two regimes ofbunch lengthening and shortening happens at around ωr ≈ 2π × 8 GHz. The two plots in Fig. 1 showthe bunch length oscillation resulting from HEADTAIL simulations in the two cases ω r = 2π × 3.4and ωr = 2π × 8.86 GHz: the two different regimes are evident, since while in the former case wesee that increasing the impedance value the bunch tends to shorten, in the latter case it clearly becomeslonger. In both cases, increasing impedances cause an undamped dipole oscillation of the bunch in thelongitudinal plane, as Figs. 2 show.

Bunch shape deformation has been studied for many values of longitudinal impedance and a widerange of ωr’s. Figures 3 and 4 summarize the results. In these figures, the vertical bars do not reallyrepresent errors, but the rms amplitude of the length oscillation, which increases as we move towardmore unstable situations. Proper microwave instability with micro-bunching occurs somewhere aroundthe predicted threshold in the higher ωr cases and at somewhat higher values for lower ωr’s. For the


-2

0

2

4

6

8

10

12

14

16

0 0.002 0.004 0.006 0.008 0.01 0.012

<z>

(m

m)

t (ms)

ωr=2π 3.4 GHz

Z||/n=0.98ΩZ||/n=1.96ΩZ||/n=3.26Ω

Z||/n=9.8Ω

0

5

10

15

0 0.002 0.004 0.006 0.008 0.01 0.012

<z>

(m

m)

t (ms)

ωr=2π 8.86 GHz

Z||/n=2.75ΩZ||/n=1.5Ω

Z||/n=0.75ΩZ||/n=0.15Ω

Figure 2: Bunch centroid evolution (longitudinal) of an ALBA bunch subject to the action of different impedances(different Rs as labelled and ωr = 2π × 3.4 GHz (left) or 8.86 GHz (right).

lower values of the impedance, in fact, the bunch shape remains very close to Gaussian, even when ahigher amplitude dipole oscillation appears, whereas for the higher values the bunch shape gets stronglydistorted. From both figures, the transition from the bunch shortening to the bunch lengthening regime,determined by how much the bunch spectrum overlaps with the capacitive or the inductive part of theimpedance respectively, appears clearly for a value of ωr between 6.5 and 8.9 GHz.In order to show how turbulent bunch lengthening sets in, we take the case ωr = 2π10 GHz and displaythe mountain range plots for different longitudinal impedance values .From the unperturbed situation,there is a transition through a regime of induced dipole motion to end up into a complete bunch defor-mation (turbulent bunch lengthening) regime.

4

4.5

5

5.5

6

6.5

7

7.5

8

0 1 2 3 4 5

σz

(mm

)

Z||/n (Ω)

ωr=2π 3.4 GHzωr=2π 5.0 GHzωr=2π 6.5 GHzωr=2π 8.9 GHzωr=2π 15 GHzωr=2π 21 GHz

The

oret

ical

mic

row

ave

inst

abili

ty th

resh

old

Figure 3: Bunch length as a function of the longitudinal impedance seen by the ALBA bunch, and for differentvalues of the ωr parameter. The error bars represent the r.m.s. values of the bunch length oscillation.


4.2

4.4

4.6

4.8

5

5.2

5.4

0 1 2 3 4 5 6 7 8 9 10

σz

(mm

)

Z||/n (Ω)

ωr=2π 3.4 GHzωr=2π 5.0 GHzωr=2π 6.5 GHzωr=2π 8.9 GHz

The

oret

ical

mic

row

ave

inst

abili

ty th

resh

old

Figure 4: Bunch length as a function of the longitudinal impedance seen by the ALBA bunch, and for the lowerconsidered values of ωr. Same as Fig. 3, but it extends to Z||/n = 10Ω and zooms on the transition regionbetween the shortening and the lengthening regime.

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

z (mm)

Z||/n=0.1Ω

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

z (mm)

Z||/n=0.5Ω

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

z (mm)

Z||/n=0.9Ω

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

z (mm)

Z||/n=1.8Ω

Figure 5: Mountain range plots of the bunch shapes for ωr = 2π × 10 GHz and different values of Rs =Z

||0 (ωr)/n. The initial condition (Gaussian bunch) is the red track appearing at the bottom of each plot.


3 Transverse plane

In the transverse plane, head-tail interaction through a wake field can drive the bunch unstable. Regularhead-tail instability can be avoided by running the ring at some slightly positive value of chromaticity.Strong head-tail instability, also known as Transverse Mode Coupling Instability (TMCI), can neverthe-less affect the ALBA bunch and cause rapid beam loss if the transverse impedance is strong enough.For a round beam in a round chamber, a criterion to find the threshold of TMCI, obtained from theapplication of the kinetic theory and the mode coupling condition, applicable in the two limits of shortor very long bunch, reads [5]:

ξ <Qs

ωrσtif ωrσt ≤ 1 (7)

ξ <√

2QQs(ωrσt)2 if ωrσt 1 (8)

where

ξ =ωr/2π < βy > RT Nbe

3.75 QE/e

If we consider ωr ranging between 2π×3.4 and 10 GHz, the discriminating number ωrσt ranges between0.3 and 1. Consequently, the ALBA bunch is in the regime of short bunch and Eq. (7) should be appliedto predict the threshold for the TMCI. In terms of broad-band resonator parameters and in convenientunits, we can simply recast this equation in the following form:

RT [kΩ/m] f 2r [GHz]

Q≤ 0.6

E [GeV] Qs

< βy > [m] Qb [C] σt [ps], (9)

where Qb = Nbe represents the bunch charge in Coulomb and fr = ωr/2π.Using the ALBA parameters from Table 1, Eq. (9) yields

RT [kΩ/m] f 2r [GHz]

Q≤ 1.2 × 105 ,

which translates into thresholds between 1 and 15 MΩ/m for the RT parameter of the transverse broadband resonator in the frequency range of interest (3 to 10 GHz) and for a quality factor Q = 1.

Nevertheless, simulations done with the HEADTAIL code using a realistic ALBA bunch inside a flatchamber, show that the thresholds are in general somewhat higher than the values predicted by usingZotter’s formula. Because of the flatness of the chamber, horizontal TMCI disappears whereas in thevertical plane a factor up to about 2 is gained on the threshold value. The beneficial effect of having a flatbeam pipe on the TMCI threshold had been already pointed out previously for long bunches [6]. Herethe result has been confirmed also in the short bunch regime. Figure 6 shows thresholds for differentvalues of the RT parameter of the broad band resonator. In the short bunch regime the f−2

r law is clearlyconfirmed, even if the thresholds sit a little higher than those expected for a round beam in a roundchamber. The transition to the long regime is also visible in the plot but, as only frequencies up to about20 GHz are considered, ωrσt comes to amount to no more than few units, and the long bunch regime isnot reached.

4 Conclusions and outlook

Single bunch collective phenomena driven by a broad band impedance can affect the ALBA bunch, andtherefore the impedance values need to be kept under control.

• The threshold for longitudinal microwave instability lies around Z||0 (ωr)/n = 0.75 Ω but, de-

pending on the value of the resonant frequency of the impedance within a reasonable range, it cancause significant bunch deformation also below threshold.


0

5

10

15

20

25

30

35

40

0 5 10 15 20

RT/Q

(MΩ

/m)

fr (GHz)

Simulated thresholds (flat chamber)Fitting curve ∝ fr

-2

Formula for short bunch (round chamber)

Toward the long bunchregime

Short bunch regime

Figure 6: Threshold RT as function of the broad band resonator frequency ωr/2π, as resulting from the HEAD-TAIL simualtions (red points). The theoretical line from Zotter’s formula (blue line) and a fitting curve for thenumbers from simulations (green line) are also plotted.

• In the transverse plane, even if the expected threshold for TMCI lies around 1–15 MΩ/m, macropar-ticle simulations taking into account of dipole and quadrupole wake fields deriving from a flatbeam pipe have shown that realistic threshold values are up to a factor 2 higher and span thereforebetween 2 and 30 MΩ/m.

• Inclusion of 3-D radiation damping in the HEADTAIL simulations as well as study of the (possiblybeneficial) effect of chromaticity on the instability thresholds are planned as future work. The sin-gle bunch effects of known lower frequency narrow-band resonators (like HOM’s from cavities)are also worth considering to have a more complete picture of this class of phenomena.

• A detailed calculation of the single contributions to the global impedance of the ALBA ring isrecommended and could be used to simulate the bunch evolution in more realistic conditions.

References

[1] A. W. Chao, “Physics of Collective Beam Instabilities in High Energy Accelerators”, John Wiley &Sons, Inc. 1993

[2] G. Rumolo and F. Zimmermann, CERN-SL-Note-2002-036-AP

[3] T. Weiland, “Summary of the Panel Discussion on Impedance Codes”, in Proc. of the CARE-HHH-APD Workshop on Beam Dynamics in Future Hadron Colliders and Rapidly Cycling High IntensitySynchrotrons (8-11 November 2004, CERN, Geneva, Switzerland)

[4] F. Perez, private communication

[5] B. Zotter, CERN/ISR-TH/82-10 (1982)


[6] E. Metral et al., “Transverse Mode-Coupling Instability in the CERN Super Proton Synchrotron”,in Proc. of the ICFA Workshop on High Brightness and High Intensity Beams, 18-22 October,Bensheim, Germany (2004)

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