Landau damping via the harmonic sextupole

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS, VOLUME 6, 054401 (2003)

Landau damping via the harmonic sextupole

Lidia Tosi, Victor Smaluk, and Emanuel KarantzoulisSincrotrone Trieste, Trieste, Italy

(Received 25 November 2002; published 14 May 2003)

054401-1

Multibunch instabilities of a storage ring electron beam occur due to coherent particle oscillationsgenerated through a bunch to bunch coupling via the impedances, deteriorating the beam quality. Onecure for multibunch instabilities is Landau damping, i.e., introducing a spread in the oscillationfrequencies among the particles of the individual bunches in order to destroy the coherence of thecoupled multibunch oscillation. Measurements at ELETTRA have shown that the harmonic sextupoleprovides Landau damping capable of suppressing transverse multibunch instabilities. The damping isinduced by the nonlinear tune spread with amplitude among the electrons within the individualbunches.

DOI: 10.1103/PhysRevSTAB.6.054401 PACS numbers: 29.20.–c, 29.27.Bd

also tune shifts with amplitude as a higher order effectthat may nevertheless induce an efficient spread.

The sextupole normalized strength is defined as K2L ��L=BR��@2By=@x2�. From the figure, one immediately

I. INTRODUCTION

The presence of multibunch instabilities is one of themost common reasons for beam quality deterioration andcurrent limitation in storage rings. The instability occursdue to coherent oscillations generated through a bunch tobunch coupling via a long memory structure in the stor-age ring (e.g., via an rf cavity). Every bunch may leave,while it passes through the structure, an electromagneticfield which perturbs the motion of the following bunches.When the system gets this loop closed (generally aftersome revolutions), all bunches execute a coupled multi-bunch oscillation coherently with a certain phase differ-ence between them. One cure for multibunch instabilitiesis to introduce a spread in the oscillation frequenciesamong the particles of the individual bunches in orderto destroy the coherence of the coupled multibunchoscillation. This mechanism is well known in manybranches of physics as Landau damping, whereby a col-lective motion performed by a certain number of particlesis damped by an increasing spread in the oscillationfrequencies of the individual particles. As the coherentoscillations build up a center of mass motion, the latterstarts decreasing as particles go out of phase with respectto each other and decohere. Individual particles may bestill oscillating but the center of mass motion is damped.If the driving force depends on the center of mass, then ittoo will disappear and in synchrotron storage rings theamplitudes of the individual particles of the bunches willdecrease with the betatron damping rate.

The necessary spread in the transverse plane may beprovided by introducing octupole magnets in the lattice(which are not present in ELETTRA [1]) or by settingnonzero chromaticities via sextupoles in the dispersivearcs. While the sextupoles give rise to tune spreads withmomenta, the octupoles generate tune shifts with ampli-tude as a first-order effect. However sextupoles induce

1098-4402=03=6(5)=054401(8)$20.00

This paper is about to examine and to present measure-ments based on Landau damping induced by the non-linear behavior of the sextupole magnets. Theoreticallythe tune shift with amplitude can be obtained by applyingperturbation theory to the Hamiltonian describing thedynamics in the action and angle variables [2]. Only thetune shifts with amplitude due to second order effects ofthe sextupoles will be considered here. These may beexpressed according to

��x � C11 � 2Jx � C12 � 2Jy;

��y � C12 � 2Jx � C22 � 2Jy;(1)

where the subscripts x and y denote the horizontal andvertical planes, respectively, ��u � x; y� are the non-linear tune shifts, Ju the perturbed action variables, andC11; C12; C22 are coefficients which depend on the sextu-pole strengths, on the beta functions at their locations,and on the relative phase advances between them [2,3]. Asthe units of Ju are meters, those of the coefficients aremeters�1. The lattice of the Italian third generation lightsource ELETTRA has been provided with three sextu-pole families. While two of these families are placed inthe dispersive arcs to compensate for the fairly largenatural chromaticities (defined as � � ��d��=�dp=p��),the third family S1 is a harmonic sextupole located inthe dispersive free regions. It has been introduced toenlarge the dynamic aperture by compensation of thegeometric aberrations generated by the chromatic sextu-poles. Thus, it is possible to change the nonlinear config-uration of the machine (hence the tune shifts withamplitude) by acting on its strength, maintaining at thesame time constant values for the chromaticities. Figure 1shows the theoretical behavior of the three coefficientsas functions of the harmonic sextupole normalizedstrength K2L at 2.0 GeV (nominal value is 2:11 m�2).

2003 The American Physical Society 054401-1

0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

1.5x 10

4

S1 K2L (m− 2)

Cm

n (m

− 1)

C11

C12

C22

FIG. 1. C11, C12, and C22 vs the harmonic sextupole normal-ized strength at 2.0 GeV.

PRST-AB 6 LANDAU DAMPING VIA THE HARMONIC SEXTUPOLE 054401 (2003)

sees that the effect of the harmonic sextupole on the tunesmay be rather strong. One may expect an enhancement ofhorizontal instabilities at the strength corresponding tothe minimum of the absolute value of C11 and of thevertical ones at the zero crossing of C22. Observationsand measurements confirm that in ELETTRA the har-monic sextupole S1 plays an essential role in dampingcoherent transverse multibunch instabilities, to the extentthat they can even be totally suppressed when the sextu-pole is appropriately set [4]. Many measurements havebeen performed and they all indicate a strong correlationwith the behavior of the coefficients C11 and C22 [5–7].

As a basic tool for the measurements, the digital trans-verse multibunch feedback (TMBF) [8] of ELETTRAhas been used. It consists of a wide-band bunch-by-bunchsystem where the positions of each of the 432 bunches canbe acquired. The wide-band signals of the bunches, sep-arated by 2 ns, are demodulated into a base-band signalbetween 0 and 250 MHz. The memory of the system issuch that it allows the acquisition of the bunch positions inreal time over a time span of 192 ms and thus over200 000 turns, yielding a precious tool for the analysisof the details of the behavior of the single bunches. It hasto be however underlined that at the moment the acquiredsignals cannot be converted to the actual beam positionbecause the system is still not calibrated.

This paper presents the main results of the observa-tions, of the measurements performed, and of computersimulations of the different Landau damping rates in-duced by the variations of the strength of the harmonicsextupole. Although similar observations have been madefor the vertical plane, due to the relatively simpler mech-anism, both measurements and simulations presented arerelated to the horizontal plane. In this case, the couplingamong the two planes due to the sextupoles is eliminatedand the tune shift with amplitude may be reduced to

054401-2

��x � C11 � 2Jx provided that the collective motion isprincipally in the horizontal plane. Section II discussesthe fast coherent damping observation when kicking asingle bunch at the injection energy of 0.9 GeV. Someresults regarding the vertical plane done at a macroscopiclevel, namely, not using the TMBF, are presented inSec. III, where the effects of the strength of S1 on generalmachine parameters are presented. Section IV insteadpresents the results of the damping effects measuredwith the TMBF compared to simulations when coupledhorizontal multibunch instabilities are present.

II. LANDAU DAMPING OBSERVATION

Effects of nonlinearities induced by the harmonic sex-tupole on horizontal betatron motion have been measuredin a single bunch and compared with computer simulation.Coherent betatron motion was excited using the injectionkickers and turn-by-turn data were taken using the TMBFsystem. There is a theoretical analysis [9] of nonlinearbeam dynamics in the case of free betatron oscillationexcited by a short kick and observed using a photomulti-plier tube with a blind placed in the image plane. Withsome modifications this analysis can be also applied to abeam position monitor (BPM) which is the pickup ofthe TMBF.

A single bunch in the absence of coherent betatron andsynchrotron oscillations has a particle distribution func-tion in betatron phase space represented in the action-phase variables as

f�J; ’; t� �X1

n��1

fn�J; t�e�in�!t�’�; (2)

where J � �a2=2�� is the perturbed action, a is the os-cillation amplitude, ’ is the phase, ! � 2��frev is thebetatron frequency, and � is the beta function. If G�x;!�is the transfer function of a diagnostic device, the outputsignal is proportional to

X �X1

k��1

X1n��1

e�i�k!0�n!�tZ 1

0fn�J; t�KkndJ; (3)

where!0�2�frev and Kkn�R��G�acos’;k!0�e

in’d’.Note that fn�J;t� is a slow function of time. Supposingthat the transfer function of the BPM electronics is linearand the amplitude-phase characteristics are constant inthe frequency band where the betatron harmonics haveconsiderable amplitudes (usually n<5 to 10), then Kkn�K� const, and the signal of the nth betatron harmonic isproportional to

Xn�Ke�i�!0�n!�tZ 1

0fn�J;t�dJ�c:c:; (4)

with the envelope as

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PRST-AB 6 LIDIA TOSI, VICTOR SMALUK, AND EMANUEL KARANTZOULIS 054401 (2003)

An�K

��Z 1

0fn�J;t�dJ

��: (5)

If the beam current is small and collective effects arenegligible, as in the case of zero chromaticity, a first-orderapproximation gives

fn�J; t� � fn�J; 0�e�in�@!=@J�Jt: (6)

054401-3

Here �@!=@J� � 2!0C, where C is the nonlinear coeffi-cient C11 (1) for the horizontal plane or C22 for thevertical one. Placing (6) into (5) gives

An � K

��Z 1

0fn�J; 0�e

�in�@!=@J�JtdJ

��: (7)

In the case of nonzero chromaticity, instead of (6) thedistribution function is

fn�J; t� � e�in�@!=@J�JtZ 1

0

Z 2�

0fn�J; "�e

�in�@!=@E��"=��sin� ��t��sin �"d"d

� e�in�@!=@J�Jt � 2�Z 1

0fn�J; "�J0

�2n@!@E

"�

sin�t2

�"d"; (8)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−40

−20

0

20

40

t (ms)

x (a

rb. u

nits

)

Measurement

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−6

−3

0

3

6

t (ms)

x (m

m)

Simulation

beam size

FIG. 2. Gaussian damping of the coherent oscillation.

where �@!=@E� � �!0=E��, J0 is the Bessel function, and�, ", are the frequency, amplitude, and phase of thesynchrotron oscillation. If fn�J; "� can be represented as aproduct fn�J; "� � Fn�J�(�"�, then, to take the chroma-ticity into account, fn�J; t� in (6) should be multiplied by

Mn � 2�Z 1

0(�"�J0

�2n@!@E

"�

sin�t2

�"d": (9)

In this case An�t� is modulated with the modulation pa-rameter Mn�t�.

In our experiments a coherent betatron oscillation wasexcited by a short kick. If F�J� is the distribution functionof oscillation amplitudes before the kick, then after thekick the distribution function becomes

f�J; ’; "� � F�J� 2��J*J

psin’� 2*J�(�"�; (10)

where *J � �*a2=2�� is the action perturbation due tothe kick with *a amplitude. Calculating the first betatronharmonic A1�t� with the conditions *a� +? and t �!0C+

2?=��

�1 where +? is the transverse beam sizegives

A1�t� � M1�t�

��Z 1

0F�J�J0�2!0C

��J*J

pt�dJ

��: (11)

For a stable bunch, the distribution functions F�J� and(�"� are Gaussian and, according to (11), the envelope is

A1�t� / e��t2=2,2�e��@!=@E��+E=��2�1�cos�t�; (12)

where , � �2!0C+?*a=��1 and +E is the rms energyspread. Thus, if the chromaticity is zero, the envelope ofbetatron oscillation is Gaussian.

Figure 2 shows an example of measured data in com-parison with simulation. The measurement was per-formed on a single bunch at 0.9 GeV energy with a beamcurrent of 1 mA and a chromaticity of 0.1. Since theemittance at 0.9 GeV is 1.415 nm rad and in the simulationC11 is �2835 m�1, the tune spread of the initial beam is8� 10�6. The simulation was carried out for 1000 par-ticles and the kick amplitude of almost 6 mm was fit to getthe damping time close to the measured one because the

measurement system is not calibrated. In the simulationplot one can see that the coherent oscillation damps andthe incoherent one (i.e., beam size) grows. This effect isdue to the mismatching of the particles’ phases caused bythe nonlinear tune spread. The initial beam size due to thedecoherence grows up to approximately 0.7 of the initialoscillations when the motion becomes completely inco-herent. At a later stage, the size will slowly decrease dueto the natural damping. This is not shown in the figuresince the damping time is 112 ms.

The damping time was measured with the variousharmonic sextupole strength in the 1:4–3:6 m�2 range.Figure 3 shows the measured damping rate versus theharmonic sextupole strength in comparison with the sim-ulation data and with the theoretical curve calculatedusing the , � �2!0C11+?*a=��

�1 formula. One cansee that the damping rates obtained by the measurements,the simulations, and the theory all agree very well. Theparabolic shape of the damping rate graph repeats thebehavior of a cubic nonlinearity parameter and, namely,the harmonic sextupole acts as an octupole. The Landaudamping of coherent betatron oscillation is rather strong,

054401-3

1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

S1 K2L (m−2)

1/τ

(ms− 1

)theorysimulationmeasurement

FIG. 3. Damping rate versus harmonic sextupole strength.


with , � 0:2 to 1 ms with the nominal small positivechromaticity ( � 0:1).

III. MACROSCOPIC INSTABILITYMEASUREMENTS

The energy loss per turn of the electrons is compen-sated in ELETTRA by four rf cavities located in thedispersive arcs. The higher order modes (HOM) of thecavities are controlled by setting the temperatures of thecavities [10]. In particular, the behavior of the horizontalcoupled bunch (dipole) mode HCBM 318, driven by thecavity HOM T3 [11], versus the harmonic sextupolestrength was investigated, as well as that of the HCBMs414-417 whose origin is still a matter of investigation.Figure 4 compares the theoretical values of the coefficientC11 with the amplitude of HCBM 318 measured with aspectrum analyzer as a function of the harmonic sextu-pole strength. The mode could be totally suppressed at2.0 GeV by setting the sextupole below 1:3 m�2 or above

1.5 2 2.5 3 3.5 4−4500

−4000

−3500

−3000

−2500

−2000

−1500

−1000

S1 K2L (m− 2)

C11

(m

− 1)

C11

A

80

75

70

65

60

55

A (

dBm

)

FIG. 4. Comparison of the amplitude of HCBM 318 with C11

as functions of the harmonic sextupole strength.

054401-4

4:1 m�2. The same parabolic behavior as C11 was foundfor the excitation levels of HCBM 414, although in thiscase the two values of the sextupole strength for whichthe mode was totally suppressed were found to be differ-ent. While the peak of the excitation level of the modeswas always found to be around the sextupole strength of2:75 m�2 (depending on the optics), deeper investigationsrevealed that the two boundary values of the sextupole atwhich the modes can be suppressed depend on the beamcurrent, on the modes excited, on the filling pattern, andon the presence or less of longitudinal instabilities. It wasnoticed that the stronger the mode, the higher the beamcurrent filling percentage and the lower the longitudinalexcitations were, the wider was the range of the sextupolestrength for which the modes were excited. This, namely,translates into requiring higher absolute values for C11 inorder to achieve the suppression of the horizontal modes.While the dependence on the beam current and on theparticular mode excited reflects the ‘‘strength’’ of thedriving force, the one on the filling pattern and on thepresence or less of longitudinal modes is due to how wellthe driving force manages to couple to the beam’s modes.In fact the coupling efficiency of a transverse drivingforce may be diminished by a gap within the bunch train,by the different arrival times of the bunches, and possiblyalso on the longitudinal density of the bunches.

It has to be strongly underlined that even in the totalabsence of longitudinal modes, confirmed by measure-ments of the full longitudinal spectrum, it was found thatthe amplitudes of the excited transverse modes follow thesame trend of C11 with the harmonic sextupole strength,excluding thus any interference of longitudinal modes inthe mechanism by which the sextupole may be influenc-ing the dynamics. This was a particularly important pointto confirm, because the cavities are located in the dis-persive arc. Furthermore, it was found, by measuring fulllongitudinal spectra as a function of the harmonic sextu-pole strength, that the excitation level of any longitudinalinstability did not depend on the sextupole.

The major beam parameters, such as tunes, closedorbit, dispersion, and chromaticities, were measured asa function of the harmonic sextupole strength. No sig-nificant changes were noticed in these quantities, with theexception of the horizontal chromaticity, which for themeasurement of Fig. 4 was found to gradually increasefrom 0.1 for the sextupole strength of 1:3 m�2 to 1.6 for4:1 m�2. Both settings are values for which the mode wassuppressed. This variation of the chromaticity is not ex-pected to be the major cause for the suppression of themode. In fact, no change in the excitation level of themode was noticed when the horizontal chromaticity wasset to 1.6, by using the chromaticity correcting sextu-poles. Since however the value of C11 also depends onthe strengths of the chromatic sextupoles, investigationswere carried out in order to confirm the leading role ofC11 in the phenomenon against that of the horizontal

054401-4

TABLE I. Amplitudes of vertical modes versus harmonicsextupole strength.

K2L VCBM 167 VCBM 246 VCBM 344 VCBM 422�m�2� (dB) (dB) (dB) (dB)

1.38 16.7 16.3 17.0 37.31.66 18.6 10.8 11.0 38.74.17 6.1 � � � � � � 25.4


chromaticity. The results revealed that the suppression ofthe excited modes using the chromaticity correcting sex-tupoles required a horizontal chromaticity greater than 7.The values instead of C11 found for these settings of thechromatic sextupoles were computed to be identical tothose at which S1 suppresses the mode.

Associated with the presence of the transverse modes,low frequencies in the spectrum (typically below 100 Hz)have also been observed.Whereas in some situations thereappears a well-defined peak at low frequency, in othersthe whole background noise level increases by 1–2 ordersof magnitude, according to whether only one or moremodes were present, respectively. Whenever well-definedlow frequencies were noticed, measurements on the spec-trum analyzer in zero span mode of the unstable sidebandshowed periodicities corresponding to the peaks mea-sured in the low frequency spectrum range. Detailedinvestigations in a narrow frequency range around theunstable sideband with an appropriate sweeping timemultiple of the associated low frequency showed a finestructure indicating that the sideband was moving withthe measured low frequency. As it will be shown in thefollowing section, the apparent movement of the unstablesideband is related to an increasing spread in the frequen-cies of the particles within the single bunches. Thestrength of the harmonic sextupole also influences thevalues of the low frequencies. Figure 5 shows the depend-ence of the low frequencies both on the harmonic sextu-pole strength and on the beam current for a fixed settingof the sextupole at 2:75 m�2. The low frequencies fitextremely well to a second order polynomial as a functionof the sextupole, presenting a maximum where the abso-lute value of C11 is minimum.

Another measurement that was considered necessarywas to confirm that effectively the horizontal tune shiftwith amplitude ��x follows the same behavior asC11. Forthis purpose, the horizontal tune spread of a 1 mA beamin single bunch mode was measured as a function of thesextupole strength, by exciting the beam with a constantamplitude using the spectrum analyzer of the tune meas-urement system. A good parabolic agreement was foundwith the minimum at K2L � 2:75 m�2.

2 2.5 3 3.530

35

40

45

50

55

60

65

S1 K2L (m−2)

f (H

z)

datafit

100 150 200 25040

50

60

70

80

Ibeam

(mA)

f (H

z)

datafit

FIG. 5. Low frequencies associated with transverse instabil-ities as functions of the harmonic sextupole strength (leftpanel) and of beam current (right panel) for a fixed strengthof the sextupole (K2L � 2:75 m�2).

054401-5

Measurements at a macroscopic level were carried outalso for the vertical plane. Assuming the same hypothesisas for the horizontal plane, some preliminary studies havebeen made on the dependence of vertically excited modeson C22. As can be seen in Fig. 1, C22 presents very lowvalues in the range of strength for the harmonic sextupolebetween 0:9 m�2 and 1:8 m�2, becoming fairly large forstrength above 2:75 m�2. Thus, if the hypothesis that thecoefficients in the tune shift with amplitude is the leadingmechanism by which excitation levels of the instabilitiesdepend on the harmonic sextupole, one would expect anenhancement of the vertical modes for lower strength ofthe sextupole and suppression for higher ones. Table Iillustrates the amplitudes of the modes as a function ofthe strength. In particular, during the measurement, itwas noticed that trying to set the harmonic sextupole to1:25 m�2 (C22 � 0 for 1:20 m�2), the vertical instabilitygrew extremely strong with an evident vertical beam sizeblowup on the synchrotron light beam profile monitor andsubsequent loss of 100 mA of beam current. Missing datafor the vertical amplitudes are due to the coexistence inthis range for the sextupole strengths of horizontal modeswhich were suppressing the vertical ones. The suppressionof the vertical modes during the existence of the hori-zontal ones is in agreement with an influence of thecoupling term C12.

IV. INSTABILITY MEASUREMENTS WITHTHE TMBF

Coupled multibunch instabilities can be viewed asconsisting of two collective motions: a macroscopic oneamong the centroids of the individual bunches and amicroscopic one among the single particles within theindividual bunches. Thus, in the past, based on the resultsof the previous section when observations were not facili-tated by the TMBF, an important question arose: whetherthe tune spread induced by the harmonic sextupole actedon different bunches or on the particles within theindividual bunches. Simulations done in 1999 using amacroscopic model of the bunches could reproduce onlyqualitatively the main features of the phenomena, but notquantitatively. When the TMBF became operational, al-lowing the visualization of the center of mass motion ofindividual bunches, further exhaustive measurementswere performed for the understanding of the mechanisminvolved.

054401-5

FIG. 6. Horizontal instability for K2L � 2:3 m�2.


Figure 6 shows the measured center of mass motion,together with its spectrum (tune resolution is 5� 10�4),of one of the bunches undergoing the horizontal coupledmultibunch instability HCBM 318 with S1 set notfar from the minimum value for the tune shift withamplitude. Setting S1 to 2:75 m�2 brought a significantbeam loss.

The damping of the center of mass motion is not due toa simple detuning of the HOM. In fact, observing Fig. 7where the same analysis is shown for the same bunch butwith the sextupole set to a stronger tune shift with am-plitude, one can note that the damping occurs at a muchsmaller tune shift of the center of mass. This occursbecause of the higher damping efficiency due to theincrease in the spectral density of the betatron tunedistribution.

Although the two motions are qualitatively differentthey have one common feature: during the growth of theinstability there is a broadening of the tune distribution

FIG. 7. Horizontal instability for K2L � 1:8 m�2.

054401-6

width. This broadening cannot be due to the spread inamplitudes of the center of mass, because during thedamping the spread in amplitudes is much larger, butthe width is noticeably smaller. The only interpretationcan be that there is an increasing tune spread within thecoherent particles of the bunch, resulting as a tune shift ofthe center of mass together with a broadening. In fact,since amplitudes can be only positive, the tune shift withamplitude is unidirectional. Thus, if there is an increasingspread during the collective motion, then the tune of thecenter of mass will seemingly move at a rate that willgenerate a broadening in its spectrum. The same analysishas also been performed on all the other bunches for thesame two sextupole strengths and they all simultaneouslypresent similar features with identical tune shifts andbroadenings. The low frequency oscillations mentionedin the previous section have been found to be related tothe repetition of excitation and damping of the coherentmotion. The two figures also show the presence of syn-chrotron sidebands which are due to a not completelylongitudinal stable situation. These however do not growin intensity as the betatron frequency does, but theyremain constant during the motion. This is retained tobe a further confirmation of the absence of longitudinaldynamics in the mechanism.

To check the above-mentioned conclusions, a simula-tion of the instability has also been done using multibunchmultiparticle tracking with an interaction between thebeam and the single rf cavity mode T3. The results forK2L � 2:3 m�2 are shown in Fig. 8. Although the detailsof the center of mass motion may differ from reality, thetune behavior is similar to the one of the measured datashown in Fig. 6 with a broadening of the spectrum whilethe instability rises. The details of the center of massmotion in the simulations have been found to depend onthe number of particles per bunch that were being simu-lated. The present is the result of 100 particles per bunch

FIG. 8. Simulation of the horizontal instability for K2L �2:3 m�2.

054401-6

0 20 40 60 80 1000

5

10

15x 10

4

f (MHz)

ampl

itude

(ar

b.un

its)

FIG. 10. Horizontal multibunch spectrum.


and is surely still not sufficient to reproduce the realityfaithfully, but the main essence of the phenomenon isthere.

Knowing the measured rise time (0.45 ms) given by theHOM of the cavity, the betatron radiation damping timeat 2.0 GeV (10.21 ms) and extracting the rise time of thebunch mode (6 ms) from Fig. 1, the Landau dampingrequired to kill the instability is ,L � 0:512 ms, providedby a tune spread of �� frev,L�

�1 � 5:37� 10�4.Assuming a constant chromatic tune spread, estimatedfor a stable beam and knowing the beam size at rest, therequired amplitude for Landau damping results to be1.8 times the beam size at rest, consistent with mea-surements from the synchrotron radiation beam profilemonitor.

Multibunch analysis with the TMBF has been carriedout also in the case in which several horizontal modeswere present contemporaneously. Figure 9 shows a typicalmotion of an individual bunch in the presence of a hori-zontal multibunch instability with the spectrum shown inFig. 10. The multibunch spectrum is obtained over 2000turns in correspondence to the increase of the amplitudeof the bunch oscillation. The main unstable modes arecentered at 0.832 and 26.277 MHz which correspond tothe horizontal coupled mode numbers 417 and 394.Neither of the two modes correspond to known cavityhigher order modes. Varying the harmonic sextupolechanges the excitation level of the unstable modes withthe maximum at 2:80 m�2 and a clearance of the modesbelow 1:65 m�2 and above 3:72 m�2.

The motion of individual bunches is very irregular andseemingly chaotic, due to the large number of modesinvolved, rendering its analysis as a function of theharmonic sextupole complicated. Thus the extraction ofthe rise times of an individual bunch motion is not suffi-cient to give an understanding of the mechanics involved,since these will depend heavily on the phases of the

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−60

−40

−20

0

20

40

60

80

t (ms)

x (a

rb.u

nits

)

FIG. 9. Individual bunch motion.

054401-7

modes and on which one is most strongly influencingthe motion at the moment. In this situation, an analysisof the evolution in time of the amplitudes of the unstablemodes for various harmonic sextupole strength is moremeaningful. It goes without saying that this was per-formed always during a time span in which horizontalbeam motion was found to have increasing amplitudes.The rise and damping times of the unstable modes as afunction of the harmonic sextupole strength are shownin Fig. 11. It can be observed that there is a concentra-tion of unstable modes with positive rise times at thosevalues for which the absolute value of C11 is near itstheoretical minimum. Furthermore, as the two sextupolestrengths where the beam is stable are approached, alarger number of unstable modes decrease in amplitudeand are damped.

1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6−70

−60

−50

−40

−30

−20

−10

0

10

20

30

40

S1 K2L (m−2)

τ (m

s)

FIG. 11. Rise and damping times of the unstable modes as afunction of the harmonic sextupole strength.

054401-7


V. CONCLUSIONS

At ELETTRA it has been found and shown that sub-stantial damping against instabilities can be obtained byvarying its harmonic sextupole. This by itself was notknown. In fact, the reason of installing such a lens was toreduce the induced nonlinearities produced by the chro-matic sextupoles and thus to enlarge the dynamic aper-ture. However it was not suspected that when varied, thechange in the nonlinear tune shift with amplitude couldbe strong enough as to induce a sufficient damping toeven suppress transverse coupled multibunch instabilities.Its damping efficiency is comparable to that of octupolesthat generally are used for this purpose. Although set-tings of the harmonic sextupole which are differentfrom its nominal one inevitably bring a reduction of thedynamic aperture of the machine and therefore its life-time, the lifetime reduction is small compared to thebenefits associated with the stabilization effects of thedetuning.

Coming to the mechanism itself, it has been provenbeyond any doubt that it is a pure Landau damping effectoriginating from the tune spread of the particles inside theindividual bunches in the machine. The mechanism cantherefore be effective both for multibunch and for singlebunch operation. It is interesting to note that it has beenobserved as the typical Landau damping behavior.Namely, the damping efficiency increases with the in-crease of the frequency spectral density and the dampingoccurs when the center of mass frequency lies within thespread of the single particle frequencies.

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At ELETTRA the harmonic sextupole is effectivelyused to damp efficiently horizontal coupled multibunchinstabilities during operation for the users.

[1] ELETTRA Conceptual Design Report, Trieste, 1989.[2] R. Nagaoka et al., Nucl. Instrum. Methods Phys. Res.,

Sect. A 302, 9–26 (1991).[3] K.Y. Ng, Fermilab Report No. TM-1281, 1984.[4] E. Karantzoulis, C. J. Bocchetta, A. Fabris, F. Iazzourene,

M. Svandrlik, L. Tosi, and R. P. Walker, in Proceedings ofthe EPAC-98, Stockholm (IOP, Bristol, U.K., 1998).

[5] L. Tosi and E. Karantzoulis, in Proceedings of the PAC-99, New York (IEEE, Piscataway, NJ, 1999).

[6] L. Tosi, V. Smaluk, D. Bulfone, E. Karantzoulis, and M.Lonza, in Proceedings of the PAC-2001, Chicago (IEEE,Piscataway, NJ, 2001).

[7] L. Tosi, V. Smaluk, and E. Karantzoulis, in Proceedingsof the EPAC-2002, Paris (EPS-IGA/CERN, Geneva,2002).

[8] D. Bulfone et al., in Proceedings of the DIPAC-2001,Grenoble (http://www.wsrf.fr/conferences/DIPAC/DIPAC2001Proceedings.html).

[9] N. Vinokurov, V. Korchuganov, G. Kulipanov, andE. Perevedentsev, Chromaticity and Cubic Non-Linearity Effects on a Dynamics Of BetatronOscillation (Budker Institute of Nuclear Physics,Novosibirsk, 1976), pp. 76–87 (in Russian).

[10] M. Svandrlik et al., in Proceedings of the PAC-95,Dallas (IEEE, Piscataway, NJ, 1995).

[11] M. Svandrlik et al., in Proceedings of the EPAC-96,Sitges (IOP, Bristol, U.K., 1996).

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Landau damping via the harmonic sextupole