Analysis of transverse beam stabilization with radio frequency quadrupoles

M. Schenk,1,2,* A. Grudiev,1 K. Li,1 and K. Papke11CERN, CH-1211 Geneva, Switzerland2EPFL, CH-1015 Lausanne, Switzerland

(Received 7 June 2017; published 13 October 2017)

A radio frequency (rf) quadrupole has been considered as a potential alternative device for Landaudamping in circular hadron colliders. The objective of this study is to benchmark and confirm its stabilizingeffect predicted by stability diagram theory by means of numerical tracking simulations. To that end, twocomplementary models of the device are implemented in PyHEADTAIL, a 6D macroparticle tracking codedesigned to study the formation and mitigation of collective instabilities. The rf quadrupole model isapplied to a slow head-tail instability observed experimentally in the Large Hadron Collider to show thatsuch a device can in principle provide beam stability similarly to magnetic octupoles. Thereafter, alternativeusage schemes of rf quadrupoles also in combination with magnetic octupoles are proposed, discussed, andbenchmarked with simulations.

DOI: 10.1103/PhysRevAccelBeams.20.104402

I. INTRODUCTION

To improve the luminosity of future machines like theHigh Luminosity Large Hadron Collider (HL-LHC) orthe Future Circular Collider (FCC), the brightness of theparticle beams will be significantly increased [1,2]. This isrealised by operating the machine with a higher bunchintensity and lower transverse emittances. These parame-ters and therefore the maximum achievable luminosity willbe strongly limited by the presence of transverse collectivebeam instabilities induced by the transverse beam couplingimpedance of the accelerator ring.A powerful stabilizing mechanism against this type of

instabilities is the effect of Landau damping which ispresent when there is an incoherent spread in the betatronfrequencies, or tunes, of the particles in the beam [3,4]. Thetune spread is a result of nonlinearities in the machine.Partially, they are of parasitical nature, e.g., originatingfrom non-linear space-charge forces, nonlinearities in themagnetic focusing systems, beam-beam interactions atcollision, etc. In addition, however, they are often intro-duced by design through dedicated nonlinear elements forbetter control and efficiency. Magnetic octupoles arecommonly used for the latter purpose. In the LargeHadron Collider (LHC), families of 84 focusing and 84defocusing, 0.32 m long superconducting magnetic octu-poles are installed to produce an incoherent betatron tunespread that depends on the transverse amplitudes, oractions, ðJx; JyÞ of the particles in the beam [5]. These

so-called Landau octupoles are an integral part of the LHCinstability mitigation toolset and are extensively used forbeam stabilization during operation [6,7].Future hadron colliders will operate with beams of

smaller transverse emittances making the Landau octupolessignificantly less effective due to the reduced spread inðJx; JyÞ. This effect is even more pronounced at higherbeam energies as a result of adiabatic damping and mayeventually lead to a loss of Landau damping of potentiallyperformance-limiting collective instabilities. Alternativeapproaches, such as betatron detuning with longitudinalaction Jz are currently under study. Longitudinal actionprovides a much larger handle for introducing a betatrontune spread due to the orders of magnitude larger spread inJz compared to ðJx; JyÞ of the beams of high energy hadroncolliders [8]. The basic formalism for this particularstabilizing mechanism has been developed by J. ScottBerg and F. Ruggiero [9]. With their work they demonstratethat the tune spread introduced as a function of Jz improvesthe stability of particle beams. The underlying mechanismis similar to the Landau damping effect from magneticoctupoles.A radio frequency (rf) quadrupole has been considered as

a potential device to realise betatron detuning with longi-tudinal amplitude [8]. Analytical calculations predict thatthe maximum RMS betatron tune spread generated by theLHC Landau octupoles can theoretically be achieved with afew metres long superconducting rf device operating in atransverse magnetic quadrupolar mode. This is for LHCnominal beam and machine parameters at top energy of7 TeV. For comparison, the total active length of the LHCLandau octupoles is about 56 m.The main purpose of this study is to confirm the

stabilizing effect of an rf quadrupole predicted by stabilitydiagram theory with numerical simulations using the

*michael.schenk@cern.ch

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW ACCELERATORS AND BEAMS 20, 104402 (2017)

2469-9888=17=20(10)=104402(15) 104402-1 Published by the American Physical Society

PyHEADTAIL 6D macroparticle tracking code [10].Section II summarizes the working principle of an rfquadrupole, describes possible designs and the maincharacteristics of the device, potential benefits over conven-tional stabilizing methods, as well as the existing theory ofthe stabilizing mechanism. Section III briefly introduces thebasic principles of the PyHEADTAIL code and details onthe implemented rf quadrupole models. Section IV containsthe numerical proof for beam stabilization with an rfquadrupole. The study is based on a slow head-tailinstability that is observed in the LHC and proven to beLandau damped by means of the magnetic octupoles bothexperimentally and in simulations [11]. Here, trackingstudies show that an rf quadrupole manages to stabilizethe beam in a similar manner. Section V discusses the beamdynamics in presence of an rf quadrupole in more detail.Furthermore, it introduces various schemes for the use of rfquadrupoles, among them a two-family configuration aswell as the combination of an rf quadrupole with magneticoctupoles. The schemes are partially motivated by stabilitydiagram theory and are benchmarked with trackingsimulations.

II. RADIO FREQUENCY QUADRUPOLE

This section gives a brief overview on the main aspects ofan rf quadrupole device. It discusses first two cavity designsexplaining the main differences between the two geometriesas well as the relevant parameters that have been optimized[8,12]. The basic working principle of the device is thenexplained by deriving an analytical formula for the incoher-ent betatron detuning that it generates. Subsequently, thefeatures and potential advantages over conventional stabiliz-ing methods are discussed in more detail in terms of themitigation of collective instabilities. Finally, an existing,approximate, stability diagram theory is applied for aqualitative characterization of the Landau damping [9].

A. Device description

The purpose of an rf quadrupole is to produce transversequadrupolar kicks on the particles in the bunch with astrength that depends on the longitudinal coordinate. Everyparticle feels a different focusing (defocusing) force as itpasses through the device, and hence experiences a changein the betatron tunes depending on its longitudinal positionin the bunch. This results in an incoherent betatron tunespread which can produce Landau damping. The frequencyof the rf quadrupole described in this study is chosen to be800 MHz for two reasons. First, the bunches of theHL-LHC and FCC-hh (hadron-hadron) are foreseen tohave a length of the order of σz ≈ 0.1 m [1,2]. An rfquadrupole operating at 800 MHz has an rf wave lengththat matches the bunch length and provides the best beamstabilizing efficiency according to numerical simulations[12]. Second, for practical reasons, the rf quadrupole

frequency is limited to harmonics of the main rf system(400 MHz) of the aforementioned machines.In [12], two different cavity designs for a superconduct-

ing rf quadrupole have been proposed and thoroughlyoptimized primarily for quadrupolar field strength, but alsoin terms of transverse and longitudinal beam couplingimpedance, and peak electric Epk and magnetic Bpk surfacefields. While the first type is an elliptical cavity operating ina transverse magnetic (TM) quadrupolar mode, the secondone is a four-vane cavity operating in a transverse electric(TE) quadrupolar mode. The geometry of the latter wasmotivated by the rf quadrupole (RFQ) linac [13]. Its designwas chosen to reach higher quadrupolar field strengths atsmaller cavity size. A cross section of the quadrupolarfields for the two cavity types is given in Fig. 1. Thetransverse kicks Δp⊥ on a positively charged particlepassing the device off-axis at a specific moment in timeare indicated by the black arrows.The goal of the cavity optimization procedure is mainly to

find a geometry that provides the largest quadrupolar fieldstrength at the lowest surface fields and impedance. For theelliptical cavity, the TM210 mode is the best for that purpose.An illustration of the normalized magnetic (left) and electric(right) field distributions for this particular mode is given inthe upper part of Fig. 2 showing one octant of the ellipticalcavity. The geometry and field distributions of the four-vanecavity are depicted in the lower part of Fig. 2. The mainadvantages of the four-vane cavity compared to an ellipticalone are [12]: (i) the quadrupolar field strength per cavity isup to two to five times larger than that of the elliptical one,given that the aperture has a radius <50 mm, and (ii) thefour-vane cavity is more compact and requires hence smallercryomodules and less cooling power. As a result, the numberof cavities can be reduced and therewith the overallimpedance and cost of the system.

B. Working principle

In the thin-lens approximation, an ultrarelativistic par-ticle of index i, electric charge q, and momentum p

FIG. 1. Qualitative description of the quadrupolar fields leadingto a transverse kick on a charged particle (cross-section). Thedirection of the momentum change Δp⊥ is indicated for apositively charged particle. Left: Elliptical cavity type operatingin a TM mode. Right: Four-vane cavity operating in a TE mode.

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traversing an rf quadrupole along the z-axis experiencesboth transverse and longitudinal kicks

Δp⊥i ¼ pk2ðyiey − xiexÞ · cos ðωti þ φ0Þ; ð1Þ

Δp∥i ¼ −

ω

2βcpk2ðx2i − y2i Þ · sin ðωti þ φ0Þ; ð2Þ

where ω denotes the rf quadrupole angular frequency, exand ey are the unit vectors along the x and y coordinaterespectively, φ0 is a constant phase offset, and ti is the timewhen the particle traverses the rf quadrupole. The latter isdefined with respect to the particle which is at the zero-crossing of the rf quadrupole voltage, and ti ¼ 0 coincideswith the bunch center. This assumption is made throughout.The substitution ti ¼ zi=βc gives the longitudinal depend-ence of the quadrupolar kick strength along the particlebunch, where β is the relativistic beta and c the speed oflight. The parameter k2 refers to the amplitude of thenormalized integrated quadrupolar gradient, i.e., it accountsfor the transit time factor,

k2 ¼q

πrpc

Z2π

0

����Z

L

0

ðEx − cByÞeiωz=cdz���� cosφdφ; ð3Þ

where L is the length of the rf quadrupole cavity, and½r;φ; z� are the cylindrical coordinates. Throughout thisarticle k2 is expressed in magnetic units [Tm/m] usingthe conversion bð2Þ ¼ B0ρk2, where B0ρ is the magneticrigidity of the beam.

Depending on their longitudinal position, particlestraversing the rf quadrupole experience a different quad-rupolar (de-)focusing force and hence a change in theirbetatron tunes

ΔQix;y ¼ �βx;y

bð2Þ

4πB0ρcos

�ωziβc

þ φ0

�; ð4Þ

where βx;y are the transverse beta functions of the machinelattice at the location of the quadrupole kicks in Eq. (1).Equation (4) describes the detuning that a particle iexperiences from a single passage through the rf quadru-pole according to its current longitudinal position zi in thebunch. As the particle undergoes synchrotron motion, itslongitudinal position changes turn after turn. Hence, eachtime it passes through the device, it experiences a differentkick and hence a different detuning. Given enough time, thelongitudinal turn-by-turn position of the particle will beevenly distributed over the interval ½−zi; zi�, where zi is itsmaximum synchrotron oscillation amplitude. This is true aslong as the synchrotron tune is not a rational number.Figure 3 illustrates the situation for two different rfFIG. 2. Normalized electric (left) and magnetic (right) fields for

an octant of the elliptical (top) and for half of the four-vane(bottom) cavity respectively.

FIG. 3. Illustration of the detuning that a particle (red) expe-riences as it passes through an rf quadrupole (orange) turn afterturn. Given enough time, the longitudinal position of the particlewill be evenly distributed over the interval ½−zi; zi� due to thesynchrotron motion. It will hence experience a changing detuningevery time it passes through (black dashed curve). If the rfquadrupole phase is φ0 ¼ 0 (top), the average detuning will notvanish and there is a net incoherent tune spread. If φ0 ¼ −π=2there will be no effective tune spread.

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quadrupole phase offsets φ0 ¼ 0 (top) and φ0 ¼ −π=2(bottom). The grey background represents the particledistribution (Gaussian) in the longitudinal plane. Oneparticle (red) is observed over time as it passes throughthe rf quadrupole (orange) turn after turn. At every passageit experiences a detuning according to its current longi-tudinal position. The black dashed line corresponds to theinterval ½−zi; zi� that the particle eventually traces out as itpasses through the rf quadrupole for many turns, i.e.,considering a long enough time (several synchrotronperiods). It is worth noting that the situation is analogousfor detuning with transverse action, such as from magneticoctupoles [14]. The only difference is that the betatronrather than the synchrotron motion is relevant. It is clearfrom Fig. 3 that for φ0 ¼ 0, the average detuning of theparticles is nonzero which corresponds to a nonvanishingincoherent tune spread. For φ0 ¼ −π=2, however, theaverage detuning vanishes for all the particles and thereis no net tune spread. To maximize hence the achievableincoherent betatron tune spread and to prevent it fromaveraging out over time, the rf quadrupole is operated witha phase of φ0 ¼ 0 or φ0 ¼ π. This is an essential require-ment for the Landau damping mechanism to work againstthe slow head-tail instabilities which develop over timescales of many synchrotron periods [4]. Selecting φ0 ¼ 0means that the device is focusing (defocusing) in thehorizontal (vertical) plane for particles in the bunch centerzi ¼ 0. The situation is inverse for φ0 ¼ π. Equivalently,operating the cavity in one of these two modes means thatparticles located in the bunch center enter the device (anti-)on-crest of the rf wave. This is different from earlier studies[15,16] where an rf quadrupole was proposed to increasethe threshold of the transverse mode coupling instability(TMCI) [4]. In that case, the cavity would have to beoperated at the zero-crossing of the rf voltage, i.e. withφ0 ¼ �π=2. This has an effect similar to the odd orders ofchromaticity.In the following we set φ0 ¼ 0. Equation (4) can be

expanded into a Taylor series in zi

ΔQix;y ¼ �βx;y

bð2Þ

4πB0ρ

�1 −

1

2

�ωziβc

�2

þOðz4i Þ�: ð5Þ

Given that the wavelength of the rf wave is much largerthan the bunch length σz, i.e. ωσz=βc ≪ 1, the higher ordertermsOðz4i Þ can be neglected. Assuming linear synchrotronmotion and taking the average detuning over a long periodof time (several synchrotron periods), again in analogy tothe derivation of detuning with transverse action frommagnetic octupoles [14], it is easy to show that the rfquadrupole indeed changes the betatron tunes as a functionof the longitudinal action Jiz of a particle

hΔQix;yi ≈ �βx;y

bð2Þ

4πB0ρ

�1 −

1

2

�ω

βc

�2

βzJiz

�: ð6Þ

βz has been defined as βz ≐ ηR=Qs, where η denotes theslip factor, R is the accelerator ring radius, and Qs is thesynchrotron tune. The longitudinal action is defined asJiz ¼ ðz2i þ β2zδ

2i Þ=2βz, where δi is the relative momentum

deviation. The constant quadrupolar detuning term inEq. (6) does not contribute to the effective tune spreadas it affects all the particles in the same manner. The pureincoherent detuning of the particles with respect to eachother is hence approximatively given by

ΔQx;yðJzÞ ≈ ∓βx;ybð2Þ

8πB0ρ

ω2

β2c2βzJz

¼ ∓ax;yz Jz; ð7Þ

where all the constants are combined in the parameter ax;yz

called detuning coefficient. The approximation is valid aslong as ωσz=βc ≪ 1. Furthermore, it assumes that the rfquadrupole is operating with a phase φ0 ¼ 0. By changingthe phase to φ ¼ π, the signs flip, meaning that the tunespreads between the two transverse planes are swapped.The longitudinal kick in Eq. (2), in a similar manner

leads to an incoherent synchrotron detuning. This effect,however, can be neglected given the parameters of theparticle colliders under consideration [8].For the following discussion and in comparison to

Eq. (7) it is useful to recall the transverse detuning withamplitude introduced by magnetic octupoles

ΔQxðJx; JyÞ ¼ axxJx þ axyJy

ΔQyðJx; JyÞ ¼ ayyJy þ ayxJx; ð8Þ

where aij, i; j ∈ fx; yg, are the detuning coefficients. Theydepend on the integrated octupolar field strength weightedwith the beta functions at the location of the magnetsaround the accelerator ring and are inversely proportional tothe magnetic rigidity of the beam [17]. Note that the twocross-detuning coefficients are identical, i.e. axy ¼ ayx.

C. Features, potential advantages, and applications

Figure 4 compares two different incoherent betatrontune distributions in the transverse tune space Qx vs. Qy,averaged over several synchrotron periods. First, frommagnetic octupoles which generate detuning with trans-verse action (blue) as in Eq. (8), and second, from a singlerf quadrupole with φ0 ¼ 0 which generates detuning withlongitudinal action (red) given by Eq. (7). Here, themagnetic octupoles are powered in a two-family schemewith focusing and defocusing elements. This scheme isapplied for example in the LHC to optimize the beamstabilizing efficiency in both transverse planes [17]. Whenpowered in such a way, magnetic octupoles produce a two-dimensional tune footprint in ðQx;QyÞ-space as shown inFig. 4. The distribution is obtained assuming negative(positive) polarity of the focusing (defocusing) family.

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The projection of the tune distribution onto the Qx- (Qy-)axis is nearly symmetric with respect to the bare machinetune Qx;0 (Qy;0) identified with the green lines.For the rf quadrupole the tune changes in Qx and Qy are

fully correlated. This is explained by the fact that thedetuning is dependent on only one variable Jz. The tilt ofthe distribution from the rf quadrupole is determined by theratio of βx vs. βy at the location of the device [see Eq. (7)].Here, the transverse beta functions are assumed to beidentical. The tune projections for a single rf quadrupole arestrongly asymmetric. This has important consequences forthe stabilizing mechanism in the two transverse planes asdiscussed in more detail in Sec. II D. In addition to theincoherent tune spread, the rf quadrupole also introduces aconstant, quadrupolar tune shift which is the same for allthe particles. This is understood formally from the presenceof the zeroth-order term in Eq. (5). Such an effect can becompensated with a dedicated quadrupole magnet ifneeded.Figure 5 compares the transverse RMS tune spreads

produced by an rf quadrupole (red) and the LHC Landauoctupoles (blue) as a function of the beam energy for LHCdesign beam parameters [5,17]. The comparison is madeusing an 800 MHz rf quadrupole installed at βx;y ¼ 200 m.The strength is fixed at bð2Þ ¼ 0.22 Tm=m and is chosen insuch a way as to provide the same RMS tune spread as theLHC Landau octupoles powered at their maximum currentof 550A at top beam energy of 7 TeV (green dashed line).

The strengths of both the rf quadrupole and the magneticoctupoles are held constant as a function of the beamenergy. While the LHC Landau octupoles have a totalactive length of 56 m, the equivalent rf quadrupole strengthcan theoretically be achieved with just one four-vane cavitywith a total active length of 0.2 m to 0.3 m [12]. For thisparticular case, the active length of the rf quadrupolecompared to magnetic octupoles is hence shorter by afactor 180 to 280. The reason behind the much largerefficiency of an rf quadrupole in terms of detuning is thefact that the spread in Jz is several orders of magnitudelarger than that in ðJx; JyÞ. This is generally true for thebeams of high energy hadron colliders like LHC, HL-LHC,or FCC-hh. Specifically, for LHC design beam parametersat an energy of 7 TeV there is roughly a factor 104

difference in the transverse and the longitudinal actionspreads. The main reason why the difference in activelengths between the rf quadrupole and magnetic octupolesdoes not reach a factor 104, but rather is limited to 180 to280 at the beam energy of 7 TeV, is that the transversedeflecting kicks in an rf cavity are much smaller than thosein a superconducting magnetic octupole. This becomesevident when comparing the detuning coefficients of an rfquadrupole with those of magnetic octupoles. In thisparticular case and for the given beam energy, the formeris ax;yz ≈ 16, while the latter are of the order of 105 for theLHC Landau octupoles at maximum strength [18]. Theratio between the two detuning coefficients amounts to afactor ≈ 104 which, as expected, corresponds to the differ-ence in the action spreads between the longitudinal and thetransverse planes.The advantages of producing a tune spread with an rf

quadrupole rather than with magnetic octupoles becomemore pronounced when the beam energy increases asdemonstrated by Fig. 5. Two effects play an importantrole here. First, due to the increased beam rigidityaccounted for by the detuning coefficients, the effect of

FIG. 4. Comparison of incoherent betatron tune distributions inthe ðQx;QyÞ-space introduced by magnetic octupoles (blue)powered in a two-family scheme with negative (positive) polarityin the focusing (defocusing) family, and a single rf quadrupole(red). The lines and the cross in green mark the bare machinetunes ðQx;0; Qy;0Þ. Histograms on the top and on the side show theprojections of the tune distributions onto the Qx- and the Qy-axisrespectively.

FIG. 5. Dependence of the RMS tune spread on the beamenergy for both LHC magnetic octupoles at maximum strength(blue) and for an rf quadrupole (red), designed in such a way as toproduce the same RMS tune spread at 7 TeV (green dashed line).

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the kicks from both magnetic octupoles and rf quadrupoleis reduced by 1=γ (Lorentz factor). This directly translatesinto a reduction of the RMS tune spread by the same factor.Second, the spreads in respectively the transverse and thelongitudinal actions exhibit a different dependence on thebeam energy. The longitudinal beam stability in a machinelike the LHC strongly relies on Landau damping in thelongitudinal plane [19]. To keep enough stability marginduring the energy ramp, the longitudinal emittance needs tobe blown up. As a result, the longitudinal action spread isroughly held constant as the beam energy increases. On theother hand, the transverse action spreads suffer fromadiabatic damping and decrease as 1=γ. To summarize,the RMS tune spread produced by the rf quadrupole is onlyreduced as 1=γ (beam rigidity) while the one introduced bymagnetic octupoles decreases as 1=γ2 (beam rigidity andadiabatic damping). These considerations are based on theassumption that the machine optics remain unchangedduring the energy ramp.In addition to the improved detuning efficiency com-

pared to magnetic octupoles, the performance of the rfquadrupole also remains unaffected by beam manipulationsin the transverse plane, such as beam halo cleaning throughcollimation. During this process, the large transverseamplitude particles are removed from the beam in acontrolled manner for reasons of machine protection [1,5].The rf quadrupole finds a potential application predomi-

nantly in future high energy hadron colliders, such asHL-LHC or FCC-hh, where the differences between thetransverse and longitudinal emittances are most pro-nounced. Due to the small transverse emittance of thebeams expected in these machines, the number of magneticoctupoles required to provide the tune spread for sufficientLandau damping is large, and a solution relying only onmagnetic octupoles may be less cost-effective than an rfquadrupole, or a combination of the two elements.However, it is worth pointing out that the size of theincoherent betatron tune spread alone does not make acomplete statement about the effectiveness of the Landaudamping itself. The stabilizing mechanisms from detuningwith transverse and longitudinal action respectively aredifferent as seen from stability diagram theory explained inthe following. Generally speaking, tracking simulations arethe most accurate way at present to study the effectivenessof beam stabilization with an rf quadrupole.

D. Stability diagram theory

A practical approach to quantify the Landau dampinggenerated by an incoherent tune spread is to use stabilitydiagram theory. A stability diagram marks the areain the complex coherent tune shift plane ReðΔQcohÞ vs.−ImðΔQcohÞ for which Landau damping is effective. Thereal part of the complex coherent tune shift is obtained froma frequency analysis of the bunch centroid motion. Theimaginary part corresponds to the growth rate of the

instability at hand. For magnetic octupoles, the stabilitydiagram theory is well established (see, e.g., [9]). The goalof this section is to summarize the stability diagram theorythat exists for detuning with longitudinal amplitude, asintroduced for instance by an rf quadrupole.The top plot in Fig. 6 shows the incoherent tune

distributions in the two transverse planes obtained analyti-cally for a Gaussian beam for a single rf quadrupole withφ0 ¼ 0 assuming identical beta functions. In accordancewith Eq. (7), the particles experience negative (positive)detuning in the horizontal (vertical) plane. The distributionsin the two planes are strictly asymmetric and are mirroredwith respect to each other. This is a direct consequence ofthe quadrupolar nature of the kicks introduced by thedevice. To obtain the distributions for φ0 ¼ π, the labels ofthe two curves can simply be exchanged.The stability diagram for a given tune distribution is

obtained by solving the corresponding dispersion relation.An approximate dispersion relation for betatron detuningwith longitudinal amplitude has been derived by J. ScottBerg and F. Ruggiero [9]

ΔQ−1coh ¼

1

N

ZZZ∞

0

Jjmjz Ψ0ðJÞ

Q−QuðJzÞ−mQsd3J; u ∈ fx; yg;

N ¼ZZZ

∞

0

Jjmjz Ψ0ðJÞd3J: ð9Þ

J ¼ ðJx; Jy; JzÞ is the vector of particle actions in all threedimensions, QuðJzÞ ¼ Q0;u þ ΔQuðJzÞ describes the tune

FIG. 6. Top: Incoherent tune distributions obtained analyticallyfor a Gaussian beam with a single rf quadrupole with a phase ofφ0 ¼ 0 assuming βx ¼ βy. Bottom: Corresponding stabilitydiagrams determined by solving Eq. (9) numerically for anazimuthal mode m ¼ 0 head-tail instability.

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dependence on Jz, m is the azimuthal mode number of theinstability under consideration, and Ψ0 represents theunperturbed particle distribution. The stability diagram isobtained by numerically computing the tune shifts ΔQcohfor different values of Q. According to the Landau bypassrule, a small complex part iϵ is added to the denominator ofEq. (9) to perform the integration [3]. An example ofstability diagrams in the two transverse planes is shown inthe bottom plot of Fig. 6 for the corresponding tunedistributions displayed in the top plot. An azimuthal modem ¼ 0 head-tail instability is assumed here. The way toread the plots is that all the coherent instabilities with anunperturbed ΔQcoh (i.e., in the absence of any tune spread)situated below the line traced out by the stability diagramwill be Landau damped. In absence of Landau dampingonly the modes with −ImðΔQcohÞ ≤ 0 would be stable, i.e.the stability diagram in that case would be represented by aconstant function with −ImðΔQcohÞ ¼ 0. It is clear that theasymmetry of the tune spreads is reflected in the shape ofthe stability diagrams. This has important consequences forthe stabilizing efficiency of the rf quadrupole in the twotransverse planes as further discussed in Sec. V.The dispersion relation in Eq. (9) is based on two main

assumptions and does hence not describe all the beamdynamics aspects in presence of an rf quadrupole. First, inthe general case it would depend on the specific impedanceof the machine under consideration. In the derivation madeby Scott Berg and Ruggiero, this dependency has beenneglected under the assumption that the frequency range ofthe impedance is small compared to the frequency spectrumof the beam [9]. This condition holds for instance for astrongly peaked impedance, i.e., a narrow-band resonatorwith a very high quality factor. However, for the impedancemodel and the beam parameters of a hadron collider like theLHC for example, it is no longer valid. One reason is thatseveral machine components have an impedance thatextends over a large range of frequencies [18]. Second,an rf quadrupole acts on the betatron tunes as a function ofthe longitudinal coordinate and thus introduces a correla-tion between the betatron and the synchrotron motion. Byconsequence, it directly affects the interaction between thebeam and the impedance with the result that the effectiveimpedance changes [4]. This effect is similar to a chro-maticity which can be accounted for by the head-tail phaseparameter describing the frequency shift of the beamspectrum with respect to the impedance. A change ofthe effective impedance leads to a modification of thecomplex coherent tune shift of the instability under con-sideration. This manifests itself for example as a change ofthe instability rise time which can become faster or slowerdepending on the impedance of the machine and therespective unstable head-tail mode. At large enough rfquadrupole strengths it is even possible that a mode isexcited on a different synchrotron side band (change of theazimuthal mode number) [20,21].

III. PYHEADTAIL

A. Basic concepts

PyHEADTAIL is a 6D macroparticle tracking code underdevelopment at CERN [10]. It is the successor of thewell-established HEADTAIL code [22]. The purpose of thesoftware is to accurately and effectively model the for-mation of collective instabilities in circular particle accel-erators to make the study of the involved mechanismspossible, and to develop and evaluate appropriate mitiga-tion techniques.The basic PyHEADTAIL model represents a particle bunch

as a collection of macroparticles, each of which is describedby a mass, an electric charge, and its generalised coor-dinates and canonically conjugate momenta with twolongitudinal and four transverse dimensions. The particlescan be initialised in various distributions, amongst othersGaussian, waterbag, or airbag, also including the possibilityfor longitudinal matching to a multiharmonic rf bucket.The accelerator ring is divided into a number of seg-

ments. At the beginning and at the end of every segmentthere is an interaction point where the macroparticlesexperience collective effects (wake fields/beam couplingimpedance, electron-cloud, space-charge, etc.) or kicksfrom a specific accelerator component, such as a transversefeedback or an rf quadrupole. The betatron motion betweentwo consecutive interaction points is modeled with linearmaps which take into account the Twiss parameters and thedispersion at the start and at the end of the connectingsegment. Non-linear tracking features such as chromaticity,i.e. the variation of the betatron tune with the relativemomentum deviation, are modeled as a change in the phaseadvance of each individual macroparticle in the beam. Thebetatron detuning with transverse amplitude introduced bymagnetic octupoles for example is parametrized by theeffective machine anharmonicities and is implemented inthe same manner. The synchrotron motion is either linear ornonlinear, potentially including particle acceleration andthe effects of multiharmonic rf systems.

B. Rf quadrupole models and benchmarks

The rf quadrupole is implemented in PyHEADTAIL withtwo different models: (i) as a detuning element, and (ii) aslocalized kicks. The first model directly applies Eq. (4) andis implemented equivalently to other effective nonlineartracking features, such as chromaticity and magnetic octu-poles. Every macroparticle experiences a change in itsphase advance depending on its longitudinal position in thebunch. In the second, more accurate localized kick model,every macroparticle receives kicks to its transverse andlongitudinal momenta computed according to the thin-lensapproximation defined by Eqs. (1) and (2). Betatrondetuning then follows as a result after transverse tracking.The use of the thin-lens approximation is justified as thefocal length of an rf quadrupole is much larger than the

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length of the device itself [23]. For example, the strengthbð2Þ required to stabilize the most unstable head-tail modein the HL-LHC at 7 TeV is of the order of 0.3 Tm=m (seeSec. V C). This translates into a focal length f>7.7×104mwhich is orders of magnitude larger than the length of thedevice.Both the detuning and the localized kick model of the rf

quadrupole account for the constant phase offset φ0

introduced in Sec. II. That way, different operational modesof the device can be studied. In terms of collective effectsand Landau damping in particular, the two models areexpected to produce identical results. The main differenceis that the detuning model does not apply kicks to theparticle momenta, meaning that the transverse and longi-tudinal action of each individual macroparticle remainunchanged. By consequence, resonances potentially intro-duced by an rf quadrupole cannot be excited and mustinstead be analyzed with the localized kick model. The twomodels are hence particularly useful when coherent andincoherent effects from an rf quadrupole are to be studiedseparately.The results of a benchmark comparing the two rf

quadrupole models in PyHEADTAIL are displayed inFig. 7. The rf quadrupole phase is set to φ0 ¼ 0. Thetwo histograms show the incoherent vertical tune distribu-tions established after tracking a 6D Gaussian bunch with500000 macroparticles over five synchrotron periodsassuming linear synchrotron motion. The frequency analy-sis is performed by means of the SUSSIX code [24]. Thetwo models yield consistent results within expectationsgiven the limited number of macroparticles. The bincolored in green represents particles at 1σ, i.e. with alongitudinal action of Jz¼σ2z=2βz. The blue dashed linemarks the corresponding detuning expected from analytical

calculations [Eq. (6)] and shows an excellent agreementwith both tracking models. The approximation of Eq. (6)holds as the beam and rf quadrupole parameters are chosensuch that ωσz=βc ≪ 1. The blue solid line marks theaverage betatron detuning determined analytically forparticles at Jz ¼ 0, i.e., particles residing in the center ofthe longitudinal phase space. As expected, it coincides withthe lower bound of the distribution. Since for these particlesz remains zero over time, the (average) detuning is givensimply by the constant quadrupolar term −βybð2Þ=4πB0ρ.This result is exact independent of whether or not thecondition ωσz=βc ≪ 1 holds.

IV. NUMERICAL PROOF OF CONCEPT

The numerical validation of the stabilizing effect of an rfquadrupole is based on a horizontal single-bunch instabilityoriginally observed experimentally in the LHC at 3.5 TeVduring machine commissioning in 2010 [11]. Dedicatedstudies in the LHC demonstrated that this particularinstability is characterized as a slow head-tail mode withan azimuthal mode number m ¼ −1. Furthermore, it wasclearly shown experimentally that it can be suppressed bymeans of the Landau damping mechanism introduced bypowering the Landau octupoles installed in the machine.The focusing and defocusing octupole families arepowered with the same current, but opposite signs, i.e.,Ioct;f ¼ −Ioct;d, if not stated otherwise. The thresholdcurrent in the Landau octupoles required to Landau dampthe head-tail mode was determined to be IExpoct;f ¼−15�5A.The corresponding instability rise time was measured to beτ ≈ 9.8 s at Ioct;f ¼ −10A. The corresponding rise time atIoct;f ¼ 0 A is expected to be significantly faster, but is notaccessible from experimental measurements.To confirm the validity of the numerical modeling, the

observations made in the LHC are reproduced using twodifferent accelerator physics models. First, the 6D macro-particle tracking code PyHEADTAIL, and second, a circulantmatrix model (CMM) [25] implemented in the BimBim code[26]. The main input for both codes is an accurateimpedance model of the LHC as well as the beam andmachine parameters at the time of the measurement. Theyare summarized in Table I.

FIG. 7. Comparison of the incoherent betatron tune distribu-tions introduced by an rf quadrupole for the detuning (red) andthe localized kick (grey) model implemented in PyHEADTAIL.Both models give consistent results. The detuning obtained fromPyHEADTAIL for particles with respectively Jz ¼ σ2z=2βz (green)and Jz ¼ 0 is in good agreement with analytical calculations(blue).

TABLE I. Main parameters used in PyHEADTAIL and BimBim toreproduce the LHC experimental machine setup at 3.5 TeV [11].

Parameter Symbol Value

Bunch intensity Nb 1.0 × 1011 pþ=bBeam energy E 3.5 TeVChromaticity Q0

x;y 6Transverse normalized emittance ϵx;y 3.75 μm radBunch length σz 0.06 m

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A. Stabilization from Landau octupoles

PyHEADTAIL simulations make a full characterization ofthe observed unstable mode possible. The main results aresummarized in Fig. 8 where 106 macroparticles have beentracked over 3 × 105 turns. The first plot (left) shows thetime evolution of the horizontal bunch centroid in absenceof incoherent tune spread. The amplitude growth is fittedwith an exponential function fðtÞ ¼ exp ðt=τÞ to obtain therise time τ ¼ 3.7 s of the unstable mode. For comparisonwith the above-mentioned experimental measurements, therise times from simulations are also determined for twodifferent nonzero Landau octupole currents. At Ioct;f ¼−10 A the rise time is τ10 ¼ 4.5 s and at Ioct;f ¼ −15 A itis τ15 ¼ 12.8 s. The values are consistent with the experi-ments. A spectral analysis of the bunch centroid oscillationat different times during the evolution of the instability isshown in the middle plot. The spectral curves correspond tothe vertical dashed lines in the first plot and are obtainedfrom a fast-Fourier-transform with a window size of160384 turns. The spectra clearly show that the azimuthalmode m ¼ −1 is the fastest growing mode, which is againconsistent with experimental observations. The correspond-ing head-tail mode pattern is a standing wave with a singlenode in the bunch center (right plot) indicating a radialmode number 1. The characterization of the head-tailinstability at hand is completed by the complex coherenttune shift ΔQcoh of the unperturbed mode, i.e., in absenceof Landau damping. The real part is obtained from spectralanalysis (SUSSIX), and the imaginary component is calcu-lated from the instability rise time fit. The results are

ReðΔQPyHTcoh Þ ¼ −3.6 × 10−6;

ImðΔQPyHTcoh Þ ¼ −9.2 × 10−5:

To determine the amount of Landau damping required tosuppress this particular mode, the LHC Landau octupolesare added to the PyHEADTAIL tracking model and a scan inIoct;f is performed with a stepsize of 5 A. A selection of the

results is presented in Fig. 9. The instability is suppressedwith a Landau octupole current of IPyHToct;f ¼ −17.5� 2.5 Awhich is in excellent agreement with experimentalmeasurements.Complementary to the studies carried out with

PyHEADTAIL, the second accelerator physics model usedhere is the CMM implemented in the BimBim code. Itincludes amongst others the effects of impedance andchromaticity, but neglects for instance multiturn wakefields and Landau damping from magnetic octupoles[27]. The former is a valid approximation for the wakefields present in the LHC as they decay on a time scalefaster than the revolution period [18]. The absence ofLandau damping in the CMM is a consequence of thelinearization of the system in the transverse planes. Thisapproximation excludes the possibility to account for thenonlinear effects required to model detuning with trans-verse amplitude. BimBim computes directly the unperturbedΔQcoh by numerically solving an eigenvalue equation. Forthe LHC instability, the most unstable mode in BimBim isagain the azimuthal mode m ¼ −1 with a coherent tuneshift of

FIG. 8. Left: Horizontal bunch centroid evolution over time with an exponential fit of the rise time (red). Middle: Frequency spectra ofthe bunch centroid at different times during the simulation (see dashed lines in the first plot). The most unstable mode is an azimuthalmode m ¼ −1 in agreement with the experiments. Right: Horizontal slice centroid vs. longitudinal position for 80 consecutive turns.The head-tail motion is characterized by one node in the center of the bunch.

FIG. 9. PyHEADTAIL results of the bunch centroid motion over3 × 105 turns in the LHC for different values of the currents in theLandau octupole magnets.

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ReðΔQBBcohÞ ¼ −3.9 × 10−6;

ImðΔQBBcohÞ ¼ −9.6 × 10−5:

The values are in good agreement with PyHEADTAIL

tracking simulations.As the CMM does not model Landau damping from

magnetic octupoles, a different method must be choseninstead to determine the current Ioct;f required for beamstability. A popular one is to use stability diagram theoryfor Landau octupoles [9]. The Python solver for stabilitydiagrams (PySSD) is used here for that purpose [26,28].The stability diagrams from LHC Landau octupoles areshown in Fig. 10 in the complex tune space ReðΔQcohÞ vs.−ImðΔQcohÞ for two different Landau octupole currents,Ioct;f ¼ −15� 5 A (green), and Ioct;f ¼ −17.5� 1 A(orange). The colored areas are respectively the corre-sponding uncertainties or the resolution of the scan inLandau octupole current. The former corresponds to theexperimental measurements, while the latter is determinedfrom the unperturbed complex coherent tune shifts (BimBimand PyHEADTAIL). The values of ΔQcoh of the unperturbedmode as obtained from BimBim and PyHEADTAIL are givenby the red and blue markers respectively. For this particularcase, the theory predicts stability from the Landau octu-poles at ISDToct;f ¼ −17.5� 1 A as illustrated in Fig. 10 by theorange dashed line and area. This is again fully consistentwith tracking simulations and experiments (green lineand area).The studies presented above clearly show that the

impedance model of the LHC used for numerical simu-lations with PyHEADTAIL and BimBim is reliable. Bothaccelerator physics models produce the same outcomeand manage to reproduce the observations made in themachine to a high degree of accuracy. The observed

head-tail mode is Landau damped in experiments, trackingsimulations, and in theory with fully consistent resultson the required Landau octupole current. The fact thatstability diagram theory predicts the same Landau octupolecurrent as obtained from experiments and tracking under-lines that the stabilization of this mode is indeed a result ofLandau damping. Furthermore, the Landau dampingmechanism is accurately modeled in PyHEADTAIL and thisforms the basis for the numerical proof of concept of thestabilizing effect from an rf quadrupole discussed in thefollowing.

B. Stabilization from an rf quadrupole

The PyHEADTAIL study presented above is repeated withnow an rf quadrupole (localized kick model) instead of theLHCLandauoctupoles.A scan is performed in the integratedquadrupolar field strength bð2Þ with the Landau octupolesswitched off. The results of the scan are summarized inFig. 11. They show a picture that is very similar to Fig. 9.With increasing quadrupolar strength, and hence increasingtune spread, the growth rate of the unstable mode is reducedand eventually suppressed entirely. The stability threshold isbð2Þ ¼ 0.105� 0.005 Tm=m. Although not displayed here,an identical result is obtained with the detuning model ofthe rf quadrupole in PyHEADTAIL. The studies confirm theeffect expected from the theory discussed in Sec. II D, anddemonstrate that an rf quadrupole for beam stabilization canwork conceptually. According to preliminary cavity designstudies, the stabilizing quadrupolar strength required herecould be achieved with a single cavity with an equivalentactive length of about 0.1m [12]. For comparison, to stabilizethe samemodewith the LHCLandauoctupoles an equivalentactive length of about 1.5m is required. This is assuming thatthey are powered at their maximum current of 550 A.In analogy to the analysis performed above for the

Landau octupoles, the stabilizing strength of an rf quadru-pole is analyzed also by means of stability diagram theory.For that purpose, the approximate dispersion relation in

FIG. 10. The dashed lines are stability diagrams obtainedwith the PySSD code for two different Landau octupole currentsIoct;f ¼ −15� 5 A (green) and Ioct;f ¼ −17.5� 1 A (orange).The colored areas are the corresponding uncertainties. The formercorresponds to the values obtained experimentally, while thelatter is determined from the unperturbed complex coherent tuneshifts (red circle: BimBim, blue cross: PyHEADTAIL).

FIG. 11. PyHEADTAIL results of the bunch centroid motion over3 × 105 turns in the LHC for different values of the rf quadrupolestrength bð2Þ.

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Eq. (9) is solved numerically for the parameters of thePyHEADTAIL simulation. The outcome is presented inFig. 12. The red and blue markers are again the complexcoherent tune shifts obtained with the BimBim code andPyHEADTAIL respectively. The dashed line marks thestability diagram for an rf quadrupole strength ofbð2Þ ¼ 0.0235� 0.0010 Tm=m. The colored area corre-sponds to the resolution of the scan in bð2Þ. There is a cleardiscrepancy of a factor four in the required stabilizingstrength between theory and PyHEADTAIL tracking simu-lations. The main reason for this difference is that thedispersion relation does not take into account the imped-ance model of the LHC, as explained in Sec. II D. Itassumes instead that there is only a single peak in theimpedance spectrum that is responsible for driving thegiven instability [9]. Clearly, this is not the case here as(i) the LHC impedance has several broad-band compo-nents, and (ii) the overlap between the bunch spectrum andthe impedance determines the coherent tune shift (andgrowth rate) of the instability [4,18]. Furthermore, thetracking simulation takes into account the full cosinedependence of the detuning from the rf quadrupole, whilethe analytical formula only considers the first order term inJz [see Eq. (7)]. For the given rf quadrupole frequency andbunch length, the detuning of the particles at largeamplitudes Jz, and thus the overall tune spread and theLandau damping, are overestimated compared to thetracking simulations.Comparing the amounts of RMS tune spread required

from magnetic octupoles and an rf quadrupole helpsto estimate the Landau damping efficiency of the twomitigation techniques. The RMS tune spreads requiredto suppress slow head-tail instabilities are typicallymuch smaller than the synchrotron tune Qs sinceReðΔQcohÞ ≪ Qs. For the unstable mode studied here,they are (from PyHEADTAIL)

ΔQoctrms ¼ ð2.4� 0.3Þ × 10−5 ≈ 0.012Qs;

ΔQrfqrms ¼ ð3.4� 0.5Þ × 10−5 ≈ 0.017Qs;

with Qs ≈ 2 × 10−3. Although the two values are of asimilar order of magnitude, the discrepancy is significantand emphasizes that the amount of incoherent tune spreadalone does not give the entire picture about whether anunstable mode is Landau damped or not. The imaginarypart of ΔQcoh as well as the shape of the stability diagramare both essential ingredients to address this question. Thisbecomes clear by comparing Figs. 10 and 12. The reasonfor the difference of the required tune spreads is aconsequence mainly of the way the Landau dampingmechanism works for detuning with transverse and longi-tudinal amplitude respectively. The dispersion integralequations for the two approaches differ significantly fromeach other [9]. As a result, the corresponding stabilitydiagrams can have a different shape and the tune spreadsrequired to suppress a particular instability will in generalbe different as well.

V. ALTERNATIVE SCHEMES

Stability diagram theory for detuning with longitudinalamplitude predicts amongst others a strong asymmetry inthe stabilizing efficiencies between the two transverseplanes. This is studied in the following by qualitativelycomparing theory against tracking simulations. To over-come the asymmetry, a two-family scheme for rf quadru-poles is proposed and the performance is validated withtracking simulations. The idea is motivated by the two-family scheme employed for beam stabilization withmagnetic octupoles [17]. Finally, the synergy between rfquadrupoles and magnetic octupoles is studied in terms ofstabilizing efficiency for an HL-LHC machine setup.

A. Qualitative comparison withPyHEADTAIL simulations

The comparisons between stability diagram theory andtracking simulations shown here are only of qualitativenature due to the simplifications made in the derivation ofthe dispersion relation (see Sec. II D). Nevertheless, theygive insight into the mechanisms and are used to motivate amore advanced scheme of rf quadrupoles to improve theperformance presented in Sec. V B.The horizontal and the vertical complex coherent

tune shifts of the unstable head-tail modes are typicallyvery similar for a hadron collider like the LHC, i.e.,ΔQx

coh ≈ ΔQycoh. For LHC operational beam and machine

parameters for instance the most unstable impedance-drivenhead-tail modes are characterized by ReðΔQx;y

cohÞ < 0. For asingle rf quadrupole, the stability diagrams in Fig. 6 illustratethat the stable regions in the two transverse planes havedifferent shapes.Given thatReðΔQx;y

cohÞ < 0, one plane (here:horizontal) is expected to exhibit a much better stabilizing

FIG. 12. The dashed line is the stability diagram obtainedby numerically solving the approximate dispersion relation inEq. (9) for an rf quadrupole with bð2Þ ¼0.0235�0.0010Tm=m.The colored area is the corresponding resolution of the scan inbð2Þ. The unperturbed complex coherent tune shifts from BimBim(red circle) and from PyHEADTAIL (blue cross) are shown as well.

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efficiency than the other one for an rf quadrupolewith a phaseof φ0 ¼ 0. The situation is inverse for φ0 ¼ π. This asym-metry has first been observed in PyHEADTAIL simulations forthe LHC instability at an energy of 3.5 TeV, introduced inSec. IV. For this particular case, by changing the phase φ0

from 0 to π, the stabilizing quadrupolar strength increases

roughly by a factor three from bð2Þ0 ¼ 0.105� 0.005 Tm=m

to bð2Þπ ¼ 0.35� 0.02 Tm=m.The asymmetric nature of the rf quadrupole kicks is

impractical, and a systematic study is made withPyHEADTAIL to further examine this particular observationand to find possible solutions to circumvent it. To simplifythe simulation set-up, a pure dipolar analytical broad-bandresonator impedance (quality factor of Q ¼ 1) is used withidentical components in the horizontal and the verticalplanes. For further simplification of the setup, the chro-maticity is set to Q0 < 0 to induce a weak azimuthal modem ¼ 0 head-tail instability (note that the study is doneabove transition energy) [4]. This has no impact on theunderlying stabilizing mechanism described by stabilitydiagram theory. Both the real and the imaginary compo-nents of the coherent tune shift ΔQx;y

coh are controlled byvarying the beam intensity. With increasing intensity, thecoherent tune shift becomes larger due to the strongerinteraction of the beam with the impedance. It is henceexpected from theory that more rf quadrupole strength willbe required for stabilization. The two transverse planes areaffected equivalently by the intensity increase. However,stability diagram theory predicts that the stabilizing rfquadrupole strength must be different in the two planesdue to the strong asymmetry of the stability diagrams (seeFig. 6, bottom).The results of the simulation studies are summarized in

Fig. 13. 800000 macroparticles have been tracked over105turns and a scan in bð2Þ has been performed for eachbeam intensity setting. The top plot corresponds to an rfquadrupole with φ0 ¼ 0 and clearly represents the asym-metry expected from the theory. The horizontal planebecomes stable already at much lower quadrupolar strengththan the vertical one. By changing φ0 from 0 to π, thebehaviour in the two planes is swapped (bottom plot), againin accordance with expectations given the quadrupolarnature of the device. At the maximum beam intensity,the difference between the threshold quadrupolar strengthsin the two planes reaches almost a factor five for thisparticular case.Figure 14 underlines the qualitative agreement between

tracking simulations and theory and is an explanation forthe observations made. The plot shows the complexcoherent tune space with the dependence of the unperturbedΔQx;y

coh on the beam intensity, represented by the coloredmarkers. They are normalized to the absolute value of themaximum real coherent tune shift measured at the highestintensity of the scan. The real and the imaginary parts

depend approximately linearly on the beam intensity.Furthermore, ReðΔQx;y

cohÞ ≤ 0. Stability diagrams for twodifferent quadrupolar strengths of an rf quadrupole operat-ing with φ0 ¼ 0 are overlaid for the horizontal (solid lines)and the vertical plane (dashed lines). It is obvious that thestabilizing behavior in the horizontal plane is expected tobe significantly better than in the vertical one. Additionally,comparing the stability diagrams for the two quadrupolarstrengths, the stable area for the horizontal plane grows

FIG. 13. PyHEADTAIL tracking simulations illustrating thestabilizing efficiency in the horizontal (red) and the vertical(blue) planes with an rf quadrupole with φ0 ¼ 0 (top) and φ0 ¼ π(bottom) respectively.

FIG. 14. Evolution of the unperturbed coherent tune shift in thecomplex tune space as a function of intensity. Stability diagramsfor the horizontal (solid lines) and the vertical (dashed lines)planes are overlaid for two different strengths for an rf quadrupoleoperating with φ0 ¼ 0.

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much faster than that in the vertical plane. This explains theincrease in the difference between the stability thresholds asa function of beam intensity shown in Fig. 13.

B. Two-family scheme

The asymmetric stabilizing behavior in the two trans-verse planes can be overcome by installing two indepen-dent rf quadrupole families that operate with oppositephases and are placed at two different locations in themachine lattice. One of them is placed at high βx, low βy toimprove beam stability mainly in the horizontal plane,while the other one is installed at a location with low βx andhigh βy for stability mainly in the vertical plane. Naturally,the difference between the local beta functions must be aslarge as possible to avoid (significant) compensation of thedetuning effect from the other rf quadrupole family.Figure 15 illustrates the stabilizing efficiency in the two

transverse planes for the two-family rf quadrupole schemefor the broad-band resonator driven instability introducedin Sec. VA. For illustrative purposes, values of ten havebeen assumed for respectively the ratio βx=βy and βy=βx atthe two rf quadrupole locations. The vertical axis representsthe total quadrupolar strength, i.e., the sum of the strengthsof the two families, to allow a better comparison with theone-family schemes depicted in Fig. 13. The two familiesare powered with the same strengths. There are two mainobvious advantages. First, the strong asymmetry betweenthe two planes is removed and the required stabilizingstrength is now equal for both planes. Second, the totalquadrupolar strength to achieve stability in both planes issignificantly lower overall. However, since the two familiescounteract each other to a small amount, the stabilizingstrength in the two-family scheme cannot reach the valuesof the respective favored plane in the one-family scheme.

C. Synergy with magnetic octupoles

To understand the practicality and usefulness of an rfquadrupole in a future hadron collider, it is important toassess its interplay with magnetic octupoles. The study

presented here is based on the HL-LHC. The stabilizingperformance of an ensemble of 800 MHz superconductingrf quadrupole cavities is evaluated in presence of the LHCLandau octupoles. The latter will remain an important partof the instability mitigation toolset also for the HL-LHC. Asingle-bunch instability driven by the dipolar acceleratorimpedance for foreseen operational flat-top beam andmachine parameters serves as a study case. The mainvalues of the machine set-up are summarized in Table II.An idealised bunch-by-bunch transverse feedback systemwith a damping time of 50 turns is included in thesimulation model. At a chromaticity of Q0

x;y ¼ 10 the mostunstable mode is a slow head-tail instability with azimuthaland radial mode numbers of 0 and 2 respectively [29]. Thisis consistent with experimental observations made in theLHC at 6.5 TeV [6].

PyHEADTAIL simulations with 8 × 105 macroparticlestracked over 6 × 105 turns predict that beam stabilityfrom LHC Landau octupoles alone is achieved forIoct;f ¼ 170� 10 A. This is about one third of their maxi-mum current. By adding an rf quadrupole to the simulationmodel, the stabilizing current in the Landau octupoles can besignificantly lowered as demonstrated in Fig. 16. The betafunctions at the potential location of the rf quadrupole in theHL-LHC are set to βx;y ¼ 200 m (conservative). Althoughthere is a minor effect on the threshold current Ioct;f up to acavity strength of about bð2Þ ¼ 0.1 Tm=m, the simulationspredict that already for bð2Þ ≥ 0.27 Tm=m the Landau

FIG. 15. PyHEADTAIL tracking simulations illustrating thestabilizing efficiency in the horizontal (red) and the vertical(blue) planes with the two-family rf quadrupole scheme.

TABLE II. Main machine and simulation parameters used forthe HL-LHC studies with PyHEADTAIL.

Parameter Symbol Value

Bunch intensity Nb 2.2 × 1011 pþ=bBeam energy E 7 TeVChromaticity Q0

x;y 10Transverse normalized emittance ϵx;y 2.5 μm radBunch length σz 0.082 m

FIG. 16. PyHEADTAIL tracking simulations showing the com-bined stabilizing effect from LHC Landau octupoles and an rfquadrupole in the HL-LHC.

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octupoles are no longer required to mitigate the instability.Given the preliminary cavity design, the latter strength can beprovided with about one to two rf cavities [12].

VI. SUMMARY

The working principle of an rf quadrupole for Landaudamping has been recapped and potential advantages overconventional stabilizing methods, in particular magneticoctupoles, have been pointed out. The major one is thatthe incoherent betatron tune spread is dependent on thelongitudinal action spread which is several orders ofmagnitude larger than the transverse ones for the beamsof LHC, HL-LHC, and FCC-hh. This method provideshence a much larger handle for producing the incoherenttune spread needed for Landau damping. The requiredactive length of rf quadrupole compared to that of magneticoctupoles is greatly reduced, even if the field strengths in anrf quadrupole are much lower than those in magneticoctupoles. The advantage is particularly pronounced athigher beam energies as the RMS tune spread from an rfquadrupole decays only with 1=γ compared to 1=γ2 formagnetic octupoles.Two models of an rf quadrupole have been implemented

in the PyHEADTAIL 6D macroparticle tracking code,and were successfully benchmarked against each otherand against analytical calculations. The validation of thePyHEADTAIL model, i.e., the formation of impedance-driveninstabilities as well as the Landau damping mechanism,was based on a slow head-tail mode studied experimentallyat 3.5 TeV in the LHC. The comparison of the LHC Landauoctupole current required for beam stability revealed anexcellent agreement between experiment, numerical sim-ulations from particle tracking and the circulant matrixmodel, and stability diagram theory. The Landau octupoleswere replaced with an rf quadrupole in a second step toshow that such a device can provide stability in a similarmanner. The comparison with stability diagram theory forrf quadrupoles shows, however, that more work is neces-sary to improve the theoretical predictions.Due to the quadrupolar nature of the device, the

stabilizing efficiency in the two transverse planes isstrongly asymmetric. This is predicted by stability diagramtheory and is confirmed by tracking simulations. Based onthese results, a two-family scheme for rf quadrupoles wasproposed and tested numerically. Not only could theasymmetry be cleared away, but also the overall quad-rupolar strength required for beam stability was reduced.Finally, based on an HL-LHC operational scenario, thecombined stabilizing effect frommagnetic octupoles and anrf quadrupole was studied and shown to be successful.

VII. OUTLOOK AND CLOSING REMARKS

While the results are promising in terms of the mitigationof impedance-driven collective instabilities, further studies

are required to assess the usefulness and practicality of an rfquadrupole. Among them are (i) incoherent effects includ-ing resonances and dynamic aperture, (ii) the efficiency ofthe device against other types of collective instabilities,e.g., induced by the presence of an electron-cloud, and(iii) the cavity alignment tolerances and the effects of abeam entering the cavity with a transverse offset. Inparallel, methods are under study for an experimentalvalidation of the collective effects simulations presentedhere. Although the construction and test of a prototype rfquadrupole cavity in an existing accelerator would be themost direct way to achieve this goal, it is associated withlarge costs and a time-scale of the order of years. As analternative, stabilization from second order chromaticityQ00is under study. Q00 provides betatron detuning with longi-tudinal amplitude, equivalently to an rf quadrupole in a firstorder approximation [21]. The main advantage of Q00 overthe rf quadrupole is that it can be introduced in an existingmachine like the LHC for instance by powering theinstalled main sextupole magnets in a specific scheme.Measurements have already been performed in the LHCwith single nominal bunches at an energy of 6.5 TeV [20].The preliminary analysis of the experiments shows that (i)Q00indeed contributes to beam stability and (ii) PyHEADTAIL

accuratelymodels the involvedmechanisms.Amore detailedanalysis will be reported at a later date.

ACKNOWLEDGMENTS

The authors would like to acknowledge GianluigiArduini, Xavier Buffat, Lee Carver, Antoine Maillard,Elias Métral, and Giovanni Rumolo for important contri-butions to these studies.

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[5] O. S. Brüning, P. Collier, P. Lebrun, S. Myers, R. Ostojic, J.Poole, and P. Proudlock, CERN Yellow Report No. CERN-2004-003-V-1, 2004.

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