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Longitudinal instabilities of bunched beams caused by short-range wake fields Dʹyachkov, Mikhail 1995

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LONGITUDINAL INSTABILITIES OF BUNCHED BEAMSCAUSED BY SHORT-RANGE WAKE FIELDSByMikhail D’yachkovM.Sc, Moscow State University, RussiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1995© Mikhail D’yachkov, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of_____________The University of British ColumbiaVancouver, CanadaDate i/Ie’,DE-6 (2188)AbstractThis thesis investigates the effects of short-range wake fields on the collective longitudinal motion of charged particle bunches in circular accelerators, especially theonset of instability. At high intensity, a short-range wake field can distort the bunchpotential well and thereby change the stationary distribution. It is shown that if thisis not taken into account, instability thresholds will be incorrectly predicted.An integral equation derived from the linearized Vlasov equation is used to findthe instability thresholds in the case of space-charge impedance alone for variousdistribution functions. The thresholds for instability caused by the coupling betweenthe m ±1 azimuthal modes have been obtained analytically for several commondistributions. The criterion determining these thresholds appears to be the same asthat for thresholds beyond which no stationary distribution can be found.A numerical method is also used to solve the linearized Vlasov equation for theself-consistent case, including distortion of the stationary distribution, and to findthe thresholds. Physical explanations are provided for the eigenmodes and instabilitythresholds predicted by this method. This results in a much simpler stability criterion,which depends only on the stationary distribution and does not require solution ofthe linearized Vlasov equation.The behaviour of electron bunches beyond the instability threshold has also beenstudied using multiparticle tracking. Some interesting phenomena have been seenwhen the bunch length is smaller than that of the wake field. Under some conditionsthe bunch length oscillates in sawtooth fashion, i.e. slow relaxation is followed by fastUblow-up. It has also been found that the bunch may split into two equal sub-buncheswhich oscillate around each other in binary star fashion. These effects may explainsome recent observations in electron storage rings at SLAC and CERN.111Table of ContentsAbstract iiList of Tables viiList of Figures viiiAcknowledgements xiiixiv1 Introduction 11.1 Outline of the Thesis 62 Essentials of Longitudinal Motion 82.1 Equations of Longitudinal Motion 82.1.1 Phase Focusing 112.2 Induced Forces 142.2.1 The Longitudinal Space Charge Field 152.2.2 Wake potential 182.2.3 Impedance 202.3 Collective Effects 202.3.1 The Vlasov Equation 222.3.2 Potential Well Distortion 222.3.3 Stability Analysis 23iv2.3.4 Collective Modes. 242.3.5 Turbulent Bunch Lengthening 253 Longitudinal Stability under Space-Charge Forces 273.1 Stationary Self-Consistent Distribution 273.1.1 Threshold 323.1.2 Discussion 343.1.3 Recovering the Distribution Function 353.2 Stability 383.2.1 Integral Equation 383.2.2 Comparison with Mode-Coupling Theory 423.2.3 Self-Consistent Case 463.3 Summary 483.3.1 Stationary Distribution 483.3.2 Stability 484 Longitudinal Stability of Electron Bunches 494.1 HaIssinski Equation 504.2 Stability 514.2.1 Integral Equation 514.2.2 Rigid Dipole Mode 544.2.3 Analysis of the Integral Equation 544.2.4 Oide-Yokoya Method 554.3 Results 574.3.1 Inductive Wake 574.3.2 Capacitive Wake 574.3.3 Resistive Wake 63V4.3.4 Broad-band Resonator Wake.4.3.5 New Criterion ..,,., .4.4 Conclusions4.4.1 Oide-Yokoya Method4.4.2 Landau damping4.4.3 New Criterion5 Beyond the Threshold5.1 Numerical Simulations5.2 Sawtooth instability5.2.1 Analysis of sawtooth instability5.3 ‘Binary star’ instability5.4 Conclusion6470737374756 Summary and Conclusions6.1 Longitudinal Space-Charge Forces6.2 Longitudinal Stability of Electron Bunches6.3 Beyond the ThresholdBibliography 100A Distribution FunctionsA.1 Binomial FamilyA.1.1 Case m = 1: Hofmann-Pedersen DistributionA.1.2 Case m = 2A.1.3 Case m = 3A.2 Hollow DistributionsA.3 ‘Thermal’ Distribution .77788087889495969799104104105105105106107viList of Tables1 Threshold parameters for different distribution functions 33viiList of Figures1 The coordinate system of a circular accelerator 92 Particle trajectories in the (q, W) phase plane 123 Circular beam in a cylindrical pipe 164 Two charged particles travelling along the axis in an accelerator. . . 195 Different modes of oscillations in a bunched beam [13] 256 Line densities (top) and density plots (bottom) of stationary distributions for the gaussian case at intensities (left to right) I = -1.552, 0 and20. The left-most plot is at threshold: for I < -1.552, no stationarydistributions exist [19] 297 Line densities (top) and density plots (bottom) of stationary distributions for the ‘hollow’ distribution f(H) = exp[—(H— Hb)2/2] atintensities (left to right) I = -8, 0 and 20. The left-most plot corresponds to negative mass and the right-most to positive mass thresholds. 298 The dependence of I on the parameter t for the binomial distributionsH—1 H ‘\‘+1/2)9 Line densities at the ‘negative mass’ threshold calculated by using (3.68) 3410 Line densities at the ‘positive mass’ threshold calculated by using (3.68) 3411 Coherent mode frequencies as a function of the intensity for a hollowbeam distribution. Matrix size: 40 x 40 40viii12 Family of the distribution functions given by (3.99) for different valuesof the parameter c 4313 Extreme eigenvalues of equation (3.100) (plotted as squares) and extreme values of A’(U) (solid lines) 4514 The eigenfunctions found by solving the matrix equation using themode coupling method, corresponding to (a) negative mass thresholdand (b) positive mass threshold 4615 U0 vs. q at threshold intensities for the ‘hollow-gaussian’ distribution.U = q2/2 is shown by the dashed curve 4716 Snapshots of the longitudinal profile of the beam at LEP made witha streak camera. Horizontal ellipses show the peak positions of earlierpulses 5217 The beam profiles at different intensities found from the HaIssinskiequation for a capacitive wake field above transition 5818 Bunch oscillation frequencies in the case of purely capacitive impedance,plotted for discrete intensities I. The upper plot shows eigenfrequencies [Lk calculated by the O+Y method, and the lower the incoherentfrequencies mw(J), where m is an integer. The eigenmode with frequency independent of intensity is the rigid dipole mode 6019 Comparison of the theoretical rigid dipole mode (4.120) (continuouscurve) with the eigenvector of the u = 1 mode (D). The calculationisfori=120,r=3 6120 RMS value of the kth eigenvector versus its eigenvalue at intensityI = 2, in the case of purely capacitive impedance with ñ = 40 (x) and= 120 (0) 62ix21 Stationary longitudinal profiles at different intensities for a resistivewake 6422 Resonator impedance with k0 = 0.6 and Q = 1: (a) the wake field and(b) longitudinal beam profiles at different intensities 6623 Eigenfrequencies versus intensity for a broad-band resonator (Q = 1,= 0.6). Unstable modes are indicated by solid symbols 6724 RMS value of the ktI eigenvector versus its eigenvalue at intensity I = 3in the case of a broad-band resonator with Q = 1, k0 = 0.6, for twodifferent matrix sizes: i = 40 (x) and t = 120 (1). Note that thereare two modes whose X is independent of ñ; these correspond to aquadrupole and a sextupole mode respectively, at the location wherew(J) is a minimum 6925 Normalized synchrotron frequency versus action J and intensity I forthe resonator impedance with Q = 1, k0 = 0.6 7026 Comparison of the eigenvector for the ‘coherent’ quadrupole mode atthe synchrotron frequency minimum (j = 2j) and a nearby ‘incoherent’ quadrupole mode, for two different matrix sizes: ñ = 120(continuous curve) and ñ = 40 (dashed curve) 7127 Comparison of thresholds obtained by different methods: new method(continuous curve), O+Y method (dashed curve) and numerical tracking (symbols). The latter two cases are taken from O+Y[25] 7228 Sawtooth instability in bunch length observed at the SLC dampingring [40] 8129 Sawtooth instability observed in the simulations 82x30 RMS bunch length (a) and rms energy spread (b) in the case of resonator impedance (Q = 1, ko = 0.5) at I = 30. Radiation dampingtime is Te = 5T8 8331 RMS bunch length and energy spread for the same parameters asFig. 30, except that I = 45 8432 RMS bunch length and energy spread for an increased damping rate:I = 30 and Te = 1.5T8 Compare with Fig. 30 8433 A complete cycle of the sawtooth instability in phase space for thecase shown in Fig. 30: I = 30 and Te = ST8. The time sequence isanticlockwise 8534 Dynamics of the sawtooth instability. Each frame shows the particledistribution in phase space and the potential well. The snapshots onthe left illustrate the diffusion process, showing every 100th turn; thoseon the right the collapse, showing every turn 8635 Above: Green function wake field with three rf waveform slopes; (a) isstable, (b) is just above threshold, and (c) is in the sawtooth regime.Looking from left to right, there is a stable fixed point if the wakefield crosses the rf waveform from below, and an unstable fixed pointif it crosses from above. Below: the separatrices created by the wakefields corresponding to cases (b) and (c). In case (a), there is only onestable fixed point so the wake field does not create a separatrix. Notethat these curves are for a Green function wake and therefore are onlysuggestive. Any accumulation of finite charge density will deform theseparatrices 8936 Simulation of the ‘binary star’ instability 90xi37 Snapshots of the phase-space distributions for the ‘binary star’ instability after each 10 turns ( 1/10 of a synchrotron period) 9138 Spectrum S, of the rms momentum spread (un) during the ‘binary star’instability. The quadrupole mode has the largest strength 9239 Spectrum of a quadrupole mode observed at LEP [6]. One can see lowfrequency sidebands which become stronger at higher intensities. . . 9340 Bunch length versus intensity at LEP[6] 93xiiAcknowledgementsI am very grateful to Michael Craddock and Richard Baartman, without whom thisthesis would have never been completed, for their guidance and support during mywork on the thesis.I would also like to thank Shane Koscielniak and Roger Servranckx for many usefuldiscussions, Fred Jones for his help with computers at TRIUMF, and Frank Curzonand Malcolm McMillan for their attention to my research and for many valuablecomments on the thesis. My special thanks go to Pat Stewart for her help andsupport during my stay at TRIUMF.Finally I want to express my deepest gratitude to Erich and Barbara Vogt, Georgeand Olga Volkoff, my wife Yulia and my son Gleb for their help, encouragement andsupport during my studies at UBC.xiiiAIXpvpIULozChapter 1IntroductionThis thesis investigates the physics of certain beam instabilities which can limit theperformance of charged particle accelerators. Accelerators are now used in a wide variety of disciplines, and the users maintain an unrelenting demand for beams of highercurrent (particles per second) or brightness (current density in phase space), whetherfor high-energy colliders and meson factories (particle and nuclear physics), for synchrotron radiation and spallation neutron sources (biology, chemistry, condensed-matter physics, materials science and medicine), or for small cyclotrons producingradioactive isotopes. The intensity and brightness of charged-particle beams are,however, limited by the disruptive effects of the electric and magnetic fields producedby the particles themselves (either directly or via the material of their enclosure) andit is some aspects of these limits to beam stability that this thesis will explore.The purpose of a particle accelerator is, of course, to accelerate a beam of chargedparticles to high energy. In order to accomplish this we have to apply external electromagnetic fields not only to accelerate particles longitudinally, but also to confine themtransversely so that they remain in the vacuum environment.In this thesis we confine the discussion to “circular” accelerators, i.e. those inwhich the particles travel along closed orbits, allowing them to pass the same electricaccelerating field many times, reducing cost and increasing effectiveness. The beam incircular accelerators is usually split into bunches because the electric radio frequency1Chapter 1. Introduction 2(rf) field used for acceleration is oscillatory. In some cases there is only a single bunch,but in most cases several regularly spaced bunches are accelerated simultaneously.The beam intensity of the first accelerators was quite low and external fields couldbe applied without taking space-charge forces into account. However, with increasingintensity collective effects due to space-charge forces become very important. Theseeffects arise not only from the internal fields within the bunch, but also indirectlyfrom the fields induced by the beam in the vacuum enclosure. Electro-magnetic fieldsinduced in rf cavities, vacuum pipes, beam position monitors and discontinuities suchas joints, bellows, etc., can affect the beam and make it unstable. The effects of thesefields can be characterized by an effective “beam-coupling impedance” ,defined as theratio of the induced voltage to the beam current.Collective effects in circular accelerators can be classified as longitudinal or transverse and also as single-bunch or multi-bunch effects. Single-bunch effects are causedby forces which are strong over the bunch length, but decay fast enough that they donot affect any other bunches in the accelerator. An excellent review of longitudinalsingle-bunch effects has been given by B. Zotter [1]. In contrast to single-bunch effectsthere are multi-bunch or ‘coupled bunch’ effects, in which the field induced by onebunch may affect later bunches (or the same bunch on a later turn).In this thesis we concentrate on longitudinal single-bunch effects. We will assumethat the transverse and longitudinal components of a particle’s motion are completelydecoupled; this assumption is valid most of the time, although there are some specialcases in which longitudinal oscillations affect transverse oscillations.As mentioned above, one of the main goals for accelerators nowadays is to obtainhigh beam brightness, implying not only a small cross-section, but also short bunches.This is especially important for a new generation of accelerators called ‘B Factories’which are under construction in both the USA and Japan [1].Chapter 1. Introduction 3Unfortunately, the length of a bunch can be changed both by ‘potential welldistortion’ and by ‘turbulent bunch lengthening’ caused by longitudinal ‘microwaveinstability’. The term ‘potential well distortion’ comes from the fact that the particlesin a bunch are oscillating in an effective potential well created by the external rffields, which can be distorted by the fields generated by the the self-forces of thebeam. Potential well distortion is dominant at low intensities when the beam isusually stable. At some intensity the beam may become unstable and this will causeturbulent bunch lengthening.Potential well distortion can lead to either lengthening or shortening of the bunch,depending on the operating conditions and the type of particles, and has to be studiedseparately for proton and electron beams.The first self-consistent equation describing the stationary distribution of particlesin accelerators was derived by HaIssinski [2]. The main idea of his theory is that theparticles in the beam should be in thermal equilibrium. This theory describes thedistribution of electrons in synchrotrons and storage rings when the radiation effectsare strong enough for the particle to reach an equilibrium during the acceleration orstorage cycle. The results of applying this theory at various conditions can be foundin [3]. A review of different techniques to find stationary self-consistent distributionsin electron bunches can be found in [4].At higher intensities turbulent bunch lengthening is always predominant. Bunchshortening has been observed at several electron synchrotrons (e.g SPEAR [5], LEP[6]), but it is observed only at very low currents and is always followed by bunchlengthening at higher intensities. The shortening observed at LEP and SPEAR wascaused by a strong capacitive component in the impedance of the rf cavities. Thiswas demonstrated at SPEAR when some of the cavities were removed and the bunchbecame longer [5].Chapter 1. Introduction 4The current understanding of turbulent bunch lengthening is that it is causedby longitudinal single-bunch instability, i.e. once the bunch becomes unstable thedistribution adjusts itself always to stay close to the instability threshold. Therefore,understanding the physics of single-bunch instability (especially finding instabilitythresholds) is very important for understanding turbulent bunch lengthening andthus finding ways to obtain shorter bunches and increased luminosity.Questions still remain regarding the mechanisms which cause this instability. Thiscan be illustrated by the case of the SLC damping ring at SLAC where the old vacuumpipe was replaced by a smoother chamber with a lower impedance, which was expectedto increase the instability threshold. That didn’t happen; instead the threshold waslowered [7]. Fortunately, the instability which causes the new threshold is not assevere as the one which limited the performance of the old structure and it is nowpossible to operate even beyond the threshold.Potential well distortion and turbulent bunch lengthening are not the only singlebunch effects occurring in circular accelerators. Recently, some different phenomenahave been observed at various electron accelerators. One interesting example has beenobserved at the SLC damping ring [8]. After injection the bunch length decreasesslowly until a threshold is reached, when its length increases sharply (in a less thena synchrotron period) and then the process repeats. This effect has been called a‘sawtooth instability’ for the sawtooth-like shape of plots of bunch length or centreof mass versus time (see Fig. 28 in Chapter 5).A unique single-bunch effect has been observed on the TRISTAN AccumulatorRing at KEK (Japan) by T. leiri [9]. At some threshold intensity the bunch experienced a very fast instability in both longitudinal and transverse directions; when thebeam current was decreased the bunch length would decrease sharply, but at a lowerthreshold.Chapter 1. Introduction 5One of the biggest problems in studying single-bunch effects is their sensitivity toimpedances at high frequencies, where impedances are difficult either to measure orcalculate. The models which are currently used in simulations do not describe theelectromagnetic fields generated in real accelerators adequately.One of the first criteria for the single-bunch instability threshold for proton beamswas proposed by D. Boussard [10] (‘Boussard criterion’), which was in fact a way toapply the simple stability criterion (‘Keil-Schilell criterion’) derived for continuousbeams to a bunched beam [11]. If the bunch is long compared to the wavelengthof oscillation (as in proton beams) the beam can be treated as continuous and theKeil-Schnell criterion can be applied locally to the bunch, where the average energyspread and current are replaced by their local values. Though the ‘Boussard criterion’was first obtained empirically, different authors have found several independent waysto derive it [12].Beam bunches can have different modes of oscillation. It is common to classifythem into dipole, quadrupole, sextupole, etc., dependillg on the nature of the oscillations. These modes may be stable or unstable and, in general, their frequenciesdepend on intensity.Coupling between two different modes of the longitudinal oscillations is oftenused to explain single-bunch instability. F. Sacherer first suggested this, interpretingturbulent bunch lengthening as the result of microwave instability [13]. G. Besnier inhis thesis [14] tried to develop the theory of bunched beam stability by consideringa more realistic situation where the nonlinearity in an rf field results in a spread ofsynchrotron frequencies. D. Neuffer [15] and independently G. Besnier and B. Zotter[16] found a self-consistent analytical solution in the case of space-charge forces forone particular (elliptic) distribution function, which is known not to change the shapeof the bunch or the internal forces [17].Chapter 1. Introduction 6R. Baartman and B. Zotter [23] have shown that in the presence of space-chargeinteractions mode coupling will also cause instability for a few other types of distributions (so-called ‘gaussian’ and ‘hollow beam’), but their calculations were donewithout taking into account potential well distortion, and thus the validity of these results is questionable. In this thesis we will try to study the problems self-consistentlyby including the effect of the potential well distortion.1.1 Outline of the ThesisIn Chapter 2 we discuss the basics of longitudinal motion in synchrotrons and storage rings. We also derive the expression for longitudinal space-charge forces, whichare typical for hadron accelerators, and the wake fields which are used to describelongitudinal forces in electron synchrotrons. Finally we use the Vlasov equation toderive equations for the potential well distortion, and also describe stability analysistechniques which will be used in the following chapters.In Chapter 3 the effects of longitudinal space-charge forces are investigated in detail. First, we describe the effects of space-charge forces on the stationary distributionof particles in a storage ring. We derive an equation which describes the potentialwell distortion due to space-charge forces, study the bunch lengthening (shortening)they cause below (above) transition, and derive a new criterion which specifies theconditions for which a stationary distribution does not exist.Next we investigate the stability of a bunched beam under longitudinal spacecharge forces. The stability analysis is done self-consistently, i.e. we study the stability of the distributions found earlier in this chapter. We compare our predictedstability thresholds with those obtained from non-self-consistent theories and alsowith the thresholds found earlier for stationary distributions.Chapter 1. Introduction 7Chapter 4 describes the longitudinal single-bunch phenomena relevant to electronsynchrotrons. We introduce the Haissinski equation, which describes a stationarydistribution of particles in the presence of synchrotron radiation, and investigate theeffects of typical wake fields: inductive, resistive, capacitive and that of a resonantcavity. We derive an integral equation which can be used to analyze the stability of thebunched beam in the presence of short-range wake fields and apply it to different wakefields and impedances. Based on analysis of stationary self-consistent distributionsfound by solving the Haissinski equation, we then introduce a new stability criterionand compare it with solutions found by numerical solution of the integral equationand particle tracking.Chapter 5 investigates the behaviour of an electron bunch beyond the instability threshold. We discuss some pathological nonlinear effects observed in modernaccelerators, such as sawtooth instability mentioned above, and describe some attempts to reproduce them using a multiparticle tracking code to gain insight into themechanisms at work.Chapter 2Essentials of Longitudinal MotionIn this chapter we define some important parameters commonly used in acceleratorphysics and derive some useful formulas which will be used later in the thesis.2.1 Equations of Longitudinal MotionTo describe the longitudinal motion we have to define a coordinate system (see Fig. 1).It is useful to describe the motion of particles in circular accelerators with respect tothe motion of a reference particle which has a designated energy E0 and travels alonga closed orbit. The angular coordinate of any other particle at time t can then bewritten(2.1)where is the angular revolution frequency of the reference particle and is theangular deviation relative to it. In addition it is useful to define the energy deviation(2.2)The charged particles in a circular accelerator are kept in orbit by a magneticfield, but a constant magnetic field does not create the conditions for stable motion.In order to provide stability for the beam in the transverse plane it is necessaryto have magnetic field gradients, and therefore the particles’ revolution frequencies8Chapter 2. Essentials of Longitudinal Motion 9Figure 1: The coordinate system of a circular acceleratormay depend on their momentum. This property of the accelerator structure can beexpressed in terms of the momentum compaction factor, which is the ratio of thefractional change in the length C of the closed orbit to the fractional change in aparticle’s momentum pSC/C (2.3)Lip/pUsually, a> 0, but it can also be negative.In order to describe the longitudinal dynamics we need equations for time derivatives of c1 and e. By differentiating (2.1) we have(2.4)where w is the revolution frequency of the particle.Introducing the slip factor i as the ratio of the fractional change in revolutionfrequency to the fractional momentum changep/pSince w Dc 3/CLwLC 6C lLpwC 3C 2p (2.6)Chapter 2. Essentials of Longitudinal Motion 10where -y = 1//1 — j32 is the relativistic factor. Taking into account (2.3) we find(2.7)Note that changes sign when passes through 7tr 1//. Often in order toemphasize this fact (2.7) is written in the form(2.8)where tr is called transition energy of the synchrotron. The transition energy playsvery important role in stability analysis and will be considered later in this chapter.Substituting (2.5) into (2.4) we finally get(2.9)The next equation we need is for the time derivative of the energy deviation e.The energy of the beam can be changed by applied rf fields, by its interaction withthe environment, and by synchrotron radiation. In this chapter we neglect the energyloss due to synchrotron radiation.Let us assume that the rf voltage at an accelerating gap oscillates with a frequencywhich is an integer multiple of the revolution frequency w0 of a particle on the closedorbitV1.f = V0 sin hw0t (2.10)where h is called the harmonic number and the rf phase angle q = h. The particlesalso experience fields caused by the beam-environment interaction but we willleave this till later in this chapter. Particles crossing the rf gap must gain energy at arate matching the magnetic field ramp in order to stay on orbit. Assuming V0 is bigenough there will be some ‘synchronous phase’ ç for which the energy gain providedby the rf system= eVo sin (2.11)Chapter 2. Essentials of Longitudinal Motion 11will be just right to keep the ‘synchronous’ particle on the closed orbit.Let us assume that the energy change is small in one turn and express the energygain relative to the synchronous particle in terms of its derivativez eVo(sinq—sillç53). (2.12)For our purposes it is useful to describe the particle’s motion in terms of a Hamiltonian. One can see thatW = — (2.13)is a co-ordinate canonically conjugate to q and thus(2.14)Curves of H = constant in the phase space (q, W) represent the particle trajectories (Fig. 2). The trajectories are closed for small oscillations from the stablesynchronous phase,but for large deviations the trajectories are not closed and themotion is therefore unstable. The boundary between the stable and unstable regionsis called the separatrix.Since the rf frequency is h times the revolution frequency w0 there are h regions(buckets) around the orbit where the motion is stable. The particles captured in eachseparate bucket are often referred to as a bunch. There may, therefore, be a maximumh bunches in the accelerator, but sometimes some of the buckets are kept empty.2.1.1 Phase FocusingTo find out whether the motion is stable or not we consider only small deviationsfrom q$sin 4 — sin q (tq) cos (2.15)Chapter 2. Essentials of Longitudinal Motion 12VW I• .• . . • ••• . ..•.• :w •• . .=oo/••I •.4. \. \..J •1 ../ 03ir 0Figure 2: Particle trajectories in the (, W) phase planeChapter 2. Essentials of Longitudinal Motion 13and combine (2.9) and (2.12) to get a simplified linearized equationd2(Lç) hwgeVocosç3 —dt2 — 2irE018 — 0.(2.16)For stable motion the coefficient in front of Lq should be positive. Therefore, thestability condition can be writtell ascos48<O (2.17)which is equivalent to the following• Below transition (-y <7tr) <0: 0 < cs <.ir/2, sin q > 0.• Above transition (‘y > > 0: 7r/2 < <‘r, sin q > 0.(The condition sin q5. <0 corresponds to deceleration and is therefore ignored).From the above conditions one can see that if y crosses 7t during accelerationthe stable phase becomes unstable and vice versa. For that reason, the rf phase mustquickly be shifted from ç to r — çb as transition is crossed.Below transition a particle with lower energy and momentum will arrive laterat the rf gap, will see a higher voltage and, therefore, gain more energy than thesynchronous particle. If a particle with lower energy is on the opposite slope, itwould lag more and more until it was lost. Above transition the situation is reversed.In electron machines the injection energy is usually above transition (though someattempts have been made to investigate the possibility of operating the acceleratorbelow transition [27]), but for proton synchrotrons transition crossing often occursduring acceleration. As we will show below, the stability of the beam due to interaction with its environment depends on whether the machine is operating above orbelow transition.Chapter 2. Essentials of Longitudinal Motion 14For small deviations (q— << 1) the force is linear, resulting in a simple harmonicmotion at the synchrotron frequencyheVocosq8= — 2irE5(2.18)Usually the synchrotron frequency wo << wo and it takes tens or hundreds ofturns to complete a synchrotron oscillation. Our previous arguments were based onthe assumption that there is only one rf cavity, but since the synchrotron frequency issmall compared to the revolution frequency it does not really matter whether there isonly one rf cavity or several and it is possible to consider eV0 sin 4 as the total energygain per turn.2.2 Induced ForcesThe synchrotron frequency is independent of the amplitude only if the forces actingon the particles are linear, but this is not usually the case. The beams in acceleratorsare surrounded by cavities, pipes, magnets, etc., most of which are metal. Sincethe beam consists of charged particles it will induce currents in these elements andgenerate electromagnetic fields which can affect the motion of particles followingbehind. Therefore we can express the total voltage gain per turn experienced by theparticles in the beam as the sum of external (rf accelerating) and induced componentsVt0t(q, ) = Vç(q,t) + 14d(,t). (2.19)The voltage 4nd modifies the effective potential well in which the beam oscillates andmakes the synchrotron frequency dependent on the amplitude. This effect is calledpotential well distortion.Chapter 2. Essentials of Longitudinal Motion 152.2.1 The Longitudinal Space Charge FieldIn proton accelerators the bunch length is usually much longer than the diameter ofthe beam pipe, so that the beam-pipe interactions are localized. The voltage inducedby longitudinal space-charge effects can then be found from a relatively simple model.The more general case of the interaction of ultrarelativistic charged particles with anaccelerator environment, which is better suited in case of electron synchrotrons, willbe introduced later.Let us consider a circular beam with the the number of particles per unit length pand radius a moving along the centre of a cylindrical pipe of radius b. We can expressp in terms of the line density )p(z,t) = N\(z,t) (2.20)so thatf (z’, t)dz’ = 1. (2.21)We assume that ) changes slowly enough along the beam so that the conditiond)/dz << )/b is always satisfied.In this case the electric and magnetic fields can be calculated from Maxwell’sequations (by applying Gauss’ and Ampere’s laws)Nq\ 1 to13cNq\ 1Er = —, Bq = —, r > a (2.22)27re0r 2ir rNq.\ r o/3cNq..\ rEr = —, B = —, r <a. (2.23)2re0a 2ir aIf we apply Faraday’s law1cdi= _J.dA (2.24)to the contour shown at Fig. 3, we can find the electric field along the beam axis.Chapter 2. Essentials of Longitudinal Motion 16‘ -, -, —-, _7z:. ::: -‘-,-,EWEwEr(z+8z) / c+ + ÷+ + ÷ + ++ + + + +1 prZBEAM._ -, -, -, -, F ‘ -, -,WALLFigure 3: Circular beam in a cylindrical pipe.The left hand side of (2.24) is— E) + --_-- (i + 21n . (2.25)The right hand side of (2.24) is1 b poNq9cãAZZ 2+ln2r (2.26)If we assume that the perturbation moves approximately with the same speed asthe particles in the beam we may substitute the derivative over time for derivativeover longitudinal coordinate_/3cW. (2.27)Combining Eqns. 2.25-2.27 together, we get= (i + 21n ) — — (2.28)where we used the fact that pofo = I/c2. The expression in square brackets is 1/72.The longitudinal field at the wall E will drive the wall current, which is equal inmagnitude, but opposite in sign, to the a.c. component of the beam current. In mostChapter 2. Essentials of Longitudinal Motion 17accelerators the walls are inductive at low and medium frequencies, i.e.B — LdI N—---—e,6c-— (2.29)where C0 is the circumference of the accelerator and L/C0 is the inductance per unitlength. Combining (2.28) and (2.29) we get the total field due to space charge andinductanceB3 = —Ne [ go 2 —,922L] (2.30)41reo7 Co 9zwhere go = (1 +2 in b/a). It is important to note that the space-charge and inductiveterms have opposite signs. The space-charge field is most significant at low energies.Integrating this field over the accelerator circumference we finally can obtain thevoltage gain per turnV = Ne3ch — LL] (2.31)where Z0 = (c0c)’ 377 !i is the so-called impedance of free space.We can expand the induced voltage into a Fourier series keeping only harmonicsof the revolution frequency w0v(t)=Z(nwo)Ine_0i (2.32)where Z(flwo) is the effective coupling impedance at the frequency w = nw0, and wecan easily show thatZ . (goZo=i\22— woL} (2.33)where Zn = Z(nwo)Therefore, the value Zn/fl is a constant independent of frequency in this model- one of the most important properties of ideal space-charge impedance. Howeverwe should always remember that this is true only within the framework of our approximations, i.e. that the perturbation length is big compared with the beam pipeChapter 2. Essentials of Longitudinal Motion 18diameter, as is usually the case in proton synchrotrons. In practice, the ratio Zn/nremains almost constant up to the so-called cut-off frequency c/b, where c isthe speed of light and b is the pipe radius.As explained above, the presence of the q factor in the longitudinal equations ofmotion make the transition energy y crucial in determining beam stability. Abovetransition, where particles with higher energy have lower revolution frequency, repulsive space-charge forces will cause the particles to move towards each other, as if theyhad a negative mass. Therefore a small perturbation will grow exponentially and acontinuous beam becomes unstable. This is called negative mass instability.In the case of a bunched beam below transition, space-charge forces decreasethe rf focusing and therefore increase the equilibrium bunch length. The oppositeis true above transition - it may happen that the self forces become so strong thatthe stationary distribution collapses. Since this corresponds to the negative massinstability regime for continuous beams it is also referred to as the negative massthreshold. Later in the thesis we will show that for some specific distributions (socalled ‘hollow beam’ distributions) a threshold due to space-charge forces also existsbelow transition; in contrast to the ‘negative mass threshold’ it is called the positivemass threshold.2.2.2 Wake potentialIn electron synchrotrons the rf frequency is usually much higher than in proton accelerators and therefore the bunch length may become comparable to the pipe diameter,making the approximation used in the previous section invalid. Also the speed of theelectrons is usually very close to that of light, so that the space-charge forces, whichvary as 1/72, become negligible.Chapter 2. Essentials of Longitudinal MotionLet us consider a particle with charge e travelling with v = c inside a vacuum pipeso that z et, generating an electric field E(z, t) along its path (Fig. 4). Anotherparticle travelling with the same speed v c at a time r behind the first particle“sees” the wake field E(z, z/c + r). The wake potential, representing the inducedvoltage per unit charge, can be defined asW(r) = fE2(z, + r)dz (2.34)where the integration is taken over the accelerator circumference.The induced voltage for a whole bunch can then be written as a convolution ofthe wake function W(r) with the line density (r, t)V(T, t) = Ne j W(r’)A(r + r’, t)dT’ (2.35)Note that since the particles are travelling with v c they cannot induce fieldsin front of themselves, i.e.Figure 4: Two charged particles travelling along the axis in an accelerator.W(r)=O, T>O (2.36)Chapter 2. Essentials of Longitudinal Motion 202.2.3 ImpedanceAnother way to describe the interactions between an accelerator beam and its environment is to use the concept of electrical impedance. The impedance may be writtenas the Fourier transform of the wake potentialZ(w) L W(T)eiLTdT (2.37)which means that by using an inverse Fourier transform one can express the wakefunction in terms of the impedance1t00W(r) =— J Z(w)e_twrdw. (2.38)2ir—ooIn other words, impedance and wake functions are equivalent, corresponding respectively to frequency- and time-domain representations. Nevertheless, one or theother method may sometime be preferable, e.g. it is often more convenient to describe long-range wake fields of high-quality resonators in impedance representation,and short-range forces, which are the subject of this thesis, using wake function terminology.In this thesis we will consider how different wake fields can affect the beam. Forexample we will study inductive (space-charge), capacitive and resistive wake fields,as well as those of a broad-band resonator, which remains a good simplified modelfor the effects of many accelerator elements (such as kickers, joints, bellows, etc.),summed together.2.3 Collective EffectsLet us choose new dimensionless canonical variables relative to the stable fixed point1 hwop— 2 (2.39)w30 E013Chapter 2. Essentials of Longitudinal Motion 21q = (2.40)where p is the normalized longitudinal momentum, q is the longitudinal coordinate.The time variable is also made dimensionless by writing t in place of wot. Assumingthat their amplitudes are small, the equations of motion becomeoHp = —q + V(q,t) = —--— (2.41)= p=- (2.42)where V t)/(V0cos ç). The Hamiltonian becomesH=-+-+UId (2.43)whereUa =— f V(q’, r’)dq’. (2.44)Typically there are 1010 — 1015 individual particles in a single bunch and thereforeit is possible to introduce a distribution function i(p, q, r) so that the approximatenumber of particles in some area in phase space can be written simply asdN=Nbdpdq (2.45)where N is the total number of particles in the bunch andJ Jb(q,p,r)dqdp= 1. (2.46)The line density is\(q,t) = fb(q,p,r)dp. (2.47)Chapter 2. Essentials of Longitudinal Motion 222.3.1 The Vlasov EquationA powerful tool for studying the motion of a very large number of particles is Liouville’s theorem, which states that the phase space density is conserved along adynamical trajectory provided the system can be described by a Hamiltonian function:=0. (2.48)Taking into account that in general is a function of p, q and t we write (2.48)in the form&b .&& ,&‘+ q--- + p-b-- = 0 (2.49)which is known as the Vlasov equation. For longitudinal motion in a circular accelerator we can substitute j and j3 from (2.41) and (2.42), obtaining:+ Pj + [—q + V(q, 1)] = 0. (2.50)To solve this equation one needs to know the form of the induced potential V(q, t).2.3.2 Potential Well DistortionLet us consider how different interactions may affect the stationary distribution ofparticles in the beam. It is easy to see from (2.49) that for the distribution to bestationary we must require= 0 (2.51)and after we express j and in terms of the Hamiltonian we findöHb aHab H —0 2528j öq Oq — — ( .Chapter 2. Essentials of Longitudinal Motion 23where [...} are Poisson brackets. This means that a stationary distribution functiono must be a function of the Hamiltonian‘zbo(p,q) f(H(p,q)) (2.53)where the Hamiltonian of the system is given by (2.43).At high intensity, a short-range wake field can distort the beam’s potential welland thereby change the stationary distribution. If this is not taken into account,instability thresholds will be incorrectly predicted. The potential well distortion inthe case of longitudinal space-charge interactions can thus be writtenUd(q) = D(q). (2.54)In the case of electrons, for which the interactions are better described by the wakefield formalism, the potential well distortion isUd(q) = —if S(q’)(q + q’) dq’, (2.55)where the step functionS(q) f W(q’) dq’ (2.56)is derived from the wake potential.2.3.3 Stability AnalysisA stationary distribution &o(p, q) (2.53), though it satisfies (2.52), may be unstablewith respect to small perturbations. These perturbations may grow exponentiallyunder some conditions.Let us assume that the distribution b is perturbed from the stationary distribution‘b0 andq, t) = o(H(p, q)) + ‘i(p, q, t) (2.57)Chapter 2. Essentials of Longitudinal Motion 24In the limit of small b1 we need to keep only terms linear in the perturbation andwe can write the linearized Vlasov equation in the form+ + [—q + Vo(q)] + V(q, t) = 0. (2.58)In the case of bunched beams it is more convenient to use a different pair ofconjugate co-ordinates. We introduce the canonical variables action J and angle 0J=—jcpdq and ê=j.=4.(J) (2.59)instead of p and q, and the linearized Vlasov equation becomes(2.60)2.3.4 Collective ModesThis equation is linear and therefore we can look for its solutions in the formf(J, 0)e_t; if Im [v] > 0 the beam is unstable.F. Sacherer has shown [13], that if one can neglect the potential well distortion,the solution of that linearized Vlasov equation can be written as a sum of eigenmodesof (2.60)anmRnm(Vi)em8e_mt (261)n,mwhere n = 1, ..., cc and m = —cc, ..., cc are the radial and the azimuthal mode numbers respectively. The Rnm are orthogonal functions specific to a particular stationarydistribution b0, and Vflm are the eigenfrequencies.It is common to classify the modes according to their azimuthal number m (m = 1is dipole, m = 2 quadrupole, m = 3 sextupole etc.). A few of the lowest order modesand line densities corresponding to them are shown in Fig. 5.Chapter 2. Essentials of Longitudinal Motion 25(I)1 ST4TION4PDISTRIBUTIONm 1DIPOLE(R,GID-BUNCH),n.2+ \}‘..?“..J QLL4DRLPJLEPHASE SPACE LINE DENSITY =Figure 5: Different modes of oscillations in a bunched beam [13]Unfortunately, there are oniy a few cases when the expressions for Rnm and 1’nmcan be found analytically [16] and in general this equation must be solved numerically.Some of the numerical methods will be discussed later in the thesis.2.3.5 Turbulent Bunch LengtheningThe behaviour of electron and proton bunches is different once the instability threshold is reached.In the case of electrons, when synchrotron radiation becomes significant, an additional diffusion-like term must be added to the Vlasov equation (2.50). The resultingequation is the Fokker-Planck equation, and its time-independent solution is no longeran arbitrary function of H, but is the Maxwell-Boltzmann distribution‘0(H) o e_H[2]. Therefore, the potential well distortion which has been considered above cancm=+ IVf( ‘‘.1 \I _P SEXTUPOLEa’ + ?..(t)Chapter 2. Essentials of Longitudinal Motion 26change the bunch length, but it does not change the energy spread of the beam.Once an instability threshold is reached, however, this is no longer the case - instability leads to modification of the distribution and this effect is called turbulent bunchlengthening. This case will be considered later in Chapter 5.Aside from its impact on the stationary distribution, the diffusion term is a smallperturbation on the Vlasov equation in most cases, where the synchrotron frequencyis large compared with the radiation damping rate.Under some conditions the behaviour of an electron bunch beyond the thresholdis more complicated. Radiation damping does not modify the distribution function tothe extent that instability disappears, but it holds the particles together and therefore one can expect to see some continuous oscillations. These phenomena will beaddressed in the last chapter of the thesis.In the case of proton bunches, for which the acceleration or storage time is smallcompared to the relaxation time due to radiation effects, the stationary distribution isusually not of Maxwell-Boltzmann form. This feature of proton beams allows specialdistributions to be created which give more even line density, smaller peak currentsand therefore reduce transverse space-charge effects. Since the radiation damping doesnot affect protons as much as electrons, an instability usually results in changing L’oso that the bunch always remains below the threshold.In the following two chapters we consider space-charge and wake-field modelsseparately using the following procedure:1. find a stationary self-consistent distribution (including potential well distortioninduced by self forces);2. investigate the stability of the self-consistent distribution with respect to a smallexponentially time-dependent perturbation.Chapter 3Longitudinal Stability under Space-Charge ForcesIn this chapter we restrict ourselves to the longitudinal space-charge forces whichwere introduced in Chapter 2. In the case of space-charge impedance the stationarydistribution changes significantly with intensity [19]. It has been shown recently [25]for a resonator impedance that the thresholds calculated ignoring the potential welldistortion differ from those obtained in self-consistent calculations; therefore, theresults obtained previously for gaussian and ‘hollow’ distributions [23], assuming anabsence of incoherent frequency spread, have to be considered critically. However,these results provide us with a clear picture of the physics of the instability and canbe used for checking any new theory.3.1 Stationary Self-Consistent DistributionThe only exact analytical self-consistent solution for a stationary distribution in thepresence of longitudinal space-charge forces is the locally elliptic distribution b cc— H. A. Hofmann and F. Pedersen [17] have shown that this distribution doesnot change the shape of the potential well, and therefore the space-charge forces arelinear if the external forces are linear. This is similar to the Kapchinsky-Vladimirsky[18] distribution used in the transverse case, which is the only distribution which doesnot change shape with intensity.27Chapter 3. Longitudinal Stability under Space-Charge Forces 28As a result, the phase-space trajectories grow longer with increasing intensity,but nevertheless remain elliptical, and the stability analysis can be done analytically.In this case individual modes of oscillation can be described in terms of orthogonalpolynomials [13].Unfortunately, an elliptic distribution is highly idealized and lacks features whichare essential for a realistic distribution: it has a very sharp edge (which results inunrealistic forces close to the boundaries), it provides no tails and it does not produceany frequency spread within the bunch - a feature known to play an important rolein Landau damping stabilization [34].The problem of finding a stationary self-consistent distribution under space-chargeforces for the exponential, or Maxwell-Boltzmann distribution (/, 6_H) has beenstudied previously by Germain and Hereward [3] by solving the “HaIssinski equation”[2] which is used to describe the stationary distribution of electrons in a synchrotron.The distribution functions for proton bunches, however, can be far from exponentialand therefore their result is not a general one.Various distribution functions have been studied by Baartman in [19] for the caseof space-charge impedance. The numerical method used in that paper is an iterativeone where the current line density A (expressed as a vector {A(qj} defined at a finitenumber of points qj) is used in (3.62) to generate a new one. However, this iterativetechnique diverges in some cases and special relaxation techniques are required fordifferent distributions.Some results obtained in [19] are shown in Fig. 6 and Fig. 7, in which the linedensities and density plots in phase space are shown for different distribution functions/‘o. One can see that the phase space trajectories are significantly deformed at thethresholds.In this chapter we describe a method which can be used to find the self-consistentChapter 3. Longitudinal Stability under Space-Charge Forces 29Figure 6: Line densities (top) and density plots (bottom) of stationary distributionsfor the gaussian case at intensities (left to right) I = -1.552, 0 and 20. The left-mostplot is at threshold: for I < -1.552, no stationary distributions exist [19].Figure 7: Line densities (top) and density plots (bottom) of stationary distributionsfor the ‘hollow’ distribution f(H) = exp[—(H— H6)2/2] at intensities (left to right) I-8, 0 and 20. The left-most plot corresponds to negative mass and the right-mostto positive mass thresholds.Chapter 3. Longitudinal Stability under Space-Charge Forces 30distribution and the line density for any distribution function (essential for protonbeams) in the case of space-charge impedance. It can also be used to determinethreshold intensities beyond which no stationary distribution exists. In the iterativemethod [19], thresholds are difficult to find because the numerical technique becomesunstable near threshold. Our method can also be inverted to recover the distributionfunction from a given stationary line density.The problem we study is how the phase-space trajectories (H = constant) and linedensity depend upon the beam intensity and on the distribution of particles in thebeam. As we have shown in Section 2.3.2. a stationary distribution can be writtenas a function of the Hamiltonian, so that o(p, q) = f(H(p, q)).If we use choose the normalization (2.39) and (2.40) introduced in Chapter 2 theHamiltonian for the particle in the bunch can be written asH(p, q) = + U(q) (3.62)whereU(q) - + I((q)- o) (3.63)where p is the normalized longitudinal momentum, q is the normalized longitudinalcoordinate and )(q) is the line density. The constant A0 = A(0) is included to keepthe value of the Hamiltonian constant at the stable fixed point ([H(0, 0)] = 0). If thisis not done the Hamiltonian (3.62) may change with intensity and some importantfeatures of the distribution may be lost (e.g. the ‘hole’ in a hollow distribution maydisappear as the intensity is raised).If the line density in (3.62) has been normalized so that f Adq = 1, the intensityparameter I may be written in terms of machine parameters as1=27rh10im() (3.64)Vcosq8 nChapter 3. Longitudinal Stability under Space-Charge Forces 31where 11, V and Jo are the harmonic number, the rf voltage and the average beamcurrent respectively. Zn/n is imaginary and constant in the case of space-chargeimpedance; I may have either sign: positive below transition and negative above.We will assume that an arbitrary distribution function f(H) is well-behaved inthe sense that it is positive, smooth, non-singular, and f(oo) — f’(oo) = 0. Theline density )(q) cx f f[H(p, q)]dp cannot be found directly, because H itself dependsupon .A (3.62). Nevertheless, we can find A(U) cx .)(q(U)) defined byA(U)= J f(U +p2/2)dp. (3.65)Proper normalization is retained if we define a parameter 1 so thatI If A(U) dU. (3.66)We can see thatL\(q) = IA(U(q)) (3.67)and from (3.62)____________________q = +/2 [u + I (A0 - A(U))] (3.68)where A0Since U(q) is a symmetric function, we need only consider the interval q 0. Asignificant feature of self-consistent stationary distributions with space charge is thatthey have only one fixed point: one can see from (3.68) that U(qi) = U(q2) only ifqi = ±q2, so the only fixed point is q = 0.Integrating (3.66) by parts and taking (3.68) into account we findi = _21f A’(U)V’2 [u + 1(A0 - A(U))] dU. (3.69)Therefore, we can find I as a function of the parameter I. Using this formalismthe integral (3.65), and therefore also q(U) (3.68), can be calculated analytically forChapter 3. Longitudinal Stability under Space-Charge Forces 32a number of different distribution functions (see Appendix A). Unfortunately, it isnot always possible to find a explicit analytical expression for U(q), but for manyapplications this is not very important. The integral (3.69) can be easily evaluatednumerically. Of especial importance is the fact that the described method allows usto determine threshold conditions, beyond which no stationary distribution exists.3.1.1 ThresholdOne can see that the solution given by (3.68) does not always exist: the thresholdcondition arises from a simple physical reason, that the potential U(q) should have afinite slope; otherwise the force V = dU/dq —* cc. We can easily find the thresholdcondition by differentiating (3.68). No stationary distribution can be found ifIA’(U) = 1. (3.70)Therefore, to define the thresholds for an arbitrary distribution function, we need tofind the extremums of A’(U) (see Appendix A).It is interesting that this method gives analytical results for the threshold parameters for one very important class of distribution functions often used in stabilityanalysis: the so-called binomial distribution functionsr Hiucc— —for H H0, b = 0 for H> H0 (3.71)i. H0iwhere H0 represents the value of H at the edge of the bunch. Results obtainedfor some frequently used distribution functions are summarized in Table 3.1.1, whereI and Ij are the ‘positive’ and the ‘negative mass’ thresholds. Here the binomialdistribution has been chosen in the form f(H) = (i—so that it convergesto e as —* cc. The dependence of I on the parameter ,tt for the binomialdistributions is shown in Fig. 8.Chapter 3. Longitudinal Stability under Space-Charge Forces 33Table 1: Threshold parameters for different distribution functions.f(H) Type I J/l— H local ell. 00 0(1—4jL) binomial 00— = —0.94(1 — if) binomial 00 64—27’/ = —1.1415/se gaussian oo —1.55H/1—H hollow J(1_)= —0.70He_H hollow 5.47 —8.82H2e’ hollow 16.72 —14.582.C — I I I I I I1.6 - -0-(I)0.8--c4-io 00.4 -0 1 2 3 4 5 6 7 8 9 10parameter jFigure 8: The dependence of I on the parameter i’ for the binomial distributionsf(H) = (i—____Chapter 3. Longitudinal Stability under Space-Charge Forces 348 • I I I I A I I I I I I I7 F(H) = F(H) Heth106IIdi’4coordinate q coordinate qFigure 9: Line densities at the ‘negative mass’ threshold calculated by using (3.68).4. I I I I I I I I I I IF(H) = H2e F(H) = HeI = 16.725 1th = 5.46791< th.4314i4 °41144coordinate q coordinate qFigure 10: Line densities at the ‘positive mass’ threshold calculated by using (3.68).The threshold found for the gaussian distribution c_H by the method describedhere (Ith = -1.551) is in good agreement with the result obtained in ref. [19] by solving (3.62) iteratively [19] (Ith = -1.552). Line densities at the thresholds for somedistributions from Table 1 are shown in Fig. 9 and Fig. 10 (dashed lines correspond tothe points where dA/dU —* oc)3.1.2 DiscussionSome qualitative conclusions can be drawn from this procedure (here and later weassume that A’(U) is a smooth function, otherwise the induced force cannot exist).Chapter 3. Longitudinal Stability under Space-Charge Forces 35• There is always a ‘negative mass’ threshold (Ith < 0) because for any realisticdistribution function there is always a region at the edge where f’(H) <0.• If f(H) is monotonic there is no ‘positive mass’ threshold (Ith> 0).• Hollow distributions with f(0) = 0 always have a ‘positive mass’ threshold.The last item is not obvious, but one can show that A’(O) > 0 for such a distribution. We illustrate it for the case of hollow distributions which have only onemaximum. We assume that f(H) and f”(H) have no breaks and singularities atH> 0, and f(O) = f(oo) = 0.Let us choose A(U) (3.65) in the formA(U) = 2f (3.72)Since the distribution has only one maximum there is a point Hm > 0 for whichf’(H)>OforO<H<Hm and f’(H)Ofor Hm<H. We can see thatf’(H) f’(H)3 73(.for any H > 0 where f’(H) 0. Differentiating (3.72) and using (3.73) we getoof() 2AI(0)=2f d> f’()d=0. (3.74)3.1.3 Recovering the Distribution FunctionIn this section we describe an algorithm which allows us to recover a stationarydistribution from the line density in the presence of space charge. This algorithm isuseful, for example, for finding an initial distribution of particles to use in a multiparticle tracking code, given the line density. It can be derived from (3.62) and (3.72).Chapter 3. Longitudinal Stability under Space-Charge Forces 36Let us choose two points Ui and U2 so that U1 < U2. We can rewrite (3.72) in theformd+2f d. (3.75)U 2( — U1) £12 — U1)If we choose a linear approximation for f(H) in the interval U1 < H < U2 thenthe first integral in (3.75) can be evaluated:/2(U- U1) {f(u1)+ f(u2)]. (3.76)If the array {qj} has been chosen so that 0 = < qi <q2 < ... < qn and A(q) = 0for q> qn, then one can define=U = +IA (3.77)where i = 0,. . . n, so the recursion formula for fi can be obtained by substituting(3.76) into (3.75)ft-1 = _Li+2 42(U1— U_1)/ pUn— 2] ‘ ‘‘ d . (3.78)U 2(—U1_) )With f, = 0 and using the recursion formula (3.78) we can find f— for i =n, n — 1, ...1. The integral in (3.78) can be evaluated by any standard numericalmethod.Using Abel transform methods [21] equation (3.72) can be inverted and f(H)written as an integral of A’(U) [22], giving a very elegant form for the recoveryalgorithm:f(H)= 1J°° A’(U)dU (3.79)7 H J2(U_H)Chapter 3. Longitudinal Stability under Space-Charge Forces 37In some cases this formula allows analytical expressions for f(H) to be found fromgiven A(q) and U(q).In order to measure the distribution function of a bunch using a beam transferfunction or Schottky spectrum one has to know how the synchrotron frequencies depend on H. Using the Abel transform method we can find the incoherent synchrotronfrequencies within the bunch from the stationary distribution function and vice versa.Let us assume that we know the dependence of incoherent synchrotron frequenciesw(H) within a bunch and we need to find the potential in which such dependenceoccurs. The period of synchrotron oscillations in a given potential well isrq2 daT(H) = 2 I (3.80)‘q J2(H— U(q))where q and q are coordinates corresponding to the same potential energy (U(qi) =U(q2) = H). Substituting T(H) = 2ir/w(H) into (3.80) we get1 — 1 H (q(U) — qU)) dU3 81(H) - Jo 2(H- U) (. )where q(U) is a known function which can be measured directly or found from (3.68).Applying an Abel transform [21] to (3.81) we getdHq2(U)—qi(U)= I . (3.82)‘0 w(H)1J2(U — H)One can use this formula to find the potential giving w’(H) = const (a case oftenused in stability analysis). Choosing the normalization in which w(0) w0 and= aw0 we havew(H) = c’ (1 — a H) (3.83)For the symmetric potential well q(U) = qi(U) = —q2(U) the self-consistentsolution isq(U) 2arcsinvU . (3.84)wo2a (1—cvU)Chapter 3. Longitudinal Stability under Space-Charge Forces 38As will be shown later the form of dependence of w on H plays a crucial role instability analysis for short-range forces.3.2 StabilityIn this section we will try to investigate the stability of a bunched beam in the presenceof space-charge forces alone in a self-consistent way. In our analysis we do not restrictourselves to any particular shape of restoring potential Uo(q)U = Uo(q) + L\(q), (3.85)although we will assume Uo(q) to be symmetric. As we will see later, the fact that wehave freedom in choosing Uo(q) will allow us to compare the results obtained in thissection with the results obtained in [23] using completely different and well studied‘mode coupling’ theory. That theory can be used only if the potential well distortioncan be neglected (often a poor approximation in the case of space-charge forces).3.2.1 Integral EquationThe Vlasov equation can be written in terms of p and q (2.58), or, more conveniently,in action-angle variables (2.60). In the case of space-charge interactions the Hamiltonian of the system can then be written asH(J,O,t) = H0(J) + I)i(q(J,O),t) (3.86)where Xi(q,t)=fb1 dp. Taking into account that dH0/dJ = w(J), the linearizedVlasov equation becomes9) dib0+ w(J) -- — I --—--=0. (3.87)Chapter 3. Longitudinal Stability under Space-Charge Forces 39It turns out to be more convenient to take 1,o and w to be functions of = H(p, q)instead of J:d’/’0 1 d’b0= (J) (3.88)We look for a solution in the form= f(, O)e’ (and = g(q)e). Thenwith the definition () = —iv/w(€), we get=0. (3.89)The solutions of this equation can be easily found if one can neglect the synchrotron frequency spread. As was mentioned above, the elliptic distribution function does not change the shape of the potential well under the influence of longitudinal space-charge forces, so linear forces will always remain linear for an ellipticdistribution at any intensity. In this case the eigenfunctions of the equation can befound analytically. This was first done by D. Neuffer [15] and later by G. Besnier andB. Zotter [16] using a different technique. Unfortunately, the elliptic distribution isthe only one with this property; all other distributions do distort the potential welland therefore it cannot be assumed that c) = constant at any intensity.It has been shown in ref. [23] that the problem of determining the m = ±1 thresholds (as well as others caused by +m coupling) in the absence of synchrotron frequencyspread (potential well distortion) can be formulated as an eigenvalue problem for theFourier components of the line density. Moreover, analytical expressions for matrixelements for some specific distributions have been found [23]:oozgk = —i —Hjg (3.90)n=—oo12where g(q) = gke’’ and the intensity parameter=/2vE/e (3.91)Chapter 3. Longitudinal Stability under Space-Charge Forces 40- :.:‘.30- (15 - ‘:7C)- 1/1/0 5- I 1f/ ; \ \ -0.0 — II ll ——6 —4—2 0 2 4 6Intensity, IFigure 11: Coherent mode frequencies as a function of the intensity for a hollow beamdistribution. Matrix size: 40 x 40.is negative above transition and positive below. As we have shown above Z/n is aconstant for space-charge forces and the matrix elements Hk in (3.90) are given byHk = () mm— Jm(kr)Jm(nr)dr. (3.92)For a general distribution function it is necessary to calculate (3.92) numerically.Since the matrix Hk is a complicated function of the unknown frequency ii, thisdoes not lead to a proper eigenvalue problem, but requires searching in frequencyspace. To find the modes, ii was stepped through a series of values and for eachvalue the eigenvalues were found [23]. The results obtained in [24] for a ‘hollow’distribution are shown in Fig. 11. Negative intensities correspond to space-chargeimpedance above transition (or inductive impedance below transition).If the intensity is small enough the eigenmodes of (3.90) cluster around integers.The mode frequencies at zero intensity give the azimuthal index (m = 1 for dipole,Chapter 3. Longitudinal Stability under Space-Charge Forces 41m = 2 for quadrupole, etc) and different frequencies for a given m correspond todifferent radial modes [13]. For larger currents the modes may couple causing instability.The thresholds found by this method were in excellent agreement with previouscalculations and with numerical simulations, provided special measures were takento avoid the frequency spread which would otherwise have been generated by spacecharge forces [24]. However, as has already been mentioned above, these results differfrom those obtained in self-consistent simulations including potential well distortion.Let us derive an integral equation which will include the effect of potential welldistortion. The periodic solution f(f, 0) = f(f, 0 + 27r) is6+2irf(, 0) =‘e21)— 1 i ) d0’. (3.93)This result is formally equivalent to that given by Krinsky and Wang [12], but itdiffers in the sense that the present treatment is a perturbation about the stationary case which includes the space-charge impedance: in [12], the stationary inducedpotential is ignored.Integrating (3.93) by parts we getf(, 0) = I(f)g(q) — I(E)e2()_ 1 e°68’g(q’)d0’. (3.94)Integrating (3.94) over momentum and taking into account that g(q) = f f(, 0)dp,we have finally— f(c) 9+2irg(q) [i — IA’(U)] = If dpb(E) 2irO()— 1 f e00 )g(q’)dO’, (3.95)where A(U) is the auxiliary function for f(H) (bo cc f(H)), introduced in (3.72)above and A’(U) = dA/dU. The parameter I is defined by (3.66).Chapter 3. Longitudinal Stability under Space-Charge Forces 42Equation (3.95) is non-linear with respect to v and is therefore not easy to solvein general. In the special case v —* 0, however, we have the simple result00 irg(q) [i_ IA’(U)] =-- J b)dpj g(q’)dO’. (3.96)As mentioned previously, the v —* 0 limit can be thought of as coupling between+m azimuthal modes [23], and since the dipole mode m = + 1 is the lowest orderantisymmetric eigenmode g(q) = —g(—q), the integral in (3.96) vanishes and we findg(q) [1 — 1A’(U)j = 0. (3.97)Since we expect to have g(q) non-zero at least at some points, the thresholdcriterion becomesIA’(U) = 1. (3.98)Surprisingly, this criterion is the same as we found in a previous section for adifferent phenomenon, i.e. stationarity (3.70). This means that the threshold corresponding to m = ±1 cannot be reached for any stationary self-consistent distribution[28]! This criterion also gives us some knowledge about the shape of g(q) at threshold:it should be zero everywhere except the point where IA’(U) = 1.3.2.2 Comparison with Mode-Coupling TheoryThe lowest threshold intensity corresponds to coupling between m = +1 azimuthalmodes. Due to the symmetry of Hk this threshold has an eigenfrequency ii = 0(Fig. 11). B. Zotter has found that in the case of ii = 0 the matrix elements Hkl canbe found analytically for ‘gaussian’ (e_H) and ‘hollow gaussian’ (H c_H) distributions[23].Chapter 3. Longitudinal Stability under Space-Charge Forces 434.3w‘--, .2.1.0Figure 12: Family of the distribution functions given by (3.99) for different values ofthe parameter a.To compare the results of the two techniques we have chosen the family of distributions defined by= [1- a(l— )) e (3.99)This family is very convenient because it contains both the ‘gaussian’ (a = 0) and‘hollow-gaussian’ (a = 1) cases (see Fig. 12) and therefore also allows the matrixelements to be found analytically.The eigenvalue problem can be written in the form= (3.100)where g(q) = ygeuI and the matrix elements areMk= [i — io(kn)je)‘hollow’(Q: = 1, L’0 €e)Energy, E3 4(3.101)Chapter 3. Longitudinal Stability under Space-Charge Forces 44- a{k2+n [1_Io(kn)} +kn[1_Ii(kn)]}.Here -Tm(Z) = 6_ZIm(Z) is the exponentially scaled modified Bessel function. Thelowest thresholds can be found from the condition = 1, where are the extremeeigenvalues (of either sign).In comparison, we have from (3.97) that 1 — A’(U) ‘th = 0. With f(H) given by(3.99), the expression for A(U) isA(U) = (i — + c u) e (3.102)In order to find the thresholds for this family of distributions we should lookfor extreme values of A’(U) for different parameters cr. Simple analysis allows us toobtain the following expressions—1+o if0<cr<=_crexp(_) if(3.103)0 if0<n<A’(U)max =—1+o if<cv<1Therefore, if one plots and A’(U) versus o on the same graph, the two curvesshould be the same. In order to solve (3.100) numerically, the matrix was truncatedat 40 x 40. The minimum and maximum eigenvalues obtained in this case are plottedas points in Fig. 13. The solid lines are A’(U) (3.103). We can see that the resultsare in good agreement.We can also find the shapes of the eigenfunctions g(q) at threshold: since thethreshold condition (3.97) is satisfied in general at oniy one specific point qth, g(q)Chapter 3. Longitudinal Stability under Space-Charge Forces 450.4-0 6 I I —?.- 0.2--o 0.0-—0.4-—0.8 -0.0 0.2 0.4 0.6 0.8 1.0Parameter aFigure 13: Extreme eigenvalues of equation (3.100) (plotted as squares) and extremevalues of A’(U) (solid lines).Chapter 3. Longitudinal Stability under Space-Charge Forces 46Figure 14: The eigenfunctions found by solving the matrix equation using the modecoupling method, corresponding to (a) negative mass threshold and (b) positive massthreshold.can be non-zero only at qth. Indeed, recovering g from the eigenfunction {gjj corresponding to = ,2 [23], we find a very sharp peak at the point where IthA’(U) = 1(see Fig. 14) and almost zero elsewhere, and the peak becomes sharper with increasingorder of the matrix used in (3.100).3.2.3 Self-Consistent CaseThe results discussed in the previous section have been obtained assuming no incoherent synchrotron frequency spread (i.e. V o q2).To satisfy this condition in the self-consistent case, the initial potential well shouldbe9 a S4b2—1-220 40-2—3—4U0 = U + I [A(0)— A(U)] (3.104)Chapter 3. Longitudinal Stability under Space-Charge Forces 47876CCwo2010—1Figure 15: U0 vs. q at threshold intensities for the ‘hollow-gaussian’ distribution.U = q2/2 is shown by the dashed curve.where U = q2/2, and A(U) is given by (3.72). The necessary Vo(q) to get a self-consistent stationary phase space distribution with U(q) = q2/2 for the ‘hollow-gaussian’ distribution L’0 = ee at threshold intensities are shown in Fig. 15. Theupper curve shows Uo(q) at the positive mass threshold and the bottom one at thenegative mass threshold. As one can see these shapes are far from ‘sinusoidal’ or‘harmonic’.Realistically, of course, the situation is different, i.e. Uo is harmonic and U(q) isdistorted by space-charge. In this case we can find the threshold beyond which nostationary distribution exists [20] and since it is the same as the stability thresholdthe modes rn = ±1 do not couple!—4 —3 —2 —1 0 1 2 3 4longitudinal coordinate, qChapter 3. Longitudinal Stability under Space-Charge Forces 48It is necessary to mention that this criterion is valid for any Uo(q) for whichdUo/d(q2) 0 (i.e. no local minima). This analysis can be extended to the case whenUo(q) is not symmetric. Unfortunately, in this case we can’t use the symmetry of theeigenfunctions to determine the thresholds as has been done earlier in this chapter.Numerical solution of the integral equation is required. This has been done for severalcases of i’0 and the results are consistent with the same threshold, namely IA’(U) = 1.However, no formal proof of the universality of this criterion has yet been found.3.3 Summary3.3.1 Stationary DistributionA method to find self-consistent distributions and line densities for any distributionfunction in the case of space-charge impedance has been derived and used to findthreshold conditions beyond which no stationary distribution exists. We have calculated the thresholds analytically for a number of different distribution functions.We have also introduced an algorithm which allows one to recover the distributionfunction from a given stationary line density or to find the incoherent synchrotronfrequencies within the bunch from the stationary distribution and vice versa.3.3.2 StabilityA simple criterion for stability thresholds in the case of space-charge impedance hasbeen derived from the linearized Vlasov equation. This criterion appears to be thesame as that for thresholds beyond which no stationary distribution exists.Chapter 4Longitudinal Stability of Electron BunchesIn the previous chapter we treated space-charge forces and found that the criterionfor bunched beam stability is simply that a stationary distribution exist. In the caseof electrons the situation is usually more complicated, because the bunch length ismuch smaller than in a proton machine and the interaction cannot be consideredlocal, as was assumed in the case of space-charge forces.The results of the previous chapter clearly indicate that in order to calculatestability thresholds correctly one must take potential well distortion into account.Attempts to substitute an exact potential with some average effect, or not to includeit at all, will result in incorrect estimates of thresholds.Oide and Yokoya have suggested a method [25] which allows one to find the eigenmodes and thresholds in the case of broad-band impedance. We call this the O+Ymethod. This method is self-consistent and includes deformation of the equilibriumdistribution due to the wake field. The results obtained are significantly differentfrom those not including the deformation.Unfortunately no rigorous analysis has been made of the convergence and stabilityof the O+Y method. Therefore its limitations and precision are not known. However,the thresholds obtained seem to be in agreement with particle tracking simulations.Another problem associated with this method is a lack of understanding of the physicsunderlying the thresholds.49Chapter 4, Longitudinal Stability of Electron Bunches 50In this chapter we derive an integral equation from the linearized Vlasov equationand then show how the O+Y method can be derived from it. Analyzing this equationwe also give physical explanations for the eigenmodes and instability thresholds whichare found in the O+Y method. As a result of this analysis we suggest a simple criterion necessary for single-bunch stability and compare this criterion with the resultsobtained by Oide and Yokoya for a broad-band resonator.The main feature of the new criterion is that the stability threshold in the case ofshort-range forces can be obtained by analyzing the stationary distribution. This issimilar to the case of space-charge impedance, investigated in the previous chapter.4.1 HaIssinski EquationStationary distributions of electrons are described by the Fokker-Planck equation,which is similar to the Vlasov equation (2.49) but has an extra term— 2— (pb +—. (4.105)apiThis term describes the effects of particle diffusion due to synchrotron radiation, asis significant in modern electron synchrotrons.HaIssinski has shown [2] that due to radiation damping and photon emission thestationary distribution in this case should have the thermal formb0(H) o e4 (4.106)where T is the equilibrium ‘temperature’ of the particles which depends on synchrotron radiation. Using a proper normalization we can always make T = 1.Substituting the HamiltonianH(p, q) = j- + + jq V(q’)dq’ (4.107)Chapter 4. Longitudinal Stability of Electron Bunches 51into (4.106) and integrating over p we finally get the Haissinski equationA(q) = Kexp [_- — if S(q’)A(q + (4.108)where K is a normalization constant and S(q) is the step function defined by (2.56).This equation can be solved numerically if the wake function is known. Exact analytical solutions have been found only for purely resistive and purely inductive impedance,though some approximate solutions have recently been found for capacitive [29] andresonator [4] impedance.Other self-consistent integral equations describing the potential well distortion forelectrons have been proposed by different authors, but as shown in [4] they can all bederived from the HaIssinski equation and differ mainly in integration limits.At low intensities, when the self forces are small compared to external rf forces,the solution to (4.108) is a gaussian: A(q) o e2/2, but with increasing intensitythe line density may differ significantly from a gaussian. At LEP for instance linedensities become noticeably double-peaked at higher intensities (Fig. 16).4.2 Stability4.2.1 Integral EquationAs we have shown earlier the solutions of the linearized Vlasov equation (2.60) are ofthe form f(J, O)e_it. We will look for solutions in the following form&1(J,O,t) = e_1t Cm(J)cosrnO+Sm(J)sinmO. (4.109)m=-Substituting (4.109) into (2.60), multiplying both sides by sinmO, integrating over0, and then separating imaginary and real parts, we get the following pair of equationsiUCm(J) = m.ü(J)Sm(J) (4.110)Chapter 4. Longitudinal Stability of Electron Bunches 52File K: 290 MV, 0.120 mAFEEDBACK ONFigure 16: Snapshots of the longitudinal profile of the beam at LEP made with astreak camera. Horizontal ellipses show the peak positions of earlier pulses.File G: 290 MV, 0.100 mA File 1: 290 MV, 0.035 mAWIGGLER ONFile J: 250 MV, 0.120 mAChapter 4. Longitudinal Stability of Electron Bunches 53i/Sm(J) —mw(J)Cm(J)— Idzboj21rQ.9öUi(q)(4.111)whereU1(q) = JW(q’))i(q’+q)dq’= f dq’S(q — q’) f dp’Cm[J(p’,q’)] cos[m’O(p’,q’). (4.112)-00 -00 mChanging variables from (p,q) to (J,O) in (4.112) givesUi(q) = J dJ’Ci(J’) J dO’S[q(J, 0) — q(J’, 0’)] cos m’O’ (4.113)m1and we finally obtain the following integral equation[2— m2w2(J)] Cm(J) = —m2w(J)I(H) fg’(J, J’)Cm’(J’) dJ’ (4.114)m’wheregmm’(J,J’)= --f dOf dO’ cosmOcosm’O’S[q(J,O) — q(J’,O’)]. (4.115)where the azimuthal mode index m = 1, ..., cc.We can also rewrite (4.114) as an integral eigenvalue equationCm(J) = J dJ’Kmmi(J, J’)Cm’(J’) (4.116)m’where the kernel Kmm’ is given byKmm’(J, J’) = m2w(J) [ömm’S(J — J’) —h/0(J)gmmi(J, J’)j . (4.117)Here 6mm’ is Kronecker’s symbol and 6(J — J’) Dirac’s delta function.To find out whether the beam is stable or not, we therefore have to solve thisintegral equation numerically. However, as we will demonstrate below, some conclusions about stability can be drawn by investigating stationary distributions withoutactually solving the integral equation.Chapter 4. Longitudinal Stability of Electron Bunches 544.2.2 Rigid Dipole ModeIt is known that there exists at least one solution of the Vlasov equation (2.49) - therigid dipole mode:q, t)= &o(p + iae_tt , q — ae_it). (4.118)If the oscillations are small one can see thatit (thbo .O’\/,i(p,q,t)=ae-———--ã-—- (4.119)is indeed a solution of the linearized Vlasov equation (2.60).At low intensities J = H(p, q) and the solution of the integral equation (4.114)for the rigid dipole mode it = 1 with the Maxwell-Boltzmann distribution is simplyC1(H) = v’7iexp(—H)Cm(H) = 0, m> 1. (4.120)It is not obvious, but it is quite easy to show by substituting (4.120) that the rigiddipole mode is indeed a solution of (4.115) for any wake potential S(q).If potential well distortion is neglected or modified from its self-consistent shape,i/ given by (4.119) is no longer a solution of (2.49) and therefore one may get unphysical modes instead. As will be shown below the rigid dipole mode plays a veryimportant role in the stability analysis and therefore using methods which do notinclude a rigid dipole mode among their solutions may lead to incorrect results. Therigid dipole mode is also a useful test of any numerical technique used to solve theintegral equation (4.114).4.2.3 Analysis of the Integral EquationAre there any other solutions of the equation (4.116) and what can we tell aboutthem? We know that in the case of a purely harmonic potential the solutions ofChapter 4. Longitudinal Stability of Electron Bunches 55(4.116) can be written in terms of orthogonal polynomials, but as we have seen in theprevious chapter the only case when the potential remains harmonic is the case ofpurely inductive (or space-charge) forces and an elliptic distribution. In the case ofelectrons neither of these conditions is satisfied: the distribution is exponential andthe wake fields are more complicated than inductive. Therefore the potential wellwill be distorted and there will always exist a synchrotron frequency spread withinthe bunch.Analyzing the dispersion integral obtained from the Vlasov equation for a one-dimensional case Van Kampen has found that a complete solution can be written asa combination of discrete modes and a continuous spectrum [33]. The discrete modesare collective modes and the continuous spectrum can be represented by S-functions.These modes, which are called Case-Van Kampen modes, are not real in the sensethat their coupling can cause an instability, but they should appear as solutions of(4.116) if we try to solve it numerically.4.2.4 Oide-Yokoya MethodA straightforward method for solving the integral equation (4.116) is to convert it toa matrix equation: if the set {J} is chosen so that 0 = Jo < J1 < ... < J, then amatrix equation can be obtained from the equations (4.114) and (4.115) by makingsubstitutions Cm(Jn)JJn “ Cmn. The resulting equation2,-iP = IVlmnm’nIL’mln#can then be solved numerically.Oide and Yokoya derived the above matrix equation by introducing a set of artificial orthogonal functions. Our analysis shows that their technique is equivalent toapproximating an integral f f(x)dx by the sum > f(x)zx. Such an approximationChapter 4. Longitudinal Stability of Electron Bunches 56is only good for a sufficiently smooth integrand. Our results presented below showthat the majority of the solutions found are not smooth and probably represent VanKampen modes. The fact that there are many non-smooth modes among the solutionsof (4.121) is alarming because the matrix elements were calculated on the assumptionthat the solution is smooth. Therefore, more rigorous mathematical analysis of theconvergence and stability of the method is required. Nevertheless, we gain confidencefrom the fact that the instability thresholds obtained with this method are in goodagreement with the particle tracking results presented by O+Y[25].Typically, to obtain stable results when the wake field is approximately equal tothe bunch length requires at least 3 azimuthal (m) modes and 120 radial subdivisions(n), resulting in matrices at least 360 x 360 in size. For a given intensity, firstthe stationary distribution has to be found, next each matrix element has to becalculated by double integration, and finally the matrix is analyzed for eigenvectorsand eigenvalues. The cpu time required for the 360 x 360 case is around 2.5 minuteson a VAX 4000. The calculation is repeated for many intensities to find a possiblethreshold [26].In the following section we will apply the methods described above to various typesof impedances and wake fields frequently used to model single-beam effects. Thoughideal inductive, capacitive and resistive wake fields do not in themselves accuratelyrepresent real machines, these models allow a better understanding of the physicsbehind single-bunch instability.Chapter 4. Longitudinal Stability of Electron Bunches 574.3 Results4.3.1 Inductive WakeWe have already met this type of wake field when we considered the interaction of longbunches in a smooth pipe. There we showed that the induced voltage is proportionalto the derivative of line densityV(q) = LI0. (4.122)We have already mentioned that for this kind of interaction the impedance Z(w)/.= constant. The wake field for this idealized interaction is simply the derivative ofthe Dirac 8-function, thusV(q) oc j 6’(x)(q + x)dx. (4.123)Since the inductive wake formulas are the same as those for space-charge forcesexcept for the sign, the results obtained in the previous chapter can easily be appliedto the thermal distribution; i.e. the distribution will remain stable above transition(the case important for electron accelerators) but has a stability threshold belowtransition. Applying the O+Y technique for this highly singular wake leads to convergence problems, which is why we used a different technique to study space-chargeand inductive forces in the previous chapter.4.3.2 Capacitive WakeIn the case of capacitance the relationship between induced voltage and current isV(q) = f (x + q)dx (4.124)Chapter 4. Longitudinal Stability of Electron Bunches 58I1.51.00.50Figure 17: The beam profiles at different intensities found from the Haissinski equation for a capacitive wake field above transition.and therefore W(q) = 1/C for q > 0. The impedance is simply Z l/(iwC). It isnatural to normalize W(q) so that W(O) = 1. The intensity parameter thus becomes:I Ib/[U(EO/e)CwO], where lb is the current per bunch, E0 beam energy and 0ethe relative rms energy spread. Some solutions of the Haissinski equation for purelycapacitive impedance are shown in Fig. 17. Note one important feature of capacitiveimpedance is that the line density will always be symmetric (just as in the cases ofpurely inductive and space-charge impedance). Above transition as I increases thebunch shortens (Fig. 17) rather than lengthens, as happens in the inductive case. Thiseffect has been observed at LEP [6] where the impedance is dominantly capacitive.Below transition the effect is opposite, i.e the bunch lengthens.Fig. 18 (upper plot) shows eigenfrequencies,i versus intensity I calculated in thecase of a purely capacitive wake field. No complex eigenfrequencies are found. Thisagrees with Burov [29], who derived an approximate analytical solution for stationary distributions at extremely large currents. In contrast to inductive wake fields,-4 -2 0 2 4‘7-Chapter 4. Longitudinal Stability of Electron Bunches 59stationary self-consistent solutions exist for capacitive wakes at any intensity, bothbelow and above transition.The rigid dipole mode is clearly distinguishable from the other eigenfrequenciesbecause, for this particular wake field, all the other frequencies are shifted upward. Wehave verified that the mode with it = 1 is indeed the rigid dipole mode by comparingC1(J) with that expected from (4.120), as shown in Fig. 19.What are the modes associated with the other eigenfrequencies in Fig. 18? Foreach of the ñ values of J into which the problem has been subdivided, one can calculatethe corresponding (incoherent) frequency (J). These and their integer multipleshave been plotted in Fig. 18 as well (lower plot). We see that they agree well with theother frequencies found by the O+Y method, excluding the rigid dipole mode. Thisindicates that these modes are not really collective modes. Further verification ofthis hypothesis comes from the fact that the eigenvector C is found to be nonzeroonly at one or two values of n. This means that if the eigenvector has any physicalinterpretation at all, it represents an eigenfunction which is extremely localized inJ, and becomes narrower, the larger the matrix size. It should also be realized thatthe O+Y method forces the existence of ñ radial modes. We conclude, therefore,that these singular modes are not real, in the sense of being physically detectable.Moreover, we do not expect them to couple and thereby cause instability. We callthese modes ‘incoherent’.In order to separate real collective modes from the other incoherent eigenmodes,we define the normalized rms value of the ktI eigenvector C,t:Xk = j (C)2. (4.125)m,riSince the eigenvectors Cmn are normalized to have a maximum value of 1, weexpect the narrow incoherent modes to have X << 1, while broad modes like the rigidChapter 4. Longitudinal Stability of Electron Bunches 60I— I I3— II I I I I! ! ! !0— 1 i ‘ 1——— —I I1—i i I I I——0 1 2 3 4 5 6Intensity IFigure 18: Bunch oscillation frequencies in the case of purely capacitive impedance,plotted for discrete intensities I. The upper plot shows eigenfrequencies k calculatedby the O+Y method, and the lower the incoherent frequencies m(J), where m isan integer. The eigenmode with frequency independent of intensity is the rigid dipolemode.Chapter 4. Longitudinal Stability of Electron Bunches 611.21.00.8‘-;:. 0.6L)0.40.20.0Figure 19: Comparison of the theoretical rigid dipole mode (4.120) (continuous curve)with the eigenvector C1 of the = 1 mode (D). The calculation is for ñ = 120,JChapter 4. Longitudinal Stability of Electron Bunches 62.5- I I-.4-.3-.2-.1.0--0 1 2 3Figure 20: RMS value of the kt eigenvector versus its eigenvalue at intensity I = 2,in the case of purely capacitive impedance with ñ = 40 (x) and ñ = 120 (C).dipole mode will have much larger X.Fig. 20 shows a plot of Xk versus 1uk for capacitive impedance. One can see thatthe point with maximum X also has R = 1, verifying that this is the rigid dipolemode. Also, this point is not sensitive to ñ, whereas the other values of X all tendto zero as , is raised.As has already been mentioned above, the capacitive impedance above transitionwill cause bunch shortening, so it seems that if one can create an environment withpurely capacitive impedance the problem of bunch lengthening will be solved. Unfortunately, the wake field in real accelerators always includes resistive and inductivecomponents and therefore the effect of bunch shortening is observed only at veryChapter 4. Longitudinal Stability of Electron Bunches 63small intensities and is then followed by ‘turbulent bunch lengthening’, as we will seelater.4.3.3 Resistive WakeResistive impedance is real and does not depend on frequency (Z = R), and sinceV(q) = RI(q) (4.126)the wake function of a resistive impedance is simply a Dirac 6-functionV(q) = RI0 j 6(x)(q + x)dx. (4.127)A solution of the HaIssinski equation for purely resistive impedance can be obtained analytically [30].( )= (4128)RI0 [coth(RIo/2) — erf(q//)]Stationary self-consistent distributions given by (4.128) at different intensities areshown in Fig. 21.The distribution is always unstable. Recently, Oide [31] has proposed an explanation of resistive instability which has its origin in a mechanism different from thecollective mode coupling which is considered in this thesis. His suggestion is thatresistive instability is produced by the mode concentrated in the area dw/dJ = 0(such an area always exists for a purely resistive wake). This mode, moreover, doesnot couple with any other modes, but instead is unstable by itself. The growth rateof this mode is quite small.Oide has also speculated [31] that this type of instability may be responsible for thelowering of the instability threshold at SLC damping ring [7] after the installation ofChapter 4. Longitudinal Stability of Electron Bunches 64I210Figure 21: Stationary longitudinal profiles at different intensities for a resistive wake.a new smooth vacuum chamber with lower impedance. He has hypothesised that thelower impedance does not automatically lead to an increased instability threshold andthat the threshold depends mainly on the ratio of the active (resistive) and reactive(inductive, capacitive) components of the impedance. Reducing the reactive part ofthe impedance (which is what happened at SLC) may only make the situation worseif the resistive component remains the same.4.3.4 Broad-band Resonator WakeThe impedances considered above are idealized and though they are very importantfor understanding single-bunch effects in electron synchrotrons it is very difficult toimagine conditions under which the interaction between the beam and environmentcan be expressed in terms of these impedances alone.A more realistic model of the many elements that make up an accelerator is aresonator with a low quality factor Q. Such an average resonator can be describedby a simple parallel LRC circuit. The impedance of this circuit is well known from-5 0 5TChapter 4. Longitudinal Stability of Electron Bunches 65electrodynamicsZ() = R (4.129)i+iQ[_-]where Q = RJiiJJ is the quality factor and r = i/\/L the resonant frequency.The corresponding wake function found by Fourier transform of (4.129) isW(r) = e_T [cos(CJJ1T) + T)i] (4.130)where w1 = Wr/1 — i/(4Q2). Interesting effects are expected to occur when theresonant frequency Wr is comparable with c/us, where o is the rms length of thebunch at low intensity. To emphasize this, it is useful to introduce a parameterk0 = w,.r/c relating bunch length to wake length and a coordinate q rc/u2 (k0>> 1corresponds to very long bunches and k0 << ito very short). Fig. 22 shows an exampleof a resonator wake field and the longitudinal beam profile at different intensities.The resonator model is also interesting because by choosing different values of L,R and C one can obtain purely resistive or purely capacitive wake fields as limitingcases, and also combinations such as L-R or R-C circuits. The latter seems to be agood model for approximating the impedance at LEP [6]. However, it has been foundthat in many accelerators, a resonator with Q 1 describes the wake field quite welland we restrict ourselves to this case.Mode coupling leading to instability should take place between collective modes,not incoherent ones. One such mode is the rigid dipole mode, and the question iswhether there are any others among the solutions of (4.121) and whether they coupleor not.The intensity parameter I is defined as above for capacitive impedance, but nowwith the resonator’s high-frequency capacitance, (rR/Q)1used in place of C. Thusour I is the same as the parameter Sr used by O+Y[25]. The eigenfrequencies calculated using the O+Y method are shown in Fig. 23. Note that there is an instabilityChapter 4. Longitudinal Stability of Electron Bunches 66ob2d115Figure 22: Resonator impedance with k0 = 0.6 and Q = 1: (a) the wake field and (b)longitudinal beam profiles at different intensities.I I I I0 2 4 6 8 10 12Intensity IFigure 23: Eigenfrequencies versus intensity for a broad-band resonator (Q = 1,= 0.6). Unstable modes are indicated by solid symbols.Chapter 4. Longitudinal Stability of Electron Bunches 67iii;•3.2-1-0-•1——II’ll •J.IIChapter 4. Longitudinal Stability of Electron Bunches 68with a threshold of I 8. As in the capacitive case, most of the frequencies correspond to incoherent modes. In fact, a plot of mw(J) is virtually indistinguishablefrom Fig. 23, except for the presence of complex eigenfrequencies corresponding tounstable modes, which are shown in the figure by solid symbols. The rigid dipolemode is not apparent because in the broad-band resonator case, incoherent frequencies are shifted both up and down, and so it is hidden among the incoherent modes.Close inspection of the eigenvectors reveals that those modes with frequencies near= 1 all have larger or smaller rigid dipole components, depending upon how farthey are from the frequency p = 1. Going to larger i is of no help, since there willalways be a couple of incoherent modes near p = 1 which will heavily contaminate it.If we plot the parameter Xk versus p, (Fig. 24) as before, the rigid dipole mode isstill difficult to distinguish. However, there appear to be a few other ‘real’ modes aswell. An investigation of the incoherent synchrotron frequency (Fig. 25) shows thatthese modes are clustered near the local minimum of w(J). The physical interpretation is that near d/dJ = 0 the particles can stay ‘in step’ longer, and so this areaconstitutes a ‘coherent band’ of action J.The collective quadrupole mode is shown in Fig. 26, where it is compared withan incoherent mode. Note the dramatic difference: the coherent mode is smooth andindependent of ñ, whereas the incoherent mode is narrow and very sensitive to ñ.The reason that the coherent mode appears so unambiguously is of course that itoccurs at the minimum of the synchrotron frequency and so is not degenerate withany incoherent modes.Chapter 4. Longitudinal Stability of Electron Bunches 69.5- I -.4-Dx3‘C% X*xxx ‘C.2-‘C ocxx x ‘C x‘C x ocxLA Li.0- I -0 1 2 3Figure 24: RMS value of the kth eigenvector versus its eigenvalue at intensity I = 3 inthe case of a broad-band resonator with Q = 1, k0 = 0.6, for two different matrix sizes:= 40 (x) and ñ = 120 (1). Note that there are two modes whose X is independentof i; these correspond to a quadrupole and a sextupole mode respectively, at thelocation where w(J) is a minimum.Chapter 4. Longitudinal Stability of Electron Bunches 7021.5(A.’8—1ws00.50Figure 25: Normalized synchrotron frequency versus action J and intensity I for theresonator impedance with Q = 1, ko = 0.6.4.3.5 New CriterionAs intensity increases, Wmjfl decreases (Fig. 25), and just at threshold it is near 1/2.This suggests that the instability arises because of coupling of the coherent quadrupolemode (whose eigenfrequency is with the rigid dipole mode whose frequencyis constant (unity). The intensity at which = 1, thus corresponds to an instability threshold. This conjecture is verified by an inspection of the eigenvector ofthe unstable mode. Also, by extrapolating the lowest frequency quadrupole mode inFig. 23, we see it crosses the rigid dipole mode near the threshold intensity.The threshold calculated from the criterion = 1/2 has been plotted versusthe bunch length in Fig. 27 on top of the data from O+Y[25]. The agreement is good:the discrepancy between the solid and dashed curves is possibly due to the truncationof the matrix in the latter case.8Chapter 4. Longitudinal Stability of Electron Bunches 710.0-II—0.5-—1.0- F I I I I I I-0 1 2 3 4 5 6 7 8 9 10JFigure 26: Comparison of the eigenvector for the ‘coherent’ quadrupole mode at thesynchrotron frequency minimum (it = and a nearby ‘incoherent’ quadrupolemode, for two different matrix sizes: ñ = 120 (continuous curve) and ‘i = 40 (dashedcurve).Chapter 4. Longitudinal Stability of Electron Bunches 7220— IU • 1/U 1/15-• //••\ ‘1,\ \•5-0- I I-0.0 0.5 1.0 1.5 2.0k0 = 0cr/cFigure 27: Comparison of thresholds obtained by different methods: new method(continuous curve), O+Y method (dashed curve) and numerical tracking (symbols).The latter two cases are taken from O+Y[25].Chapter 4. Longitudinal Stability of Electron Bunches 73In the case of resonator impedance the instability caused by collective mode coupling described here happens first for 0.5 < ko < 1,2. Outside this region anothertype of instability (much weaker than that caused by mode coupling) dominates.Mode coupling is also less probable below transition. This property has been studiedby Fang [27], who suggests that, for higher beam intensities, it may become preferable to operate electron synchrotrons with a negative momentum compaction factora, because even if the instability threshold is lower than for the same wake fieldabove transition, the instability itself is much weaker and does not create significantproblems.4.4 Conclusions4.4.1 Oide-Yokoya MethodOide and Yokoya have invented a helpful technique for solving a previously intractableproblem. However, the method should be used with caution. It always generates asmany eigenmodes as the order of the matrix. The vast majority of these modes appearto be infinitesimally narrow in the limit of infinite order, and have eigenfrequenciesequal to the incoherent synchrotron frequency at the mode location. The methoddoes find real collective modes such as the rigid dipole mode, but these are difficult toextract in cases where they are degenerate with the incoherent modes. Nevertheless,even in degenerate cases instability thresholds attributable to coupling between realcollective modes are found, and these appear to be valid.Chapter 4. Longitudinal Stability of Electron Bunches 744.4.2 Landau dampingThe fact that so few real modes are found should not be surprising. Landau-dampedexcitations do not have exponential time dependence, so the standard method ofsolving the Vlasov equation by assuming harmonic time dependence simply shouldnot find any modes which we think of as being ‘Landau-damped’. The situationis similar to the case where synchrotron frequency spread originates externally byrf wave nonlinearity. Besnier [14] originally developed a technique for solving theSacherer integral equation where the dispersion function was expanded in the sameorthogonal polynomials as the kernel. This technique suffers from the same problemas the O+Y method in that there will always be as many modes as the order of theresulting matrix. Subsequent re-analysis by Chin, Satoh, and Yokoya [32] showedthat below threshold these modes have growth rates which tend to zero as the matrixorder is increased. They developed a dispersion integral approach and found thatbelow threshold there were no modes, while above threshold their eigenfrequenciesagreed with those found by the Besnier technique.The difference between external frequency spread and that arising from the wakefield itself is that in the latter case the frequency ‘spread’ and the mode ‘shift’ changeat the same rate with intensity. In the former case, the modes which would exist inthe absence of frequency spread are one-by-one ‘freed’ from the incoherent band asthe intensity is raised. But in the latter case, the dispersion also grows with intensityand so most of the modes which would exist in the absence of dispersion do notexist (equivalently, they are Landau-damped) at any intensity. The few modes thatdo exist can be unstable on their own (for example, with a resistive wake), or canbecome unstable by coupling together when their frequencies coincide.Chapter 4. Longitudinal Stability of Electron Bunches 754.4.3 New CriterionThe simple necessary criterion for instability suggested in this chapter is in reasonableagreement with the O+Y method and numerical tracking. The instability thresholdcan be found by analyzing the stationary self-consistent distribution, without solvingthe Vlasov equation. One collective mode found by this method is the rigid dipolemode; the others are multipole ‘collective’ modes concentrated near the synchrotronamplitude where the synchrotron frequency is a minimum. Since this minimum wdecreases as intensity is raised, and the frequency of the rigid dipole mode is a constant, it may happen that mw = w80, at which point coupling between the rigiddipole and the m-th multipole mode leads to instability. The intensity at which= 1/2w8o thus corresponds to an instability threshold. This is similar to, butmore stringent than, the threshold suggested by Wilson as quoted by Bane and Oide[40]: namely that instability occurs when = 0.Our new criterion has been confirmed by numerical simulations and comparedwith the results obtained by the O+Y technique (a code which uses methods similarto those used in [25] has been written for this purpose). Collective modes have beenfound among the solutions of the Vlasov equation [26] and the shape of the modeswhich were unstable clearly indicates the coupling mechanism of the instability.In this theory the rigid dipole mode, and especially the fact that its frequencydoes not depend on intensity, plays an important role. The importance of the rigiddipole mode in instability analysis was mentioned by Oide in his ‘two-particle’ modelof longitudinal instability [35], which was constructed in analogy to the two-particlemodel used to explain single-bunch transverse instability [36]; the results obtainedusing this model, however, are only in qualitative agreement with observations andnumerical simulations. The other problem with the model is that the results dependChapter 4. Longitudinal Stability of Electron Bunches 76on the choice of the two particles. The collective quadrupole mode described in thischapter can be related to the ‘two particles’ in Oide’s model, but here it is uniquelydefined by the stationary distribution.Apparently the collective mode coupling described in this chapter is not the onlyeffect which can cause single-bunch instability. Recently, Oide [311 has proposed anexplanation of resistive wall instability which has a different origin. According tohim, a resistive instability is exhibited by a mode which is unstable by itself. In hisanalysis Oide has shown that electron bunches in a resistive environment will alwaysbe unstable, but the growth rate is quite small at low intensities.The methods described above are based on the assumption that the effects of radiation damping and photon excitation are very small and do not effect the instabilitythresholds. This assumption is often taken for granted, but, as shown by Nagaitsev[41] for a continuous beam and narrow-band impedance, the instability thresholdsfound from the Vlasov and the Fokker-Planck equations are, in fact, different evenwhen the Fokker-Planck diffusion term is very small. Another reason why one shouldnot neglect this term is that the Vlasov equation does not allow a full investigationof the beam dynamics beyond the instability threshold - the topic to be addressed inthe next chapter.Chapter 5Beyond the ThresholdSo far in this thesis we have considered stability of a beam from the point of view ofthe stability of a stationary distribution with respect to an infinitesimal perturbation.Questions remain about what happens to the beam when the threshold intensity isexceeded.In the case of protons, a turbulent bunch lengthening effect is observed, i.e. thebunch distribution alters under the self forces until the distribution becomes stable.In the case of electrons turbulent bunch lengthening is also observed, but it has different features. At the threshold intensity the bunch becomes unstable, but radiationdamping causes the particles to be confined and the instability does not necessarilycause loss of particles. The ‘sawtooth effect’ observed at SLAC SLC damping ring[8], anomalous quadrupole sidebands at CERN LEP [6] and the hysteresis effect atTRISTAN AR [9] indicate that the behaviour beyond threshold can no longer be described in terms of solutions of the Vlasov equation and a more complicated equation(such as the Fokker-Planck equation, which includes radiation damping and diffusioneffects) must be used instead. Moreover, based on some results from plasma physics,one can expect to see different phenomena such as solitons, chaos, etc. [37]. Thesephenomena are extremely difficult to treat analytically, therefore we use a multipartide tracking technique to observe and study the behaviour of the beam. As will beseen, this makes it possible to simulate pathological bunch behaviour which could77Chapter 5. Beyond the Threshold 78underlie the observations at SLAC, CERN and KEK mentioned above.5.1 Numerical SimulationsAnalytical methods usually describe only small perturbations well and are not convenient for dealing with transient processes. Sometimes, however, the perturbations arenot small and analytical methods make incorrect predictions. In these cases particletracking simulations may be the only option for studying the beam dynamics. Almostall the laboratories which are involved in accelerator design have developed computerprograms to study collective effects, in particular longitudinal effects such as bunchlengthening, beam loading, injection, etc.Computing power limits the number of particles that can be followed and thislimits the usefulness of tracking methods for stability analysis. A few years ago thetypical number of particles in simulations ranged from 100-1000, while nowadays itcan be pushed to iO — 106. This breakthrough in computational power makes itpossible to use particle tracking simulations for studying the internal motion in abunch over thousands of turns and allows one to see effects previously masked bystatistical noise.The most impressive results in this field have been obtained at SLAC. Startingwith the pioneering works of P.Wilson [5], studies have been continued by R. Siemannand K.Bane [40]. Siemann [39] has obtained very good agreement with the experimental results obtained at SPEAR by using just 100 macroparticles and calculatingthe forces between every pair of particles. In his recent simulations K. Bane [40] isusing up to 600,000 macroparticles, allowing him not only to get very good estimatesfor thresholds measured for the SLC damping rings, but also to see the structure ofthe unstable mode.Chapter 5. Beyond the Threshold 79To simulate the electron’s motion in a synchrotron we use a standard multi-particletracking scheme [40]. The beam is represented by N macroparticles each with phaseand energy coordinates (z, ) relative to the mean energy and phase. These coordinates are recalculated every turn according to the following equations= + 2ueorj + V4z + 4d(z) (5.131)cxcT0=, ( + (5.132)L/oHere T0 is the revolution period, Te the damping time, 0e0 the rms energy spreadin the absence of a wake, V4 = dVd/dz the rf voltage gradient, € the compactionfactor, and E0 the energy of the particles; r is a random number with a standardnormal distribution which is used to approximate the effects of noise and synchrotronradiation.To calculate lij we have used the same method as Bane [40], i.e. binning themacroparticles in z without smoothing. Then, approximating (2.35) the voltage 4ndinduced by the beam is given by4a(z) —e NkW(zk — z), (5.133)where Nk is the number of particles in the kth bin and W(z) is the Green functionwake field. Other methods of finding T4d [39] give smoother results for a given numberof macroparticles, but require more cpu time.The radiation damping usually takes tens or even hundreds of synchrotron oscillations. However, such long damping times require in general too much cpu timeto simulate easily. Fortunately, the damping rate does not play a significant role ininstabilities which are fast compared with synchrotron motion. This is the regimeof the present study. To optimize computation time versus simulation accuracy, weChapter 5. Beyond the Threshold 80used artificial radiation damping times of the order of 5 to 10 times the synchrotronoscillation period.We found that reasonable accuracy can be achieved with as few as 5,000 macropartides. This depends upon the wake field being fairly smooth: many times moremacroparticles are required for wake fields which have many oscillations in one bunchlength. [40]For the following work we have chosen a resonator wake field with a quality factorQ = 1 and bunch length parameter k0 = 0.5. The radiation damping time Te wasset to 500 turns and the other parameters V, a, E0 in (5.132) have been chosen toobtain a synchrotron period T8 of 100 turns.5.2 Sawtooth instabilityA so-called ‘sawtooth’ instability has been observed in the SLC damping rings [8].After injection the bunch length decreases slowly until a threshold is reached, whenthe length increases sharply (in less then a synchrotron period) and then the processrepeats. This effect has been called a ‘sawtooth instability’ for the sawtooth-likeshape of the plot of bunch length and centre of mass versus time (see Fig. 28). Similarbehaviour has been observed in other electron synchrotrons as well [38].In simulations with a very simple wake field and short bunches, we have observedthat energy spread and bunch length may oscillate in a sawtooth fashion beyonda certain threshold. We find that this is due to the double-peaked nature of thestationary distribution (see Fig. 22). Over many synchrotron oscillations, particlesdiffuse from the head peak to the tail to the point where the tail peak becomes aslarge as the head. The two resulting sub-bunches then collapse together in less thanone synchrotron oscillation, causing a net blow-up in emittance. Radiation dampingChapter 5. Beyond the Threshold 81CH I gndo_zCHI4 A 2.s 230.v? cmriiTime(ms)Figure 28: Sawtooth instability in bunch length observed at the SLC damping ring[40].__•••••___I - I IIfMP1i__________liii IiiI IIIIIIIIitii t.tii.iiIiiiiI,ii,Ii•t.. Injection- ExtractionI I I p p IChapter 5. Beyond the ThresholdFigure 29: Sawtooth instability observed in the simulations.82then gradually lowers the emittance and diffusion begins again (see Fig. 29 and 34).Rather than attempting precise simulations of an actual machine, as was done byBane and Oide [40], we have simplified the model to determine which features of thewake field lead to sawtooth behaviour.We start with a large emittance and allow the beam to damp. At low intensities the beam relaxes to a thermodynamically stationary distribution which is welldescribed by the Haissinski equation [2]. When the intensity increases, however, ittakes more time for particles to reach thermodynamic equilibrium, particularly whenthe distribution has a two-peak line-density profile.The ‘two peak’ distributions can be easily found from the Haissinski equation[2], but not until recently have they been observed experimentally by using a streakcamera (Fig. 16) [6]. In the case k0 = 0.5 the second peak in the line density appears0’10 10turn—10—6 —2 2 6q10turnChapter 5. Beyond the Threshold 8354b32Figure 30: RMS bunch length (a) and rms energy spread (b) in the case of resonatorimpedance (Q = 1, ko = 0.5) at I = 30. Radiation damping time is Te = 5T8.approximately at I = 10. This is near the stability threshold found by solving theVlasov equation (see Fig. 27).The region close to threshold is difficult to model because of the slow growth rateof the instability. Above approximately I = 20, the sawtooth instability becomesapparent. As intensity is raised, the sawtooth periodicity also increases. A typicalexample showing rms bunch length and energy spread is shown in Fig.30 for I 30.At very high intensity, the behaviour becomes irregular: the case of I = 45 is shownin Fig.31.We have found that the sawtooth repetition rate is mainly determined by thediffusion process and not by radiation damping. To illustrate this point, the case of10/3 times stronger radiation damping is shown in Fig. 32. Comparing Fig. 32 withFig. 30, one can see that the sawtooth frequency has not changed.A complete cycle corresponding to one ‘tooth’ is shown in Fig. 33:• a —* b: The trailing bunch damps down (about 5 synchrotron oscillations).• b —* C: Diffusion populates the trailing bunch until it is approximately equal tothe first (about 30 synchrotron oscillations). Note that the two bunches haveChapter 5. Beyond the Threshold 84.4 10turn/iD3I I5.4.2-1—0-CaFigure 31: RMS bunch length and energy spread for the same parameters as Fig. 30,except that I = 45.turn/iD3Figure 32: RMS bunch length and energy spread for an increased damping rate:I = 30 and Te = 1.5T3 Compare with Fig. 30.Chapter 5. Beyond the Threshold 85• dCFigure 33: A complete cycle of the sawtooth instability in phase space for the caseshown in Fig. 30: I = 30 and Te = 5T. The time sequence is anticlockwise.started to move towards each other and a third is already beginning to form.• c —÷ d In about 1/3 of a synchrotron period the two main bunches collapsetogether.• d —* a The combined bunch throws out a large cloud of particles as it executeslarge synchrotron oscillations (less than a synchrotron period).A more detailed picture of the diffusion process and collapse is shown in Fig. 34.The sawtooth behaviour was most clearly seen in the region 0.4 < ko < 0.6. Fork0 < 0.4, the diffusion process was too slow. For k0 > 0.6, where the thresholdintensity increases with bunch length (Fig. 27), sawtooth behaviour is not seen either;instead, the bunch length oscillates chaotically.a4• 4., •b4Chapter 5. Beyond the Threshold 86j 4:444i•1:“.-..4 I2.c4:f,aFigure 34: Dynamics of the sawtooth instability. Each frame shows the particle distribution in phase space and the potential well. The snapshots on the left illustrate thediffusion process, showing every lOOt” turn; those on the right the collapse, showingevery 10th turn.trriI:Nit[iI1N.[iN/1.phase space potential phase space potentialChapter 5. Beyond the Threshold 875.2.1 Analysis of sawtooth instabilityQualitatively, the instability can be understood by considering the wake of an extremely short bunch (Fig. 35, upper). In order for the energy lost by the bunch tothe wake field to be compensated by the rf cavities, the rf waveform (here drawn asa straight line, since the rf wavelength is much larger than the bunch length) mustintersect the wake voltage at half the maximum1.This is the location of the centre ofthis very short bunch, and is of course a stable fixed point. Situations for various rfvoltage values can be considered by pivoting the rf waveform (line) about this point,as indicated in Fig. 35. Situations with differing beam intensities can be simulated inthe same way, since amplifying the wake field has the same effect on the diagram asreducing the rf slope. At low intensity or large rf voltage, there is only the one fixedpoint. At high intensity or low rf voltage, the wake field intersects the rf waveformat three points; there is an unstable fixed point behind the bunch, and a stable onefarther along. The separatrices created by the extra fixed points are also shown inFig. 35.Because of the random excitation due to emission of synchrotron radiation, particles can diffuse through the unstable fixed point and collect at the downstream stablefixed point. These particles begin to create their own wakes, and will move forwardas they lose energy to their own wake field. At the same time, the remaining particles in the leading bunch will move backwards as they decrease in number and nolonger need a large energy gain from the rf fields. At some point, the potential barrierbetween the two sub-bunches becomes small enough that the diffusion turns into anavalanche and the sub-bunches suddenly coalesce. The resulting bunch is over-denseis due to the fact that the wake field is non-zero only behind the particle and if one considersinteraction between each pair of particles, every particle spends on average half its time behind theother particle, which means that the rf voltage has to be only one half of the induced wake voltageto compensate the energy loss.Chapter 5. Beyond the Threshold 88and at the wrong phase with respect to the needed energy gain. It begins to executea large synchrotron oscillation, while beginning to lose particles to diffusion again.This results in a large cloud of particles and a large rms bunch length and energyspread. The cloud condenses again at the downstream stable fixed point and diffusioncontinues.5.3 ‘Binary star’ instabilityThe sawtooth instability we have described in the previous section is a fast instability- it develops in a fraction of a synchrotron period and therefore little attention waspaid to radiation damping time; however, in order to observe weaker instabilities withslower growth rates in simulations one should have a realistic damping time (usuallytens or hundreds of synchrotron periods).For this simulation we have chosen rd = 20T3 and use the same wake field as above(k0 = 0.5), but a lower intensity, I = 20. Previously, when we had Td = 5T we sawjust chaotic oscillations of the bunch parameters, but now the picture is completelydifferent: the bunch splits into two identical sub-bunches and they oscillate shiftedapproximately 1800 to each other (see Figs. 36 and 37), their motion in phase spaceresembling that of a binary star. It can be seen that the intensity at which thisphenomenon is observed is more than twice the threshold intensity for this wake (cf.Fig. 27)It is not surprising that the spectrum of this signal has a strong line at approximately twice the synchrotron frequency (see Fig. 38), but it is interesting to note thatat some intensity two sidebands appear.Low frequency sidebands around the quadrupole frequency have been observed atCERN LEP [6] (see Fig. 39). It was suggested in [6] that these lines may correspondChapter 5. Beyond the Threshold 891.51.00.50.0—0.5Figure 35: Above: Green function wake field with three rf waveform slopes; (a) isstable, (b) is just above threshold, and (c) is in the sawtooth regime. Looking fromleft to right, there is a stable fixed point if the wake field crosses the rf waveform frombelow, and an unstable fixed point if it crosses from above. Below: the separatricescreated by the wake fields corresponding to cases (b) and (c). In case (a), there isonly one stable fixed point so the wake field does not create a separatrix. Note thatthese curves are for a Green function wake and therefore are only suggestive. Anyaccumulation of finite charge density will deform the separatrices.a-2Chapter 5. Beyond the Threshold 904 I I I —2—2—4 1111111 I— —8 —6 —4 —2 0 2 4 6 100qFigure 36: Simulation of the ‘binary star’ instability.to radial modes predicted by conventional theory derived from the linearized Vlasovequation; however, the quadrupole oscillations observed at LEP had a very largeamplitude (more then 30 percent of the bunch length, see Fig. 40) and therefore it isunlikely that these lines can be associated with any modes found using a perturbationformalism.Though the ‘binary star’ effect observed in the simulations has some qualitativefeatures of the behaviour observed at CERN, no quantitative agreement has beenfound. There are several reasons for this: first, all these effects depend stronglyon the shape of the wake function, and second, in order not to have the results ofthe numerical simulations contaminated by noise associated with a small number ofparticles, the total number of macroparticles needs to be increased significantly (from5,000 to perhaps 100,000 or even 1,000,000)- a major computational constraint.Oide once suggested a model in which the bunch is represented by two identical5000 6000 7000 80)0 9000 iodoo iicooturnq• • •• I.-.5000 6000 7000 8000 9000 )O00turnChapter 5. Beyond the Threshold 91.00100 5 —r_•0 ..{‘.&‘:°0 505o0 0:J.—A:______:°°::_0 5 0longitudinal profile phase spaceFigure 37: Snapshots of the phase-space distributions for the ‘binary star’ instabilityafter each 10 turns ( 1/10 of a synchrotron period).Chapter 5. Beyond the Threshold 92.20.16rJ).08.04.00Figure 38: Spectrum S, of the rms momentum spread (o,) during the ‘binary star’instability. The quadrupole mode has the largest strength.macroparticles which oscillate in each other’s field [35]. It is tempting to use thismodel to describe the ‘binary star’ behaviour, but unfortunately, such a straightforward procedure does not work for the following reason. In the case of linear externalforces one can rewrite the equation of motion for separate macroparticles in two equations in new coordinates, one describing the centre of mass motion and the other therelative motion q1—q. The equation describing the relative motion will not dependon the centre of mass coordinate, but since there is radiation damping q—q —* 0eventually. We therefore conclude that nonlinear phenomena, such as turbulence,play a significant role in this instability.R. Meller has proposed a thermodynamical explanation of the longitudinal singlebunch instability threshold [421. He suggests that at some intensity the stationaryIsOChapter 5. Beyond the Threshold 93X—1.4kHZ 7Ya——B7.434 d8VrmsPOWER SPECI iOAvg — O%Ovp_Han_roV-48.0——ir— ZZ_L 1 I8.0f82IDly____——---_;—112 i—- __1__ -1.4k Hz 2.4kFigure 39: Spectrum of a quadrupole mode observed at LEP [6]. One can see lowfrequency sidebands which become stronger at higher intensities.Bunch length (psec)“‘-‘ I0 p0 100 200 300 400 500Bunch current iA)Figure 40: Bunch length versus intensity at LEP[6].Chapter 5. Beyond the Threshold 94distribution becomes energetically less favourable than an oscillatory solution. A similar idea was later tried by K. Yokoya [43], who also suggested that at some intensitya dynamical solution of the Fokker-Planck equation should be more favourable thana stationary, time-independent one.5.4 ConclusionWe have developed a qualitative picture of the sawtooth instability. The wake fieldcreates its own unstable and stable fixed points, particles diffuse to the second fixedpoint, and then the resulting sub-bunch collapses into the leading sub-bunch. Thesawtooth frequency is therefore primarily determined not by radiation damping, butby the subsequent diffusion process.The sawtooth effect is most readily seen when the bunch length is comparable withthe wake field length. Qualitatively quite different behaviours can be seen when thebunch is either short or long compared with the wake. In the former case, for example, the bunch may experience either sawtooth instability or ‘binary star’ instability.These regimes, as well as various types of wake fields, are still under investigation.Rigorous analysis of the behaviour of bunched beams beyond threshold is a verybroad and complicated subject which is outside the scope of this thesis; this subjectis, probably, a fruitful arena for future developments in instability theory.Chapter 6Summary and ConclusionsIn this thesis we have studied the effects of short-range wake fields on the longitudinalmotion of protons and electrons in synchrotrons. The main purpose of the thesis hasbeen to show that the potential well distortion caused by short range wake fieldshas a significant effect on beam stability and cannot be ignored or approximatedbut must always be taken into account. We have shown that the results obtainedby different techniques where the analysis is not done self-consistently can lead toincorrect conclusions about the beam’s stability.We have studied two major effects of short-range forces on a single bunch:1. Effect of potential well distortion - deformation of stationary distributions causedby self-forces;2. Stability of self-consistent stationary distributions.Due to the fact that there are usually 1010— lO’’ particles in a bunch it is possibleto describe collective effects in the bunch using the Vlasov equation, and investigatethe behaviour of the distribution function rather than individual particles.We have investigated two major cases where short-range forces play an importantrole• longitudinal space-charge forces, which are proportional to the derivative of theline density, for bunches with an arbitrary distribution of particles. This is a95Chapter 6. Summary and Conclusions 96case important in proton synchrotrons, for which the bunch length is quite bigcompared to the size of the vacuum pipe, so the self forces are local, radiationeffects are very small and there is no constraint on the distribution function.• effect of longitudinal short-range wake fields for bunches with a Maxwell- Boltzmann distribution. This case is typical for ultrarelativistic electrons in storagerings. The bunch length in this case is comparable to the characteristic lengthof the wake field.6.1 Longitudinal Space-Charge ForcesIn the case of protons the bunch length is large compared to the vacuum pipe diameterand the short-range forces are approximately proportional to the derivative of the linedensity. We have found a way to determine stationary self-consistent distributionsand have also derived a simple criterion which allows the intensity threshold to befound beyond which there is no stationary distribution. We have applied this methodto the analysis of various distributions and obtained the following results:• In the case of the distribution j’ for which &‘(H) <0 a stationary distributionalways exists below the transition energy, but above transition there is alwaysa threshold beyond which no stationary distribution can be found.• For any distribution for which b(0) = 0 stationarity thresholds exist below aswell as above transition.We have developed an algorithm to recover the distributioll function from a knownline density in the presence of space charge. Different variations of this algorithm allowthe synchrotron frequency to be determined from a given stationary distribution andvice versa.Chapter 6. Summary and Conclusions 97We have also found the threshold for instability caused by coupling of m = +1azimuthal modes. Surprisingly, the criterion for this threshold is identical to that forthe stationarity threshold.6.2 Longitudinal Stability of Electron BunchesA typical bunch length in the case of electrons in a storage ring is comparable to thediameter of the beam pipe and therefore the wake field has a characteristic lengthcomparable with the bunch’s length. Bunch lengthening caused by short-range wakeshas been studied using the Haissinski equation, which describes a stationary distribution of ultrarelativistic particles in the presence of synchrotron radiation.The potential well distortion due to self forces creates a spread in the synchrotronfrequencies of individual particles in a bunch and has an important effect on thestability of the beam. We have found that in some cases the synchrotron frequencyhas a local minimum, i.e dw/dJ = 0, and that the stability of the bunch dependsessentially on the parameters of the distribution in this region.The existence of frequency spread makes it impossible to apply the standard stability analysis based on orthogonal polynomials, which assumes that the frequency ofsynchrotron oscillations does not depend on the amplitude; the threshold obtained inthat approximation differs significantly from those found self-consistently.We have derived an integral equation which allows the stability analysis to beformulated as an eigenvalue problem. We have shown that a numerical method forsolving the Vlasov equation proposed by Oide and Yokoya [25] can be derived fromthis equation and is, in fact, one of the ways to solve it.We have also presented an analysis of the different modes which are obtained bythe Oide-Yokoya method and found that below threshold there exist two types ofChapter 6. Summary and Conclusions 98solutions:• A discrete spectrum of collective modes, such as the rigid dipole mode and themodes which are concentrated in the region where the derivative of synchrotronfrequency dw/dJ = 0,• A continuous spectrum of ‘incoherent’, &like modes which cannot be seen individually.Analyzing the behaviour of the collective modes in the case of broad-band resonatorimpedance we have come to the conclusion that instability can be caused by couplingbetween the collective modes such as the rigid dipole mode and the modes concentrated in the dw/dJ = 0 region where the synchrotron frequency has its minimum.The mode which couples with a rigid dipole mode first is a quadrupole mode whoseeigenfrequency 2w and therefore a threshold criterion can be written in the form= w80, where‘sO is a rigid dipole mode frequency which does not depend onintensity. Coupling between other collective modes can happen only when w(J) —* 0,which corresponds to a criterion proposed by P. Wilson [5] which states that the instability will happen at the point where V’(q) = 0; this, however, occurs at higherintensities than the coupling between the quadrupole and rigid dipole modes.This hypothesis is in agreement with particle tracking simulations, and directnumerical solution of the Vlasov equation using the Oide-Yokoya technique, done fora broad-band resonator impedance over a wide range of resonator parameters. Theinstability caused by this coupling is very fast and easily seen in simulations.Chapter 6. Summary and Conclusions 996.3 Beyond the ThresholdA computer program developed for checking the threshold criteria for electron buncheshas been used to investigate electron dynamics beyond threshold. It has been foundthat the oscillations of the bunch shape have different features depending on theparameters of the wake field. Turbulent bunch lengthening and widening (increasedenergy spread) have been observed above the stability threshold.Some interesting phenomena have been seen when the bunch length is smallerthan that of the wake field. At some intensities the bunch length oscillates in sawtoothfashion, i.e. slow damping is followed by fast blow-up and then the process repeats.It was also found that the bunch may split into two equal sub-bunches which oscillatearound each other in binary star fashion.The effects observed in these simulations exhibit features of some experimentalobservations at SLAC and CERN and indicate that strong nonlinear effects do playa significant role in existing accelerators. This means that the Vlasov equation whichis often used for stability analysis nowadays may not be sufficient to describe thesephenomena and different methods need to be developed.Bibliography[1] B. Zotter, Short is Beautiful, Proc. 4th Advanced ICFA Beam Dynamics Workshop, KEK Report 90-21 (1990).[2] J. Haissinski, Exact Longitudinal Equilibrium Distribution of Stored Electrons inthe Presence of Self-Fields, Nuovo Cimento 18B, 72 (1973).[3] P. Germain and H. Hereward, Longitudinal Equilibrium Shape for ElectronBunches with Various Self-Fields, CERN/MPS/DL 75-5 (1975).[4] B. Zotter, A Review of Self-Consistent Integral Equations for the Stationary Distribution in Electron Bunches, Proc. 4th Advanced ICFA Beam Dynamics Workshop, KEK Report 90-21 (1990).[5] P. Wilson, R. Servranckx, A.P. Sabersky, J. Gareyte et al, Bunch Lengtheningand Related Effects in SPEAR II, Trans. IEEE NS-24, 1211 (1977).[6] D. Brandt, K. Cornelis and A. Hoffmann, Experimental Observations of Instabilities in the Frequency Domain at LEP, Proc. 3’’ Europ. Part. Acc. Conf., Berlin,345 (Editions Frontières, 1992); see also CERN/LEP Report MD-35 (1990).[7] K. Bane, R. Siemann et al., High Intensity Single Bunch Instability Behaviour inThe New SLC Damping Ring Vacuum Chamber, Proc. 1995 Part. Acc. Conf.,Dallas TX, (in press).[8] P. Krjcick, K. Bane, P. Corredoura F.J. Decker et al, High Intensity Bunch LengthInstabilities in the SLC Damping Ring, Proc. 1993 Part. Acc. Conf., WashingtonD.C., 3240 (IEEE, 1993).[9] T. leiri, Bunch Lengthening Observed Using Real- Time Bunch-Length Monitor inthe TRISTAN AR, Proc. 1993 Part. Acc. Conf., Washington D.C., 3333 (IEEE,1993).[10] D. Boussard, CERN Div. Report MPS/DL-75/5 (1975).[11] E. Keil and W. Schnell, CERN/ISR Div. Report 69-48 (1969).[12] S. Krinsky, J.M. Wang, Longitudinal Instability of Bunched Beams Subject to aNon-Harmonic RF Potential, Particle Accelerators 17, 109 (1985).100Bibliography 101[13] F. Sacherer, Bunch Lengthening and Microwave Instability, Trans. IEEE NS-24,1393 (1977).[14] G. Besnier, Contribution a la Theorie de la Stabilite’ des Oscillations Longitudinales d’un Faisceau Accelère en Re’gime de Charge d’Espace, Ph. D. thesis(B-282-168) Université de Rennes, France (1978).[15] D. Neuffer, Stability of a Self Consistent Longitudinal Phase-Space Distributionunder Space Charge Perturbations, Particle Accelerators, 11, 23 (1980).[16] G. Besnier and B. Zotter, Oscillations Longitudinales d’une Distribution Elliptique, Couple’es par un Resonateur: Application au Calcul de l’Allongement deFaisceaux Intenses, CERN-ISR-TH/82-17 (1982).[17] A. Hofmann and F. Pedersen, Bunches with Local Elliptic Distributions, IEEETrans. NS-26, 3526 (1979).[18] I. Kapchinsky and V. Vladimirsky, Limitations of Proton Beam Current in aStrong Focusing Linear Accelerator Associated with the Beam Space Charge,Proc. Conf. High Energy Acc. and Instr., CERN, 274 (1959).[19] R. Baartman, Stationary Longitudinal Phase Space Distributions with SpaceCharge, Proc. 1991 Part. Acc. Conf., San Francisco, 1731 (IEEE, 1991).[20] M. D’yachkov and R. Baartman, Methods for Finding Stationary LongitudinalDistributions, Proc. XV-th Tnt. High Energy Acc. Conf., Hamburg, Tnt. J. Mod.Phys. A (Proc. Suppi.) 2, 1064 (1992).[21] P.W. Krempl, The Abel-type integral transformation with the kernel (t2 — x2)_h/2and its application to density distributions of particle beams, CERN MPS/Int.Br/74-1 (1974).[22] G. Rybkin, Longitudinal Stationary Distribution and its Connection with LineDensity of Particles in Proton Synchrotrons, INR Report 733/91 (1991).[23] R. Baartman and B. Zotter, Longitudinal Stability of Hollow Beams II: ModeCoupling Theory, TRIUMF Note TRI-DN-91-K177 (1991).[24] R. Baartman, S. Koscielniak, Stability of Hollow Beams in Longitudinal PhaseSpace, Particle Accelerators, 28, 95 (1990).[25] K. Oide and K. Yokoya, Longitudinal Single-Bunch Instability in Electron StorageRings, KEK Report 90-10 (1990).Bibliography 102[26] M. D’yachkov and R. Baartman, Method for Finding Bunched Beam InstabilityThresholds, Proc. 4th Europ. Part. Acc. Conf., London, 1075 (World Scientific,1994).[27] S. Fang, K. Oide, Y. Yokoya, B. Chen and J.Q. Wang, Microwave Instabilities inElectron Rings with Negative Momentum Compaction Factor, KEK Report 94-190 (1994).[28] R. Baartman and M. D’yachkov, Computation of Longitudinal Bunched Beam Instability Thresholds, Proc. 1993 Part. Acc. Conf., Washington D.C., 3225 (IEEE,1993).[29] A.V.Burov, Bunch Lengthening - Is It Inevitable?, Part. Acc., 28, 525 (1990).[30] A. Ruggiero, Theory of Longitdinal Instability for Bunched Electron and ProtonBeams, Trans. IEEE NS-24, 1205 (1977).[31] K. Oide, A Mechanism of Longitudinal Single-Bunch Instability in Storage Rings,KEK Report 94-138 (1994).[32] Y. Chin, K. Satoh, K. Yokoya, Instability of a Bunched Beam with SynchrotronFrequency Spread, Particle Accelerators 13, 45 (1983).[33] N.G. Van Kampen, On the Theory of Stationary Waves in Plasma, Physica 21,949 (1955); also K.M. Case, Plasma Oscillations, Ann. Phys. 7, 349 (1959).[34] L. Landau, On the Vibrations of the Electronic Plasma, J. Phys. USSR 10, 25,(1946); also in Collected Papers, Pergamon, Oxford, 445 (1960).[35] K. Oide, Two Particle Model, private communication (1990).[36] R.D. Kohaupt, DESY Report M-80/19 (1980).[37] J.J. Bisognano, Solitons and Particle Beams, AlP Conf. Proc. 253, 42 (1992).[38] 0. Rakovsky and L.R. Hughey, SURF’s up at NBS: a Progress Report, Trans.IEEE NS-26, 3845 (1979).[39] R. Siemann, Computer Simulations of Bunch Lengthening in SPEAR, Nuci. Instr. Meth., 203, 57 (1982).[40] K. Bane and K. Oide, Simulations of the Longitudinal Instability in the SLCDamping Ring, Proc. 1993 Part Acc. Conf., Washington D.C., 3339 (IEEE, 1993).[41] 5. Nagaitsev, On the Longitudinal Stability of Cooled Coasting Ion Beams, Proc1993 Part. Acc. Conf., Washington D.C., 3524 (IEEE, 1993).Bibliography 103[42] R. Meller, Thermodynamic Mechanism for Bunch Lengthening, Proc. 1987 Part.Acc. Conf., 1155 (IEEE, 1987).[43] K. Yokoya, private communication (1994).Appendix ADistribution FunctionsA.1 Binomial FamilyAnalytical solutions for U(q) can be found for the family of binomial distributionsH m-1/2f(H) oc [i—(A.134)One can easily calculate A(U) in this case:1 [1_Jm_2dA(U) = 2J m (A.135)U 2(-U)and soA(U) Lm [i — U]tm (A.136)whereL -/27rmP(m+1/2) A137m I’(l+m)Writing Im = lLm, the equation for the potential well becomesU = Uo(q) + Im ([i—ujm— i) (A.138)It is easy to see that in the case of Uo(q) = q2/2 we can find the solution of (A.138)in the form q(U) for any mq= i2[U_Im ([U],n)] (A.139)104Appendix A. Distribution Functions 105We also can find the solution in the formU(q) = g[Uo(q)]. (A.140)The analytical solutions for U(q) can be easily found for m = 1, 2, 3 and 4.In this section we assume Uo = q2/2, but one can choose any potential instead.A.1.1 Case m = 1: Hofmann-Pedersen DistributionIn this case A(U) is a linear functionA(U) = L1(1— U) (A.141)andU=-+I1[(1—U)—1] (A.142)U(q)= 2(1+ I) (A.143)A.1.2 Case m = 2A(U) = L2 [i — U]2 (A.144)andU(q) =- + ‘2 ([i— (1]2— 1) (A.145)U(q) = 2(1 + 12) — /4(i +12)2— 2I2q (A.146)A.1.3 Case m = 3A(U)=[1— ] . (A.147)An expression for U(q) can be found by using the Mathematica program by executing the following scriptAppendix A. Distribution Functions 106Solve[U == q2/2 + 13 ((1- U/3Y3 - 1), U] II TeXForm;U I. ‘/.C[3]]Since the equation we have to solve is of the 3 order, there will be 3 solutions; twoof them are non-physical, and therefore the result for 13 < 0 is272k (i_)U(q) = 3 +(11664 I — 17496 132 (1 + 13) + 2916 1 q2 + 136O48896 I3 + (11664 I — 17496 i2 (1 + 13) + 2916 I q2)2)— (i + ) (11664 1 —174964 (1 + 13) + 29164 q2 + 13:o48896 4 + (116644 174964 (1 + 13) + 29164 q2)2)1223 13A similar result can be obtained for m = 4, but it is too lengthy to record here.A.2 Hollow DistributionsAnalytical solutions for U(q) can also be found for the ‘hollow distributions’ whichcan be derived from the binomial familyf(H) cc H’(1 — H)m_h/2 (A.148)where rn> 0 and n > 0 are integers and m + n 4. As example, let us consider thecasef(H) = -H\/1- H (A.149)for whichA(U) = 1 + 2 U — 3 U2 (A.150)and the solution for U(q) is1_21_\/(1_21)2±61q2U(q)= 61(A.151)Appendix A. Distribution Functions 107Analytical solutions in the form q(U) can be found for any integers m, n, and alsofor the following family of distributionsf(H) cx (A.152)where m is an integer. For example:f(H) (A.153)for whichA(U) = (1 + 2 U) e_U (A.154)and the solutionq(U) = /2 [u — I((i + 2U)e_U —1)]. (A.155)A.3 ‘Thermal’ DistributionThe solution for ‘thermal’ distributionf(H) = ‘eH (A.156)for whichA(U) = (A.157)isq(U) = 2 [U — l(e_U — 1)]. (A.158)

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