LONGITUDINAL INSTABILITIES OF BUNCHED BEAMS CAUSED BY SHORT-RANGE WAKE FIELDS By Mikhail D’yachkov M.Sc, Moscow State University, Russia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1995 © Mikhail D’yachkov, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2188) i/Ie’, Abstract This thesis investigates the effects of short-range wake fields on the collective lon gitudinal motion of charged particle bunches in circular accelerators, especially the onset of instability. At high intensity, a short-range wake field can distort the bunch potential well and thereby change the stationary distribution. It is shown that if this is not taken into account, instability thresholds will be incorrectly predicted. An integral equation derived from the linearized Vlasov equation is used to find the instability thresholds in the case of space-charge impedance alone for various distribution functions. The thresholds for instability caused by the coupling between the m ±1 azimuthal modes have been obtained analytically for several common distributions. The criterion determining these thresholds appears to be the same as that for thresholds beyond which no stationary distribution can be found. A numerical method is also used to solve the linearized Vlasov equation for the self-consistent case, including distortion of the stationary distribution, and to find the thresholds. Physical explanations are provided for the eigenmodes and instability thresholds predicted by this method. This results in a much simpler stability criterion, which depends only on the stationary distribution and does not require solution of the linearized Vlasov equation. The behaviour of electron bunches beyond the instability threshold has also been studied using multiparticle tracking. Some interesting phenomena have been seen when the bunch length is smaller than that of the wake field. Under some conditions the bunch length oscillates in sawtooth fashion, i.e. slow relaxation is followed by fast U blow-up. It has also been found that the bunch may split into two equal sub-bunches which oscillate around each other in binary star fashion. These effects may explain some recent observations in electron storage rings at SLAC and CERN. 111 Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgements xiii xiv 1 2 Introduction 1 1.1 6 Outline of the Thesis Essentials of Longitudinal Motion 8 2.1 8 Equations of Longitudinal Motion 2.1.1 2.2 2.3 Phase Focusing 11 Induced Forces 14 2.2.1 The Longitudinal Space Charge Field 15 2.2.2 Wake potential 18 2.2.3 Impedance 20 Collective Effects 20 2.3.1 The Vlasov Equation 22 2.3.2 Potential Well Distortion 22 2.3.3 Stability Analysis 23 iv 3 Collective Modes 2.3.5 Turbulent Bunch Lengthening . 24 25 Longitudinal Stability under Space-Charge Forces 27 3.1 27 3.2 3.3 4 2.3.4 Stationary Self-Consistent Distribution 3.1.1 Threshold 32 3.1.2 Discussion 34 3.1.3 Recovering the Distribution Function 35 Stability 38 3.2.1 Integral Equation 38 3.2.2 Comparison with Mode-Coupling Theory 42 3.2.3 Self-Consistent Case 46 Summary 48 3.3.1 Stationary Distribution 48 3.3.2 Stability 48 Longitudinal Stability of Electron Bunches 49 4.1 HaIssinski Equation 50 4.2 Stability 51 4.2.1 Integral Equation 51 4.2.2 Rigid Dipole Mode 54 4.2.3 Analysis of the Integral Equation 54 4.2.4 Oide-Yokoya Method 55 4.3 Results 57 4.3.1 Inductive Wake 57 4.3.2 Capacitive Wake 57 4.3.3 Resistive Wake 63 V 4.4 5 6 4.3.4 Broad-band Resonator Wake 4.3.5 New Criterion ..,,., 64 . 70 . Conclusions 4.4.1 Oide-Yokoya Method 73 4.4.2 Landau damping 74 4.4.3 New Criterion 75 Beyond the Threshold 77 5.1 Numerical Simulations 78 5.2 Sawtooth instability 80 5.2.1 87 Analysis of sawtooth instability 5.3 ‘Binary star’ instability 88 5.4 Conclusion 94 Summary and Conclusions 95 6.1 Longitudinal Space-Charge Forces 96 6.2 Longitudinal Stability of Electron Bunches 97 6.3 Beyond the Threshold 99 Bibliography 100 A Distribution Functions 104 A.1 Binomial Family 104 A.1.1 Case m = 1: Hofmann-Pedersen Distribution 105 A.1.2 Case m = 2 105 A.1.3 Case m = 3 105 A.2 Hollow Distributions 106 A.3 ‘Thermal’ Distribution . vi 107 List of Tables 1 Threshold parameters for different distribution functions vii 33 List of Figures 1 The coordinate system of a circular accelerator 9 2 Particle trajectories in the (q, W) phase plane 12 3 Circular beam in a cylindrical pipe 16 4 Two charged particles travelling along the axis in an accelerator. 5 Different modes of oscillations in a bunched beam [13] 6 Line densities (top) and density plots (bottom) of stationary distribu tions for the gaussian case at intensities (left to right) I . . 19 25 = -1.552, 0 and 20. The left-most plot is at threshold: for I < -1.552, no stationary distributions exist [19] 7 29 Line densities (top) and density plots (bottom) of stationary distri butions for the ‘hollow’ distribution f(H) intensities (left to right) I = = exp[—(H — /2] at 2 Hb) -8, 0 and 20. The left-most plot corre sponds to negative mass and the right-most to positive mass thresholds. 29 8 The dependence of I on the parameter t for the binomial distributions 1 H— H ‘\‘ +1/2) 9 Line densities at the ‘negative mass’ threshold calculated by using (3.68) 34 10 Line densities at the ‘positive mass’ threshold calculated by using (3.68) 34 11 Coherent mode frequencies as a function of the intensity for a hollow beam distribution. Matrix size: 40 x 40 viii 40 12 Family of the distribution functions given by (3.99) for different values of the parameter c 13 43 Extreme eigenvalues of equation (3.100) (plotted as squares) and ex treme values of A’(U) (solid lines) 14 45 The eigenfunctions found by solving the matrix equation using the mode coupling method, corresponding to (a) negative mass threshold and (b) positive mass threshold 15 0 vs. q at threshold intensities for the ‘hollow-gaussian’ distribution. U U 16 46 = q / 2 2 is shown by the dashed curve 47 Snapshots of the longitudinal profile of the beam at LEP made with a streak camera. Horizontal ellipses show the peak positions of earlier pulses 17 52 The beam profiles at different intensities found from the HaIssinski equation for a capacitive wake field above transition 18 58 Bunch oscillation frequencies in the case of purely capacitive impedance, plotted for discrete intensities I. The upper plot shows eigenfrequen cies [Lk calculated by the O+Y method, and the lower the incoherent frequencies mw(J), where m is an integer. The eigenmode with fre quency independent of intensity is the rigid dipole mode 19 60 Comparison of the theoretical rigid dipole mode (4.120) (continuous curve) with the eigenvector of the u = 1 mode (D). The calculation isfori=120,r=3 20 61 RMS value of the kth eigenvector versus its eigenvalue at intensity I = 2, in the case of purely capacitive impedance with ñ = 120 = 40 (x) and 62 (0) ix 21 Stationary longitudinal profiles at different intensities for a resistive wake 22 64 0 Resonator impedance with k = Q 0.6 and = 1: (a) the wake field and (b) longitudinal beam profiles at different intensities 23 Eigenfrequencies versus intensity for a broad-band resonator = 24 66 (Q = 1, 0.6). Unstable modes are indicated by solid symbols 67 RMS value of the ktI eigenvector versus its eigenvalue at intensity I = 3 in the case of a broad-band resonator with Q different matrix sizes: i 120 (1). Note that there = 40 (x) and t = = 1, k 0 = 0.6, for two are two modes whose X is independent of ñ; these correspond to a quadrupole and a sextupole mode respectively, at the location where w(J) is a minimum 25 69 Normalized synchrotron frequency versus action J and intensity I for the resonator impedance with 26 Q = 1, k 0 = 0.6 70 Comparison of the eigenvector for the ‘coherent’ quadrupole mode at the synchrotron frequency minimum (j = j) 2 and a nearby ‘in coherent’ quadrupole mode, for two different matrix sizes: ñ (continuous curve) and ñ 27 = = 120 40 (dashed curve) 71 Comparison of thresholds obtained by different methods: new method (continuous curve), O+Y method (dashed curve) and numerical track ing (symbols). The latter two cases are taken from O+Y[25] 28 29 72 Sawtooth instability in bunch length observed at the SLC damping ring [40] 81 Sawtooth instability observed in the simulations 82 x 30 RMS bunch length (a) and rms energy spread (b) in the case of res onator impedance time is 31 Te = (Q 0.5) at I = 30. Radiation damping 83 RMS bunch length and energy spread for the same parameters as = 45 84 RMS bunch length and energy spread for an increased damping rate: I 33 = 8 5T Fig. 30, except that I 32 1, ko = = 30 and Te = . Compare with Fig. 30 8 1.5T 84 A complete cycle of the sawtooth instability in phase space for the case shown in Fig. 30: I = 30 and Te = . The time sequence is 8 ST anticlockwise 34 85 Dynamics of the sawtooth instability. Each frame shows the particle distribution in phase space and the potential well. The snapshots on the left illustrate the diffusion process, showing every on the right the collapse, showing every 35 th 100 turn; those turn 86 Above: Green function wake field with three rf waveform slopes; (a) is stable, (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 wake field crosses the rf waveform from below, and an unstable fixed point if it crosses from above. Below: the separatrices created by the wake fields corresponding to cases (b) and (c). In case (a), there is only one stable fixed point so the wake field does not create a separatrix. Note that these curves are for a Green function wake and therefore are only suggestive. Any accumulation of finite charge density will deform the 36 separatrices 89 Simulation of the ‘binary star’ instability 90 xi 37 Snapshots of the phase-space distributions for the ‘binary star’ insta bility after each 10 turns 38 ( 1/10 of a synchrotron period) 91 Spectrum S, of the rms momentum spread (un) during the ‘binary star’ instability. The quadrupole mode has the largest strength 39 Spectrum of a quadrupole mode observed at LEP [6]. One can see low frequency sidebands which become stronger at higher intensities. 40 92 Bunch length versus intensity at LEP[6] xii . . 93 93 Acknowledgements I am very grateful to Michael Craddock and Richard Baartman, without whom this thesis would have never been completed, for their guidance and support during my work on the thesis. I would also like to thank Shane Koscielniak and Roger Servranckx for many useful discussions, Fred Jones for his help with computers at TRIUMF, and Frank Curzon and Malcolm McMillan for their attention to my research and for many valuable comments on the thesis. My special thanks go to Pat Stewart for her help and support during my stay at TRIUMF. Finally I want to express my deepest gratitude to Erich and Barbara Vogt, George and Olga Volkoff, my wife Yulia and my son Gleb for their help, encouragement and support during my studies at UBC. xiii AIX pvp IUL oz Chapter 1 Introduction This thesis investigates the physics of certain beam instabilities which can limit the performance of charged particle accelerators. Accelerators are now used in a wide va riety of disciplines, and the users maintain an unrelenting demand for beams of higher current (particles per second) or brightness (current density in phase space), whether for high-energy colliders and meson factories (particle and nuclear physics), for syn chrotron radiation and spallation neutron sources (biology, chemistry, condensedmatter physics, materials science and medicine), or for small cyclotrons producing radioactive isotopes. The intensity and brightness of charged-particle beams are, however, limited by the disruptive effects of the electric and magnetic fields produced by the particles themselves (either directly or via the material of their enclosure) and it 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 charged particles to high energy. In order to accomplish this we have to apply external electro magnetic fields not only to accelerate particles longitudinally, but also to confine them transversely so that they remain in the vacuum environment. In this thesis we confine the discussion to “circular” accelerators, i.e. those in which the particles travel along closed orbits, allowing them to pass the same electric accelerating field many times, reducing cost and increasing effectiveness. The beam in circular accelerators is usually split into bunches because the electric radio frequency 1 Chapter 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 could be applied without taking space-charge forces into account. However, with increasing intensity collective effects due to space-charge forces become very important. These effects arise not only from the internal fields within the bunch, but also indirectly from the fields induced by the beam in the vacuum enclosure. Electro-magnetic fields induced in rf cavities, vacuum pipes, beam position monitors and discontinuities such as joints, bellows, etc., can affect the beam and make it unstable. The effects of these fields can be characterized by an effective “beam-coupling impedance” ,defined as the ratio of the induced voltage to the beam current. Collective effects in circular accelerators can be classified as longitudinal or trans verse and also as single-bunch or multi-bunch effects. Single-bunch effects are caused by forces which are strong over the bunch length, but decay fast enough that they do not affect any other bunches in the accelerator. An excellent review of longitudinal single-bunch effects has been given by B. Zotter [1]. In contrast to single-bunch effects there are multi-bunch or ‘coupled bunch’ effects, in which the field induced by one bunch may affect later bunches (or the same bunch on a later turn). In this thesis we concentrate on longitudinal single-bunch effects. We will assume that the transverse and longitudinal components of a particle’s motion are completely decoupled; this assumption is valid most of the time, although there are some special cases in which longitudinal oscillations affect transverse oscillations. As mentioned above, one of the main goals for accelerators nowadays is to obtain high 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 3 Unfortunately, the length of a bunch can be changed both by ‘potential well distortion’ and by ‘turbulent bunch lengthening’ caused by longitudinal ‘microwave instability’. The term ‘potential well distortion’ comes from the fact that the particles in a bunch are oscillating in an effective potential well created by the external rf fields, which can be distorted by the fields generated by the the self-forces of the beam. Potential well distortion is dominant at low intensities when the beam is usually stable. At some intensity the beam may become unstable and this will cause turbulent 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 studied separately for proton and electron beams. The first self-consistent equation describing the stationary distribution of particles in accelerators was derived by HaIssinski [2]. The main idea of his theory is that the particles in the beam should be in thermal equilibrium. This theory describes the distribution of electrons in synchrotrons and storage rings when the radiation effects are strong enough for the particle to reach an equilibrium during the acceleration or storage cycle. The results of applying this theory at various conditions can be found in [3]. A review of different techniques to find stationary self-consistent distributions in electron bunches can be found in [4]. At higher intensities turbulent bunch lengthening is always predominant. Bunch shortening 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 bunch lengthening at higher intensities. The shortening observed at LEP and SPEAR was caused by a strong capacitive component in the impedance of the rf cavities. This was demonstrated at SPEAR when some of the cavities were removed and the bunch became longer [5]. Chapter 1. Introduction 4 The current understanding of turbulent bunch lengthening is that it is caused by longitudinal single-bunch instability, i.e. once the bunch becomes unstable the distribution adjusts itself always to stay close to the instability threshold. Therefore, understanding the physics of single-bunch instability (especially finding instability thresholds) is very important for understanding turbulent bunch lengthening and thus finding ways to obtain shorter bunches and increased luminosity. Questions still remain regarding the mechanisms which cause this instability. This can be illustrated by the case of the SLC damping ring at SLAC where the old vacuum pipe was replaced by a smoother chamber with a lower impedance, which was expected to increase the instability threshold. That didn’t happen; instead the threshold was lowered [7]. Fortunately, the instability which causes the new threshold is not as severe as the one which limited the performance of the old structure and it is now possible to operate even beyond the threshold. Potential well distortion and turbulent bunch lengthening are not the only single bunch effects occurring in circular accelerators. Recently, some different phenomena have been observed at various electron accelerators. One interesting example has been observed at the SLC damping ring [8]. After injection the bunch length decreases slowly until a threshold is reached, when its length increases sharply (in a less then a 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 centre of mass versus time (see Fig. 28 in Chapter 5). A unique single-bunch effect has been observed on the TRISTAN Accumulator Ring at KEK (Japan) by T. leiri [9]. At some threshold intensity the bunch experi enced a very fast instability in both longitudinal and transverse directions; when the beam current was decreased the bunch length would decrease sharply, but at a lower threshold. Chapter 1. Introduction 5 One of the biggest problems in studying single-bunch effects is their sensitivity to impedances at high frequencies, where impedances are difficult either to measure or calculate. The models which are currently used in simulations do not describe the electromagnetic fields generated in real accelerators adequately. One of the first criteria for the single-bunch instability threshold for proton beams was proposed by D. Boussard [10] (‘Boussard criterion’), which was in fact a way to apply the simple stability criterion (‘Keil-Schilell criterion’) derived for continuous beams to a bunched beam [11]. If the bunch is long compared to the wavelength of oscillation (as in proton beams) the beam can be treated as continuous and the Keil-Schnell criterion can be applied locally to the bunch, where the average energy spread and current are replaced by their local values. Though the ‘Boussard criterion’ was first obtained empirically, different authors have found several independent ways to derive it [12]. Beam bunches can have different modes of oscillation. It is common to classify them into dipole, quadrupole, sextupole, etc., dependillg on the nature of the oscil lations. These modes may be stable or unstable and, in general, their frequencies depend on intensity. Coupling between two different modes of the longitudinal oscillations is often used to explain single-bunch instability. F. Sacherer first suggested this, interpreting turbulent bunch lengthening as the result of microwave instability [13]. G. Besnier in his thesis [14] tried to develop the theory of bunched beam stability by considering a more realistic situation where the nonlinearity in an rf field results in a spread of synchrotron 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 for one particular (elliptic) distribution function, which is known not to change the shape of the bunch or the internal forces [17]. Chapter 1. Introduction 6 R. Baartman and B. Zotter [23] have shown that in the presence of space-charge interactions mode coupling will also cause instability for a few other types of dis tributions (so-called ‘gaussian’ and ‘hollow beam’), but their calculations were done without taking into account potential well distortion, and thus the validity of these re sults is questionable. In this thesis we will try to study the problems self-consistently by including the effect of the potential well distortion. 1.1 Outline of the Thesis In Chapter 2 we discuss the basics of longitudinal motion in synchrotrons and stor age rings. We also derive the expression for longitudinal space-charge forces, which are typical for hadron accelerators, and the wake fields which are used to describe longitudinal forces in electron synchrotrons. Finally we use the Vlasov equation to derive equations for the potential well distortion, and also describe stability analysis techniques which will be used in the following chapters. In Chapter 3 the effects of longitudinal space-charge forces are investigated in de tail. First, we describe the effects of space-charge forces on the stationary distribution of particles in a storage ring. We derive an equation which describes the potential well distortion due to space-charge forces, study the bunch lengthening (shortening) they cause below (above) transition, and derive a new criterion which specifies the conditions for which a stationary distribution does not exist. Next we investigate the stability of a bunched beam under longitudinal space charge forces. The stability analysis is done self-consistently, i.e. we study the sta bility of the distributions found earlier in this chapter. We compare our predicted stability thresholds with those obtained from non-self-consistent theories and also with the thresholds found earlier for stationary distributions. Chapter 1. Introduction 7 Chapter 4 describes the longitudinal single-bunch phenomena relevant to electron synchrotrons. We introduce the Haissinski equation, which describes a stationary distribution of particles in the presence of synchrotron radiation, and investigate the effects of typical wake fields: inductive, resistive, capacitive and that of a resonant cavity. We derive an integral equation which can be used to analyze the stability of the bunched beam in the presence of short-range wake fields and apply it to different wake fields and impedances. Based on analysis of stationary self-consistent distributions found by solving the Haissinski equation, we then introduce a new stability criterion and compare it with solutions found by numerical solution of the integral equation and particle tracking. Chapter 5 investigates the behaviour of an electron bunch beyond the instabil ity threshold. We discuss some pathological nonlinear effects observed in modern accelerators, such as sawtooth instability mentioned above, and describe some at tempts to reproduce them using a multiparticle tracking code to gain insight into the mechanisms at work. Chapter 2 Essentials of Longitudinal Motion In this chapter we define some important parameters commonly used in accelerator physics and derive some useful formulas which will be used later in the thesis. 2.1 Equations of Longitudinal Motion To 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 to the motion of a reference particle which has a designated energy E 0 and travels along a closed orbit. The angular coordinate of any other particle at time t can then be written (2.1) where is the angular revolution frequency of the reference particle and is the angular 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 magnetic field, 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 necessary to have magnetic field gradients, and therefore the particles’ revolution frequencies 8 Chapter 2. Essentials of Longitudinal Motion 9 Figure 1: The coordinate system of a circular accelerator may depend on their momentum. This property of the accelerator structure can be expressed in terms of the momentum compaction factor, which is the ratio of the fractional change in the length C of the closed orbit to the fractional change in a particle’s momentum p SC/C (2.3) Lip/p Usually, a> 0, but it can also be negative. In order to describe the longitudinal dynamics we need equations for time deriva tives 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 revolution frequency to the fractional momentum change p/p Since w Dc 3/C LwLC wC 6C 3C lLp p 2 (2.6) Chapter 2. Essentials of Longitudinal Motion where -y = 1//1 — j32 10 is the relativistic factor. Taking into account (2.3) we find (2.7) Note that changes sign when emphasize this fact passes through 7tr 1//. Often in order to (2.7) is written in the form (2.8) where tr is called transition energy of the synchrotron. The transition energy plays very 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 with the environment, and by synchrotron radiation. In this chapter we neglect the energy loss due to synchrotron radiation. Let us assume that the rf voltage at an accelerating gap oscillates with a frequency which is an integer multiple of the revolution frequency w 0 of a particle on the closed orbit .f 1 V = t 0 0 sin hw V where h is called the harmonic number and the rf phase angle q (2.10) = also experience fields caused by the beam-environment interaction h. The particles but we will leave this till later in this chapter. Particles crossing the rf gap must gain energy at a rate matching the magnetic field ramp in order to stay on orbit. Assuming V 0 is big enough there will be some ‘synchronous phase’ ç for which the energy gain provided by the rf system = eVo sin (2.11) Chapter 2. Essentials of Longitudinal Motion 11 will 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 energy gain relative to the synchronous particle in terms of its derivative z ). 3 eVo(sinq—sillç5 (2.12) For our purposes it is useful to describe the particle’s motion in terms of a Hamil tonian. One can see that W = (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 tra jectories (Fig. 2). The trajectories are closed for small oscillations from the stable synchronous phase , but for large deviations the trajectories are not closed and the motion is therefore unstable. The boundary between the stable and unstable regions is called the separatrix. 0 there are h regions Since the rf frequency is h times the revolution frequency w (buckets) around the orbit where the motion is stable. The particles captured in each separate bucket are often referred to as a bunch. There may, therefore, be a maximum h bunches in the accelerator, but sometimes some of the buckets are kept empty. 2.1.1 Phase Focusing To find out whether the motion is stable or not we consider only small deviations from q$ sin 4 — sin q (tq) cos (2.15) Chapter 2. Essentials of Longitudinal Motion 12 V 3ir I W • .• . w 0 . • ••• . . •• / I •.4. : ..• .• .=oo •• \. Figure 2: Particle trajectories in the \..J •1 (, W) ../ phase plane 0 Chapter 2. Essentials of Longitudinal Motion 13 and combine (2.9) and (2.12) to get a simplified linearized equation d ( 2 Lç) 2 dt 3 hwgeVocosç — 2 8 01 2irE — — (2.16) 0. For stable motion the coefficient in front of Lq should be positive. Therefore, the stability condition can be writtell as <O 8 cos4 (2.17) which is equivalent to the following • Below transition (-y <7tr) • Above transition (‘y > <0: 0 < cs <.ir/2, sin q > 0. > 0: 7r/2 < sin q > 0. <‘r, (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 acceleration the stable phase becomes unstable and vice versa. For that reason, the rf phase must quickly be shifted from ç to r — çb as transition is crossed. Below transition a particle with lower energy and momentum will arrive later at the rf gap, will see a higher voltage and, therefore, gain more energy than the synchronous particle. If a particle with lower energy is on the opposite slope, it would 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 some attempts have been made to investigate the possibility of operating the accelerator below transition [27]), but for proton synchrotrons transition crossing often occurs during acceleration. As we will show below, the stability of the beam due to inter action with its environment depends on whether the machine is operating above or below transition. Chapter 2. Essentials of Longitudinal Motion For small deviations (q— 14 << 1) the force is linear, resulting in a simple harmonic motion at the synchrotron frequency = — heVocosq 8 2 2irE5 (2.18) Usually the synchrotron frequency wo << wo and it takes tens or hundreds of turns to complete a synchrotron oscillation. Our previous arguments were based on the assumption that there is only one rf cavity, but since the synchrotron frequency is small compared to the revolution frequency it does not really matter whether there is 0 sin 4 as the total energy only one rf cavity or several and it is possible to consider eV gain per turn. 2.2 Induced Forces The synchrotron frequency is independent of the amplitude only if the forces acting on the particles are linear, but this is not usually the case. The beams in accelerators are surrounded by cavities, pipes, magnets, etc., most of which are metal. Since the beam consists of charged particles it will induce currents in these elements and generate electromagnetic fields which can affect the motion of particles following behind. Therefore we can express the total voltage gain per turn experienced by the particles in the beam as the sum of external (rf accelerating) and induced components t(q,t) 0 Vt The voltage nd 4 = Vç(q,t) + 14d(,t). (2.19) modifies the effective potential well in which the beam oscillates and makes the synchrotron frequency dependent on the amplitude. This effect is called potential well distortion. Chapter 2. Essentials of Longitudinal Motion 2.2.1 15 The Longitudinal Space Charge Field In proton accelerators the bunch length is usually much longer than the diameter of the beam pipe, so that the beam-pipe interactions are localized. The voltage induced by 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 an accelerator environment, which is better suited in case of electron synchrotrons, will be introduced later. Let us consider a circular beam with the the number of particles per unit length p and radius a moving along the centre of a cylindrical pipe of radius b. We can express p in terms of the line density ) p(z,t) = N\(z,t) (2.20) so that f (z’, t)dz’ = 1. (2.21) We assume that ) changes slowly enough along the beam so that the condition d)/dz << )/b is always satisfied. In this case the electric and magnetic fields can be calculated from Maxwell’s equations (by applying Gauss’ and Ampere’s laws) Nq\ 1 r 0 27re Nq.\ r Er = a 0 2re Er = —, —, to13cNq\ 1 r 2ir r o/3cNq..\ B = a 2ir Bq = —, —, r > a (2.22) r <a. (2.23) If we apply Faraday’s law cdi= _J.dA 1 (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 z :. 7 _ -, -‘ ::: / -, Ew EW + Er(z+8z) ÷+ + + + ÷ + + + + +1 BEAM ._ -, -, -, -, F c + prZ -, -, ‘ WALL Figure 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) is 1 b 2 + ln ZZ poNq9cãA 2r (2.26) If we assume that the perturbation moves approximately with the same speed as the particles in the beam we may substitute the derivative over time for derivative over longitudinal coordinate (2.27) _/ c 3 W. Combining Eqns. 2.25-2.27 together, we get = where we used the fact that pofo (i + 21n = ) — — . The expression in square brackets is 2 I/c (2.28) 1/72. The longitudinal field at the wall E will drive the wall current, which is equal in magnitude, but opposite in sign, to the a.c. component of the beam current. In most Chapter 2. Essentials of Longitudinal Motion 17 accelerators the walls are inductive at low and medium frequencies, i.e. B — LdI —---— N e,6c-— (2.29) where C 0 is the inductance per unit 0 is the circumference of the accelerator and L/C length. Combining (2.28) and (2.29) we get the total field due to space charge and inductance 3 = —Ne B [ go — 41reo7 2 L] 9 , 22 Co 9z (2.30) where go = (1 +2 in b/a). It is important to note that the space-charge and inductive terms have opposite signs. The space-charge field is most significant at low energies. Integrating this field over the accelerator circumference we finally can obtain the voltage gain per turn V = Ne3ch where Z 0 = (c c)’ 0 — LL] (2.31) 377 !i is the so-called impedance of free space. We can expand the induced voltage into a Fourier series keeping only harmonics 0 of the revolution frequency w v(t) = Z(nwo)Ine_0i (2.32) , and we 0 where Z(flwo) is the effective coupling impedance at the frequency w = nw can easily show that Z = . (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. However we should always remember that this is true only within the framework of our ap proximations, i.e. that the perturbation length is big compared with the beam pipe Chapter 2. Essentials of Longitudinal Motion 18 diameter, as is usually the case in proton synchrotrons. In practice, the ratio Zn/n remains almost constant up to the so-called cut-off frequency c/b, where c is the speed of light and b is the pipe radius. As explained above, the presence of the q factor in the longitudinal equations of motion make the transition energy y crucial in determining beam stability. Above transition, where particles with higher energy have lower revolution frequency, repul sive space-charge forces will cause the particles to move towards each other, as if they had a negative mass. Therefore a small perturbation will grow exponentially and a continuous beam becomes unstable. This is called negative mass instability. In the case of a bunched beam below transition, space-charge forces decrease the rf focusing and therefore increase the equilibrium bunch length. The opposite is true above transition - it may happen that the self forces become so strong that the stationary distribution collapses. Since this corresponds to the negative mass instability regime for continuous beams it is also referred to as the negative mass threshold. Later in the thesis we will show that for some specific distributions (so called ‘hollow beam’ distributions) a threshold due to space-charge forces also exists below transition; in contrast to the ‘negative mass threshold’ it is called the positive mass threshold. 2.2.2 Wake potential In electron synchrotrons the rf frequency is usually much higher than in proton accel erators 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 the electrons is usually very close to that of light, so that the space-charge forces, which vary as 1/72, become negligible. Chapter 2. Essentials of Longitudinal Motion Figure 4: Two charged particles travelling along the axis in an accelerator. Let us consider a particle with charge e travelling with v so that z = c inside a vacuum pipe et, generating an electric field E(z, t) along its path (Fig. 4). Another particle 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 induced voltage per unit charge, can be defined as W(r) f = E ( 2 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 of the wake function W(r) with the line density (r, t) V(T, t) = Ne j W(r’)A(r + r’, t)dT’ Note that since the particles are travelling with v (2.35) c they cannot induce fields in front of themselves, i.e. W(r)=O, T>O (2.36) Chapter 2. Essentials of Longitudinal Motion 2.2.3 20 Impedance Another way to describe the interactions between an accelerator beam and its envi ronment is to use the concept of electrical impedance. The impedance may be written as the Fourier transform of the wake potential L Z(w) W(T)eiLTdT (2.37) which means that by using an inverse Fourier transform one can express the wake function in terms of the impedance W(r) 1t00 = J wrdw. t Z(w)e_ 2ir—oo — (2.38) In other words, impedance and wake functions are equivalent, corresponding re spectively to frequency- and time-domain representations. Nevertheless, one or the other method may sometime be preferable, e.g. it is often more convenient to de scribe 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 ter minology. In this thesis we will consider how different wake fields can affect the beam. For example 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 model for the effects of many accelerator elements (such as kickers, joints, bellows, etc.), summed together. 2.3 Collective Effects Let us choose new dimensionless canonical variables relative to the stable fixed point p 1 hwo 30 E w 13 2 0 — (2.39) Chapter 2. Essentials of Longitudinal Motion q 21 (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. Assuming that their amplitudes are small, the equations of motion become p where V = —q + V(q,t) = p=- oH = —--— (2.41) (2.42) 0 cos ç). The Hamiltonian becomes t)/(V H=-+-+UId (2.43) where Ua = — Typically there are 1010 — 1015 f V(q’, r’)dq’. (2.44) individual particles in a single bunch and therefore it is possible to introduce a distribution function i(p, q, r) so that the approximate number of particles in some area in phase space can be written simply as dN=Nbdpdq (2.45) where N is the total number of particles in the bunch and J 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 2.3.1 22 The Vlasov Equation A powerful tool for studying the motion of a very large number of particles is Li ouville’s theorem, which states that the phase space density is conserved along a dynamical trajectory provided the system can be described by a Hamiltonian func tion: (2.48) =0. 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-- = (2.49) 0 which is known as the Vlasov equation. For longitudinal motion in a circular acceler ator 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 Distortion Let us consider how different interactions may affect the stationary distribution of particles in the beam. It is easy to see from (2.49) that for the distribution to be stationary we must require = and after we express j and (2.51) 0 in terms of the Hamiltonian we find öHb 8j öq aHab Oq — H —0 — ( 252 . Chapter 2. Essentials of Longitudinal Motion where [...} 23 are Poisson brackets. This means that a stationary distribution function o 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 well and thereby change the stationary distribution. If this is not taken into account, instability thresholds will be incorrectly predicted. The potential well distortion in the case of longitudinal space-charge interactions can thus be written Ud(q) = D(q). (2.54) In the case of electrons, for which the interactions are better described by the wake field formalism, the potential well distortion is Ud(q) = —if S(q’)(q + q’) dq’, (2.55) where the step function S(q) f W(q’) dq’ (2.56) is derived from the wake potential. 2.3.3 Stability Analysis A stationary distribution &o(p, q) (2.53), though it satisfies (2.52), may be unstable with respect to small perturbations. These perturbations may grow exponentially under some conditions. Let us assume that the distribution b is perturbed from the stationary distribution 0 and ‘b q, t) = o(H(p, q)) + ‘i(p, q, t) (2.57) Chapter 2. Essentials of Longitudinal Motion 24 In the limit of small b 1 we need to keep only terms linear in the perturbation and we 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 of conjugate co-ordinates. We introduce the canonical variables action J and angle 0 J=—jcpdq and .(J) 4 ê=j.= (2.59) instead of p and q, and the linearized Vlasov equation becomes (2.60) 2.3.4 Collective Modes This equation is linear and therefore we can look for its solutions in the form f(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 eigenmodes of (2.60) anmRnm(Vi)em8e_mt n,m where n = 1, ..., cc and m = —cc, ..., (261) cc are the radial and the azimuthal mode num bers respectively. The Rnm are orthogonal functions specific to a particular stationary distribution b , and 0 Vflm are the eigenfrequencies. It is common to classify the modes according to their azimuthal number m (m is dipole, m = 2 quadrupole, m = = 1 3 sextupole etc.). A few of the lowest order modes and line densities corresponding to them are shown in Fig. 5. Chapter 2. Essentials of Longitudinal Motion 25 (I) 1 ST4TION4P DISTRIBUTION cm = PHASE SPACE LINE DENSITY = a’ m 1 DIPOLE (R,GID-BUNCH) + \}‘..?“..J ,n.2 QLL4DRLPJLE + IVf( ‘.1 \I SEXTUPOLE + ?..(t) _P ‘ 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 nm can 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 Lengthening The behaviour of electron and proton bunches is different once the instability thresh old is reached. In the case of electrons, when synchrotron radiation becomes significant, an addi tional diffusion-like term must be added to the Vlasov equation (2.50). The resulting equation is the Fokker-Planck equation, and its time-independent solution is no longer an 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 can Chapter 2. Essentials of Longitudinal Motion 26 change 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 instabil - ity leads to modification of the distribution and this effect is called turbulent bunch lengthening. This case will be considered later in Chapter 5. Aside from its impact on the stationary distribution, the diffusion term is a small perturbation on the Vlasov equation in most cases, where the synchrotron frequency is large compared with the radiation damping rate. Under some conditions the behaviour of an electron bunch beyond the threshold is more complicated. Radiation damping does not modify the distribution function to the extent that instability disappears, but it holds the particles together and there fore one can expect to see some continuous oscillations. These phenomena will be addressed in the last chapter of the thesis. In the case of proton bunches, for which the acceleration or storage time is small compared to the relaxation time due to radiation effects, the stationary distribution is usually not of Maxwell-Boltzmann form. This feature of proton beams allows special distributions to be created which give more even line density, smaller peak currents and therefore reduce transverse space-charge effects. Since the radiation damping does not affect protons as much as electrons, an instability usually results in changing L’o so that the bunch always remains below the threshold. In the following two chapters we consider space-charge and wake-field models separately using the following procedure: 1. find a stationary self-consistent distribution (including potential well distortion induced by self forces); 2. investigate the stability of the self-consistent distribution with respect to a small exponentially time-dependent perturbation. Chapter 3 Longitudinal Stability under Space-Charge Forces In this chapter we restrict ourselves to the longitudinal space-charge forces which were introduced in Chapter 2. In the case of space-charge impedance the stationary distribution changes significantly with intensity [19]. It has been shown recently [25] for a resonator impedance that the thresholds calculated ignoring the potential well distortion differ from those obtained in self-consistent calculations; therefore, the results obtained previously for gaussian and ‘hollow’ distributions [23], assuming an absence 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 can be used for checking any new theory. Stationary Self-Consistent Distribution 3.1 The only exact analytical self-consistent solution for a stationary distribution in the presence of longitudinal space-charge forces is the locally elliptic distribution b cc — H. A. Hofmann and F. Pedersen [17] have shown that this distribution does not change the shape of the potential well, and therefore the space-charge forces are linear 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 does not change shape with intensity. 27 Chapter 3. Longitudinal Stability under Space-Charge Forces 28 As 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 orthogonal polynomials [13]. Unfortunately, an elliptic distribution is highly idealized and lacks features which are essential for a realistic distribution: it has a very sharp edge (which results in unrealistic forces close to the boundaries), it provides no tails and it does not produce any frequency spread within the bunch - a feature known to play an important role in Landau damping stabilization [34]. The problem of finding a stationary self-consistent distribution under space-charge forces for the exponential, or Maxwell-Boltzmann distribution (/, _H) 6 has been studied 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 exponential and therefore their result is not a general one. Various distribution functions have been studied by Baartman in [19] for the case of space-charge impedance. The numerical method used in that paper is an iterative one where the current line density A (expressed as a vector {A(qj} defined at a finite number of points qj) is used in (3.62) to generate a new one. However, this iterative technique diverges in some cases and special relaxation techniques are required for different distributions. Some results obtained in [19] are shown in Fig. 6 and Fig. 7, in which the line densities 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 the thresholds. In this chapter we describe a method which can be used to find the self-consistent Chapter 3. Longitudinal Stability under Space-Charge Forces 29 Figure 6: Line densities (top) and density plots (bottom) of stationary distributions for the gaussian case at intensities (left to right) I = -1.552, 0 and 20. The left-most plot is at threshold: for I < -1.552, no stationary distributions exist [19]. Figure 7: Line densities (top) and density plots (bottom) of stationary distributions for the ‘hollow’ distribution f(H) = exp[—(H H /2] at intensities (left to right) I 2 ) 6 -8, 0 and 20. The left-most plot corresponds to negative mass and the right-most to positive mass thresholds. — Chapter 3. Longitudinal Stability under Space-Charge Forces 30 distribution and the line density for any distribution function (essential for proton beams) in the case of space-charge impedance. It can also be used to determine threshold intensities beyond which no stationary distribution exists. In the iterative method [19], thresholds are difficult to find because the numerical technique becomes unstable near threshold. Our method can also be inverted to recover the distribution function from a given stationary line density. The problem we study is how the phase-space trajectories (H = constant) and line density depend upon the beam intensity and on the distribution of particles in the beam. As we have shown in Section 2.3.2. a stationary distribution can be written as 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 the Hamiltonian for the particle in the bunch can be written as H(p, q) = + U(q) (3.62) where U(q) - + I((q) - (3.63) o) where p is the normalized longitudinal momentum, q is the normalized longitudinal coordinate and )(q) is the line density. The constant A 0 = A(0) is included to keep the value of the Hamiltonian constant at the stable fixed point ([H(0, 0)] = 0). If this is not done the Hamiltonian (3.62) may change with intensity and some important features of the distribution may be lost (e.g. the ‘hole’ in a hollow distribution may disappear as the intensity is raised). If the line density in (3.62) has been normalized so that f Adq = 1, the intensity parameter I may be written in terms of machine parameters as 1= 27rh1 0 im() 8 Vcosq n (3.64) Chapter 3. Longitudinal Stability under Space-Charge Forces 31 where 11, V and Jo are the harmonic number, the rf voltage and the average beam current respectively. Zn/n is imaginary and constant in the case of space-charge impedance; I may have either sign: positive below transition and negative above. We will assume that an arbitrary distribution function f(H) is well-behaved in the sense that it is positive, smooth, non-singular, and f(oo) line density )(q) cx f f[H(p, q)]dp cannot — f’(oo) = 0. The be found directly, because H itself depends upon .A (3.62). Nevertheless, we can find A(U) cx .)(q(U)) defined by A(U) = J f(U + p /2)dp. 2 (3.65) Proper normalization is retained if we define a parameter 1 so that I If A(U) dU. (3.66) We can see that L\(q) and from = IA(U(q)) (3.67) (3.62) q = +/2 0 [u + I (A - A(U))] (3.68) where A 0 Since U(q) is a symmetric function, we need only consider the interval q 0. A significant feature of self-consistent stationary distributions with space charge is that they have only one fixed point: one can see from (3.68) that U(qi) qi = ±q2, so the only fixed point is q = = ) 2 U(q only if 0. Integrating (3.66) by parts and taking (3.68) into account we find i = _21f A’(U)V’2 0 [u + 1(A - A(U))] dU. (3.69) Therefore, we can find I as a function of the parameter I. Using this formalism the integral (3.65), and therefore also q(U) (3.68), can be calculated analytically for Chapter 3. Longitudinal Stability under Space-Charge Forces 32 a number of different distribution functions (see Appendix A). Unfortunately, it is not always possible to find a explicit analytical expression for U(q), but for many applications this is not very important. The integral (3.69) can be easily evaluated numerically. Of especial importance is the fact that the described method allows us to determine threshold conditions, beyond which no stationary distribution exists. Threshold 3.1.1 One can see that the solution given by (3.68) does not always exist: the threshold condition arises from a simple physical reason, that the potential U(q) should have a finite slope; otherwise the force V = dU/dq —* cc. We can easily find the threshold condition by differentiating (3.68). No stationary distribution can be found if IA’(U) = 1. (3.70) Therefore, to define the thresholds for an arbitrary distribution function, we need to find the extremums of A’(U) (see Appendix A). It is interesting that this method gives analytical results for the threshold pa rameters for one very important class of distribution functions often used in stability analysis: the so-called binomial distribution functions cc r i. Hiu — — i 0 H for H , 0 H b = 0 for H> H 0 (3.71) where H 0 represents the value of H at the edge of the bunch. Results obtained for some frequently used distribution functions are summarized in Table 3.1.1, where I and Ij are the ‘positive’ and the ‘negative mass’ thresholds. Here the binomial distribution has been chosen in the form f(H) to e as —* = (i — cc. The dependence of I on the parameter distributions is shown in Fig. 8. so that it converges ,tt for the binomial Chapter 3. Longitudinal Stability under Space-Charge Forces 33 Table 1: Threshold parameters for different distribution functions. f(H) /l H (1 4jL) — — (1 — if) e H/1—H He_H e’ 2 H 2.C 1.6 I — - Type local ell. binomial binomial gaussian hollow hollow hollow I I J 0 00 00 00 oo 5.47 16.72 I —0.94 —1.14 —1.55 J(1_)= —0.70 —8.82 —14.58 = — 64—27’/ = 15/s I I I 6 7 - 0 - (I) -c 0.8- 4-i 0.4 - o 0 0 1 2 3 4 5 parameter 8 9 10 j Figure 8: The dependence of I on the parameter i’ for the binomial distributions f(H) = (i — Chapter 3. Longitudinal Stability under Space-Charge Forces • 8 I I I I A I I I I I F(H) = th106 7 34 I F(H) I He I Idi’4 coordinate q coordinate q Figure 9: Line densities at the ‘negative mass’ threshold calculated by using (3.68). 4. I I I I I I I F(H) = H e 2 I th = 16.725 1< .4314i4 I I F(H) = I t1 h = I He 5.4679 °41144 coordinate q coordinate q Figure 10: Line densities at the ‘positive mass’ threshold calculated by using (3.68). The threshold found for the gaussian distribution here (Ith = c_H by the method described -1.551) is in good agreement with the result obtained in ref. [19] by solv ing (3.62) iteratively [19] (Ith = -1.552). Line densities at the thresholds for some distributions from Table 1 are shown in Fig. 9 and Fig. 10 (dashed lines correspond to the points where dA/dU 3.1.2 —* oc) Discussion Some qualitative conclusions can be drawn from this procedure (here and later we assume 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 realistic distribution 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 dis tribution. We illustrate it for the case of hollow distributions which have only one maximum. We assume that f(H) and f”(H) have no breaks and singularities at H> 0, and f(O) = f(oo) = 0. Let us choose A(U) (3.65) in the form A(U) = 2f (3.72) Since the distribution has only one maximum there is a point Hm > 0 for which f’(H)>OforO<H<Hm and f’(H)Ofor Hm<H. We can see that f’(H) for any H > 0 where f’(H) 3.1.3 (.3 73 0. Differentiating (3.72) and using (3.73) we get oof() AI(0)=2f f’(H) d> 2 f’()d=0. (3.74) Recovering the Distribution Function In this section we describe an algorithm which allows us to recover a stationary distribution from the line density in the presence of space charge. This algorithm is useful, for example, for finding an initial distribution of particles to use in a multi particle tracking code, given the line density. It can be derived from (3.62) and (3.72). Chapter 3. Longitudinal Stability under Space-Charge Forces Let us choose two points Ui and U 2 so that U 1 36 . We can rewrite (3.72) in the 2 U < form d+2f U 2( ) 1 U — d. £12 — (3.75) ) 1 U If we choose a linear approximation for f(H) in the interval U 1 < H < 2 then U the first integral in (3.75) can be evaluated: 2 /2(U - ) 1 U ) + f(u 1 {f(u )]. 2 If the array {qj} has been chosen so that 0 < = (3.76) 2 qi <q < ... < qn and A(q) = 0 for q> qn, then one can define = U where i = 0,. . . +IA = (3.77) n, so the recursion formula for fi can be obtained by substituting (3.76) into (3.75) -1 t f = _Li+ 2 1 42(U / pUn — 2] n, n — f, = ) 1 U_ ‘ ‘‘ U With — ) _ 1 2(—U d (3.78) . ) 0 and using the recursion formula (3.78) we can find f— for i = 1, ...1. The integral in (3.78) can be evaluated by any standard numerical method. 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 recovery algorithm: f(H) = J°° 1 7 H A’(U)dU J2(U_H) (3.79) Chapter 3. Longitudinal Stability under Space-Charge Forces 37 In some cases this formula allows analytical expressions for f(H) to be found from given A(q) and U(q). In order to measure the distribution function of a bunch using a beam transfer function or Schottky spectrum one has to know how the synchrotron frequencies de pend on H. Using the Abel transform method we can find the incoherent synchrotron frequencies within the bunch from the stationary distribution function and vice versa. Let us assume that we know the dependence of incoherent synchrotron frequencies w(H) within a bunch and we need to find the potential in which such dependence occurs. The period of synchrotron oscillations in a given potential well is da rq2 T(H) = I 2 J2(H ‘q — (3.80) U(q)) where q and q are coordinates corresponding to the same potential energy (U(qi) ) 2 U(q = H). Substituting T(H) 1 (H) = — - = 2ir/w(H) into (3.80) we get 1 H (q(U) Jo — qU)) dU 2(H - (.3 81 ) 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 get I ‘0 (U)—qi(U)= 2 q dH — H) One can use this formula to find the potential giving w’(H) used in stability analysis). = (3.82) . J2(U 1 w(H) = const (a case often Choosing the normalization in which w(0) 0 and w 0 we have aw w(H) = c’ (1 For the symmetric potential well q(U) — = a H) qi(U) (3.83) = (U) the self-consistent 2 —q solution is q(U) 2arcsinvU . wo2a (1 — cvU) (3.84) Chapter 3. Longitudinal Stability under Space-Charge Forces 38 As will be shown later the form of dependence of w on H plays a crucial role in stability analysis for short-range forces. 3.2 Stability In this section we will try to investigate the stability of a bunched beam in the presence of space-charge forces alone in a self-consistent way. In our analysis we do not restrict ourselves 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 we have freedom in choosing Uo(q) will allow us to compare the results obtained in this section 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 distortion can be neglected (often a poor approximation in the case of space-charge forces). 3.2.1 Integral Equation The 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 Hamil tonian of the system can then be written as H(J,O,t) where Xi(q,t) = = (J) + I)i(q(J,O),t) 0 H 1 dp. Taking into account that dH fb /dJ 0 (3.86) = w(J), the linearized Vlasov equation becomes + w(J) -- — I 0 9) dib -- —-- =0. (3.87) Chapter 3. Longitudinal Stability under Space-Charge Forces 39 It turns out to be more convenient to take 1 ,o and w to be functions of = H(p, q) instead of J: 0 d’/’ We look for a solution in the form with the definition () = = 1 d’b 0 (J) = f(, O)e’ (3.88) (and = g(q)e). Then —iv/w(€), we get =0. (3.89) The solutions of this equation can be easily found if one can neglect the syn chrotron frequency spread. As was mentioned above, the elliptic distribution func tion does not change the shape of the potential well under the influence of longi tudinal space-charge forces, so linear forces will always remain linear for an elliptic distribution at any intensity. In this case the eigenfunctions of the equation can be found analytically. This was first done by D. Neuffer [15] and later by G. Besnier and B. Zotter [16] using a different technique. Unfortunately, the elliptic distribution is the only one with this property; all other distributions do distort the potential well and 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 thresh olds (as well as others caused by +m coupling) in the absence of synchrotron frequency spread (potential well distortion) can be formulated as an eigenvalue problem for the Fourier components of the line density. Moreover, analytical expressions for matrix elements for some specific distributions have been found [23]: gk = —i ooz —Hjg n=—oo where g(q) = 12 (3.90) gke’’ and the intensity parameter = / E v 2 /e (3.91) Chapter 3. Longitudinal Stability under Space-Charge Forces :‘. :. - ( 30- 15 - 40 ‘:7 C) - 1/ 1/ 0 5 I - 0.0 1f/ ; \ II — —6 —4 —2 0 2 Intensity, I \ - ll — 4 6 Figure 11: Coherent mode frequencies as a function of the intensity for a hollow beam distribution. Matrix size: 40 x 40. is negative above transition and positive below. As we have shown above Z/n is a constant for space-charge forces and the matrix elements Hk in (3.90) are given by Hk () = For a general distribution function mm— Jm(kr)Jm(nr)dr. (3.92) it is necessary to calculate (3.92) numerically. Since the matrix Hk is a complicated function of the unknown frequency ii, this does not lead to a proper eigenvalue problem, but requires searching in frequency space. To find the modes, value the eigenvalues ii was stepped through a series of values and for each were found [23]. The results obtained in [24] for a ‘hollow’ distribution are shown in Fig. 11. Negative intensities correspond to space-charge impedance 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 m = 41 2 for quadrupole, etc) and different frequencies for a given m correspond to different radial modes [13]. For larger currents the modes may couple causing insta bility. The thresholds found by this method were in excellent agreement with previous calculations and with numerical simulations, provided special measures were taken to avoid the frequency spread which would otherwise have been generated by space charge forces [24]. However, as has already been mentioned above, these results differ from those obtained in self-consistent simulations including potential well distortion. Let us derive an integral equation which will include the effect of potential well distortion. The periodic solution f(f, 0) = f(f, 0 + 27r) is 6+2ir f(, 0) = ‘e21) — 1 ) d0’. i (3.93) This result is formally equivalent to that given by Krinsky and Wang [12], but it differs in the sense that the present treatment is a perturbation about the station ary case which includes the space-charge impedance: in [12], the stationary induced potential is ignored. Integrating (3.93) by parts we get f(, 0) = I(f)g(q) — I(E) ’g(q’)d0’. 68 e° 1 e2()_ Integrating (3.94) over momentum and taking into account that g(q) (3.94) = f f(, 0)dp, we have finally — g(q) [i — IA’(U)] = If dpb(E) f(c) 2irO() — 1 f 9+2ir 00 )g(q’)dO’, e (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 42 Equation (3.95) is non-linear with respect to v and is therefore not easy to solve in general. In the special case v 0, however, we have the simple result —* g(q) [i_ IA’(U)] J 00 = -- As mentioned previously, the v —* ir b)dpj g(q’)dO’. 0 limit can be thought of as coupling between +m azimuthal modes [23], and since the dipole mode m antisymmetric eigenmode g(q) = (3.96) = + 1 is the lowest order —g(—q), the integral in (3.96) vanishes and we find g(q) [1 — 1A’(U)j = 0. (3.97) Since we expect to have g(q) non-zero at least at some points, the threshold criterion becomes IA’(U) = 1. (3.98) Surprisingly, this criterion is the same as we found in a previous section for a different phenomenon, i.e. stationarity (3.70). This means that the threshold corre sponding 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) 3.2.2 = 1. Comparison with Mode-Coupling Theory The lowest threshold intensity corresponds to coupling between m = +1 azimuthal modes. Due to the symmetry of Hk this threshold has an eigenfrequency (Fig. 11). B. Zotter has found that in the case of be found analytically for ‘gaussian’ [23]. (e_H) ii = 0 the matrix elements ii = Hkl 0 can and ‘hollow gaussian’ (H c_H) distributions Chapter 3. Longitudinal Stability under Space-Charge Forces 43 4 e) .3 w ‘--, .2 ‘hollow’ (Q: = 1, L’ 0 €e) .1 .0 Energy, 3 4 E Figure 12: Family of the distribution functions given by (3.99) for different values of the parameter a. To compare the results of the two techniques we have chosen the family of distri butions defined by [1- a(l = — )) e This family is very convenient because it contains both the ‘gaussian’ (a ‘hollow-gaussian’ (a = (3.99) = 0) and 1) cases (see Fig. 12) and therefore also allows the matrix elements to be found analytically. The eigenvalue problem can be written in the form = where g(q) = (3.100) ygeuI and the matrix elements are Mk = [i — io(kn)j (3.101) Chapter 3. Longitudinal Stability under Space-Charge Forces - Here -Tm(Z) = 6_ZIm(Z) +n 2 a{k 44 [1_Io(kn)} +kn[1_Ii(kn)]}. is the exponentially scaled modified Bessel function. The lowest thresholds can be found from the condition = 1, where are the extreme eigenvalues (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) is A(U) = (i — + c u) e (3.102) In order to find the thresholds for this family of distributions we should look for extreme values of A’(U) for different parameters cr. Simple analysis allows us to obtain the following expressions —1+o if0<cr< _crexp(_) if = (3.103) 0 if0<n< —1+o if<cv<1 A’(U)max = Therefore, if one plots and A’(U) versus o on the same graph, the two curves should be the same. In order to solve (3.100) numerically, the matrix was truncated at 40 x 40. The minimum and maximum eigenvalues obtained in this case are plotted as points in Fig. 13. The solid lines are A’(U) (3.103). We can see that the results are in good agreement. We can also find the shapes of the eigenfunctions g(q) at threshold: since the threshold condition (3.97) is satisfied in general at oniy one specific point qth, g(q) Chapter 3. Longitudinal Stability under Space-Charge Forces I 0 6 45 I — 0.4- ?.- 0.2- -o 0.0- —0.4- —0.8 - 0.0 0.2 0.4 0.6 Parameter a 0.8 1.0 Figure 13: Extreme eigenvalues of equation (3.100) (plotted as squares) and extreme values of A’(U) (solid lines). Chapter 3. Longitudinal Stability under Space-Charge Forces 9 a S 46 b 4 2 20 —1 40 -2 -2 —4 —3 Figure 14: The eigenfunctions found by solving the matrix equation using the mode coupling method, corresponding to (a) negative mass threshold and (b) positive mass threshold. can be non-zero only at sponding to = qth. Indeed, recovering g from the eigenfunction {gjj corre ,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 increasing order of the matrix used in (3.100). 3.2.3 Self-Consistent Case The results discussed in the previous section have been obtained assuming no inco herent synchrotron frequency spread (i.e. V o 2 q ) . To satisfy this condition in the self-consistent case, the initial potential well should be 0 U = U+ I [A(0) — A(U)] (3.104) Chapter 3. Longitudinal Stability under Space-Charge Forces 47 8 7 6 C C w o2 0 1 0 —1 —3 —4 —1 —2 0 1 2 longitudinal coordinate, q 3 4 Figure 15: 0 U vs. q at threshold intensities for the ‘hollow-gaussian’ distribution. U= q /2 is shown by the dashed curve. 2 where U = q / 2 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) gaussian’ distribution L’ 0 = = q / 2 2 for the ‘hollow- ee at threshold intensities are shown in Fig. 15. The upper curve shows Uo(q) at the positive mass threshold and the bottom one at the negative 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) is distorted by space-charge. In this case we can find the threshold beyond which no stationary distribution exists [20] and since it is the same as the stability threshold the modes rn = ±1 do not couple! Chapter 3. Longitudinal Stability under Space-Charge Forces 48 It is necessary to mention that this criterion is valid for any Uo(q) for which ) 2 dUo/d(q 0 (i.e. no local minima). This analysis can be extended to the case when Uo(q) is not symmetric. Unfortunately, in this case we can’t use the symmetry of the eigenfunctions 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 several 0 and the results are consistent with the same threshold, namely IA’(U) cases of i’ = 1. However, no formal proof of the universality of this criterion has yet been found. 3.3 3.3.1 Summary Stationary Distribution A method to find self-consistent distributions and line densities for any distribution function in the case of space-charge impedance has been derived and used to find threshold conditions beyond which no stationary distribution exists. We have calcu lated the thresholds analytically for a number of different distribution functions. We have also introduced an algorithm which allows one to recover the distribution function from a given stationary line density or to find the incoherent synchrotron frequencies within the bunch from the stationary distribution and vice versa. 3.3.2 Stability A simple criterion for stability thresholds in the case of space-charge impedance has been derived from the linearized Vlasov equation. This criterion appears to be the same as that for thresholds beyond which no stationary distribution exists. Chapter 4 Longitudinal Stability of Electron Bunches In the previous chapter we treated space-charge forces and found that the criterion for bunched beam stability is simply that a stationary distribution exist. In the case of electrons the situation is usually more complicated, because the bunch length is much smaller than in a proton machine and the interaction cannot be considered local, as was assumed in the case of space-charge forces. The results of the previous chapter clearly indicate that in order to calculate stability thresholds correctly one must take potential well distortion into account. Attempts to substitute an exact potential with some average effect, or not to include it at all, will result in incorrect estimates of thresholds. Oide and Yokoya have suggested a method [25] which allows one to find the eigen modes and thresholds in the case of broad-band impedance. We call this the O+Y method. This method is self-consistent and includes deformation of the equilibrium distribution due to the wake field. The results obtained are significantly different from those not including the deformation. Unfortunately no rigorous analysis has been made of the convergence and stability of 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 physics underlying the thresholds. 49 Chapter 4, Longitudinal Stability of Electron Bunches 50 In this chapter we derive an integral equation from the linearized Vlasov equation and then show how the O+Y method can be derived from it. Analyzing this equation we also give physical explanations for the eigenmodes and instability thresholds which are found in the O+Y method. As a result of this analysis we suggest a simple crite rion necessary for single-bunch stability and compare this criterion with the results obtained by Oide and Yokoya for a broad-band resonator. The main feature of the new criterion is that the stability threshold in the case of short-range forces can be obtained by analyzing the stationary distribution. This is similar to the case of space-charge impedance, investigated in the previous chapter. 4.1 HaIssinski Equation Stationary 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) . — api This term describes the effects of particle diffusion due to synchrotron radiation, as is significant in modern electron synchrotrons. HaIssinski has shown [2] that due to radiation damping and photon emission the stationary distribution in this case should have the thermal form (H) o e 0 b 4 (4.106) where T is the equilibrium ‘temperature’ of the particles which depends on syn chrotron radiation. Using a proper normalization we can always make T = 1. Substituting the Hamiltonian H(p, q) = j- + jq + V(q’)dq’ (4.107) Chapter 4. Longitudinal Stability of Electron Bunches 51 into (4.106) and integrating over p we finally get the Haissinski equation A(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 analyti cal solutions have been found only for purely resistive and purely inductive impedance, though some approximate solutions have recently been found for capacitive [29] and resonator [4] impedance. Other self-consistent integral equations describing the potential well distortion for electrons have been proposed by different authors, but as shown in [4] they can all be derived 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 e / 2 2, but with increasing intensity the line density may differ significantly from a gaussian. At LEP for instance line densities become noticeably double-peaked at higher intensities (Fig. 16). 4.2 4.2.1 Stability Integral Equation As we have shown earlier the solutions of the linearized Vlasov equation (2.60) are of the form f(J, O)e_it. We will look for solutions in the following form (J,O,t) = e_1t 1 & m= - Cm(J)cosrnO+Sm(J)sinmO. (4.109) Substituting (4.109) into (2.60), multiplying both sides by sinmO, integrating over 0, and then separating imaginary and real parts, we get the following pair of equations iUCm(J) = m.ü(J)Sm(J) (4.110) Chapter 4. Longitudinal Stability of Electron Bunches File G: 290 MV, 0.100 mA File 1: 290 MV, 0.035 mA WIGGLER ON File J: 250 MV, 0.120 mA File K: 290 MV, 0.120 mA FEEDBACK ON 52 Figure 16: Snapshots of the longitudinal profile of the beam at LEP made with a streak camera. Horizontal ellipses show the peak positions of earlier pulses. Chapter 4. Longitudinal Stability of Electron Bunches — i/Sm(J) 53 9 1 2 Idzboj ö Ui(q) rQ. (4.111) —mw(J)Cm(J) where (q) = JW(q’))i(q’+q)dq’ 1 U = f dq’S(q — q’) -00 f dp’Cm[J(p’,q’)] cos[m’O(p’,q’). m -00 (4.112) Changing variables from (p,q) to (J,O) in (4.112) gives Ui(q) = 1 m J dJ’Ci(J’) J dO’S[q(J, 0) — q(J’, 0’)] cos m’O’ (4.113) and we finally obtain the following integral equation [2 — m2w2(J)] (J)I(H) w 2 Cm(J) = —m m’ f g’(J, J’)Cm’(J’) dJ’ (4.114) where gmm’(J,J’) = --f dOf dO’ cosmOcosm’O’S[q(J,O) where the azimuthal mode index m = 1, ..., — q(J’,O’)]. (4.115) cc. We can also rewrite (4.114) as an integral eigenvalue equation Cm(J) = m’ J (4.116) dJ’Kmmi(J, J’)Cm’(J’) where the kernel Kmm’ is given by (J) [ömm’S(J w 2 Kmm’(J, J’) = m Here 6 mm’ is Kronecker’s symbol and 6(J — — J’) — (J)gmmi(J, J’)j 0 h/ . (4.117) J’) Dirac’s delta function. To find out whether the beam is stable or not, we therefore have to solve this integral equation numerically. However, as we will demonstrate below, some conclu sions about stability can be drawn by investigating stationary distributions without actually solving the integral equation. Chapter 4. Longitudinal Stability of Electron Bunches 4.2.2 54 Rigid Dipole Mode It is known that there exists at least one solution of the Vlasov equation (2.49) - the rigid dipole mode: q, t) = &o(p + iae_tt , q — ae_it). (4.118) If the oscillations are small one can see that /,i(p,q,t)=ae it (thbo .O’\ -———--ã-—- (4.119) is indeed a solution of the linearized Vlasov equation (2.60). At low intensities J = for the rigid dipole mode it H(p, q) and the solution of the integral equation (4.114) = 1 with the Maxwell-Boltzmann distribution is simply (H) 1 C = iexp(—H) 7 v’ Cm(H) = 0, m> 1. (4.120) It is not obvious, but it is quite easy to show by substituting (4.120) that the rigid dipole 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 un physical modes instead. As will be shown below the rigid dipole mode plays a very important role in the stability analysis and therefore using methods which do not include a rigid dipole mode among their solutions may lead to incorrect results. The rigid dipole mode is also a useful test of any numerical technique used to solve the integral equation (4.114). 4.2.3 Analysis of the Integral Equation Are there any other solutions of the equation (4.116) and what can we tell about them? We know that in the case of a purely harmonic potential the solutions of Chapter 4. Longitudinal Stability of Electron Bunches 55 (4.116) can be written in terms of orthogonal polynomials, but as we have seen in the previous chapter the only case when the potential remains harmonic is the case of purely inductive (or space-charge) forces and an elliptic distribution. In the case of electrons neither of these conditions is satisfied: the distribution is exponential and the wake fields are more complicated than inductive. Therefore the potential well will be distorted and there will always exist a synchrotron frequency spread within the bunch. Analyzing the dispersion integral obtained from the Vlasov equation for a onedimensional case Van Kampen has found that a complete solution can be written as a combination of discrete modes and a continuous spectrum [33]. The discrete modes are 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 sense that 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 Method A straightforward method for solving the integral equation (4.116) is to convert it to a matrix equation: if the set {J} is chosen so that 0 = Jo < J 1 < ... < J, then a matrix equation can be obtained from the equations (4.114) and (4.115) by making substitutions Cm(Jn)JJn “ Cmn. The resulting equation P 2,-i = IVlmnm’nIL’mln# can then be solved numerically. Oide and Yokoya derived the above matrix equation by introducing a set of arti ficial orthogonal functions. Our analysis shows that their technique is equivalent to approximating an integral f f(x)dx by the sum > f(x)zx. Such an approximation Chapter 4. Longitudinal Stability of Electron Bunches 56 is only good for a sufficiently smooth integrand. Our results presented below show that the majority of the solutions found are not smooth and probably represent Van Kampen modes. The fact that there are many non-smooth modes among the solutions of (4.121) is alarming because the matrix elements were calculated on the assumption that the solution is smooth. Therefore, more rigorous mathematical analysis of the convergence and stability of the method is required. Nevertheless, we gain confidence from the fact that the instability thresholds obtained with this method are in good agreement with the particle tracking results presented by O+Y[25]. Typically, to obtain stable results when the wake field is approximately equal to the 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, first the stationary distribution has to be found, next each matrix element has to be calculated by double integration, and finally the matrix is analyzed for eigenvectors and eigenvalues. The cpu time required for the 360 x 360 case is around 2.5 minutes on a VAX 4000. The calculation is repeated for many intensities to find a possible threshold [26]. In the following section we will apply the methods described above to various types of impedances and wake fields frequently used to model single-beam effects. Though ideal inductive, capacitive and resistive wake fields do not in themselves accurately represent real machines, these models allow a better understanding of the physics behind single-bunch instability. Chapter 4. Longitudinal Stability of Electron Bunches 4.3 4.3.1 57 Results Inductive Wake We have already met this type of wake field when we considered the interaction of long bunches in a smooth pipe. There we showed that the induced voltage is proportional to the derivative of line density V(q) = . 0 LI (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 of the Dirac 8-function, thus V(q) oc j 6’(x)(q + x)dx. (4.123) Since the inductive wake formulas are the same as those for space-charge forces except for the sign, the results obtained in the previous chapter can easily be applied to the thermal distribution; i.e. the distribution will remain stable above transition (the case important for electron accelerators) but has a stability threshold below transition. Applying the O+Y technique for this highly singular wake leads to con vergence problems, which is why we used a different technique to study space-charge and inductive forces in the previous chapter. 4.3.2 Capacitive Wake In the case of capacitance the relationship between induced voltage and current is V(q) = f (x + q)dx (4.124) Chapter 4. Longitudinal Stability of Electron Bunches 58 I 1.5 1.0 0.5 0 -4 -2 0 2 4 ‘7- Figure 17: The beam profiles at different intensities found from the Haissinski equa tion for a capacitive wake field above transition. and therefore W(q) = 1/C for q > 0. The impedance is simply Z natural to normalize W(q) so that W(O) I Ib/[U(EO/e)CwO], where lb = l/(iwC). It is 1. The intensity parameter thus becomes: is the current per bunch, E 0 beam energy and e 0 the relative rms energy spread. Some solutions of the Haissinski equation for purely capacitive impedance are shown in Fig. 17. Note one important feature of capacitive impedance is that the line density will always be symmetric (just as in the cases of purely inductive and space-charge impedance). Above transition as I increases the bunch shortens (Fig. 17) rather than lengthens, as happens in the inductive case. This effect 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 the case of a purely capacitive wake field. No complex eigenfrequencies are found. This agrees with Burov [29], who derived an approximate analytical solution for station ary distributions at extremely large currents. In contrast to inductive wake fields, Chapter 4. Longitudinal Stability of Electron Bunches 59 stationary self-consistent solutions exist for capacitive wakes at any intensity, both below and above transition. The rigid dipole mode is clearly distinguishable from the other eigenfrequencies because, for this particular wake field, all the other frequencies are shifted upward. We have verified that the mode with it = 1 is indeed the rigid dipole mode by comparing (J) with that expected from (4.120), as shown in Fig. 19. 1 C What are the modes associated with the other eigenfrequencies in Fig. 18? For each of the ñ values of J into which the problem has been subdivided, one can calculate the corresponding (incoherent) frequency (J). These and their integer multiples have been plotted in Fig. 18 as well (lower plot). We see that they agree well with the other frequencies found by the O+Y method, excluding the rigid dipole mode. This indicates that these modes are not really collective modes. Further verification of this hypothesis comes from the fact that the eigenvector C is found to be nonzero only at one or two values of n. This means that if the eigenvector has any physical interpretation at all, it represents an eigenfunction which is extremely localized in J, and becomes narrower, the larger the matrix size. It should also be realized that the 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 call these 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 . 2 (C) m,ri (4.125) Since the eigenvectors Cmn are normalized to have a maximum value of 1, we expect the narrow incoherent modes to have X << 1, while broad modes like the rigid Chapter 4. Longitudinal Stability of Electron Bunches I I 3— 0— I I I I I ! ! ! 1 i ‘ 60 I — I ! 1——— — I I i 1— i I I I — 0 — 1 2 3 Intensity I 4 5 6 Figure 18: Bunch oscillation frequencies in the case of purely capacitive impedance, plotted for discrete intensities I. The upper plot shows eigenfrequencies k calculated by the O+Y method, and the lower the incoherent frequencies m(J), where m is an integer. The eigenmode with frequency independent of intensity is the rigid dipole mode. Chapter 4. Longitudinal Stability of Electron Bunches 61 1.2 1.0 0.8 ‘-;:. 0.6 L) 0.4 0.2 0.0 J Figure 19: Comparison of the theoretical rigid dipole mode (4.120) (continuous curve) with the eigenvector C 1 of the = 1 mode (D). The calculation is for ñ = 120, Chapter 4. Longitudinal Stability of Electron Bunches .5- 62 I I 1 2 - .4.3.2 - .1 .00 - 3 Figure 20: RMS value of the kt eigenvector versus its eigenvalue at intensity I in the case of purely capacitive impedance with ñ = 40 (x) and ñ = 120 (C). = 2, dipole mode will have much larger X. Fig. 20 shows a plot of Xk versus 1 u k the point with maximum X also has R for capacitive impedance. One can see that = 1, verifying that this is the rigid dipole mode. Also, this point is not sensitive to ñ, whereas the other values of X all tend to zero as , is raised. As has already been mentioned above, the capacitive impedance above transition will cause bunch shortening, so it seems that if one can create an environment with purely capacitive impedance the problem of bunch lengthening will be solved. Un fortunately, the wake field in real accelerators always includes resistive and inductive components and therefore the effect of bunch shortening is observed only at very Chapter 4. Longitudinal Stability of Electron Bunches 63 small intensities and is then followed by ‘turbulent bunch lengthening’, as we will see later. 4.3.3 Resistive Wake Resistive impedance is real and does not depend on frequency (Z V(q) = R), = RI(q) and since (4.126) the wake function of a resistive impedance is simply a Dirac 6-function V(q) = RI 0 j 6(x)(q + x)dx. (4.127) A solution of the HaIssinski equation for purely resistive impedance can be ob tained analytically [30] .( )= 0 [coth(RIo/2) RI — erf(q//)] (4128) Stationary self-consistent distributions given by (4.128) at different intensities are shown in Fig. 21. The distribution is always unstable. Recently, Oide [31] has proposed an expla nation of resistive instability which has its origin in a mechanism different from the collective mode coupling which is considered in this thesis. His suggestion is that resistive 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, does not couple with any other modes, but instead is unstable by itself. The growth rate of this mode is quite small. Oide has also speculated [31] that this type of instability may be responsible for the lowering of the instability threshold at SLC damping ring [7] after the installation of Chapter 4. Longitudinal Stability of Electron Bunches 64 I 2 1 0 -5 0 5 T Figure 21: Stationary longitudinal profiles at different intensities for a resistive wake. a new smooth vacuum chamber with lower impedance. He has hypothesised that the lower impedance does not automatically lead to an increased instability threshold and that the threshold depends mainly on the ratio of the active (resistive) and reactive (inductive, capacitive) components of the impedance. Reducing the reactive part of the impedance (which is what happened at SLC) may only make the situation worse if the resistive component remains the same. 4.3.4 Broad-band Resonator Wake The impedances considered above are idealized and though they are very important for understanding single-bunch effects in electron synchrotrons it is very difficult to imagine conditions under which the interaction between the beam and environment can be expressed in terms of these impedances alone. A more realistic model of the many elements that make up an accelerator is a resonator with a low quality factor Q. Such an average resonator can be described by a simple parallel LRC circuit. The impedance of this circuit is well known from Chapter 4. Longitudinal Stability of Electron Bunches 65 electrodynamics Z() where Q = = R (4.129) i+iQ[_-] RJiiJJ is the quality factor and r = i/\/L the resonant frequency. The corresponding wake function found by Fourier transform of (4.129) is W(r) = where w 1 = Wr/1 resonant frequency — e_T [cos(CJJ1T) + T)i] (4.130) ). Interesting effects are expected to occur when the 2 i/(4Q Wr is comparable with c/us, where o is the rms length of the bunch at low intensity. To emphasize this, it is useful to introduce a parameter 0 k = w,.r/c relating bunch length to wake length and a coordinate q 2 (k rc/u >> 1 0 corresponds to very long bunches and k 0 << ito very short). Fig. 22 shows an example of 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 limiting cases, and also combinations such as L-R or R-C circuits. The latter seems to be a good model for approximating the impedance at LEP [6]. However, it has been found that in many accelerators, a resonator with Q 1 describes the wake field quite well and 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 is whether there are any others among the solutions of (4.121) and whether they couple or not. The intensity parameter I is defined as above for capacitive impedance, but now with the resonator’s high-frequency capacitance, (rR/Q) , used in place of C. Thus 1 our I is the same as the parameter Sr used by O+Y[25]. The eigenfrequencies calcu lated using the O+Y method are shown in Fig. 23. Note that there is an instability Chapter 4. Longitudinal Stability of Electron Bunches 66 ob2d 115 Figure 22: Resonator impedance with k 0 = 0.6 and longitudinal beam profiles at different intensities. Q = 1: (a) the wake field and (b) Chapter 4. Longitudinal Stability of Electron Bunches •1— —I 67 J. 3. 2- iii;• 1- I’ll 00 2 • II I I 4 I 6 Intensity I I 8 10 12 Figure 23: Eigenfrequencies versus intensity for a broad-band resonator = 0.6). Unstable modes are indicated by solid symbols. (Q = 1, Chapter 4. Longitudinal Stability of Electron Bunches with a threshold of I 68 8. As in the capacitive case, most of the frequencies corre spond to incoherent modes. In fact, a plot of mw(J) is virtually indistinguishable from Fig. 23, except for the presence of complex eigenfrequencies corresponding to unstable modes, which are shown in the figure by solid symbols. The rigid dipole mode is not apparent because in the broad-band resonator case, incoherent frequen cies 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 far they are from the frequency p = 1. Going to larger i is of no help, since there will always 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 is still difficult to distinguish. However, there appear to be a few other ‘real’ modes as well. An investigation of the incoherent synchrotron frequency (Fig. 25) shows that these modes are clustered near the local minimum of w(J). The physical interpreta tion is that near d/dJ = 0 the particles can stay ‘in step’ longer, and so this area constitutes a ‘coherent band’ of action J. The collective quadrupole mode is shown in Fig. 26, where it is compared with an incoherent mode. Note the dramatic difference: the coherent mode is smooth and independent 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 it occurs at the minimum of the synchrotron frequency and so is not degenerate with any incoherent modes. Chapter 4. Longitudinal Stability of Electron Bunches .5- 69 I - .4D x 3 ‘C % X *xxx ‘C .2‘C LA .00 ocxx x ‘C x ‘C x ocx Li I 1 2 - 3 Figure 24: RMS value of the kth eigenvector versus its eigenvalue at intensity I = 3 in the case of a broad-band resonator with Q = 1, k 0 = 0.6, for two different matrix sizes: = 40 (x) and ñ = 120 (1). Note that there are two modes whose X is independent of i; these correspond to a quadrupole and a sextupole mode respectively, at the location where w(J) is a minimum. Chapter 4. Longitudinal Stability of Electron Bunches 70 2 1.5 8 (A.’ —1 ws0 0.5 0 8 Figure 25: Normalized synchrotron frequency versus action J and intensity I for the resonator impedance with Q = 1, ko = 0.6. 4.3.5 New Criterion As 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 quadrupole mode (whose eigenfrequency is with the rigid dipole mode whose frequency is constant (unity). The intensity at which = 1, thus corresponds to an in stability threshold. This conjecture is verified by an inspection of the eigenvector of the unstable mode. Also, by extrapolating the lowest frequency quadrupole mode in Fig. 23, we see it crosses the rigid dipole mode near the threshold intensity. The threshold calculated from the criterion = 1/2 has been plotted versus the 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 truncation of the matrix in the latter case. Chapter 4. Longitudinal Stability of Electron Bunches 71 0.0I I —0.5- —1.0- 0 F 1 2 I I I I I 3 4 5 J 6 7 I 8 9 - 10 Figure 26: Comparison of the eigenvector for the ‘coherent’ quadrupole mode at the synchrotron frequency minimum (it = and a nearby ‘incoherent’ quadrupole mode, for two different matrix sizes: ñ = 120 (continuous curve) and ‘i = 40 (dashed curve). Chapter 4. Longitudinal Stability of Electron Bunches 72 I 20— U U • 1/ 1/ 15- • // • • \ \ ‘1, \• 5- 00.0 I 0.5 I 1.0 0 = 0 k cr/c - 1.5 2.0 Figure 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 73 In the case of resonator impedance the instability caused by collective mode cou pling described here happens first for 0.5 < ko < 1,2. Outside this region another type of instability (much weaker than that caused by mode coupling) dominates. Mode coupling is also less probable below transition. This property has been studied by Fang [27], who suggests that, for higher beam intensities, it may become prefer able to operate electron synchrotrons with a negative momentum compaction factor a, because even if the instability threshold is lower than for the same wake field above transition, the instability itself is much weaker and does not create significant problems. 4.4 4.4.1 Conclusions Oide-Yokoya Method Oide and Yokoya have invented a helpful technique for solving a previously intractable problem. However, the method should be used with caution. It always generates as many eigenmodes as the order of the matrix. The vast majority of these modes appear to be infinitesimally narrow in the limit of infinite order, and have eigenfrequencies equal to the incoherent synchrotron frequency at the mode location. The method does find real collective modes such as the rigid dipole mode, but these are difficult to extract in cases where they are degenerate with the incoherent modes. Nevertheless, even in degenerate cases instability thresholds attributable to coupling between real collective modes are found, and these appear to be valid. Chapter 4. Longitudinal Stability of Electron Bunches 4.4.2 74 Landau damping The fact that so few real modes are found should not be surprising. Landau-damped excitations do not have exponential time dependence, so the standard method of solving the Vlasov equation by assuming harmonic time dependence simply should not find any modes which we think of as being ‘Landau-damped’. The situation is similar to the case where synchrotron frequency spread originates externally by rf wave nonlinearity. Besnier [14] originally developed a technique for solving the Sacherer integral equation where the dispersion function was expanded in the same orthogonal polynomials as the kernel. This technique suffers from the same problem as the O+Y method in that there will always be as many modes as the order of the resulting matrix. Subsequent re-analysis by Chin, Satoh, and Yokoya [32] showed that below threshold these modes have growth rates which tend to zero as the matrix order is increased. They developed a dispersion integral approach and found that below threshold there were no modes, while above threshold their eigenfrequencies agreed with those found by the Besnier technique. The difference between external frequency spread and that arising from the wake field itself is that in the latter case the frequency ‘spread’ and the mode ‘shift’ change at the same rate with intensity. In the former case, the modes which would exist in the absence of frequency spread are one-by-one ‘freed’ from the incoherent band as the intensity is raised. But in the latter case, the dispersion also grows with intensity and so most of the modes which would exist in the absence of dispersion do not exist (equivalently, they are Landau-damped) at any intensity. The few modes that do exist can be unstable on their own (for example, with a resistive wake), or can become unstable by coupling together when their frequencies coincide. Chapter 4. Longitudinal Stability of Electron Bunches 4.4.3 75 New Criterion The simple necessary criterion for instability suggested in this chapter is in reasonable agreement with the O+Y method and numerical tracking. The instability threshold can be found by analyzing the stationary self-consistent distribution, without solving the Vlasov equation. One collective mode found by this method is the rigid dipole mode; the others are multipole ‘collective’ modes concentrated near the synchrotron amplitude where the synchrotron frequency is a minimum. Since this minimum w decreases as intensity is raised, and the frequency of the rigid dipole mode is a con stant, it may happen that mw = 80 at which point coupling between the rigid w , dipole and the m-th multipole mode leads to instability. The intensity at which = o thus corresponds to an instability threshold. This is similar to, but 8 1/2w more 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 compared with the results obtained by the O+Y technique (a code which uses methods similar to those used in [25] has been written for this purpose). Collective modes have been found among the solutions of the Vlasov equation [26] and the shape of the modes which were unstable clearly indicates the coupling mechanism of the instability. In this theory the rigid dipole mode, and especially the fact that its frequency does not depend on intensity, plays an important role. The importance of the rigid dipole mode in instability analysis was mentioned by Oide in his ‘two-particle’ model of longitudinal instability [35], which was constructed in analogy to the two-particle model used to explain single-bunch transverse instability [36]; the results obtained using this model, however, are only in qualitative agreement with observations and numerical simulations. The other problem with the model is that the results depend Chapter 4. Longitudinal Stability of Electron Bunches 76 on the choice of the two particles. The collective quadrupole mode described in this chapter can be related to the ‘two particles’ in Oide’s model, but here it is uniquely defined by the stationary distribution. Apparently the collective mode coupling described in this chapter is not the only effect which can cause single-bunch instability. Recently, Oide [311 has proposed an explanation of resistive wall instability which has a different origin. According to him, a resistive instability is exhibited by a mode which is unstable by itself. In his analysis Oide has shown that electron bunches in a resistive environment will always be unstable, but the growth rate is quite small at low intensities. The methods described above are based on the assumption that the effects of ra diation damping and photon excitation are very small and do not effect the instability thresholds. This assumption is often taken for granted, but, as shown by Nagaitsev [41] for a continuous beam and narrow-band impedance, the instability thresholds found from the Vlasov and the Fokker-Planck equations are, in fact, different even when the Fokker-Planck diffusion term is very small. Another reason why one should not neglect this term is that the Vlasov equation does not allow a full investigation of the beam dynamics beyond the instability threshold the topic to be addressed in - the next chapter. Chapter 5 Beyond the Threshold So far in this thesis we have considered stability of a beam from the point of view of the stability of a stationary distribution with respect to an infinitesimal perturbation. Questions remain about what happens to the beam when the threshold intensity is exceeded. In the case of protons, a turbulent bunch lengthening effect is observed, i.e. the bunch 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 dif ferent features. At the threshold intensity the bunch becomes unstable, but radiation damping causes the particles to be confined and the instability does not necessarily cause loss of particles. The ‘sawtooth effect’ observed at SLAC SLC damping ring [8], anomalous quadrupole sidebands at CERN LEP [6] and the hysteresis effect at TRISTAN AR [9] indicate that the behaviour beyond threshold can no longer be de scribed in terms of solutions of the Vlasov equation and a more complicated equation (such as the Fokker-Planck equation, which includes radiation damping and diffusion effects) 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]. These phenomena are extremely difficult to treat analytically, therefore we use a multipar tide tracking technique to observe and study the behaviour of the beam. As will be seen, this makes it possible to simulate pathological bunch behaviour which could 77 Chapter 5. Beyond the Threshold 78 underlie the observations at SLAC, CERN and KEK mentioned above. 5.1 Numerical Simulations Analytical methods usually describe only small perturbations well and are not conve nient for dealing with transient processes. Sometimes, however, the perturbations are not small and analytical methods make incorrect predictions. In these cases particle tracking simulations may be the only option for studying the beam dynamics. Almost all the laboratories which are involved in accelerator design have developed computer programs to study collective effects, in particular longitudinal effects such as bunch lengthening, beam loading, injection, etc. Computing power limits the number of particles that can be followed and this limits the usefulness of tracking methods for stability analysis. A few years ago the typical number of particles in simulations ranged from 100-1000, while nowadays it can be pushed to iO — 106. This breakthrough in computational power makes it possible to use particle tracking simulations for studying the internal motion in a bunch over thousands of turns and allows one to see effects previously masked by statistical noise. The most impressive results in this field have been obtained at SLAC. Starting with the pioneering works of P.Wilson [5], studies have been continued by R. Siemann and K.Bane [40]. Siemann [39] has obtained very good agreement with the experi mental results obtained at SPEAR by using just 100 macroparticles and calculating the forces between every pair of particles. In his recent simulations K. Bane [40] is using up to 600,000 macroparticles, allowing him not only to get very good estimates for thresholds measured for the SLC damping rings, but also to see the structure of the unstable mode. Chapter 5. Beyond the Threshold 79 To simulate the electron’s motion in a synchrotron we use a standard multi-particle tracking scheme [40]. The beam is represented by N macroparticles each with phase and energy coordinates (z, ) relative to the mean energy and phase. These coordi nates are recalculated every turn according to the following equations + = = 0 cxcT , L/o + V4z + (5.131) 4d(z) (+ Here T 0 is the revolution period, in the absence of a wake, V4 2ueorj = Te (5.132) the damping time, e0 0 the rms energy spread dVd/dz the rf voltage gradient, € the compaction factor, and E 0 the energy of the particles; r is a random number with a standard normal distribution which is used to approximate the effects of noise and synchrotron radiation. To calculate lij we have used the same method as Bane [40], i.e. binning the macroparticles in z without smoothing. Then, approximating (2.35) the voltage nd 4 induced by the beam is given by 4a(z) where Nk is —e NkW(zk — (5.133) z), the number of particles in the kth bin and W(z) is the Green function wake field. Other methods of finding T4d [39] give smoother results for a given number of macroparticles, but require more cpu time. The radiation damping usually takes tens or even hundreds of synchrotron os cillations. However, such long damping times require in general too much cpu time to simulate easily. Fortunately, the damping rate does not play a significant role in instabilities which are fast compared with synchrotron motion. This is the regime of the present study. To optimize computation time versus simulation accuracy, we Chapter 5. Beyond the Threshold 80 used artificial radiation damping times of the order of 5 to 10 times the synchrotron oscillation period. We found that reasonable accuracy can be achieved with as few as 5,000 macropar tides. This depends upon the wake field being fairly smooth: many times more macroparticles are required for wake fields which have many oscillations in one bunch length. [40] For the following work we have chosen a resonator wake field with a quality factor Q = 1 and bunch length parameter k 0 = 0.5. The radiation damping time Te was set to 500 turns and the other parameters V, a, E 0 in (5.132) have been chosen to obtain a synchrotron period T 8 of 100 turns. 5.2 Sawtooth instability A 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, when the length increases sharply (in less then a synchrotron period) and then the process repeats. This effect has been called a ‘sawtooth instability’ for the sawtooth-like shape of the plot of bunch length and centre of mass versus time (see Fig. 28). Similar behaviour has been observed in other electron synchrotrons as well [38]. In simulations with a very simple wake field and short bunches, we have observed that energy spread and bunch length may oscillate in a sawtooth fashion beyond a certain threshold. We find that this is due to the double-peaked nature of the stationary distribution (see Fig. 22). Over many synchrotron oscillations, particles diffuse from the head peak to the tail to the point where the tail peak becomes as large as the head. The two resulting sub-bunches then collapse together in less than one synchrotron oscillation, causing a net blow-up in emittance. Radiation damping Chapter 5. Beyond the Threshold A CHI4 81 2.s I _ • • •_ _I o_z cm 230.v? rii II - fMP1i__________ CH I gnd liii IiiI t.tii.iiIiiiiI,ii,Ii IIIIIIIIitii •t . . Injection I Extraction - I I p p I Time(ms) Figure 28: Sawtooth instability in bunch length observed at the SLC damping ring [40]. Chapter 5. Beyond the Threshold 82 0’ 10 10 turn —10 —6 —2 2 6 10 q turn Figure 29: Sawtooth instability observed in the simulations. then 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 by Bane and Oide [40], we have simplified the model to determine which features of the wake field lead to sawtooth behaviour. We start with a large emittance and allow the beam to damp. At low intensi ties the beam relaxes to a thermodynamically stationary distribution which is well described by the Haissinski equation [2]. When the intensity increases, however, it takes more time for particles to reach thermodynamic equilibrium, particularly when the 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 streak camera (Fig. 16) [6]. In the case k 0 = 0.5 the second peak in the line density appears Chapter 5. Beyond the Threshold 83 5 4 3 b 2 Figure 30: RMS bunch length (a) and rms energy spread (b) in the case of resonator impedance (Q = 1, ko = 0.5) at I = 30. Radiation damping time is Te = 8 5T . approximately at I = 10. This is near the stability threshold found by solving the Vlasov equation (see Fig. 27). The region close to threshold is difficult to model because of the slow growth rate of the instability. Above approximately I = 20, the sawtooth instability becomes apparent. As intensity is raised, the sawtooth periodicity also increases. A typical example showing rms bunch length and energy spread is shown in Fig.30 for I At very high intensity, the behaviour becomes irregular: the case of I = 30. 45 is shown in Fig.31. We have found that the sawtooth repetition rate is mainly determined by the diffusion process and not by radiation damping. To illustrate this point, the case of 10/3 times stronger radiation damping is shown in Fig. 32. Comparing Fig. 32 with Fig. 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 —* C: b: The trailing bunch damps down (about 5 synchrotron oscillations). Diffusion populates the trailing bunch until it is approximately equal to the first (about 30 synchrotron oscillations). Note that the two bunches have Chapter 5. Beyond the Threshold I 84 I 5. 4. 21— 0C a .4 10 3 turn/iD Figure 31: RMS bunch length and energy spread for the same parameters as Fig. 30, except that I = 45. 3 turn/iD Figure 32: RMS bunch length and energy spread for an increased damping rate: I = 30 and Te = 1.5T . Compare with Fig. 30. 3 Chapter 5. Beyond the Threshold 85 • a • 4., • d 4 b C 4 Figure 33: A complete cycle of the sawtooth instability in phase space for the case shown 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 collapse together. • d —* a The combined bunch throws out a large cloud of particles as it executes large 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 0 k < 0.4, the diffusion process was too slow. For k 0 > < ko < 0.6. For 0.6, where the threshold intensity increases with bunch length (Fig. 27), sawtooth behaviour is not seen either; instead, the bunch length oscillates chaotically. Chapter 5. Beyond the Threshold j 86 4:4 44i •1 : c4 1N. trriI :Nit [iI [i : “.-..4 I f,a N/1. 2. phase space potential phase space potential Figure 34: Dynamics of the sawtooth instability. Each frame shows the particle distri bution in phase space and the potential well. The snapshots on the left illustrate the diffusion process, showing every lOOt” turn; those on the right the collapse, showing th every 10 turn. Chapter 5. Beyond the Threshold 5.2.1 87 Analysis of sawtooth instability Qualitatively, the instability can be understood by considering the wake of an ex tremely short bunch (Fig. 35, upper). In order for the energy lost by the bunch to the wake field to be compensated by the rf cavities, the rf waveform (here drawn as a straight line, since the rf wavelength is much larger than the bunch length) must intersect the wake voltage at half the maximum . This is the location of the centre of 1 this very short bunch, and is of course a stable fixed point. Situations for various rf voltage 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 in the same way, since amplifying the wake field has the same effect on the diagram as reducing the rf slope. At low intensity or large rf voltage, there is only the one fixed point. At high intensity or low rf voltage, the wake field intersects the rf waveform at three points; there is an unstable fixed point behind the bunch, and a stable one farther along. The separatrices created by the extra fixed points are also shown in Fig. 35. Because of the random excitation due to emission of synchrotron radiation, parti cles can diffuse through the unstable fixed point and collect at the downstream stable fixed point. These particles begin to create their own wakes, and will move forward as they lose energy to their own wake field. At the same time, the remaining par ticles in the leading bunch will move backwards as they decrease in number and no longer need a large energy gain from the rf fields. At some point, the potential barrier between the two sub-bunches becomes small enough that the diffusion turns into an avalanche and the sub-bunches suddenly coalesce. The resulting bunch is over-dense is due to the fact that the wake field is non-zero only behind the particle and if one considers interaction between each pair of particles, every particle spends on average half its time behind the other particle, which means that the rf voltage has to be only one half of the induced wake voltage to compensate the energy loss. Chapter 5. Beyond the Threshold 88 and at the wrong phase with respect to the needed energy gain. It begins to execute a 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 energy spread. The cloud condenses again at the downstream stable fixed point and diffusion continues. ‘Binary star’ instability 5.3 The 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 was paid to radiation damping time; however, in order to observe weaker instabilities with slower growth rates in simulations one should have a realistic damping time (usually tens or hundreds of synchrotron periods). For this simulation we have chosen rd 0 (k = 0.5), but a lower intensity, I = = 3 and use the same wake field as above 20T 20. Previously, when we had Td = 5T we saw just chaotic oscillations of the bunch parameters, but now the picture is completely different: the bunch splits into two identical sub-bunches and they oscillate shifted approximately 1800 to each other (see Figs. 36 and 37), their motion in phase space resembling that of a binary star. It can be seen that the intensity at which this phenomenon 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 approxi mately twice the synchrotron frequency (see Fig. 38), but it is interesting to note that at some intensity two sidebands appear. Low frequency sidebands around the quadrupole frequency have been observed at CERN LEP [6] (see Fig. 39). It was suggested in [6] that these lines may correspond Chapter 5. Beyond the Threshold 89 1.5 1.0 0.5 0.0 —0.5 a- 2 Figure 35: Above: Green function wake field with three rf waveform slopes; (a) is stable, (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 wake field crosses the rf waveform from below, and an unstable fixed point if it crosses from above. Below: the separatrices created by the wake fields corresponding to cases (b) and (c). In case (a), there is only one stable fixed point so the wake field does not create a separatrix. Note that these curves are for a Green function wake and therefore are only suggestive. Any accumulation of finite charge density will deform the separatrices. Chapter 5. Beyond the Threshold 90 5000 q 4 I 6000 7000 80)0 9000 iodoo iicoo turn I I — • • •• I. 2 -. —2 —4 — 1111111 —8 —6 —4 —2 0 q 2 4 I 6 5000 6000 7000 8000 9000 turn )O00 100 Figure 36: Simulation of the ‘binary star’ instability. to radial modes predicted by conventional theory derived from the linearized Vlasov equation; however, the quadrupole oscillations observed at LEP had a very large amplitude (more then 30 percent of the bunch length, see Fig. 40) and therefore it is unlikely that these lines can be associated with any modes found using a perturbation formalism. Though the ‘binary star’ effect observed in the simulations has some qualitative features of the behaviour observed at CERN, no quantitative agreement has been found. There are several reasons for this: first, all these effects depend strongly on the shape of the wake function, and second, in order not to have the results of the numerical simulations contaminated by noise associated with a small number of particles, the total number of macroparticles needs to be increased significantly (from 5,000 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 identical Chapter 5. Beyond the Threshold 91 — .00 100 5 r _• 0 ..{‘.& ‘:° 0 o0 : 505 0 J.—A: :°° : :_ 0 longitudinal profile 5 0 phase space Figure 37: Snapshots of the phase-space distributions for the ‘binary star’ instability after each 10 turns ( 1/10 of a synchrotron period). Chapter 5. Beyond the Threshold 92 .20 .16 rJ) .08 .04 .00 IsO Figure 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 this model to describe the ‘binary star’ behaviour, but unfortunately, such a straightfor ward procedure does not work for the following reason. In the case of linear external forces one can rewrite the equation of motion for separate macroparticles in two equa tions in new coordinates, one describing the centre of mass motion and the other the relative motion q 1 — q. The equation describing the relative motion will not depend on the centre of mass coordinate, but since there is radiation damping q — q —* 0 eventually. 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 single bunch instability threshold [421. He suggests that at some intensity the stationary Chapter 5. Beyond the Threshold 7 X—1.4kHZ Ya——B7.434 d8Vrms POWER SPECI -48.0 8.0 93 ——ir— iOAvg IDly — O%Ovp_Han_ 1 r I oV ZZ_L 2 8 f —— -- - ; —112 i — - 1.4k Hz __1__ - 2.4k Figure 39: Spectrum of a quadrupole mode observed at LEP [6]. One can see low frequency sidebands which become stronger at higher intensities. Bunch length (psec) “‘ 0 -‘ I p 0 100 200 300 Bunch current iA) 400 Figure 40: Bunch length versus intensity at LEP[6]. 500 Chapter 5. Beyond the Threshold 94 distribution becomes energetically less favourable than an oscillatory solution. A sim ilar idea was later tried by K. Yokoya [43], who also suggested that at some intensity a dynamical solution of the Fokker-Planck equation should be more favourable than a stationary, time-independent one. 5.4 Conclusion We have developed a qualitative picture of the sawtooth instability. The wake field creates its own unstable and stable fixed points, particles diffuse to the second fixed point, and then the resulting sub-bunch collapses into the leading sub-bunch. The sawtooth frequency is therefore primarily determined not by radiation damping, but by the subsequent diffusion process. The sawtooth effect is most readily seen when the bunch length is comparable with the wake field length. Qualitatively quite different behaviours can be seen when the bunch is either short or long compared with the wake. In the former case, for exam ple, 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 very broad and complicated subject which is outside the scope of this thesis; this subject is, probably, a fruitful arena for future developments in instability theory. Chapter 6 Summary and Conclusions In this thesis we have studied the effects of short-range wake fields on the longitudinal motion of protons and electrons in synchrotrons. The main purpose of the thesis has been to show that the potential well distortion caused by short range wake fields has a significant effect on beam stability and cannot be ignored or approximated but must always be taken into account. We have shown that the results obtained by different techniques where the analysis is not done self-consistently can lead to incorrect 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 caused - by 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 possible to describe collective effects in the bunch using the Vlasov equation, and investigate the behaviour of the distribution function rather than individual particles. We have investigated two major cases where short-range forces play an important role • longitudinal space-charge forces, which are proportional to the derivative of the line density, for bunches with an arbitrary distribution of particles. This is a 95 Chapter 6. Summary and Conclusions 96 case important in proton synchrotrons, for which the bunch length is quite big compared to the size of the vacuum pipe, so the self forces are local, radiation effects are very small and there is no constraint on the distribution function. • effect of longitudinal short-range wake fields for bunches with a Maxwell- Boltz mann distribution. This case is typical for ultrarelativistic electrons in storage rings. The bunch length in this case is comparable to the characteristic length of the wake field. 6.1 Longitudinal Space-Charge Forces In the case of protons the bunch length is large compared to the vacuum pipe diameter and the short-range forces are approximately proportional to the derivative of the line density. We have found a way to determine stationary self-consistent distributions and have also derived a simple criterion which allows the intensity threshold to be found beyond which there is no stationary distribution. We have applied this method to the analysis of various distributions and obtained the following results: • In the case of the distribution j’ for which &‘(H) <0 a stationary distribution always exists below the transition energy, but above transition there is always a threshold beyond which no stationary distribution can be found. • For any distribution for which b(0) = 0 stationarity thresholds exist below as well as above transition. We have developed an algorithm to recover the distributioll function from a known line density in the presence of space charge. Different variations of this algorithm allow the synchrotron frequency to be determined from a given stationary distribution and vice versa. Chapter 6. Summary and Conclusions We have also found the threshold for instability caused by coupling of m 97 = +1 azimuthal modes. Surprisingly, the criterion for this threshold is identical to that for the stationarity threshold. 6.2 Longitudinal Stability of Electron Bunches A typical bunch length in the case of electrons in a storage ring is comparable to the diameter of the beam pipe and therefore the wake field has a characteristic length comparable with the bunch’s length. Bunch lengthening caused by short-range wakes has been studied using the Haissinski equation, which describes a stationary distri bution of ultrarelativistic particles in the presence of synchrotron radiation. The potential well distortion due to self forces creates a spread in the synchrotron frequencies of individual particles in a bunch and has an important effect on the stability of the beam. We have found that in some cases the synchrotron frequency has a local minimum, i.e dw/dJ = 0, and that the stability of the bunch depends essentially on the parameters of the distribution in this region. The existence of frequency spread makes it impossible to apply the standard sta bility analysis based on orthogonal polynomials, which assumes that the frequency of synchrotron oscillations does not depend on the amplitude; the threshold obtained in that approximation differs significantly from those found self-consistently. We have derived an integral equation which allows the stability analysis to be formulated as an eigenvalue problem. We have shown that a numerical method for solving the Vlasov equation proposed by Oide and Yokoya [25] can be derived from this 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 by the Oide-Yokoya method and found that below threshold there exist two types of Chapter 6. Summary and Conclusions 98 solutions: • A discrete spectrum of collective modes, such as the rigid dipole mode and the modes which are concentrated in the region where the derivative of synchrotron frequency dw/dJ = 0, • A continuous spectrum of ‘incoherent’, &like modes which cannot be seen indi vidually. Analyzing the behaviour of the collective modes in the case of broad-band resonator impedance we have come to the conclusion that instability can be caused by coupling between the collective modes such as the rigid dipole mode and the modes concen trated 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 whose eigenfrequency = 2w and therefore a threshold criterion can be written in the form 80 where w , ‘sO is a rigid dipole mode frequency which does not depend on intensity. 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 in stability will happen at the point where V’(q) = 0; this, however, occurs at higher intensities than the coupling between the quadrupole and rigid dipole modes. This hypothesis is in agreement with particle tracking simulations, and direct numerical solution of the Vlasov equation using the Oide-Yokoya technique, done for a broad-band resonator impedance over a wide range of resonator parameters. The instability caused by this coupling is very fast and easily seen in simulations. Chapter 6. Summary and Conclusions 6.3 99 Beyond the Threshold A computer program developed for checking the threshold criteria for electron bunches has been used to investigate electron dynamics beyond threshold. It has been found that the oscillations of the bunch shape have different features depending on the parameters of the wake field. Turbulent bunch lengthening and widening (increased energy spread) have been observed above the stability threshold. Some interesting phenomena have been seen when the bunch length is smaller than that of the wake field. At some intensities the bunch length oscillates in sawtooth fashion, 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 oscillate around each other in binary star fashion. The effects observed in these simulations exhibit features of some experimental observations at SLAC and CERN and indicate that strong nonlinear effects do play a significant role in existing accelerators. This means that the Vlasov equation which is often used for stability analysis nowadays may not be sufficient to describe these phenomena and different methods need to be developed. Bibliography [1] B. Zotter, Short is Beautiful, Proc. shop, KEK Report 90-21 (1990). th 4 Advanced ICFA Beam Dynamics Work [2] J. Haissinski, Exact Longitudinal Equilibrium Distribution of Stored Electrons in the Presence of Self-Fields, Nuovo Cimento 18B, 72 (1973). [3] P. Germain and H. Hereward, Longitudinal Equilibrium Shape for Electron Bunches with Various Self-Fields, CERN/MPS/DL 75-5 (1975). [4] B. Zotter, A Review of Self-Consistent Integral Equations for the Stationary Dis th tribution in Electron Bunches, Proc. 4 Advanced ICFA Beam Dynamics Work shop, KEK Report 90-21 (1990). [5] P. Wilson, R. Servranckx, A.P. Sabersky, J. Gareyte et al, Bunch Lengthening and Related Effects in SPEAR II, Trans. IEEE NS-24, 1211 (1977). [6] D. Brandt, K. Cornelis and A. Hoffmann, Experimental Observations of Instabili ties 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 in The 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 Length Instabilities in the SLC Damping Ring, Proc. 1993 Part. Acc. Conf., Washington D.C., 3240 (IEEE, 1993). [9] T. leiri, Bunch Lengthening Observed Using Real- Time Bunch-Length Monitor in the 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 a Non-Harmonic RF Potential, Particle Accelerators 17, 109 (1985). 100 Bibliography 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 Longitu dinales 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 Distribution under Space Charge Perturbations, Particle Accelerators, 11, 23 (1980). [16] G. Besnier and B. Zotter, Oscillations Longitudinales d’une Distribution Ellip tique, Couple’es par un Resonateur: Application au Calcul de l’Allongement de Faisceaux Intenses, CERN-ISR-TH/82-17 (1982). [17] A. Hofmann and F. Pedersen, Bunches with Local Elliptic Distributions, IEEE Trans. NS-26, 3526 (1979). [18] I. Kapchinsky and V. Vladimirsky, Limitations of Proton Beam Current in a Strong 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 Space Charge, Proc. 1991 Part. Acc. Conf., San Francisco, 1731 (IEEE, 1991). [20] M. D’yachkov and R. Baartman, Methods for Finding Stationary Longitudinal Distributions, 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 (t 2 x2)_h/2 and 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 Line Density of Particles in Proton Synchrotrons, INR Report 733/91 (1991). [23] R. Baartman and B. Zotter, Longitudinal Stability of Hollow Beams II: Mode Coupling Theory, TRIUMF Note TRI-DN-91-K177 (1991). [24] R. Baartman, S. Koscielniak, Stability of Hollow Beams in Longitudinal Phase Space, Particle Accelerators, 28, 95 (1990). [25] K. Oide and K. Yokoya, Longitudinal Single-Bunch Instability in Electron Storage Rings, KEK Report 90-10 (1990). Bibliography 102 [26] M. D’yachkov and R. Baartman, Method for Finding Bunched Beam Instability th Thresholds, Proc. 4 Europ. Part. Acc. Conf., London, 1075 (World Scientific, 1994). [27] S. Fang, K. Oide, Y. Yokoya, B. Chen and J.Q. Wang, Microwave Instabilities in Electron Rings with Negative Momentum Compaction Factor, KEK Report 94190 (1994). [28] R. Baartman and M. D’yachkov, Computation of Longitudinal Bunched Beam In stability 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 Proton Beams, 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 Synchrotron Frequency 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. In str. Meth., 203, 57 (1982). [40] K. Bane and K. Oide, Simulations of the Longitudinal Instability in the SLC Damping Ring, Proc. 1993 Part Acc. Conf., Washington D.C., 3339 (IEEE, 1993). [41] 5. Nagaitsev, On the Longitudinal Stability of Cooled Coasting Ion Beams, Proc 1993 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 A Distribution Functions A.1 Binomial Family Analytical solutions for U(q) can be found for the family of binomial distributions H f(H) oc [i m-1/2 (A.134) — One can easily calculate A(U) in this case: [1_Jm_ d 2 m 2(-U) 1 A(U) = 2J U (A.135) and so A(U) Lm [i — tm U] (A.136) where Lm -/27rmP(m+1/2) I’(l+m) A137 Writing Im = lLm, the equation for the potential well becomes U = Uo(q) + Im ([i m uj — — i) (A.138) It is easy to see that in the case of Uo(q) = q /2 we can find the solution of (A.138) 2 in the form q(U) for any m q= i2[U_Im ([U],n)] 104 (A.139) Appendix A. Distribution Functions 105 We also can find the solution in the form U(q) = g[Uo(q)]. (A.140) The analytical solutions for U(q) can be easily found for m In this section we assume Uo A.1.1 = = 1, 2, 3 and 4. /2, but one can choose any potential instead. 2 q Case m = 1: Hofmann-Pedersen Distribution In this case A(U) is a linear function A(U) = (1 1 L U) — (A.141) and [(1—U)—1] 1 U=-+I U(q) (A.143) 2(1+ I) = A.1.2 (A.142) Case m = 2 A(U) 2 [i L = — 2 U] (A.144) and — U(q) = U(q) A.1.3 Case m = = - + 2(1 + 12) ‘2 — (1]2 ([i — /4(i + 12)2 — 1) 2 2I2q (A.145) (A.146) 3 — A(U) [1 = ] . (A.147) An expression for U(q) can be found by using the Mathematica program by exe cuting the following script Appendix A. Distribution Functions Solve[U U == q2/2 + 106 13 ((1 - U/3Y3 - 1), U] II TeXForm; I. ‘/.C[3]] Since the equation we have to solve is of the 3 order, there will be 3 solutions; two of them are non-physical, and therefore the result for 13 < 0 is 272k U(q) = 3 (11664 I — (i + (i_) + ) (11664 — 17496 132 (1 + 13) + 2916 1 q 2 + 136O48896 I3 1 —174964 (1 + 13) + 29164 q 2 + 13:o48896 + 4+ (11664 I — 17496 i2 (1 (116644 174964 (1 + 13) + 2916 I q2)2) q2)2) + 13) + 29164 1223 13 A similar result can be obtained for m = 4, but it is too lengthy to record here. A.2 Hollow Distributions Analytical solutions for U(q) can also be found for the ‘hollow distributions’ which can be derived from the binomial family f(H) cc H’(1 — 2 H)m_h/ where rn> 0 and n > 0 are integers and m + n (A.148) 4. As example, let us consider the case f(H) = -H\/1 H (A.149) 3U 2 (A.150) - for which A(U) = 1 +2U — and the solution for U(q) is 1_21_\/(1_21)2±61q2 U(q) = 61 (A.151) Appendix A. Distribution Functions 107 Analytical solutions in the form q(U) can be found for any integers m, n, and also for the following family of distributions f(H) cx (A.152) where m is an integer. For example: f(H) (A.153) for which A(U) = (1 + 2 U) e_U (A.154) and the solution q(U) = /2 A.3 [u — I((i + 2U)e_U —1)]. (A.155) ‘Thermal’ Distribution The solution for ‘thermal’ distribution f(H) = ‘eH (A.156) for which A(U) = (A.157) is q(U) = 2 [U — l(e_U — 1)]. (A.158)
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Longitudinal instabilities of bunched beams caused by short-range wake fields Dʹyachkov, Mikhail 1995
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Title | Longitudinal instabilities of bunched beams caused by short-range wake fields |
Creator |
Dʹyachkov, Mikhail |
Date Issued | 1995 |
Description | This thesis investigates the effects of short-range wake fields on the collective longitudinal motion of charged particle bunches in circular accelerators, especially the onset of instability. At high intensity, a short-range wake field can distort the bunch potential well and thereby change the stationary distribution. It is shown that if this is not taken into account, instability thresholds will be incorrectly predicted. An integral equation derived from the linearized Vlasov equation is used to find the instability thresholds in the case of space-charge impedance alone for various distribution functions. The thresholds for instability caused by the coupling between the m = ±1 azimuthal modes have been obtained analytically for several common distributions. The criterion determining these thresholds appears to be the same as that for thresholds beyond which no stationary distribution can be found. A numerical method is also used to solve the linearized Vlasov equation for the self-consistent case, including distortion of the stationary distribution, and to find the thresholds. Physical explanations are provided for the eigenmodes and instability thresholds predicted by this method. This results in a much simpler stability criterion, which depends only on the stationary distribution and does not require solution of the linearized Vlasov equation. The behaviour of electron bunches beyond the instability threshold has also been studied using multiparticle tracking. Some interesting phenomena have been seen when the bunch length is smaller than that of the wake field. Under some conditions the bunch length oscillates in sawtooth fashion, i.e. slow relaxation is followed by fast blow-up. It has also been found that the bunch may split into two equal sub-bunches which oscillate around each other in binary star fashion. These effects may explain some recent observations in electron storage rings at SLAC and CERN. |
Extent | 2251866 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085657 |
URI | http://hdl.handle.net/2429/7226 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-11 |
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UBCV |
Scholarly Level | Graduate |
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