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Single-particle and collective effects of cubic nonlinearity in the beam dynamics of proton synchrotrons Tran, Hy J. 1998

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Single-Particle and Collective Effects of Cubic Nonlinearity in the Beam Dynamics of Proton Synchrotrons Hy J. Tran B.Sc, The University of British Columbia, 1991 M . S c , The University of British Columbia, 1993 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 August 1998 © Hy J. Tran, 1998 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. The University of British Columbia Vancouver, Canada Department Date Q C 4 - \S ; DE-6 (2/88) A b s t r a c t This thesis describes some new studies of the effects of cubic nonlinearities arising from image-charge forces and octupole magnets on the transverse beam dynamics of proton synchrotrons and storage rings, and also a study of the damping of coherent oscillations using a feed-back damper. In the latter case, various corrective algorithms were modeled using linear one-turn maps. Kicks of fixed amplitude but appropriate sign were shown to provide linear damping and no coherent tune shift, though the rate predicted analytically was somewhat higher than that observed in simulations. This algorithm gave much faster damping (for equal power) than conventional proportional kicks, which damp exponentially. Two single-particle effects of the image-charge force were investigated: distortion of the momentum dispersion function and amplitude-dependence of the betatron tunes (resulting in tune spread). The former is calculated using transfer maps and the method of undetermined coefficients, the latter by solving the cubic nonlinear equation of motion with the smooth approximation and time av-eraging. In the case of the C E R N Large Hadron Collider, neither effect was found to be of serious concern. Two collective effects caused by tune spread, decoherence and Landau damping, were studied, using bounded binomial (rather than gaussian) den-sity distributions to make the results valid for proton beams. The spread arises from a cubic term in the restoring force. To study decoherence, the motion of a transversely displaced beam bunch was determined using the Vlasov equation, and its centroid found by ensemble averaging; the displaced density distribution was Taylor-expanded in terms of the original one to simplify the integration boundary. The decoherence rate ii I l l was found to depend primarily on the average density gradient. The centroid motion is also amplitude modulated at the synchrotron tune. To study Landau damping of the weak head-tail instability, a perturbation technique was applied to the Vlasov equa-tion, and the dispersion-relation concept was augmented to handle many coupled radial modes by equating the inverse of the beam transfer function to the set of eigenvalues of the interaction matrix. The matrix elements are the overlaps of the impedance and the mode spatial spectra, while the eigenvalues are the mode frequencies. A new and rigorous derivation is given for the head-tail mode spectra for binomial distributions. In the case of an LHC-type bunch in the C E R N PS, several modes were predicted to be unstable, with growth rates compatible with the rather imprecise measurements. The octupole strength needed to Landau damp these unstable modes was estimated by mapping out the stability region in the complex frequency plane. C o n t e n t s Abstract ii Table of Contents iv List of Tables ix List of Figures x Acknowledgements xii Dedication xiii 1 Introduction 1 1.1 Synopsis 3 2 Single-Particle Motion 5 2.1 Transverse motion • 6 2.1.1 Transverse coordinate system 6 2.1.2 Strong focusing . 6 2.1.3 The Courant-Snyder parameters 8 2.1.4 Transfer matrix 11 2.1.5 Dispersion 12 2.2 Longitudinal Motion 13 2.2.1 Phase focusing 13 iv CONTENTS v 2.2.2 Synchrotron oscillations 14 2.2.3 Chromaticity 17 3 A Feedback D a m p i n g System 19 3.1 Introduction 19 3.1.1 Transverse damping system 20 3.2 Decrement and tune 21 3.2.1 Beam model 23 3.2.2 Kick functions 23 3.3 The proportional kick function 25 3.3.1 Eigenvector 28 3.4 Interpolated feedback 29 3.5 The constant magnitude kick function 30 3.6 Damping simulations using A C C S I M 35 3.6.1 Proportional kick function 35 3.6.2 Constant magnitude kick function 40 3.6.3 Amplifier kick function 42 3.7 Discussion 42 3.7.1 Critique 43 3.7.2 Performance comparison 43 4 The Image—Charge Force 45 4.1 Introduction 45 4.1.1 Images vs direct space charge 45 4.1.2 Tune and dispersion shifts 47 4.2 Historical survey 48 4.3 The image field 48 4.3.1 The pencil beam model 49 4.3.2 Image field in a circular beam pipe 49 CONTENTS vi 4.3.3 Image field between parallel plates 49 4.4 Linear image force 52 4.4.1 The dispersion function 52 4.4.2 Linear perturbation 54 4.4.3 Dispersion function of the FODO cell 56 4.5 Cubic image force 60 4.5.1 Incoherent tune spread 60 4.5.2 The smooth focusing approximation 61 4.6 Equation of motion with image force 62 4.6.1 Nonlinear image force in a circular beam pipe 62 4.6.2 Nonlinear image force between a pair of parallel plates 63 4.6.3 Elliptical equation of motion 63 4.6.4 Tune-shift and tune-spread calculations 64 4.7 Discussion 65 5 Descr ipt ion of Col lect ive B e a m M o t i o n 67 5.1 Introduction 67 5.2 The phase-space representation 68 5.3 Oscillation modes 70 5.3.1 Coasting beam modes 70 5.3.2 Single-bunch modes 71 5.3.3 Coupled-Bunch Modes 76 5.4 Head-tail instability 76 5.4.1 Weak head-tail 76 5.4.2 Strong head-tail 77 5.5 Transverse wake field and impedance 77 5.6 The Vlasov equation 78 5.7 The stationary distribution 80 5.7.1 The binomial distribution 81 CONTENTS vii 6 Decoherence and Recoherence 83 6.1 Introduction 83 6.2 Nonlinearity and chromaticity 85 6.3 Phase space solution 86 6.4 Dipole moment 87 6.5 Decohering dipole signal 89 6.6 Amplitude modulated signal 95 7 Landau Damping of the Weak Head-Tai l Instability 101 7.1 Introduction 101 7.2 Beam instability theory 102 7.3 Historical survey 103 7.3.1 Weak versus strong head-tail 104 7.3.2 Longitudinal development 105 7.4 The head-tail instability mechanisms 107 7.5 Landau damping mechanism 108 7.6 Derivation of the dispersion relation 109 7.6.1 The Vlasov equation 109 7.6.2 The dispersive integral equation I l l 7.6.3 The air bag distribution 115 7.7 The binomial distribution 117 7.7.1 Commentary 120 7.8 Landau damping by octupoles 120 7.8.1 Growth rates and frequency shifts of the LHC bunch 121 7.8.2 Landau damping of the LHC bunch 124 7.9 Discussion 125 8 Conclusions 128 8.1 Damping system 129 CONTENTS viii 8.2 Image-charge force 129 8.3 Decoherence 130 8.4 Landau damping of weak head-tail instability 130 B ib l iography 132 A p p e n d i x A : C u b i c equation of mot ion 140 A p p e n d i x B : B i n o m i a l mode spectra 142 L i s t o f T a b l e s 3.1 Damping constants and tune shifts of the proportional kick function with k = 0.003 mr ad/mm for various phase advances 36 3.2 Damping constants and tune shifts of the proportional kick with k = 0.01 mrad/mm for various phase advances 37 3.3 Linear damping rates and tune shifts of the constant magnitude kick function with Ay' = 0.005 mrad for various phase advances. 40 7.1 Transverse head-tail frequency shifts and growth rates of the L H C bunch. 124 ix L i s t o f F i g u r e s 2.1 Reference orbit and local transverse coordinate system 7 2.2 Representation of ID beam by phase space ellipse 10 2.3 Example trajectories in longitudinal phase space 16 3.1 A schematic transverse damper system 22 3.2 The kick function of a typical kicker driven by linear power amplifier. . 24 3.3 A synthesized damper system 29 3.4 The displacement of a bunch while its oscillations are being damped by the proportional kick function 38 3.5 The displacement of a bunch while its oscillations are being damped by the constant magnitude kick function 39 3.6 The displacement of a bunch while its oscillations are being damped by an amplifier kick function 41 4.1 Cross section of a circular beam pipe and location of beam and its image. 50 4.2 Cross section of a parallel plates pipe with beam 51 4.3 Schematic of a FODO cell 57 4.4 Dispersion of a typical FODO cell of the LHC 58 4.5 Change in the dispersion due to the image charge 59 5.1 Orbits of single particle motion 69 5.2 The lowest few transverse modes in phase space 72 x LIST OF FIGURES xi 5.3 Line density profiles of the lowest few longitudinal modes 74 5.4 Dipole signals of the lowest few head-tail modes 75 5.5 The binomial distribution for a =1/2, 1,2,3,4 82 6.1 Beam centroid from turns 0 to 1000 91 6.2 Beam centroid from turns 1000 to 2000 93 6.3 Beam centroid for a = 1/2 94 6.4 Amplitude modulation of the beam centroid signal 96 6.5 Amplitude modulation function for various widths of tune spread . . . 98 6.6 Amplitude modulation function for various sharpness 99 6.7 Amplitude modulation of the dipole signal with decoherence 100 7.1 Contour lines and stability boundary diagram for the L H C bunch. . . . 121 Acknowledgements I am particular grateful to Shane Koscielniak, who is my Triumf research supervisor, for his guidance and subtle encouragement throughout my research. His genuine interest in the research and constant enthusiasm for sharing his knowledge had inspired me. He had given generous assistance and many helpful suggestions which transformed this thesis into a far better one. I would also like to sincerely thank Michael Craddock, who is my faculty research supervisor, for his valuable advice and supervision. He had also given a lots of useful suggestions in the preparation and presentation of this thesis. I am fortunate to have Misha Lachinov and Drobin Kaltchev for colleagues. I appreciate their moral support and, above all, their friendship, which had made my time at Triumf an enriching experience. Drobin had been very generous in sharing his knowledge and experience, from which my research benefited and for which I am very thankful. My wife Donna deserves my deepest gratitude for her support, understanding, and, above all, for her love. I am indebted to my parents for giving me the opportunity and unfailing support throughout my studies. Words alone cannot describe my appreciation and admiration for the many sacrifices they had made for their children. I dedicate this work to them for all that they had done. xn Dedication To my parents xm C h a p t e r 1 I n t r o d u c t i o n In this thesis we describe some new studies of the effects of cubic nonlinearities on the transverse beam dynamics of proton synchrotrons and storage rings, and also of the damping of transverse coherent oscillations. In such machines the proton beam travels within an evacuated pipe, being guided by a series of dipole magnets, and contained or "focused" by quadrupole magnets. The beam also passes through radiofrequency (rf) resonant cavities, which are used either to accelerate the protons to high energy, or, in storage rings, to group them in regularly spaced "bunches", with the time structure desired for experiments. The individual particles in the beam are in constant motion even if it is in equilib-rium and showing no outward signs of change. When these motions maintain a definite phase relation, the beam can execute coherent motion. The beam leaves an electro-magnetic field in its path termed the wake field, which can produce a variety of effects ranging from benign (increased frequency spread) through minor perturbation of the guiding, focusing and accelerating fields, to destructive. For example, if the wake field due to coherent motion enhances that motion, then feedback leads to a coherent insta-bility, and possibly subsequent beam loss. Transverse instabilities, such as that driven by the "resistive-wall" impedance, are cured either by installing octupole magnets to promote Landau damping via increased frequency spread or by using active "dampers" 1 2 to kick the beam back on axis. A n example is the "head-tai l " instability in which there is a periodic variation of the transverse centroid along the bunch. Cubic nonlinearities may be caused by induced charge on the pipe wall (image-charge forces) or by octupole magnetic fields, and give rise to an amplitude-dependent spread in the "incoherent betatron tunes" (single-particle transverse oscillation fre-quencies). Frequency spread can, through the mechanism of de-phasing, dilute the beam phase-space density and hence reduce the beam quality, but it can also be used to counter instabilities by providing Landau damping. Another effect of the image-charge forces is to alter the "momentum dispersion", distorting the closed orbits for off-momentum particles. This thesis is organized in two parts. The first (Chapters 2 through 4) introduces the single-particle equations of motion and deals with three topics for which they are sufficient: • feedback damping systems. • distortion of the momentum dispersion function • amplitude-dependent oscillation frequency (tune spread) The second part (Chapters 5 through 7) deals with collective effects, for which a mult i -particle picture is required: the beam is modelled by a phase-space density distribution which is governed by the Vlasov equation. Two major effects are discussed, both pro-duced by tune spread: • decoherence and recoherence • Landau damping of the weak head-tail instability. Previous studies of these effects have almost al l been confined to beams of electrons, which, because of the quantum emission of synchrotron radiation, assume gaussian 1.1. SYNOPSIS 3 density distributions with long tails. To investigate proton beams it has been nec-essary to consider binomial distributions with a sharp boundary - which tend to be mathematically harder to handle. A major impetus for these studies has come from TRIUMF' s involvement with C E R N , the European Laboratory for Particle Physics, in providing the Canadian con-tribution to the 7 TeV x 7 TeV Large Hadron Collider (LHC), the world's highest energy accelerator, now under construction near Geneva. Although not an official part of the collaboration, it has been natural to apply our theoretical models to the C E R N accelerator complex, particularly the LHC and the 25 GeV Proton Synchrotron (PS), not only because of the connection, but because of the interesting scientific challenges posed by the very high brightness (phase-space density) required in the beams. As this thesis reports studies of several rather specialized effects, we delay a detailed account of previous investigations to the appropriate chapters, and confine ourselves to an introductory survey here. 1.1 Synopsis We begin in Chapter 2 with a brief introduction to the transverse and longitudinal mo-tions in circular accelerators, and the single-particle equations that govern them. The effects of momentum deviation, such as dispersion (orbit distortion) and chromaticity (change in the betatron tune), are also described here. Chapter 3 describes the working of a damper system for coherent transverse oscilla-tions in terms of transfer maps tracking the beam centroid turn by turn. Two different algorithms for the corrective kick - one proportional to the displacement, the other of fixed amplitude but appropriate sign - are analyzed and compared for effectiveness by both analytical methods and computer simulations. Proportional dampers have been studied previously, but this is believed to have been the first analytical study of fixed-amplitude dampers. 1.1. SYNOPSIS 4 In Chapter 4 we determine the linear and cubic perturbations due to image-charge forces for two geometries of metallic vacuum pipe. From this we determine the amplitude-dependence of the betatron tunes, an effect that has been extensively studied in the linear case, but for which the cubic terms have previously been neglected. We also calculate, for the first time, the distortion of the dispersion function by image-charge forces. The results are applied to give upper bounds for these effects in the L H C . In Chapter 5, we introduce the phase-space description of collective motion, describe the possible coherent oscillation modes of a beam, and introduce the Vlasov equation. In Chapter 6, the de-phasing of an initially coherent dipole oscillation due to amplitude-dependent frequency spread is studied. Formulae are derived for the de-cay of the coherent amplitude for the binomial family of particle distributions needed to describe proton beams, a case not previously treated. The modulation arising from coupling to the longitudinal motion is also included in this account. In Chapter 7, the theory of the weak head-tail instability is formulated so as to include Landau damping, and applied to find thresholds for high-order modes driven by the resistive-wall impedance. In particular, we investigate the octupole-magnet strength required to Landau damp the weak head-tail instability observed in the C E R N Proton Synchrotron when accelerating bunches suitable for the L H C . Again we consider binomial phase-space density distributions for both longitudinal and transverse planes, whereas previous studies have considered only gaussian distributions, which are not appropriate for proton beams. Chapter 8 summarizes the results of these studies and the major conclusions. C h a p t e r 2 S i n g l e — P a r t i c l e M o t i o n In this chapter we describe the essential single-particle motion of a beam in a storage ring for use in the following chapters. We use the term 'single-particle motion', to describe the motion of a particle under externally applied forces only, such as the guiding and restoring forces. We neglect its interactions with other particles in the beam, and we neglect wake-field effects that arise from objects that comprise and/or surround the vacuum chamber. We restrict our considerations to a beam in a storage ring. The particles are kept in 'circular' orbits by dipole magnets and focused transversely by quadrupole magnets. In the case of a coasting beam, the beam current is uniform around the ring and there are no longitudinal electric fields. In the case of a bunched beam, there are longitudinal fields but no overall acceleration; rf cavities provide focusing, and may also restore any energy losses through dissipative processes. In Section 2.1 we describe transverse motion and in Section 2.2 longitudinal motion. 5 2.1. TRANSVERSE MOTION 2.1 Transverse motion 6 2.1.1 Transverse coordinate system The transverse coordinate system of a storage ring is defined with respect to the ideal orbit. This is a closed curve inside the ring defined by the dipole bending magnets and straight sections. A particle whose energy is equal to the design energy, also known as the ideal particle, will follow this curve if its initial velocity is tangential to it. Location along the curve is parameterized by the arc distance s with respect to some reference point. At every point along this curve a right-handed coordinate system can be defined with unit vectors x, y and s. The directions of the unit vectors are illustrated in Figure 2.1: x points outward radially, y points upward perpendicular to the plane, and s points tangentially to the the ideal orbit in the direction of travel. The divergence or the slope of a particle's trajectory with respect to the ideal orbit is given by the derivative of the displacement with respect to s: x' = dx/ds and y' = dy/ds. The complete transverse phase space coordinates of a particle are given by (x,x') and M). 2.1.2 Strong focusing Without some means of transverse focusing, particles which stray from the ideal or-bit would eventually hit the vacuum wall and be lost. Most accelerators today use quadrupole magnets to provide strong focusing, also known as alternating gradient fo-cusing. This concept was independently discovered by Christofilos[2] and by Courant & Snyder[l]. When a particle strays from the ideal orbit, it is made to feel a restoring force and undergoes oscillations instead of straying farther away. For historical reasons, these oscillations are called betatron oscillations. The field gradients of quadrupole magnets provide a force which is proportional to the transverse displacement. However, there is a complication: these field gradients focus in one plane and defocus in the orthogonal 2.1. TRANSVERSE MOTION 7 2.1. TRANSVERSE MOTION 8 plane. When a quadrupole is rotated through 90 degrees, the focusing and defocusing planes are swapped. Net focusing in both planes is achieved by using a series of quadrupoles with alternating polarity, so that, on average, particles are further from the optic axis in focusing than in defocusing magnets. Hence the term alternating gradient focusing. 2.1.3 The Courant-Snyder parameters A storage ring is made up of many different elements. As far as transverse motion is concerned, the basic elements are the dipole and quadrupole magnets, and the drift spaces between them. Assuming that the motions of the two transverse planes are decoupled, one can consider motion in each plane independently and separately. The governing equation of motion is a second-order differential equation known as Hill's equation, + m * = o, (2.i) where K(s) represents the restoring force and varies from element to element. For a ring K(s) is periodic. The repeat distance may be as large as the circumference but is often less, since a ring is usually constructed of identical sections. The general solution of Hill's equation can be expressed in a form that is similar to the harmonic oscillator solution. It is customary in accelerator physics to write the general solution in the form x(s) = A,JpWcos[th(s) + 8], (2.2) where A and S are the two integration constants reflecting the initial conditions and the (3(s) function is one of the Courant-Snyder parameters a,(3,j derived in detail in e.g. [22]. A\ffi~ is similar to the amplitude of a harmonic oscillator and 5 is the initial phase. The phase advance >^(s) is given by the integral Jo s dt Wr  (2'3) 2.1. TRANSVERSE MOTION 9 where t is the path length. Hence, the 8(s) function may be interpreted as the local wavelength of the oscillation divided by 2TY. An important quantity associated with phase advance is the tune, usually denoted v in the U.S. and Q in Europe. It is the number of betatron oscillations per one complete turn around the ring, and given by ( 2 ' 4 ) Z7T where ift is the phase advance per turn. In the foregoing discussion we assumed that the restoring forces were linear. In practice, the magnets have field imperfections and moreover, sextupole and octupole magnets may be deliberately incorporated into the ring. Consequently, the betatron oscillation frequency depends on amplitude and there will be a spread of tunes. It should be noted that if the tunes approach a quotient of integers, magnetic imperfections in the ring will be encountered repeatedly at the same betatron phase, allowing perturbations to grow indefinitely. The physical significance of the constant A is that it is related to the amplitude of the betatron oscillation and the size of a beam. This can be seen if we look in the (x, x') phase plane. An equation for x'(s) can be obtained by differentiating Equation 2.2. Now consider the two equations for x and x' at fixed azimuth S\ and notice that these two equations represent a parametric tilted ellipse in the (x, x') phase plane (Figure 2.2). A particle returning on successive turns will trace out points on the ellipse as ij>(s) increases. The ellipse can also be thought of as representing a group of particles with a given A but various initial phases 5. It can be shown that the area of the ellipse is irA 2. For different locations around the ring, the ellipse will have different shapes and orientation, but it will have the same area. A beam of particles with all phases of oscillation and all amplitudes up to a maximum, say, A , can therefore be described by points filling an ellipse of area IT A 2 . In accelerator parlance the beam is said to have an emittance e = 7 r A 2 . From Figure 2.2, the width of the beam at s is given by yjd(s)e. The envelope of the beam is obtained by plotting the width as a function of s. For later use, we introduce the other Courant-Snyder parameters ct(s) and 7(5) TRANSVERSE MOTION Figure 2.2: Representation of ID beam by phase space ellipse. 2.1. TRANSVERSE MOTION 11 which may be defined in terms of (3(s): a(s) 7 ( 5 ) = IdS ~ 2 d J ' 1 + a 2 (2.5) (2.6) a(s) has a simple physical interpretation, being related to the local divergence or convergence of the beam envelope. 2.1.4 Transfer matrix An alternative, but equivalent approach, to the Courant-Snyder formulation is to prop-agate particle motions using transfer matrices. The solution of any linear differential equation can be expressed as a transport matrix which 'transfers' the initial coordi-nates (in the phase plane) to the final coordinates. If in each element K is constant, the motion of a particle is easily solved and the motion around the ring can be com-puted piece-wise. This can be systematized by introducing the column vector {x,x') and propagating this by operating on it with a transfer matrix, (2.7) The transfer matrix for each individual element is formed by solving for the element output given the intial conditions (1,0) and (0,1). The determinant of the matrix is unity because the restoring force is conservative. To find the overall transfer matrix for a segment of the ring, one then multiplies the matrices together, in order. Indeed, a one-turn map can be constructed at any point of interest in the ring to track the coordinates of a particle turn-by-turn. The elements of a transfer matrix M from Si to s2 can also be expressed in param-eterized form in terms of the Courant-Snyder parameters [6]: ^•(cos Aip + a?i sin Aip) X a b X x' 2 c d x' Pi v7?i& sin AV> M^r sin Atp + ° ^ 2 = cos Aip \P§-(cos Aip - ax sin A^ >) (2.8) 2.1. TRANSVERSE MOTION 12 where the subscripts 1 and 2 refer to the entrance and exit locations of the segment, respectively. Computation of the periodic Courant-Snyder parameters for a complete ring is obtained by setting si = s and s2 — s + C, where C is the circumference of the ring, cos ip + a(s) sin ip 3(s)s'mip —7(3) sin ^ ) cos 1}) — a(s) sin tp and comparing the resulting matrix element by element with the one-turn map com-puted by matrix multiplication of the individual ring elements. The phase advance ip for a complete turn about the ring is related to the trace of the one-turn matrix by 2cos^> = Tr{M} . Note that if the focusing is too strong, and the trace exceeds 2, then the phase advance becomes complex, and the motion is unstable; this is called 'over-focusing'. (2.9) 2.1.5 Dispersion The dispersion function describes the closed orbit of an off-momentum particle relative to the ideal orbit. Momentum dispersion is a consequence of the fact that a particle of higher momentum is deflected through a smaller angle in a bending magnet. For a particle whose momentum p deviates from the ideal momentum po, the equation of motion is an inhomogeneous Hill's equation: £ + (2.10) ds 2 p(s) po where p(s) is the local radius of curvature of the ideal orbit, and Ap = p — po- i n dipole magnets, p is their bending radius, and elsewhere (such as drifts and quadrupoles) p is infinite. The dispersion function, d(s), defined by x(s) = d(s)^-, (2.11) Po is the periodic solution of this equation such that d(s + C) = d(s) where the period length C is the ring circumference. A procedure for finding the periodic dispersion in 2.2. LONGITUDINAL MOTION 13 terms of the Courant-Synder parameters is described in Ref. [6]. Here we only quote the final expression: W*) = / cos t s ' ) - ~ ™W (2.12) z sin nv Js p[s ) which is suitable for numerical evaluation. Note that the path length L along this dispersed orbit is not necessarily equal to that of the reference orbit. In Chapter 4 we will introduce a Green's function method for obtaining the dispersion using a vector formulation. 2.2 Longi tudina l M o t i o n 2.2.1 Phase focusing Acceleration of particles to very high energies involves the use of high-frequency reso-nant cavities for the production of accelerating fields. In a ring, there is typically more than one cavity, but since the particles are accelerated on each turn, only moderate accelerating voltages and few cavities are required. This is the major advantage of circular accelerators over linear ones, where many cavities are needed. In a storage ring, there is no overall acceleration; individual cavities accelerate or decelerate the particles (depending on arrival rf phases) to provide longitudinal or 'phase focusing'. In contrast to the transverse motion with alternating gradients, all cavities are focusing. Phase focusing was independently discovered by Veskler[4] and McMillan [5] . To illustrate phase focusing, it is useful to define a synchronous particle as one which follows the ideal orbit and is so timed that it stays at the exact design energy in the presence of the rf cavities. For simplicity assume that there is only one cavity in the ring and that it runs with a sinusoidal voltage V(t) = Vo s'm(hu>0t) across the accelerating gap - where UIQ is the angular revolution frequency and h is an integer. A synchronous particle arrives at the cavity with 'synchronous phase' either 0 or ir 2.2. LONGITUDINAL MOTION 14 so that the particle is not accelerated and stays on the ideal orbit. The voltage slope has opposite signs for 0 and ix. Now consider a general particle whose energy may deviate from the design energy, so that its arrival at the cavity may precede or follow the synchronous particle. This particle will not be lost in the long run, if its next arrival time is corrected rather than allowed to deteriorate. This is the principle of phase focusing. The correction is done by accelerating or decelerating the particle as required using the sinusoidal variation of the cavity voltage. 'As required' depends on the properties of the ring: a particle with extra kinetic energy will travel faster but on a different path from the synchronous particle, and so the orbital period may decrease or increase. The reference energy at which these competing effects are equal, is known as 'transition' and is determined by the particulars of the magnet lattice. Below transition the 'speed-up' wins over changes in path length, and the correct voltage slope ensures that a late particle gets a little extra energy so that it will arrive a little earlier the next time round; and an early particle loses a little excess energy so that it will arrive a little later. In effect, what we have is a restoring force. A displacement in phase or energy relative to the synchronous particle does not continue to grow but causes the particle to undergo oscillations about it. These oscillations are called synchrotron oscillations. 2.2.2 Synchrotron oscillations In this section, we introduce the longitudinal coordinate system and outline the deriva-tion of the equation of synchrotron motion. A synchronous particle whose energy is Es arrives at the same r.f. phase (j)s (modulo 2n) each time and hence receives the same increment of energy (0 for a proton storage ring). For a particle with some deviation in energy or, equivalently, Ap in momentum, the time taken between successive arrivals will differ by A r from the synchronous time r = 2n/u>o. The time difference is due in part to path-length difference and in part due to velocity difference. The net effect is 2.2. LONGITUDINAL MOTION 15 described by — = " — , (2-13) T p V=-2-~, (2-14) It 7 where 7 has the usual relativistic meaning; rj is called the slip factor and changes sign at the transition energy when 7 = j t . Now we have all the preliminaries needed to write down the difference equations that govern the phase and energy of an arbitrary particle. Suppose a particle arrives at the nth revolution with energy En and phase (f>n, then at the (n + l ) th revolution the phase and energy would be Ap 4>n+\ = 4>n + hu0r)T , (2.15) Pn En+i = En + eV0sm(f)n, (2.16) w here Vo is the amplitude of the e.m.f. across the cavity gap. Since E = ^mc and p = j m v , one can show that p E where v is the particle velocity, c is the speed of light and AE = E — Es. Hence the two difference equations for motion relative to the synchronous particle are 4>n+l = <Pn ^ ^—AEn+i (2.18) andAjG n + i = AEn + eVo(sin</>n - sin<^s). (2.19) These equations can be iterated numerically and the results of each iteration can be plotted to track the phase-space trajectory of a particle. A traditional analytical approach is to approximate the difference equations by differential ones. The legitimacy of treating phase and energy as continuous variables is justified by the fact that these dynamical variables change by rather small amounts 2.2. LONGITUDINAL MOTION 16 4 > ( r a d i a n s ) Figure 2.3: Example trajectories in longitudinal phase space. 2.2. LONGITUDINAL MOTION 17 from turn to turn. Substituting one equation into the other, we obtain a single second-order nonlinear equation, d2<b 2rjTrhTeV0c 2 . d^ = v2Ea  (2" 20) assuming Vo is constant and taking s'md)s = 0 for a storage ring. Solutions of this equation give the particle trajectories in d> — AE phase space shown in Figure 2.3. Assuming that the displacement from the stable fixed point at <bs = 0 is small, we can linearize the above equation, putting it in the form ^ + ( 2 ^ ) ^ = 0, (2.21) where us is the number of synchrotron oscillations per turn, more often referred to as the synchrotron tune. This quantity is given by U'  = y-2^ET' ( 2 - 2 2 ) Figure 2.3 shows a few trajectories for different initial (d>, AE) when <j>3 = 0 or 7r. This is the case in which the synchronous particle is unaccelerated as in a storage ring. There is a well-defined boundary around each fixed point, called the separatrix, between regions of stable and unstable motion. The figure displays the circumstance that the harmonic number is generally greater than one and so there can be many stable fixed points. Three are shown. For an unaccelerated beam, these regions are called stationary buckets. For the case of accelerating buckets, a particle outside the separatrix diverges in both energy and phase and ultimately will depart from the accelerator. A l l buckets need not be populated by particles. A collection of particles of various energies and phases sharing a particular bucket is called a bunch and a bunched beam is made up of a train of such bunches. As for transverse motion, the area of phase space occupied by particles is known as the emittance. 2.2.3 Chromaticity Chromatic effects are caused by the momentum dependence of the focal properties of the magnetic quadrupole lenses. A more energetic particle is less focused than a 2.2. LONGITUDINAL MOTION 18 less energetic one. This effect leads in turn to momentum dependence of the betatron frequency. To first order, this dependence is quantified by the linear chromaticity £ according to the definition (2.23) where vx<y is the betatron tune of a particle with zero momentum deviation (Ap/p0). Because, for a bunched beam, the momentum deviation undergoes synchrotron oscil-lations, the betatron oscillations are modulated. The modulation is very slow because the synchrotron frequency is much smaller than the betatron frequency. Po Ap Po C h a p t e r 3 A F e e d b a c k D a m p i n g S y s t e m 3.1 Introduct ion Beam oscillations are often excited inadvertently. An injection error causes a beam to undergo coherent betatron oscillations. These are also known as dipole oscillations as the beam moves from side to side. If the oscillation amplitude is not quickly reduced, a tune spread will cause the individual particle oscillations to de-phase (or decohere[7]) leading to phase-space filamentation. This accumulates with time, and will increase the effective emittance area, leading to reduced beam quality. In fact, if fuamentation is great enough the coherent motion 'washes out', the dipole signal vanishes and damping stops. Hence a damping system must act quickly if it is to preserve beam quality. Another problem is seeding of coherent instability. For a bunched beam, the bunch train leaves a series of wake fields in the surrounding structures which can act back on the train so as to couple the bunches' motions. This coupling may drive any incipient oscillation. When conditions are right for a particular multi-bunch oscillation mode, it can quickly develop into an instability. These oscillations are seeded by injection errors and magnet power-supply ripple, etc.. The growth rate of the instability is proportional to the initial excitation and so the initial growth can be fast. For this reason also, it is important to provide fast damping. 19 3.1. INTRODUCTION 20 To summarize, dampers are used for two tasks: (i) to correct injection errors and (ii) to combat coherent instability. Ideally, two dampers should be used, with each optimized for a single task, but more often one damper serves both purposes. A problem arises if a damper intended to combat instability (later in acceleration) is used to reduce injection errors, because its response will saturate for large amplitudes. The principal subject of this chapter is to investigate the damping rate when an instability damper is operated in the saturated, nonlinear regime, so that its performance as a corrector of injection errors is pushed to the limit. 3.1.1 Transverse damping system Damping systems have been discussed for many years. Indeed there are descriptions of complete systems for several machines, such as the ISR[7], CPS[13], SPS[12] and PETRA[11], etc.. Figure 3.1 shows the essential components of a typical damping system: a beam pick-up monitor and a kicker at a betatron phase advance ipi down-stream. The 'kicker' is a device which imparts a transverse impulse to the charged particles using electric and/or magnetic fields. Some possible designs are discussed by Cappi[13]. As the bunch passes the pick-up, it induces a signal which is proportional to its displacement. The signal provides the necessary feedback information to adjust the polarity and kick strength that is subsequently applied to the bunch. The kick changes the beam angle with the optic axis but not its transverse position. A kick in the right direction reduces the divergences of all the particles in the bunch by a small amount. The result is that the bunch oscillation amplitude is decreased or 'damped'. Typically, many kicks are required to completely damp the oscillation because of power amplifier limitations. 3.2. DECREMENT AND TUNE 3.2 Decrement and tune 21 The damping rate depends on the 'kick function' which relates the beam signal to the applied kick strength. Part of this kick function is the amplifier response. The two most common functions are the proportional kick and the constant magnitude kick; the latter offers the possibility of faster damping. In addition to damping, a feedback system also perturbs the coherent tune of a ring. The tune shift depends on the kick function. Because the tune must avoid structural resonances, a small shift can have important consequences. For the case of the proportional kick function, previous analytical treatments[8] [9] have found the damping rate and tune shift. We propose our own variation[17] on this theme in Section 3.3, based on the one-turn-matrix: the solution of the damped oscillations is obtained by solving the eigenvalue problem of a modified one-turn map that includes the kick. The equations of motion have also been solved numerically in multi-particle simulations by several authors[7],[9],[14], [16]. Damping with digital filtering has been discussed in Ref. [15]. For the case of constant-magnitude kicks, or 'bang-bang' damping as it is sometimes called, one must avoid temporary over-kicking and anti-damping. In this scheme, only the polarity of the kick depends on (the sign of) the beam displacement. Given that the betatron phase of the beam centroid changes from turn to turn in a pseudo-random pattern (because of the insistence on non-integer tunes) it is difficult to formulate a simple analytic expression for the cumulative effect of the 'bang-bang' damping, and Willie[9] and Ivanov[10] both resorted to numerical recursion. In Section 3.5 we give an analytic treatment [17] of this problem based on expanding the motion in terms of eigenvectors of the one-turn map (i.e. the transfer matrix for one complete turn around the ring without kicking) and leading to an approximate expression for the decrement. Finally, in Section 3.6 we compare our results against computer simulation. Subsequent to our initial publication[17], we found that Xu[18] gave an expression for damping decrement assuming phase averaging. Later, Sagan[19] re-derived the 3.2. DECREMENT AND TUNE 22 Figure 3.1: A schematic transverse damper system. same result: he gave expressions for damping decrement and tune-shift of a 'bang-bang' damper under the assumption of perfectly random betatron phases; in which case phase averaging may be used to determine the average reduction in the oscillation amplitude per kick. This method gives satisfactory results, but cannot display any of the resonant phenomena that occur when the tune is near a ratio of integers. Nor can it give a detailed picture, turn by turn, of the evolution of the centroid vector x = (x, #'). We start by introducing a single-particle model of the bunch, followed by discussion of possible kick functions. The major part of the chapter is devoted to the derivations of the damping rates and tune shifts. The chapter concludes with a critique of our simple analytic expression for the progression of 'bang-bang' damping. For the purposes of this chapter and Figure 3.1, the quantities x, y, z are not orthogonal coordinates, rather they refer to displacements in a single plane but at three consecutive locations around the ring. 3.2. DECREMENT AND TUNE 23 3.2.1 Beam model We need a simple but adequate model of a charged particle bunch. Though a bunch consists of a collection of particles undergoing betatron and synchrotron oscillations about its centroid, we shall focus solely on the centroid motion. This can be represented by a single macro-particle at the centroid and (in the absence of wake fields) described by the single-particle equations of motion. For simplicity, we assume that the betatron and synchrotron oscillations are decoupled from each other, as are motions in the two transverse planes. Hence, each can be considered separately and independently. This reduces the problem to one dimension. We assume that the beam pick-up is linear and that the dipole signal induced in the pick-up is proportional to the centroid displacement. If there is linear coupling between planes, the analysis below (of proportional kick-ing) can be extended by considering 2 pick-ups and 2 kickers (one horizontal and one vertical set) and adopting 4x4 transfer matrices to describe the complete 2-dimensional motion and the kick strategy. In general, one anticipates that the eigenvectors will be rotated with respect to pure horizontal and vertical directions. 3.2.2 Kick functions The rate of damping and the tune shift depend on the kick function being used. In this section, we introduce two idealized kick functions that are the limits of a linear amplifier's kick function. A kick produces a change in the bunch mean divergence y' according to the kick function K(x) A y ' = K(x), (3.1) where x is the bunch displacement at the pick-up. Figure 3.2 shows a realistic kick function for a system in which the kicker is connected to the output of a linear amplifier and the pick-up is connected to its input. The range between ± x m a x corresponds to 3.2. DECREMENT AND TUNE 24 A kick strength Ay' Figure 3.2: The kick function of a typical kicker driven by linear power amplifier. the linear range of the amplifier in which the kick strength Ay' is proportional to the displacement. For displacements greater than ±xmax, the amplifier response saturates and the bunch receives the maximum kick strength ±Ay'max, but no more. For the kick function of Figure 3.2, a bunch with initial oscillation amplitude larger than xmax will experience a combination of proportional and constant magnitude kicks in a pseudo-random sequence. The motion in this case is not simple and does not have an analytic solution. However, there are two cases where the kick function is simple. If the oscillation amplitude is smaller than xmax, then all kicks are proportional and the kick function can be written: Ay' = kx, (3.2) where A; is a proportionality constant. A second case results if the dependence on the magnitude of the displacement is removed but the sign dependence is retained. A l l 3.3. THE PROPORTIONAL KICK FUNCTION 25 kicks are then constant magnitude: Ay' Ay'max x > 0 , (3.3) This is called the constant-magnitude kick function. 3.3 The proport ional kick function In order to obtain the damping rate and tune shift, we first have to derive an expression for the displacement of a bunch at a point of observation as function of time or turn number. This is done by finding the one-turn transfer matrix which includes the kicking. The damping rate and coherent tune are given by the eigenvalues of this matrix. For completeness sake, in Section 3.5 we give also an expression for the turn-by-turn beam displacement in terms of the eigenvectors of the matrix. We use the vector notation x to denote the coordinates (x, x') of the bunch. In Figure 3.1, the vectors x, y and z are the coordinates of the bunch at the pick-up, before the kicker and after. On each turn the bunch is kicked with a strength proportional to its displacement at the pick-up: A y l = kxn, (3.4) where the subscript n denotes the turn number. The coordinates of the bunch at the pick-up from the nth to (n + l)th turn are be obtained by mapping x n plus the kick y'n. After the kick, the coordinates of the bunch are given by z„ = M i x „ + K x n , (3.5) where M i is the transfer matrix from the pick-up to the kicker and K the kick matrix, K = I I . (3.6) 3.3. THE PROPORTIONAL KICK FUNCTION 26 To obtain x n + i , multiply zn with the rest of the one turn map M 2 which defines the section of the ring from the kicker to the pick-up, V i = {M 2(M 1 + K) }x„ (3.7) The result is a linear one-turn map of the previous coordinates. Notice that the one-turn map M without the kicker is equal to M 2 M i . Thus starting from the initial coordinates x0, we have M m x 0 (3.8) where M'(= M + M 2 K) is the new one-turn map including the damper system. We have a matrix equation which can be solved by finding its eigenvalues and eigenvectors. If e is an eigenvector of M ' with eigenvalue A then and M'e = Ae, M e = Ane. (3.9) (3.10) From the form of the above expressions, exponential damping occurs when |A| < 1. Let * denote the complex conjugate. The eigenvalues A and A* are the roots of the characteristic equation of the matrix M ' det(M' - AI) = 0, and the corresponding eigenvectors satisfy (M' - AI)e = 0. (3.11) (3.12) The elements of matrix M ' are / M' = y c + k (81'/9i)1/2 (cos *02 - ai sin ip2) / a + a' c + c' d (3.13) (3.14) 3.3. THE PROPORTIONAL KICK FUNCTION 27 where a, 6, c, and d are elements of the one-turn transfer matrix M . As shown in Figure 3.1, (3 and f3\ are the betatron functions at the pick-up and kicker respectively, and ip2 is the phase advance from the kicker to the pick-up. This is related to the one-turn phase advance I/J of the ring by ip = ipi + tyi- Taking the determinant according to Equation 3.11 gives the eigenvalue equation 1+a'd-bc' - (2cos?/> + a')A + A 2 = 0, (3.15) where we have made use of det(M) = l and 2 cos V> = (a+d). Substituting A as exp(—a+ i fi) and separating the equation into its real and imaginary parts gives 1 + g - e~ a cos fi (2 cos %b + a') + e~ 2a cos 2fi = 0, (3.16) -e~ a sin fi (2 cos xj} + a') + e~ 2a sin 2\i = 0 , (3.17) where g = a'd — be'. We use a negative exponent (-a) in anticipation of damping. Solving Equation 3.17 gives the real part of A, 2 cos V> + a' e cos fi = , (3.18) and substituting it into Equation 3.16 we have e~2a = i + a'd-bc', (3.19) = 1 - f c ( / ? / ? i ) 1 / 2 sin V>i , (3.20) after some simplifications. It is clear from the above expression that the damping constant is an increasing function of (PPi)1/2 sin tpi. This means that in order to obtain the maximum damping constant, for a given k, the pick-up and the kicker should ideally both be placed at [ 3 m a x , , and also 7r/2 apart in betatron phase. Such a severe constraint on location is usually difficult to satisfy in practise. Substituting Equation 3.20 into Equation 3.18 and solving for the one-turn phase advance fi, we have 2cos^-fc(/?/? 1 ) 1 / 2 sin^ 2 C O S » = 2 [ l - M W / 2 s i n ^ ] 1 / 2 " ( } 3.3. THE PROPORTIONAL KICK FUNCTION 28 This equation correctly gives the one-turn phase advance ip of the ring when there is no damping, that is, when k = 0. When the damper is on, the one-turn phase advance \x of the bunch differs from and the difference divided by 27t is called the tune shift Au. If all other parameters are held constant, the least tune shift occurs when ipi = ir/2. This phase advance also gives the maximum damping constant. Thus the most desirable tune shift and damping constant can be achieved with the same phase advance. 3.3.1 Eigenvector Because M ' is a two-by-two real matrix, it has two eigenvalues and its eigenvectors are complex conjugates of each other. The two independent complex solutions of the matrix equation are x l n = Ane, (3.22) x 2 n = A*ne*. (3.23) The general real-valued solution of xn can be constructed from a linear combination of the real part and imaginary part of either x i n or x 2 n because the real and imaginary vectors of e form a complete set. Let us write e = u + i v, where u and v are the real and imaginary components, respectively; then we have x l n =e"an(ucosn/i - vs inn^) + ie~an( u s i n n ^ + vcosn/i). (3.24) Forming a linear combination of the real and imaginary parts gives the general solution, which can be written as an exponential factor times an oscillatory term to give xn = e~an{c1( u cos n\i — vsinn/i) + c2( usinn/^ + vcosn/i} , (3.25) where c\ and c2 are real coefficients. Setting the solution to the initial condition n = 0 gives x0 = ciu + c 2 v, (3.26) which can be used to find the coefficients. 3.4. INTERPOLATED FEEDBACK 29 C a l c u l a t e d Figure 3.3: A synthesized damper system. 3.4 Interpolated feedback Ideally one would like to place the components of a damper system such that the product QQ\ obtains its maximum value and ipi is 7r/2. This gives the largest damping constant for a given k with the smallest tune shift. In practice, this may not be possible due to lack of space or location in the ring. One solution is to infer the bunch displacement at the desired pick-up location using measurements at two pick-ups placed either side of the ideal location, where space permits, as shown in Figure 3.3. However, when possible, the two pick-ups should be placed close to the ideal location so as to minimize errors because the actual (rather than theoretical) transfer matrices are usually not known with great precision. With the constraint on pick-up location removed, one strategy to maximize the damping constant is to place the kicker at as large a /?i value as possible. 3.5. THE CONSTANT MAGNITUDE KICK FUNCTION 30 Two pick-ups are needed because a single device measures only one of the two dy-namical variables, namely the displacement. Referring to Figure 3.3, the displacement x of the bunch at the desired location can be calculated from its displacements xi and X2 at two other locations, x = P J X J and x = P 2 xx2 (3.27) where Pi is the transfer matrix from pick-up # 1 to the desired location and likewise P2 1 from pick-up # 2 . Equating the two equations above to eliminate x, we have P2-1x2 = P1X1 . (3.28) This is a system of two linear equations with two unknown variables x[ and x'2. To obtain x, first solve the above system for x[ or x'2 and substitute their values into either of Equations 3.27. 3.5 The constant magnitude kick function We need first to obtain an expression for the beam displacement in terms of the eigen-vectors of the one-turn map. The constant magnitude kick function contains the sign (sgn) function of the displacement, which is not analytic in the usual sense. The com-plete one-turn map, including the kick, contains the one-turn transfer matrix and a summation over all previous revolutions (indexed by k) of the function sgn(x^). The result is that the eigen-expansion of the displacement includes a summation of signed terms which can only be put in closed form by making some approximations. The one-turn advance of x„ can be written: x n + i = M(xTl + sgn(xn)A) , (3.29) where A = Mj"^ 0, Ay'max)T is the kick back-propagated to the pick-up. Using the above relation recursively n times, xn can be obtained from the initial coordinates x0. 3.5. THE CONSTANT MAGNITUDE KICK FUNCTION 31 The displacement at the nth turn is: n-1 x n = M n x 0 + YI s g n ( x * ) M n - * A . (3.30) fc=0 The first term is the one-turn map of the ring applied n times without the kick. The second term is the sum over the n — 1 kicks propagated around the ring the appropriate number of times. Expressing x 0 and A in terms of the eigenvectors ei and e 2 (= e*) of M , we write: x 0 = c;e,-, (3.31) A = dtei, i = {l,2}. (3.32) Equation 3.30 becomes n-1 <n = A? ' (3.33) c;e; + Y \ ksgn(xk)diei k=0 where A i i 2 are the eigenvalues of M . A i i 2 can be written in terms of the one-turn phase advance ip as A l j 2 = exp(±i^>). For x n to be real, both C i i 2 and dl:2 must be complex conjugate pairs, respectively, because e i ) 2 are complex conjugate pairs. In the phasor representation, the conjugate pairs C i ) 2 and d i ; 2 can be written as ci = C exp (id)), c2 = C exp (—id)), (3.34) di = D exp (i6), d2 = D exp (-i6), (3.35) where C and D are real. Using the results above, Equation 3.33 can be written: n-1 x n = exp{z(m/> + <£)}[C + Dj2exp{-i(k^ + (f)-0)}}(u-riv) (3.36) A;=0 n-1 + exp {-i(nij> + d))}[C + Dj2 exp {i(k^ + d> - 0)}](u - iv ) , k=0 where ei = u + iv and e 2 = u — iv . Expressing the complex exponential function in terms of sin and cos, and rearranging we have x „ = 2C{ucos(mp + 4>) — vs'm(nip + </>)} (3.37) 3.5. THE CONSTANT MAGNITUDE KICK FUNCTION 32 n - 1 + 2/J{ucos (nip + </>) — vsin(w0 + 0)} X/  s&n(xk) cos(kip + A >^) k=0 n - 1 + 2/){u sin(nf/' + 4>) + v cos(nip + </>)} ^ s g n ^ ) sin (fc^ > + A</>), A:=0 where A(p = <p — 9. Given the known eigenvalues, the eigenvectors e i | 2 can be obtained by solving ( M - A l i 21)e l i 2 = 0, (3.38) which gives ( h \ ( 0 \ (3.39) b u = | I and cos ip — a y sin^> j The eigenvectors are not orthogonal and cannot be rotated because they form a con-jugate pair. They are linearly independent of each other and are unique to within a real-valued scaling factor. In the representation shown above, the imaginary displace-ment is zero and, because the vectors cannot be rotated, it w i l l always remain zero. This reduces the number of terms in the eigen-expansion of the displacement by half. Substituting the components of the eigenvectors into Equation 3.37 and collecting al l the terms in the first row, we have for the displacement xn xn = (1, 0) • x „ (3.40) = A cos {nip + <f>) n - 1 + A cos (nip + 4>) ^2 sgn(xk) cos (kip + A<p) n-1 + A sin (nip + 4>)  sEn(xk) sin (kip + A<p), k=0 where A = 2Cb is the ini t ia l oscillation amplitude, and A = 2Db is the resultant displacement of the bunch due to the kick backward-propagated. We assume both A and A are known. When the kick is independent of amplitude, one can profitably think of the damping as occurring not by changing the divergence, but by changing the closed orbit (co.) each turn to bring it closer to the displaced beam. Given that it is only the c o . which changes we should expect no tune shift. Hence to first order, 3.5. THE CONSTANT MAGNITUDE KICK FUNCTION 33 we can substitute the unperturbed oscillation xm = C cos(mip + cp) in sgn(xm) on the right hand side of Equation 3.40. In order to evaluate the sum, we expand sgn[a;m] by the cosine series O O A sgn(xk) = £ n { 2 . _ 1 } cos{(2j - l)(hb + 4>)} . (3.41) Even though the amplitude of successive terms falls only inversely as j , we will approx-imate the series by only the first term. The amplitude of the first term can be used as a scaling constant to fit experimental or simulation data. For convenience, we will set it to unity and so we have sgn(x^) « cos(kip + <p). (3.42) Substituting the above into Equation 3.40, we obtain A cos (nip + d>) (3.43) n-1 + A cos (nip + <p) c o s ( ^ + 0^ c o s + A0) k=0 n-1 + A sin (nip + d>) ^ cos (fc^ > + c/>) sin (kip + A</>). Summing over k and after some simplification, we get xn & A cos(nip + (b)+ A (n - 1) cos(n^ + 0 + 0) (3.44) sin (TI — + A cos0cos(nV> + 0) + A — c o s ( < f > - 9). 2 sm ip The solution above contains four sinusoidal terms with equal phase advance per turn, but each with a different initial phase and amplitude. Hence to first order there is no coherent tune shift of the damped oscillation and so the trial solution is self-consistent. The resultant motion is the superposition of all four terms. For small n, the first term dominates because A, which is the initial displacement, is much larger than A . For large n , however, the amplitude of the second term, which grows linearly with n , becomes comparable with that of the first term. The last two terms, which are independent of n , remain of order A , provided the tune is far from an integer. Thus 3.5. THE CONSTANT MAGNITUDE KICK FUNCTION 34 when n is large, the motion is approximately the superposition of the first two cosine terms whose phases differ by 9; and the last two terms can be neglected, leading to: x n « A cos(nV> + </>)+ A (n - 1) cos(mp + cp + 9). (3.45) The angle 9 can be determined from its definition as the phase angle of the coefficients d\,2 (Equation 3.35), which is equal to 9 = (3.46) For the special case that the pick-up to kicker phase advance ip\ is 7r/2, then 9 is zero and Equation 3.45 becomes: xn = [A- (n- l)A]cos(nV> + <£), (3.47) for 0 < n < A/A. The expression shows that the amplitude is damped linearly at a rate of A per turn. Damping stops when the amplitude is approximately A and turns instead into a pseudo-random excitation. This has been confirmed with simulations. The value of A is given by A = A y ^ ^ s i n i h . (3.48) Note that damping is optimal when the phase advance ip\ is 7r/2, which is the same condition as for the proportional kick function. The divergence x'n of the bunch is given by x'n = (0, 1) • x„ (3.49) P = P cos (nip -f <f>) cos ((n + 1)^ + 4>) Q cos (nip + (p) cos ((n + \)ip + (p) ^ sgn(x^) cos (kip + A<p>) a k=o Q sin (nip + 4>) sin ((n + l)ip + (p) Yl sS n( xfc) s m + A<p), a k=o where P = —2Ca and Q = —2Da. The two summation series above are identical to those in Equation 3.40. Using previous results, it can be shown that the divergence envelope also decays linearly. 3.6. DAMPING SIMULATIONS USING ACCSIM 35 As more terms of the cosine series expansion of s g n ^ ) are used, the solution (Eqns. 3.40 and 3.49) would become more accurate but the number of terms would become unmanageable. In general, the contribution of the j t h term is proportional to 1/j and so terms later in the Fourier series (Eqn. 3.41) can be neglected. Furthermore, when the phases differ from kip + <p there is some self-cancellation within the sums appearing in Equation 3.44 and so amplitudes are proportional to either 1/n or 1/n 2, provided the tune is a non-rational number. This condition is always observed in order to avoid resonances. 3.6 D a m p i n g simulations using A C C S I M Using a damper subroutine in the accelerator simulation computer program ACCSIM[20], we can simulate the effect of a damper system. We have studied damping by three dif-ferent kick functions. (1) We simulated damping by the proportional kick function so as to validate the computer code, since we can make a comparison to analytic results. (2) We simulated damping with the constant magnitude kick function to compare with the approximate analytic results. (3) Finally, we simulated damping with the kick function depicted in Figure 3.2. For the simulations we chose to use the proposed KAON[21] Factory Accumulator ring's optics with a non-integer partial tune of 0.790. In all cases, the bunch was deliberately injected with a displacement error of 10 mm from the ideal orbit and also with a divergence of 10 mrad, giving an initial oscillation amplitude of 93.4 mm. 3.6.1 Proportional kick function Figure 3.4 shows the simulated bunch displacement during damping with propor-tional kicks. Notice that the envelope is an approximately exponential decay curve, as expected. We considered two different values of the kick proportionality constant 3.6. DAMPING SIMULATIONS USING ACCSIM 36 Table 3.1: Damping constants and tune shifts of the proportional kick function with k = 0.003 mrad/mm for various phase advances. Phase 8 Pi Au Au a a advance analytic simulation analytic simulation (degrees) (m) M X l O " 3 x l O - 3 x l 0 ~ 2 / t u r n X l O - 2 / t u r n 84.49 9.145 5.262 -0.158 -0.160 ±0.008 -1.046 -1.046 90.00 9.145 6.033 0.000 0.000±0.008 -1.126 -1.126 113.0 9.145 15.891 1.15 1.15 ±0.02 -1.69 -1.69 162.2 9.145 9.303 2.110 2.110 ±0.004 -0.422 -0.422 fc, and for each we used five different phase advances between the pick-up and the kicker. To change the phase advance, the pick-up was held fixed and only the kicker was moved. The damping constants (a) and tune shifts (Au) from the simulations with k — 0.003 mrad/mm and with k = 0.01 mrad/mm are given in Table 3.1 and Table 3.2, respectively. These values and their r.m.s. deviations were obtained by fitting Equation 3.25 to the displacement data of a few hundred simulated turns using the method of least-sum-of-squares. The goodness of the fit may be gauged from the r.m.s. deviation \jfhyi-y(*iW> ( 3- 5°) from the theoretical values y(xi) over approximately 500 data points. For all fits, the deviations are a small fraction of a millimeter, as may be expected when both theory and simulation involve no approximations. According to standard practice, this error estimate may be used to find probable values for the r.m.s. deviations of the fit parameters such as Au appearing in the tables. Where there is no indication of error, as in the damping constants, the accuracy is ± 1 in the least significant digit. Overall, the agreement between the calculations and simulations is very good, especially for the 3.6. DAMPING SIMULATIONS USING ACCSIM 37 Table 3.2: Damping constants and tune shifts of the proportional kick with k — 0.01 mrad/mm for various phase advances. Phase P Pi Au Au a a advance analytic simulation analytic simulation (degrees) (m) (m) X l O - 3 X l O " 3 x l 0 - 2 / t u r n x l 0 _ 2 / t u r n 84.49 9.145 5.262 -0.52 -0.52 ± 0.05 -3.57 -3.57 90.00 9.145 6.033 0.030 0.030±0.006 -3.85 -3.85 113.0 9.145 15.891 4.07 4.07 ±0.02 -5.87 -5.87 162.2 9.145 9.303 7.14 7.14 ±0.03 -1.42 -1.42 damping constants. The coherent tune shift for the phase advance of 90 degrees is the smallest, as anticipated. 3.6. DAMPING SIMULATIONS USING ACCSIM 38 Figure 3.4: The displacement of a bunch while its oscillations are being damped by the proportional kick function. 3.6. DAMPING SIMULATIONS USING ACCSIM 39 Figure 3.5: The displacement of a bunch while its oscillations are being damped by the constant magnitude kick function. 3.6. DAMPING SIMULATIONS USING ACCSIM 40 Table 3.3: Linear damping rates and tune shifts of the constant magnitude kick function with A y ' = 0.005 mrad for various phase advances. Phase Au A A advance simulation analytic simulation (degrees) (m) (m) x l O - 5 x l 0 - 3 m m / t u r n x l O _ 3 m m / t u r n 84.49 9.145 5.262 -1 -21.93 ± 0 . 0 5 90.00 9.145 6.033 0 -37.14 -23.63 ± 0 . 0 5 113.0 9.145 15.891 4 -35.34 ± 0 . 0 5 162.2 9.145 9.303 5 -8.81 ± 0 . 0 5 3.6.2 Constant magnitude kick function For the case of correcting kicks which only change polarity, Figure 3.5 shows the bunch displacement at the pick-up while the oscillations are being damped. A damping rate comparable to that in Figure 3.4 has been chosen for comparison of the decay envelopes. Notice the approximately linear decay of the envelope which confirms the analytic re-sults. The damping stops when the amplitude falls to approximately A , because at this point the bunch is alternately damped and antidamped on consecutive turns. In effect, the kicker becomes a source of noise, keeping the amplitude approximately con-stant at A . However, the data reported in Table 3.3 are for a somewhat different case. To collect sufficient data points for least sum-of-squares fitting, we chose a constant magnitude kick of 0.005 mrad, which damps an oscillation amplitude of 100 mm in about 2000 turns. Table 3.3 shows the linear damping rates and tune shifts for four different phase advances, ifti, including the special case of 90 degrees. A l l four cases exhibit linear damping and the damping rates are obtained by fitting the displacement to Equation 3.47. The r.m.s. deviations for the four fits are 0.089 mm, 0.039 mm, 0.95 mm, and 3.6. DAMPING SIMULATIONS USING ACCSIM 41 6 = 0 . 0 0 1 a n d x = 3 0 m m max - i o o H 1 1 1 h 0 2 0 0 4 0 0 6 0 0 8 0 0 N o . s o f t u r n s Figure 3.6: The displacement of a bunch while its oscillations are being damped by an amplifier kick function. 0.34 mm, in the order listed in the table for N w 2000 data points. It is interesting to note that one obtains the best fit when the phase advance is 7r/2. Generally, the closer the phase advance is to 7r/2, the better is the fit. This is an indication that the tune shift is not exactly zero for the simulation data. The tune shifts for four different phase advances range from zero to 5 x 1 0 - 5 . These values are smaller, by two orders of magnitude, than the tune control that is achievable in a storage ring. We therefore conclude that tune shifts of the constant magnitude kick function can be assumed to be zero for all practical purposes. When the phase advance is 7r /2 the damping rate can be calculated for comparison using Equation 3.47: —37.14 x 1 0 - 3 from calculation and (—23.63 ± 0.05) x 1 0 ~ 3 from simulation. The ratio of these numbers is close to 7r/2, and the discrepancy is indicative of a systematic error. This is discussed further in Section 3.7 3.7. DISCUSSION 3.6.3 Amplifier kick function 42 We can simulate damping by any kick function, even when analytic results are not possible, and so the kick function of Figure 3.2 can be considered. Figure 3.6 shows the displacement of the bunch at the pick-up as a function of the turn number during damping. Because the intial amplitude (93.4 mm) of the oscillations is much larger than xmax (30 mm), initially the majority of kicks are constant magnitude kicks, with a very small number of proportional kicks. As expected the damping is approximately linear during this time period. A transition to exponential damping can be seen as the amplitude is damped below 30 mm; all subsequent kicks are proportional. 3.7 Discussion There is a significant discrepancy (a factor of 1.57) between the damping rate given by Equation 3.47 and the simulations. This is presumably due to the approximation of sgn(o?fc) by just the first term of the Fourier expansion. Given that |sgn(xfc)| > cos (kip -\- cf)) and kicks in the simulations are larger than the approximation, it may seem surprising that the simulation damping rate is smaller. However, with a constant kick, there will be times when the kick is too large, resulting in temporary anti-damping. If the kick is artificially scaled as cos(kip + </>) the resulting damping is more effective because there is less over-kicking. Under the assumption that the tune is a 'very irrational' number and not close to a quotient of integers, the set of turn-by-turn bunch displacements forms a pseudo-random series. Hence, instead of mapping the displacement after each kick as we have done to determine the damping decrement, the change in the amplitude can be calculated by phase averaging. Subsequent to the original publication of our simulation and analytic work, described in Sections 3.5 and 3.6, Sagan [19] derived a formula for the average damping decrement using the phase-averaging method, which in our notation 3.7. DISCUSSION 43 may be written as, 6x = --Ccos^ib + cb). (3.51) 7T This is therefore in good agreement with our simulation results and we can conclude that the factor of 2/IT is needed to correct the approximations made in our analytic results. The revised Equation 3.47 is 2 xn = A cos (nib + (j>) + A - (n - 1) cos(nib + <f> + 6). (3.52) 7T Note that, though the phase averaging method gives the correct average damping rate and tune shift, it cannot give the detailed progression of displacements nor cope with a rational tune. 3.7.1 Critique We have concentrated on the case that 9 = ibi — TY/2 is adjusted to zero, i.e. the pick-up to kicker phase advance ib\ — ir/2. When this condition is not fulfilled, the leading terms in the expression for bunch displacement, Equation 3.45, become a sum over sinusoids with time-varying amplitudes. In this case, the concepts of damping rate and tune shift become ill-defined. However, this does not imply we have nothing useful so say about the oscillation amplitude. Rather, combining terms in quadrature, we find the oscillation damps as: \JA2 + (n - l ) 2 A 2 - 2A(n - 1) A cos 9 (3.53) which implies the system is damping for \9\ < n/2 and anti-damping for \9\ > n/2. Hence, the decrease is still approximately linear. 3.7.2 Performance comparison For a given power level, the constant-magnitude kick function produces faster damping than the proportional kick function. To see this, consider an oscillation with an ampli-tude xmax, which corresponds to the amplifier power limit, and compare the number 3.7. DISCUSSION 44 of turns needed to reduce the original amplitude by a factor of e = 2.71828... for the two cases. The number of turns (np) required with proportional kicks is Accordingly, 'bang-bang' damping is ~ (4e)/[(e — 1)TT] = 2.014 faster than exponential damping for the same peak power. As the bang-bang damping proceeds its superiority increases because the damping rate of the proportional damper diminishes in relation to the amplitude. A drawback of bang-bang damping, however, is that it does not progressively reduce the amplitude to zero but stops at some small constant amplitude. To overcome this, some final stage exponential damping is necessary. (3.54) and with constant magnitude kicks is (3.55) C h a p t e r 4 T h e I m a g e — C h a r g e F o r c e 4.1 Introduct ion High-energy storage rings require close control of the betatron tune in order to avoid beam blow-up and particle loss by nonlinear resonances. On the other hand, some spread in the tune is required to help combat coherent instabilities through the mech-anism of Landau damping. Any force which can shift the tune or give rise to a tune spread needs to be investigated as it can affect beam stability. Whereas the guiding and focusing forces are external forces which can be accurately controlled and compen-sated so as to remain linear, the space-charge and image-charge forces originating in the beam are nonlinear forces which causes amplitude-dependent tune shifts. 4.1.1 Images vs direct space charge The term "space-charge force" is sometimes used in cavalier fashion in the accelerator literature. We adhere to the following distinctions. The space-charge force is the direct, mutual Coulombic repulsion of like charges in free space assuming an open boundary condition. However, a closed boundary in the form of the vacuum chamber (or pipe) is close by and will modify the fields experienced by the particles. In the case of a 45 4.1. INTRODUCTION 46 perfectly conducting boundary, the beam will induce an equal and opposite charge over the pipe surface. In general, the distribution of wall charge and current will be quite complicated, but for certain simple, symmetrical conductor geometries (such as infinite flat plates or circular or elliptical cylinders), the field distribution produced will be exactly the same as that given by much simpler arrangements of line charges or currents placed at 'image positions' (generally outside the beam pipe), according to the well-known "method of images". We will refer to the fields and forces produced by these images as 'image fields' and image forces. These forces act back on the beam in a way that usually adds to the defocusing caused by the direct space-charge. The space-charge and the image-charge fields have quite different characteristics. Let E and H denote the electric and magnetic field, respectively. Firstly, the space-charge field satisfies V -E = p/eo, where p is the free charge density, and is repulsive everywhere inside the beam. Here tQ the permittivity of free space. Inside the chamber, the image charge field satisfies V - E = 0, and so if the field gradient is defocusing in one direction, it may be focusing in another. In the laboratory frame, the beam is moving and constitutes a current source. Consequently, there will be a space-charge related magnetic field. Whereas the electric field points radially outward and perpendicular to the beam axis, the magnetic field wraps around the beam. Due to the orientation of this field, the v x H force tends to pinch the beam, counteracting the repulsive electric effect. The net space-charge force varies as I/7 2, where 7 is the usual relativistic kinematic parameter, and falls to zero in the ultra-relativistic limit, where the two effects just balance. On the other hand, the image currents, at least for coasting beams, do not generate a magnetic field in the pipe interior because V x H = 0; and so the image-charge force is independent of kinetic energy. For a bunched beam, the situation is slightly modified because the changing longitudinal electric field implies a non-zero magnetic component according to V x H = e0dE/dt. To summarize, the internal or self-force in a beam can be separated into two com-4.1. INTRODUCTION 47 ponents: the direct space-charge component, and the image component due to surface charges and/or currents in the surrounding chamber walls. Whereas the direct space charge component decreases with energy, the image component remains constant and hence becomes dominant at high energies. 4.1.2 Tune and dispersion shifts In this chapter we look at some linear and nonlinear effects of the image force. This force depends strongly on the geometry of the beam pipe and the beam cross-section. We will only study two simple geometries: the circular beam pipe and the parallel plates 'pipe'. Consider an elliptical beam, on axis, in a vacuum pipe of elliptical cross-section. By symmetry, there is no force on the beam centroid. However, the space-charge and image-charge forces will both modify the single-particle motions so as to change the incoherent betatron and synchrotron tunes. Suppose now, the same beam is displaced, breaking the symmetry. The space-charge force on the centre of charge is still zero, but the re-distribution of the induced surface charges will now give a net image-charge force on the beam centroid, resulting in modification of the coherent betatron tune. In previous treatments, only the linear tune shifts were calculated and the image-effect on the off-momentum closed orbit was not considered. We extend the linear theory by deriving the new dispersion function as modified by image forces. As a demonstration of the procedure, we calculate the dispersion of a simple FODO cell (focus-drift-defocus-drift). We also modify the theory of betatron oscillations to ob-tain the amplitude-dependent tune shift due to third-order nonlinearity in the image force. The smooth focusing approximation is used together with the time averaging approximation to simplify the equation of motion. Finally we estimate the image-charge-induced tune spread of the LHC and compare with other authors. 4.2. HISTORICAL SURVEY 4.2 His to r ica l survey 48 Early investigations[23] of the space-charge force focused on its limiting effect on injec-tion intensity. The effects of image-charge forces were observed by the M U R A staff in 1959 and prompted the comprehensive theory of both space-charge and image-charge limitations on intensity published by Laslett[24] in 1963. Laslett defined the direct and image coefficients such that the incoherent and coherent betatron tune shifts were proportional to them. He derived incoherent coefficients for the parallel plate geome-try with the beam offset, and for the elliptic chamber with beam centred. Subsequent works by Zotter[25][26] derived incoherent and coherent tune shift coefficients for lam-inar beams in circular pipes, and for offset 'pencil beams' in elliptic pipes. The image field of a laminar beam is found by integrating over many parallel 'pencil beams'. It was Zotter who drew attention to the variation of the coefficients across the vacuum pipe for high-intensity stacked-beam applications in the C E R N Intersecting Storage Rings. 4.3 The image field In this section, we will reproduce the electric image fields of a beam in a circular beam pipe and in a pair of infinite parallel plates from Laslett's work. These two geometries represent the two extremes for a typical elliptical beam pipe, for which the aspect ratio can range for unity (circular) to infinity (parallel plates). As we will see, the the circular pipe gives rise to a weak image force whereas the parallel plates yields a strong force. We assume that the chamber walls are non-magnetic and the beam is unbunched. Hence only the electric image field is present. 4.3. THE IMAGE FIELD 49 4.3.1 The pencil beam model For the purposes of finding the image charge distribution, the beam is approximated by a 'pencil', i.e. a longitudinal filament with length but no breadth. This simplification means that we have a 2-dimensional electrostatics problem, and that we can adopt the fields of line charges rather than point charges. For the case of a coasting beam, the only condition needed to justify this approximation is that the distances between the beam and its image-charge distribution are much greater than the transverse beam size. For the case of a bunched beam, the approximation remains quite good provided that the bunch's length is much greater than its cross-section. 4.3.2 Image field in a circular beam pipe Figure 4.1 shows the cross section of a circular beam pipe of radius r confining a circular particle beam. For this geometry, if the beam is on centre the electric image field is zero. If the beam is displaced horizontally a distance x from the centre, the effect of the induced surface charge distribution can be mimicked by an image line charge at a distance r2/x from the origin as shown, giving the image field where A is the charge per unit length and x the field point. We can Taylor expand the image field: 4.3.3 Image field between parallel plates For the parallel plates geometry we will consider the image field when the beam is dis-placed perpendicular to the plates and obtain the field along this direction. Figure 4.2 shows the orientation of the plates and a beam displaced a distance x from the centre. Unlike the circular beam pipe, there is an image field even if the beam is on centre. (4.1) 4.3. THE IMAGE FIELD 50 Image beam X Figure 4.1: Cross section of a circular beam pipe and location of beam and its image. The image line charges are determined by repeated reflection and consist of an infinite series with alternating polarity above and below the parallel plates. The field Em at point x is the sum of contributions from all the line charges: - A 2ne0 1 1 + 1 1 (4.3) .2h — x — x 2h + x + x Ah — x + x 4h + x — x where 2h is the separation of the plates. Let v — x + x and w = x — x; then the field can be written ^ 0 n = 0 + w la2 — v2 b2 — w2 (4.4) where a = 2h(2n — 1) and b = Ahn. When »/o < 1 and w/b <§C 1, one can Taylor expand the functions as 4.3. THE IMAGE FIELD 5 1 Bee 1 m centr » y oid X ft • Fie ld point 2h w • X Figure 4.2: Cross section of a parallel plates pipe with beam Reordering the terms gives: E. -A E + w (2hy „ t l (2n - 1 ) 2 W n t i ri T-,3 oo + 1 + 3 oo E i + (4.6) K(2hy^ (2n - l ) 4 ( 4 / 0 \ t > 4 , This expression is correct up to order of / i ~ 4 and contains all the dominant cubic terms. The two series can be summed using the results number 0.233 and 0.234 in [80]. After some simplification, the field expansion is -A 7re 0 (4.7) Due to the linear term in x, which is absent for the circular pipe, a particle in the beam will feel an image force even if the beam is centred, i.e. x = 0. 4.4. LINEAR IMAGE FORCE 4.4 Linear image force 52 As can be seen from the respective expressions for the circular pipe and parallel plates, the image-force perturbation on a particle depends on its location as well as that of the beam centre. In addition to changing the tune, the perturbation also modifies the closed orbit of off-momentum particles. We will derive an expression for the new dispersion function due to a generic linear perturbation of the restoring force, but with the image force in mind. The method will be developed first to obtain the unperturbed dispersion function, and later applied to the distortion caused by the linear component of the image forces. 4.4.1 The dispersion function Because the source of momentum dispersion is an independent driving term the non-homogenous equation can be solved by the method of Green's functions[27]. We choose to construct the Green's-type solution by the technique of "undetermined coefficients". The Hill's type equation of motion (Eqn. 2.1) has two independent solutions, known as the principal trajectories C and S. C(s) represents the cosine-like trajectory, since C(0) = 1 and C"(0) = 0, whereas S(s) represents the sine-like trajectory S(0) = 0 and S'(0) = 1. Written with matrix notation, the Hill's equation can be reduced to two coupled first-order differential equations: x' + K(s)x = 0, (4.8) where K(s) \ I x = (4.9) 0 -1 \K(s) 0 J y and x' = dx/ds. We can easily construct the general solution in terms of the principal trajectories C(s) and S(s): x(s) = *(0,s)x o , (4.10) 4.4. LINEAR IMAGE FORCE 53 where x 0 = x(0) is the vector notation of the initial coordinates and *(0,s) 1 C(s) S(s) ^ (4.11) V C'(s) S'(s) J is the same transfer matrix (M) as defined in Chapter 2 but expressed in terms of the principal trajectories, instead of the related Courant-Synder parameters. A particle whose momentum p deviates from the ideal momentum po, obeys the inhomogeneous Hill's equation (2.10) of motion. Again with vector notation, this second-order equation can be reduced to two coupled first-order equations: x' + K ( s ) x = ^ g ( 5 ) , Po (4.12) (4.13) where / 0 1/P(s) Here p(s) is the local radius of curvature of the ideal orbit and A p = p — p0. The right-hand side indicates that the bending is the source of momentum dispersion. It is customary to write the solution of this equation in a re-dimensioned form: Po where the dispersion function d(s) obeys the equation d '+ K(s )d = g(s). (4.14) (4.15) Using the method of undetermined coefficients[28], we can easily write down the formal solution in terms of transfer matrices: d(s) = * (0,s) 1 0 + / V ^ W ) ds' Jo (4.16) where <I>-1(0,s), which is the inverse matrix of 4>(0,s), plays the role of Green's func-tion. This general solution consists of the homogenous solution, with initial condition do, plus the particular integral whose source is the bending magnets. 4.4. LINEAR IMAGE FORCE 54 In a storage ring g(s) is periodic. The closed orbit xc_0_(s) of an off-momentum particle (relative to that of the ideal orbit) is proportional to the periodic dispersion function which satisfies dp(s + L) = dp(s), (4.17) where L is the period length equal to the ring circumference. Hence the dispersion-solution also satisfies: rs+L d(s) = * (0 , s + L) (4.18) 1 0 + f Q-'&s'teWds' Jo We may use the condition of equality to eliminate the initial coordinate do from the two expressions (4.16 and 4.18). To achieve this, we partition the integral into two separate contributions which contain terms from 0 to s and from s to s + L. The transfer matrices can also be partitioned: expressed as the product of matrices of separate path segments: $(0 ,5 + L) = $(s,s + L)3>(0,s). Hence d(s+L) = $(s, s+L)$(0, s) | d 0 + jT * _ 1 ( 0 , s')g(s')ds' + J°+L $ - ^ 0 , s')g(s')ds' (4J9) We pre-multiply Equation 4.16 by &(s,s + L), compare this against the last equation, and after some manipulations obtain: dp(s) = [ I - S ^ s + L ) ] - 1 * ( 5 , 6 + L ) £ + Z ' * - 1 ( 5 , 5 ' ) g ( 5 , ) ^ ^ (4-20) where I is the identity matrix. Note, we used the property that $ _ 1 ( 0 , 3 ' ) = $ _ 1 ( 0 , 5 ) $ _ 1 ( To demonstrate the calculation procedure prescribed in Equation 4.20, we will obtain an explicit expression for the dispersion of the simple FODO cell in Section 4.4.3. 4.4.2 Linear perturbation The equation of motion for an off-momentum particle with linear perturbation a(s)x, as in the presence of the linear component of the image force, can be written as two first-order coupled differential equations: x' + K ( 5 ) x = A(s)x + ^lg(s), (4.21) Po 4.4. LINEAR IMAGE FORCE 55 A(s) = ( \ (4.22) where / 0 0 a(s) 0 Because this is a linear equation, the principle of superposition applies, and we can write the solution as x = ^ [ d p + e] (4.23) Po where e the perturbation to the dispersion satisfies: e' + K(s)e = A(s)[e + d p ] . (4.24) To use the method of Green's functions, the driving terms appearing on the right should be independent of the response terms on the left side of Equation 4.24. However, as-suming convergence, one can take a recursive approach and substitute the unperturbed solution into the right hand side: namely e(s) = (0,0). Using the method of "undeter-mined coefficients", the first order solution is obtained: e(s) = * ( 0 , s) L + f * _ 1 ( 0 , s')A{s')dp{s') ds'} , (4.25) L Jo where d p(s) is the periodic dispersion and eo is the intial coordinates. Because we are trying to find the perturbation to the closed orbit of an off-momentum particle, so it follows we need a periodic solution that satisfies e(s) = e(s + L). Starting from the displacement at s, we can propagate Equation 4.25 one period length: e(s + L) = $(0,s + L) e 0 + r + L * - 1 ( 0,5 , ) A ( 5 , ) d P(5') ds' Jo (4.26) Following the same steps as were outlined for the periodic dispersion function, we may use the condition of periodicity to eliminate the intial coordinates, leading to a formal expression for the change to the dispersion function, e(s) = [I - * ( s , s + s + L) J*+L $ " 1 ( 6 , s')A(s')dp(s')ds' . (4.27) The total dispersion is d(s) = d(s) p + e(s). The above expression is similar to that of the original periodic dispersion function (Eqn. 4.20). Despite its formal simplicity, one must remember that d p is itself defined by an integral equation (Eqn. 4.20). 4.4. LINEAR IMAGE FORCE 56 4.4.3 Dispersion function of the FODO cell A magnet system consisting of focusing (F) and defocusing (D) quadrupoles separated by drift spaces or dipoles is called a FODO cell. The whole structure of a ring can be built with this basic cell and the ring is said to have a FODO lattice. In this section we use the procedures outlined above to evaluate the original and image-force-perturbed dispersion function of a regular FODO cell, using Equation 4.20 and Equation 4.27, respectively. The results are applied to the specific case of the L H C lattice. First we will describe the thin lens approximation, which is used in constructing the transfer matrices of quadrupole magnets. When the lengths of the quadrupole magnets are small compared to the drift spaces separating them, the thin lens approximation can be used: the length of a quadrupole is taken to be infinitely thin while conserving the total bending power. Figure 4.3 shows the schematic of a FODO cell. The drift spaces are of length /, making the total length of the cell 21. The F and D quadrupoles are usually of similar strength. For simplicity, their strengths are assumed to be equal with value f/2. We further assume the spaces between them are filled with dipoles of equal bending power, giving rise to a uniform radius of curvature po- The transfer matrices needed to calculate the dispersion function are T), the drift space of length /; T/, the F quadrupole; and Td, the D quadrupole: T,= / i / \ 11 o \ I i o\ 0 1 ' [Vf i j (4.28) We will first calculate the unperturbed dispersion given by Equation 4.20. Because in a FODO cell there are symmetry planes at the F or D quadrupoles, the function needs only to be evaluated over half of the cell, say from the D to the F quadrupole; and the remainder can be obtained by reflection. The procedure is as follows: first, find the periodic <&(s,s + 21) and running &(s,s') transfer matrices, respectively, by multiplying the matrices of different elements. Then form the integral piece-wise and 4.4. LINEAR IMAGE FORCE 57 F D Unit cell F Unit cell D Figure 4.3: Schematic of a FODO cell. multiply the resultant matrices together giving the dispersion function: P dJs) = — 2/ + l / 2/2 s + 2 / 2 (4.29) which is identical to the formula obtained by a different method, given in [6]. The same procedure is used to evaluate the perturbation to the dispersion (Eqn. 4.27). In this case, the integrand now contains the unperturbed dispersion and the matrix A . The result for e(s) in the half cell from the D to the F quadrupole is: e(s) = op 3 (J  13 1 24/ 24/2 r + \ 2 r 4/ , s2 + 1 6/ 12/2 24/ 2 (4.30) To first order, the LHC lattice[31] is a FODO one. The typical cell length is 106.92 m, the half-power integrated focal length of the F and D quadrupoles is 75.56 m, and the bending radius in the arc is 2784.32 m. A plot of the dispersion is shown in Figure 4.4. The beam in the LHC is surrounded by a 'beam screen' consisting of a circular pipe 4.4. LINEAR IMAGE FORCE 58 LHC FODO Cell ' _ 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 Distance [m] Figure 4.4: Dispersion of a typical FODO cell of the L H C . 44 mm in diameter, flattened horizontally to 36 mm, between superconducting coils with an inner diameter of 56 mm and ferromagnetic poles separated by 50 mm. This is close to the ideal situation of a circular beam at the centre of a circular beam pipe, for which the image force is zero. To obtain a simple (but very pessimistic) upper limit on the image effect, we will consider a beam pipe with parallel-plate geometry. Here we take the plate separation to be 50 mm. Inspection of the image-field expansion (Eqn. 4.7) yields a linear perturbing force with a coefficient equal to e A 7T 2 a — (4.31) m 0 c 2 7 B 2wc0 24/i2 Here the bunching factor B, which is the ratio of the average to the peak charge density, takes into account that the image force is enhanced for a bunched beam. For the LHC at injection [29] with an energy of 450 GeV, B = 3.5 x I O - 2 , and A = 1.7 x I O - 9 C /m. 4.4. LINEAR IMAGE FORCE 59 Figure 4.5: Change in the dispersion due to the image charge. 4.5. CUBIC IMAGE FORCE 60 Hence a = 1.3 x 1 0 _ 6 m _ 1 . This linear perturbation gives rise to a linear tune shift of —0.25 using a formula given by [24], which is considerable. This is expected since the parallel geometry is extreme. A more accurate linear tune shift could be obtained using a more realistic geometry but here we are interested in the upper limit change in the dispersion. Figure 4.5 shows the change in the dispersion due to the image force in the FODO cell obtained from Equation 4.30. The maximum change is at the F quadrupole and it is approximately 20 mm, which is small compared to a dispersion value of 2.7 m at the same location. One consequential effect is the increase in the transverse beam size, which can be estimated by multiplying the one-sigma momentum deviation (10 - 3) by the change in the dispersion function. We find the upper limit on the increase of transverse beam size to be 0.02 mm, which is negligible compared to the average bunch radius of 1 mm at injection. 4.5 C u b i c image force 4.5.1 Incoherent tune spread The presence of a perturbing self-force, such as the image force, changes the betatron tune. We have to distinguish between the coherent tune shift, the frequency change when the beam oscillates as a whole, and the incoherent shift which expresses the change of the single-particle betatron frequency. If the force is nonlinear, the tune shift will depend on oscillation amplitude and a distribution of incoherent amplitudes will give rise to an incoherent tune spread. To estimate the spread, we need to obtain the amplitude dependence. 4.5. CUBIC IMAGE FORCE 61 4.5.2 The smooth focusing approximation Because the effects of interest are integrated over complete turns in the machine, it is reasonable to build the theory on the smooth focusing approximation for the single-In fact, these elements are usually arranged in a periodic sequence of identical unit cells. This results in betatron oscillations that have a constant wavelength and frequency throughout the ring. Hence the unperturbed single-particle motion in the horizontal and vertical planes can be approximated as simple harmonic oscillations governed by the equation: where u is the average betatron tune, R is the average radius and the primes denote derivatives with respect to arc-distance 5. The influence of the other particles in the beam can be included (in a non self-consistent way) by inserting an effective transverse Lorentz force F on the right-hand side of the equation. A self-consistent approach uses the phase space formalism and impedance and solves for the evolution of the collection of particles. This method will be adopted in the next part of the thesis to studies some collective effects. Here we are only interested in an estimate of the nonlinear tune spread due to the image-charge force and so a non self-consistent approach will suffice. The self force F(x,x,s) a particle sees depends on its position, the charge centroid x of the beam, and, if the beam pipe is not of uniform cross section, the location s in the ring. Let us suppose the image-force component of the self-force dominates over the space-charge component. If the image field of the beam for a particular geometry of a beam pipe has been obtained, then the modified equation of motion is where mo7 is the relativistic mass of the test particle, and we have dropped the s dependence of the force by taking a beam pipe of uniform cross section. To first particle model. In most rings, the placement of the focusing elements is quite repetitive. (4.32) (4.33) 4.6. EQUATION OF MOTION WITH IMAGE FORCE 62 order, the image force induces a tune shift which is independent of the oscillation amplitude. This has been treated thoroughly in previous works[24] [25] [26]. Because we are interested in the tune shift and not in the detailed motion, we can forgo the exact solution of motion in favour of a much simpler time-averaged one where we can easily extract the amplitude-dependent tune shift. The time-averaged tune shift is adequate for a rough estimate of the tune spreads which determine decoherence time and Landau damping power. We shall perform the time averaging under the assumption that the beam undergoes coherent betatron oscillations. 4.6 Equa t ion of motion w i t h image force 4.6.1 Non l inea r image force in a c ircular b e a m pipe Substituting the image force in a circular beam pipe into the equation of motion (Eqn. 4.33) we have ,. f v \ 2 e A /x xx2 x2x3 x s x 4 \ ,AnAs X +[-B) x  = ^ O — -2 + — + — + — 8 ~ ' 4 - 3 4 \RJ mc2 2ire0 \ r 2 r 4 r 6 r 8 / where e is the electron charge. We have truncated the expansion after the third order in x. Taking the time average of the beam centre's oscillation gives x><+(l)\ = A; — (^ + ^ j R • (4.35) \RJ mc2 2ne0 \ r4 r8 J y J The linear term shifts the tune by the same amount for all amplitudes and, as we have shown previously, is also responsible the changing the dispersion function. Since we are interested in the amplitude-dependent tune shift, we can move the linear term to the left of the equality and renormalize the base tune to give \Rj mc2 2-KCQ r8 where u' is the renormalized tune. This formula is particularly relevant to injection; this is a time when there can be potentially large-amplitude coherent betatron os-cillations. This expression indicates that the Laslett formulae (for a circular beam 4.6. EQUATION OF MOTION WITH IMAGE FORCE 63 in a circular pipe) which predict no image-charge induced tune shift or spread are completely inadequate when the beam is in coherent motion. 4.6.2 Nonlinear image force between a pair of parallel plates Similarly, for the parallel-plates, we can write down the single-particle equation of motion as modified by image forces. Substituting Equation 4.7 into Equation 4.33, we obtain Assuming xrms/r <C 1, which is the usual case, the third-order term for the parallel plates is four orders of magnitude greater than that for the circular pipe. In the next section, we will solve the nonlinear equation of motion with a generic third-order term. 4.6.3 Elliptical equation of motion With time averaging of the coherent motion, the single-particle equation of motion reduces to a second-order differential equation with linear and third-order forcing terms. The equation has exact solutions in terms of the Jacobian elliptic functions. We quote the results here and consign the details of the solution to Appendix A . Consider the equation (4.37) Again, taking the time average and renormalizing the base tune, we have (4.38) (4.39) where C is the third-order coefficient. The solution can be written as x = A sn[9, k], 0 = a x (s — SQ) , (4.40) 4.6. EQUATION OF MOTION WITH IMAGE FORCE 64 where A is the amplitude of the oscillation, and k is the usual parameter of the elliptic functions [80]. a is similar to the wave number for the trigonometric functions. Ex-tracting the tune from the solution and isolating the amplitude-dependent tune shift, we have Aisa = --~ — , . 4.41 16 v' where v' is the small amplitude tune. 4.6.4 Tune-shift and tune-spread calculations In this section we calculate the image-charge-induced tune spread for the LHC as an application of the theory. Although the extreme rigidity of a multi-TeV particle makes it somewhat impervious to errors on a single pass, the beam re-circulates many millions of times and so the long-term cumulative effect of even small perturbations could have an influence. As we have noted, for the circular beam pipe, the image force is zero if the beam is not displaced from the pipe centre. Even if the beam is off centre, the image force is about four orders of magnitude weaker than that for the case of parallel plates. To take the worst-case scenario, we use the parallel plates as beam pipe. We take the parallel plates separation, 2h, to be 5 cm. In the same spirit, we shall take the worst-case conditions at the centre of the bunch where fields are enhanced by a factor of 1/B. We will need the following information taken from [31] for the calculations. The injection energy is 450 GeV, the number of protons per bunch, A^, is 10 1 1, the av-erage radius, R, is 4242 m, the r.m.s. bunch length, arms, is 13 cm, the number of bunches is 2835, and the average betatron function, J3, is 85 m. First we demonstrate that the parallel-plate pipe geometry gives the upper bound of image-charge effects by calculating the linear incoherent tune shift and comparing it to that obtained by Ruggiero[29] who used a more realistic elliptical pipe geometry. The linear tune shift 4.7. DISCUSSION 65 is given by the formula [24], aR2 where the constant a is given in Equation 4.31. For the LHC at injection, the image-induced tune shift for the parallel plates geometry is -0.25 using the average tune of v = 50. This is much larger than the value of -0.017 obtained by Ruggiero[29] and shows that the parallel-plate geometry gives upper bound image effects. Now we calculate the upper bound tune spread using the parallel-plate pipe ge-ometry. The amplitude-dependent tune spread Aua for a bunched beam is given by Equation 4.41 enhanced with the bunching factor: A 5 A2CR2 ,. ._x A " - = - l 6 ^ i T ' ( 4 4 3 ) The third-order coefficient, C , is given by C = (4.44) m 0 7 27re0 8n 4 Taking the r.m.s. amplitude in the bunch to have the same value as the r.m.s beam size of 1.2 mm at injection, we obtain from Equation 4.43 an r.m.s. tune spread of 1.4 x 10 - 4 inside the bunch. This is equivalent to an r.m.s. frequency spread of 77 Hz. So even with an unrealistically pessimistic assumption for the pipe geometry the image-charge-induced tune spread is negligible. 4.7 Discussion As the energy of a beam increases, it becomes more more rigid transversely. This effect is made evident by the presence of the beam energy as an inverse factor in all the transverse quantities. For the L H C , where the beam energy is great, transverse effects such as dispersion distortion and tune shift due to the image-charge force are very small. However, this does not mean image force is unimportant elsewhere, since the majority of machines do not operate anywhere near 5 TeV. 4.7. DISCUSSION 66 The distortion of the dispersion function due to the image force in the LHC pro-duces a negligible effect on the transverse size of the beam. Even if the tune shift is considerable, as is the case for the LHC (Au = 0.25), the distortion is still small because the tune shift is a cumulative effect over the entire ring whereas the dispersion distortion is a local effect. Only at low energies and high beam currents, would we expect the image force to have an non-negligible effect on the beam size and the closed orbit. In general, the image force induces a small tune spread because the nonlinearity is inherently small. For the LHC, the spread is further reduced due to the high beam rigidity. For the L H C , the majority of the beam pipe has a geometry which is closer to circular than the parallel-plates approximation we have used. Because of the overly cautious approximations, we may have considerably over-estimated the tune spread, which could be negligible compared to other sources such as Landau damping octupoles and high order multipoles in the lattice. Such a very small spread cannot be counted on to substantially supplement other sources for Landau damping nor does it contribute much to beam decoherence when compared to other sources. C h a p t e r 5 D e s c r i p t i o n o f C o l l e c t i v e B e a m M o t i o n 5.1 Introduct ion In the first part of this thesis we investigated the nonlinear dynamics of the individual particle motion due to image-charge forces and calculated two quantities of interest, namely the dispersion function and the amplitude-dependent betatron tune shift. In the second half of this thesis, our focus is on the transverse collective motion of the beam in the presence of a cubic nonlinear perturbing force such as the image force, which is but one of many possible sources of nonlinearity. Other sources include focusing and bending field errors and Landau damping octupoles. Specifically, in the next chapter, we will calculate the decoherence of a beam with binomial density distribution and in the following chapter, we will derive the Landau damping criteria for the weak head-tail instability of the same beam. These two phenomena are multi-particle problems and so we require a collective description of the beam and additional mathematical tools for analysis. Because a beam consists of a very large number of particles sometimes (101 3 or more), to describe the collective motion by tracking the individual (but coupled) mo-67 5.2. THE PHASE-SPACE REPRESENTATION 68 tions would be totally impractical. A high-level approach which focuses on the collec-tive motion is needed. The phase-space approach was developed in classical mechanics to solve multi-particle problems and has been used extensively in the physics of particle beams. The beam is represented by a particle density distribution. In this approach, the individual motions follow orbits in phase space which are determined by the in-coherent equation of motion and collective motions are associated with macroscopic density fluctuations. The formalism allows one to deal with the fluctuations directly and the individual motions are taken care of by the mathematics. With some provisos, the distribution satisfies Liouville's theorem, and Vlasov's equa-tion can be applied to determine its evolution (see below, Section 5.6). The beam interacts with the surroundings electromagnetically and this interaction excites wake forces which in turn act back on the motion. Self-consistent solutions are obtained by solving Vlasov's equation. We will describe the interaction and the wake force in more detail in Section 5.5. In this chapter we will describe and classify the coherent oscillation modes of a coasting beam and of a single bunch using the phase-space picture. More attention is given to the so-called head-tail modes, in which we have a particular interest. We will derive the Vlasov equation for use in the following two chapters and introduce the concept of the stationary distribution which is the starting point of perturbation analysis of beam stability. 5.2 The phase-space representation A particle is represented by its coordinates in a phase space with six degrees of freedom: four for the vertical and horizontal planes and two for the longitudinal plane. As we pointed out in Chapter 2, the displacements are with respect to the synchronous and ideal particle trajectory. For simplicity, we assume that the motions in the three planes are decoupled and consider each separately. In the case of chromaticity, we assume 5.2. THE PHASE-SPACE REPRESENTATION 69 that the betatron oscillation is modulated by the synchrotron oscillation but not vice versa. The betatron and synchrotron motions are periodic oscillatory motions, and hence stable particle orbits in phase space are closed curves about the origin. Take for example the motion in the horizontal plane with linear restoring force. Using normalized coordinates x and px = /3(s)x' + a(s)x, the tilted elliptical orbits become circles centred about the origin, as shown in Figure 5.1. A beam consists of a collection of many particles, even at low intensities, and this collection can be represented by a smooth distribution function. For example, in the transverse horizontal phase space, a function f(x,px, t) can be used to describe a distri-bution. For distributions in which there is rotational symmetry, it is more convenient A Px=0(s)x ' + a(s)x ^ X Figure 5.1: Orbits of single particle motion 5.3. OSCILLATION MODES 70 to use the polar coordinates (r, (b) with radius r and azimuth <b m the phase plane. If each orbit is uniformly populated, the distribution is completely static and is described by some function f(r), known as a stationary distribution. Later we will show how this distribution is a time-independent solution of Vlasov's equation. 5.3 Osci l la t ion modes It is instructive to give a phenomenological picture of the possible coherent modes of oscillations in a beam. For each possible azimuthal mode em^ (m being an integer) there are an infinite number of possible radial distributions / (r) and in the absence of wake fields different modes do not couple to each other because there is no medium of interaction. The presence of the wake forces imposes self-ordering and only certain distributions will be consistent. These are solutions of the Vlasov equation, which we discuss in chapter 7. Further, as soon as we include wake fields, the modes become coupled to each other and they are no longer considered pure modes. However, we shall take the weak-coupling limit. Although we do not deal with coasting beams, the oscillation modes of these beams will be briefly described for completeness. For bunched beams, the modes are more numerous and more complicated. The bunched structure supports oscillation modes within individual bunches in addition to coupled-bunch modes, which are characterized by a definite phase relation between the oscillations from one bunch to the next. Our focus will be on the transverse and head-tail modes of a single bunch. We will discuss each type of mode in a separate section. 5.3.1 Coasting beam modes In a coasting beam, the particles populate the whole circumference of the ring. Under the influence of wake fields, the particles can bunch longitudinally with alternate seg-ments of high and low density. The number of wavelengths around the ring must be 5.3. OSCILLATION MODES 71 an integer and is known as the azimuthal mode number. The beam can also have transverse dipole, quadrupole, etc., moments distributed around the ring. Again the number of the wavelengths around the ring must be an integer. Thus, the complete description of these modes of the coasting beam is given by the transverse moment and the wave number. 5.3.2 Single-bunch modes First we describe the single-bunch transverse and longitudinal modes. The phase-space descriptions of both are simple and identical. Next we describe the transverse head-tail modes, which have a two-dimensional structure and are more complicated. Imagine being in a reference frame that is co-moving with a bunch. Let us consider the transverse modes by taking transverse slices through the bunch. Figure 5.2 shows the motions of the dipole, quadrupole, and sextupole perturbations of that slice relative to the stationary distribution. The distributions rotate at the betatron frequency. The projections of these motions in real space give the slice oscillating transverse moments. The transverse mode index is m and the mode frequency is m times the betatron frequency (mup) where cj/j = VLO0. The dipole mode (m = 1) shown is a displaced stationary distribution. In real space, the bunch oscillates rigidly back and forth about the closed orbit. For the quadrupole mode (m = 2), the bunch contracts and expands periodically at twice the betatron frequency. In the sextupole mode, it exhibits throbbing motion and the mode is sometimes referred to as the throbbing mode. When all the slices from head to tail of the bunch oscillate in phase, we have the / = 0 head-tail mode. For longitudinal modes, the phase-space description of the whole bunch is identical to that of a single transverse slice. A density perturbation in longitudinal phase space with /-fold symmetry is enumerated by the mode index / and the mode frequencies are /-multiples of the synchrotron frequency. The longitudinal modes are usually observed with a current monitor. Figure 5.3 shows the density profiles of a bunch executing 5.3. OSCILLATION MODES 72 | m=±3 sextupole Figure 5.2: The lowest few transverse modes in phase space. dipole and quadrupole oscillations. In the dipole mode, the profiles oscillate back-and-forth about the synchronous particle. In the quadrupole mode, the bunch centre oscillates in anti-phase with the tails. For transverse oscillations, the betatron phase of each slice can have a longitudinal dependence, with phase varying in a periodic manner along the length of the bunch. Hence different parts of the bunch may have different transverse displacements. Due to boundary condition, only an integer number (/) of wavelengths of phase variation is allowed. For example, all slices being in-phase at the head and in anti-phase at the tail gives the / = 1 mode. These transverse oscillations with longitudinal correlation of the 5.3. OSCILLATION MODES 73 betatron phase are called transverse head-tail (H-T) modes. In one respect, they are much like the coasting beam modes: an integer number of wavelengths along the beam. In another respect, the bunched-beam transverse modes are different: the synchrotron motion will carry the transverse dipoles around inside the length of the bunch. Given that we rarely consider anything other than the transverse dipole1, the longitudinal variation is often used to categorize the transverse H-T modes; and for that reason, coupling of modes with different longitudinal index I is often termed "transverse mode coupling". Figure 5.4 shows the dipole moment along the length of the bunch for the first few transverse H-T modes. Finally, in this classification of within-bunch modes, we note that a bunch can undergo simultaneous oscillations involving both transverse and longitudinal charge-density variations. These are coherent synchro-betatron modes and arise when both transverse and longitudinal wake fields are present. Due to their much weaker wake fields, the higher multipoles are negligible. OSCILLATION MODES Figure 5.3: Line density profiles of the lowest few longitudinal modes. OSCILLATION MODES Figure 5.4: Dipole signals of the lowest few head-tail modes. 5.4. HEAD-TAIL INSTABILITY 76 5.3.3 Coupled-Bunch Modes In a bunched beam, there are usually several bunches spaced evenly around the ring. Suppose the bunches undergo transverse or longitudinal oscillations. The beam can now exhibit coupled-bunch modes in which the phase difference between consecutive bunches is sustained by long-range wake fields excited in electromagnetic devices with high quality factors. The wake fields are considered long if they extend to other bunches. The coupled-bunch mode number n defines the phase difference 9 between consecutive bunches, where M is the number of bunches. Due to the periodic boundary condition, the mode index is an integer and there are only M modes allowed. 5.4 H e a d - t a i l instabi l i ty There are, in fact, two types of head-tail instability. The "weak head-tail" depends on chromaticity and has a growth rate proportional to beam current. The "strong (or fast) head-tail" has a current threshold, above which the growth rate is faster than the synchrotron frequency. Because of the threshold, the more slowly developing "weak head-tail" was discovered earlier, whereas the "fast head-tail" which relies upon coupling of modes with different / indices (and is largely independent of chromaticity) was discovered later. Within-bunch H-T modes can become unstable in the presence of short-range wake fields, such as are generated by electromagnetic devices with low quality factor, such as the vacuum pipe. 5.4.1 Weak head-tail Let us assume no /-mode coupling. Without chromaticity, different parts of the bunch (in a single /-mode) can either be precisely in-phase or exactly in anti-phase. Moreover, 5.5. TRANSVERSE WAKE FIELD AND IMPEDANCE 77 the short-range wake field left behind the head of the bunch must be in anti-phase with the head. Given that it requires a quadrature component to drive an oscillation2, it follows that there is no instability unless the wake field lasts longer than one turn. With chromaticity, there can be a progressive phase-shift between head and tail, including some parts of the bunch oscillating in quadrature, and hence the possibility of within-bunch or single-bunch instability. 5.4.2 Strong head—tail Suppose now that we consider coupling of the / = 0 and the / = — 1 modes. The wake field will couple these modes. If the coupling is strong enough, these two / components can become 90° out of phase; and this quadrature allows the modes to be self-driving without invoking the mechanism of chromaticity. The threshold is the intensity at which the phase shift between the two modes becomes 90°. 5.5 Transverse wake field and impedance A particle moving in a beam pipe generates an electromagnetic field that propagates with it. The nature of this field depends on the structures and materials enclosing the vacuum space. The superposition of all such fields due to all the particles in a bunch constitutes the wake field. This field can be separated into the transverse and longitudinal components. The magnitude of the transverse force felt by a particle in the bunch depends on both the bunch offset and that of the particle. For a small magnitude of the bunch offset ro, the transverse wake field is proportional to ro- Let r and v be the transverse coordinates and velocity, respectively. The transverse wake function, t « i , is defined as the integrated transverse kick caused by the transverse 2 F o r a harmonic osci l lator dr iven at resonance, the drive is in-phase w i t h the velocity response but in-quadra ture w i t h the displacement response. 5.6. THE VLASOV EQUATION 78 component of the wake field divided by the bunch offset ro, wL(s,r) 1 / dz E + - x H (*,r , t ) f°° , v (5.2) :t=(«+s)/v With this convention, the dependence on ro is removed. The wake field is a time-domain quantity and may be calculated from convolution of the beam current pulse and the impulse response of the structures bounding the vacuum space. The wake field may also be calculated in the frequency domain by integrating over the product of the beam's frequency spectrum and the structure's impedance. The impedance is the Fourier transform of the wake field due to a current impulse. For example, Z±(u) is defined as the u;th Fourier harmonic of —iw±, The advantages of using impedance are essentially all the benefits of the frequency domain: no time ordering, products rather than convolutions, Nyquist analysis of sta-bility, etc.. Moreover, there are standard engineering equipment and practices for bench measurement of impedances using swept-frequency excitation and network analyzers, etc.. In this section, we will outline a non-rigorous derivation of the Vlasov equation which governs the time evolution of the density distribution of an assembly of many particles. It is a powerful tool for finding self-consistent solutions for the behaviour of such an assembly, in which each particle feels the sum of the external forces and the collective force of all other particles. If the particles do interact with each other individually, then the evolution of the distribution function is governed instead by a series of coupled equations known as the B B G K Y hierarchy [33]. A rigorous derivation begins with this hierarchy and by induction reduces to the Vlasov equation under the assumption of no (5.3) 5.6 The Vlasov equation 5.6. THE VLASOV EQUATION 79 collisions. The starting point for finding self-consistent solutions of coherent problems begins with this equation, stated in the most appropriate coordinates. Basically, the Vlasov equation is a mathematical expression of Liouville's theorem which states that the phase-space density of an elemental area along a phase-space trajectory is invariant provided only purely Hamiltonian forces are acting[34]. Two points should be noted here before we proceed any further; the individual particles of a beam are not in detail the same as the continuous phase space, and not all forces in accelerators are Hamiltonian. For example, the phase-space damping that occurs when an electron beam radiates its energy is an example of a non-Hamiltonian force at work. Also, due to 'coarse-graining', conservation of phase-space area will not preserve effective beam emittance if empty phase space becomes mixed in. A striking example is beam decoherence where the effective emittance grows while microscopically the emittance is conserved. Let tp(q, p,t) be the density function describing the particle density around the point (q, p). The vector variables q and p are the canonically conjugate position and momentum variables of the six-dimensional phase space required to describe the motion of a particle. Liouville's theorem says that the density is constant in the vicinity of a trajectory, that is the total derivative is zero, ^ ( q , p , i ) = 0. (5.4) Carrying out the derivative gives the Vlasov equation it its simplest form, d d d ^ ( q , p) + « ^ ^ ( q , P ) + Pi-fy^fa P ) = °> ( 5 - 5 ) where q = dq/dt and p = dp/dt. p represents the forces acting on a particle and, in our case, where the forces are electromagnetic, it is derivable from a Hamiltonian, as is q: •  d H • dH Pi = -jrr- (5-6) dpi'  1 dqi ' In the derivation of the Vlasov equation, we have assumed that there is no sig-nificant diffusion or external damping. These are usually good approximations for 5.7. THE STATIONARY DISTRIB UTION 80 proton beams. However, for electron beams, synchrotron radiation contributes to both damping and diffusion, and one needs to modify the Vlasov equation to obtain another equation called the Fokker-Planck equation[36]. 5.7 The stationary dis t r ibut ion When there are no time-dependent forces and when the bunch is in perfect equilibrium, there are still the individual particle betatron and synchrotron motions, but because of the detailed balance in the particle distribution, the spatial distribution is static. Such a distribution is known as a stationary distribution. In this section, we will outline the derivation of the stationary distribution for the case of a time-independent Hamiltonian. We will also introduce the binomial distribution, which will be used as the stationary distribution in the next two chapters. The detailed balance which enables a bunch to be static overall can be easily illus-trated with the use of the phase-space diagram. Figure 5.1 shows a few particle orbits. These are also contours of the Hamiltonian function. Suppose we have a distribution where particles are distributed uniformly along the contours, then as the distribution evolves, as many particles leave any region as enter. Hence the density does not change and the projection or profile remains static. What we have is a stationary distribution. In the absence of time-dependent forces such as wake forces, the system is described by a Hamiltonian H(q, p). For this special case the Vlasov equation (Eqn. 5.5) can be solved exactly for stationary solutions tb0. By trial, we can substitute any function of the Hamiltonian, ip(H), into Equation 5.5. Using the chain rule of differentiation and Equation 5.6, ibo(H) can be shown to satisfy the Vlasov equation. Such a function assigns a uniform particle density around each Hamiltonian contour, and this property produces a stationary distribution. In the next section, we will indicate how to construct a binomial distribution that is of the form ip0(H). 5.7. THE STATIONARY DISTRIBUTION 81 5.7.1 The binomial distribution Our primary interest in the binomial distribution family is that it can approximate the particle density distributions of proton beams quite closely because there is usually no significant synchrotron radiation damping. When there is damping, as in the case of electrons, the Gaussian distribution is more appropriate. For protons because of their heavier mass [60], radiation effects are many orders of magnitude smaller and the density distribution is mainly determined by the injection conditions, hardware, steering and focusing errors. For most proton accelerators, the beam distribution in all degrees of freedom at equilibrium can be approximated by a binomial distribution. As we have shown, a stationary distribution can be given by any function of the Hamiltonian. In the absence of wake forces, the Hamiltonian for each degree of freedom can be written as a function of the action J of the oscillation, which is defined by an action-angle transformation. Take for example, the transverse horizontal motion. If we define the action-angle transformation, x/y/8 = ^2J t cos0, px/y/p = y/2Jtsm8, (5.7) then the Hamiltonian can be written as H = u>pjt, where up is the betatron frequency and Jt is unit of length. Therefore any function ipo(Jt) is a stationary distribution. In this thesis, we define the binomial distribution by MJ) = f i-i J (5.8) 0 for J > J , where J is the maximum action, N the normalization factor, and a is a real number defining the sharpness. Figure 5.5 shows the density profiles for a few different values of a. They range from roughly uniform like (a < 1/2) to something approaching Gaussian for a > 4. Further, the limit a —> 0 gives the 'water bag' distribution, and the case a < 0 gives the 'air-bag' distribution. This flexibility allows the binomial 5.7. THE STATIONARY DISTRIB UTION 82 2 . 5 2 . 0 c q .-1 1.5 CO Q >, 1.0 'w c <D Q 0 . 5 H 0 . 0 " " " • . \ " • . \ a=2 a = 1 « = i / 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 Norma l i z ed Amp l i t ude ( A / 2 J ) 1.0 Figure 5.5: The binomial distribution for a =1/2, 1,2,3,4. distribution to model many different types of density profiles3. Incidentally, just as the projection of a 2D-Gaussian is also gaussian distributed, so the projection of a 2D binomial is binomial distributed. The most striking difference between the binomial distribution and the Gaussian is that the binomial is finite in extent whereas the Gaussian has tails which extend to infinity 4 . The difference is significant in terms of calculating collective quanti-ties since different mathematical methods are required for these two different types of distribution. The next chapter will illustrate this point. 3 One of the virtues of the binomial distribution is that it can model quite nicely a truncated gaussian. 4 Strictly speaking, a gaussian distribution is unphysical and should be truncated. However, the mathematical convenience of this distribution usually outweights the implausibility of the very long tails. C h a p t e r 6 D e c o h e r e n c e a n d R e c o h e r e n c e 6.1 Introduct ion When a stationary bunch is displaced transversely, it begins making dipole oscillations about the closed orbit. This can happen due to excitations such as injection error and deliberate kicking. Much information about bunched beam parameters can be extracted from turn-by-turn measurements[35] of the centroid displacement. In con-junction with other measurements[37], parameters which can be determined include (but are not limited to) betatron tune, synchrotron tune, chromaticity and energy spread. Measurements are made using pick-ups which sense the electric dipole mo-ment and generate an electric signal. Depending on the time constant, the pick-up signal may give the dipole moment as a function of distance along the bunch, or just the average or integrated displacement, which we shall refer to as the centroid. The act of displacement or kicking correlates the initial values of the individual particle betatron phases (which are otherwise randomly distributed) so as to effect a collective betatron motion. Whether or not the motion stays coherent depends on whether there is a betatron tune spread that can de-phase the individual particle motions. Possible sources of tune spread are nonlinearity of the transverse restoring force and momentum spread through the mechanism of chromaticity. 83 6.1. INTRODUCTION 84 In the absence of nonlinear effects and wake fields, the pick-up signal is similar to that of a single particle undergoing betatron oscillations but modulated by the beam current variation along the bunch. In practice these perturbing effects are always present to some degree. Nonlinearities in the restoring force give rise to decoherence of the dipole motion and, consequently, a decaying signal. Because of chromaticity the betatron motion is modulated by the synchrotron motion, and the dipole motion recoheres periodically while decohering. This can be seen in the modulation of the decaying signal envelope. Hence, it is of importance to understand how this affects the dipole motion and to be able to accurately model the dipole signal in order to extract useful information. Such knowledge is required in the design of a feedback damper system which uses the dipole signal as feedback, as we have described in Chapter 3. In this chapter, we consider the transverse decoherence and recoherence of a dis-placed bunch with binomial transverse and longitudinal distributions. Previous works by Meller[38], Hsu[41] and Shi[39] all used the Gaussian distribution, which is only appropriate for electron beams. The boundary conditions are very different for the two distributions. Whereas the binomial is finite, the Gaussian is infinite in extent and this property greatly facilitates the integrations involved in forming the ensem-ble averages. Correspondingly, taking the dipole moment of the displaced binomial is not a straight-forward integration due to the non-trivial boundary. Nevertheless, the binomial case has some very desirable features. Unlike the Gaussian distribution which extends indefinitely, the binomial distribution has well defined limits. Hence a binomial has a limited tune spread due to nonlinearity, while a Gaussian distribution has an infinite spread which may lead to over-estimation of the rate of decoherence. We begin with an outline of the mechanisms of decoherence and recoherence, which are respectively amplitude-dependent betatron frequency and synchrotron modulation of the betatron motion due to chromaticity. After writing down the Hamiltonian, we solve the Vlasov equation for the phase space solution. We then take the ensemble-average displacement of the solution to derive an integral formula for the motion of 6.2. NONLINEARITY AND CHROMATICITY 85 the centroid, using a Taylor-like expansion. Expressions describing the centroid are calculated from the formula for a few cases, and plots of the centroid signal are carefully examined for the dependence of the signal on the sharpness of the binomial distribution and on the width of the chromatic tune spread. 6.2 Nonl inear i ty and chromatici ty One source of frequency spread is the nonlinear restoring force provided by octupole magnets or image charges. In this case, the betatron frequency depends on amplitude, as was shown in Chapter 4. Hence a bunch with a distribution of oscillation amplitudes acquires a tune spread. There is an additional frequency spread due to the energy spread of the bunch because, due to chromaticity, the betatron oscillation is modulated by the synchrotron motion. In the absence of nonlinearity, the effect of individual-particle synchrotron motion, on a beam performing collective betatron oscillations, is to make the beam periodically decohere and recohere, through the mechanism of chromatic tune shift. The net effect is that the signal-envelope of the collective betatron oscillations is modulated at the synchrotron frequency. In the absence of chromatic effects, an amplitude-dependent spread in the individual particle betatron tunes produces decoherence and a progressive reduction of the signal-envelope of the collective betatron oscillations. Though re-coherence is possible, it depends on the initial particle distribution, and is unlikely. When the chromatic and nonlinear effects are combined, the net effect is for the shrinking envelope (of collective oscillations) to be modulated by the synchrotron mo-tion, resulting in periodic, partial re-coherence but with an overall trend of decoher-ence. 6.3. PHASE SPACE SOLUTION 6.3 Phase space solution 86 The transverse motion of a displaced bunch satisfies the Vlasov equation. Only motion in one transverse plane, say (x,px), will be considered. For simplicity, the transverse and longitudinal distributions are treated as uncorrelated. We assume that the lon-gitudinal motion is linear and the perturbing transverse nonlinearity is restricted to an octupole-like force. In this section, we outline the formal solution of the Vlasov equation for any general transverse distribution, displaced or not. In the following sec-tion, we will apply this general solution to the case of a displaced transverse binomial distribution. The transformation of the transverse coordinates (x,px) to action-angle variables (Jt, 9) is given by Equation 5.7. Similarly, the transformation of the longitudinal coor-dinates (Ad), AE), to the polar coordinates (Js,d>) is defined as Ad) = nh—Jjs cos cf), (6-1) AE = -ES8 2^ JJS sin cb, (6.2) where 8 = v/c and us = UQ x vs is the angular synchrotron frequency. When the chro-maticity £ is non-zero, the betatron angular frequency up is coupled to the synchrotron motion as follows up(Js,d>) - up\Y + i\[Js cos(ust + </>)] (6.3) where time / is equal to path length divided by the beam velocity. The transverse motion of a particle in the presence of an octupole-like force and chromaticity is governed by the Hamiltonian: H = up(Js,<b)Jt + ]- + ^(4cos26> + cos AB) bJ2, (6.4) where b is the octupole strength averaged over one periodic length of the lattice. The derivative of the two trigonometric terms with respect to 9 gives the rate of change of Jt and leads to a periodic (in 9) distortion of the closed orbits of the particles in betatron 6.4. DIPOLE MOMENT 87 phase space. However, this perturbation averages to zero and the trigonometric terms can be dropped from the Hamiltonian. Using Equation 5.6 and substituting the results into Equation 5.5, we obtain the Vlasov equation for the transverse distribution ipt' | + W J s i ) + i J t ] | = 0 . (6.5) The evolution of a distribution ipt{Jt, 9, t) that satisfies the equation is easily found[39] to be ibt(Jt,e,t) = rbt[Jue-(u}P + b Jt)t - X(t)], (6.6) where x(0 is given by xW = 2^yXsin^cos(^ + ^), (6.7) LOS z z which is the betatron phase advance modulation due to chromaticity. Note that the solution has the same functional dependence on Jt as the initial distribution ipt(Jt, 0,0). This means that the density distribution in terms of Jt is constant, which is consistent with the fact that Jt is a constant of motion. By contrast, the density distribution in terms of cb is dispersive as well as modulated. 6.4 Dipole moment In the last section, the phase space solution for a general distribution was found. Now we can apply it to the binomial distribution. By itself the solution is not very useful because it does not give any observable macroscopic quantities directly. To find the motion of the centroid, an ensemble average of the displacement is performed over both the longitudinal and transverse distributions. When these distributions are uncorrelated, the ensemble averaging can be performed separately and independently, which is what we have assumed. In this section, we will derive an integral expression for the centroid motion from the phase space solution. A stationary bunch can be displaced [{x) ^ 0] or kicked [(x1) ^ 0] or a combination. In phase space, this is represented by a translation of the stationary distribution away 6.4. DIPOLE MOMENT 88 from the origin at some angle. Without loss of generality, assume at time t = 0 taht the distribution is displaced horizontally a distance d in phase space. This corresponds to a displaced bunch in real space. The stationary transverse distribution is binomial. After displacement it becomes ibt(Jt,0,0), a function of both Jt and 9, since it lacks rotational symmetry. The time evolution of the centroid of the bunch is calculated by taking the ensemble average. This is most conveniently done using the action-angle coordinates defined by Equa-tion 5.7. The transverse displacement is \/2Jt cos 9. The ensemble average is given by the expression the transverse distribution are the boundary of the distribution displaced from the origin. Though this boundary is circular, it is a complicated function of Jt,9 because of the displacement. Note that the difficulty of describing the boundary does not arise for distributions which extend to infinity, such as the Gaussian, because the shifted distribution still extends to infinity and over all angles. However, the integration limits can be simplified, if we Taylor expand the displaced distribution in terms of the stationary distribution whose boundary is circular. We first Taylor expand ibt(x — d, px,0) in the (x,px) coordinates because the displacement is defined in this coordinate system and then make a change of variables to (Jt,9) as required by Equation 6.8. The result up to third order in d is x 2Jt cos 9 ibt[Ju 9- {u0 + bJt)t-x{t)\4>o{J.) dJtd9dJsdcb, (6.8) where ib0(Js) is the stationary longitudinal distribution. The integration limits of (6.9) where xb0(Jt) is the transverse stationary distribution and the primes denote derivatives with respect to Jt. Substituting the Taylor expansion into Equation 6.8, we can consider 6.5. DECOHERING DIPOLE SIGNAL 89 the contribution of each order to the integration. It is clear that none of the even-order terms contribute because of the odd powers of cos 9. The next highest order that contributes is the third, which is two orders of magnitude smaller if d \^ X-Consequently, for displacements which are small compared to the half width of the distribution, the contributions of the third and higher orders can be neglected. To first order then, the beam centroid is given by x(t) = d J J 2ip'0(Jt)Jtcos9cos[9 - (io0 + bJt)t - x{t)}M JsWtd9dJsd<f>. (6.10) We can separate the transverse and longitudinal variables by making use of the cosine identity for the difference of two angles, and after some rearrangement we have fjt f2w x(t) = d / 2^0(Jt)Jtcos9cos[9 - {LOp + bJt)t]dJtd9 Jo Jo f2v X / / cos Jo Jo /j^sin ^ c o s ( ^ + d>) us v " 2 v 2 where Js is the maximum synchrotron action and Jt is the maximum betatron action. The centroid motion is now given by a product of two integral expressions, where there is no mixing of the transverse and longitudinal variables. This is a consequence of the fact that the two distributions are uncorrelated. The first integral expression describes the decoherence and the second the amplitude modulation. Note that if the chromaticity is zero, the modulation factor is unity and there is no recoherence, as expected. We will consider each effect separately in the next two sections. 6.5 Decohering dipole signal In this section we will omit the chromatic effect. The decohering motion of the beam centroid due to octupole nonlinearity is given by the first integral expression of Equa-tion 6.11. We can now substitute in the binomial distribution (Eqn. 5.8) and evaluate it for any parameter a. In general, the integration can be evaluated analytically only for positive integer values of the parameter a, since the integrand is just a Jt polyno-mial times cos 9. For fractional a, there may be only a few cases where the integral 6.5. DECOHERING DIPOLE SIGNAL 90 can be evaluated analytically. Specifically, we have obtained expressions for the beam centroid x(t) with a = 1/2, 1, 2, 3, and 4. For a — 1, we have x(t) = 2d (Auty cos(upt) — cos(uOt) —Au t s'm(ut) (6.12) where Au = bJo, which is the difference between the unperturbed angular frequency and the maximum/minimum frequency u at the edge of the distribution. For a = 2, 3, and 4, we have respectively 6d (Auty Au t cos(u/3t) + Aa; t cos(ut) +2 s'm(upt) — 2s'm(ut) (6.13) x(t) 12d [Auty 4Au t sm(upi) + 2Au; t sm(ui) +6 cos(u)r.) — 6 cos(upt) + (Aut)2 cos(upt) (6.14) X(t): 20d (Auty (Au t)3 cos(upt) + 6(Au t)2 s'm(upt) — 18Au t cos(upt) — 6Au t cos(ut) 24s'm(ut) - 24sm(upt) (6.15) These expressions, which were reported in reference[42], show that the oscillation fre-quency of the centroid contains components of up and u and is time dependent. Had we included the next higher order in the expansion, other frequency components would be present. For a = 1/2, the expression involves the generalized hypergeometric function iF2 defined in Chapter 9.14 of [80]. The displacement is: x(t) = ^d^^-^-^Autfsmupt) (6.16) c o s ^ t { ,F2 ( l ; | , 7-; ~\(Aut)2) + ^(Aut)2 ,F2 (2; 1 ^ ; -\(Aut)2) 6.5. DECOHERING DIPOLE SIGNAL 91 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 N u m b e r o f T u r n s Figure 6.1: Beam centroid from turns 0 to 1000. For illustration and comparison with experimental results, we will plot the dipole signals of these expressions, as they would be recorded by a pick-up at some location in the ring. This is simply accomplished by evaluating the expressions at consecutive integer multiples of the revolution period. Plots of the dipole signals for all the five values of a are shown for two time "windows" in Figures 6.1, 6.2 and 6.3. The initial dipole moments for all cases are normalized to unity for comparison. In the following discussion we will refer to the betatron tune v instead of u>p, as this is more common in experimental accelerator physics. To exaggerate the effect of decoherence, the octupole strength is set such that the 6.5. DECOHERING DIPOLE SIGNAL 92 tune spread Aw = 0.01. For a = 1, the signal envelope decreases initially as 1/n 2, where n is the number of revolutions; and then, at later times, as 1/n. This is evident in Equation 6.12 where t is equivalent to n. There is some beating of the envelope but this is not completely destructive. For a = 2, the envelope decreases as 1/n 2 at all times, and the envelope beating is periodic; this can be seen from Equation 6.13, where the different frequency components have equal amplitudes. For a = 3, the envelope decreases as 1/n 3 initially and as 1/n 2 at later times; there is some beating but it is irregular. For a = 4, the envelope starts to decrease at a slower rate than for a = 2,3 ; and at longer times it falls as 1/n 2. There is no prominent beating. In summary, for the same transverse tune spread, the rate of decoherence of the dipole signal depends on the sharpness of the distribution through the parameter a. In general the sharper the distribution, the weaker the modulation, as can be seen in Figure 6.6, and in the limit of a delta-function distribution, there is no modulation, as expected. To understand this qualitative behaviour, one must realize that the centroid is proportional to the ensemble average of ib'0(Jt) x Jt, and that there are competing effects. For small values of a the distributions are "flat" and the derivative of ipo(Jt) is small, but there are proportionately more particles at large amplitudes. For large values of a, the magnitude of the derivative is large, but there are rather fewer large amplitude particles. One can imagine two extreme cases that show no decoherence: (i) when o. —y oo (and we have a ^-function distribution), and (ii) when a = 0 (and we have a ^-function derivative). The effect of distribution gradient on the rate of decoherence was previously noted by Hereward[40]; and is the reason why the distribution with a = 1 decoheres more quickly than for a = 1/2, despite the latter case having more large-amplitude particles. Further, according to Equation 6.10, a uniform distribution (a —> 0) cannot decohere (x = DcosQt) because there is no gradient. 6.5. DECOHERING DIPOLE SIGNAL 93 Number of Turns Figure 6.2: Beam centroid from turns 1000 to 2000. 6.5. DECOHERING DIPOLE SIGNAL 94 Number of Turns Figure 6.3: Beam centroid for cc = 1/2 6.6. AMPLITUDE MODULATED SIGNAL 95 6.6 A m p l i t u d e modulated signal As we have indicated previously, the second integral expression of Equation 6.11 con-tains the recoherence effect of chromaticity. We call this expression the amplitude modulation factor A(t) because of its modulating effect on the envelope of the dipole signal. In this section we evaluate this modulation factor for a longitudinal binomial distribution with positive sharpness parameter ft} We begin by substituting the formula (Eqn. 5.8) of the binomial distribution into the integral expression fj3 f2-K Mt) = I 7 Jo Jo COS 2 yJs sin — cos (— + 0) u ; , Z Z k i - i " . J.. 10 dJsdcb. (6.17) sin \k 2 J J, dJs, (6.18) Integrating with respect to cb first, we obtain A(t)= I'* Jo[2^JIs u Jo U>s where Jn(x) is the nth order Bessel function of the first kind. This integral can be evaluated in general for all positive values of ft using the formulae 6.567 in the integral tables of [80]; the result is . . . T. . Jr(+W (6.19) A(t) = ( l + f t ) 2 ^ T ( l + f t ) ^ ^ , where T(x) is the standard gamma function, and c(t) is defined as c(t) = — sin —t (6.20) The expression for c{t) is a sinusoidal function with an oscillation frequency at half the synchrotron frequency. However, the modulation frequency is at the synchrotron frequency because A(t) is an even function of c(t). It is useful to calculate the smallest value of A(t), which is not necessarily zero. From Equation 6.20, the smallest value of A{t) occurs at odd multiples of n/us and it is equal to {2Auc (1 + ft)2^T{\ + ft)J1+p V U s (6.21) XP is used to distinguish it from the a used for the transverse distribution. 6.6. AMPLITUDE MODULATED SIGNAL 96 1.2 12 —| i i I i |— 0 200 400 600 800 1000 N u m b e r of T u r n s Figure 6.4: Amplitude modulation of the beam centroid signal where Au>c = £ujp\f~Js, which is the spread in the betatron angular frequency due to chromaticity from a particle with zero energy error to one with maximum energy error. Now we can examine the effect of chromaticity on the dipole signal, assuming a perfectly linear transverse restoring force. Again, it is more convenient to use tune and turn number instead of angular frequency and time when describing the dipole signal. Figure 6.4 shows a plot of the dipole signal of a bunch with a chromatic tune spread Avc = 0.01, a betatron tune v = 6.7, and no tune spread due to nonlinearity. A slow oscillation in the envelope of the signal due to the chromatic tune spread at the synchrotron tune vs = 0.01 can clearly be seen. After 100 turns (one synchrotron period) the dipole signal returns to its original magnitude. This is the reason why the phenomenon is referred to as recoherence: the return of a temporary signal loss. It is interesting to observe qualitatively the relationship between the width of tune spread and the amplitude of the modulation envelope. This relationship can be used to determine the chromaticity [41]. Figure 6.5 shows the modulation factor for a 6.6. AMPLITUDE MODULATED SIGNAL 97 distribution with 0 = 2 for various tune spreads. It clearly shows that the wider the spread, the stronger the modulation. If the width is sufficiently great, it can lead to temporary total loss of the signal; for the binomial with 8 = 2, this happens when Auc/us is larger than 3, as shown in Figure 6.5, where the factor crosses the zero line. In the limit of no tune spread, Equation 6.19 gives the correct result of no modulation, that is, A(t) = 1. The amplitude of the modulation also depends on the sharpness parameter 0. In general, for a fixed tune spread, the sharper the distribution, the weaker the modu-lation, as can be seen in Figure 6.6, and in the limit of a delta-function distribution, there is no modulation. Figure 6.7 shows the combined effect of chromaticity and transverse nonlinearity on the dipole signal. As can be seen, the modulation of the signal envelope is independent of the decay of the signal. It is strictly an amplitude modulation effect, which does not change the decoherence rate on average. The amplitude of the envelope modulation depends strongly on the ratio of the chromatic tune spread to synchrotron tune, and a sufficiently large ratio, which is easily achieved, can cause a periodic complete loss of the dipole signal. The initial rate of fall of the envelope is proportional to Avc/us, as can be expected for a de-phasing phenomenon. It also depends on the distribution: the sharper the distribution, the weaker is the modulation. This can be understood if we note that particles with large synchrotron action contribute most to the modulation. For a sharp distribution, there are relatively fewer particles at the edge than at the centre, and because of this depopulation at the edge (Figure 5.5) as sharpness index increases, the modulation is reduced. AMPLITUDE MODULATED SIGNAL B i n o m i a l D i s t r i b u t i o n w i th /?=2 A i / c A ,=5 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 N u m b e r of T u r n s Figure 6.5: Amplitude modulation function for various widths of tune spread AMPLITUDE MODULATED SIGNAL A m p l i t u d e Modu la t i on F u n c t i o n A ( t ) j L 1000 2 0 0 0 3 0 0 0 N u m b e r of T u r n s 4 0 0 0 Figure 6.6: Amplitude modulation function for various sharpness. 6.6. AMPLITUDE MODULATED SIGNAL 100 Number of Turns Figure 6.7: Amplitude modulation of the dipole signal with decoherence C h a p t e r 7 L a n d a u D a m p i n g o f t h e W e a k H e a d — T a i l I n s t a b i l i t y 7.1 Introduct ion The simple theory of weak head-tail instability is formulated under the assumption of no intrinsic spread in the synchrotron or betatron frequencies[50]. The modulation of the betatron frequency caused by synchrotron motion via chromaticity, as described in the previous chapter, is not an intrinsic spread. Only an intrinsic frequency spread can provide Landau damping (see below, Section 7.5). When Landau damping is ignored, the instability fails to exhibit a beam current threshold. In this chapter, we extend the theory to include Landau damping: in particular, that which arises from a spread caused by a quadratic dependence of frequency on amplitude, as occurs when octupoles or image forces are present. A binomial density distribution, which is appropriate for proton beams, is chosen as the stationary distribution for both the transverse and longitudinal phase-space distributions. Section 7.3 outlines the development of beam instability theory, concentrating mostly on work related to our formulation. Section 7.4 gives a simple physical explanation for the head-tail instabilities in terms of the "two-particle" model. The physical origin 101 7.2. BEAM INSTABILITY THEORY 102 of Landau damping of coherent oscillations is explained qualitatively in Section 7.5. Section 7.6.1 gives a brief outline of the steps in deriving the general dispersion integral for any stationary distribution from the Vlasov equation using first-order perturbation methods. The dispersion relation for the Dirac delta-function distribution is calcu-lated first, and then that of the more realistic binomial distribution. For the binomial distribution, Section 7.7, this is done by transforming the integral equation into an eigenvalue problem using a set of orthogonal functions. Solving the eigenvalue problem yields the dispersion relation. In Section 7.8, we make use of the dispersion relation to obtain the stability diagram to determine the octupole strength needed to damp the reported high-order head-tail modes observed in the C E R N PS for bunches with LHC characteristics. 7.2 Beam instability theory Beam instabilities can be divided into two types: transverse and longitudinal. For bunched beams, further classification into two broad categories is warranted: one in which the growth rate is proportional to beam intensity and the other in which it is not. Transverse weak head-tail and coupled-bunched instabilities are examples of the former case, whereas mode-coupling instability such as the strong head-tail and turbulence are examples of the latter case. In the case of thresholding, a beam is stable up to a certain intensity where the frequencies of two modes coincide, then suddenly it becomes unstable above this threshold. The rate of increase in the growth rate beyond the threshold intensity is often very sharp. Yet another class of thresholding is that due to Landau damping; in this case there is no mode-coupling, but there is an intrinsic tune spread. For this case, the rate of increase of growth-rate above threshold is not particularly sharp. The question of beam stability can be answered by performing a perturbation analy-sis on the stationary phase space distribution. The perturbation method is satisfactory 7.3. HISTORICAL SURVEY 103 because the primary interest is whether or not the beam is stable, and not to find the detailed long-term evolution. A small disturbance of the stationary distribution is assumed. The disturbance excites wake fields in the surroundings which act back on it creating a new disturbance. If the new disturbance enhances the original one, then the oscillation will grow. The beam's longitudinal motion will provide the energy through coupling to sustain growth of the instability. In practice, any initial noise in the beam will provide the seed from which the disturbance may start. The disturbance is gov-erned by the Vlasov equation. If found, self-consistent solutions will determine the dispersion relation and whether each frequency component is stable or unstable. 7.3 His to r ica l survey There exists a great deal of theory developed over the past five decades devoted to the explanation of particle beam instabilities. The theory now seems to give reasonable agreement with experiments for many (possibly most) of the observed instabilities. Here we will try to give a brief history of the development. Key figures only will be mentioned, but many other authors have also made valuable contributions to the field. Sessler pioneered the use of the perturbation method in his treatment of longitudinal[43] and transverse[44] instabilities of coasting beams. He included Landau damping due to the spread in the oscillation frequency and derived frequency dispersion relations from which stability diagrams[46] [47] can be constructed. These diagrams map out the region in the impedance plane within which a beam is stable. Sessler[45] began to extend the theory to bunched beams, but it was Sacherer[50] who founded the modern formalism for all bunched-beam instabilities (both longitudinal and transverse) and derived an integral equation which now bears his name. 7.3. HISTORICAL SURVEY 104 7.3.1 Weak versus strong head-tail Pellegrini[48] first proposed the weak head-tail effect (depending on chromatic phase shift) and this was analyzed for a longitudinal "air-bag" distribution by Sands[49]. Sacherer's extension[50] [53] of this work produced a beautiful theory[51] with good agreement with experiments[52]. Kohaupt[64] first proposed the strong head-tail effect in terms of a two-macro-particle model, with the particles corresponding to the head and tail of the bunch. This concept was quickly taken up and formulated[63] in terms of coupling between modes with different longitudinal index /. For both the strong and the weak instability, it is customary to take a very simple dipole form (m = 1) for the transverse structure of the beam perturbation, and so the main task is to elaborate its (possibly complicated) longitudinal pattern. Before recounting this elaboration, we mention how the qualitative differences between "weak" and "strong" head-tail have directed their respective mathematical developments. For "weak" head-tail, the coherent betatron frequency shifts are typically small. Hence it is possible[52] to Landau-damp the instability with octupoles; this idea has been pursued by Chin[69], Tran[78] and Berg[73]. Berg[73] extended the theory to include the case where both horizontal and vertical betatron frequencies are amplitude dependent, and demonstrated that the stability region is dramatically increased when there is frequency spread in both transverse planes. The other development in "weak" head-tail has been the quest for exact solutions with ever more realistic longitudinal stationary distributions: starting with the air-bag[49, 50], the binomial[50] with a = —0.5, the gaussian[61, 68, 73] and finishing with the general binomial[78]. For "strong" head-tail, the coherent betatron frequency shift is typically large: at the instability threshold it is equal to the synchrotron frequency. This implies that modes with different / will couple strongly. Fortunately, there is a well-established longitudinal mode-coupling theory[54, 61] which may be borrowed for use in the head-tail[63, 66, 68, 69]. Because the frequency shifts are so large, octupole damping is ineffective, but 7.3. HISTORICAL SURVEY 105 Landau damping by a large synchrotron frequency spread is possible[66]. Probably more effective is to shift the I = —I mode frequency (but not the / = 0 mode) by a reactive feedback[65] and so reduce the mode coupling. 7.3.2 Longitudinal development As we have said, despite the fact that "head-tail" is a transverse instability, the math-ematical difficulty stems from the longitudinal variation of the phase of the betatron oscillations. Let us write the perturbation as the product of transverse (/i) and longi-tudinal (gi) functions: fi(q,0)gi(r,(b). Much of the longitudinal instability theory can be taken over, with the only difference being that for the longitudinal case gi cc dgo/dr whereas for the transverse case gi oc go{r), where go(r) is a stationary distribution. The Sacherer[50] integral equation relates the normal modes to the stationary dis-tribution and the impedance of the accelerator. Originally, this equation was solved by making an educated guess for the eigen-functions, for particularly simple stationary distributions. Soon it was discovered that the kernel of the equation could be sim-plified by means of the Hankel transform, and this formulation was quickly adopted. Laclare[57] transformed the integral equation to an eigenvalue problem for the Fourier components of the perturbation. Besnier[56] and Zotter[60], looking for a formulation with better convergence (for broad-band impedances) expanded the radial function in orthogonal polynomials and also reduced the integral equation to an eigenvalue prob-lem for the coefficients of the polynomials. This method was adopted by Satoh[59] for the general binomial distribution. During these developments, go{f) became more realistic: air/water bags, elliptic (i.e binomial with a = ± 1 / 2 ) , gaussian and general binomial. At high beam intensities the mode-frequency shifts are comparable to the syn-chrotron frequency so that longitudinal mode coupling of the azimuthal modes must be considered[54]. Though Pellegrini[58] showed how to do "infinite mode-coupling", Zotter's[62] formulation (with a finite number of modes) proved more fruitful. The 7.3. HISTORICAL SURVEY former method was used by Satoh[67] and the latter by Chin[68]. 106 Frequency spread Starting from the "no azimuthal mode coupling" theory, there has been some progress to include the Landau damping effect of a synchrotron frequency spread. Sacherer[55] was probably the first to make such an attempt. Besnier[56] and Zotter[61] initially added a "dispersion matrix" to the interaction matrix which defines the eigenvalue problem. But this was flawed: because they avoided dispersion integrals, so the coher-ent solutions are only valid outside the incoherent band of frequencies and cannot be used to determine the stability boundary. When there is a frequency spread, one is faced with the prospect of integrating a dispersion-integral which is singular at the coherent frequency. One performs the inte-gration as a principal value and a residue, according to the prescription of Landau[74]. Assuming no coupling (at all) and known radial modes, Zotter[61] performed the dis-persion integral for a single eigenvalue. Chin[66] extended this approach: he assumed the orthonormal basis does not change when frequency dispersion is introduced, and used this to set up a matrix eigenvalue problem where the matrix elements are disper-sion integrals. Chin considered both longitudinal instability and the strong head-tail mode; and showed that the threshold is shifted by Landau damping. In order to find solutions inside the stability diagram, one must consider the "ini-tial value problem" of Van Kampen[75] and complement the coherent modes with a continuous set of eigenvalues and functions. Chin[66] gave a brief indication of how this might be done, and detailed working for a narrowband resonator and the Laclare formalism is given by Koscielniak[76]. 7.4. THE HEAD-TAIL INSTABILITY MECHANISMS 107 7.4 The head-tail instability mechanisms Head-tail instabilities involving the / = 0, —1 modes can be rather intuitively explained in terms of the "two-particle model" [64, 70] with one particle representing the head and the other the tail of the bunch. Both macro-particles oscillating in-phase correspond to the / = 0 mode, and in anti-phase to theJ = — 1 mode. The head-tail instability mechanisms involve, as the name suggests, the driving of the trailing particles in a bunch by the transverse wake force of the particles ahead. Due to the synchrotron motion, the particles exchange their relative longitudinal positions. This allows all particles in turn to be driven for half a synchrotron period by the wake force of the particles ahead while they are at the back of the bunch. Suppose for a moment that the chromaticity is zero. The response of the tail particle to the wake force will consist of a growing quadrature resonant term with respect to the drive by the head. Half a synchrotron oscillation later, what was the head (and has now become the tail) will be driven in the same way. The net effect of these quadrature drives is to effect an additional phase-space rotation of the coherent betatron oscillation, so that the coherent motion receives an additional focusing. If the wake field is too strong, the motion can become over-focused, leading to a "strong" head-tail instability. Suppose now the chromaticity is not zero: the betatron phase of the head and tail • particles is modulated at the synchrotron frequency. As a consequence, the response of the tail particle consists of the previous resonant quadrature term and an addi-tional in-phase1 but non-resonant term which is proportional to the head oscillation. This chromatic term alone is responsible for the weak head-tail instability. Half a synchrotron oscillation later, what was the head (and has now become the tail) will experience the same multiplicative effect. Though it is a small effect, it accumulates as a geometric series and the head and tail particles will slowly spiral out on ever larger O r anti-phase, depending on the sign of £. 7.5. LANDAU DAMPING MECHANISM orbits in betatron phase space; this is the weak head-tail instability. 108 7.5 Landau damping mechanism Experimentally, the weak head-tail instability has a threshold; this suggests that there is some readily available stabilizing mechanism against collective instabilities. When particles in a bunch have an intrinsic frequency spread in their incoherent synchrotron or betatron frequencies, one such mechanism is Landau damping. The frequency spread can comes from several sources. An important source is nonlinearity in the focusing system. This causes a dependence of the betatron frequency on the particle's oscillation amplitude. A distribution of betatron amplitude then leads to a frequency spread. This is the type of frequency spread we will be considering for the Landau damping of the weak head-tail instability. In practice, a frequency spread is obtained by introducing nonlinear magnetic elements in the lattice such as octupole magnets. Buried within the mathematics of the dispersion relation is the stabilizing mecha-nism of Landau damping. Here we give a simplified picture of the mechanism. More details with simple illustrations are given by Hereward [40] and Edwards and Syphers (Chapter 6.6) [22]. Essentially, Landau damping works because it is not possible to persuade a collection of particles with an intrinsic frequency spread to respond both coherently and resonantly to a driving force of single frequency - even if some of the particles response resonantly. The collection's net response remains at a constant mag-nitude. The energy is mostly absorbed by the incoherent motions of those particles whose incoherent frequencies are close to the driving frequency. The incoherent am-plitudes of these particles grow in time but the actual number responding resonantly diminishes, as more and more particles find that their resonance frequency does not coincide precisely with that of the drive. Hence their net response is constant. In the case of a coherent instability, the driving force is not external but is due to the displacement of the beam itself by way of a feedback mechanism. The net response 7.6. DERIVATION OF THE DISPERSION RELATION 109 of the beam is proportional to the displacement and inversely proportional to the fre-quency spread. Hence for a sufficiently large spread the feedback mechanism that is the source of instability is not strong enough to produce exponential growth. Below such a threshold, the coherent instability is said to be Landau damped. 7.6 Der iva t ion of the dispersion relat ion 7.6.1 The Vlasov equation In this section we will set up the Vlasov equation to include the weak head-tail mech-anism and the quadratic dependence of the betatron frequency on the betatron ampli-tude. The first step in writing down the Vlasov equation is to determine the dimensions of the phase space necessary to describe the physics of the problem. For simplicity, the motions in the two transverse planes are assumed to be decoupled from each other and hence can be considered independently. This reduces the number of dimensions needed to describe the phase space distribution of a bunch from six to four, two for the vertical or horizontal and two for the longitudinal. We choose to work with the vertical plane. There would be no difference if we chose the horizontal plane. The coordinates of a par-ticle in the vertical phase space are the displacement y and the conjugate momentum py, and in the longitudinal phase space the relative longitudinal displacement z and the relative fractional energy deviation 6. The later are not a conjugate pair, but it is more convenient to work with z than with the conjugate time coordinate. As explained earlier, in the betatron or synchrotron phase planes particles follows elliptical orbits at betatron frequency up and synchrotron frequency us, respectively. The elliptical orbits can be made circular with the right scaling, permitting the use of polar coordinates, which are more convenient. The transformation of the transverse coordinates into their polar forms is defined by y = qcos6, (7.1) 7.6. DERIVATION OF THE DISPERSION RELATION 110 P y = --^.qsmO, (7.2) c where q is the betatron amplitude and 9 is the betatron phase angle. Likewise for the longitudinal coordinates, z = rcosd> (7-3) 8 = —rs'mcf), (7-4) r\c where r is the synchrotron amplitude, d> the synchrotron phase angle, q the slip factor and c the speed of light. The head-tail mechanism relies on the coupling of the betatron and synchrotron motions. The dependence of the betatron frequency up on momentum deviation S and the chromaticity if is given by u>p(l + from Eqn. 2.23, assuming kinematic 8 —> 1. The betatron frequency is modulated at the synchrotron frequency since 5 undergoes synchrotron oscillations. For now the amplitude dependence of the betatron frequency will simply be denoted implicitly as wp(q). Following references[69, 70], which use the justifiable approximation that the wake has no effect on the longitudinal motion, we can write the Vlasov equation for the evolution of a distribution ip(r,(f),q,9): f + " ' W + fflg + + = 0, (7 .5) ds c 89 Es dpy c d<f> where s parameterizes the time t, s = ct. The quantity Fy is the transverse wake force generated by the dipole moment of the beam, and Es = jm0c2 is the beam energy. If there is no wake force Fy, and the chromaticity £ is zero, then there exists a trivial distribution which satisfies the Vlasov equation; this can be written as a product, because the longitudinal and transverse motions become decoupled = fo(q)go(r). (7.6) This is called the stationary distribution and it is a function of r and q only. 7.6. DERIVATION OF THE DISPERSION RELATION 111 7.6.2 The dispersive integral equation In this section, we allow the stationary distribution to be perturbed by a small distur-bance and derive the equation which determines self-consistency of the perturbation and its wake. It is not necessary at this stage to specify the stationary distribution; the results will be of general application. In the following section we will solve the integral equation first, when the stationary distribution is a Dirac delta function and second, when it is a binomial distribution. In the derivation of Sacherer[50], where there is no Landau damping, the coherent mode frequencies f i of the perturbation are obtained by solving the integral equation. The solution contains a discrete set of complex mode frequencies of which the real part gives the oscillation frequency and the imaginary part gives the growth/decay rate. A mode is unstable if there is a positive imaginary component in its complex frequency. As we have mentioned, with Landau damping, coherent modes can exist only above a certain threshold current. We will call these mode frequencies Landau mode frequencies A to differentiate them from Sacherer's mode frequencies f i , for the case of no Landau damping. Below the threshold, no discrete modes exist since the beam is in the decoherence regime. A brief description of the beam below the threshold is given by Chin et al [66] and a more lengthy one by Koscielniak [76]. Assuming that the principle of superposition applies, we can treat one single mode of coherent oscillation at a time. The perturbed stationary distribution with a single coherent mode of oscillation can be written as where fi(q, 9) and gi(r, ci) are the respective phase space distributions of the perturba-tion. Substituting the perturbed distribution into the Vlasov equation and linearizing with respect to the perturbation, keeping in mind that Fy is already first order, we find the linearized equation (7.7) d f f l -iAs/c dcb\ sin 9Fygo dfo dq 0. (7.8) 7.6. DERIVATION OF THE DISPERSION RELATION 112 As mentioned previously, we have already fixed the transverse structure to be a dipole moment. The transverse perturbation then contains only two transverse modes (m — ± 1 ) -h(q,0) = fl+(q)e+te + fr(q)e-id, (7.9) where fi(q) is the radial density distribution of the +1 mode and / f (q) is that of the —1 mode. They are not necessarily identical. Both modes must be present for completeness. If there is no wake force, then / f = 0 and = 1. Consequently, for weak wakes the positive azimuthal mode dominates. Moreover, for the weak head-tail instability, the mode frequency shift is much less than the betatron frequency, and so coupling between the two modes can be completely ignored. Hence the solution can be approximated by the dominant mode fi(q)elS, where we have dropped the + superfix for convenience. This treatment is referred to as no-transverse-multipole coupling and is the standard treatment of all previous authors. Substituting fi(q)ete into Equation 7.8 we obtain |«[A - W / J fo) ( l + (5)}9l - u , s | | j h{qy6e-^'c + ^ ^ / v ^ d o = 0 , (7.10) where the exponential representation of the s'mO function has be used and e~ld is dropped since there is no mode coupling. To solve the above equation, we follow the method originated by Sacherer[50]: the longitudinal perturbation gi(r,d>) is expanded in a Fourier series (of individual longitudinal modes), and this transforms the Vlasov equation into a set of coupled integral equations for which the longitudinal radial function Ri(r) of mode / can be solved. First, the dipole force Fy(z, s) must be evaluated in terms of the dipole moment De~zAs/cpi(z) of the beam and the transverse wake Wi(z), where D is the dipole displacement of the transverse distribution: D = (y) = I" [ +* fi(q)ei9qcos8 ^qdqdO. (7.11) J0 J--K c 7.6. DERIVATION OF THE DISPERSION RELATION 113 Note that the dipole moment is not a constant along the length of the bunch but de-pends upon the longitudinal factor that carries the coherent betatron phase information Pi(z) of the perturbation. Summing the wake field of all previous revolutions and making a Fourier transform into the frequency domain, we obtain Fy(z,s) = i ^ e - ^ Y U ^ Z ^ V " ' * 1 ' , (7-12) where u' = pujo + A, e is the electron's charge, Zi(ui) the total transverse impedance, too the revolution frequency, and To the revolution period. The Fourier series expansion of gi(r,4>) is given by 9l(r,cb) = e^lc\Y,aiRi{r)e il* (7.13) i where a\ are the Fourier coefficients, and / is the longitudinal mode number (/ = 0, ± 1 , ± 2 , . . . ) . The last term accounts for the additional head-tail phase factor due to the chromaticity. Substituting Equations 7.12 and 7.13 into Eqn. 7.10, multiplying by e~%1^ and integrating with respect to (b, we get a set of coupled equations for the radial functions Ri(r), [ A - c ^ ) - / c ^ / ^ ) / i ( ^ = - i J ^ £ p i ( " W » V i (jr - goe"^, (7.14) where 7 is the usual relativistic parameter, ro the classical electron or proton radius, Ji(x) the Bessel function of the first kind of order /, and the chromatic frequency shift tot = (uip/n. Furthermore, Fourier transforming the longitudinal function pi(z), we obtain p^u') = 2TT— £ rdr aiRi{r)i~ l Ji (U'~^r\ . (7.15) If we substitute the above expression for f>i(u>') into Equation 7.14, we see there is longitudinal mode coupling; i.e. all modes are present in the equation. This means that we have to solve a linear set of integral equations to find the radial function Ri(r). For the "weak" head-tail instability, we assume that the mode frequency shift 7.6. DERIVATION OF THE DISPERSION RELATION 114 ( A A = A — up — lus) is small compared with us. In this case, modes with different / will only couple weakly and this coupling can be neglected. If fi(q)e l9 and D are removed from Equation 7.14, only longitudinal functions and variables remain. This can be done by dividing throughout by [A — up(q) — lus], multiplying by q 2 cos 8up/c, and integrating with respect to 6 and q, and using Equation 7.11 to eliminate D, giving Ri(r) . nr0us = -» ^2 #o(r) ( 7 - 1 6 ) / (A) + ig(A) 7upT0\ x j T r'dr'R^r') £ Zi(u')Ji J, . This is the dispersive integral equation. The function / (A) + ig(A) is known as the beam transfer function (BTF), and measures the response of the beam to a driving sinusoidal force of frequency A. Dispers ion integral / is defined by the dispersion integral as / ( A ) + # ) i r « M i i . (7.17) 2 Jo A - up{q) — lcos  V ; As an example, let us evaluate the dispersion integral for the specific case that the transverse stationary distribution is the binomial form: a ( l + a ) ( I Y 2q 2 v q 2 where 0 < q < q, q is the maximum betatron amplitude and a is a parameter that determines the profile of the distribution. Also, let us suppose that the betatron frequency depends quadratically on the amplitude, ojp(q)=up + S^ (7.19) where S is the frequency spread in the distribution. Let us write A A = A — up — lus and assume that the coherent frequency shift A A is pure real and lies in the incoherent band of frequencies S. The integral cannot be evaluated for a in general; as an example /o = ^ P ' l - ^ , C I S ) 7.6. DERIVATION OF THE DISPERSION RELATION 115 we choose a value of 4. The singular integral is evaluated according to the prescription of Landau[74] and leads to a principal value (formed by real, symmetrical integration about the singularity) and an imaginary residue from a semi-circular indentation in the complex plane. The results are given below. /*/ A A \ 20 / - 3 S AA (AA-S)3 3S2 - 3SAA + A A 2 = ss[— + ^ + + 2S A A ( A A - S)3 log\S - A A | (S - A A ) 3 A A l o g | A A | \ S3 S3 j ' Not only can the B T F be calculated, but it can also be measured in the limit of zero growth rate, i.e. Im(A)=0, by driving the beam with a sinusoidal force of frequency Re(A) and measuring the phase response. This response can be decomposed into i n -phase and quadrature components, and these are related to the real part / (A) and imaginary part g(A), respectively. In the absence of a frequency spread, i.e. wp(q) = up, the dispersion integral becomes = f) — up — l u s , (7-21) / ( A ) + iflf(A) W / 3 ( 9 )=W / 3 and the dispersive integral equation (Eqn. 7.17) reduces to the Sacherer integral Eqn. 6.179 of [70], as expected: (il-up-lu^R^r) = _ z ^ ^ - 5 o ( r ) (7.22) / ~U(_ A fu' - u^ x / rdr'Kdr') > ,Z-i(u')Ji I V J, ~r Whether it is Equations 7.17 or 7.23 that we study, in the first approximation the impedance can be evaluated at the unperturbed coherent frequency u' = puo-\-up-\-lus. 7.6.3 The air bag distribution In this section we will solve the dispersive integral equation for the case an "air-bag" longitudinal distribution. This is the easiest distribution to consider because the radial 7.6. DERIVATION OF THE DISPERSION RELATION 116 function Ri(r) will be a delta function. In the following section we consider the more realistic binomial distribution. In the air bag distribution all the particles have the same synchrotron oscillation amplitude r. The phase space distribution is a thin ring around the origin and can be defined by a Dirac delta function, 9o(r) = ^ 5(r - rl (7.23) where N is the number of particles in the bunch. The constants in the distribution function come from the normalization, N = I g0(r)— rdrdcb. (7.24) J nc It is easy to see that the radial function Ri(r) = cti8(r — r) because the perturbation can happen only where there are particles. Substituting the distribution and the radial function into the dispersive integral equation and integrating with respect to r and then r', we obtain the dispersion relation, The right hand side of Equation 7.25 contains information about wake while the left hand side contains information about the beam response. If the impedance Z\{io) is known, then in principle, the dispersion relation can be used to calculate the Landau mode frequency A by inverting the B T F . However, obtaining A for some particular impedance function does not give us much of a "picture" of the beam stability. To find the "big picture", we use the dispersion relation as a map relating A to some familiar quantity. Ideally we want to map A to impedance. This is not possible since Equation 7.25 cannot (in general) be inverted to obtain Zi(u'). Fortunately, there is an alternative. To see this, suppose the impedance is known. Evaluating the right hand side of Equation 7.25 then gives the Sacherer frequency shift of the /th head-tail mode (Eqn. 6.188 of [70]), J2 - U 0 - lu>. = - i ^ - E Zx{J)Jf l ^ ^ z ] . (7.26) 7.7. THE BINOMIAL DISTRIB UTION 117 Let us write the shift ($1 — up — lus) = U + iV and A = P + iQ where U, V, P, C} are pure real. Hence we obtain the map: 1 U + iV. (7.27) f(P + iQ) + ig(P + iQ) Essentially, if the simpler quantity fl is calculated, then A can be found using this mapping, and whether there is enough Landau damping for stability can be determined. A graphical plot which shows the mapping (Eqn. 7.27) is called the stability diagram. This is constructed by tracing out the locus of points in the complex ([/, V) plane as the Landau frequency P is scanned while the Landau growth rate Q is held constant. This defines a contour of constant Landau growth rate. Figure 7.1 shows a few of these contours for different growth rates. Relation (Eqn. 7.27) does not map the Landau (P, Q) plane onto the entire Sacherer (U, V) plane. There is a region around the (£7, V)-origin to which no (P, Q) points are mapped. This means that there are ([/, V) points for which there are no corresponding Landau solutions. These are {U, V) coherent oscillations which are Landau damped when there is a frequency spread. The area bounded by the U axis and the contour line of zero growth rate defines the stable area. 7.7 The binomial distribution In this section we consider the case where the longitudinal stationary distribution is the binomial distribution. This time the radial function Ri(r) is much more complicated and cannot be found by simple inspection. The method[60] we use is expansion of the radial function in terms of a set of orthogonal functions. This transforms the integral equation into a set of linear equations. Solving for the eigenvalue then yields a set of dispersion relations, one for each eigenvalue. First, we introduce the orthogonal functions, then we outline the transformation and the steps in obtaining a dispersion relation. 7.7. THE BINOMIAL DISTRIBUTION 118 We start by rewriting Equation 7.17 with two new symbols in anticipation of trans-forming it into an eigenvalue problem: Mr) fCO W(r) r'dr'Ri(r')Gi(r,r'). (7.28) Jo f(A) + ig(A) Here the weight function W(r), which is essentially the stationary transverse distribu-tion function multiplied by some constants, is W(r) = ^ f l b ( r ) , (7.29) and Gi(r,r') is the kernel function of the integral, which is defined as G.(r, O = -i^THL. E Zl(J)J, ( , ^ r ) J, (^V) . (7.30) Suppose we have a set of basis functions {/fc(r), k = 0,1,2,3...} that satisfies the orthonormal condition Jroo ' rdrW(r)fk{r)fk,{r) = 8kk,, (7.31) o where 8kki is the Kronecker delta function. The radial function can then be expanded as Ri(r) = W(r)J2akfk(r), (7.32) k where ak are the coefficients of R\{r) with respect to the orthogonal set {fk{r)} and weight function W(r). The orthogonal set is given in Appendix B. Substituting the above expression into the dispersive integral equation (Eqn. 7.28) and using the or-thonormality condition (Eqn. 7.31), we obtain a set of linear equations k> ak = Y2,Mkk'ak'-, for k = 0,1, 2 ,3 , . . . (7.33) [f(A) + ig(A)_ where A;' = 0,1,2,3, . . . , and the matrix elements Mkk' are given by /•oo r co Mkk, = Jo rdrW(r)fk(r)JQ r'dr'f^G^r'). (7.34) The matrix elements contain information about the beam intensity and the total trans-verse impedance through the kernel Gi(r,r'). The expression for the matrix elements 7.7. TEE BINOMIAL DISTRIB UTION 119 may seem complicated, but it can be given a simple physical interpretation if we in-troduce hlk{u) = l°°rdr W(r)/ f c (r) J, ( ^ ^ ) . (7.35) These are the Fourier spectra of the orthogonal functions times the phasor distribution, fik(r)e iltt,e iuj^ c. The spect ra are derived in Appendix B, and the result is: hlk{u) = KiJi+olJr2k+i ( ^ y ^ ) / ( ^ r ^ ) 1 + * ' ( 7 - 3 6 ) Using the above definition of hik, the matrix elements can be written as Mkk, = ^J^YsM^hi^h^u'). (7.37) This means that the coupling between two different functions of indices k and k' de-pends on the overlapping of the corresponding Fourier spectra with each other and also with the impedance. For a nontrivial solution to exist, the reciprocal of the. beam response function / (A) + i g(A) must satisfy ^hushm)1-"1}^ (7'38) where I is the identify matrix and M is the interaction matrix defined previously. This is true if and only if - — = A n , n = 0 , l , 2 , 3 . . . , (7.39) / (A) + ig{A) where Xn is an eigenvalue of M , and n is the eigenvalue index. There are an infinite set of such dispersion relations (Eqn. 7.39) for each /-index; one for each and every n-index. On the other hand, for the air bag distribution, there is only one dispersion relation for each / because there is only one possible radial function. As pointed out in the last section, the most important use of the dispersion relation is as a complex map. Now Xn is, in fact, just the Sacherer frequency shift: Xn = Vln - lus - top = U + iV (7AO) Hence we can utilize the same mapping as given in Equation 7.27. In the next section we make use of this map. 7.8. LANDAU DAMPING BY OCTUPOLES 120 7.7.1 Commentary Zotter [60] had derived the spectrum (Eqn. 7.36) earlier using the same set of orthogonal basis functions but using the unjustifiable assertion that the Weber-Schafheitlin integral (B.9) is identically zero beyond the radial limit of the phase space distribution. This allowed the integrals to be easily evaluated using standard techniques and the properties of the Bessel functions. In our new derivation of the spectrum, we have confirmed that this "sleight of hand" gives identical results to performing the rigorous contour integration required. Subsequent to initial publication of this work (Tran and Koscielniak [78]), we learnt that Satoh, in an earlier unpublished note [59], had also reported a rigorous derivation of the spectrum. Satoh also found the orthonormal basis and calculated its spectrum using a different (non-standard) representation of the Jacobi polynomial (Eqn. 22.2.2 of [81]). This choice allowed him to adopted a different procedure, integrating the series expansion of the Jacobi polynomial term-by-term and recombining these terms into a single Bessel function using a recurrence relation. This method is more direct but, does not closely follow the standard route mapped by Zotter. 7.8 Landau damping by octupoles With ever increasing intensity and more stringent conditions, Landau damping of in-stabilities by intrinsic nonlinearities in the machine lattice and image-charge forces cannot be assumed to be adequate. The weak head-tail instability is likely to be the first transverse instability to be encountered. This is clearly demonstrated by the recent observation of high-order modes at the C E R N - P S in an experimental run of the PS as an LHC injector (Cappi Ref.[71]). During the test run, just one bunch with the LHC characteristics was injected. This test bunch was four times longer than normal, mak-ing it much more susceptible to transverse instabilities. The resistive-wall impedance has been identified[71] as responsible. In this section, we will make practical use of 7.8. LANDAU DAMPING BY OCTUPOLES 121 the theory presented in Section 7.7 to calculate growth rates of the unstable head-tail modes and compare them with measurements. Then we will use the stability bound-ary diagram to obtain the octupole strength required to Landau damp the observed unstable modes. The Sacherer Frequency Plane 1 4 0 0 ^ ' ' 1 1 1 1 L • 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 Coherent Frequency Shift [Hz] Figure 7.1: Contour lines and stability boundary diagram for the LHC bunch. 7.8.1 Growth rates and frequency shifts of the L H C bunch The longitudinal amplitude distribution of the LHC bunch can be approximated by the binomial distribution with a = 4 (Eqn. 7.18) because this gives a profile which is close to that of the Gaussian but without the tails. For this distribution we can calculate the growth rates and frequency shifts of the head-tail modes using the spectra in 7.8. LANDAU DAMPING BY OCTUPOLES 122 Appendix B. These quantities are given by the eigenvalues (Eqn. 7.40) of the interaction matrix (Eqn. 7.37). Approximate values for the eigenvalues can be easily found, if the cross-coupling terms of M are neglected. As we can see from Equations B.21 and 7.37, the cross-coupling terms, which are the off-diagonal elements, differ by two Bessel orders for each column away from the diagonal. This not only shifts the peaks of the spectra away from each other but also reduces their relative magnitudes, effectively decreasing the overlap. The result is that the cross-coupling terms are approximately two orders of magnitude smaller for each column away, compared with the diagonal term of the same row. Thus they can be neglected with little error. This gives us a very simple expression for the approximate eigen-frequencies, For each longitudinal head-tail mode /, the eigen-frequency of (n = 0) gives the largest frequency shift and growth rate because the peak of its spectrum is the most dominant and is approximately two orders of magnitude larger than that of the next. The matrix element Mnn' (Eqn. 7.37) is given as a summation over index p. To evaluate the summation, we can transform it into an integration, provided the impedance Zi(u>') satisfies the following two conditions: it is a smooth function, except for u' = 0, with no local peaks narrower than the revolution frequency UJQ (i.e. it is a broad-band impedance); and \Z[(LJ')UJ0/2\ <C ZI(U>'), where Z[(LO') is the derivative of Z\ with respect to UJ' . When these two conditions are satisfied, Mkk' is given by the integral The resistive-wall impedance satisfies the two conditions above, permitting the summation to be transformed approximately to an integral. We use the following formula [77] for the impedance of the resistive wall, tin = Up + lujs + M , nn • (7.41) (7.42) (7.43) 7.8. LANDAU DAMPING BY OCTUPOLES 123 where (for the C E R N PS) the machine radius R — 100 m, the vacuum chamber radius b — 3.5 cm, the vacuum wall resistivity p = 9 x 10 - 7 fim and the permittivity of free space e0 = 8.85 x 10 - 1 2 F /m. We substitute the formula into Equation 7.42, and perform the integration numerically. Table 7.1 lists the growth rates and frequency shifts for modes / —1 to 8. For each mode we only calculate for n = 0 because this index gives the largest growth rate and frequency shift, at least one order of magnitude greater than that of the next index. The following beam parameters were used for the calculations: UJQ = 2TT x 0.417 MHz Energy E = 1.93 GeV us = 2TT x 0.9 kHz = 2TT x 12.5 MHz vx and uy — 6.25. (7.44) Here uXiV are the horizontal and vertical tune and are defined by ux>y = u>pXty/uo. According to our calculations, modes 5, 6, and 7 should be unstable as shown in Table 7.1. The predicted rise times for these three modes range from 200-285 ms using the spectrum hikUJ of a binomial distribution, which are comparable to the observed rise times of 100-200 ms by Cappi [71]. He does not specify the individual growth rates probably because all modes were present during measurements. We are not able to compare the frequency shifts with measurements as there are no data available. The agreement of the rise times suggests that the resistive-wall resistance is primarily responsible [71]. There may be other sources of reactance which only cause coherent frequency shifts. Cappi assumed the distribution to be parabolic since it is a proton bunch. He used the approximate mode spectrum to calculate the theoretical growth rates since the exact binomial spectrum was not yet available. The approximate spectrum was 7.8. LANDAU DAMPING BY OCTUPOLES 124 Mode no.s Frequency shift Growth rate Growth rate Growth rate Our calc. Our calc. Cappi calc. Cappi observed / (Hz) (s-1) 2 -160 -120 - -3 -109 -45 - -4 -67 -7 - -5 -36 5 10 5-10 6 -18 5.6 8 5-10 7 -10 3.6 2 5-10 8 -6 2.1 - -Table 7.1: Transverse head-tail frequency shifts and growth rates of the L H C bunch. formulated by Sacherer and Laclare [53, 57] and is given by kt \ n i i \2 1 + (-l)'cos(a;r 6) = {l + 1) [ ( u r b / , y - ( i + i y r  (7 5) where u is the total bunch length. Cappi [71] obtained growth rates of 10, 8 and 2 s _ 1 for modes 5, 6 and 7 respectively, as shown in the table. These results are discussed further in Section 7.9 below. 7.8.2 Landau damping of the LHC bunch In the previous section we calculated the frequency shifts and growth rates of the un-stable modes. In this section we estimate the upper bound integrated octupole strength required to damp these modes. To do this, we have to find a sufficient frequency spread S such that all unstable mode frequencies fl = U + iV are inside the stable area of the stability diagram. To obtain a stability diagram, we choose a value of S, evaluate the B T F and draw contour lines of constant P and Q in the U, V plane. If not all modes are inside the stability region, then we increase S and re-draw the diagram; and repeat 7.9. DISCUSSION 125 as necessary. Let us suppose that the transverse betatron amplitude distribution of the LHC bunch is the binomial distribution for a = 4. The B T F of this distribution is given by Equation 7.20. An upper bound on the octupole strength is obtained from the mode whose coor-dinates U, V give the largest growth rate and coherent frequency shift. As reported in [71], the largest observed growth rate was 10 s _ 1 . The frequency shifts of the unstable modes are not available. Fortunately, that of the dipole mode (/=0) is reported and it is the upper limit, which would give a conservative estimate. Guided by the well-known rule of thumb for Landau damping, which says that the spread should be at least as wide as the growth rate, a frequency spread of 1000 Hz should be sufficient. Figure 7.1 shows the location of the point. Note that it is well inside the stable region. This frequency spread requires an integrated octupole strength of 63 m~ 3 , where the integrated strength is defined as Comparison of growth rates calculated using the binomial spectrum with those of Cappi [71] who used the approximate spectrum (Eqn. 7.45) shows significant differences; the 5th mode is twice as slow, the 6th about one and half and the 7th is almost twice as fast. Since the same impedance is used for both spectra, these differences indicate that the approximate spectrum is not a good approximation of the binomial spectrum. Both our growth rates and those of Cappi are in general agreement with the experimentally observed rates - but since these cover a broad range and are not quoted for individual modes, it is not possible to make a more detailed comparison. The experimental observation that the 5th mode was the "most common" tends to support Cappi's prediction that it is the strongest of the three, rather than our prediction that 5th (7.46) and / is the total length of the octupole magnet. 7.9 Discussion 7.9. DISCUSSION 126 and 6th modes are almsot equally strong. More measurements with better precision as well as accurate data on the bunch properties are required in order to tell definitively whether the binomial or the approximate spectrum gives better results. Discrepancies between the predicted and observed growth rates may be due to factors such as the actual distribution of the bunch shape, the real impedance of the ring and the assumptions of the theory. One has to keep in mind that the theory is linear and uses perturbation analysis and hence the predicted growth rates are for a linear system and at zero oscillation amplitude. Experimentally it is only possible to measure the growth of an oscillation when it has some amplitude. In addition, the real system is never completely linear, and the observed growth rate is one which has already overcome some Landau damping. An other important factor is the true impedance of the ring. Both Cappi and we use the resistive-wall impedance as the sole impedance to calculate growth rates. In reality, the C E R N PS ring is a complex impedance environment, consisting of many impedances, both resistive and reactive, other than the resistive-wall. However, it must be noted that the resistive-wall gives, by far, the largest contribution. Therefore, a more accurate calculation must include other impedances as well. Finally, in order to ensure a meaningful comparison between the calculated and observed growth rates, the bunch shape needs to be accurately fitted to a binomial distribution to infer the a parameter that gives the best match; the precise bunch length is also required. The growth rates of the weak head-tail instability are slow compared to other instabilities such as the strong head-tail. Landau damping of the instability could be accomplished with a moderate octupole strength which does not adversely affect the emittance of the beam due to decoherence caused by the introduction of the required nonlinearity, as discussed in the previous chapter. For the L H C bunch in particular, only a moderate octupole strength of 63 m - 3 is required to damp the most unstable mode. Though Landau damping has some influence, to first order the growth rate and fre-7.9. DISCUSSION 127 quency shift depend on the beam intensity; and small changes in the bunch distribution only contribute small differences to rates provided that the stationary distribution is not pathological (i.e. provided it already has smooth tails). However, when the beam is close to the instability boundary, the shape of the distribution can be important. For example, if the beam is just outside the boundary, then a small reduction in the sharpness of the tails (higher a in the binomial case) can move the beam inside the boundary and restore stability. A much neglected component in assessing the Landau damping requirement - but as important - is the coherent frequency (tune) shift. As can be seen from the stability diagram (Figure 7.1), for a given growth rate, a larger coherent tune shift requires more stable area (octupole strength). Hence for a ring with lots of reactive impedances, the coherent tune shift could be large. If the octupole strength is fixed, then to maintain Landau damping, the coherent tune must be kept within the incoherent frequency band or equivalently within the stable area in the stability diagram. The benefit of keeping the coherent tune shift small is that the required octupole strength can be significantly reduced, which in turn helps maintaining beam quality due to less decoherence. The coherent tune can be adjusted through the use of a feed-back damper described in Section 3, but in a reactive mode. C h a p t e r 8 C o n c l u s i o n s This thesis has described some new studies of the effects of cubic nonlinearities arising from image-charge forces and octupole magnets on the transverse beam dynamics of proton synchrotrons and storage rings, and also a study of the damping of transverse coherent oscillations. Three single-particle topics were first dealt with: • feedback damping systems • distortion of the momentum dispersion function • amplitude-dependent oscillation frequency (tune spread) Two important collective effects caused by tune spread due to cubic nonlinearities were then described: • decoherence and recoherence • Landau damping of the weak head-tail instability. A novel feature of these studies was that binomial (rather than gaussian) density dis-tributions were considered, in order to make the results valid for proton beams. For practical examples, we have applied our results to the LHC complex at C E R N , where the very high beam brightness required makes collective effects a concern, particularly 128 8.1. DAMPING SYSTEM 129 in the lower energy accelerators. ;indeed the head-tail instability has been observed at the PS for LHC beam bunches. The results of the various studies are summarized below. 8.1 D a m p i n g system We have derived the characteristics of a transverse damping system using a simple matrix formalism. This method was applied to the well-known case of a correcting kick proportional to the displacement and showed the expected exponential damping (Eqn. 3.20) and induced coherent betatron tune shift (Eqn. 3.21). The method was extended to the case of constant magnitude kicks, but with polarity depending on the sign of the displacement, by expanding the motion in terms of the eigenvectors of the one-turn matrix. We have shown (under certain conditions) that this "bang-bang" scheme should produce linear damping (Eqn. 3.47) and no coherent tune shift. Simu-lations confirmed the linearity, although the observed rate was slower than predicted, presumably because of the rather drastic approximation used. We have compared the performance of the proportional and fixed-magnitude kick schemes, under the con-straint of equal maximum power, and found the latter scheme to be substantially faster. 8.2 Image—charge force An integral formula (Eqn. 4.27) for the change in the dispersion function due to the linear1 component of the image-charge force has been derived using transfer matri-ces and the method of Green's functions. Using this formula we have obtained an expression (Eqn. 4.30) for the orbit distortion for a simple FODO cell. We have also found the cubic component of the image-charge forces for circular xThe method is, in fact, valid for any type of linear perturbation. 8.3. DECOHERENCE 130 and parallel-plate beam pipe geometries. This component gives rise to an amplitude-dependent tune shift. We have solved the nonlinear equation of particle motion in terms of Jacobian elliptic functions and obtained a simple formula (Eqn. 4.41) for the quadratic dependence of betatron tune on oscillation amplitude. The results were applied to give upper bounds for the image-charge tune spread and distortion of the dispersion in the LHC. 8.3 Decoherence We have obtained expressions (Eqns. 6.12) describing the centroid motion of an ini-tially transversely displaced beam bunch by performing ensemble averaging over the transverse and longitudinal phase-space density distributions. These distributions were assumed to be uncorrelated and to have binomial dependences on amplitude. The dis-placed transverse distribution was Taylor-expanded in terms of the original stationary distribution to make use of its simple boundary when averaging over the ensemble; to first order, the rate of decoherence depends on the average gradient of the stationary distribution (Eqn. 6.10). An additional tune spread in the bunch due to longitudinal energy spread causes the the centroid motion to be amplitude-modulated at the syn-chrotron tune The strength of this modulation depends strongly on the ratio of the chromatic tune spread to the synchrotron tune (Eqn. 6.19); a sufficiently large ratio, which is easily achieved, can cause a periodic loss of the centroid displacement signal. 8.4 Landau damping of weak head- ta i l instabi l i ty We have obtained the dispersion relation (Eqn. 7.17) for Landau damping of the weak head-tail instability by equating the inverse of the beam transfer function (BTF) to the eigenvalues of the interaction matrix. Whereas the B T F depends solely on the beam and its incoherent frequency spread, the interaction matrix depends on the Fourier 8.4. LANDAU DAMPING OF WEAK HEAD-TAIL INSTABILITY 131 spectra of the coherent modes and the perturbing impedance. As an example, we have calculated analytically the B T F for a binomial distribution with sharpness parameter a = 4 (Eqn. 7.20). The dispersion relations map the "Sacherer mode frequencies" for the case of no frequency spread, to the "Landau mode frequencies" when there is a spread of incoherent frequencies. In general the mode frequencies are complex, and can be represented as points in the complex plane. For modes at the threshold of instability, the Landau frequencies are pure real, whereas the Sacherer frequencies are complex. By using the dispersion relations as maps, we can find (for a given incoherent frequency spread) the area in the Sacherer coherent-frequency plane that is made stable by Landau damping. Hence stability may be tested by calculating the Sacherer frequency and checking whether it falls inside or outside the inverse-BTF plotted in the Sacherer frequency plane. We have derived the head-tail mode spectra (Eqn. 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A p p e n d i x A C u b i c e q u a t i o n o f m o t i o n v-2 Consider the cubic equation X"+ [R~)  x = + C x ^ (a-V where ' = djds and C is constant. This equation is solved in [30, 79] and the solution consists of Jacobian elliptical functions whose definitions are given in Chapter 8.14 of [80]. Since we are mostly interested in the amplitude-dependent tune shift we can use the particular solution sn with intial conditions x = 0 and x' = x'0 instead of the more complicated general solution. The sn solution can be written as x = Asn[6,k], 0 = a x (s - s0) , (A.2) where A is the amplitude of the oscillation, and k is the parameter. Our task is to relate the parameters k, a to the coefficients of the differential equation. Substituting the trial solution, we have i £ - a 2 f l + k 2 - 2k2sn2e] = CA 2sn 29. (A.3) Equating the coefficients of the above equation to those of Eqn. A . l , we find the following expressions: Q=(ITA0' (A-4) 72 ICA 2R 2 k = 2 ^ ^ ' (A.5) 140 1 4 1 provided « 1, which is true in this case since the nonlinear coefficient C is much smaller than the linear restoring force (v/R)2. The new amplitude-dependent tune ua is given by the total phase advance per turn divided by the period of the sn function which is 4K(k), a2nR u« = 4KWy  ( A - 6 ) where K(k) is the complete elliptic integral of the first kind. The new tune can also be written as va = u + A i / a , where v is the small-amplitude tune and Aua is the amplitude-dependent tune shift. Squaring the above equation and substituting the expression for a (Eqn. A . 4 ) , we have [u + Az/ a ) 2 = v (l + k)K 2(k) ' For k << 1 , the function K(k) can be Taylor expanded as (A.7) (2n)! k 2n + (A.8) ( 2 2 " ( n ! ) 2 Truncating the series to second-order in k and substitution of Eqn. 4 . 5 2 in Eqn. 4 . 5 4 gives the tune shift approximation: 5 CA 2R 2 Aua -1 6 ( A . 9 ) A p p e n d i x B B i n o m i a l m o d e s p e c t r a In this appendix, we will derive the frequency spectra of the the orthonormal basis fk(r) for the binomial distribution. Our route follows that of Zotter [60] but makes good some deficiencies of that work. Using a contour integration method, which we shall demonstrate, we give a rigorous derivation of earlier results. We start with a binomial distribution of the form (1 +a) r a nr" 2 \ f 2 where r is the maximum synchrotron amplitude (r < r) and a is a real number that de-fines the sharpness of the binomial distribution. Substituting the binomial distribution into Equation 7.31, which defines the orthonormal condition, we have „2' a f rrdr {-^^-(l- r-) fk(r)fk,(r) = Skk, (B.2) Jo nr z \ r z J From a standard list[81] of weighted orthonormal functions we obtain, after normal-ization, Mr) 2n(l + a + 2k + l)T(l + a + k + l) k\ (r_\> . „ / _ rfV \J (l + a)T(l+k + l)T(l + a + k) \r)  k \ r 2 / ' 1 J where the radial function index k = 0,1,2,3..., T(x) is the gamma function and P1k' a(x) defined by Equation 8.960 in [80] is the Jacobi polynomial of order k and index / equal 142 143 to the the H-T mode number. We can now evaluate the spectrum hik(u>). Substituting Equation B.3 into the definition for the Fourier spectrum (Eqn. 7.35), we have hlk(u) = jT rdr K (1 - y2)a y l ^ ( l - 2y2) J, ( ^ ^ r ) , (B.4) where y = r/r and K is a function of a, k and /, K 2(1 + a)( l + a + 2k + l)T(l + a + k + I) k\ ^ r ( l + fc + / ) r ( l + a + fc) { " j The integrand is very complicated, but can be evaluated with a trick discovered by Zotter [60]. It involves transforming the single integral into a double one using the Weber-Schafheitlin integral formula (Eqn. 6.574(3) of [80]), r°° v lT(l 4- k 4-1) I W + , ( « W ( r t ) r » dt « w - . ? ( 1 ( + a + ^ ( 1 + l ) f d+*+'. — * : 1+'. A (B .6 ) where F(l + k + /, —a — A;; 1 + /, y 2) is the hypergeometric series, 1 + A: + / > 0, and a > — 1. The Jacobi polynomial is defined in terms of the hypergeometric series as follows (Eqns. (8.960) and (8.962) of [80]), P * ' a ) { x ) = THT(l + l ) F { 1 + * + ' + " M + (B-7) After substituting 1 — 2y2 for x into the above formula and using the transformation formula of Equation 9.131(1) in [80]), we obtain Pt\l - 2y 2 )(l - y 2 )« = ^ { ^ n i + k + I, -a - k; 1 + /, y 2 ) . (B.8) Substituting the above hypergeometric series into the integral formula (Eqn. B.6), we have I J1+a+2k+lWMrt)r° dt = 2 ^ _ a r ( 1 + Q + fc)y'n(;'a)(i - 2y 2 )( i - y2T- (B.9) Using the above formula, we transform the single integral (Eqn. B.4) into a double integral for the evaluation of the spectrum, hlk(u) = f r rdrdtK, Ju-cw+iirtWrtM ( ^ ^ r ) t~a, (B.10) Jo Jt=o \ c / 144 where K\ is Kx = 2 2(1 + a){\ + a + 2k + l)T(l + a + k + l)T(l + a + k) 7T ( B . l l ) \TT r ( i + fc + / ) f c ! The integration above can be performed in two steps: with respect to r, we can make use of an integration formula; however with respect to t, a specific contour integration is required. The integration formula1 is Equation 11.3.29 of [81], r UJ / JA—r)Ji(tr)rdr Jo c t2 ~ ( 7 ) 2 However, [60] essentially used the result tJi+i(tr)Ji(-r) - -Ji{tr)J,+1(-r) c c c (B.12) f ° J,(-r)J,(*r) rdr = \s(t - u/c) Jo c t (B.13) by assuming, erroneously, that (B.9) is zero for y > 1. Using the formula above (B.12), the double integral reduces to poo / dtt~ aJ1+a+2k+i(rt)-Jo i Ui+1(tf)Ji(-r) --Ji(tr)Jl+1(-r) c c c (B.14) * 2 - ( 7 ) 2 Next we consider the difference of the contour integration of the following integrals in the upper half of the complex plane: x_a J„(rt)^r1+/(rz) / z2 — s2 dz J z s (B.15) (B.16) where s = LO/C, Hu(Z) = Ju{z) + iYu(z) and Yu(z) is the Bessel function of the sec-ond kind, also known as Hankel's function. The contour is a large semi circle with indentations on the real axis over the two simple poles (z = ± 5 ) and the branch point (z = 0). In the limit of the radius of the semi-circle becoming infinite, the contributions along the circumference is negligible. In the case that / + 1 — a > 0, the contribution 1Special case of Bessel functions of equal order. 145 from the indentation at z = 0 tends to zero. Because z = 0 is a branch point, the functions change their phases around the indentation according to: Jv{ze iir) = e+iv"J,,(z), Hv(ze iv) = e~ iurHu(z)-2e+ivirJ„(z), and {ze ivya = z ^ e " * ™ (B.17) This leads to some cancellation between the contribution on the positive and negative real axes. Equating these to the semi-residues at z = ± 5 leads to the results: jf J'(%>y = -Is-MH)YW(U) (B.18) /" sM*t)J'(!t)t-°dt = -ls-°MH)Y,(rs) ( B . 1 9 ) Jo tz — sz Z If we take the difference of these last two relations, we obtain (B.14) on the left and (7r /2 ) s - a [ J / + 1 ( f 5 )^ ( f s ) - [ J ; ( f 5 )K / + 1 ( f 5 ) ] (B.20) on the right hand side. But using the identity relation (pg.77 Theory of Bessel Func-tions, Watson) the terms in square brackets are just equal to 2/[7rrs]. Hence, finally, the result is: hlk{u) = Kx Jl+a+2k+i ( ^ ^ ) / ( ^ T ^ ) • (B.21) 


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