Master course in Accelerator physics Impedance and Instabilities T.F. Günzel, CELLS 10 June 2010

Introduction 1. speed of light c = 0.3 GHz·m The concepts will be introduced by basing them on first principles, consequently it is a more theoretical approach. advantage: less confusing, less phenomenology, but better view on the heart of the theory. disadvantage: some concepts need a good understanding of abstract concepts (but not necessarily more formulas) units and constants : SI , 1. speed of light c = 0.3 GHz·m cgs-units 4πε0=1 2. impedance of free space 3. transverse coordinates and ang. frequency

Introduction In literature many talks are based on the concept of coasting beam, a beam without synchrotron motion. This is a concept which can be used for proton machines, but is not applicable for electron machines with many bunches. Here effects will be treated which are of concern for electron machines, in particular for synchrotron light sources. Furthermore effects based on space-charge forces are often presented, but those also are more important in proton machines.

Literature A.W.Chao, Physics of collective beam instabilities in high energy accelerators, J.Wiley& Sons (1993) B.Zotter and S.A.Kheifets, Impedances and Wakes in High Energy Accelerators,World Scientific (1998) K.Y.Ng, Physics of Intensity dependent Beam Instabilities, World Scientific (2006) G. Stupakov, Lecture notes on Classical mecanics and Electromagnetism in Accelerator physics, The US Accelerator School, June 2007, Michigan CERN Accelerator School (CAS) gives introductory talks into impedance and instabilities www.jacow.org keywords: impedance, instability in general, the World Wide Web is an excellent source in this field

Basics of about fourier transforms(FT) oscillator motion: admitting Keep in mind that other definitions of FT exist which vary by where to put the 2π frequency time bunch length

Fourier transforms con’t convolution:

Example : beam spectrum revolution time (@ALBA 0.9μs) revolution frequency *2π this motion is modulated with synchrotron motion the contribution +ωs comes from the synchrotron motion, note ωs is much smaller than ω0 res. pω0 if it’s a transverse motion, on top it’s modulated by the betatron oscillation

Concept of impedance --- Introduction An ultrarelativistic electron moving in a perfectly conducting uniform vacuum chamber B only a radial electrical field E, fieldlines of B are circles around the electron c E No longitudinal electric field component No transverse (radial) force since Lorentz and Colomb force at speed of light cancel out No impedance If the beam velocity is not c,the resulting effect from the forces is called space charge impedance

Concept of impedance --- Introduction an electron on-axis in a non-uniform chamber Excitation of a longitudinal electric field longitudinal impedance (on+ off axis) an electron off-axis in a non-uniform chamber longitudinal electric field which varies in transverse direction transverse impedance (only off axis)

Wakefields (like Einstein riding on an electron) test particle exciting r0 r0 ┴ Add up the electric field sampled by a test particle of charge at a distance s from the leading particle : transverse wake field : 2-dimensional vector field: depending on

Typical wakes longitudinal wake of a pointlike particle test particle exciting particle transverse wake of a pointlike particle exciting particle test particle

Properties of wakefields for a pointlike source as an electron for s<0 for s<0 Wakes of a whole bunch with a charge distribution l(s) number of particles in a bunch momentum, energy loss, angle longitudinal loss factor transverse kick factor

Machine impedance Fourier transforms of the wake fields are called impedance: longitudinal Zl(ω) is a genuine impedance with units of an impedance [Ω] transversal The i was introduced to correct for phase difference between longitudinal and transverse wake. Z┴(ω) has units [Ω/m], is responsible for transverse kicks, strictly speaking it is not an impedance anymore. Symmetry properties:

Loss and kick factor in frequency domain Computation of the loss factor in frequency space: longitudinal with gaussian bunch profile with transversal The first quantity is important for the power loss of n bunches (ignoring each other) of current Ib the second one for transverse kicks of a single bunch transversal transverse kick longitudinal

intuitive meaning of impedance bunch profile wake bunch longer bunch + bunch = corresponding to this image, impedance creates voltage as additional effect, bunch lengthening can be observed very wellknown effect generated by inductive longitudinal impedance if the long. impedance is a broadband resonator, above some threshold it can drive the strong microwave instability. This instability can increase the energy dispersion of the beam.

Example of a wakefield and its associated impedance computed by GdfidL at low f the impedance is inductive the charge is not point-like the obtained wake is a convolution of a wake for a point charge with the charge distribution the impedance is obtained by fourier transform of the wake and deconvolution of the charge in frequency space the wake mainly has inductive character its shape equals to the derivative of the charge distribution

broadband impedance a model for the geometrical impedance of a vacuum chamber element Rs shunt impedance, Q quality factor, ωr resonance freq. broadband resonator (reduced) impedance at low frequencies this impedance is inductive up to a given resonant frequency. Above this frequency the impedance shows the behaviour of a resonator can be brought in the form above with the beampipe is like a resonator fields excited by the beam current to ring allows a more intuitive meaning of the notion machine impedance

Interaction of the beam with the impedance Effective impedance(reduced longitudinal or transverse) most important slide !! this spectrum can be of reduced longitudinal impe- dance or of transverse impedance Essential is the overlap of the beam spectrum with the impedance have both same symmetry properties for longitudinal reduced impedance : for transverse impedance :

Types of impedances Geometrical impedance (longitudinal + transverse) due to cross section change of the beam pipe is numerically computed by the simulation of the beam passage through the beampipe by programs like GdfidL, MAFIA, NOVO etc. The computation of of many vacuum chamber elements requires a big effort. The obtained spectra are added up to get the total Z(w) Resistive wall impedance (longitudinal + transverse) due to the finite conductivity of the vacuum chamber wall covers ~2/3 of the total impedance of a modern synchrotron is computed analytically (see below) impedance related to space charge(longitudinal + transverse) due to the repelling forces inside a bunch, at speed of light it’s zero impedance related to radiation due to delay of the particles on a curved orbit

transition to a low-gap chamber Vacuum chamber geometries which create geometrical broadband impedance beam vertical scraper bellow (RF-finger) transition to a low-gap chamber (vertical taper) bellow followed by vertical taper

Example: impedance of a Beam Position Monitor vacuum chamber with BPMs button r = 5.5mm Numerical calculation of the wakefield and its fourier transform Re(Zl(ω)) n bunches, ignoring each other: Ploss = 3.22W (ALBA)

Resistive wall impedance Etan a B c E diffusion of the magnetic field (B or H) in the wall solution: with: Leontovich condition (linking the undisturbed field to the wake field): more general:

Resistive wall impedance (con’t) Etan a B c impedance! E Leontovich condition: the longitudinal resistive wall impedance: Panofsky-Wenzel theorem: in frequency space

spectrum of the transverse resistive wall impedance at ω<0 the RW-impedance behaves according to the symmetry rules beam spectrum strongly depends on the beampipe radius , for a small pipe it’s large ! also depends on the wall material. it has a long-range component, but also has a short-range component at in fact does not go to infinity at 0 at very low ω returns to zero and to a finite value these details depend on the structure of the wall (homogeneous, multilayer, thickness) the derived formula is only valid in a circular vacuum chamber, in rectangular chambers form factors and the depen- dence of the test particle position have to be considered.

Collective motion in phase space the beam is an excellent example of an particle ensemble in phase space Longitudinal phase space : synchrotron motion rest frame of the synchronous particle phase space density Description of an ensemble of N particles: Corollar of the Liouville theorem Assumption : the ensemble converses its energy

The Vlasov equation a differential equation for particle motion in phase space as the Schrödingers equation for an electron in a H-atom Longitudinal phase space : if the equation of motion for a single particle is linear and the motion takes place in only one plane, it is easy to obtain solutions However, if the motion couples 2 planes and is nonlinear, it is difficult In accelerator physics we only look for linear solutions However, for an electron beam the condition of energy conservation is only approximately fulfilled. Radiation damping and other diffusion effects produce energy loss (Plank-Fokker-Vlasov equation)

Beam modes as solutions of the Vlasov equation Longitudinal phase space : Collective motion can be described in a perturbative approach: We won’t solve the equation with this ansatz, I immediately show you the solutions:

Beam modes as solutions of the Vlasov equation It is very difficult to observe these longitudinal modes, however, much more interesting are the frequencies these modes oscillate in phase space. They can be easily measured. Again we use the analogy with the Schrödingers equation governing the H-atom we are most interested in the energy levels of a H-atom, in longitudinal motion we are interested in the frequencies at which the modes oscillate

The concept of an instability in general complex frequencies are possible : if tp>0 the distribution is increasing exponentially. This is what we call in this formalism instability. Nevertheless in case of radiative damping (e-machines) if the beam is stable. if tp<0 the distribution is decreasing exponentially. Then the perturbation ψp is going to be damped. F. Sacherer developed the theory of beam modes in the 70’s at CERN

Head - Tail modes: Collective transverse motion depending on longitudinal phase space these modes also are solution of the Vlasov equation, but at it is a combined motion in longitudinal and tranverse phase space, it’s 4-dimensional the head-tail modes in frequency space Ph. Kernel’s thesis

Interaction of the beam with the impedance generalisation (reduced)longitudinal or transverse We can learn two things (transverse case): imaginary part of Z┴ creates tune shift real part of Z┴ creates growth or damping if Re(Z┴) < 0, an instability develops

a typical transverse impedance in interaction with the beam modes(ξ=0) zoom & rotation lines of the same colour have larger distance: ω0 all beam modes cover equal part of negative and positive Re(Z┴), for that reason the beam modes are stable However, if a pair and odd mode meet by detuning, they couple and have same frequency, one of them becomes instable: transverse-mode-coupled INSTABILITY (TMCI)

Head-tail instability can act if the chromaticity of the machine optics is non-zero: Head-tail phase : beam spectrum: ξ>0 ξ>0 ξ<0 overlap of the beam mode with negative real part : Instability! Head-tail instability

Overview on the different impedance driven instabilities Instabilities driven by the short range part of the wake (high ω of impedance) transverse-mode coupled instability (TMCI) transverse Head-Tail instability Microwave instability (longitudinal) fast transverse blow-up Instabilities driven by the long range part of the wake (low ω of impedance) longitudinal and transverse Robinson instability (driven by the fundamental mode of the cavity) longitudinal and transverse coupled-bunch instability (driven by higher order modes of the cavity) resistive wall instability (longitudinal and transverse) other effects: bunch lengthening, energy dispersion increase , diverse tune shifts, tune spread, power loss etc.

Long-range interaction of the beam with the impedance the wake of a strong resonator can subsist until the next bunch, even over one or several turns difference to the precedent slides: before interaction with 1 single bunch now interaction of several, but mostly pointlike bunches the consequence is a possible collective motion of the whole bunch train if there is no high-ω component, each bunch can be considered pointlike. it is like a chain of points which execute a collective motion. If this effect is considered in the heatload calculation, Ploss=nкI2bT0 has to be modified.

Long-range interaction of the beam with the impedance the wake of a high Q resonator can subsist until the next bunch, even over one or several turns The bunch chain can swing in M(=number of bunches in the bunch train) different modes with the frequencies (tacitly we asume no gap in the filling of the ring): 4 different modes with 4 bunches now the repetition frequency is Mω0 instead of ω0 modulated by the swing mode of the bunches within the bunch train longitudinal motion transverse motion

Transverse Resistive wall instability zero chromaticity ξ=0 m=0 m=1 all lines (all p) are to be taken into account. However, a line on the negative side nearly cancels out with a line on the positive side. Except the line closest to 0, it samples a huge impedance and therefore drives the instability for m=0. The overlap of mode m=1 at ω~0, however, is very small, the threshold of this mode is very high

Robinson instability (longitudinal) Interaction of the beam with the cavity zoom instable stable the frequency of the cavity (RF-system) has to be tuned in a way that the lines of the beam spectrum are on the right side of the fundamental RF-impedance resonance. it also depends on the sign of the slipping factor η, but at std. electron rings always η>0

longitudinal coupled bunch instability (HOM) Cavities have apart from the fundamental modes higher order modes (HOM) which are represented peaks of high value in the impedance spectrum. Their frequency zoom Different from the Robinson instability, the excited mode is not , consequently if is fulfilled, there is no other p=q which fulfills if . In that case the damping term can be neglected. Contrary to the Robinson instability which can be overcome by correct tuning, the HOM’s represent a serious problem to many accelerators. Only “by chance” the RF-system can be tuned so that no beam spectrum line falls on a HOM-peak.

Resumée Longitudinal instabilities are caused by the positive part of the real part of the impedance. Transverse instabilities are caused by the negative part of the real part of the transverse impedance. Of course the subject of this presentation is not at all completely covered, many details have not been mentioned. Several instabilities were not mentioned neither. Furthermore, many simplications have been applied. I hope I could give you a interesting insight on one of the most fascinating parts of accelerator physics. Acknowledgements: R.Nagaola, Ph.Kernel, G.Stupakov(RW-impedance)

Homework Calculate the instability current threshold for 6 cavities each having a higher order mode (HOM) at 676MHz of a shunt impedance of 10.8kW.Assume that the HOMs of the cavities add up coherently. Parameters: due to radiative damping Calculate the power loss generated by homogeneous bunch filling pattern passing the 2.5m long wiggler vacuum pipe made of copper σ=5.8x107(Wm)-1, m=1. The pipe is assumed to be circular. The radius of the pipe is 4mm. The bunch length in time units is Help: First compute the long. loss factor for Resistive wall impedance using only variables, no numerical values, plug the resulting formula in the heat power formula and evaluate it numerically. For the first part the following integral is needed: Don’t mix up s (conductivity) with st (bunch length) !