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This paper deals with the determination of the wake-function, impedance and energy loss due to surrounding accelerator components of two countermoving beams, i.e., two beams moving in opposite directions. Differently from the classical case, the distance between source and test charges cannot be considered constant. The paper defines a generalized wake-function, independent of the source and test particle’s direction of motion. From this generalized wake-function a wake-potential is derived. The paper focuses in particular on the longitudinal direction. The expression of the energy dissipated by two beams transiting in the same vacuum chamber is obtained. This expression is valid both in the comoving and countermoving beam scenario. Subsequently, the proposed model is benchmarked by calculating the longitudinal co and countermoving wake-functions, the corresponding impedances and wake-potentials for two simple geometries (a circular pipe and a pillbox cavity) traversed by two countermoving beams, and by comparing the results with the ones of the electromagnetic code CST Studio Suite. Finally, for the pillbox geometry, the energy loss of the two countermoving beams and their rf-heating are investigated.

The CERN Proton Synchrotron Booster (PSB) has been known to suffer from horizontal instabilities since its early operation. These instabilities appear at specific beam energies and range of working points. The source of the instability and the reason why the instabilities appear at specific energies remained unidentified. In routine operation, the instabilities have not been limiting the performance reach thanks to the horizontal feedback system. Recently, the interest in these instabilities has been sparked by the on-going LHC Injectors Upgrade (LIU) program, as well as, the Physics Beyond Colliders (PBC) study group. Their systematic characterization has been carried out through measurements. Macroparticle simulations and analytical modeling have been applied to explain the measurements and the dependence on the kinetic energy. Finally, the extraction kicker has been unambiguously identified as the source of the instability., CERN Yellow Reports: Conference Proceedings, Vol. 9 (2020): Proceedings of the ICFA mini-Workshop on Mitigation of Coherent Beam Instabilities in Particle Accelerators, Zermatt, Switzerland, 23–27 September 2019

At the beginning of the 2016 run, an anomalous beam instability was systematically observed at the CERN Large Hadron Collider (LHC). Its main characteristic was that it spontaneously appeared after beams had been stored for several hours in collision at 6.5 TeV to provide data for the experiments, despite large chromaticity values and high strength of the Landau-damping octupole magnet. The instability exhibited several features characteristic of those induced by the electron cloud (EC). Indeed, when LHC operates with 25 ns bunch spacing, an EC builds up in a large fraction of the beam chambers, as revealed by several independent indicators. Numerical simulations have been carried out in order to investigate the role of the EC in the observed instabilities. It has been found that the beam intensity decay is unfavorable for the beam stability when LHC operates in a strong EC regime.

The Future Circular Collider (FCC) study aims at designing different options of a post-LHC collider. The high luminosity electron-positron collider FCC-ee based on the crab waist concept is considered as an intermediate step on the way towards FCC-hh, a 100 TeV hadron collider using the same tunnel of about 100 km. Due to a high intensity of circulating beams the impact of collective effects on FCC-ee performance has to be carefully analyzed. In this paper we evaluate beam coupling impedance of the FCC-ee vacuum chamber, estimate thresholds and rise times of eventual single- and multibunch beam instabilities and discuss possible measures to mitigate them., Talk at the LaPlas-2018 Conference, NRNU MEPhI, 30 January-01 February 2018, Moscow, Russia. 6 pages, 3 figures, 2 tables

Due to its effective pumping ability, nonevaporable getter (NEG) coating is considered for the vacuum chambers of the Compact Linear Collider (CLIC) electron damping rings (EDR). The aim is to suppress fast beam ion instabilities. The electromagnetic (EM) characterization of the NEG properties up to ultra-high frequencies is required for the correct impedance modeling of the damping ring (DR) components. The properties are determined using rectangular waveguides which are coated with NEG. The method is based on a combination of complex transmission coefficient S-21 measurements with a vector network analyzer (VNA) and 3D simulations using CST Microwave Studio (R) (CST MWS). The frequency ranges discussed in this paper are 220-330 and 500-750 GHz.

In this report a method to measure the machine chromaticity is presented. This method consists in applying a longitudinal kick to the beam and computing the Fourier transform of the transverse turn-by-turn position. Analytical expressions for the Fourier transform are derived, which show how the chromaticity can be inferred from the synchrotron sidebands. Multiparticle simulations are also performed to validate the analytical expressions.

Within the Bethe diffraction theory, the impedance of a small circular hole has been calculated for particle beams of arbitrary β and finite size via two approaches. In the first approach we define the impedance in terms of the total work done by the fields excited in the beam pipe, where it finally reduces to a surface integral over the hole region. In the second approach, the hole has been treated as a radiating electric and magnetic dipole with effective electric and magnetic moments resulting from fictitiously introduced surface charge and current densities. The above two approaches lead to exactly the same result for the hole impedance which is consistent with the predictions made by the Bethe theory for wavelengths that are much larger than the hole size.

In the ESR heavy-ion cooler storage ring at GSI the exponential growth and the subsequent saturation phase of the longitudinal instability in space charge dominated ion beams can be monitored with high resolution. Kinetic simulations together with the experimental data lead to a new insight into the effects of space charge and electron cooling on the long-time evolution of the instability. In the simulations we observe the continuous excitation of long-lived collective modes generated by particle trapping in the self-excited potential, which suggest that previous ``overshoot'' concepts need revision.

During the beam commissioning of the Large Hadron Collider (LHC) [LHC Design Report No. CERN-2004-003-V-1, 2004 [http://cds.cern.ch/record/782076?ln=en]; O. Brüning, H. Burkhardt, and S. Myers, Prog. Part. Nucl. Phys. 67, 705 (2012)PPNPDB0146-641010.1016/j.ppnp.2012.03.001] with 150, 75, 50, and 25-ns bunch spacing, important electron-cloud effects, like pressure rise, cryogenic heat load, beam instabilities, or emittance growth, were observed. Methods have been developed to infer different key beam-pipe surface parameters by benchmarking simulations and pressure rise as well as heat-load observations. These methods allow us to monitor the scrubbing process, i.e., the reduction of the secondary emission yield as a function of time, in order to decide on the most appropriate strategies for machine operation. To better understand the influence of electron clouds on the beam dynamics, simulations have been carried out to examine both the coherent and the incoherent effects on the beam. In this paper we present the methodology and first results for the scrubbing monitoring process at the LHC. We also review simulated instability thresholds and tune footprints for beams of different emittance, interacting with an electron cloud in field-free or dipole regions.

The Large Hadron Collider (LHC) is one of the largest scientific instruments ever built. Since opening up a new energy frontier for exploration in 2010, it has gathered a global user community working in fundamental particle physics and the physics of hadronic matter at extreme temperature and density. To sustain and extend its discovery potential, the LHC will undergo a major upgrade in the 2020s. This will increase its rate of collisions by a factor of five beyond the original design value and the integrated luminosity by a factor ten. The new configuration, known as High Luminosity LHC (HL-LHC), will rely on a number of key innovations that push accelerator technology beyond its present limits. Among these are cutting-edge 11–12 T superconducting magnets, including Nb(3)Sn-based magnets never used in accelerators before, compact superconducting cavities for longitudinal beam rotation, new technology and physical processes for beam collimation. The dynamics of the HL-LHC beams will be also particularly challenging and this aspect is the main focus of this paper.

The aim of our study is the numerical computation of the wakefield and energy loss per unit length for relativistic, short ($l10\text{ }\text{ }\mathrm{ns}$) proton bunches interacting with an electron cloud inside the beam pipe. We present analytical expressions for the energy loss in the impulse kick approximation. For the simulation of the wakefields a 2D self-consistent, electrostatic particle-in-cell (PIC) code is employed. Results for the energy loss and for the wakefields are presented for the parameter scope of the CERN LHC and SPS. For selected parameters the results are compared to a three-dimensional (3D) electromagnetic PIC code.

To achieve high luminosity at the collision point of the Compact Linear Collider (CLIC) the normalized horizontal and vertical emittances of the electron and positron beams must be reduced to 500 nm and 4 nm before the beams enter the 1.5TeV linear accelerators. An effective way to accomplish ultra-low emittances with only small effects on the electron polarization is using damping rings operating at 2.86 GeV equipped with superconducting wiggler magnets. This paper describes a technical design concept for the CLIC damping wigglers.

Electron cloud buildup is a major limitation for high-energy particle accelerators such as the CERN Super Proton Synchrotron (SPS). Amorphous carbon thin films with low initial secondary electron yield (SEY ffi 1:0) have been applied as a mitigation material in the SPS vacuum chambers. This paper summarizes the experimental setups for electron cloud monitoring, coating procedures, and recent measurements performed with amorphous carbon coated vacuum chambers in the SPS. The electron cloud measured by dedicated monitors is completely suppressed for LHC-type beams. Even after more than one year’s exposure in the SPS with the machine in operation, the coating does not show any increase in the secondary electron yield. The study of coated vacuum chambers for the SPS dipole magnets is in progress; the correlation between electron cloud reduction and pressure rises is not yet fully understood. Some prototypes have already been installed in the accelerator and plans for the implementation of an optimized coating technique are under development.

The electron cloud (EC) can be formed in the beam pipe of a circular accelerator if the secondary emission yield (SEY) of the inner surface is larger than 1, and it can detrimentally affect the circulating beam. Understanding the underlying physics and defining the scaling laws of this effect is indispensable to steer the upgrade plans of the existing machines and the design of new ones. The single bunch EC instability (ECI) is shown to be strongly affected by the transverse beam size. Transversely, smaller beams going through an electron cloud generate higher electron peak densities and lower the intensity threshold to make the beam unstable. In particular, since higher energy beams have smaller transverse sizes (for equal normalized transverse emittances), the scaling of the ECI threshold with the beam energy turns out to be surprisingly unfavorable.

Incoherent electron effects could seriously limit the beam lifetime in proton or ion storage rings, such as LHC, SPS, or RHIC, or blow up the vertical emittance of positron beams, e.g., at the B factories or in linear-collider damping rings. Different approaches to modeling these effects each have their own merits and drawbacks. We describe several simulation codes which simplify the descriptions of the beam-electron interaction and of the accelerator structure in various different ways, and present results for a toy model of the SPS. In addition, we present evidence that for positron beams the interplay of incoherent electron-cloud effects and synchrotron radiation can lead to a significant increase in vertical equilibrium emittance. The magnitude of a few incoherent e+e- scattering processes is also estimated. Options for future code development are reviewed.

Following the first observation of a fast vertical instability for a single high-brightness bunch at injection in the SPS in 2002, a series of detailed measurements and simulations has been performed in order to assess the resulting potential intensity limitations for the SPS, as well as possible cures. During the 2006 run, the characteristics of this instability were studied further, extending the intensity range of the measurements, and comparing the experimental data with simulations that take into account the latest measurements of the transverse machine impedance. In this paper, we summarize the outcome of these studies and our understanding of the mechanisms leading to this instability. The corresponding intensity limitations were also determined.

The nominal LHC beam has been produced successfully in the CERN proton synchrotron since 2003. However, after having restarted the CERN PS in spring 2006, the LHC beam was set-up and observed to be unstable on the 26 GeV/c extraction flat top. An intensive measurement campaign was made to understand the instability and to trace its source. This paper presents the observations, possible explanations and the necessary measures to be taken in order to avoid this instability in the future.

For many years, a horizontal head-tail instability has been observed at the CERN Proton Synchrotron during the long 1.2 s injection flat-bottom. This slow instability has been damped using linear coupling only, i.e. with neither octupoles nor feedbacks. Using the nominal machine and beam parameters for LHC, the sixth head-tail mode number is usually observed. Several other modes were also observed in the past by tuning the chromaticity, and these observations were found to be in good agreement with Sacherer's formula. The purpose of this paper is to present the results of assessing the effect of chromaticity and linear coupling on this slow head-tail instability using the HEADTAIL simulation code, and to compare these simulations with both measurements performed over the last few years, and theoretical calculations.

The generation of a quasistationary electron cloud inside the beam pipe through beam-induced multipacting processes has become an area of intensive study. The analyses performed so far have been based on heavy computer simulations taking into account photoelectron production, secondary emission, electron dynamics, and space charge effects, providing a detailed description of the electron-cloud evolution. Iriso and Peggs [U. Iriso and S. Peggs, Phys. Rev. ST Accel. Beams 8, 024403 (2005)PRABFM1098-440210.1103/PhysRevSTAB.8.024403] have shown that, for the typical parameters of RHIC, the bunch-to-bunch evolution of the average electron-cloud density at a point can be represented by a cubic map. Simulations based on this map formalism are orders of magnitude faster compared to those based on standard particle tracking codes. In this communication we show that the map formalism is also applicable to the case of the Large Hadron Collider (LHC), and that, in particular, it reproduces the average electron-cloud densities computed using a reference code to within ∼15% for general LHC bunch filling patterns. We also illustrate the dependence of the polynomial map coefficients on the physical parameters affecting the electron cloud (secondary emission yield, bunch charge, bunch spacing, etc.).

After the first experimental observations compatible with the presence of the e-cloud effect in the DAΦNE positron ring, a more systematic study has been performed regarding the e-cloud build-up. The measured field map of the magnetic field has been taken into account in the simulation for elements present in the four 10 m long bending sections, representing 40% of the whole positron ring. The obtained simulation results are presented together with the recent experimental observations performed on the vacuum behaviour of the positron ring.

The electron cloud may cause transverse single-bunch instabilities of proton beams such as those in the Large Hadron Collider (LHC) and the CERN Super Proton Synchrotron (SPS). We simulate these instabilities and the consequent emittance growth with the code HEADTAIL, which models the turn-by-turn interaction between the cloud and the beam. Recently some new features were added to the code, in particular, electric conducting boundary conditions at the chamber wall, transverse feedback, and variable beta functions. The sensitivity to several numerical parameters has been studied by varying the number of interaction points between the bunch and the cloud, the phase advance between them, and the number of macroparticles used to represent the protons and the electrons. We present simulation results for both LHC at injection and SPS with LHC-type beam, for different electron-cloud density levels, chromaticities, and bunch intensities. Two regimes with qualitatively different emittance growth are observed: above the threshold of the transverse mode-coupling (TMC) type of instability there is a rapid blowup of the beam, while below this threshold a slow, long-term, emittance growth remains. The rise time of the TMC instability caused by the electron cloud is compared with results obtained using an equivalent broadband resonator impedance model, demonstrating reasonable agreement.

Several simulation codes have been adapted so as to model the single-bunch electron-cloud instability including a realistic variation of the optical functions with longitudinal position. In addition, the electron cloud is typically not uniformly distributed around the ring, as frequently assumed, but it is mainly concentrated in certain regions with specific features, e.g., regions which give rise to strong multipacting or suffer from large synchrotron radiation flux. Particularly, electrons in a dipole magnet are forced to follow the vertical field lines and, depending on the bunch intensity, they may populate two vertical stripes, symmetrically located on either side of the beam. In this paper, we present simulation results for the CERN SPS and LHC, which can be compared with measurements or analytical predictions.

A vertical single‐bunch instability has been observed in 2003 right after injection at 26 GeV/c in the CERN Super Proton Synchrotron (SPS). High‐intensity proton bunches (∼1.2 1011 p/b) with low longitudinal emittance (∼0.2 eVs) are affected by heavy losses after less than one synchrotron period. Such phenomenon has already been observed with leptons in many machines, e.g. in the SPS, or with protons at transition, e.g. in the CERN Proton Synchrotron (PS). However, to the authors’ knowledge, it is the first time with protons far from transition. The absence of transverse mode‐coupling instability in hadron machines is generally explained by three mechanisms: (i) the intensity threshold for the longitudinal microwave instability is generally lower than for the transverse mode‐coupling instability, (ii) the intensity threshold due to mode‐coupling between the two lowest azimuthal modes increases with space charge, and (iii) the intensity threshold increases with bunch length (in the long‐bunch regime). In this talk measurements performed in the SPS are compared to analytical and simulation predictions.

Recently, resonance driving terms were successfully measured in the CERN SPS and the BNL RHIC from the Fourier spectrum of beam position monitor (BPM) data. Based on these measurements a new analysis has been derived to extract truly local observables from BPM data. These local observables are called local resonance terms since they share some similarities with the global resonance terms. In this paper we derive these local terms analytically and present experimental measurements of sextupolar global and local resonance terms in RHIC. Nondestructive measurements of these terms using ac dipoles are also presented.

Coherent tune shifts with current have been measured in the SPS with single bunches and two bunches at varying distances between them. The measurements showed the usual negative slope with intensity in the vertical plane and a small positive one in the horizontal direction. The results are compared with theoretical predictions which include estimates of tune shifts due to the impedance of resistive vacuum chambers with non-circular geometry and finite wall thickness.

This paper studies dipolar and quadrupolar decoherence of a bunch in presence of space charge. The centroid of a bunch displaced transversely or longitudinally decoheres due to nonlinearities that cause phase space filamentation and mixing. Here we show that space charge can inhibit decoherence and keep the beam centroid oscillations undamped over long times. This feature complicates the detection of echoes for diagnostics purposes. An echo requires in fact a fully decohered beam to show up as a coherent signal at later times than the excitations. Results are qualitatively compared with experiments in the GSI synchrotron SIS.

Recently a method to measure chromaticity has been proposed by applying a transverse and a longitudinal kick to the beam. Assuming a Gaussian bunch in the 6-dimensional phase space, analytical expressions were derived, which relate the synchrotron sidebands to the machine chromaticity. To assess limitations and extensions of this technique, a more realistic accelerator model is considered including dispersion, transverse nonlinearities and second order chromaticity, which modify the Fourier spectrum. Tracking and analytical studies are performed to describe these effects.

We simulate the long-term emittance growth of a proton beam due to an electron cloud of moderate density. This emittance growth is sometimes characterized by a rapid blow up of the bunch tail, and it appears to be different from the strong head-tail instability, which is observed at higher electron densities. We study whether this instability can occur in the absence of transverse dipole motion along the bunch, and its sensitivity to various simulation parameters, such as the number of beam-electron interaction points (IPs) and the phase advances between them. Using a frozen-potential model, we compute tune footprints, which reveal the resonances contributing to the incoherent part of the emittance growth.

The properties of the electrons participating in the build up of an electron cloud (EC) inside the beam-pipe have become an increasingly important issue for present and future accelerators whose performance may be limited by this effect. The EC formation and evolution are determined by the wall-surface properties of the accelerator vacuum chamber. Thus, the accurate modeling of these surface properties is an indispensible input to simulation codes aimed at the correct prediction of build-up thresholds, electron-induced instability or EC heat load. In this letter, we present the results of surface measurements performed on a prototype of the beam screen adopted for the Large Hadron Collider (LHC), which presently is under construction at CERN. We have measured the total secondary electron yield (SEY) as well as the related energy distribution curves (EDC) of the secondary electrons as a function of incident electron energy. Attention has been paid, for the first time in this context, to the probability at which low-energy electrons (< 20 eV) impacting on the wall create secondaries or are elastically reflected. It is shown that the ratio of reflected to true-secondary electrons increases for decreasing energy and that the SEY approaches unity in the limit of zero primary electron energy. PACS numbers: 79.20.Hx ; 29.27.Bd; 41.75.Lx

The interaction between a low-density electron cloud in a circular particle accelerator with a circulating charged particle beam is considered. The particle beam's space charge attracts the cloud, enhancing the cloud density near the beam axis. It is shown that this enhanced charge and the image charges associated with the cloud charge and the conducting wall of the accelerator may have important consequences for the dynamics of the beam propagation. The tune shift due to the electron cloud is obtained analytically and compared to a new numerical method (QUICKPIC) that is described here. Sample numerical results are presented and their significance for current and planned experiments is discussed. (23 refs).

Space charge effects generally play a rather minor role in high energy machines like the SPS. Rather high space charge tune shifts may however become unavoidable in the SPS for the heavy ion beams required by the LHC. We describe recent measurements performed with intense proton beams in the SPS. The space charge effects were enhanced by a reduction of the injection energy from 26 to 14 GeV.

The Gamow-Teller(GT) states are investigated in relativistic models. The Landau-Migdal(LM) parameter is introduced in the Lagrangian as a contact term with the pseudo-vector coupling. In the relativistic model the total GT strength in the nucleon space is quenched by about 12% in nuclear matter and by about 6% in finite nuclei, compared with the one of the Ikeda-Fujii-Fujita sum rule. The quenched amount is taken by nucleon-antinucleon excitations in the time-like region. Because of the quenching, the relativistic model requires a larger value of the LM parameter than non-relativistic models in describing the excitation energy of the GT state. The Pauli blocking terms are not important for the description of the GT states., Comment: REVTeX4, no figures

Experimental observations on the electron cloud have been collected at the CERN Proton Synchrotron machine throughout the last two years. At the same time, an intense campaign of simulations has been carried out to understand the observed electron cloud buildup and the related instability. In this paper, the results of the numerical simulations are presented and discussed. (19 refs).

During the 1999 SPS run, strong transverse instabilities were observed with the LHC beam [1]. Both the instability characteristics and the identical threshold current as for beam-induced electron multipacting led to consider the interaction of the beam with the electron cloud as a likely source. In 2000, we have measured the dependence of beam motion, beam loss, and emittance growth on bunch intensity, number of bunches, octupole strength, chromaticity, and gaps in the bunch train. We report on these recent studies and compare the beam observations with simulations of electron cloud build up and electron-induced single-bunch instabilities.

Photoemission, or gas ionization, and secondary emission can give rise to a quasi-stationary electron cloud inside the beam pipe through a beam-induced multipacting process. We investigate single bunch instabilities driven by a quasi-stationary electron cloud by means of a computer simulation. The model that we apply makes use of two sets of macroparticles for both the bunch particles and for the electrons, which interact at one or more locations along the beam orbit. Two different schemes have been implemented for the electron cloud field calculation (PIC and soft-Gaussian), and their efficiencies are compared. The code is used to simulate possible instability mechanisms in the SPS. The options of a broad-band wake-field and space charge includuced tune spread have ben also introduced in order to follow the bunch evolution under the combined effect of the elctron-cloud and a broad-band impedance.

The beam-induced electron cloud buildup is one of the major concerns for the SPS and the design of the future LHC. During the 2000 run, this effect has also been observed in the PS with the nominal LHC- type beam. The electron cloud induces a baseline distortion in electrostatic pickup signals, both during the last turns in the PS, when the full bunch length is reduced to less than 4 ns, and in the transfer line between the PS and the SPS rings. In the year 2001, modifications in the rf hardware allowed us to study the properties of the beam instability related with the electron cloud phenomenon for a total bunch length of about 10 ns. The complete set of experimental observations carried out in the PS machine is presented and discussed in detail. (20 refs).

HEADTAIL is a simulation program developed at CERN which is aimed at studying the single-bunch instability arising from the interaction on successive turns of a single bunch with the cloud generated by the previous bunches. The code includes chromaticity, space charge tune spread, broad-band impedance, and detuning with amplitude for more realistic simulation. Examples of application are shown. Transverse and longitudinal wake functions are also outputs of the HEADTAIL code. (20 refs).

Electron-cloud build up, incoherent tune spread and electron-induced beam instability are likely to be responsible for the vertical beam-size increase observed at the KEKB Low Energy Ring (LER). We report on recent simulations and analytical estimates, addressing the electron-cloud evolution for various magnet configurations, comparing the beam size blow up predicted by three different simulation models (micro-bunches, multi-particle tracking with soft Gaussian approximation, particle-in-cell), and finally discussing analytical expressions for the instability threshold.

Photoemission and secondary emission are known to give rise to a quasi-stationary electron cloud inside the LHC beam pipe through a beam-induced multipacting process. We investigate the phenomena of electron-cloud build up and related effects via computer simulation. In our model, macroparticles representing photoelectrons are emitted synchronously with the passing proton bunch, and are subsequently accelerated in the field of the beam. As they hit the beam pipe, new macroelectrons are generated, whose charges are determined by the energy of the incoming particles and by the secondary emission yield of the beam pipe. A quasi-stationary state of the electron cloud is eventually reached due to space charge. The equilibrium distribution of the electron cloud is used as an input parameter for a program that analyses the electron-cloud driven single-bunch instability. The electron cloud simulation also allows the evaluation of the heat load on the cold beam screen, which must stay within the available cooling capacity, and the electron charge deposited on or emitted from the electrodes of the beam-position monitors.

The longitudinal space charge and resistive wall impedances have been investigated in a smooth cylindrical beam pipe. At any point from the beam axis, we obtained an expression for the total impedance, which at the beam surface $r=a$ for infinite pipe wall conductivity gives the expression for the total impedance that was derived by Zotter and Kheifets in studying the impedance of uniform beams in concentric cylindrical wall chambers, when a single cylindrical chamber is considered [B. W. Zotter and S. A. Kheifets, Impedances and Wakes in High-Energy Particle Accelerators (World Scientific, Singapore, 1998), Chap. 6]. A fitting formula for the space-charge impedance at the beam surface $(r=a),$ which is valid for arbitrary wavelengths, is given. Rather than calculating the impedance with the field on the axis [Joseph J. Bisognano, Fifth European Particle Accelerator Conference (EPAC96), edited by S. Myers, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, and J. Poole (Institute of Physics, Bristol, 1996), Vol. 1, p. 328], the fitting formula is obtained by averaging over the transverse beam distribution. We also give another approach for the calculation of the resistive wall impedance using the flux of the Poynting vector at the pipe wall and then compare it with the expression obtained from the volume integral over the beam distribution.

It is shown that energy conservation in the longitudinal envelope equation can be used to derive analytic expressions to model fast bunch compression and the effect of space charge in terms of a dimensionless Coulomb parameter $\ensuremath{\Sigma}$ $(\ensuremath{\propto}1/\ensuremath{\eta}$, with $\ensuremath{\eta}$ the slip factor). For small $|\ensuremath{\eta}|$ (below transition, hence $\ensuremath{\Sigma}\ensuremath{\gg}1$), the rf voltage required is nearly independent of $\ensuremath{\eta}$ and dominated by space charge repulsion. The extra voltage generates the coherent momentum spread $\ensuremath{\delta}\ensuremath{\propto}1/\sqrt{|\ensuremath{\eta}|}$ required to compensate the increasing space charge force gradient during compression. This sets a clear limit to the useful approach to transition. An $\ensuremath{\eta}$ jump scheme is discussed to minimize this effect. Particle-in-cell computer simulation confirms the validity of our results also for more realistic beam distributions. Noticeable tails in momentum space due to the nonlinear space charge force are found for Gaussian line density bunches and $\ensuremath{\Sigma}\ensuremath{\gg}1$.

International audience; Beam instabilities cover a wide range of effects in particle accelerators and they have been the subjects of intense research for several decades. As the machines performance was pushed new mechanisms were revealed and nowadays the challenge consists in studying the interplays between all these intricate phenomena, as it is very often not possible to treat the different effects separately. The aim of this paper is to review the main mechanisms, discussing in particular the recent developments of beam instability theories and simulations.

The detailed study of the low energy secondary electron yield (LE-SEY) of technical Cu for low electron energies (from 0 to 20 eV) is very important for electron cloud build up in high intensity accelerators and in many other fields of research. Different devices base their functionalities on the number of electrons produced by a surface when hit by other electrons, namely its SEY, and, in most cases, on its very low energy behavior. However, LE-SEY has been rarely addressed due to the intrinsic experimental complexity to control very low energy electrons. Furthermore, several results published in the past have been recently questioned, allegedly suffering from experimental systematics. Here, we critically review the experimental method used to study LE-SEY and precisely define the energy region in which the experimental data can be considered valid. By analyzing the significantly different behavior of LE-SEY in clean polycrystalline Cu (going toward zero at zero impinging energies) and in its as received technical counterpart (maintaining a significant value in the entire region), we solve most, if not all, of the apparent controversy present in the literature, producing important inputs for better understanding the device performances related to their LE-SEY. Simulations are then performed to address the impact of such results on electron cloud predictions in the LHC.

We discuss the predicted electron cloud build up in the arcs and the long straight sections of the LHC, and its possible consequences on heat load, beam stability, long-term emittance preservation, and vacuum. Our predictions are based on computer simulations and analytical estimates, parts of which have been benchmarked against experimental observations at the SPS.