The concepts of wake fields and impedance are introduced to describe the electromagnetic interaction of a bunch of charged particles with its environment in an particle accelerator. The wake fields can act back on the beam and lead to instabilities, which may limit the achievable current per bunch, the total current, or even both. Some typical examples are used to illustrate the wake function and its basic properties. Then the frequency-domain view of the wake field or impedance is explained, and basic properties of the impedance are derived. The impedance of a cavity mode is illustrated using an equivalent circuit model. The relation of wake field effects to important beam parameters is treated in the rigid beam approximation. Several examples are employed to illustrate the impact of the geometry and the material properties of the vacuum chamber on the impedance. Finally, a basic introduction to beam instabilities based on a head tail model of the beam is given.
In this paper, we study single-bunch instabilities observed in the CERN Super Proton Synchrotron (SPS). According to the linearized Vlasov theory, radial or azimuthal mode-coupling instabilities result from a coupling of bunch-oscillation modes, which belong to either the same or adjacent azimuthal modes, respectively. We show that both instability mechanisms exist in the SPS by applying the Oide-Yokoya approach to compute van Kampen modes for the realistic longitudinal impedance model of the SPS. The results agree with macroparticle simulations and are consistent with beam measurements. In pa…
The electromagnetic wakefields excited by the bunches of charged particles in the particle accelerators may cause instability in the longitudinal and transverse motion of these bunches leading to the beam loss. These multibunch effects need to be analyzed carefully in an accelerator design to ensure the stability of the charged particle beams. This paper presents an analytical method describing the multibunch transverse instabilities for ultrarelativistic electron bunches whose energy changes linearly through the accelerating section of interest. Fast algorithmic implementations of this analyt…
Currently, Fermilab Booster accelerates ~4.5E12 protons per pulse (ppp) in 81 bunches from 400 MeV to 8 GeV at 15 Hz to provide beam to multiple HEP experiments and is being upgraded to handle higher beam intensity >6.7E12 ppp at a repetition rate of 20 Hz. In the current mode of operation, we control longitudinal coupled bunch instabilities (LCBI) using combination of passive and active dampers. The issues with LCBI are expected to worsen at higher beam intensities. Here, we would like to investigate at what time in the beam cycle a particular mode is going to originate and how much it contri…
Updates on Impedance Studies for the FCC-ee High Energy Booster
GANIL；CEA（法国原子能和替代能源委员会）；CERN（欧洲核子研究组织）；IPM（伊朗数学物理研究所）；University of Rome La Sapienza（罗马大学）；INFN - Roma1（意大利国家核物理研究所罗马第一分部）；INFN - LNF（意大利国家核物理研究所弗朗卡蒂实验室）；University of Caen Normandy（诺曼底卡昂大学）；Paris-Saclay University（巴黎萨克雷大学）
Recently a mechanism of storage ring operation based on the steady-state microbunching has been proposed and investigated, which contains a laser cavity modulator providing the longitudinal focusing for the circulating microbunches. In this paper we analyze the impact of the coherent undulator radiation on the longitudinal single-bunch multi-turn collective dynamics, exploring a new possible instability mechanism. We formulate the multi-turn equations of motion for the single microbunch as two sets of difference equations in the modulator and in the remaining storage ring. The dispersion equat…
Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameter-less integral equation. For any multipolarity, the discrete part of the spectrum is found to consist of infinite number of modes with real tunes, which limit point is the incoherent zero-amplitude frequency. In other words, notwithstanding the RF bucket nonlinearity and potential well distortion, the Landau damping is lost. Hence, even a tiny c…
Coherent transverse oscillations of a bunch in a ring accelerator are considered. Three different models of the bunch are used: a hollow bunch in a square potential well, a square bunch in a parabolic potential well, and a parabolic bunch in a parabolic well. The bunch space-charge field, wakefield, and chromaticity are taken into account in all the cases. The bunch eigenmodes and various instabilities are studied. The results are compared and analyzed to establish the range of applicability of the models. Role of the Landau damping due to dependence of the SC field on the longitudinal coordin…
Van Kampen模是Jeans-Vlasov方程的本征模。它们的光谱由连续部分和可能的离散部分组成。离散范坎彭模式的出现意味着出现了没有任何朗道阻尼的相干模式;因此，即使是微小的耦合束尾流也足以驱动不稳定性。在Tevatron，RHIC和SPS上观察到的纵向不稳定性可以解释为朗道阻尼（LLD）的损失，这在这里显示为发生在相当低的阻抗。对于排斥性唤醒和单谐波RF，LLD被发现是非常敏感的陡峭度的束分布函数在小幅度。在此基础上，提出了一种光束稳定的方法。
Shanghai Institute of Applied Physics（上海应用物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；High Energy Accelerator Research Organization (KEK)（高能加速器研究机构）；Zhangjiang Laboratory（张江实验室）；Shanghai Advanced Research Institute（上海高等研究院）
The FCC-ee impedance model is being constantly updated closely following the vacuum chamber design and parameters evolution. In particular, at present, a thicker NEG coating of 150 nm (instead of previous 100 nm) has been suggested by the vacuum experts, and a more realistic impedance model of the bellows has been investigated. Moreover, also the transverse impedance has been updated by considering the same sources as for the longitudinal case. Therefore, the FCC-ee impedance database is getting more complete and the impedance model is being refined. In this paper we describe the presently ava…
The first mention of the impedance concept appeared on November 1966 in the CERN internal report "Longitudinal instability of a coasting beam above transition, due to the action of lumped discontinuities" by V.G. Vaccaro. This was the beginning of many studies, which took place over the last five decades, and today, impedances and wake fields continue to be an important field of activity, as concerns theory, simulation, bench and beam-based measurements. Building a reliable impedance or wake field model of a machine is the first necessary step to be able to evaluate the machine performance lim…
