Introduction to Collective Effects in Particle Accelerators

The beam intensity and the beam brightness of particle accelerators or colliders operated for high - energy physics were, and are, often severely limited by “collective effects” (e.g.[1]). By contrast, new light sources, such as linac - based free electron lasers, may even rely on collective instabilities to accomplish their mission! The term “collective effects” refers to the interaction of beam particles with each other through a variety of processes, e.g. (1) non-delayed self-fields and image fields present even for constant perfectly conducting and magnetic boundaries (direct and indirect “space - charge effects”), (2) longer - lived electro-magnetic “wake fields” due to a finite chamber resistivity or geometric variation in the beam - pipe cross section, which typically affect later parts of the beam, (3) coherent synchrotron radiation, which on a curved trajectory may even influence earlier parts of the beam, giving rise to “non-causal” wake fields, otherwise not normally encountered for ultra - relativistic beams, (4) beam - beam collisions, (5) particle - particle scattering inside the beam (single scattering called “Touschek effect” and multiple scattering known as “intrabeam scattering’), (6) gas ionization (“trapped - ion” or “fast - ion” instability), and (7) ionization electrons, photoelectrons and secondary electrons (“electron cloud effects”). Arguably also the appearance of (8) micron - size “dust” particles near the beam (“UFO effect”) could be considered a collective effect, as it is not observed, or does rarely happen, at low beam current. Half a century ago, collective effects were often overlooked or could not be well computed. The design of the storage ring collider SPEAR, for example, seems to have considered beam currents of up to 40 A [2], but it only achieved 30 mA [3]. By contrast, the Intersection Storage Rings (ISR) at CERN were constructed with a careful assessment and minimization of the “impedance” for all their components, and, as a result, the ISR reached maximum (coasting) beam currents around 50 A. Indeed the first solid theories of wake - field induced beam instabilities, by Neil and Sessler [4], and even the term “impedance,” introduced in the accelerator field by Vaccaro [5], date from about this era. Nowadays, the impedance or wake fields of most accelerator components can be calculated fairly reliably, using modern simulation codes run on powerful computers. Probably the first such code was developed by Weiland [6]. The impedance of special elements or for particular situations (e.g. two - beam impedance) still require care, however. Our understanding is rapidly evolving for other types of collective effects such as those driven by electron cloud or ions. The formation of beam tails and the required beam collimation also are important subject of active research. And so is the interplay between the optical lattice and collective effects. Micro - bunching instability, free - electron lasing or other types of coherent photon - beam interactions, e.g. beam interactions with FEL “seed laser” beams passing through undulators, as well as the harmful hose instability and the desired self - modulation instability in plasma acceleration (relevant for the AWAKE experiment at CERN), are further tantalizing manifestations of collective effects in modern and future particle accelerators.