Non-linear Dynamics in Accelerators

Non-linear effects in accelerator physics are important both during the design stage and for successful operation of accelerators. Since both of these aspects are closely related, they will be treated together in this overview. Some of the most important aspects are well described by methods established in other areas of physics and mathematics. Given the scope of this handbook, the treatment will be focused on the problems in accelerators used for particle physics experiments. Although the main emphasis will be on accelerator physics issues, some of the aspects of more general interest will be discussed. In particular to demonstrate that in recent years a framework has been built to handle the complex problems in a consistent form, technically superior and conceptually simpler than the traditional techniques. The need to understand the stability of particle beams has substantially contributed to the development of new techniques and is an important source of examples which can be verified experimentally. Unfortunately the documentation of these developments is often poor or even unpublished, in many cases only available as lectures or conference proceedings.

Accelerators, Colliders and Their Application

Accelerators are modern, high precision tools with applications in a broad spectrum that ranges from material treatment, isotope production for nuclear physics and medicine, probe analysis in industry and research, to the production of high energy particle beams in physics and astronomy. At present about 35,000 accelerators exist world-wide, the majority of them being used for industrial and medical applications. Originally however the design of accelerators arose from the request in basic physics research, namely to study the basic constituents of matter.

Design and Principles of Synchrotrons and Circular Colliders

Lattice design in the context we will describe it here is the design and optimization of the principle elements—the lattice cells—of a circular accelerator, and it includes the dedicated variation of the accelerator elements (as for example position and strength of the magnets in the machine) to obtain well defined and predictable parameters of the stored particle beam. It is therefore closely related to the theory of linear beam optics that has been described in Chap. 10.1007/978-3-030-34245-6_2 [1].

Interactions of Beams with Surroundings

With the exceptions of Synchrotron Radiation sources, beams of accelerated particles are generally designed to interact either with one another (in the case of colliders) or with a specific target (for the operation of Fixed Target experiments, the production of secondary beams and for medical applications). However, in addition to the desired interactions there are unwanted interactions of the high energy particles which can produce undesirable side effects. These interactions can arise from the unavoidable presence of residual gas in the accelerator vacuum chamber, or from the impact of particles lost from the beam on aperture limits around the accelerator, as well as the final beam dump. The wanted collisions of the beams in a collider to produce potentially interesting High Energy Physics events also reduces the density of the circulating beam and can produce high fluxes of secondary particles.

Outlook for the Future

The charge separation between electrons and ions that exists within an electron plasma density wave can create large electric fields. In 1979 Tajima and Dawson first recognized that the longitudinal component of the field of a so-called “relativistic” wave (one propagating with a phase velocity close to c), could be used to accelerate charged particles to high energies in a short distance [1]. The accelerating gradient of such a plasma wave, Eo, can be approximated—assuming a total separation of electrons and ions in such a wave with wavelength λp = 2πc/ωp—as

Cosmic Particle Accelerators

In the century since the measurements of Victor Hess [1]—considered as the discovery of cosmic rays—the properties of cosmic rays, as they arrive on Earth, have been studied in remarkable detail; we know their energy spectrum, extending to 1020 eV, their elemental composition, their angular distribution, and we understand the basic energetic requirements of cosmic ray production in the Galaxy.

Accelerator Operations

The cost of building a particle accelerator is a major capital investment. Commissioning should be swift and the subsequent exploitation of a facility must provide an effective return. This return may be difficult to quantify unambiguously but generally acceptable measures of performance can be established. These measures might include: machine availability; integrated luminosity; protons on target; beam hours to users and so on.

Impedance and Collective Effects

As the beam intensity increases, the beam can no longer be considered as a collection of non-interacting single particles: in addition to the “single-particle phenomena”, “collective effects” become significant. At low intensity a beam of charged particles moves around an accelerator under the Lorentz force produced by the “external” electromagnetic fields (from the guiding and focusing magnets, RF cavities, etc.). However, the charged particles also interact with themselves (leading to space charge effects) and with their environment, inducing charges and currents in the surrounding structures, which create electromagnetic fields called wake fields. In the ultra-relativistic limit, causality dictates that there can be no electromagnetic field in front of the beam, which explains the term “wake”. It is often useful to examine the frequency content of the wake field (a time domain quantity) by performing a Fourier transformation on it. This leads to the concept of impedance (a frequency domain quantity), which is a complex function of frequency. The charged particles can also interact with other charged particles present in the accelerator (leading to two-stream effects, and in particular to electron cloud effects in positron/hadron machines) and with the counter-rotating beam in a collider (leading to beam–beam effects). As the beam intensity increases, all these “perturbations” should be properly quantified and the motion of the charged particles will eventually still be governed by the Lorentz force but using the total electromagnetic fields, which are the sum of the external and perturbation fields. Note that in some cases a perturbative treatment is not sufficient and the problem has to be solved self consistently. These perturbations can lead to both incoherent (i.e. of a single particle) and coherent (i.e. of the centre of mass) effects, in the longitudinal and in one or both transverse directions, leading to beam quality degradation or even partial or total beam losses. Fortunately, stabilising mechanisms exist, such as Landau damping, electronic feedback systems and linear coupling between the transverse planes (as in the case of a transverse coherent instability, one plane is usually more critical than the other).

Application of Accelerators and Storage Rings

It is well known from Maxwell theory that electromagnetic radiation is emitted whenever electric charges are accelerated in free space. This radiation assumes quite extraordinary properties whenever the charged particles move at ultrarelativistic speed: The radiation becomes very powerful and tightly collimated in space, and it may easily cover a rather wide spectrum ranging from the THz into the hard X-ray regime. When generation of such radiation is intended rather than being a side effect, the charged particles are normally electrons, thus kinetic energies are then typically in the multi-MeV range.

Accelerator Engineering and Technology: Accelerator Technology

Magnets are at the core of both circular and linear accelerators. The main function of a magnet is to guide the charged particle beam by virtue of the Lorentz force, given by the following expression:where q is the electrical charge of the particle, v its velocity, and B the magnetic field induction. The trajectory of a particle in the field depends hence on the particle velocity and on the space distribution of the field. The simplest case is that of a uniform magnetic field with a single component and velocity v normal to it, in which case the particle trajectory is a circle. A uniform field has thus a pure bending effect on a charged particle, and the magnet that generates it is generally referred to as a dipole.