Correction to: Measurement and Control of Charged Particle Beams

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Transverse Optics Measurement and Correction

In order to preserve the beam quality, accurate knowledge of the transverse optics and its correction is most often mandatory. For example, if the distribution of a beam injected into a storage ring is not matched to the ring optics, the emittance will grow due to filamentation. Or, if there is a significant optics error, e.g., induced by a strength error in a quadrupole magnet, the beam envelope may vary strongly. The resulting reduction in dynamic aperture may then lead to enhanced beam loss.

Longitudinal Phase Space Manipulation

In this chapter we describe various techniques used to control the longitudinal properties of particle beams We concentrate on the manipulation of the second moments of the longitudinal distribution; that is, on the bunch length and energy spread. As will be shown, the bunch length can be varied using accelerating cavities to compress, coalesce, split, and lengthen stored bunches. The energy spread of the beam can also be adjusted (usually to be a minimum) by proper phasing of the rf, by invoking cancellations between the applied and beam-induced rf, and by more sophisticated techniques for the case of long bunch trains. A practical application of the use of rf systems to affect the beam’s transverse emittance is presented lastly.

Transverse Beam Emittance Measurement and Control

The beam emittance ∈xyz represents the volume of the beam occupied in the six dimensional phase space (x, x′, y, y′, φ, δ), where x and y are the transverse positions, x′ and y′ are the transverse angles, φ is the time-like variable representing the relative phase of the beam, and δ is the relative beam momentum error. Using the notation of the beam matrix Σbeam introduced in Chap. 1, the 6-dimensional emittance is $${\varepsilon _{xyz}} = \det \Sigma _{beam}^{xyz}.$$ Considering now only the horizontal plane, the corresponding 2-dimensional horizontal emittance is obtained from $${\varepsilon _x} = \sqrt {\left\langle {{x^2}} \right\rangle \left\langle {{{x'}^2}} \right\rangle- {{\left\langle {xx'} \right\rangle }^2}} ,$$ where the first moments have been subtracted, and the average (〈…〉) is taken over the distribution function of the beam; recall also (1.27–1.29). An analoguous expression holds for the vertical plane. For a coupled system, the general form of (4.1) must be taken.

Beam Manipulations in Photoinjectors

The design of an electron source is a challenging task. The designer must reconcile the contradictory requirements for a small emittances, a high charge, a high repetition rate, and, possibly, a high degree of beam polarization.

Longitudinal Optics Measurement and Correction

Longitudinal focusing for a bunched beam is provided by both the change in path length with particle energy and by the time-dependent accelerating voltage. Usually one employs a smooth approximation, i.e., one ignores the discrete locations of the rf cavities, in describing the particle motion. The longitudinal motion can then be modelled by second order differential equations. For small oscillation amplitudes these equations simplify to those of harmonic oscillators.

Introduction

Particle accelerators were originally developed for research in nuclear and high-energy physics for probing the structure of matter. Over the years advances in technology have allowed higher and higher particle energies to be attained thus providing an ever more microscopic probe for understanding elementary particles and their interactions. To achieve maximum benefit from such accelerators, measuring and controlling the parameters of the accelerated particles is essential. This is the subject of this book.

Orbit Measurement and Correction

In practice, there are many uncertainties whose presence must be appreciated when correcting the beam orbit in both linear and circular accelerators. Such uncertainties include the variations in the electronic and/or mechanical centers of the beam position monitors (BPMs), in the magnetic center of the quadrupoles (inside which the position monitors are often mounted), or in the electromagnetic center of accelerating structures. Consider the case illustrated in Fig. 3.1.

Collimation

Particles at large betatron amplitudes or with a large momentum error constitute what is generally referred to as a beam halo. Such particles are undesirable since they produce a background in the particle-physics detector. The background arises either when the halo particles are lost at aperture restrictions in the vicinity of the detector, producing electro-magentic shower or muons, or when they emit synchrotron radiation that is not shielded and may hit sensitive detector components. In superconducting hadron storage rings, a further concern is localized particle loss near one of the superconducting magnets, which may result in the quench of the magnet, i.e., in its transition to the normalconducting state.

Cooling

Many applications of particle accelerators require beam cooling, which refers to a reduction of the beam phase space volume or an increase in the beam density via dissipative forces. In electron and positron storage rings cooling naturally occurs due to synchrotron radiation, and special synchrotron-radiation damping rings for the production of low-emittance beams are an integral part of electron-positron linear colliders. For other types of particles different cooling techniques are available. Electron cooling and stochastic cooling of hadron beams are used to accumulate beams of rare particles (such as antiprotons), to combat emittance growth (e.g., due to scattering on an internal target), or to produce beams of high quality for certain experiments. Laser cooling is employed to cool ion beams down to extremely small temperatures. Here the laser is used to induce transitions between the ion electronic states and the cooling exploits the Dopper frequency shift. Electron beams of unprecedentedly small emittance may be obtained by a different type of laser cooling, where the laser beam acts like a wiggler magnet. Finally, designs of a future muon collider rely on the principle of ionization cooling. Reference [1] gives a brief review of the principal ideas and the history of beam cooling in storage rings; a theoretical dicussion and a few practical examples can be found in [2].