PHYSICS OF PARTICLE ACCELERATORS

An introduction to the physics of particle accelerators
View Description Hide DescriptionThis article is based on our lecture notes prepared for a two‐week intensive summer course given at Cornell University in 1988. This was the second year in which the U.S. Particle Accelerator School teamed with a university to arrange courses for which students could receive credit from the host institution. Our course was intended to be introductory in character, emphasizing the basic physics underlying the field of particle accelerators. Hence, no subject was treated in depth; rather a variety of topics was introduced and pursued far enough, we hope, to illustrate the physics at work. (AIP)

Transverse motion of single particles in accelerators
View Description Hide DescriptionAny accelerator must have focusing elements, usually magnetic quadrupoles, which hold individual particles close to an ideal central orbit. The purpose of this chapter is to analyse the transverse motion of particles and beams in such systems, with particular emphasis on circular machines. A particle not precisely on the central orbit is subject to a restoring force tending to reduce the deviation and, as a result, the particle oscillates around the central orbit. These oscillations are usually called betatron oscillations when they occur in a circular machine, and we will use the same terminology even for motion in any beam transport system. (AIP)

Fundamentals—longitudinal motion
View Description Hide DescriptionI will cover acceleration and longitudinal motion in a synchrotron. What I hope to add (to the current literature) is a unified overview of the related rf subjects in an accelerator and a close coupling between accelerator physics and engineering practices, which is essential for the major progress in areas such as high intensity synchrotrons, a multistage accelerator complex, and anti−proton production cooling, made possible in the past 20 years. I also hope that after this summer school, you will have the basic knowledge to let you understand the discussion and to ease the way for you to dip into the field if you so choose. (AIP)

RF system considerations for a large hadron collider
View Description Hide DescriptionThe radio‐frequency (rf) system of a large hadron collider is require to operate in three different modes. It must capture and accumulate in stationary buckets (closed trajectories in longitudinal phase space) successive groups of bunches from its injector; accelerate these bunches in moving buckets up to the design energy; and finally store them, for several hours, while maintaining a minimum ratio of bunch‐to‐bucket area. In the storage mode it provides the nominal energy gain per turn required to make up for synchrotron radiation losses and for the power loss due to voltages induced in the impedances presented to the beam by the vacuum chamber and accelerating structures. In what follows then, we shall discuss how we arrive at a particular choice of voltage and frequency; the type of accelerating structure that would be suitable for obtaining the required voltage and resonant impedance; static beam loading including a simplified beam stability criterion involving the beam current and total rf system shunt impedance; the basic principle of rf phase and frequency control loops; and the effect of rf noise and its interaction with these loops. Finally, we shall consider the need for and design of rf systems to damp independently coherent oscillations of individual bunches or groups of bunches. (AIP)

Beam observation and the nature of instabilities
View Description Hide DescriptionIn this manuscript ... we will rather dwell on the various means which have been invented to observe the beams, to track and record their most intimate behavior in order to assess their margin of stability, and finally through a better understanding of their properties to devise ways of increasing the machine performance. Although in this exercise good theories are a necessary ingredient, we will put more emphasis on the almost ‘intuitive’ understanding of the nature of the fundamental phenomena at work. The first chapter will describe the observable manifestations of an ensemble of particles with no interactions between them. The second and third chapters will study the effect of wake fields on the collective behavior of the beam particles. The basic nature of beam instabilities will be shown with the help of simple examples. (AIP)

Bunched beam diagnostics
View Description Hide DescriptionThe emphasis of this paper is ... on frequency domain analysis of beam generated signals in storage rings. The goal is to connect spectrum analyzer observations to what the beam is doing. In addition, understanding beam spectra is essential for understanding coherent effects and instabilities in storage rings, and this is discussed extensively. (AIP)

Fields, impedances, and structures
View Description Hide DescriptionBeam particles interact directly with other beam particles or indirectly through the electromagnetic fields reflected back from the walls of the vacuum chamber. The interaction between the fields and a bunch may lead to coherent motions of the particles inside the bunch resulting in instabilities. In the frequency domain, we say that the beam current sees longitudinal and transverse impedances due to space charge and the discontinuities of the vacuum chamber. In this chapter, we first discuss the electromagnetic fields left by a particle after traversing some structures of the vacuum chamber. Next, coupling impedances are defined in terms of the wakefields. They are then estimated for various discontinuities in the chamber. (AIP)

Introduction to wakefields and wake potentials
View Description Hide DescriptionWhat are wakefields and wake potentials, and why are these concepts useful in the physics of linear accelerators and storage rings? We approach this question by first reviewing the basic physical concepts which underlie the mathematical formalism. We then present a summary of the various techniques that have been developed to make detailed calculations of wake potentials. Finally, we give some applications to current problems of interest in accelerator physics. No attempt at completeness can be made in an introductory article of modest length. Rather, we try to give a broad overview and to list key references for more detailed study. It will also be apparent that the last chapter on this subject, with all the loose ends neatly tied up, has yet to be written. There are subtle points, there are controversial questions, and active calculations to resolve these questions are continuing at the time of this writing. (AIP)

Characteristics of synchrotron radiation
View Description Hide DescriptionIn this paper, we review the principle of various synchrotron radiation sources and their characteristics. Spectral characteristics, the angular distribution, the polarization, and the frequency integrated power of the bending‐magnet radiation is discussed. Undulator radiation topics include the undulator harmonics, the spectrum at a given angle, the angular distribution at a given frequency, the effect due to the electron beam distribution, the angle‐integrated power, and the polarization. In addition to the mathematical analysis, we emphasize physical understanding based on the properties of the apparent trajectories. The wiggler as a limiting case of an undulator for a large orbit excursion. We establish the conditions under which a wiggler may be regarded as a sequence of bending magnets. A general discussion of the properties of synchrotron at finite, how it propagates through optical medium, how it forms interference patterns, etc., is given. (AIP)

Review of linear collider beam‐beam interaction
View Description Hide DescriptionThree major effects from the interaction of e ^{+} e ^{−} beams — disruption, beamstrahlung, and electron‐positron pair creation — are reviewed. For the disruption effects we discuss the luminosity enhancement factor, the maximum and rms disruption angles, and the kink instability. All the results are obtained from computer simulations. Scaling laws for the numerical results and theoretical explanations of the computor acquired phenomena are offered wherever possible. For the beamstrahlung effects we concentrate only on the final electron energy spectrum resulting from multiple photon radiation process, and the deflection angle associated with low energy particles. For the effects from electron−positron pair creation, both coherent and incoherent processes of beamstrahlung pair creation are discussed. In addition to the estimation on total number of such pairs, we also look into the energy spectrum and the deflection angle.

The linear collider beam‐beam problem
View Description Hide DescriptionIn this paper, I would like to briefly review some of the early results of beam‐beam simulations, point out some of the main features of that work, and add a few comments which have thus far only been given in seminars and talks. The first two sections deal with the mechanics of doing the simulation including some tricks which are employed. This is followed by a general discussion of beam disruption, the question of stability, how the luminosity behaves, and finally a discussion of some particular special cases which are of interest. (AIP)

A quantum treatment of bremstrahlung and its application to ribbon pulses
View Description Hide DescriptionIn Part I a straightforward high energy expansion is discussed and applied to the problem of radiation from an extended target. In particular, we discuss bremsstrahlung from an electron‐pulse collision. A full quantum treatment of the power spectrum and the average energy loss is given. Scaling laws that smoothly join the quantum regime to the classical limit are derived. In Part II the high energy expansion introduced in Part I is used to describe the problem of beamstrahlung due to an extended pulse with an elliptical cross section of arbitrary eccentricity. We show that the transverse geometry of the pulse enters in a remarkably simple scaling manner. This case is of interest because the radiative energy loss can be markedly reduced while simultaneously keeping a fixed luminosity if the beam pulses are very thin in one transverse direction, i.,e. shaped like ribbons. Effects of other types of beam shaping are briefly discussed, and the physics of the process is emphasized.

Methods of beam optics
View Description Hide DescriptionIn the following report we give a survey of linear machine theory. Our investigations are restricted to coasting‐beam betatron motion but coupling is taken into account in a general way. The equations of motion for on‐ and off‐momentum particles are derived and written in canonical form. From the canonicity it follows that all transfer matrices are automatically symplectic. Eigenvector methods are introduced to study the stability behavior. To investigate the influence of coupling, generalized lattice functions are defined and canonical perturbation techniques are applied.

Nonlinear dynamics
View Description Hide DescriptionFollowing an introduction on analytical mechanics, canonical transformations, and perturbation theory, a description is given of nonlinear effects for near‐integrable systems associated with resonances, amplitude variations with multiple crossings, adiabatic changes of the parameters, and time variation of the perturbation. Examples of applications coming from observations and calculations related to CERN accelerators concern coupling measurements, intra‐beam scattering, beam‐beam effects magnetic imperfections, two‐beam overlap knockout, and synchro‐betatron resonances. Finally, the notions of invariant distortion and aperture of bounded motion in the presence of multipoles are introduced.

Phase space concepts
View Description Hide DescriptionIn these lectures we shall look at the geometric approach to the study of Hamiltonian dynamical systems, especially in connection with the kinds of problems which arise in accelerator orbit theory. This is a vast subject, and we certainly shall not be able to treat it as fully or as carefully as it deserves. In recent years a number of books have been published on dynamics, and the reader who wants to learn more will find some of these titles included in the bibliography. Our own relatively modest goals will be to delineate the idea of invariant tori in phase space, to define and illustrate the importance of resonant orbits and their separatrices as structures for organizing dynamics, and to touch upon the meaning of chaotic orbits in nonintegrable systems. (AIP)

Comments on nonlinear dynamics studies in storage rings
View Description Hide DescriptionNonlinear dynamics of storage rings are discussed. Canonical perturbation theory, resonances, the Hamilton−Jacobi Equations, tracking simulations, explicit canonical integration, Taylor and Lie maps, and differential algebra are topics discussed. Results from the Fermilab experiment E778 which studied nonlinear dynamics in the Tevatron at 150 GeV are presented. (AIP)

The description of particle accelerators using high‐order perturbation theory on maps
View Description Hide DescriptionThe perturbative description of particle accelerators using Taylor series maps is discussed. A new technique is presented which allows a very efficient computation of high order maps. It is shown how this method can be used to determine maps of the action of various electromagnetic fields including fringe fields, measured fields, and space‐charge fields.
Once the map of the system is known, it can be used for several purposes. Firstly, it allows an exact, non‐numerical calculation of quantities of interest like tune shifts and chromaticities. Secondly, it can be tranformed into different coordinates in which the motion of the particles has a particularly simple form and which is helpful for the search of invariants and KAM surfaces.
Finally the map can also be used for tracking, if desired also in combination with symplectification procedures. Here the major advantage of the map is the gain in speed compared to direct tracking. As with any tracking method, however, care should be exercised not to overestimate the results of such simulations, in particular for large numbers of iterations.

Methods of stability analysis in nonlinear mechanics
View Description Hide DescriptionWe review our recent work on methods to study stability in nonlinear mechanics, especially for the problems of particle accelerators, and compare our ideas to those of other authors. We emphasize methods that (a) shows promise as practical design tools, (b) are effective when the nonlinearity is large, and (c) have a strong theoretical basis.

Advanced nonlinear theory: Long‐term stability at the SSC
View Description Hide DescriptionThe SSC is supposed to be one of the most stable dynamic systems. The luminosity lifetime of one day means that particles have to be in a regular, stable, bounded motion for 10^{8} revolutions. This is only an order of magnitude less than the lifetime of the solar system. The problem of beam stability, one of the main problems in the theory of accelerators, has many aspects: one‐particle dynamics, and intrabeam, beam‐environment, and beam‐beam interactions. We address here only the first problem—the dynamics of a single particle. (AIP)

A review on the lattice design of large hadron colliders
View Description Hide DescriptionThe conceptual evolution of the accelerator lattice design is discussed. Indicated are aspects of IR design. We emphasize the cancellation of stop‐band width in the cluster design. The case of symmetric vs antisymmetric design is also discussed. The SSC lattice is used as an example.