The derivations given in Chapters 2 and 3 for the transverse motion appear very simple and easy to understand, which was the motivation for doing them in the way shown. The final results are valid and are universally applied, but behind this economy of the truth there are some pitfalls.

As often happens in applied physics, the final expressions are relatively simple, but only as a result of making a number of approximations, which are justified as having very small effects. Usually this is fully satisfactory compared to the desired accuracy, but in some cases, after very many oscillations of the system, the consequences may no longer be negligible. In such cases the approximations are violating, however slightly, some fundamental principle of physics, which for accelerators will be seen later to be the conservation of phase space. The same problem reappears in celestial mechanics where again expressions are required that will accurately represent the motion of a planetary system through extremely large numbers of oscillations. In both cases the conservation of phase space must be rigorously observed. At present there is a lot of research activity motivated by the design of new accelerators to determine the limit between stability and instability for very large numbers of oscillations in the presence of non-linear fields. It is therefore imperative to be sure that at least the equations for the simple linear motion are stable and to find a safe method for treating this and more complicated cases.

The first goal of any accelerator project is to reach the design energy. This battle is usually won rather quickly, only to be replaced by a longer struggle to reach the highest intensity possible. This latter phase can last for the whole of the accelerator's working life and, for the most part, will end up as a struggle against coherent instabilities. Both theoretically and experimentally the study of coherent instabilities is too advanced to be covered exhaustively in this book, but in line with the extreme importance of this subject an introduction is given in this chapter. More emphasis is put on physical description than the mathematical treatment and the latter is restricted to coasting beams. For more complete analyses, including bunched beams, the reader can refer to Hofmann, Zotter and Sacherer, Laclare and Chao. Many of the same phenomena were already known in electron tubes, where they were of interest as amplification mechanisms. A good account of the early work in this field can be found in Beck. For those readers interested in the historical aspects, Maxwell had already analysed the stability of a ring of weakly interacting particles in a prize-winning essay on Saturn's rings in 1856.

GENERAL DESCRIPTION

In all accelerators the beams are surrounded by cavities, pipes, plates, grids, etc., which for the most part are metal.

A charge that is being accelerated radiates photons. In any kind of accelerator, the radiation due to the longitudinal acceleration is negligible. However, this is not always true for the radiation due to the transverse acceleration which is imposed by the guide field in circular machines. High-energy electron storage rings, for example, suffer severely from this fundamental energy loss by radiation and it is this fact that makes linear colliders the favoured electron machines of the future. While energy loss by radiation is a problem when considering the efficiency of an accelerator or the background caused in experimental physics detectors, it nevertheless does have certain redeeming features. Being a dissipative force, it is not Liouvillian and it is possible to exploit this to decrease the longitudinal and transverse phase space areas occupied by the beam. It can also provide some stabilisation against coherent oscillations. Best of all, this unwanted energy loss, or synchrotron radiation as it is called, can be turned into an intense source, which is typically in the visible to X-ray region with good directional properties and a well-defined polarisation. Accelerators dedicated to producing this radiation are called synchrotron light sources or simply light sources. Light sources have proved to be extremely valuable research tools and are a rapidly growing branch of the accelerator family.

The history of synchrotron radiation, with the theoretical and experimental development starting with Maxwell, can be found in a very readable account by Blewett.

In Chapters 2 and 3 the analysis of the particle motion is developed in the environment of sharply defined regions of constant dipole and quadrupole fields. Real-world magnets are more complicated.

Multipole expansion of a 2-dimensional magnetic field

First consider a purely 2-dimensional field, which is a rather reasonable approximation since the majority of accelerator magnets are long compared with their aperture. In the source-free region of the magnet gap, the field can be derived from a scalar potential φ. In the local cylindrical coordinate system for the magnet, the general Fourier expansion of this scalar potential will be,

where, Am and Bm are constants. The zero order term is a constant and can be disregarded, since it will make no contribution to the fields derived later. Equation (1B) contains two orthogonal sets of multipoles;

• the sine terms are designated normal or right multipoles, and

• the cosine terms are designated skew multipoles.

An alternating-gradient lattice has two normal modes for particle oscillations and by using only normal lenses these modes are made horizontal and vertical. As discussed in Chapter 5 the inclusion of skew lenses in a normal lattice will cause coupling between these modes. It is clear from (1B) that the skew multipoles can be made by simply rotating the normal multipoles by π/(2m).

Equation (1B) is a natural starting point in a mathematical sense, but is slightly inconvenient for certain applications.

Machine designers now think almost exclusively in terms of alternating-gradient focusing, or strong focusing as it is also known. Their conversion from weak to strong focusing was rapid and decisive following the publication of Ref. 1 by Courant, Livingston and Snyder from the Brookhaven National Laboratory in 1952. CERN, for example, immediately abandoned its already-approved project for a 10 GeV/c weak-focusing synchrotron in favour of a 25 GeV/c strong-focusing machine, which it was estimated could be built for the same price. Strong focusing had broken through a cost–size–tolerance barrier. Both the betatron amplitude and the momentum compaction functions are compressed and the required aperture is typically reduced from tens of centimetres to centimetres. The momentum compaction is the more strongly affected, but in accelerators the momentum spread is usually small and the betatron amplitude reduction dominates. The stronger gradients alleviate the tolerance problem and the alternating structure leads naturally to a modular design for the lattice. It took several years, however, before this latter point was fully exploited for special optics modules.

Alternating-gradient focusing was an attractive idea, but it was not entirely new. It is based on a long-known result in classical optics and, in fact, an American-born engineer, Christofilos, living in Athens, had already filed a USA patent on the same scheme in 1950.

There is much truth in the statement that a machine is only as good as its diagnostic equipment, but paradoxically it often happens that this is the first area to feel the cold draught of budget cuts. The careful preparation of algorithms for controlling machine parameters and plentiful instrumentation saves time and avoids the random chance of trial-and-error adjustments. Time spent in observation and understanding is always well spent and leads to future improvements in performance.

When commissioning an accelerator ring the first problem is to get a circulating beam. Apart from the jubilant press release announcing the first injection (of importance for funding agencies) little can be done until the beam makes at least a full circuit of the machine. This is primarily a problem of the closed orbit. The first step is taken during the machine design when a prognosis is made of the expected closed-orbit distortion, which is then used to set tolerances. Later algorithms are developed for measuring and correcting the orbit under various conditions as well as some tools for diagnosing the errors to guide the survey team. Orbit measurements can also be used for checking many optics parameters. Next on the list is the tune of the machine. The tune can be deduced from orbit measurements by noting the beam positions on four consecutive passages, but unless the beam is being lost after a few turns, it is more convenient and accurate to apply other techniques.

The Vlasov equation is a powerful tool for finding a self-consistent solution for the behaviour of an assembly of many particles (1010–1020), in which each particle feels the sum of the external forces and the collective force of all particles. The collective forces should not be affected by the interactions between neighbouring particles, which is often expressed as: collisions can be neglected. The equation is an expression of Liouville's Theorem, which states that phase space is conserved along a dynamical trajectory when only Hamiltonian forces are acting, and a simple, but non-rigorous, derivation will be based on this. As with the Hamiltonian formalism described in Appendix A, the Vlasov equation provides a standard approach to problems and ensures the physics is sound leaving only mathematical difficulties.

First, however, it is interesting to make a small digression to discuss phase-space area and phase-space density. In Appendix A and elsewhere care has been taken to use the former term in order to imply a sort of continuous incompressible fluid flowing through phase space. In contrast to this, a beam is an assembly of discrete particles embedded in phase space. Under the conditions specified for the Vlasov equation the forces cannot manoeuvre single particles, so the particle distribution moves as phase space moves. In a given element of phase space there will be a certain density of particles and since the area is conserved so also is the phase-space density (of the particles).

By the early forties, machine designers understood weak focusing and phase focusing well enough for their attention to be drawn away from single-particle phenomena towards the next limitation on performance, that of space charge. For example, Kerst in 1941 was already discussing the self, or direct, space-charge force inside the beam, which was limiting the injection current and changing the optimum injection voltage into his betatron. Some years later the effects of image space-charge forces arising from the proximity of the conducting vacuum chamber walls were discussed by the MURA staff in 1959. A comprehensive theory of these two mechanisms was published by Laslett in 1963. Subsequent work has taken into account the penetration of a.c. fields through the vacuum chamber walls, the solution of image coefficients for more complicated boundary conditions, and some practical applications. These ideas have led on naturally to the new fields of study of coherent instabilities and space-charge dominated beams.

THEORETICAL CONTEXT

The overall scheme for investigating space-charge effects is summarised in Table 8.1. This chapter will concentrate on how the local self and image fields manifest themselves as an incoherent tune shift for the motion of individual particles and as a coherent tune shift for the beam as a whole.

Since the effects of interest are integrated over complete turns in the machine, it is reasonable to build the theory on the smooth or weak approximation for the single-particle unperturbed betatron motion.

Nuclear physics research was the birth-place of charged-particle accelerators and for many decades their main ‘raison d'être’. This has given them a somewhat specialised and academic image in the eyes of the general public and indeed accelerators and storage rings do provide an extremely rich field for the study of fundamental physics principles. However, this academic image is fast changing as the applications for accelerators become more diversified. They are already well established in radiation therapy, ion implantation and isotope production. Synchrotron light sources form a large and rapidly growing branch of the accelerator family. The spallation neutron source is based on an accelerator and there are many small storage rings for research around the world relying on sophisticated accelerator technologies such as stochastic and electron cooling. In time accelerators may be used for the bulk sterilisation of food and waste products, for the cleaning of exhaust gases from factories, or as the drivers in inertial fusion devices.

During the first third of our century, natural radioactivity furnished the main source of energetic particles for research in atomic physics. Let us mention a famous example. At McGill University, Montreal, Canada, in 1906, Rutherford bombarded a thin mica sheet with alpha particles from a natural radioactive source. He observed occasional scattering, but most of the alpha particles traversed the mica without deviating or damaging the sheet.