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The development of charged particle accelerators and it’s underlying principles has its basis on the theoretical and experimental progress in fundamental physical phenomena. While active particle accelerator experimentation started seriously only in the twentieth century, it depended on the basic physical understanding of electromagnetic phenomena as investigated both theoretically and experimentally mainly during the nineteenth and beginning twentieth century. In this introduction we will recall briefly the history leading to particle accelerator development, applications and introduce basic definitions and formulas governing particle beam dynamics.

Before we address the physics of beam dynamics in accelerators it seems appropriate to discuss briefly various methods of particle acceleration as they have been developed over the years. It would, however, exceed the purpose of this text to discuss all variations in detail. Fortunately, extensive literature is available on a large variety of different accelerators and therefore only fundamental principles of particle acceleration shall be discussed here. A valuable source of information for more detailed information on the historical development of particle accelerators is Livingston’s collection of early publications on accelerator developments [8].

Parallel with the development of electrostatic and linear rf accelerators the potential of circular accelerators was recognized and a number of ideas for such accelerators have been developed over the years. Technical limitations for linear accelerators encountered in the early 1920s to produce high-power rf waves stimulated the search for alternative accelerating methods or ideas for accelerators that would use whatever little rf fields could be produced as efficiently as possible.

Based on d’Alembert’s principle, we formulate Hamilton’s integral principle by defining a function such that for any mechanical system the variation of the integral d vanishes along any real path (Fig. 4.1) so that

The most obvious components of particle accelerators and beam transport systems are those that provide the beam guidance and focusing system. Whatever the application may be, a beam of charged particles is expected by design to follow closely a prescribed path along a desired beam transport line or along a closed orbit in case of circular accelerators. The forces required to bend and direct the charged particle beam or provide focusing to hold particles close to the ideal path are known as the Lorentz forces and are derived from electric and magnetic fields through the Lorentz equation.

Beam dynamics is effected by electromagnetic fields. Generally, magnetic fields are used for relativistic particle guidance and focusing while electric fields are mostly used in the form of electro-static fields or microwaves for acceleration of the particles.

The general equations of motion, characterized by an abundance of perturbation terms on the right-hand side of, for example, (), () have been derived in the previous chapter. If these perturbation terms were allowed to become significant in real beam transport systems, we would face almost insurmountable mathematical problems trying to describe the motion of charged particles in a general way. For practical mathematical reasons it is therefore important to design components for particle beam transport systems such that undesired terms appear only as small perturbations. With a careful design of beam guidance magnets and accurate alignment of these magnets we can indeed achieve this goal.

The solution of the linear equations of motion allows us to follow a single charged particle through an arbitrary array of magnetic elements. Often, however, it is necessary to consider a beam of many particles and it would be impractical to calculate the trajectory for every individual particle. We, therefore, look for some representation of the whole particle beam.

In previous chapters we have concentrated the discussion on the interaction of transverse electrical and magnetic fields with charged particles and have derived appropriate formalisms to apply this interaction to the design of beam transport systems. The characteristics of these transverse fields is that they allow to guide charged particles along a prescribed path but do not contribute directly to the energy of the particles through acceleration. For particle acceleration we must generate fields with nonvanishing force components in the direction of the desired acceleration.