ABSTRACT
The motion in the longitudinal phase space has to be understood. Although there are many different ways of accelerating charged particles, this chapter focuses on circular accelerators (synchrotrons, to be specific), where the acceleration is done using radio frequency (RF) cavities. It is devoted to a brief description of a typical RF cavity used in a ring. The time-of-flight as the function of energy is derived. The map of the longitudinal phase space is obtained and the longitudinal motion is studied in detail. The chapter discusses the transverse effect of the RF cavities. Most RF cavities used in synchrotrons are variations of the cylindrical pillbox cavity, which consists of two circular metallic plates of radius that are separated by the distance and that are connected with a cylindrical mantel of radius, resulting in a geometry reminiscent of a circular pill box.
