Fundamentals - longitudinal motion

There are many ways to accelerate charged particles to high energy for physics research. Each has served its purpose but eventually has encountered fundamental limitations of one kind or another. Looking at the famous Livingston curve, the initial birth and final level-off of all types of accelerators is seen. In fact, in the mid-80s we personally witnessed the creation of a new type of collider - the Stanford Linear Collider. Also witnessed, was the resurgence of study into novel methods of acceleration. This paper will cover acceleration and longitudinal motion in a synchrotron. A synchrotron is a circular accelerator with the following three characteristics: (1) Magnetic guiding (dipole) and confinement (quadrupole) components are placed in a small neighborhood around the equilibrium orbit. (2) Particles are kept in resonance with the radio-frequency electric field indefinitely to achieve acceleration to higher energies. (3) Magnetic fields are varied adiabatically with the energy of the particle. D. Edwards described the transverse oscillations of particles in a synchrotron. Here the author talks about the longitudinal oscillations of particles. The phase stability principle was invented by V. Veksler and E. McMillan independently in 1945. The phase stability and strong focusing principle, invented by Courant and Livingston in 1952, enabled the steady energy gain of accelerators and storage rings witnessed during the past 30 years. This paper 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 and cooling, made possible in the past 20 years