Chapter 10
Synchrotron Motion
Up to now we have been primarily concerned with the motion in the transverse planes. Yet, for particle accelerators, as the name implies, acceleration
is the primary interest. Therefore the motion in the longitudinal phase space
has to be understood. Although there are many different ways of accelerating
charged particles, we restrict ourselves mostly to circular accelerators (synchrotrons, to be specific), where the acceleration is done using radio frequency
(RF) cavities. The only exception is the last section, where the transverse dynamics of RF cavities is discussed, which is of great significance mainly for
linacs.
The chapter is organized as follows. First, a section is devoted to a brief
description of a typical RF cavity used in a ring. Second, the time-of-flight
as the function of energy is derived. Next, combining the results from the
previous sections, the map of the longitudinal phase space is obtained and
the longitudinal motion is studied in detail. Last, the transverse effect of the
RF cavities is discussed briefly.
10.1 RF Fundamentals
Most RF cavities used in synchrotrons are variations of the cylindrical pillbox cavity, which consists of two circular metallic plates of radius R c that are
separated by the distance l and that are connected with a cylindrical mantel
of radius R c , resulting in a geometry reminiscent of a circular pill box.
The field distribution in the interior of such a kind of a metal box can be
written in a simple analytical form. Following Wangler [70], the electromagnetic field of such a cavity of length l and radius R c with transverse magnetic
241
DOI:10.1201/b12074-10
Synchrotron Motion
Up to now we have been primarily concerned with the motion in the transverse planes. Yet, for particle accelerators, as the name implies, acceleration
is the primary interest. Therefore the motion in the longitudinal phase space
has to be understood. Although there are many different ways of accelerating
charged particles, we restrict ourselves mostly to circular accelerators (synchrotrons, to be specific), where the acceleration is done using radio frequency
(RF) cavities. The only exception is the last section, where the transverse dynamics of RF cavities is discussed, which is of great significance mainly for
linacs.
The chapter is organized as follows. First, a section is devoted to a brief
description of a typical RF cavity used in a ring. Second, the time-of-flight
as the function of energy is derived. Next, combining the results from the
previous sections, the map of the longitudinal phase space is obtained and
the longitudinal motion is studied in detail. Last, the transverse effect of the
RF cavities is discussed briefly.
10.1 RF Fundamentals
Most RF cavities used in synchrotrons are variations of the cylindrical pillbox cavity, which consists of two circular metallic plates of radius R c that are
separated by the distance l and that are connected with a cylindrical mantel
of radius R c , resulting in a geometry reminiscent of a circular pill box.
The field distribution in the interior of such a kind of a metal box can be
written in a simple analytical form. Following Wangler [70], the electromagnetic field of such a cavity of length l and radius R c with transverse magnetic
241
DOI:10.1201/b12074-10