2
An Introduction to Beam Physics
x
p x
Z 0
Ensemble
of particles
FIGURE 1.1: A beam — an ensemble of particles in the vicinity of a
reference particle with phase space coordinate
Z 0 .
the cloud of particles in phase space has a special name. It is called emittance. As we shall see later, in many systems the emittance is conserved and
hence plays a special role.
Because all particles are close together, it is often useful to pick one of
these particles, typically one that is somewhere in the middle, and describe
the motion of the others relative to this reference particle. So if the
reference particle has coordinates
Z 0 , then the motion of the particles would
be described in the relative coordinates Δ
Z =
Z −
Z 0 .
In many cases, the density of particles is so low that their interaction can
be neglected or expressed by simple collective models. In other cases, it is
necessary to include the study of the self-interaction, i.e., we have to take into
account the fields due to the space charge.
If the fields are electromagnetic, then the motion is described by the Lorentz
force law, which in SI units is
dd p
dt
= q
E + v ×
B
.
(1.1)
Here
E and
B are the electric and magnetic fields, respectively. These fields
are connected to the scalar potential V and the vector potential
A via the
relations
B =
∇ ×
A,
E = −
∂
A
∂t
−
∇V.
Although this may not be directly relevant in this book, we want to note here
for the sake of completeness that the equations of motion in the form of the