Beam Properties

So far, we have considered exclusively the propagation of single particles in external electromagnetic fields. However, in many cases, one is also interested in the behavior of the charged-particle beam, which represents an ensemble of trajectories. In most cases, one characterizes the beam by its current and mean energy E₀. However, these quantities do not suffice to describe its focusing properties, which strongly depend on the emission characteristics of the source. If we neglect the interaction of the particles, we can conceive the beam as a bundle of independent rays. In order that one can neglect the effect of space charge forces, the current density or trajectory density must stay sufficiently small along the entire course of the beam.

We assume in the following that this condition is fulfilled. Then, we can represent the properties of each particle at any point along its trajectory by a point in the six-dimensional phase space with coordinates x,p _x ,y,p _y , z, E. Instead of the energy, one uses generally the energy deviation ▵E = E − E₀ or the relative energy deviation ? = ▵E/ E₀. If we set ▵E = 0, we can represent the properties of each particle by a point in the five-dimensional state space [105]. At a given plane z in this space, the beam intersects a certain area, which is known as hyperemittance. We can project this four-dimensional area onto the two-dimensional phase planes x,p _x and y,p _y . The sum of these projections forms the total transverse emittance. In orthogonal systems, the motion of the particle in the vertical principal section decouples from that in the horizontal section. In the absence of coupling between these degrees of freedom, we may split the total transverse emittance into two independent two-dimensional emittances: one for the x-section and the other for the y-section. For a real beam, these emittances are defined by the extension of the source and/or apertures, which limit the maximum width and the maximum lateral momentum of the beam along the optic axis.