Subatomic Physics

16 Accelerators
Figure 2.2: An incident monoenergetic beam is scattered by a target; the counter observing the
scattered particles makes an angle θ with respect to the incident beam direction, subtends a solid
angle dΩ, and records dN particles per unit time.
2.2 Cross Sections and Luminosity
Before we describe accelerators we need to understand two quantities that are of
interest to describe their power. Collisions are the most important processes used
to study structure in subatomic physics. The behavior of a collision is usually
expressed in terms of a cross section. To define cross section, a monoenergetic
particle beam of well-defined energy is assumed to impinge on a target (Fig. 2.2).
The flux F of the incident beam is defined as the number of particles crossing a unit
area perpendicular to the beam per unit time. If the beam is uniform and contains
ni particles per unit volume, moving with velocity v with respect to the stationary
target, the flux is given by
F = niv. (2.11)
In most calculations, the number of incident particles is normalized to one particle
per volume V.Thenumberni is then equal to 1/V . Particles scattered by the target
are observed with a counter that detects all particles scattered by an angle θ into
the solid angle dΩ. The number dN recorded per unit time is proportional to the
incident flux F , the solid angle dΩ, and the number N of independent scattering
centers in the target that are intercepted by the beam (1) :
dN = FNσ(θ)dΩ. (2.12)
The coefficient of proportionality is designated by σ(θ); it is called the differential
scattering cross section, andwealsowrite
σ(θ)dΩ =dσ(θ) or σ(θ) = dσ(θ)
. (2.13)
dΩ
1It is assumed here that each particle scatters at most once in the target and that each scattering
center acts independently of each other one.