ABSTRACT
The terms cross section and luminosity are synonymously used to quantitatively indicate the probability that a nuclear or particle interaction occurs. The definition of cross section has its origin in classical physics. Second, unlike the conventional terminology where probability is simply a number (0≤ probability ≤ 1), a cross section is simply a measure of probability and it has dimensions of area (length square).
In classical physics, we notice that when two particles collide, the outcome is well defined if we know the direction and momentum of a projectile relative to the target particle. In a pool game, the directions and momenta of billiard balls are uniquely determined if the impact angle and speed of a striking ball relative to the stationary ball are specified. A good pool player takes advantage of physics to score well in a competition. The concept of cross section is an extension of this knowledge to the microscopic world. We immediately face a few complications. First, when we shoot particles or radiation, all we know is a beam of finite dimensions propagating in the laboratory; the smallest beam size we can determine is about a few microns, while particles themselves are nanometers (charged ions) or femtometers (protons) or even smaller. We make the stationary target of either solid materials or fluids (liquids or gases). Again, atoms and nuclei are invisibly small and we do not know which atom or nucleus is the partner in an interaction. Furthermore, unlike the contact interaction in a billiard ball scattering, the interaction is affected by fields, either electromagnetic fields of infinite range or nuclear fields of finite range. This means that projectiles do not necessarily undergo one instantaneous, local interaction with a target atom or nucleus, but they traverse through a field region over a small but finite amount of time. However, these regions and time in-
FIGURE C.1: Particles 1 and 3 approaching the target A at an impact parameter b are scattered at an angle θ. Particles 2 and 4 approach it at a corresponding value of b+db and are scattered at an angle θ−dθ. Particles with impact parameters between b and b+ db emerge at angles between θ and θ−dθ. In the target plane perpendicular to the direction of propagation of particles, the spread of the impact parameters corresponds to an areal cross section 2pibdb. Here 2pib is the circumference of a circle of radius b.
