Physics:Cross-section

The cross-section is a Lorentz invariant measure of the probability of interactions in a two-particle initial state. It has dimension of area (unit or barn ), and is defined such that the expected number of interactions (events) in a small volume and a time interval is

with

and are the number densities of the two particle species (the number of particles per volume), while and are their velocities. and u describe the particle flux and relative direction, respectively, and can be summarily expressed by F. The cross-section can be visualized as the area presented by the target particle, which must be hit by the pointlike projectile particle for an interaction to occur.

To specify what is meant by interaction, one must specify the final state. For example, in the case of elastic scattering, if particle 1 is scattered into the solid angle , the cross-section for the process is denoted , and by definition the differential cross-section is . Cross-sections in colliding beam experiments: The beams in storage rings travel in bunches or continuously, and collide either head-on or at a small angle. The time average of the quantity is called the luminosity L of the collider, and describes the achieved intensity. The average event rate (counts per unit of time) is simply .

Cross-sections in fixed target experiments: For beam particles incident upon a fixed target the expected number of events is, if the attenuation of the beam along the target is neglected,

is the number of target particles, T is the area of the target perpendicular to the beam direction and l is the length (thickness) of the target along the beam direction. is the number density of target particles, which is related to the mass density by

Here is the mass of one particle, mole is Avogadro's number and A is the atomic weight [g/mole].

Due to attenuation of the beam, events are exponentially distributed along the target, and one way to take this effect into account is to write

where

is the effective number of beam particles and is the interaction length.

The observed number of events, , allows one to estimate the cross-section. is subject to observational losses; it has expectation value and variance

The factor includes all effects that cause loss of events in an experiment; it may be called the acceptance, although the term ``acceptance is often used in the more restricted sense of ``geometric acceptance. The exponential distribution of events along the target is only one of the many effects that must be taken into account in calculating the acceptance.

An unbiased estimator for is

with the (estimated) variance

If experimental conditions, like beam intensity, geometry, etc., vary with time, then the factor is replaced by , summed over periods in time such that conditions do not vary during one period. In this way all events are assigned equal weight, whereby ) is minimized.

Formulae for cross-sections of specific processes are given in Barnett96.