Beam emittance

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The beam emittance of a particle accelerator is the extent occupied by the particles of the beam in space and momentum phase space as it travels. A low emittance particle beam is a beam where the particles are confined to a small distance and have nearly the same momentum. A beam transport system will only allow particles that are close to its design momentum, and of course they have to fit through the beam pipe and magnets that make up the system. In a colliding beam accelerator, keeping the emittance small means that the likelihood of particle interactions will be greater resulting in higher luminosity.

Emittance has units of length, but is usually referred to as "length/angle", for example, "millimeter/milli-radians". It can be measured in all three spatial dimensions. The dimension parallel to the motion of the particle is called the longitudinal emittance. The other two dimensions are referred to as the transverse emittances.

The arithmetic definition of a transverse emittance is:

$emittance = \frac{6\pi ( width^2 - D(\frac{dp}{p})^2)}{B}$

Where:

width is the width of the particle beam
dp/p is the momentum spread of the particle beam
D is the value of the dispersion function at the measurement point in the particle accelerator
B is the value of the beta function at the measurement point in the particle accelerator

Since it is difficult to measure the full width of the beam, either the RMS width of the beam or the value of the width that encompasses a specific percentage of the beam (for example, 95%) is measured. The emittance from these width measurements is then referred to as the "RMS emittance" of the "95% emittance", respectively.

Lenses can focus a beam, reducing its size while increasing its angular spread, but cannot change the total emittance. Ways of reducing the beam emittance include radiation damping, stochastic cooling, and electron cooling.