Beam emittance
Emittance is a property of a charged particle beam in a particle accelerator. It is a measure for the average spread of particle coordinates in position-and-momentum phase space and has the dimension of length (e.g., meters) or length times angle (meters times radians). As a particle beam propagates along magnets and other beam-manipulating components of an accelerator, the position spread may change, but in a way that does not change the emittance. If the distribution over phase space is represented as a cloud in a plot (see figure), emittance is the area of the cloud. A more exact definition handles the fuzzy borders of the cloud and the case of a cloud that does not have an elliptical shape.
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. In a synchrotron light source, low emittance means that the resulting x-ray beam will be small, and result in higher brightness.
Emittance has units of length, but is usually referred to as "length x angle", for example, "millimeter x 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:
Where:
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" or the "95% emittance", respectively.
One should distinguish the emittance of a single particle to that of the whole beam. The emittance of a single particle is the value of the invariant quantity
where x and x' are the position and angle of the particle respectively and are the Twiss parameters. (In the context of Hamiltonian dynamics, one should be more careful to formulate in terms of a transverse momentum instead of x'.) This is the single particle emittance. In the case of a distribution of particles, one can define the RMS (root mean square) emittance as the RMS value of this quantity. The Gaussian case is typical, and the term emittance in fact often refers to the RMS emittance for a Gaussian beam.
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