Beam emittance

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.

Definition

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:

emittance =  6π ( width ² - D²( (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" or the "95% emittance", respectively.

Acceptance

The acceptance (also called admittance[1])is the maximum emittance that a beam transport system or analysing system is able to transmit. This is the size of the chamber transformed into phase space and does not suffer from the ambiguities of the definition of beam emittance.

Conservation of emittance

Lenses can focus a beam, reducing its size in one transverse dimension while increasing its angular spread, but cannot change the total emittance. This is a result of Liouville's theorem. Ways of reducing the beam emittance include radiation damping, stochastic cooling, and electron cooling.

Normalised emittance

The emittance so far discussed is inversely proportional to the beam momentum; increasing the momentum of the beam reduces the emittance and hence the physical size of the beam. This reduction is called adiabatic damping. It is often more useful to consider the normalised emittance:[2]

ε * = βγε

where β and γ are the relativistic functions. The normalised emittance does not change as a function of energy and so can track beam degradation if the particles are accelerated. If β is close to one then the emittance is approximately inversely proportional to the energy and so the physical width of the beam will vary inversely to the square root of the energy.

Emittance and brightness

Emittance is also related to the brightness of the beam, in microscopy brightness is very often used because it includes the current in the beam and most systems are circularly symmetric.

B= η I /( ε_x ε_y )

with η = 1/ 8π²

Notes

1 ^ [1] Accelerator Physics by Shyh-Yuan Lee.
2 ^ Wilson, Edmund (2001). An Introduction To Particle Accelerators.