The beam emittance of a particle accelerator

Particle accelerator

A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...

 is the extent occupied by the particles of the beam in space and momentum phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the 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

Luminosity

Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...

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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:

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)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

Quadrupole magnet

Quadrupole magnets consist of groups of four magnets laid out so that in the multipole expansion of the field the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows...

 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

Liouville's theorem (Hamiltonian)

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...

. Ways of reducing the beam emittance include radiation damping

Radiation damping

Radiation damping in accelerator physics is a way of reducing the beam emittance of a high-velocity beam of charged particles.There are two main ways of using radiation damping to reduce the emittance of a particle beam—damping rings and undulators—and both rely on the same principle...

, stochastic cooling

Stochastic cooling

Stochastic cooling is a form of particle beam cooling. It is used in some particle accelerators and storage rings to control the emittance of the particle beams in the machine. This process uses the electrical signals that the individual charged particles generate in a feedback loop to reduce the...

, and electron cooling

Electron cooling

Electron cooling is a process to shrink the size, divergence, and energy spread of charged particle beams without removing particles from the beam. Since the number of particles remains unchanged and the space coordinates and their derivatives are reduced, this means that the phase space occupied...

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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:


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.


with