Beam emittance

Samples of a bivariate normal distribution, representing particles in phase space, with position horizontal and momentum vertical.

In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space.^[1]

Each particle in a beam can be described by its position and momentum along each of three orthogonal axes, for a total of six position and momentum coordinates. When the position and momentum for a single axis are plotted on a two dimensional graph, the average spread of the coordinates on this plot are the emittance. As such, a beam will have three emittances, one along each axis, which can be described independently. As particle momentum along an axis is usually described as an angle relative to that axis, an area on a position-momentum plot will have dimensions of length × angle (for example, millimeters × milliradian).^[1]^: 78–83

Emittance is important for analysis of particle beams. As long as the beam is only subjected to conservative forces, Liouville's theorem shows that emittance is a conserved quantity. If the distribution over phase space is represented as a cloud in a plot (see figure), emittance is the area of the cloud. A variety of more exact definitions handle the fuzzy borders of the cloud and the case of a cloud that does not have an elliptical shape. In addition, the emittance along each axis is independent unless the beam passes through beamline elements (such as solenoid magnets) which correlate them.^[2]

A low-emittance particle beam is a beam where the particles are confined to a small distance and have nearly the same momentum, which is a desirable property for ensuring that the entire beam is transported to its destination. In a colliding beam accelerator, keeping the emittance small means that the likelihood of particle interactions will be greater resulting in higher luminosity.^[3] In a synchrotron light source, low emittance means that the resulting x-ray beam will be small, and result in higher brightness.^[4]

^ ^a ^b Edwards, D. A.; Syphers, M. J. (1993). An introduction to the physics of high energy accelerators. New York: Wiley. ISBN 978-0-471-55163--8.
^ Conte, Mario; MacKa, W (2008). An introduction to the physics of particle accelerators (2nd ed.). Hackensack, N.J.: World Scientific. pp. 35–39. ISBN 9789812779601.
^ Wiedemann, Helmut (2007). Particle accelerator physics (3rd ed.). Berlin: Springer. p. 272. ISBN 978-3-540-49043-2.
^ Minty, Michiko G.; Zimm, Frank (2003). Measurement and Control of Charged Particle Beams. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 5. ISBN 3-540-44187-5.

[Syphers-1] Edwards, D. A.; Syphers, M. J. (1993). An introduction to the physics of high energy accelerators. New York: Wiley. ISBN 978-0-471-55163--8.

[Conte-2] Conte, Mario; MacKa, W (2008). An introduction to the physics of particle accelerators (2nd ed.). Hackensack, N.J.: World Scientific. pp. 35–39. ISBN 9789812779601.

[Wiedemann-3] Wiedemann, Helmut (2007). Particle accelerator physics (3rd ed.). Berlin: Springer. p. 272. ISBN 978-3-540-49043-2.

[minty-4] Minty, Michiko G.; Zimm, Frank (2003). Measurement and Control of Charged Particle Beams. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 5. ISBN 3-540-44187-5.

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