Abstract
The beam emittance ∈ xyz represents the volume of the beam occupied in the six dimensional phase space (x, x′, y, y′, φ, δ), where x and y are the transverse positions, x′ and y′ are the transverse angles, φ is the time-like variable representing the relative phase of the beam, and δ is the relative beam momentum error. Using the notation of the beam matrix Σ beam introduced in Chap. 1, the 6-dimensional emittance is
Considering now only the horizontal plane, the corresponding 2-dimensional horizontal emittance is obtained from
where the first moments have been subtracted, and the average (〈…〉) is taken over the distribution function of the beam; recall also (1.27–1.29). An analoguous expression holds for the vertical plane. For a coupled system, the general form of (4.1) must be taken.
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Minty, M.G., Zimmermann, F. (2003). Transverse Beam Emittance Measurement and Control. In: Measurement and Control of Charged Particle Beams. Particle Acceleration and Detection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-08581-3_4
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