Beam acceptance (admittance) in an accelerator

The maximum phase space area that particles can survive in an accelerator is called acceptance (admittance), or the dynamical aperture [1].


This discussion is related by the space demanded by the beam and the space provided by the accelerator [2]. Considering a perfect accelerator (without the field imperfections), the solution for the betatron oscillations,

A can be expressed in terms of x and x' as the following:

The  Courant-Snyder invariant A^2 defines an ellipse in the x-x' space. According to the analytic geometry, the general equation and the area of an ellipse are,

Accordingly, the area in the phase space (x-x') occupied by the beam can be deduced as,

--

The admittance is associated with the maximum phase space area that can be accepted by the accelerator. At any point in an accelerator, the maximum beam size can be Asqrt(\beta). If the half aperture available to the beam is a(s), then somewhere there will be a minimum in a(s) / sqrt(\beta(s)).  

Then the acceptance (admittance) will be:

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[1] Accelerator Physics, S. Y. Lee
[2] An Introduction to the Physics of High Energy Accelerators, D. A. Edwards, M. J. Syphers

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OM.