The transverse motion in phase space in a circular accelerator is elliptical.

One can transform this to a circle using the Twiss parameters.

Let $(x,p)$ be the phase space coordinates. We define the normalized coordinates by

The Twiss parameters are related by

(2)In terms of these coordinates, the Courant Snyder invariant

(3)becomes

(4)Now, let us use this mathematics in a slightly different context. Consider the full beam distribution, and its projection onto the x-y plane.

Suppose we know the second moments in this plane $\sigma_x^2= <x^2>, \sigma_y^2= <y^2>, \sigma_xy= <xy>$. Now, we may define an "emittance" and Twiss parameters in this plane such that:

One may then show that the emittance is given by

(6)