Courant–Snyder parameters

In accelerator physics, the Courant Snyder parameters (also known as Twiss Parameters) are a set of quantities used to describe the distribution of positions and velocities of the particles in a beam. When the positions along a single dimension and velocities (or momenta) along that dimension of every particle in a beam are plotted on a phase space diagram, an ellipse enclosing the particles can be given by the equation:

$\gamma x^{2}+2\alpha xx'+\beta x'^{2}=\epsilon$

where $x$ is the position axis and $x'$ is the velocity axis. In this formulation, $\alpha$, $\beta$, and $\gamma$ are the Courant Snyder parameters for the beam along the given axis, and $\epsilon$ is the emittance. Three sets of parameters can be calculated for a beam, one for each orthogonal direction, x, y, and z.