Courant–Snyder parameters

In accelerator physics, the Courant–Snyder parameters (frequently referred to as Twiss parameters or CS 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:

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

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