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.

For a single particle in the presence of only conservative forces, the trajectory of that particle in phase space will lie on the contour of an ellipse defined by the CS parameters at each point along the beamline. Such a particle will have an emittance ${\displaystyle \epsilon }$ which depends only on the amplitude of its motion (which is determined by its initial conditions).