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 (CS) 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.

History

The use of these parameters to describe the phase space properties of particle beams was popularized in the accelerator physics community by Ernest Courant and Hartland Snyder in their 1953 paper, "Theory of the Alternating-Gradient Synchrotron". They are also widely referred to in accelerator physics literature as "Twiss parameters" after British astronomer Richard Q. Twiss, although it is unclear how his name became associated with the formulation.