Beta function (accelerator physics)

The beta function in accelerator physics is a function related to the transverse size of the particle beam at the location s along the nominal beam trajectory.

It is related to the transverse beam size as follows:^[1]

${\displaystyle \sigma (s)={\sqrt {\epsilon \cdot \beta (s)}}}$

where

• ${\displaystyle s}$ is the location along the nominal beam trajectory
• the beam is assumed to have a Gaussian shape in the transverse direction
• ${\displaystyle \sigma (s)}$ is the width parameter of this Gaussian
• ${\displaystyle \epsilon }$ is the RMS geometrical beam emittance, which is normally constant along the trajectory when there is no acceleration

Typically, separate beta functions are used for two perpendicular directions in the plane transverse to the beam direction (e.g. horizontal and vertical directions).

The beta function is one of the Courant–Snyder parameters (also called Twiss parameters).