The beta function in
accelerator physics
Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
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:
[https://www.cockcroft.ac.uk/wp-content/uploads/2014/12/Neil_3.pdf ]
where
*
is the location along the nominal beam trajectory
* the beam is assumed to have a Gaussian shape in the transverse direction
*
is the width parameter of this
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
*
is the RMS geometrical
beam emittance
In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space.
Each particle in a beam can be described by its position and momentum along each of t ...
, 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
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 positi ...
(also called Twiss parameters).
Beta star
The value of the beta function at an interaction point is referred to as beta star.
The beta function is typically adjusted to have a local minimum at such points (in order to
minimize the beam size and thus maximise the interaction rate). Assuming that this point is
in a drift space, one can show that the evolution of the beta function around the
minimum point is given by:
:
where z is the distance along the nominal beam direction from the minimum point.
This implies that the smaller the beam size at the interaction
point, the faster the rise of the beta function (and thus the beam size) when going away from the interaction point.
In practice, the
aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane.
An opt ...
of the beam line elements (e.g. focusing magnets) around the interaction point
limit how small beta star can be made.
References
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Accelerator physics