Sadiq Proposal Defense

Emittance

What is Emittance

In accelerator physics, Cartesian coordinate system was used to describe motion of the accelerated particles. Usually the z-axis of Cartesian coordinate system is set to be along the electron beam line as longitudinal beam direction. X-axis is set to be horizontal and perpendicular to the longitudinal direction, as one of the transverse beam direction. Y-axis is set to be vertical and perpendicular to the longitudinal direction, as another transverse beam direction.

For the convenience of representation, we use [math]z[/math] to represent our transverse coordinates, while discussing emittance. And we would like to express longitudinal beam direction with [math]s[/math]. Our transverse beam profile changes along the beam line, it makes [math]z[/math] is function of [math]s[/math], [math]z~(s)[/math]. The angle of a accelerated charge regarding the designed orbit can be defined as:

[math]z'=\frac{dz}{ds}[/math]

If we plot [math]z[/math] vs. [math]z'[/math], we will get an ellipse. The area of the ellipse is an invariant, which is called Courant-Snyder invariant. The transverse emittance [math]\epsilon[/math] of the beam is defined to be the area of the ellipse, which contains 90% of the particles

File:Sadiq phd emittance phase space ellipse.eps

File:Emittance.tex

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