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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 <ref name="MConte08"> M. Conte and W. W. MacKay, “An Introduction To The Physics Of Particle Accelera tors”, World Scientifc, Singapore, 2008, 2nd Edition, pp. 257-330. </ref>.

Fig.1 Phase space ellipse <ref name="MConte08"></ref>.

Measurement of Emittance with Quad Scanning Method

In quadrupole scan method, a quadrupole and a Yttrium Aluminum Garnet (YAG ) screen was used to measure emittance. Magnetic field strength of the quadrupole was changed in the process and corresponding beam shapes were observed on the screen. Transfer matrix of a quadrupole magnet under thin lens approximation:

[math] \left( \begin{matrix} 1 & 0 \\ -k_{1}L & 1 \end{matrix} \right)=\left( \begin{matrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{matrix} \right) [/math]

Here, [math]k_{1} L[/math] is quadrupole strength, [math]L[/math] is quadrupole magnet thickness, and f is quadrupole focal length. [math] k_{1} L \gt 0 [/math] for x-plane, and [math] k_{1} L \lt 0 [/math] for y-plane. Transfer matrix of a drift space between quadrupole and screen:

[math] \mathbf{S} = \left( \begin{matrix} S_{11} & S_{12} \\S_{21} & S_{22} \end{matrix} \right)=\left( \begin{matrix} 1 & l \\ 0 & 1 \end{matrix} \right) [/math]

Here, [math]l[/math] ([math]S_{12}[/math]) is the distance from the center of the quadrupole to the screen. Transfer matrix of the scanned region is:

[math] \mathbf{M} = \mathbf{SQ} = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}= \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -k_1L & 1 \end{pmatrix}= \begin{pmatrix} S_{11} - k_1LS_{12} & S_{12} \\ S_{21} - k_1L S_{22} & S_{22} \end{pmatrix} [/math]

[math]\mathbf{M}[/math] is related with the beam matrix [math]\mathbf{\sigma}[/math] as:

[math] \mathbf{ \sigma_{screen}} = \mathbf{M \sigma_{quad} M^T} = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} \begin{pmatrix} \sigma_{quad, 11} & \sigma_{quad, 12} \\ \sigma_{quad, 21} & \sigma_{quad, 22} \end{pmatrix} \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} [/math]

Since:

[math] \sigma_{x}=\sqrt{\epsilon_x\beta},~\sigma_{x'}=\sqrt{\epsilon_x\gamma},~\sigma_{xx'}={-\epsilon_x\alpha} [/math]

[math] \mathbf{ \sigma} = \begin{pmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{pmatrix} = \begin{pmatrix} \sigma_{x}^2 & \sigma_{xx'} \\ \sigma_{xx'} & \sigma_{x'}^2 \end{pmatrix} [/math]

So, [math]\mathbf{\sigma}[/math] matrix can be written as: [math] \mathbf{\sigma}_{quad} = \begin{pmatrix} \sigma_{quad, x} & \sigma_{quad, xx'} \\ \sigma_{quad, xx'} & \sigma_{quad, x,} \end{pmatrix} = \epsilon_{rms, x} \begin{pmatrix} \beta & -\alpha \\ -\alpha & \gamma \end{pmatrix} [/math]

Substituting this give:

[math] \mathbf{ \sigma_{screen}} = \mathbf{M \sigma_{quad} M^T} = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} \epsilon_{rms, x} \begin{pmatrix} \beta & -\alpha \\ -\alpha & \gamma \end{pmatrix} \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix} [/math]

References

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Using APA reference style.

File:Emittance.tex

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