Sadiq Proposal Defense

Introduction

We are going to use a High Repetition Rate Linac (HRRL) S-band electron linear accelerator, with frequency of 2856 MHz, to create positrons. HRRL cavity is located at the Beam Lab of the Physics Department, at Idaho State University (ISU).

The room HRRL cavity located is divided into two parts by a L-shaped cement wall, we will call them accelerator cell (where cavity located) and experimental cell (where nuclear experiments take palce). Previously cavity was located at the center of the accelerator cell, as shown in figure below. To adapt positron project, it was relocated to new position, as shown in same figure.

Fig. HRRL experiment cell .

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]

Dropping off subscript "rms" on emittance [math]\epsilon_{rms, x}[/math]: [math] \sigma_{screen, 11}=\sigma_{screen, x}^2=\epsilon_x (m_{11}^2\beta - 2m_{12}m_{11}\alpha+m_{12}^2\gamma) [/math]

Using [math]\mathbf{\sigma}[/math] matrix relations:

[math] \sigma_{x}={\epsilon_x\beta},~\sigma_{12}={-\epsilon_x\alpha},~{\epsilon_x\gamma}=\epsilon_x \frac{1+\alpha^2}{\beta}=\frac{\epsilon_x^2+\sigma_{12}^2}{\sigma_{11}} [/math]

Here [math]\sigma_{x}[/math] is [math]\sigma_{screen, x}[/math]. We got:

[math] \sigma_{x}^2=m_{11}^2 \sigma_{11} + 2m_{12}m_{11} \sigma_{12} + m_{12}^2 \frac{\epsilon_x^2+\sigma_{12}^2}{\sigma_{11}} [/math]

[math] \sigma_{x}^2=\sigma_{11} \left(m_{11}^2 + 2m_{11}m_{12}\frac{\sigma_{12}}{\sigma_{11} }+ m_{12}^2\frac{\sigma_{12}^2}{\sigma_{11}^2}\right) + m_{12}\frac{\epsilon_x^2}{\sigma_{11}} [/math]

[math] \sigma_{x}^2=\sigma_{11}\left(m_{11} + m_{12}\frac{\sigma_{12}}{\sigma_{11} }\right)^2 + m_{12}\frac{\epsilon_x^2}{\sigma_{11}} [/math]

Recall:

[math] m_{11} = S_{11}-kLS_{12}~~~~~~m_{12}=S_{12} [/math]

Substituting and reorganizing result in:

[math] \sigma_{x}^2=\sigma_{11} {S_{12}^2}\left(kL - \left( \frac{S_{11}}{S_{12}} + \frac{\sigma_{12}}{\sigma_{11}} \right) \right)^2 + S_{12}^2\frac{\epsilon_x^2}{\sigma_{11}} [/math]

Introducing constants [math]A[/math],[math]B[/math], and [math]C[/math]

[math] A = \sigma_{11} {S_{12}^2},~~B = \frac{S_{11}}{S_{12}} + \frac{\sigma_{12}}{\sigma_{11}},~~C = S_{12}^2\frac{\epsilon_x^2}{\sigma_{11}} [/math]

This will simplify equation to:

[math] \sigma_{x}^2=A(kL - B)^2 + C = A(kL)^2 - 2AB(kL)+(C+AB^2) [/math]

It is easy to see that:

[math] \epsilon = \frac{\sqrt{AC}}{S_{12}^2} [/math]

By changing quadrupole magnetic field strength [math]k[/math], we can change beam sizes [math]\sigma_{x,y}[/math] on the screen. We make projection to the x, y axes, then fit them with Gaussian fittings to extract rms beam sizes, then plot vs [math]\sigma_{x,y}[/math] vs [math]k_{1}L[/math]. By Fitting a parabola we can find constants [math]A[/math],[math]B[/math], and [math]C[/math], and get emittances.

Emittance Measurement Experiment

In July 2010y, Emittance measurement of HRRL was conducted at Beam Lab, at Physics Department of ISU. We installed a YAG crystal on the HRRL beam line to see electron beam. A quadrupole magnet was installed between HRRL gun and the YAG screen. We changed current on the quadrupole to control magnetic field strength of the quadrupole magnet, thus we changed electron beam shape on the YAG screen.

Experimental Setup

We did quadrupole scan to measure emittance of the electron beam in HRRL. In quadrupole scan method, the strength of the quadrupole magnet was changed by changing the current go through quadrupole coils. The electron beam were coming out of the gun went through quadrupole (), then beam would enter a 3-way cross. Two end of the 3-way cross was installed on the beam line. The third end of the 3-way cross was placed upward and there was a actuator installed to it. The YAG crystal was mounted in the actuator, which can put the YAG in the beam line or take it out of the beam line. A camera was placed inside the actuator to look through vacuum a window and to capture the image on the YAG crystal created by electron beam. A Faraday cup was mounted at the end of the beam line to measure the transmission of the charge.

Setup and beam line and are shown in figures 1.2 and 1.3:

Fig.2 Experiment set up of HRRL 2010 July emittance test.

Fig.3 Beam Line of HRRL 2010 July emittance test.

Figures 4, 5, and 6 show Faraday cup, Quadrupole Magnet, and YAG Chrystal used in the test:

Fig.4 Faraday cup used for HRRL 2010 July emittance test.

Fig.5 Quadrupole Magnet used for HRRL 2010 July emittance test.

Fig.6 YAG Christal used for HRRL 2010 July emittance test.

Experiment

Emittance measurement was carried out on HRRL on July of 2010 under the experimental setup discussed in previous section.

References

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File:Emittance.tex

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