Sadiq Thesis

Introduction

Positron

Positron is the antimatter of electron. Positrons have same mass as electron ( [math]511~\frac{keV}{c^2}[/math]), carries positive charge, and it is noted as "[math]e^+[/math]".

Positrons predicted by Paul Dirac in 1928, <ref name="Dirac1928"> The Quantum Theory of the Electron, P. A. M. Dirac, Proc. R. Soc. Lond. A February 1, 1928 117 778 610-624;</ref>, and experimentally observed by Dmitri Skobeltsyn in 1929 and by Carl D. Anderson in 1932 <ref name="e+_discover"> General Chemistry, Taylor and Francis. p. 660. </ref>. Anderson also coined the term positron and he won the Nobel Prize for Physics in 1936.

<ref name="name"> BOOK_Gernal, Auther, Month_Year,issue,page </ref>

Apparatus

HRRL Positron Beamline

HRRL beamline layout is given in the figure below, and the item descriptions are given in the table below.

HRRL beamline layout.

Item	Description
T1	Positron target
T2	Annihilation target
EnS	Energy Slit
FC1, FC2	Faraday Cups
Q1,...Q10	Quadrupoles
D1, D2	Dipoles
NaI	NaI Detecotrs
OTR	Optical Transition Radiaiton screen
YAG	Yttrium Aluminium Garnet screen

Som parameters of HRRL is given by:

Parameter	Unit	Value
maximum energy	MeV	16
peak current	mA	100
repetition rate	Hz	300
absolute energy spread	MeV	2-4
macro pulse length	ns	>50

Beam properties

Emittance Measurement

Emittance is an important parameter in accelerator physics. If emittance with twiss parameters are given at the exit of the gun, we will be able to calculate beam size and divergence any point after the exit of the gun. Knowing the beam size and beam divergence on the positron target will greatly help us study the process of creating positron. Emittance with twiss parameters are also key parameters for any accelerator simulations. Also, energy and energy spread of the beam will be measured in the emittance measurement.

What is Emittance

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

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>.

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 Measurements with an OTR

A beam emittance measurement of the HRRL was performed at Idaho State University's IAC with YAG (Yttrium Aluminium Garnet) was performed. Due to the high sensitivity of YAG, the beam image was saturated and beam spot pears rather big. Comparison study shows that for same beam YAG screen shows bigger beam size than Optical Transition Radiation (OTR) screen. An OTR based viewer was installed to allow measurements at the high electron currents available using the HRRL.

The OTR screen is 10 [math]\mu m[/math] thick aluminium. Beam images were taken with digital CCD camera, a MATLAB tool were used to extract beamsize and emittance.

The visible light from the OTR based viewer is produced when a relativistic electron beam crosses the boundary of two mediums with different dielectric constants. Visible radiation is emitted at an angle of 90 degree with respect to the incident beam direction<ref name="OTR"> Tech. Rep., B. Gitter, Los Angeles, USA (1992). </ref> when the electron beam intersects the OTR target at a 45 degree angle. These backward-emitted photons are observed using a digital camera and can be used to measure the shape and intensity of the electron beam based on the OTR distribution.

Quadrupole Scanning Method

Fig. Simplified setup configuration for HRRL beamline at 2011 March measurement.

Fig. on the right illustrates the apparatus used to measure the emittance using the quadrupole scanning method. A quadrupole is positioned at the exit of the linac to focus or de-focus the beam as observed on a downstream view screen. The 3.1 m distance between the quadrupole and the screen was chosen in order to minimize chromatic effects and to satisfy the thin lens approximation.

Assuming the thin lens approximation, [math]\sqrt{k_1}L \lt \lt 1[/math], is satisfied, the transfer matrix of a quadrupole magnet may be expressed as

[math] \mathrm{\mathbf{Q}}=\Bigl(\begin{array}{cc} 1 & 0\\ -k_{1}L & 1 \end{array}\Bigr)=\Bigl(\begin{array}{cc} 1 & 0\\ -\frac{1}{f} & 1 \end{array}\Bigr), [/math]

where [math] k_{1}[/math] is the quadrupole strength, [math]L[/math] is the length of quadrupole, and [math]f[/math] is the focal length. A matrix representing the drift space between the quadrupole and screen is given by

[math] \mathbf{\mathbf{S}}=\Bigl(\begin{array}{cc} 1 & l\\ 0 & 1 \end{array}\Bigr), [/math]

where [math]l[/math] is the distance between the scanning quadrupole and the screen. The transfer matrix of the scanning region is given by the matrix product [math]\mathbf{SQ}[/math]. In the horizontal plane, the beam matrix at the screen ([math]\mathbf{\sigma_{s}}[/math]) is related to the beam matrix of the quadrupole ([math]\mathbf{\sigma_{q}}[/math]) using the similarity transformation

[math] \mathbf{\mathbf{\sigma_{s}=M\mathrm{\mathbf{\mathbf{\sigma_{q}}}}}M}^{\mathrm{T}}. [/math]

where the [math]\mathbf{\sigma_{s}}[/math] and [math]\mathbf{\sigma_{q}}[/math] are defined as <ref name="SYLee"> Accelerator Physics, S.Y. Lee, (Singapore: World Scientific, 2004), 61. </ref>

[math] \mathbf{\mathbf{\sigma_{s,{x}}=}}\Bigl(\begin{array}{cc} \sigma_{{s},x}^{2} & \sigma_{{s},xx'} \\ \sigma_{{s},xx'} & \sigma_{{s},x'}^{2} \end{array}\Bigr) ,\; \mathbf{\mathbf{\sigma_{q,{x}}}}=\Bigl(\begin{array}{cc} \sigma_{{q},x}^{2} & \sigma_{{q},xx'}\\ \sigma_{{q},xx'} & \sigma_{{q},x'}^{2} \end{array}\Bigr). [/math]

By defining the new parameters <ref name="quad-scan"> D.F.G. Benedetti, et al., Tech. Rep., DAFNE Tech. Not., Frascati, Italy (2005). </ref>, [math]A \equiv \sigma_{11},~B \equiv \frac{\sigma_{12}}{\sigma_{11}},~C \equiv \frac{\epsilon_{x}^{2}}{\sigma_{11}}[/math]

[math] A \equiv l^2\sigma_{{q},x}^{2},~B \equiv \frac{1}{l} + \frac{\sigma_{{q},xx'}}{\sigma_{{q},x}^{2}},~C \equiv l^2\frac{\epsilon_{x}^{2}}{\sigma_{{q},x}^{2}}, [/math]

the matrix element [math]\sigma_{{s},x}^{2}[/math], the square of the rms beam size at the screen, may be expressed as a parabolic function of the product of [math]k_1[/math] and [math]L[/math]

[math] \sigma_{{s},x}^{2}=A(k_{1}L)^{2}-2AB(k_{1}L)+(C+AB^{2}). [/math]

The emittance measurement was performed by changing the quadrupole current, which changes [math]k_{1}L[/math], and measuring the corresponding beam image on the view screen. The measured two-dimensional beam image was projected along the image's abscissa and ordinate axes. A Gaussian fitting function is used on each projection to determine the rms value, [math]\sigma_{s}[/math] in Eq.~(\ref{par_fit}). Measurements of [math]\sigma_{s}[/math] for several quadrupole currents ([math]k_{1}L[/math]) is then fit using the parabolic function in Eq.~(\ref{par_fit}) to determine the constants [math]A[/math], [math]B[/math], and [math]C[/math]. The emittance ([math]\epsilon[/math]) and the Twiss parameters ([math]\alpha[/math] and [math]\beta[/math]) can be found using Eq.~(\ref{emit-relation}).

[math] \epsilon=\frac{\sqrt{AC}}{l^2},~\beta=\sqrt{\frac{A}{C}},~\alpha=\sqrt{\frac{A}{C}}(B+\frac{1}{l}). [/math]

Fig. HRRL beamline: Optical lenses lay out for optical cage system.

Fig. HRRL beamline: Cage system for camera. Camera is at the bottom.

The OTR Imaging System

The OTR target is 10 [math]\mu[/math]m thick aluminum foil with a 1.25 inch diameter. The OTR is emitted in a cone shape with the maximum intensity at an angle of [math]1/\gamma[/math] with respect to the reflecting angle of the electron beam<ref name="OTR"></ref>. Three lenses, 2 inches in diameter, are used for the imaging system to avoid optical distortion at lower electron energies. The focal lengths and position of the lenses are shown in Fig. on the right. The camera used was a JAI CV-A10GE digital camera with a 767 by 576 pixel area. The camera images were taken by triggering the camera synchronously with the electron gun.

Fig. OTR image of 0 Amp Q1 coil current.

Fig. OTR image of +1 Amp Q1 coil current.

Fig. OTR image of +2 Amp Q1 coil current.

Data Analysis and Results

The observed image profiles were not well described by a single Gaussian distribution. The profiles may be described using a Lorentzian distribution, however, the rms of the Lorentzian function is not defined. The super Gaussian distribution seems to be the best option <ref name="BeamDisBeyondRMS"> F.-J Decker, “Beam Distributions Beyond RMS”, BIW94, Vancouver,CA,Sep 1994, . </ref>, because rms values may be directly extracted.

I used the MATLAB to analyze the data. The results shows that:

[math] \sigma_x^2= (3.678 \pm 0.022) + (-4.17 \pm 0.22)k_1L + (5.55 \pm 0.42)(k_1L)^2 [/math]

[math] \epsilon_x = 0.417 \pm 0.023~mm*mrad ~\Rightarrow~ \epsilon_{n,x} = 11.43 \pm 0.64~mm*mrad[/math]

[math] \beta_x=1.385 \pm 0.065, \alpha_x=0.97 \pm 0.07 [/math]

[math]\sigma_y^2 = (2.843 \pm 0.044) + (1.02 \pm 0.52)k_1L + (3.8 \pm 1.2)(k_1L)^2 [/math]

[math] \epsilon_y = 0.338 \pm 0.065~mm*mrad ~\Rightarrow~ \epsilon_{n,y} = 9.3 \pm 1.8~mm*mrad[/math]

[math] \beta_y=1.17 \pm 0.19, \alpha_y=0.22 \pm 0.10 [/math]

Energy Spread Measurement

Energy scan was done to measure the energy profile of HRRL at nominal 12 MeV. A Faraday cup was placed at the end of the 45 degree beamline to measure the electron beam current bent by the first dipole. Dipole coil current were changed by 1 A increment and the Faraday cup currents were recorded. The scan results with corresponding beam energies are shown in the table below.

The energy distribution of HRRL can be described by two skewed Gaussian fit overlapping.

Dipole    FC        En
Coil      Current   
Current   Reading
(A)       (mA)      (MeV)
8         0.5       4.3119
9         0.6       4.8896
10        0.67      5.4570
11        0.716     6.0141
12        0.752     6.5610
13        0.828     7.0975
14        0.896     7.6238
15        1.112     8.1399
16        1.328     8.6456
17        1.624     9.1411
18        1.896     9.6262
19        2.448     10.1012
20        3.36      10.5658
21        4.82      11.0201
22        6.58      11.4642
22.2      7.88      11.5518
22.5      9.24      11.6824
22.7      10.0      11.7690
23        10.2      11.8980
23.1      11.2      11.9408
23.2      11.56     11.9836
23.3      13.04     12.0262
23.5      13.2      12.1111
23.8      13.04     12.2377
24        13.48     12.3216
24.2      12.36     12.4050
24.5      11.24     12.5295
24.7      10.2      12.6119
25        10.28     12.7348
25.1      8.6       12.7756
25.2      6.92      12.8162
25.5      6.2       12.9376
25.7      4.84      13.0180
26        3.64      13.1378
26.5      1.82      13.3354
27        1.28      13.5305
28        0.56      13.9129
29        0.364     14.2850
30        0.364     14.6469

2 Skewed Gaussian Fit

Beam Distributions Beyond RMS: Media:Beam_Distributions_Beyond_RMS.pdf

Amplitude = 2.13894, mean = 12.07181, sigma_L = 4.46986, sigma_R = 1.20046

Sigma = 2.83516, Es = -0.57658

rms = 2.56472, skew = -0.94853

Amplitude2 = 10.88318, mean2 = 12.32332, sigma_L2 = 0.69709, sigma_R2 = 0.45170

Sigma2 = 0.57440, Es2 = -0.21360

rms2 = 0.56719, skew2 = -0.34231

Positron detection

DAQ setup

Trigger for DAQ

NaI detector dynode signal amplified and delayed to initiate trigger logic. Then

Data Analysis

Signal extraction

For 3 MeV and on detector show all the steps

1. Raw counts target in and out (calibrated energy)
2. Normalized counts
3. background subtracted
4. Integral (zoomed in and with error)

Example of error propagation for the above

Raw counts target in and out

Lets take example of run#3735 for the data analysis.

The ingeral shown in read is from the background is subtracted spectrum.

run in: 3735

run out: 3737

Some run parameters in 3735:

run number	Reprate	Run time	Pulses	Events	e+ Counts NaI Detectors
3735	300 Hz	1002 s	301462	9045	256 +- 16

Electron Rate Calculation

A scintillator placed between quadrupole 9 and quarupole 10 (shown as in the following figures) was used to estimate electron current:

Scintillator Charge Calbiration

The electron beam current was changed to calibrate of the scintillator. The mean of the counts in scintillator decreased in the experiment as the current decreased, the change is linear.

run#	Scope Pics	FC1 area nVs	Scint L area nVs	Scint Left (Ch9)	ADC9 mean	events in scaler
3703		[math]1201.1 \pm 10.1[/math]	[math]4.4 \pm .3[/math]	1127.75 [math] \pm[/math] .7	[math]1126 \pm 0.8 [/math]	first measurement at full current	7853	7855
3705		[math]776.8 \pm 112[/math]	[math]7.26 \pm 4.5[/math]	782.5 [math] \pm[/math] 1	[math]791.8 \pm 0.6[/math]	cut the current in half		7591
3706		[math]367.7 \pm 2.32[/math]	[math]6.5 \pm .34[/math]	239.1 [math] \pm[/math] 0.6	[math] 242.1 \pm 0.3 [/math]	cut the current to its 1/4 th		8154

The data was fitted with three methods to find calbiration, and the average and variance them are used to find the calibration.

The first method is fit the three data points with root.

The second method, the point (0,0) introduced, and the line was forced to pass through this point. The point is introduced based on physics that when there is no electron beam, scintillator should have no counts. The slope was fitted by root. The fit is done in root.

In the second method, the curved is far off from the two data points when ADC channel number is large. The third fitting method is introduced to compensate it, in which fits are done manually. Two fits were done by this method, and average and variance of two fits were obtained.

The average and variance of these three methods were used for the calibration of the scintillator.

Fit Methods	Fit Method 1	Fit Method 2	Fit Method 3	Average of Three Methods
Fit Curve
Calibration	[math] 1.063 \pm 0.013 [/math] nV s /ADC channel	[math]0.789 \pm 0.004[/math] nV s /ADC channel	[math] 0.94 \pm 0.02 [/math] nV s /ADC channel	[math] 0.93 \pm 0.14 [/math] nV s /ADC channel

Electron Current Calculation for Run 3735

Total number of electrons in this run [math]N_{e^-}= (2.99 \pm 0.45 ) \times 10^{16}.[/math]

Total charge of electrons in this run: [math]Q_{e^-} = (4.78 \pm 0.72 ) \times 10^{-3}~C.[/math]

Electron rate: [math] e^-_{rate} = (2.97 \pm 0.45) \times 10^{13}~Hz. [/math]

Average electron current: [math]I_{e^-} = (4.8 \pm 0.8 ) \times 10^{-6}~(A).[/math]

Positron Rate Calculation

This a normalized spectrum (no background subtraction).

run in: 3735

positron rate: [math] e^+_{rate} = 0.259 +- 0.016 ~ Hz.[/math]

Positron to Electron Ratio

positron to electron ratio: [math]R = \frac{N_{e^+}}{N_{e^-}} = (8.7 \pm 1.4)\times10^{-15}.[/math]

positron to electron current ratio: [math] \frac{N_{e^+}}{I_{e^-}} = \frac{0.26 ~ Hz ~ e^+}{4.8 ~ \mu ~ A ~ e^-}[/math]

Hand Calculation of Errors

[math]R = \frac{N_{e^+}}{N_{e^-}} = \frac{ 0.259 \pm 0.016 Hz}{ (2.98 \pm 0.45) \times 10^{13} Hz} = 0.087 \times 10^{-13} = 8.7 \times 10^{-15}[/math]

[math]\Delta R = \sqrt{\frac { \Delta N_{e^+}^2 }{ N_{e^-}^{2} } + (- \frac{ N_{e^+} }{ N_{e^-}^{2} })^2 \Delta N_{e^-}^{2} } = \frac{\sqrt{ \Delta N_{e^+}^{2} + \frac{ N_{e^+}^{2} }{ N_{e^-}^{2} } \Delta N_{e^-}^{2} }}{ N_{e^-}} = \frac{\sqrt{ \Delta N_{e^+}^{2} + R^2 \Delta N_{e^-}^{2} }}{ N_{e^-}}[/math]

[math]\Delta R = \frac{\sqrt{ 0.016^{2} + ( 8.7 \times 10^{-15} \times 4.49 \times 10^{12})^{2} }}{ 2.98\times 10^{13}} [/math]

[math] = \frac{\sqrt{ 0.016^{2} + ( 8.7 \times 4.49 \times 10^{-3})^{2} }}{ 2.98\times 10^{13}} [/math]

[math] = \frac{\sqrt{ 0.000256 + 0.001526}}{ 2.98\times 10^{13}} [/math]

[math] = \frac{\sqrt{ 0.001782}}{ 2.98\times 10^{13}} [/math]

[math] = 0.014 \times 10^{-13} [/math]

[math] = 1.4 \times 10^{-15} [/math]

Background subraction

Run Number	T2 Position	[math]R = \frac{N_{e^+}}{N_{e^-}} [/math]
3735	in	[math](8.7 \pm 1.4)\times 10^{-15}[/math]
3736	out	[math](1.5 \pm 0.8)\times 10^{-16}[/math]
3737	in	[math](8.1 \pm 1.3)\times 10^{-15}[/math]

[math]\bar{R}_{in}= (8.40 \pm 0.96)\times 10^{-15} [/math]

[math]\bar{R}_{out}= (1.5 \pm 0.8)\times 10^{-16} [/math]

[math]{R}_{3MeV} = ( 8.25 \pm 0.96 )\times 10^{-15}[/math]

All Energies

[math]{R}_{5MeV} = ( 6.2 \pm 1.6 )\times 10^{-16}[/math]

[math]{R}_{4MeV} = ( 4.2 \pm 0.8 )\times 10^{-15}[/math]

[math]{R}_{3MeV} = ( 8.25 \pm 0.96 )\times 10^{-15}[/math]

[math]{R}_{2MeV} = ( 6.9 \pm 2.4 )\times 10^{-16} [/math]

[math]{R}_{1MeV} = ( 1.9 \pm 0.9 )\times 10^{-16} [/math]

References

<references/>