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>

Theory

Positron creation from Bremsstrahlung

A charge particle passes through a nuclear field generates photon by breamsstrahlung process.

Bremsstrahlung.

There are three competing processes that light can interact with matter. At low electron volt energy range, light interact with matter via photoelectric effect. As energy of light increases up to to kilo electron volt range, Compton scattering phenomena starts to be more dominant. While the energy of a photon in the photoelectric effect is totally absorbed by an electron, in the Compton effect the energy is partially absorbed. As energy of light reaches twice rest mass energy of electron, 1.22~MeV, the pair production process starts to become the main interaction process. In this process, photon decays into an electron and positron pair.

Pair production.

Discuss other processes an if pair production is dominant process.

HRRL Beamline

https://wiki.iac.isu.edu/index.php/Beamline_Drawing_2011_J.Ellis

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

HRRL beamline layout.

Label	Beam Line Element
T1	positron target
T2	annihilation target
EnS	energy elit
FC1, FC2	Faraday cups
Q1,...Q10	quadrupoles
D1, D2	dipoles
NaI	NaI detecotrs
OTR	optical transition radiaiton screen
YAG	yttrium aluminium garnet screen

Positron production target (T1), front and back detectors.

Positron production target is tungsten foil with 1.25 inch (31.75 mm) diameter, 0.04 inch (1.016 mm) thickness, and placed with 45 degree angle leaning forward as shown in figure. Two virtual detectors with 25.4 mm radius located 26.52 mm away from the center of the T1.

Simulation Result

G4beamline simulation of the experiment using electron beam: 3160571 electrons on T1 is given in red. 2385 positrons escaped downstream of T1 is given in blue.

incident electron beam position profiles (X,Y) and energy distribution.

Positron momentum distributions (Px,Py,Pz,E)

electron after the target (Px,Py,Pz,E)

positron distributions right before first quad and the positrons that will be transported

generate ASCII file to use as positron event generator.  Use root to draw histograms of the ASCII file contents to check them and put them in the wiki.

Electron beam spacial distribution on the T1.

Electron beam energy distribution on the T1.

Positron Momentum Projection on x-axis.	Positron Momentum Projection on y-axis.
Positron Momentum Projection on z-axis.	Positron energy distribution.

Electron Momentum Projection on x-axis after T1.	Electron Momentum Projection on y-axis after T1.
Electron Momentum Projection on z-axis after T1.	Electron energy distribution after T1.

Apparatus

The HRRL (High Repetition Rate Linac) ia a 16~MeV S-band linac located at the Beam Lab of the Department of the Physics, Idaho State University. The HRRL linac structure was upgraded beyond the capabilities of a typical medical linac so it can achieve a repetition rate of 1~kHz.

Some parameters of the 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]

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

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

Quadrupole Scanning

The quadrupole current is changed to alter the strength and direction of the quadrupole magnetic field such that a measurable change in the beam shape is seen by the OTR system. Initially, the beam was steered by the quadrupole indicating that the beam was not entering along the quadrupole's central axis. Several magnetic elements upstream of this quadrupole were adjusted to align the incident electron beam with the quadrupole's central axis. First, the beam current observed by a Faraday cup located at the end of beam line was maximized using upstream steering coils within the linac nearest the gun. Second, the first solenoid nearest the linac gun was used to focus the electron beam on the OTR screen. Steering coils were adjusted to maximize the beam current to the Faraday cup and minimize the deflection of the beam by the solenoid first then by the quadrupole. A second solenoid and the last steering magnet, both near the exit of the linac, were used in the final step to optimize the beam spot size on the OTR target and maximize the Faraday cup current. A configuration was found that minimized the electron beam deflection when the quadrupole current was altered during the emittance measurements.

The emittance measurement was performed using an electron beam energy of 15~MeV and a 200~ns long, 40~mA, macro pulse peak current. The current in the first quadrupole after the exit of the linac was changed from $-$~5~A to 5~A with an increment of 0.2~A. Seven measurements were taken at each current step in order to determine the average beam width and the variance. Background measurements were taken by turning the linac's electron gun off while keep the RF on. Background image and beam images before and after background subtraction are shown in Fig.~\ref{bg}. A small dark current is visible in Fig.~\ref{bg}b that is known to be generated when electrons are pulled off the cavity wall and accelerated.

(a) a beam with the dark current and background noise.

(b) a background image.

(c) a beam image when dark background was subtracted.

In Fig. below, the images of the electron shown for 0, +1 and +2 Q1 coildcurrents.

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.

The electron beam energy was measured using a dipole magnet downstream of the quadrupole used for the emittance measurements. Prior to energizing the dipole, the electron micro-pulse bunch charge passing through the dipole was measured using a Faraday cup located approximately 50~cm downstream of the OTR screen. The dipole current was adjusted until a maximum beam current was observed on another Faraday cup located just after the 45 degree exit port of the dipole. A magnetic field map of the dipole suggests that the electron beam energy was 15~[math]\pm[/math]~1.6~MeV. Future emittance measurements are planned to cover the entire energy range of the linac.

Data Analysis and Results

Images from the JAI camera were calibrated using the OTR target frame. An LED was used to illuminate the OTR aluminum frame that has a known inner diameter of 31.75~mm. Image processing software was used to inscribe a circle on the image to measure the circular OTR inner frame in units of pixels. The scaling factor can be obtained by dividing this length with the number of pixels observed. The result is a horizontal scaling factor of 0.04327~[math]\pm[/math]~0.00016~mm/pixel and vertical scaling factor of 0.04204~[math]\pm[/math]~0.00018~mm/pixel. Digital images from the JAI camera were extracted in a matrix format in order to take projections on both axes and perform a Gaussian fit. 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="sup-Gau"> F.J. Decker, SLAC-PUB-6684, Sep, 1994. </ref>, because rms values may be directly extracted.

Fig.~\ref{par-fit} shows the square of the rms ([math]\sigma^2_{s}[/math]) vs k_1L for x (horizontal) and y (vertical) beam projections along with the parabolic fits using Eq.~\ref{par_fit}. The emittances and Twiss parameters from these fits are summarized in Table~\ref{results}. An automatic MATLAB based emittance measurement tool is under development~\cite{emit-mat}.

<ref name="emit-mat"> C.F. Eckman et al., in Proc. IPAC2012, New Orleans, USA. </ref>.

Fig. Square of rms values and parabolic fittings. .

Fig. Square of rms values and parabolic fittings. .

Parameter	Unit	Value
projected emittance [math]\epsilon_x[/math]	[math]\mu[/math]m	0.37 [math]\pm[/math] 0.02
projected emittance [math]\epsilon_y[/math]	[math]\mu[/math]m	0.30 [math]\pm[/math] 0.04
[math]\beta_x[/math]-function	m	1.40 [math]\pm[/math] 0.06
[math]\beta_y[/math]-function	m	1.17 [math]\pm[/math] 0.13
[math]\alpha_x[/math]-function	rad	0.97 [math]\pm[/math] 0.06
[math]\alpha_y[/math]-function	rad	0.24 [math]\pm[/math] 0.07
micro-pulse charge	pC	11
micro-pulse length	ps	35
energy of the beam E	MeV	15 [math]\pm[/math] 1.6
relative energy spread [math]\Delta E/E[/math]	%	10.4

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 relation between dipole current and beam energy is given in the appendix.

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

2 Skewed Gaussian Fit

The beam energy spread does not follow Gaussian distribution, but the overlapping of two skewed Gaussian found to be the best description of beam energy profile 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

parameter	amplitude	mean	[math]\sigma[/math] left	[math]\sigma_R[/math]
A	[math]\mu[/math]		[math]\sigma_R[/math]
	[math]\mu[/math]	[math]\sigma_L[/math]	[math]\sigma_R[/math]

parameter	Notation	First Gaussian	Second Gaussian
amplitude	A
mean	[math]\mu[/math]
sigma left	[math]\sigma_L[/math]
sigma right	[math]\sigma_R[/math]

Positron detection

DAQ setup

NaI Detectors

The NaI detectors have two outputs, one is at second last dynode and one anode signal. PMT base configuration of the NaI detectors is shown in the the figure. It takes ADC 5.7 [math]\mu[/math]s to convert analog signal to digital signal. The signal from anode was delayed 6 [math]\mu[/math]s by long cable and sent to the ADC.

Trigger for DAQ

The trigger for DAQ required a coincidence between one or more NaI detectors and the electron accelerator gun pulse. The last dynode signals from left and right NaI detectors were inverted using a Ortec ??? amplifier and sent to a Constant Fraction Discriminator (CFD Model specs).

RF noise from the accelerator is as large as the signal from the NaI detector. Since it is correlated in time with the gun pulse, the gun pulse was used to generate a VETO pulse that prevent the CFD from triggering on this RF noise.

After this discrimination and RF noise rejection, the discriminated dynode signals were sent to an Octalgate Generator (Model) that increased the width of the logic signals to prevent multiple pulses during a single electron pulse. Then the signals were sent to Quad Coincidence to generate AND logic between electron gun and dynode signals. The logic is set as: (Na Left && Gun Trigger) && (NaI Rgiht && Gun Trigger). This is to make sure we have trigger when photons back to back scatter to the NaI detectors when electron gun is on.

Then this trigger was sent to ORTEC Gate & Delay Generator. One of the out from gate generator was used to generate a gate to read analog signal from anode. Another output was delayed by 6 [math]\mu[/math]s, necessary time to convert the analog signal from anode to digital signal, and used as trigger for the DAQ.

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:

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 Electron Rate Calculation for Run 3735

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

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

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

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

Electron rate error is [math]\Delta e^-_{rate} = \sqrt{(\frac{\partial e^-_{rate}}{\partial N_e^-})^2 \Delta N_{e^-}^2}[/math]

Here the error on electron counts comes from the combination of the error ADC channel to Faraday cup charge area calibration factor and the error on ADC 511 keV photon peak counts.

[math]\Delta N_{e^-}=\frac{\Delta Q_{e^-}}{e^-} =\frac{ \sqrt{ (\mbox{error on ADC peak count}) (cabliration) + (ADC~peak~count) (error~on~cabliration)} }{e^-} [/math]

Here, To find the average charge two methods were used. One method

Positron Rate Calculation

This a normalized spectrum (no background subtraction).

run in: 3735

Positron rate calculated by: [math] e^+_{rate} = \frac{N_{e^+}}{t}[/math].

Error on positron rate calculated by: [math] e^+_{rate,~er} = \sqrt{\frac{(N_{e^+, er})^2}{t^2}}= \sqrt{\frac{N_{e^+}}{t^2}}[/math].

positron rate: [math] e^+_{rate} = 0.259 \pm 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]

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

The background is taken when the annihilation target, T2, is out. The difference in T1 run and T2 out run is the measure of positrons annihilated at T2.

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

Ratio of Positron Rate to Electron Rata

The measured ratio of positron to electron ratio is given the following table and figure.

Energy (MeV)	Positron to Electron Ratio
1	[math]{R}_{1MeV} = ( 1.9 \pm 0.9 )\times 10^{-16} [/math]
2	[math]{R}_{2MeV} = ( 6.9 \pm 2.4 )\times 10^{-16} [/math]
3	[math]{R}_{3MeV} = ( 8.25 \pm 0.96 )\times 10^{-15}[/math]
4	[math]{R}_{4MeV} = ( 4.2 \pm 0.8 )\times 10^{-15}[/math]
5	[math]{R}_{5MeV} = ( 6.2 \pm 1.6 )\times 10^{-16}[/math]

Sources of Systematic Errors

Determine sources of systematic errors

Error on Energy (x-axis)

To find Error on the energy, the electron beam is directed to the phosphorous screen at the end of the 90 degree beamline. The beam centered then steering away from the center. The current change on the dipole [math]\Delta I[/math] when beam is center and at the edge is 0.2 A. This is corresponding to 0.13 MeV in beam energy.

Error on Ratio (y-axis)

Error on electron beam is derived from:

Error on positron beam rate is derived from: [math]\sqrt{\frac{positron~rate}{run~time}}[/math]

Appendix

1. Dipole Coil Current vs. Beam energy

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

References

<references/>