Abhilash Mathews › AbhilashMathews... · Created Date: 8/17/2015 5:04:16 PM
1
Optimization of CVD Diamond Detector for ALPHA Experiment at CERN Abhilash Mathews Supervisor: Dr. Scott Menary Department of Physics & Astronomy Detector Dimensions Tested (mm) CVD Diamond Detector Schematic z = 100 2πσ x σ y e -[ (r cos θ -r o cos θ o ) 2 2σ x 2 + (r sin θ -r o sin θ o ) 2 2σ y 2 ] S 3= 5π 4 R2 RR 3π 4 R1 z × rdrdθ (An example computation of the signal [in mV] in Region 3) x = r cos θ y = r sin θ Antiproton Beam Antiproton (¯ p) beams at CERN have a roughly Gaussian distribution with elliptical symmetry. These beams are pulsed and produced every 100 seconds. There are approximately 3 × 10 7 antiprotons in each pulse and its relative intensity was simulated as the z-component in cylindrical coordinates. When antiprotons hit a CVD diamond detector, an ¯ p may ionize a carbon atom, ultimately resulting in the emission of an electron. In the presence of an external electric field, freed electrons move away from the diamond lattice (Tapper 2000 Rep. Prog. Phys. 63: 1286) and are detected through the metal plates on the diamond’s surface. These signals proportional to ¯ p flux were simulated and further investigated to identify the antiproton beam’s physical characteristics (namely centroid position, σ x , and σ y ). Noise Simulation • Instrumental uncertainty and cosmic noise disrupts readings and introduces errors of Gaussian nature. • The various levels of error (i.e. σ N oise ) tested in the detector signals were: 0, 1, 2, and 5 mV. (Note: the maximal total signal is 100 mV due to normalization of the beam.) • Total noise from all sources was accounted by using Monte Carlo methods. • Signals with noise were within a range of ±2σ N oise from its corresponding error-free signal. Comparison of Detector Designs • To be as general as possible initially when comparing the 10 different detectors, the same variables were used for each detector when computing the centroid radius, centroid angle, σ x , and σ y . • The beam pipe’s radius is 7mm—the necessary outer radius for the design that will be installed—but alternatives were also tested to better understand general trends in optimizing the dimensions. • A quintic function was used to fit the actual data based on the variables for the centroid radius, σ x , and σ y ; θ output is the final angular output. • 1000 random antiproton beams were generated for each test trial when comparing different beam characteristics and detector dimensions. • Each simulated beam’s actual centroid radius, centroid angle, σ x , and σ y randomly ranged from 0 - 7 mm, 0 - 2π radians, 2 - 7 mm, and 2 - 7 mm, respectively. • Based on the following results, the R1=3mm/R2=7mm detector appeared to be the optimal detector design, particularly based on its good angular resolution and higher relative resistance to noise. θ output = arctan( S 2-S 4 S 5-S 3 ) (between 0 - 2π ) σ x output = √ 1 3 ∑ 2 n=0 (S 2n+1 - ¯ S 1,3,5 ) 2 S 3+S 5+1 σ y output = √ 1 3 ∑ 2 n=0 (S 2 n - ¯ S 1,2,4 ) 2 S 2+S 4+1 R X 1 output = √ (S 2-S 4) 2 +(S 5-S 3) 2 S 1+1 R X 2 output = √ (S 2-S 4) 2 +(S 5-S 3) 2 (S High )(S 1+1) *R X 2 output was tested only for the optimal detector Testing the Optimal Detector • Orthogonal Distance Regression (ODR) was used to account for errors in the independent variable on the data points—this method reduces to a Least Squares (LSQ) fitting when noise approaches 0. • f 1 = e -(ax-b) + 2 (exponential decay) modelled beam width while f 2 = -e -(ax-b) + 7 (reflected exponential decay) modelled the beam’s centroid radius (this time using R X 2 output ). • The histograms below are based on ODR results, but LSQ fitting resulted in models with lower overall standard deviation (and μ ≈ 0) between the computed and actual radius and/or beam width. • Errors on the parameters a and b also helped assess the modelling function’s capacity to fit the data. Summary of Results • The R1=3mm/R2=7mm detector is the best design for antiproton beam detection. • After numerous variables were tested, θ output , R X 2 output σ x output , and σ y output were found to be the optimal variables (a [reflected] exponential decay function used the latter 3 variables as input to compute the beam’s characteristics). • Angular resolution is 0.18 (10.3 ◦ ), 0.38 (21.8 ◦ ), 0.56 (32.1 ◦ ), and 0.82 radians (47.0 ◦ ), for noise levels of 0, 1, 2, and 5 mV, respectively. ODR Analysis: • Radial computation precision is 0.64, 0.72, 0.85, and 1.61 mm, for noise levels of 0, 1, 2, and 5 mV, respectively. • σ x and σ y computation precision is 0.76, 0.80, 0.98, and 2.08 mm, for noise levels of 0, 1, 2, and 5 mV, respectively. LSQ Analysis: • Radial computation precision is 0.64, 0.72, 0.84, and 1.41 mm, for noise levels of 0, 1, 2, and 5 mV, respectively. • σ x and σ y computation precision is 0.76, 0.80, 0.91, and 1.27 mm, for noise levels of 0, 1, 2, and 5 mV, respectively. *For beams outside of the tested ranges (i.e. 0 ≤ R ≤ 7 [mm], 0 ≤ θ ≤ 2π [rad], 2 ≤ σ x ,σ y ≤ 7 [mm]), the algorithms are untested and thus not necessarily suitable or as precise as stated.
Transcript of Abhilash Mathews › AbhilashMathews... · Created Date: 8/17/2015 5:04:16 PM
Optimization of CVD DiamondDetector for ALPHA Experiment at
CERNAbhilash Mathews
Supervisor: Dr. Scott MenaryDepartment of Physics & Astronomy
Detector DimensionsTested (mm)
CVD Diamond Detector Schematic
X
201510505101520Y
20 15 10 5 0 5 10 15 20
Inte
nsity
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Model of Antiproton Beam
z = 1002πσxσy
e−[(r cos θ−ro cos θo)2
2σx2 +(r sin θ−ro sin θo)2
2σy2 ]
S3 =
5π4 R2∫∫
3π4 R1
z × rdrdθ
(An example computation of the signal [in mV] in Region 3)
x = r cos θy = r sin θ
Antiproton BeamAntiproton (p̄) beams at CERN have a roughly Gaussian distribution with elliptical symmetry. Thesebeams are pulsed and produced every 100 seconds. There are approximately 3 × 107 antiprotons ineach pulse and its relative intensity was simulated as the z-component in cylindrical coordinates. Whenantiprotons hit a CVD diamond detector, an p̄ may ionize a carbon atom, ultimately resulting in theemission of an electron. In the presence of an external electric field, freed electrons move away from thediamond lattice (Tapper 2000 Rep. Prog. Phys. 63: 1286) and are detected through the metal plateson the diamond’s surface. These signals proportional to p̄ flux were simulated and further investigatedto identify the antiproton beam’s physical characteristics (namely centroid position, σx, and σy).
Noise Simulation• Instrumental uncertainty and cosmic noise disrupts readings and introduces errors of Gaussian nature.
• The various levels of error (i.e. σNoise) tested in the detector signals were: 0, 1, 2, and 5 mV.(Note: the maximal total signal is 100 mV due to normalization of the beam.)
• Total noise from all sources was accounted by using Monte Carlo methods.
• Signals with noise were within a range of ±2σNoise from its corresponding error-free signal.
1 0 1 2 3 4 5 6 7 8
Radius Input (mm)
10
0
10
20
30
40
50
60
70
Dete
ctor
Sig
nal
Detector Signal vs. Radius Input (Noise=0)
S1
S2
S3
S4
S5
1 0 1 2 3 4 5 6 7 8
Radius Input (mm)
10
0
10
20
30
40
50
Dete
ctor
Sig
nal
Detector Signal vs. Radius Input (Noise=5)
S1
S2
S3
S4
S5
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Radius Variable
0.0
0.2
0.4
0.6
0.8
1.0
Rela
tive F
requency
Combined Probability Histogram Radius (Noise = 0) σ = 0.0
σ = 0.5
σ = 1.0σ = 1.5
σ = 2.0
σ = 2.5
σ = 3.0
σ = 3.5
σ = 4.0σ = 4.5
σ = 5.0σ = 5.5
σ = 6.0
σ = 6.5
σ = 7.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Radius Variable
0.0
0.2
0.4
0.6
0.8
1.0
Rela
tive F
requency
Combined Probability Histogram Radius (Noise = 5) σ = 0.0
σ = 0.5
σ = 1.0σ = 1.5
σ = 2.0
σ = 2.5
σ = 3.0
σ = 3.5
σ = 4.0σ = 4.5
σ = 5.0σ = 5.5
σ = 6.0
σ = 6.5
σ = 7.0
Comparison of Detector Designs• To be as general as possible initially when comparing the 10 different detectors, the same variables
were used for each detector when computing the centroid radius, centroid angle, σx, and σy.
• The beam pipe’s radius is 7mm—the necessary outer radius for the design that will be installed—butalternatives were also tested to better understand general trends in optimizing the dimensions.
•A quintic function was used to fit the actual data based on the variables for the centroid radius, σx,and σy; θoutput is the final angular output.
• 1000 random antiproton beams were generated for each test trial when comparing different beamcharacteristics and detector dimensions.
• Each simulated beam’s actual centroid radius, centroid angle, σx, and σy randomly ranged from 0−7mm, 0− 2π radians, 2− 7 mm, and 2− 7 mm, respectively.
• Based on the following results, the R1=3mm/R2=7mm detector appeared to be the optimal detectordesign, particularly based on its good angular resolution and higher relative resistance to noise.
θoutput
= arctan(S2−S4S5−S3) (between 0− 2π)
σxoutput =
√13
∑2n=0(S2n+1−S̄1,3,5)2
S3+S5+1 σyoutput =
√13
∑2n=0(S2n−S̄1,2,4)2
S2+S4+1
RX1output =
√(S2−S4)2+(S5−S3)2
S1+1 RX2output =
√(S2−S4)2+(S5−S3)2
(SHigh)(S1+1)*RX2output was tested only for the optimal detector
Testing the Optimal Detector•Orthogonal Distance Regression (ODR) was used to account for errors in the independent variable
on the data points—this method reduces to a Least Squares (LSQ) fitting when noise approaches 0.
• f1 = e−(ax−b) + 2 (exponential decay) modelled beam width while f2 = −e−(ax−b) + 7 (reflectedexponential decay) modelled the beam’s centroid radius (this time using RX2output).
• The histograms below are based on ODR results, but LSQ fitting resulted in models with lower overallstandard deviation (and µ ≈ 0) between the computed and actual radius and/or beam width.
• Errors on the parameters a and b also helped assess the modelling function’s capacity to fit the data.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Radius Variable
0
2
4
6
8
10
Act
ual R
adiu
s (m
m)
fleastsq=−e−((15.60)x−(1.98)) +7,
σ1 =0.22,σ2 =0.01,σy =0.64
fodr=−e−((15.60)x−(1.98)) +7,σ1 =0.22,σ2 =0.01,σy =0.64
Actual Radius vs. Variable (Noise = 0)
LSQ
ODR
0.0 0.1 0.2 0.3 0.4 0.5
Radius Variable
0
2
4
6
8
10
Act
ual R
adiu
s (m
m)
fleastsq=−e−((15.50)x−(1.97)) +7,
σ1 =0.25,σ2 =0.01,σy =0.72
fodr=−e−((15.76)x−(1.97)) +7,σ1 =0.25,σ2 =0.01,σy =0.72
Actual Radius vs. Variable (Noise = 1)
LSQ
ODR
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Radius Variable
0
2
4
6
8
10
Act
ual R
adiu
s (m
m)
fleastsq=−e−((14.85)x−(1.96)) +7,
σ1 =0.29,σ2 =0.01,σy =0.84
fodr=−e−((16.10)x−(1.98)) +7,σ1 =0.29,σ2 =0.01,σy =0.85
Actual Radius vs. Variable (Noise = 2)
LSQ
ODR
0.0 0.2 0.4 0.6 0.8 1.0
Radius Variable
0
2
4
6
8
10
Act
ual R
adiu
s (m
m)
fleastsq=−e−((11.25)x−(1.83)) +7,
σ1 =0.47,σ2 =0.02,σy =1.41
fodr=−e−((21.81)x−(2.07)) +7,σ1 =0.61,σ2 =0.02,σy =1.61
Actual Radius vs. Variable (Noise = 5)
LSQ
ODR
2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of Radius difference : µ=0.0113, σ=0.6371
3 2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of Radius difference : µ=−0.0226, σ=0.7184
3 2 1 0 1 2 3 4
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of Radius difference : µ=−0.0772, σ=0.8479
8 6 4 2 0 2 4 6
Difference (Input-Output)
0
20
40
60
80
100
120
140
160
Frequency
Histogram of Radius difference : µ=−0.5761, σ=1.5072
0.2 0.4 0.6 0.8 1.0 1.2 1.4
σx Variable
0
2
4
6
8
10
Act
ual σx
(m
m)
fleastsq=e−((3.77)x−(1.70)) +2,
σ1 =0.10,σ2 =0.02,σy =0.76
fodr=e−((3.77)x−(1.70)) +2,
σ1 =0.10,σ2 =0.02,σy =0.76
Actual σx vs. Variable (Noise = 0)
LSQ
ODR
0.2 0.4 0.6 0.8 1.0 1.2
σx Variable
0
2
4
6
8
10
Act
ual σx
(m
m)
fleastsq=e−((3.65)x−(1.68)) +2,
σ1 =0.10,σ2 =0.02,σy =0.77
fodr=e−((4.09)x−(1.76)) +2,
σ1 =0.10,σ2 =0.02,σy =0.77
Actual σx vs. Variable (Noise = 1)
LSQ
ODR
0.2 0.4 0.6 0.8 1.0 1.2
σx Variable
0
2
4
6
8
10
Act
ual σx
(m
m)
fleastsq=e−((3.19)x−(1.61)) +2,
σ1 =0.10,σ2 =0.02,σy =0.91
fodr=e−((4.58)x−(1.84)) +2,
σ1 =0.13,σ2 =0.02,σy =0.99
Actual σx vs. Variable (Noise = 2)
LSQ
ODR
0.5 1.0 1.5 2.0 2.5 3.0
σx Variable
0
2
4
6
8
10
Act
ual σx
(m
m)
fleastsq=e−((1.38)x−(1.31)) +2,
σ1 =0.11,σ2 =0.03,σy =1.28
fodr=e−((7.40)x−(2.40)) +2,
σ1 =0.36,σ2 =0.07,σy =2.14
Actual σx vs. Variable (Noise = 5)
LSQ
ODR
3 2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of σx difference : µ=−0.0233, σ=0.7605
3 2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of σx difference : µ=−0.0123, σ=0.7745
4 3 2 1 0 1 2 3 4
Difference (Input-Output)
0
20
40
60
80
100
120
140
160
Frequency
Histogram of σx difference : µ=0.0373, σ=0.9936
8 6 4 2 0 2 4 6
Difference (Input-Output)
0
20
40
60
80
100
120
140
160
Frequency
Histogram of σx difference : µ=0.2648, σ=2.1212
0.2 0.4 0.6 0.8 1.0 1.2
σy Variable
0
2
4
6
8
10
Act
ual σy
(m
m)
fleastsq=e−((3.67)x−(1.70)) +2,
σ1 =0.10,σ2 =0.02,σy =0.77
fodr=e−((3.67)x−(1.70)) +2,
σ1 =0.10,σ2 =0.02,σy =0.77
Actual σy vs. Variable (Noise = 0)
LSQ
ODR
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
σy Variable
0
2
4
6
8
10
Act
ual σy
(m
m)
fleastsq=e−((3.52)x−(1.68)) +2,
σ1 =0.10,σ2 =0.02,σy =0.82
fodr=e−((3.95)x−(1.76)) +2,
σ1 =0.11,σ2 =0.02,σy =0.83
Actual σy vs. Variable (Noise = 1)
LSQ
ODR
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
σy Variable
0
2
4
6
8
10
Act
ual σy
(m
m)
fleastsq=e−((3.18)x−(1.64)) +2,
σ1 =0.10,σ2 =0.02,σy =0.91
fodr=e−((4.48)x−(1.84)) +2,
σ1 =0.13,σ2 =0.02,σy =0.98
Actual σy vs. Variable (Noise = 2)
LSQ
ODR
0 2 4 6 8
σy Variable
0
2
4
6
8
10
Act
ual σy
(m
m)
fleastsq=e−((1.65)x−(1.40)) +2,
σ1 =0.11,σ2 =0.03,σy =1.26
fodr=e−((7.18)x−(2.38)) +2,
σ1 =0.33,σ2 =0.07,σy =2.01
Actual σy vs. Variable (Noise = 5)
LSQ
ODR
4 3 2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
160
180
Frequency
Histogram of σy difference : µ=−0.0179, σ=0.7732
3 2 1 0 1 2 3
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of σy difference : µ=−0.0145, σ=0.8266
4 3 2 1 0 1 2 3 4
Difference (Input-Output)
0
20
40
60
80
100
120
140
Frequency
Histogram of σy difference : µ=0.0497, σ=0.9792
10 8 6 4 2 0 2 4 6
Difference (Input-Output)
0
20
40
60
80
100
120
140
160
Frequency
Histogram of σy difference : µ=0.2951, σ=1.9903
Summary of Results• The R1=3mm/R2=7mm detector is the best design for antiproton beam detection.
•After numerous variables were tested, θoutput, RX2output σxoutput, and σyoutput were found to be the
optimal variables (a [reflected] exponential decay function used the latter 3 variables as input tocompute the beam’s characteristics).
•Angular resolution is 0.18 (10.3◦), 0.38 (21.8◦), 0.56 (32.1◦), and 0.82 radians (47.0◦), for noise levelsof 0, 1, 2, and 5 mV, respectively.
ODR Analysis:
•Radial computation precision is 0.64, 0.72, 0.85, and 1.61 mm, for noise levels of 0, 1, 2, and 5 mV,respectively.
• σx and σy computation precision is 0.76, 0.80, 0.98, and 2.08 mm, for noise levels of 0, 1, 2, and 5mV, respectively.
LSQ Analysis:
•Radial computation precision is 0.64, 0.72, 0.84, and 1.41 mm, for noise levels of 0, 1, 2, and 5 mV,respectively.
• σx and σy computation precision is 0.76, 0.80, 0.91, and 1.27 mm, for noise levels of 0, 1, 2, and 5mV, respectively.
*For beams outside of the tested ranges (i.e. 0 ≤ R ≤ 7 [mm], 0 ≤ θ ≤ 2π [rad], 2 ≤ σx, σy ≤ 7[mm]), the algorithms are untested and thus not necessarily suitable or as precise as stated.
