Certificate of Committee Approval

Transcription

1 Certificate of Committee Approval Brigham Young University-Idaho Department of Physics July 19, 2011 We hereby recommend that the thesis by: Jacopo Lafranceschina Entitled: POSITRON ANNIHILATION DEFECT IMAGING (PADI) be accepted in partial fulfillment of the requirements for the degree of: Bachelor of Science Signatures: Department Chair - Stephen Turcotte Advisor, Thesis Coordinator - Evan Hansen Committee Member - Richard Hatt Committee Member Jon Paul Johnson

2 Copyright 2011 Jacopo Lafranceschina All Rights Reserved

3 POSITRON ANNIHILATION DEFECT IMAGING (PADI) BY JACOPO LAFRANCESCHINA SENIOR THESIS Submitted in partial fulfillment of the requirements for the degree of Bachelors of Science in Physics in the Department of Physics of Brigham Young University-Idaho, 2011 Rexburg, Idaho Advisor: Evan D. Hansen

4 ABSTRACT Imaging is a technique that is done in many different ways. One way of imaging defects at the atom scale is done by using positron annihilation. This senior thesis will briefly give an overview of the history behind positrons, and how they annihilate with electrons. The S-parameter and its usefulness will also be presented in the introduction of the thesis. In order to image using positron annihilation three main parameters need to be taken into account: the spot size of the source of positrons, the size of the object that is going to be imaged, and the error bar of the S-Parameter. A careful choice of a combination of these three parameters will result in a clear and recognizable image.

5 to Rachel and Brooklyn ii

6 ACKNOWLEDGMENTS I would like to acknowledge the BYU-Idaho Physics department for the opportunity given to me to work in their labs using their machinery. A thanks should go to Idaho State University for the equipment provided to us for this research. Additional recognition should be given to Dr. Marcus Gagliardi for his dissertation work. I would also like to recognize the contribution of Dr. Richard Hatt, and Dr. Jon Paul Johnson for the time spent in revising and helping with this senior thesis. Special appreciation goes to my advisor and research coordinator, Dr. Evan Hansen, for the many hours of time spent on the project, and the various teaching and counsel shared with me. A special thanks goes to my wife Rachel for her patience and her support given while carrying on this research. iii

7 TABLE OF CONTENTS CHAPTER 1: INTRODUCTION Positron Annihilation.1 Shape Parameter.3 CHAPTER 2: IMAGING Introduction 5 Spot size.5 Object size.7 Step size....8 Error bar size CHAPTHER 3: EXPERIMENTAL DATA Introduction.. 13 Spatial Resolution Experiment 15 Reproducibility Experiment 19 1-D Scan Experiment D Scan Experiment of the Y..21 CONCLUSIONS REFERENCES APPENDIX iv

8 CHAPTER 1 INTRODUCTION Positron Annihilation In 1928, Paul Dirac was the first to introduce the idea [1] that electrons have both a positive charge and negative energy, thus indicating the existence of positrons. A positron has an electric charge of +1e (elementary charge C), a spin of ½, and the same mass as an electron. It wasn t until four years later, in 1932, that Carl D. Anderson confirmed the existence of Dirac s theoretical prediction by discovering the positron, the first antiparticle. The annihilation of a positron with an electron in matter was first studied in the 1940s (figure 1.1). When a positron and an electron collide, they both are annihilated, and the outcome is the production of, usually, two gamma ray photons. It was discovered early that energy and momentum conservation during the annihilation process could be utilized to study properties of solids. Figure Feynman diagram showing Space-Time vectors of Electron-Positron Annihilation and the formation of the 2 γ rays. The process of positron annihilation produces γ rays that contain important information related to the electron density and the electron momenta in the region from which the positron 1

9 annihilates [2]. When the positron enters the sample material it undergoes three main stages (figure 1.2). First, it will thermalize, meaning the positron will reach thermal equilibrium with the particles in the sample. This usually happen within a few micrometers of the surface. Second, it will go through the diffusion process where the positron will be repelled by positively charged nuclei (protons) in the sample. Lastly it will annihilate with a nearby electron. Figure 1.2 Stages leading to Annihilation: Thermalization, Diffusion, Annihilation. Figure 1.3 (A) Shown is the positron and the annihilation with the core electron in the defect free sample. (B) Shown is the positron and the annihilation with the valence electron in the sample with mono-vacancies. 2

10 The positron will annihilate either with the core electrons (high momentum) or the valence electrons (low momentum). The positron will have a higher probability of annihilating with the core electrons if the material is defects free (figure 1.3.A). Otherwise if the sample has defects in the lattice structure, such as vacancies or dislocations, the positron will preferentially be trapped in these defects and therefore annihilate with the valence electrons (figure 1.3.B). Shape-Parameter Since the information is now contained in the two γ rays we will be talking about those. When we detect the γ rays we will have photons resulting from annihilation of high and low momentum electrons, thus influencing the Doppler broadening of the 511 kev peak in the detector (figure 1.4). The more the defects in the sample the higher and the sharper the peak will be. Those peaks can be analyzed by looking at their width and their area and a parameter can be derived to describe the defects in the sample. This parameter is called the S-Parameter or lineshape parameter. The S-Parameter was first introduced by Mackenzie[3] as the ratio of the area of the central part of the annihilation line to the total area of the peak (figure 1.5). A detailed analysis of the Doppler broadening and derivation of the S-parameter has been done by Marcus Gagliardi[4], therefore it will not be treated in this paper. We will be using his results and the software he developed for the S-parameter for our experiment in imaging. 3

11 Figure 1.4 Doppler broadening effect of gamma rays detected from sample core electrons and valence electrons. Figure 1.5 The S-parameter will be given by the ratio of the central area(red) and the total peak area (pink) [3]. 4

12 CHAPTER 2 IMAGING Introduction The S-Parameter can be a useful tool when detecting defects. For this thesis, however, we have analyzed a sample in which a change in material can be detected by looking at a change in the S-Parameter. We will be using this fact to recognize shapes and recreate a 2-D map of a writing of the letters BYU-I. Imaging can be done in many different ways. Some of the most common way to image things are done by using photons, radioactive isotopes, x-rays, magnetic fields, ultrasound and positrons. Since the late 1950s positrons have been widely used as a technique for imaging. One of their applications consists of injecting in the bloodstream of a person a radiolabeled tracer. This tracer will emits gamma rays that can be detected by the Positron Emission Tomography (PET) scanner. Spot Size The most important part of imaging with Positron Annihilation is the spot size of the scanner. The spot size is a measure of how much of the sample is active and emitting gamma rays when the data acquisition is on. The spot size is determined by the size of the positron source being used for our specific experiments. We will attempt to measure the spot size of our sodium source in this paper. 5

13 Figure 2.1 Sketch of the comparison between the spot size (yellow) and the sample to be scanned. The spot size is much bigger than the sample (left), the spot size is smaller than the sample size(right). As we can see from figure 2.1 the object can be either bigger or smaller than the spot size. The spot size is assumed to have a Gaussian profile. We create a 2-D Gaussian using the following equation: ( ) ( ) where µ is the center of the spot, and σ is the variance. Figure 2.2 shows the results of the plotting of a 2D Gaussian. Figure 2.2 3D view of a 2D Gaussian (Spot size). 6

14 Object Size We continue our simulation by changing the size of the object and looking at the effect it will produce on the image, while keeping the spot size the same. Figure 2.3 Letter Y convolved with the Spot size having a length of: 2, 12, 20, 36, 50 % of the length of the image length. 7

15 In figure 2.3 are shown few images calculated with different spot size to object size ratios. Our spot size is fixed, so this is the same as reducing the object size. The object will eventually so small compared with the spot size that we will not be able to see anything, like in the last picture of figure 2.3, where the spot size is 50% of the length of the object. Otherwise we have the object being much bigger than the spot size than we are able to clearly see what it is, and the edges of the object will not influence much of our final image. Best Images are obtain if the spot size is less than 20 % of the length of the object. Step Size Every time we talk about a digital image we have to understand that it is made out of finite number of points. The smallest unit of an image is commonly called a pixel. Pixels can be made of different shapes and sizes. They are usually made out of small squares or small circles. A combination of pixels create the final picture. The size of the image is given by two positive integers, and they stand for the number of columns of pixels and the number of rows of pixels. This idea of pixels is related to the step size we are using in imaging. The step size is how far apart from each other are the points where we take the data, or if we want to talk about pixels, it is how many pixels we are going to have for a given length (pixels per length, ppl). If our step size is small compare to the object we are trying to imaging, then we will have a higher ppl, therefore a better quality picture. On the other hand if the step size is big, hence a lower ppl, we might miss some important data or the complete picture. An example of high ppl and low ppl is given in figure

16 As we can see (figure 2.4) there is a point when the spot size is too small compared to the spacing of the sampled points and we completely missed the object (this refers to the last image). In that case the object will not be seen in the image. Figure 2.4 Letter Y with different step size. 9

17 Error bar size The interval of confidence in the S-Parameter is also very important in imaging. Let us assume two values for the S-Parameter like 0.52 (background) and 0.55 (object) (Gagliardi [4]). As the error bar in the data increases compared to the gap of.03 (in this case) between the object and background, there would be a point at which we cannot distinguish the object any more. This point of confidence will be changing based on the number of data points forming the image. As we can notice from figure 2.5, the two points in the left image are clearly different having an error bar of 33% the size of the gap. On the other hand the points in the right image are not clearly distinguishable any more, they have an error bar that is as big as the gap between points. Figure 2.5 Difference between error bar in the S-Parameter. (Left) Distinguishable points error of 33% size gap. (Right) Indistinguishable points, error of 100% size gap. If we have a large sample of data the above case will be different. We can divide our standard deviation by the square root of the number of samples. This gives us an uncertainty if the mean of many samples. We continue the simulation by applying error bars to the letter Y having a spot size of 15% the size of the Y. For the S-Parameter we use the values of 0.55 for the object and the value 0.52 for the background values. The results are show in figure

18 Figure 2.6 Y at different stages of error bars in the S-Parameters. The error percentage % after being divided by the square root on number of samples is respectively: 3, 15, 30, 60,

19 As we can see from figure 2.6, when the error bar increases our image will become less and less recognizable. When the error in the S-Parameter become bigger than 30% of the gap between the background and the object, then the letter will be unrecognizable. We will be using positron annihilation as a tool to scan a prepared surface and create a 3- D digital image of the surface. We will be operating techniques commonly applied in imaging processing and imaging filtering. Some of the techniques used are: Convolution, Filtering, Deconvolution, Fast-Fourier-Transform, Inverse-Fast-Fourier-Transform. 12

20 Experimental Data Introduction In order to take data for a long period of time, we had to create software that will allow us to continuously take data by remotely controlling the experiment. Our biggest challenge was that we needed to switch back and forth between the software taking the data called GAMMA, and the software moving the motors MOTORS. A figure of the two programs used is given in figure 3.1. We let the GAMMA software take data for the chosen amount of time. As soon as the GAMMA software was done taking data, we had to move the motor to a new position and then restart the GAMMA software. The program was developed in Matlab. I personally carried out the writing and the testing of the program. I used the java class called Robot to move remotely the mouse and execute keyboard commands. All of the experiment that were carried out in this paper, were possible thanks to the Matlab code developed. A guide to the Matlab code developed can be found in the Appendix. The positron source used in the following experiments is a 10 Ci Sodium-22. This source produces mainly beta-plus (positrons) radiation. 13

21 Figure 3.1 (Top) - MOTOR Software, (Bottom) GAMMA Software. 14

22 Spatial Resolution Experiment Because we did not have a two dimensional defect distribution to image we conducted experiments with different materials to characterize the imaging capability of the system. We decided we wanted to measure the resolution of our system, therefore we ran another scan. We prepared the Cu plate with half of the plate covered in tape and the other left as Cu, and we scanned the sample with a step size of 1/20 of a mm. The line scan took about 75 hours of machine time. After that we were able to analyze the data (figure 3.2). From figure 5 we can see that the stepping from the Cu plate to the tape took approximately 80 points. Having a step size of 1/20 mm, the transition was 4 mm long. Figure error bar plot of the tape line scan Data Analysis By looking at this data we conclude that the data is a convolution of two functions. We will called the observed data h(x), the sample size f(x), and the sample function g(x). The convolution of the sample with the target data, is mathematically shown as: 15

23 ( ) ( ) ( ) Another way of rewriting the convolution theorem is the following: ( ) ( ) ( ) where f(x) is a Point spread function (Gaussian), g(x) a step function, and h(x) is the error function. ( ) ( ), By making the substitution of f(x) and g(x) we obtain: ( ) ( ) ( ) the first term is 0 so we can rewrite the above as follow: ( ) ( ) By making the proper substitution: ( ) ( ) We can identify: 16

24 We can simplify the integral in: ( ) ( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ( ( ) )) ( ) With the calculation above we just prove that if we take the derivative of h(x) we have back f(x) the Gaussian, this has been done according to Gagliardi [4]. So we proceed by taking the derivative of our data and we obtain the result in figure 3.3. Figure 3.3 Discrete derivative of the data in figure 3.2 We realized that our original data contained too much noise and we could not see anything after taking the derivative. So we decided to smooth the data using a moving average with 30 nearest neighbor averaged (figure 3.4) and then we took the derivative (figure 3.5). 17

25 Figure 3.4 Smoothed data of figure 3.2 (red line) Figure 3.5 Discrete derivative of smoothed data of figure 3.4, and Gaussian Fit of the resultant data. 18

26 Now we can clearly identify the Gaussian. We were able to fit it with the following parameter. ( ) ( ) Where: We then look at the FWHM of the Gaussian and we saw that it was according to Bevington [6]: σ Our step size is 1/20 of a mm, therefore: Thus making the spot size associated with our source 3.47 mm in size. Reproducibility Experiment Since we wanted to scan a large portion of a Cu plate we wanted to see if the data was reliable even after a week of taking data continually. We took to data at the same point by using the Matlab code in the Appendix. We repeated the experiment for 35 data points. We also compared the reliability of the software with the data previously taken [5]. 19

27 Table 1. Standard deviation of the S-Parameter in different type of experiments STD of S-Parameter Short term Long Term Manually moved Software moved In table 1 we can see the Standard deviation of the S parameter as we run different experiments. The short term experiment refers to the taking data of the same spot on a Cu plate after every 15 minutes of counting time. The long term experiment refers to the taking data of the same spot on a Cu plate in this case after a day of wait. In the manually moved experiment the sample has been taken off and then put it back on and analyze again. The last data in Table 1 has been newly added for comparison. We used the Matlab code to move the sample every 15 minutes of counting time. We observed that the standard deviation in the S-parameter is in agreement with what has been previously observed, and much smaller than the manually moved experiment, thus confirming the reliability of using the software to control the moves of the sample. 1-D Scan Experiment The first experiment we ran was across a strip of tape on a copper plate of area of one square inch. We decided to make the step size 1 mm wide. In the line scan we decided to start taking data off the plate, and moving towards it. We wanted to observed what S-parameters we were able to detect in stepping from air to the Cu plate, and from the Cu plate to the tape. This would give us an idea of the resolution of the system. 20

28 Figure error bar plot of the tape line scan The line scan took about 13 hours of machine time. After that we were able to analyze the data (figure 3.6). We clearly could detect the tape strip and the copper plate. The data from point 1 to 12 and 38 to 50 are not the plate but the air. From point 24 to 30 we can clearly see a change in the S-parameter and therefore recognize the tape there. There are clearly three distinguishable point on the tape, point 26, 27 and D Scan Experiment of the Y We decided to take a 2-D scan of the letter Y. The previous experiment gave us confidence that we could construct a letter Y 8 mm wide and have three points on the letter if we take steps of 2 mm in size. Since we wanted to have an error less than 8% of the gap between values, we used a data collection time of 900 seconds for each point. The 2-D scan took approximately 150 hours of machine time. From the plot of the 2-D data displayed in figure 3.7, we can visibly see and recognize the letter Y. 21

29 Figure 3.7 2D plot of the S-parameter of the letter Y Statistics We analyzed the variation in the data of the letter Y to see if the data exhibited the same error as the previous results. We separately analyzed data in the blue region of the Y (background), and the red region of the Y (the object). The data and the errors matched the reproducibility experiment done previously (Table 1). 22

30 Table 2. Mean and standard deviation of the S-parameter in the letter Y Mean (S- Parameter) STD Sample of 77 points in the blue area Sample of 90 points in the red area Image Analysis We wanted to create a smoothed image of the Y. Therefore we first expanded our 27x28 matrix into a 216x224 matrix by using the original points. An example of what we did is given below: 2x2 matrix transform into a 4x4 matrix * + [ ] Then we convolved the spatial frequency of new matrix with the filter disk in Matlab. The disk filter is a circular averaging filter, using a default radius of 5. The result of the frequencies before and after is display in figure

31 Figure 3.8 (left) frequency plot of the letter Y before convolution, (right) frequency plot of the letter Y after being convolved. Then we used the Inverse Fast Fourier Transform to look at the smoothed image of the Y. The result is shown in figure 3.9. Figure D plot of the convolved Y from experiment data. 24

32 We decided to recreate the letter Y by using the same parameter that we found in using our experiment, like so: Figure D plot of simulated Y given specific spot size, error bar and object size. (Top) 28x28 (Bottom) 250x250 25

33 We concluded our experiment by scanning the complete writing of BYU-I. The result of the complete scan can be seen in figure The filtered version is shown in figure Figure D plot of the S-Parameter of the BYU-I writing. Figure D plot of the convolved data of the BYU-I writing.

34 Conclusion This paper clearly demonstrates the possibility of using positron annihilation to image two dimensional distributions of defects using the S-Parameter. We were able to see the effect of: object size, step size, and interval of confidence in the S-Parameter, before trying to image the writing BYU-I. We concluded that with a given spot size, the object size that we are trying to image should not be any smaller than 3 times the spot size. We also determined that in order to see the object the step size of the scanning should be frequent enough such that the size of the step is smaller than the size of our object, so we don t miss it. Finally we decided that the error in the S-Parameter divided by the square root of the number of samples needs to be smaller than 30% the size of the gap between the values of the background and of the object. By using the above guidelines for parameters, we can successfully image an object by using positron annihilation. 27

35 References [1] The Quantum Theory of the Electron, P. A. M. Dirac, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 117, No. 778 (Feb. 1, 1928), pp [2] Positron Annihilation Spectroscopy, R.W. Siegel, Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439, Ann. Rev. Mater. Sci., (1980),10: [3] Mackenzie I.K., Eady J.A. and Gingerich R., R. Phys. Lett., v. 33A, (1970). [4] Gagliardi, Marcus A., Positron Defect Mapping, Dissertation, Idaho State University, (2008), Print. [5] Perry, David, Undergraduate Research Conference, Brigham Young University, Fall (2010). [6] Bevington, Philip R., Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, (1969), 28, Print. 28

36 Appendix A-1 Matlab code to simulate the letter Y. 29

37 A-2 Matlab code to simulate error in the letter Y. 30

38 A-3 Matlab code to simulate the letter Y and the spot size. A-4 Matlab code to initialize the Motor Software. 31

39 A-5 Matlab code to open the Motor Software. A-6 Matlab code to initialize the Board. A-7 Matlab code to open the port of the Motor Software. 32

40 A-8 Matlab code to view the address of the Motor Software. 33

41 A-9 Matlab code to set the winding current for the Motor Software. 34

42 A-10 Matlab code to set up parameter in motor 1. 35

43 36

44 A-11 Matlab code to set up parameter in motor 2. This code is the same as the one for motor 1 in A-10, the only difference is this top part. A-12 Matlab code to set up parameter in motor 3. This code is the same as the one for motor 1 in A-10, the only difference is this top part. A-13 Matlab code to close all of the port in the Motor Software. 37

45 A-14 Matlab code to close the Motor Software. A15 Matlab code to switching from the Gamma and the Motor Software, using two part of the screen. 38

46 A-16 Matlab code to start the Jacopo sequence in Gamma Software. A-17 Matlab code to send to my account. 39

47 A-18 Matlab code to image 2-D. 40

48 41

49 A-19 Matlab code to filter images. 42