DETERMINING THE ISOTOPIC CONCENTRATION OF URANIUM FROM VECTOR REPRESENTATION OF THE GAMMA SPECTRUM

Transcription

1 DETERMINING THE ISOTOPIC CONCENTRATION OF URANIUM FROM VECTOR REPRESENTATION OF THE GAMMA SPECTRUM A Thesis submitted to the Faculty of the Graduate School of Arts and Sciences of Georgetown University in partial fulfillment of the requirements for the degree of Master of Science in Health Physics By Tristan Glover White, B.S. Washington, DC July 10, 014

2 Copyright 014 by Tristan Glover White All Rights Reserved ii

3 DETERMINING THE ISOTOPIC CONCENTRATION OF URANIUM FROM VECTOR REPRESENTATION OF THE GAMMA SPECTRUM Tristan G. White, B.S. Thesis Advisor: Gary W. Phillips, Ph.D. ABSTRACT Gamma emissions from Uranium-35 in a source of interest were compared to gamma emissions from Protactinium-34m (which is in equilibrium with Uranium-38) in order to determine the isotopic composition of the source. The 144 kev gamma ray from U-35 was compared with 1001 kev gamma ray from Pa-34m. Two analytical methods were compared: the relative activity method and the vector representation method. The relative activity method is similar to the (standard) relative intensity method, but accounts for more variables. Calculations were performed using both methods in order to evaluate precision and accuracy. Relative activity compares the number of counts under one gamma-ray peak from a reference source to the number of counts under another peak from an unknown source. This method is sensitive to systematic errors in the efficiency calibration of the detector when two different peaks with different energies are used. Vector representation compares the count ratio of two gamma-ray peaks from one source to the count ratio of the same two gamma-ray peaks from another source. Vector representation was found to be practical for analyzing depleted uranium, but not highly enriched uranium (HEU), due to different branching ratios and detector efficiency. iii

4 TABLE OF CONTENTS Chapter 1: Introduction...1 Statement of the Problem....1 Rationale...3 Chapter : Methodology...5 Equipment...5 Cabling diagram...6 Procedure...7 Chapter 3: Results...9 HEU data...9 Uranyl acetate data...11 Uranium ore data...15 Chapter 4: Relative Activity Method Equation and algorithm...17 Activity of U-35 in the uranyl acetate source...19 Moles of uranium in the uranyl acetate source...19 Moles of U-35 and U-38 in the uranyl acetate source...0 Activity of uranium ore source...1 Conclusions about the relative activity method... Chapter 5: Vector representation...4 Choosing the best gamma-ray peaks...4 iv

5 Defining the orthogonal vector basis....6 Counts per minute under each peak for each source....7 Count ratios Concentration of U-35 in the uranyl acetate source...9 linear fit standard deviation 95% Confidence Interval Graphical representation of vectors...31 Uranyl acetate 0 hours Uranium ore 0 hours Optimization, extrapolation, and limitations...33 Chapter 6: Conclusion...36 APPENDIX A: Energy Calibration...39 A1 -- Low-energy calibration...39 A -- High-energy calibration...44 APPENDIX B: Efficiency Calibration...47 B1 Manufacturer s specifications...47 B Efficiency calibration using spectra from sources...49 B3 Efficiency calibration using the background spectrum...5 B4 Combined efficiency calibration from sources and background...54 v

6 B5 Detector efficiency for gamma rays from uranium sources...57 APPENDIX C: Calculations for the Relative Activity Method...59 C1 Data and equation used for relative activity calculations...59 C Analysis of the HEU spectrum...60 C3 Relative intensity of two sources containing U C4 Gamma rays from Thorium C5 Relative activity of nuclides in the uranyl acetate source...66 C6 Specific activities of U-35 and U APPENDIX D: Interference from Radium D1 Spectrum from uranyl acetate source with Ra-6 source...69 D Tabulated data from spectrum D3 Calculating the activity and concentration of U APPENDIX E: Discussion of Alpha Spectroscopy...73 APPENDIX F: Background...75 REFERENCES...78 vi

7 CHAPTER 1: INTRODUCTION STATEMENT OF THE PROBLEM: The purpose of this investigation was to determine the most accurate method for calculating the concentration of Uranium-35 from the gamma spectrum of a uranium source. Natural uranium has a U-35 component of 0.7% and a U-38 component of approximately 99.3%. Uranium enrichment results in two products: an enriched product and a depleted product. In the enriched product, the U-35 component has been enriched to a value of 3-5% for use in light water reactor fuel. In weapons grade uranium it has been enriched to more than 90%. In the depleted product the U-35 component is less than the natural value, typically between 0.1% and 0.3%. A high-purity Germanium (HPGe) detector was used to obtain high-resolution spectra of gamma-ray emissions from 3 uranium sources: a highly enriched uranium (HEU) source, a uranium ore source, and a uranyl acetate source. (The activities and isotopic compositions of these sources are listed in Chapter.) A multichannel analyzer (MCA) coupled to a computer sorted the data into pulse-height spectra containing peaks corresponding to gamma rays omitted from each source. The detector was calibrated for detection efficiency versus gamma-ray energy using spectra from known sources (see Appendix B). MAESTRO-3 software (MAESTRO-3 software user s manual) was used to analyze the pulse-height spectra. Peak information was tabulated in a Microsoft Excel spreadsheet. 1

8 Two analytical methods were compared: the relative activity method and the vector representation method. The relative activity method compares the number of counts under one peak from one source to the number of counts under another peak from another source. The branching ratios of each gamma ray and the efficiency of the detector for each gamma ray must be included in the calculation. The relative activity method is sensitive to systematic errors in the efficiency calibration of the detector when two different peaks with different energies are used. The vector representation method compares the count ratio of two gamma-ray peaks from a reference source to the count ratio of the same two gamma-ray peaks from an experimental source. Vector representation avoids errors in the efficiency calibration of the detector as long as the same detector and geometry are used for each measurement. In order to optimize the level of precision, the gamma-ray spectra were first evaluated using the relative activity method to identify the peaks with the best statistics and the least interference. Different combinations of gamma-ray peaks were exhaustively tested, including peaks from Thorium-34 and Protactinium-34m. These are both short-lived decay products of U-38 and are therefore in secular equilibrium with the U-38 activity. The vector representation method is similar to the method used by Forney in 01 (Forney, 01), although different gamma-ray peaks are analyzed here to determine the most accurate way to represent the gamma spectrum as a vector. A background radiation spectrum was acquired for a live time of 0 hours, and this spectrum was used as a reference vector. Pa-34m was detected in the background spectrum and U-35 was not detected in the background spectrum. An orthogonal vector basis was defined

9 with the 1001 kev peak intensity from Pa-34m on the horizontal axis and the 144 kev peak intensity from U-35 on the vertical axis. The vector algebra is different from that used by Forney in 01. For each source analyzed, the angle φ between the vector and the positive horizontal axis was calculated by taking the inverse tangent of the count ratio of the two gamma-ray peaks. RATIONALE: U-35 has a higher probability for gamma-ray emission than Pa-34m, so gamma spectroscopy can detect a smaller concentration of U-35 relative to U-38. This is the case with depleted uranium. Uranium enrichment is usually done using a cascade of centrifuge enrichment units, taking advantage of the difference in mass between U-35 and U-38. The enriched output of each centrifuge is fed up to the next unit in the cascade while the depleted output is fed down into the preceding unit. Each unit results in a slight enrichment. Typically a large number of units are connected in a cascade (Wilson, 1996). The International Atomic Energy Agency s verification efforts are now focused on evaluating a country s entire nuclear program; including the entire fuel cycle from ore to enrichment to waste (IAEA 007). Accountancy involves keeping track of all the isotopes during every stage of the fuel cycle (Baeckmann, et. al. 1995). If the element being enriched is uranium, then the fissile isotope of interest is Uranium-35. Nuclear forensics is a timeconsuming process that must account for all of the U-35 going into an enrichment facility and coming out of an enrichment facility. 3

10 When access to the enriched product is denied or deception is suspected, the concentrations of U-35 in the feed and the tails can be used to calculate the concentration of U- 35 in the enriched product. (More samples of depleted uranium may be needed if the concentration of U-35 in the tails is not constant.) The feed is the input to the enrichment process, and the tails are the (depleted) waste product. Analyzing the mass and isotopic composition of the feed and tails is most important when a country is not honest about its enrichment program. The feed is usually natural uranium containing 0.704% U-35. The same amount of feed material can be enriched to a different level, depending on the number of separative work units (SWU). Separative work units are a measure of the amount of energy required to separate isotopes in a cascade of centrifuges (Federation of American Scientists website). The concentration of U-35 in the tails can be used to calculate the number of separative work units and the concentration of U-35 in the enriched product (assuming the masses are known). Reactor fuel is typically enriched to 4% U-35, and the tails are typically depleted to 0.3% U-35 (Wilson, 1996). Weapons grade uranium (WGU) is enriched to at least 90% U-35, and the tails may be depleted to as low as 0.1% U-35. It takes more separative work units to deplete the tails down to 0.1% U-35. 4

11 CHAPTER : METHODOLOGY EQUIPMENT: ORTEC GEM10P4-70 Coaxial Germanium Detector System ORTEC 459 Detector Bias Supply ORTEC 67 Spectrosocpy Amplifier ORTEC 480 Pulser Digital Oscilloscope NIM Bin and Power Supply coaxial cables and BNC Tee Connector Easy-MCA-8K System MAESTRO-3 software sources: o HEU source 97.7% Uranium nCi (3707Bq) Isotope Products Laboratories o uranyl acetate powder 17.44μCi, unknown isotopic composition o uranium ore sample 0.704% U-35, unknown activity o Cesium-137 1μCi Spectrum Techniques o Cobalt-60 1μCi Spectrum Techniques o Americium μCi (45Bq) Isotope Products Laboratories o Cobalt-57 1μCi Spectrum Techniques o Cadmium-109 1μCi o Radium-6 0.9μCi Amersham/Searle model (The cabling diagram is shown below in Figure a.) 5

12 Figure a Cabling diagram used in the measurements. 6

13 PROCEDURE: The HPGe detector was cooled to the correct operating temperature with liquid nitrogen. The detector output pulse was shaped with a pre-amplifier. The pre-amplifier output was connected to a spectroscopy amplifier. The output of the amplifier was sent to the multichannel analyzer (Easy MCA). The MCA output was sent to the computer running MAESTRO-3 analysis software. A digital pulse-height spectrum was accumulated, with the counts in each channel equal to the number of gamma rays detected with the corresponding amplitude. The amplifier gain was set to 50 for the first part of the experiment. Cesium-137 and Cobalt-60 were used for energy calibration. With the same amplifier setting, the energy calibration varied by less than 0.3 kev over two weeks. Most of the data were gathered with this amplifier setting. Energy calibration was repeated every day. Efficiency calibration was done once. Spectra were acquired for all the gamma sources listed above in the equipment section. After weeks, the amplifier gain was reduced to 0. This adjustment was necessary to identify the background sources which had high-energy peaks. The calibration was repeated with the lower gain setting on the amplifier. The spectra were analyzed using MAESTRO-3 software (MAESTRO-3 software user s manual). Raw data were saved in native MAESTRO-3 file format. Backup data were saved in ASCII format, which is compatible with Microsoft Office. The data were transferred into a Microsoft Excel spreadsheet and graphed. Peak channels were found in MAESTRO-3 and double-checked in Microsoft Excel. The same algorithm was used to analyze each peak. The net areas of each peak were calculated in MAESTRO-3, which has an algorithm for fitting a peak to a Gaussian curve 7

14 (MAESTRO-3 software user s manual). MAESTRO-3 overestimates the standard deviation, so these calculations are a conservative estimate of precision. To ensure correct interpretation of data, the results were analyzed manually as described below and in more detail in Chapter 4 and in Appendix C. Each spectrum consisted of a continuum due to scattered gamma rays in the environment and to gamma rays which did not deposit all their energy in the detector. The peaks in the spectrum were observed rising above this continuum. To obtain the net peak area, the continuum under the peak had to be subtracted. In some cases, systematic errors were identified with the software algorithm used in MAESTRO-3 to calculate net area. The calculations were repeated manually in Microsoft Excel. Manual calculations of net area require choosing the correct region of interest (ROI), calculating the gross area of the ROI, and subtracting the average continuum calculated on either side of the peak from the gross area. 8

15 counts CHAPTER 3: RESULTS HIGHLY ENRICHED URANIUM DATA (97.7% U-35): This section discusses gamma-ray spectra and peak information. Spectra were analyzed using the calibration discussed in Appendices A and B. The 100. nci HEU source is inside a plastic disc 0.5 cm thick and.5 cm diameter. The bottom of the source was placed approximately 5.5 cm above the detector, so that the center of the source was approximately 5.75 cm above the detector. Its spectrum is shown in Figures 3a.i.and 3.a.ii U-35 spectrum Energy (kev) Figure 3a.i 100. nci HEU source full spectrum for 10 hours at 5.75cm (on a logarithmic scale) 9

16 counts U-35 spectrum 186keV keV 163keV 05keV Energy (kev) Figure 3a.ii 100. nci HEU peak channels for 10 hours at 5.75 cm The data from the U-35 peaks are tabulated in Table 3a. Table 3a -- Peak data from HEU spectrum: measured Energy (kev) predicted Energy (kev) branching ratio (%) gross area net area σ σ (%) 10 FWHM (kev) de/e (%) ,847 65, , , ,38,339 1,160,668 1, ,8 38, ,601 97, n/a n/a Table 3a peak data from the HEU spectrum for 10 hours at 5.75 cm MAESTRO-3 incorrectly calculated the net area for the 05 kev peak (see below). The incorrect data are shown in italics in row 4. The calculation was repeated manually. The corrected data are shown in row 5. The first column of Table 3a is the measured gamma-ray energy using that day s calibration. The second column is the expected energy according to the Table of Isotopes. The third column lists the branching ratio for each gamma ray. The fourth column lists the gross area under the peak calculated by the MAESTRO-3 spectral analysis program. A wide region of interest leads to a large gross area. The fifth column

17 lists the net area under the peak. MAESTRO-3 lists the net area plus or minus the standard deviation σ. The sixth column is the standard deviation expressed as integer counts. It is important to note that Poisson statistics predict the standard deviation as the square root of the gross area. MAESTRO-3 chooses an arbitrarily wide region of interest and overestimates the gross area. The standard deviation calculated in MAESTRO-3 is greater than the standard deviation would be if one chose only the peak channels in the region of interest. The seventh column of Table 3a is the percent standard deviation (the standard deviation divided by the net area) expressed in units of per cent. The eighth column is the FWHM of the peak. The ninth column is the energy resolution of the peak (the FWHM of the peak divided by the measured energy given as a percent). The 144 and 163 kev gamma-ray peaks from U-35 had no interference. The 186 kev gamma-ray peak from U-35 is convoluted with a 183 kev peak from the same nuclide (and another 186 kev peak from Ra-6). The detector could not resolve these two peaks. The 05 kev gamma-ray peak from U-35 is close to a 0 kev peak from the same nuclide. The detector was able to fully resolve both of these gamma rays from U-35, but the MAESTRO-3 software erroneously included both peaks in the same ROI. The net area of the 05 kev peak had to be calculated manually, as discussed in Appendix C. URANYL ACETATE DATA: The uranyl acetate source was a yellow powder in a petri dish approximately 7. cm in diameter. The handwritten label read uranyl acetate µci. The manufacturer was not listed on the label. This source was provided by Georgetown University s Environmental Health 11

18 & Safety Office (Forney, 01). The isotopic composition of the uranyl acetate source was not listed on the label. Calculations are required in order to determine the concentration of U-35 in this source (see the following chapter on relative activity). The label does not indicate whether the source contains natural uranium or depleted uranium. Figure 3b.i shows the decay scheme for uranium-38, which is in equilibrium with its short-lived decay products Thorium-34 and Proactinium-34. These undergo beta decay accompanied by gamma-ray emission. Note that 99.87% of the decay of Pa-34 comes from the isomeric state 75 kev above the ground state. This state is referred to as Pa-34m. A spectrum was acquired for the uranyl acetate source. The 7.3 cm diameter source was placed 30 cm away from the detector in order to use the point source approximation. A live time of 0 hours was necessary for good statistics. Its spectrum is shown in Figures 3b.ii and 3b.iii. 1

19 Figure 3b.i decay scheme for Uranium-38, Thorium-34, Protactinium-34m, and Protactinium-34. (Browne, et. al. 1986) 13

20 counts counts Uranyl acetate spectrum Energy (kev) Figure 3b.ii Uranyl acetate μci source for 0 hours at 30 cm, full spectrum (on a log scale) Uranyl acetate spectrum 1001keV keV keV 786keV Energy (kev) Figure 3b.iii -- Expanded region of above spectrum showing characteristic gamma-rays from Pa-34m The peak data from the spectrum in Figures 3b.i and 3b.ii are tabulated in Table 3b. 14

21 Table 3b uranyl acetate peak data nuclide measured Energy (kev) predicted Energy (kev) branching ratio (%) gross area net area σ σ (%) FWHM (kev) de/e (%) Th , ,79 1, Th ,555 7, N/A N/A U ,787 16, U ,136 8, U ,135 95, U ,589 3, Pa-34m ,46, Pa-34m ,99 9, Pa-34m ,764 1, Pa-34m ,399 17, Table 3b -- tabulated data from the µci uranyl acetate spectrum for 0 hours at 30 cm URANIUM ORE DATA: The uranium ore sample was a gray rock with yellow discoloration. The widest dimension of the rock was approximately 11 cm. The center of the rock was placed approximately 45 cm above the detector. The activity of the ore sample was unknown, but the container (the plastic bag) labeled the dose rate as 0. mr/hr. The spectrum from the uranium ore sample is shown in Figure 3c. 15

22 counts Uranium ore spectrum Energy (kev) Figure 3c Uranium ore sample full spectrum (on a logarithmic scale) at 45 cm for 0 hours The data from Figure 3c are tabulated in Table 3c. Table 3c -- uranium ore peak data nuclide measured Energy (kev) predicted Energy (kev) branching ratio (%) gross area net area σ σ (%) FWHM (kev) Th ,941 9, Ra ,133 7, Pa-34m , U ,039, Table 3c -- tabulated data from the uranium ore spectrum at 45cm for 0 hours de/e (%) 16

23 CHAPTER 4: RELATIVE ACTIVITY METHOD This chapter has two purposes: to calculate the concentration of U-35 in unknown samples; and to determine which peaks are most useful for these calculations. EQUATION AND ALGORITHM: Let A equal the activity of the unknown source and A 1 equal the activity of the reference source. Let %I equal the branching ratio of the gamma ray. Let r equal the distance between the source and the detector. The activity of an unknown source can be calculated from the activity of the reference source using the following equation: A cpm % I1 cpm 1 % I efficiency efficiency 1 r r 1 A 1 Table 4a compares the gamma-rays observed from the uranyl acetate source to those from the HEU source. The column labeled branching ratio (%) is the tabulated branching fraction for the decay of the given nucleus leading to that particular gamma ray (Table of Radioactive Isotopes). The last column is the relative efficiency of the detector at a given energy compared to the efficiency of a standard 3 diameter by 3 thick sodium iodide scintillation detector at 133 kev. Efficiency calibration is discussed in more detail in Appendix B. 17

24 Gamma Rays from the uranyl acetate source, acquired for 100 minutes nuclide Energy (kev) cpm above background σ (%) branching ratio (%) relative efficiency (%) Th Th Th U U U U U Pa-34m Pa-34m Pa-34m Pa-34m Gamma rays observed from the HEU source, acquired for 600 minutes nuclide Energy (kev) cpm above background σ (%) branching ratio (%) 18 relative efficiency (%) U U U , U U Table 4a data for relative activity calculations Rows in italics are incorrect due to a systematic error in the software algorithm. The corrections are the rows underneath the italicized row. Exhaustive calculations using different combinations of peaks as the reference source and unknown source are detailed in Appendix C. The calculations using the 1001 kev gamma ray from Pa-34m as a reference point are the most accurate. The calculations using the 113 kev gamma-ray peak from Th-34 are not accurate, because this peak is convoluted with uranium characteristic x-rays. Calculations using the HEU source have systematic errors arising from the point-source approximation.

25 ACTIVITY OF URANIUM-35 IN THE URANYL ACETATE SOURCE: The activity of U-35 in the uranyl acetate source was calculated using the 1001 kev gamma ray from Pa-34m. This gamma ray was compared to the 144 kev, 163 kev, and 186 kev gamma rays from U-35. The results from these calculations are in Table 4b. Table 4b activity of U-35 Activity of U-35 (µci) σ (µci) σ (%) 144 kev peak % 163 kev peak % 186 kev peak % average % Table 4b average estimate of activity of U-35 in the uranyl acetate source. The isotopic composition of the uranyl acetate source was calculated from the activities of U-35 and U-38. The calculation had to be repeated because the Table of Isotopes lists the relative abundance for natural uranium, not depleted uranium. The only isotopes that should be considered are U-38 and U-35. MOLES OF URANIUM IN THE URANYL ACETATE SOURCE: The specific activity of U-35 is 508 µci/mol, and the specific activity of U-38 is 80 µci/mol (as calculated in Appendix C). The natural abundance of U-38 is 99.3%. (This number will not be accurate after separation of isotopes.) The number of moles of uranium can be calculated as follows, assuming natural abundance. 18 mmol U mmolu The abbreviation mmol denotes millimoles of uranium. The activity of U-35 in the uranyl acetate source was determined from the relative activity calculations in Table 4b. The 19

26 number of moles of U-35 was calculated. For natural uranium, the concentration of U-35 should be 0.704%. The molar ratio of U-35 to all uranium was calculated. These calculations are shown in Table 4c.i. Table 4c.i calculating the number of moles of U-35 and U-38 reference source reference nuclide A(U-35) in acetate A(U-38) in acetate mmol U-35 mmol U-38 mmol U total % U-35 acetate Pa-34m 0.1μCi 17.44μCi % ± % Table 4c.i molar ratio of U-35/U-38 in the uranyl acetate sample The notation ±0.0056% is used to express the standard deviation, σ. The 95% confidence interval is within standard deviations of the mean. The 95% CI for the molar ratio of U-35 in the uranyl acetate source is calculated in Table 4c.ii. Table 4c.ii calculating the 95% CI reference [U-35] - σ % U-35 [U-35] + σ nuclide Pa-34m % % 0.108% Table 4c.ii 95% confidence interval for the concentration of U-35 in the uranyl acetate sample MOLES OF URANIUM-35 AND URANIUM-38 IN THE URANYL ACETATE SOURCE: The natural abundance of U-35 is 0.7%. The calculations in Tables 4c.i and 4c.ii show that the uranyl acetate sample is depleted uranium. Assume that the concentration of U-38 is 100% minus the concentration of U-35. The calculations in Tables 4d.i and 4d.ii assume the sample contains 18 millimoles (mmol) of uranium instead of mmol. The results are slightly different because there are no other isotopes besides U-38 and U-35. Table 4d.i calculating the molar ratio of uranium isotopes in a depleted uranium source reference reference A(U-35) A(U-38) mmol mmol % U-35 source nuclide in acetate in acetate U-35 U-38 acetate Pa-34m 0.1μCi 17.44μCi % ± Table 4d.i repeated calculations of molar ratio of U-35/U-38 0

27 The notation ±0.0056% is used to express the standard deviation, σ. The 95% confidence interval is within standard deviations of the mean. The 95% CI for the molar ratio of U-35 in the uranyl acetate source is calculated in Table 4d.ii. Table 4d.ii calculating the 95% CI [U-35]-σ % U-35 [U-35]+σ Pa-34m % % 0.118% Table 4d.ii repeat 95% confidence interval for the concentration of U-35 in the uranyl acetate sample The results in Tables 4c.ii and 4d.ii are in agreement within two standard deviations. The uranyl acetate sample contains depleted uranium, i.e., tails from the enrichment process. Tails from enrichment of reactor fuel have approximately 0.3% U-35 in order to minimize the cost and the number of separative work units (SWU). It takes a lot of SWU s to get down to 0.11% U-35. The number of SWU s is one of the variables used for calculating the U-35 concentration in the enriched product. ACTIVITY OF URANIUM ORE SOURCE: The data in Table 4e were used to calculate the activity of the uranium ore source. The net area of the 1001 kev peak from the uranium ore source is compared to the net area of the 1001 kev peak from the uranyl acetate source. Table 4e comparing the 1001 kev peak from natural uranium and depleted uranium counts per minute σ (%) distance (cm) Activity (μci) (1001 kev peak) uranyl acetate % uranium ore % 45 unknown Table 4e data used to calculate the activity of the uranium ore source 1

28 The activity A of the uranium ore can be calculated from the known activity A 1 of the uranyl acetate as follows: A cpm r A1 cpm 1 r 1 Using the data from the Table, A cpm 45cm cpm 30cm 17.44Ci Ci The overall standard deviation is given by: 0.98% 16.5% 16.8% Ci 16.8% Ci A 95% confidence interval is plus or minus : μCi < Activity < μCi CONCLUSIONS ABOUT THE RELATIVE ACTIVITY METHOD The relative activity method gives a lot of information about the source, but is very time consuming. A quicker method is needed. Table 4f list of gamma rays measured from U-35 gamma ray energy (kev) advantages disadvantages 144 good statistics 163 small branching ratio, poor statistics 186 good statistics convoluted with 186 kev gamma ray from Ra-6 05 convoluted with 0 kev gamma ray from U-35 Table 4f evaluating the advantages and disadvantages of each gamma ray measured from U-35

29 The 186 kev peak from U-35 has good statistics, but it is convoluted with the 186 kev gamma ray from Ra-6 in background. Convolution would be even worse should someone deliberatively use a Ra-6 source to trick the detector (as discussed in Appendix D). The 144 kev peak from U-35 is not convoluted with background radiation gamma rays. This peak has the second-best statistics after the 186 kev peak. The relative branching ratios for decay of U-35 are 57% for the 186 kev gamma ray and 11% for the 144 kev gamma ray (Table of Isotopes). The following chapter will discuss the vector representation. 3

30 CHAPTER 5: VECTOR REPRESENTATION Vector representation expresses the counts from two gamma-ray peaks as a vector with the counts in each peak projected along orthogonal axes. This is a graphical representation of the count ratios of the two peaks. In the case of uranium, one of the gamma rays is from Uranium- 38 progeny, and one of the gamma rays is from Uranium-35. CHOOSING THE BEST GAMMA RAY PEAKS: There are several possible gamma rays to choose for the axes. The chosen peaks must have good statistics and no interference from other gamma rays nearby in energy. As shown in Chapter 4, Appendix C, and Appendix D, the 144 kev gamma ray from Uranium-35 best satisfies these requirements. The 186 kev gamma ray from U-35 was not chosen due to interference from a background Radium-6 gamma ray at 186 kev. No background peaks were found at 144 kev, and the resolution of the HPGe detector is high enough to resolve this peak above the continuum. The advantages and disadvantages of analyzing the gamma-ray peaks from U-35 are listed in Table 5a.i. Table 5a.i list of gamma rays measured from U-35 gamma ray energy (kev) branching ratio (%) advantages disadvantages good statistics small branching ratio convoluted with 186 kev gamma ray 186 good statistics from Ra convoluted with 0 kev gamma ray from U-35 Table 5a.i evaluating the advantages and disadvantages of each gamma ray measured from U-35 4

31 Uranium-38 is in equilibrium with its decay products Thorium-34 and Protactinium- 34m. The calculations in Appendix C show that large errors were found in calculations using the 63.3 kev peak from Thorium-34; and that the 113 kev gamma-ray peak from Th-34m is convoluted with the 111 kev characteristic x-ray peak from uranium. The disadvantages of using the gamma rays from Th-34 are listed in Table 5a.ii. Table 5a.ii list of gamma rays measured from Th-34 Energy (kev) branching ratio (%) advantages disadvantages good statistics unknown detector response at low energies good statistics convoluted with 111 kev x-ray from uranium Table 5a.ii evaluating the disadvantages of the gamma rays measured from Th-34 Despite the low branching ratio for the 1001 kev gamma ray from Pa-34m, this peak is still useful because the background continuum is very low at energies above 1001 kev. There is a small background peak at 1001 kev due to natural uranium in the environment with a net count rate of cpm. This background count rate can be subtracted from the count rate of the Pa- 34m source being analyzed. The advantages and disadvantages of analyzing the gamma-ray peaks from Pa-34m are listed in Table 5a.iii. Table 5a.iii list of gamma rays measured from Pa-34m Energy (kev) branching ratio (%) advantages disadvantages poor statistics convoluted with 768 kev gamma ray from Bi poor statistics and convoluted with 786 kev gamma ray from Bi good statistics and low interference background Pa-34m Table 5a.iii evaluating the advantages and disadvantages of each gamma ray measured from Pa-34m 5

32 Thus, the 144 kev peak from U-35 and the 1001 kev peak from Pa-34m are the most useful peaks for vector analysis to determine the concentration of U-35 in a uranium sample. DEFINING THE ORTHOGONAL VECTOR BASIS: Let a -dimensional vector v have a horizontal component and a vertical component. The horizontal component of the vector is the count rate of the 1001 kev peak from Pa-34m, and the vertical component of the vector is the count rate of the 144 kev peak from U-35. v iˆ, cpm ˆ) ( cpm1001 kev 144keV j The unit vectors i and j form an orthonormal vector basis on the horizontal and vertical axes, respectively. The count ratios are expressed as vectors with the count rate of the 1001 kev peak as the horizontal component and the count rate of the 144 kev peak as the vertical component, using the net counts above the background continuum. Two feature vectors form an orthogonal basis. The background spectrum has a peak at 1001 kev, and no peak at 144 kev above the continuum. The HEU spectrum has no peak at 1001 kev above the continuum, and a peak at 144 kev above the continuum. The peak counts rates from these vectors lie along the orthogonal axes as shown in Table 5b. Vector representation is another way of expressing the count ratio. The vector space is the first quadrant of a Cartesian coordinate plane. Let φ equal the angle between a vector v and the horizontal axis. The count rate for each peak is expressed in counts per minute (cpm). Table 5b defining the basis for the vector space ORTHOGONAL BASIS horizontal component vertical component background spectrum greater than 0 0 HEU spectrum 0 greater than 0 Table 5b orthogonal basis for vector representation 6

33 The background spectrum is on the horizontal axis, and the HEU spectrum is on the vertical axis. The ordered pairs are expressed as (i, 0) for the background spectrum and (0, j) for the HEU spectrum. The letters i and j are used to represent the horizontal and vertical axes respectively. The unit vectors i and j are defined as the feature vectors divided by their magnitude. The unit vectors i and j form an orthonormal basis. Vector components from all other uranium spectra are expressed as the count rates multiplied by the unit vectors. Using this orthogonal basis, the positive axes of the Cartesian coordinate plane can be represented by real data from the spectra. Vectors from all other uranium spectra are in the first quadrant of the Cartesian coordinate plane. tan( ) cpm cpm 144keV 1001keV COUNTS PER MINUTE UNDER EACH PEAK FROM EACH SOURCE: Background radiation in the environment causes interference in the spectra from experimental sources. A background spectrum was acquired for a live time of 0 hours when there were no sources near the detector. The net area of the background 1001 kev peak must be subtracted from the net area of the 1001 kev peak from each of the uranium sources, as shown in Tables 5c.i and 5c.ii. This calculation must be done manually. Tables 5c.i and 5c.ii list the data from the 1001 kev and 144 kev peaks from the uranyl acetate source and the uranium ore source, respectively. 7

34 Table 5c.i peak data from the depleted uranyl acetate spectrum Gamma ray net area minutes cpm background cpm Net cpm Count ratio σ (%) overall σ (%) overall σ (cpm) 144 kev 16, % 4.97% kev 17, % Table 5c.i -- uranyl acetate 0-hour spectrum data Table 5c.ii peak data from the natural uranium ore spectrum Gamma ray net area minutes cpm background cpm Net cpm Count ratio σ (%) overall σ (%) overall σ (cpm) 144 kev, % 30.5% kev % Table 5c.ii -- ore 0-hour spectrum data COUNT RATIOS: The count ratio is the ratio of net counts under the 144 kev and 1001 kev peaks. MAESTRO-3 calculates the σ (standard deviation) for each peak. Dividing the standard deviation by the net area expresses the standard deviation as a percentage. The percent standard deviations for each peak are summed in quadrature to give the overall σ in units of percent. The far right column of Table 5c.ii is the absolute of the count ratio (expressed as a decimal, not a percentage). The 95% confidence intervals for these count ratios are calculated in Tables 5d.i and 5d.ii. Table 5d.i 95% CI for the count ratio of the uranyl acetate spectrum acetate ratio - σ count ratio ratio + σ σ (%) σ σ 0 hours % Table 5d.i count ratio with 95% confidence interval for uranyl acetate 0-hour spectrum Table 5d.ii 95% CI for the count ratio of the uranium ore spectrum ore ratio - σ count ratio ratio + σ σ (%) σ σ 0 hours % Table 5d.ii count ratio with 95% confidence interval for uranium ore 0-hour spectrum 8

35 ratio (144keV/1001keV) CONCENTRATION OF URANIUM-35 IN THE URANYL ACETATE SOURCE: The count ratio is directly proportional to the concentration of U-35. The count ratio from the natural uranium ore spectrum was used to make a linear equation relating count ratio to U-35 concentration. Here the concentration of U-35 in the uranyl acetate sample is treated as an unknown variable. Figure 5e shows a linear fit using the uranium ore count ratio from Table 5d.ii as a reference point. Ore spectrum linear fit 8 7 y = x % 0.0% 0.40% 0.60% 0.80% %U-35 Figure 5e linear fit using only the natural uranium ore data as a reference point (0.704% U-35) Using the linear equation from the ore sample, one can double-check the concentration of U-35 in the depleted uranyl acetate sample. The calculation was already performed using the relative activity method. Comparing the count ratios is another way to calculate the concentration of U-35. In Figure 5e, x is the concentration of U-35 and y is the count ratio. A count ratio of 0.97 (from the 0-hour uranyl acetate spectrum) corresponds to a U-35 concentration of %. 9

36 There is only one data point (other than the origin) used for the linear fit calculation in Figure 5e, so the percent standard deviation of the linear fit is the same as the percent standard deviation for that one data point. The percent standard deviation of the ore count ratio is 30.5%. The percent standard deviations from the ore count ratio and the uranyl acetate count ratio are summed in quadrature as follows to calculate the overall σ: 30.5% 4.97% 30.66% This is a high percent standard deviation because data for the uranium ore source had poor statistics. A problem with the count ratio method and the vector representation method is the statistical uncertainty of the reference source spectrum. The calculations in Table 5e.ii determine the concentration of U-35 in the depleted uranyl acetate source by comparing the count ratio to that of the natural uranium ore source. The high percent standard deviation in the estimate of the concentration of U-35 comes from the high percent standard deviation in the net areas of the peaks from the natural uranium ore source. Table 5e.ii count ratio with standard deviation Count ratio %[U-35] σ σ uranyl acetate 0 hours % 0.098% % Table 5e.iii calculating the standard deviation for the count ratio method The high standard deviation from Table 5e.ii leads to a wide confidence interval in Table 5e.iii. Table 5e.iii calculating the 95% CI for the concentration of U-35 %[U-35] - σ %[U-35] %[U-35] + σ uranyl acetate 0 hours % % % Table 5e.iii 95% confidence intervals for the concentration of U-35 in the uranyl acetate source 30

37 cpm under 144 kev peak GRAPHICAL REPRESENTATION OF VECTORS: Figure 5f shows graphical representations of vectors for the depleted uranium and natural uranium spectra. 15 Vectors from uranium gamma spectra 10 depleted uranium 5 natural uranium cpm under 1001 kev peak Figure 5f vectors from depleted uranyl acetate and natural uranium ore spectra depleted uranium: % U-35; μci U-38; 30 cm from detector natural uranium: 0.704% U-35; μci U-38; 45 cm from detector The vectors in Tables 5f.i and 5f.ii are not normalized. The magnitude of each vector depends on the activity of the source and the distance between the source and the detector. The uranyl acetate vector is longer than the uranium ore vector because of the activity and distance. Let φ equal the angle between a vector and the positive horizontal axis. The angle φ is calculated in Table 5f as the inverse tangent of the count ratio from Tables 5d.i and 5d.ii. The count ratio has standard deviation. Uncertainty in the angle φ cannot be expressed as standard deviation because the inverse tangent function is non-linear. The lower bound of φ is the inverse 31

38 tangent of the count ratio minus standard deviations. The upper bound of φ is the inverse tangent of the count ratio plus standard deviations. The first row of Table 5f is theoretical because there are no data from a source containing 0% U-35. Table 5f angle φ for each uranium source measured %[U-35] lower bound φ (degrees) upper bound 0.00% n/a 0.00 n/a 0.11% % % n/a n/a Table 5f dependence of φ on the concentration of U-35 Forney defined θ as the angle between the reference vector (control variable) and the other vector (experimental variable). This angle was calculated by taking the dot product of the reference vector and the other vector: Using the background spectrum as the reference vector simplifies this calculation because the reference vector is on the horizontal axis, so that =. It is not necessary to take the dot product. 3

39 OPTIMIZATION, EXTRAPOLATION, AND LIMITATIONS: The purpose of this section is to show the limitations of vector representation, and to show that vector representation is not practical for enriched uranium. The vector should be in the first quadrant of the Cartesian coordinate plane. An angle of 45 degrees is in the middle of the first quadrant; therefore, vectors close to 45 degrees would optimize this method. Vector representation is effective when the count ratio is on the order of magnitude one. cpm cpm 144keV 1001keV tan tan o 45 1 The previous chapter on relative activity relates the activities of two sources to the count ratio as: A cpm % I1 cpm 1 % I efficiency efficiency 1 r r 1 A 1 For a uranium source containing U-35 and U-38 both uniformly distributed throughout the source, the distance between source and detector is constant. That means the activity and count ratio are independent of distance when both gamma rays come from the same source. If only one vector is used with no reference vector, then a systematic error in the efficiency calibration could affect the accuracy of the relative activity calculations. For the sake of argument, this calculation is attempted using the efficiency calibration: A A U U 35 A U cpm 144keV % I1001 kev 1001 kev 34m Pa cpm1001 kev % I144 kev 144keV A The branching ratios, relative efficiencies, and standard deviations for the 0-hour uranyl acetate spectrum in Table 5g.i come from Table 4a. The data from Table 5g.i could be substituted into the above equation if the count ratio were one. 33

40 Table 5g.i nuclide Energy (kev) σ (%) branching ratio (%) relative efficiency (%) U Pa-34m Table 5g.i data for optimization calculations A count ratio of 1 indicates a relative activity of: A A U 0.837% 11.5% 3 Ci U U % 145.0% Ci U The molar specific activity of U-35 is 508 µci per mole, and the molar specific activity of U-38 is 80 µci per mole. The specific activities for U-35 and U-38 calculated in Appendix C are used for the following calculation: Ci U mol U Ci U 1 Ci U Ci U mol U % % 100% % 4 mol mol U U % Thus, an angle of 45 degrees corresponds to a count ratio of one, which corresponds to a U-35 concentration of %. The natural concentration is 0.7%. Therefore, the vector space is optimized for very depleted uranium. The count ratio is directly proportional to the concentration of U-35. The following algebraic expression can be used to extrapolate the count ratios for different concentrations of U- 35: % 35 U tan % 34

41 Table 5g.ii lists the U-35 concentrations, count ratios, and vector angles for hypothetical spectra from uranium sources. Statistical uncertainty is not included in Table 5g.ii because the calculations are theoretical. Table 5g.ii theoretical count ratios and vector angles for uranium spectra count %[U-35] φ (degrees) classification ratio 0.1% depleted uranium 0.3% depleted uranium 0.7% natural uranium 1.0% % LEU (reactor fuel) 10.0% % LEU 0.0% HEU 90.0% weapons grade uranium Table 5g.ii extrapolation to enriched Uranium The difference between reactor grade uranium (4% enrichment) and weapons grade uranium (90% enrichment) is 1.37 degrees. The difference between 0% U-35 and 90% U-35 is only 0.1 degrees. Thus, this method cannot easily distinguish between LEU, just below 0%, and weapons grade uranium, 90%. The limitation of this method is that it only works well for natural uranium and for depleted uranium tails leftover from the enrichment process, but not for enriched uranium. However, measuring the tails is an important part of the isotopic accountancy process in an enrichment facility (Baeckmann, et. al. 1995). This process is very time consuming because there are thousands of kilograms of tails (Federation of American Scientists website). Vector analysis of the gamma spectrum could make it easier. 35

42 CHAPTER 6: CONCLUSIONS Vector representation of the count ratio is sensitive to statistical uncertainty in the net areas of the peaks. A reference source with poor statistics (i.e., high standard deviation) leads to high uncertainty in the estimate of the Uranium-35 concentration. The natural uranium ore source has poor statistics due to its low activity, self-absorption, and geometry. Future research would require another depleted uranium source (0.1% - 0.3% U-35) as a reference point. The relative activity method is sensitive to systematic errors in the efficiency calibration when different gamma rays with different energies are analyzed. Future research could utilize more sources and more precise geometry for a better efficiency calibration. The vector representation method is independent of detector efficiency as long as the same detector and geometry are used for each measurement, so the efficiency calibration may be unnecessary. In principle, one could use data taken with different detectors for the relative activity method as long as the efficiencies for each detector are well determined. The weaknesses of the relative activity method are that it is time consuming and that it requires a good efficiency calibration. However, the relative activity method is still a widely accepted technique. Any other method for analyzing gamma spectra must be compared to the relative activity method. It is important to note that the relative activity method has more potential data points. The peaks with the best statistics and the least interference can be used. Uranium-35 has 4 gamma-ray peaks with good statistics. Protactinium-34m has 4 gamma-ray peaks, and one of these peaks has good statistics. Thorium-34 gamma rays were not found to be accurate indicators of Uranium-38 concentration, due to convolution with uranium x-rays and due to the uncertainty in the response of the detector at low energies. 36

43 The 144 kev gamma ray from Uranium-35 is the most useful gamma ray from this nuclide because it has good statistics and is not convoluted with Radium-6. The 1001 kev gamma ray from Protactinium-34m is a good indicator of the Uranium-38 concentration because these two nuclides are in the same decay chain and are in equilibrium. Including additional peaks from U-35 could improve the precision of the calculation. Taking the ratio of counts in only two peaks simplifies the data by ignoring most of the peaks. Information is lost for the sake of simplicity. Vector representation is algebraically equivalent to the count ratio method and has the same percent standard deviation as the count ratio method. One reason to represent the count ratio as a vector is to fit in a standardized system. Methods exist for using vector representation in thermoluminescent dosimetry (TLD) to distinguish between proton and photon irradiation (Skopec, 006). Applying this method to gamma spectroscopy requires choosing the best peaks to distinguish between Uranium-35 and Uranium-38. Another argument for vector representation is that it is a way to graphically represent the count ratio. In other words, it may be quicker to look at a graph than to read the numbers. The disadvantage is that the graph does not provide useful information when the vector is too close to the vertical axis. Vectors from enriched uranium gamma spectra have high angles between 89 degrees and 90 degrees. Vector representation of uranium gamma ray spectra is optimum for depleted uranium. The angle φ is close to 45 degrees when the count ratio is close to one. Different detectors with different efficiency calibrations may yield a different count ratio for the same uranium sample. It is safe to assume that the relative efficiency of most detectors at 144 kev is higher than the 37

44 relative efficiency at 1001 kev because higher energy gamma rays are more likely to escape the detector (Knoll, 1989). Regardless of the efficiency of the detector, the branching ratio of the 144 kev gamma ray from Uranium-35 is two orders of magnitude higher than the branching ratio of the 1001 kev gamma ray from Protactinium-34m. (However, this is offset by the much lower background continuum at 1001 kev.) Also, the specific activity of Uranium-35 is higher than the specific activity of Uranium-38 (and its progeny in equilibrium). The implication of the different branching ratios and specific activities is that gamma spectroscopy detects U-35 more easily than U-38 progeny. 38

45 APPENDIX A: ENERGY CALIBRATION There are two types of calibrations for the gamma detector: energy calibration and efficiency calibration. Energy calibration determines the energy represented by each channel. Efficiency calibration determines how many incident gamma rays are counted by the detector as a function of energy. The horizontal scale on the MCA depends on the energy calibration, and the vertical scale on the MCA depends on the efficiency calibration. LOW-ENERGY CALIBRATION: Before analyzing the spectrum from an unknown source, the MCA must be calibrated using known sources. The energy calibration spectrum and the unknown spectrum must use the same detector and the same electronics at the same settings for the energy calibration to be relevant to the results. MAESTRO-3 has a calibration function that fits data points to a calibration curve using a non-linear algorithm. The same data points were entered into an Excel spreadsheet and analyzed using linear regression. The linear regression from the Excel spreadsheet is more straightforward than the MAESTRO-3 algorithm. Both energy calibration functions were in agreement for the channels in between the calibration channels. The MAESTRO-3 calibration curve extrapolated the lowest and highest channels differently from the Excel trendline. The energy calibration was repeated every day for 5 minutes each day. Figure A1.I shows the spectrum from one of these energy calibrations. Cesium-137 and Cobalt-60 were the sources used for energy calibration. These sources are commonly used for calibration because of 39

46 Energy (kev) their intense peaks. The gamma ray from Cs-137 was at kev, and the gamma rays from Co-60 were at 1173 kev and 133 kev Calibration kev kev 133 kev channel Figure A1.I Energy calibration spectrum taken on January 7, 014 using Cs-137 and Co-60 for a live time of 300 seconds. The data from the spectrum shown in Figure A1.I were tabulated in a Microsoft Excel spreadsheet in Table A1.I. The known energies of the gamma rays from the calibration source were compared to their respective channels on the MCA. Table A1.I tabulated data from an energy calibration spectrum channel Energy (kev) Table A1.I peak channels for the energy calibration spectrum taken on January 7, 014 using Cs-137 and Co-60 for a live time of 300 seconds and a real time of 39.5 seconds The data from Table A1.I were graphed on an X-Y scatter plot in Microsoft Excel. The trendline is shown in Figure A1.II. 40

47 Energy (kev) Calibration trendline y = x channel Figure A1.II The calibration points were fitted to a linear regression trendline in Microsoft Excel. The variable x in the linear equation represents the peak channel on the MCA, and the variable y represents the energy of the gamma ray in the peak channel. The slope of the line has units of kev per channel. This equation was used to calculate the energies of gamma rays from unknown sources. The energy calibration was repeated every day, and the trendlines were almost the same (i.e., with a level of precision within 0.keV) until the amplifier gain was adjusted. For brevity, the similar calibration trendlines from the rest of the week need not be included. Each day s calibration was used to analyze the spectra that were acquired on the same day. The Results chapter shows that gamma rays from Thorium-34 and Uranium-35 have energies below 66 kev. The energy calibration with Cesium-137 and Cobalt-60 is accurate only for gamma rays with energies between 66 kev and 133 kev. The energy calibration was repeated one more time using the 0-hour background spectrum shown in Figure A1.III. 41

48 counts Low Energy Background Energy (kev) Figure A1.III 0-hour background spectrum when the amplifier gain was 50 (Data from this spectrum were used for the energy and efficiency calibration calculations.) The peak data from this spectrum are tabulated in Table A1.II. Background peaks at lower energies make the energy calibration more accurate at lower energies. Table A1.II tabulated data from the low energy background spectrum nuclide channel Energy (kev) Ra Pb Pb e Tl Bi Cs Co Co Table A1.II peak channels from the background spectrum and from the energy calibration that was acquired on January 7, 014. The data from Table A1.II were graphed on a scatter plot in Excel. The trendline is shown in Figure A1.IV. 4

49 Energy (kev) 1400 Energy calibration with background y = 0.191x channel Figure A1.IV The data points were fitted to a linear regression trendline in Microsoft Excel. The previous calibration spectra were acquired when the amplifier coarse gain was 50. Depending on energy, this amplifier setting has approximately 8 channels per peak; enough channels per peak to fit a Gaussian curve. This amplifier setting was optimized for the low energy U-35, Th-34, and Pa-34m gamma rays, but not for the higher energy background gamma rays, which were off scale. The amplifier coarse gain was reduced to 0 in order to analyze the high energy gamma rays from the background spectrum shown in Figure A.I. The same calibration sources were used for the same live time at the same distance. 43

50 counts HIGH-ENERGY CALIBRATION: High Energy Background Spectrum Energy (kev) Figure A.I 0-hour background spectrum when the amplifier gain was 0 With this amplifier setting, the peaks were only 3 channels wide. MAESTRO-3 could not fit a Gaussian curve to the peak. The only parameter that can be measured is the number of counts in the peak channel. Calibration spectra were acquired with the lower amplifier gain setting. One of these spectra is shown in Figure A.II. 44

51 counts per channel Re-calibration for high energy kev Cs kevco kev Co channel Figure A.II calibration spectrum that was acquired on January 9, 014 after the amplifier gain was reduced from 50 to 0 The data from this spectrum were tabulated in a Microsoft Excel spreadsheet and are shown in Table A. The known energies of the gamma rays from the calibration source were compared to their respective channels on the MCA. Table A tabulated data from the energy calibration spectrum with lower amplifier gain channel Energy (kev) Table A peak channels from the calibration spectrum that was acquired on January 9, 014 (after the amplifier gain was adjusted) for a live time of 300 s and a real time of s The data from Table A were entered graphed on a scatter plot in Excel. Its trendline is shown in Figure A.III. 45

52 Energy (kev) Calibration with low gain y = x channel Figure A.III The calibration points were fitted to a linear regression trendline in Microsoft Excel. The calibration was repeated every day after January 9, 014, and the trendline equations were almost the same. As long as the amplifier gain remains constant, calibration with Cs-137 and Co-60 has a high level of precision. Each day s calibration trendline was used to analyze the spectra that were acquired on the same day. 46

53 APPENDIX B: EFFICIENCY CALIBRATION Efficiency calibration is a time consuming process requiring several different sources with different gamma rays at different energies. The areas under the peaks are analyzed. The area is the number of counts in the channels containing the peak. A high number of counts is needed for good statistics. The standard deviation σ is calculated in MAESTRO-3 for each peak. A longer live time is needed to get more counts and a lower percent standard deviation (i.e., good statistics ). The standard deviation does not account for systematic errors. Source-detector geometry includes the shape of the source, the size of the detector, and the distance between the source and the detector. Errors in geometry are not counted as statistical error. Error in geometry is treated as a systematic error, which is difficult to quantify. MANUFACTURER S SPECIFICATIONS: The manufacturer s specifications in Figure B1 give the efficiency of the HPGe detector at 133 kev relative to the efficiency of a 3 X 3 NaI(Tl) detector at 133 kev. The specifications do not mention the relative efficiency at 100 kev, where there is a sharp increase in efficiency. Each individual detector has a different efficiency curve; even the same model from the same manufacturer. 47

54 Figure B1 manufacturer s specifications for the HPGe detector 48

55 EFFICIENCY CALIBRATION USING SPECTRA FROM CALIBRATION SOURCES: The vendor specified relative efficiency for this detector at 133 kev is 10% of the efficiency of a 3 X 3 NaI(Tl) detector at 133 kev. The intensities of the other gamma-ray peaks may be compared to the intensity of the 133 kev gamma-ray peak, which is the reference point. Data were gathered from efficiency calibration spectra and entered into an Excel spreadsheet, as shown in Table B.I. Table B.I tabulated data for efficiency calibration nuclide Energy (kev) Branching ratio (%) Activity (μci) Distance (cm) Gross counts Net counts σ (%) FWHM (kev) de/e (%) Am ,38 6, Cd ,305 11, Co , , Co ,774 48, Ra , , Cs ,010 41, Co ,056 16, Co ,041 16, Table B.I raw data used for efficiency calculations from sources The third column of Table B.I contains the branching ratios listed in the Table of Isotopes reference. The fourth column is the activity of the source, calculated from the calibrated activity and date listed on the label and decayed to the current date. A( t) t A 0 e where λ is defined in terms of the half-life T ½ as ln T 1 49

56 The fifth column is the source to detector distance. The second column from the right is the full width at half maximum (FWHM) of the peak. The last column is the FWHM divided by the energy. The net areas were converted to counts per minute (cpm) in Table B.II. Table B.II comparison of count rates from different gamma-ray peaks from different spectra nuclide Energy (kev) Net counts Count time (min) Count rate (cpm) Am , Cd , Co , Co , Ra , Cs , Co , Co , Table B.II Net areas were divided by live times to calculate count rates. These raw data were used to calculate the efficiency of the detector at each energy relative to the 133 kev reference point. cpm Activity % I efficiency r where I is the branching ratio and r is the source to detector distance (cm). Solving this relationship for efficiency ε gives cpm r Activity % I The proportion sign can be replaced by an equals sign by comparing each gamma ray to the 133 kev reference point with a specified efficiency of 10% relative to a 3 x3 NaI(Tl) detector. The counts for each data point are divided by the counts for the 133 kev data point. 50

57 relative efficiency Calculations are adjusted for activity, branching ratio, and distance. Table B.III shows the relative efficiencies calculated for each gamma ray. Table B.III efficiencies relative to 3 X 3 NaI(Tl) at 133 kev nuclide Energy (kev) Relative efficiency Relative efficiency % compared to NaI(Tl) Am Cd Co Co Ra Cs Co Co Table B.III relative efficiencies of gamma rays from calibration sources Relative efficiency versus energy was graphed as shown in Figure B.I Efficiency calibration curve Energy (kev) Figure B.I attempted efficiency calibration using all data points The 186 kev peak in Figure B.I is an outlier. The long, cylindrical shape of the Ra-6 calibration source causes errors in geometry. This outlier is excluded from Figure B.II. 51

58 Relative efficiency Efficiency Calibration excluding outlier Energy (kev) Figure B.II The 186keV gamma ray from Ra-6 was excluded from the calibration. Instead of a continuous function, this chart is a series of line segments with constant slopes. Extrapolating the slope of the line segment between the last two data points (between 1173 kev and 133 kev) gives the relative efficiency at 110 kev. EFFICIENCY CALIBRATION USING THE BACKGROUND SPECTRUM: The calibration curve in Figure B.II has no data between 136 kev and 66 kev. This portion of the curve is non-linear. More data points must be included to fill in the gaps. The efficiency calibration was repeated using the background sources Lead-14 and Bismuth-14, which are in equilibrium. The data from the background spectrum are tabulated in Table B3.I. Table B3.I raw data from the low-energy background spectrum nuclide Energy (kev) Branching ratio (%) Net counts Standard deviation () Standard deviation (%) Pb , Pb , Bi Bi Bi Table B3.I Tabulated data from the 0-hour background spectrum when the amplifier gain was 50 5

59 relative efficiency Propagation of error is not included in the efficiency calibration. Error in the efficiency calibration is treated as a systematic error, not a statistical error. Table B3.II shows the relative efficiencies of the background peaks compared to the 110 kev gamma ray from Bi-14. Table B3.II relative efficiencies of background peaks compared to 110 kev. Relative E (kev) efficiency Table B3.II background peaks compared to the 110 kev gamma ray from Bi-14 reference point Using only the 5 data points from Table B3.II gives an incomplete efficiency calibration, as shown in Figure B3.I..50 Background efficiency Energy (kev) Figure B3.I relative efficiencies of the background peaks compared to the 110keV peak from Bi-14 The relative activity of the 110 kev gamma ray from Bi-14 can be compared to the relative activity of the 1173 kev gamma ray from Co-60. Interpolating the efficiency of the 53

60 1173 kev and 133 kev gamma rays from Co-60 gives an estimate of the 110 kev efficiency as 10.5%. All of the background data points can be converted to relative efficiency compared to 3 X 3 NaI(Tl) at 133 kev. Table B3.III shows the conversion of the relative efficiency to this reference point. Table B3.III comparing the efficiencies of background peaks to the calibration point Energy (kev) Relative efficiency (%) Table B3.III relative efficiencies of the background peaks compared to 3 X 3 NaI(Tl) at 133 kev COMBINED EFFICIENCY CALIBRATION FROM SOURCES AND BACKGROUND: The data from the background gamma rays can be combined with the previous efficiency calibration from the sources, as shown in Table B4.I. Table B4.I Combined relative efficiencies from both calibrations. nuclide Energy (kev) Relative efficiency (%) Am Cd Co Co Pb Pb Bi Cs Bi Bi Co Co Table B4.I relative efficiencies of gamma rays from background and from sources 54

61 efficiency relative to NaI(Tl) at 133keV The data from Table B4.I were graphed as shown in Figure B4.I. 180% Efficiency Calibration 160% 140% 10% 100% 80% 60% 40% 0% 0% Energy (kev) Figure B4.I efficiency curve approximated with line segments between data points The outlier in Figure B4.I is the 66 kev gamma ray from Cesium-137. This discrepancy may be due to an error in geometry, because the Cs-137 source was placed underneath the Co-60 source when the calibration spectrum was acquired. The ubiquitous background radiation is assumed to have no error in geometry. The 609 kev and 768 kev gamma rays from Bismuth-14 fit better when the 66 kev outlier from Cesium-137 is excluded, as shown in Figure B4.II. 55

62 ε relative to NaI(Tl) at 133keV 180% Efficiency Calibration 160% 140% 10% 100% 80% 60% 40% 0% 0% Energy (kev) Figure B4.II efficiency calibration omitting the outliers at 186keV and 66keV The efficiency calibration in Figure B4.II agrees with the efficiency of a typical detector shown in Figure B4.III (ORTEC detector specifications). Every HPGe detector has a different response, but the shapes of the efficiency curves should be similar. Figure B4.III a typical efficiency curve described in the manufacturer s specifications 56