ENERGY RESPONSE STANUTY OF A MULTKHANNEL GAMMA scmn'LLAnoN spzcmomersafl ‘ Thesis for the Domed of M. S. MICHIGAN STATE. unfwmsm . LaWrence J, Perez, Jr; - 1966 ' w 'J LIBRARY Michigan State ' University THESIS ABSTRACT ENERGY RESPONSE STABILITT’OF A HULTICHANNEL GAMMA SCINTILLATION SPECTROMETER by Lawrence J. Perez, Jr. The study deals with the development of mathematical methods for describing and determining energy response shifts due to the extrinsic variables of line voltage, temperature, count rate and photo- ‘multiplier tube fatigue; and with the effects of energy response shifts upon the results of the simultaneous equations method for resolving composite spectra. Basic methods developed were peak location using a fourth degree logarithmic fitting equation, an energy response model incorporating a correction for non-linearity and for a non-x-axis bias pivot point, and spectrum normalisation using a special seven point average of exact polynomial fitting equations with a restriction on spectrum slepe at the channel points. Peak locations can be determined to within the order of one-hundredth of a channel. Gain and bias shifts on the order of 0.01 percent and 0.01 channels can be detected. Using the above methods, energy response stability was determined by varying the parameter under investigation and holding all others constant. Temperature and line voltage coefficients for gain and bias were 0.055 Kev per channel per degree Fahrenheit and 0.010 channels per degree Fahrenheit, and 0.0014 Kev per channel per volt and 0.000 channels per volt. The fatigue rate found was 8.67 x 10'3Kev per channel per hour per count per minute. The count rate dependence was expressed mathematically as the sum of two exponential functions. The effect of energy response shift upon the simultaneous equa- tions method for spectrum resolving was determined by artificially constructing composite spectra of simulated milk samples. The composite spectra were then shifted varying amounts by the previously developed spectrum normalization procedure. An error in nuclide concentration 'was mathematically expressed as the sum of errors due to each component of the composite spectrum. The resolving method was judged to be stable for small gain and bias shifts. ENERGY RESPONSE STABILITY'OF A HDLIICHANNEL GAMMA SCINTILLATION SPECTROMETER By Lawrence J. Peres, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Master of Science Department of Civil and Sanitary Engineering 1966 ACKNOWLEDGEMENTS The author expresses his gratitude to his major professors, Dr. Shosei Serata, Associate Professor, and Dr. Richard Neff, Assistant Professor, Department of Civil Engineering, Michigan State University, for their guidance and review of the manuscript. Thanks are also extended to‘Hr. Richard Jaquish, Chief, Training Branch Activities, Southwestern Radiological Health Laboratory, United States Public Health Service, for supplying reference gamma spectra; and to the staffs of the Radiological Health Research Activities and Environmental Health Training Section of the R. A. Taft Sanitary Engineering Center, united States Public Health Service, for the many discussions while the author was associated with their institution. Special appreciation is expressed to the author's wife,‘Haryln, for her encouragement, and assistance in typing and editing. Financial assistance, in the form of an outside-the-service training assignment from the Division of Radiological Health, United States Public Health Service, is greatly appreciated. ii TABLE OF CONTENTS Chapter Page p—I I. manION.0.00.00.00.00000000000000.00.00.0000000000.0... Statement of Problem .................. .. ..... ............... Objectives......................... ..... ..... ..... .......... Scope of Study.............................................. Historicial Perspective of Gamma Spectroscopy............... Applications of Gamma Spectroscopy.... ..... ................. UIUNNH II. BACKGROUND INPOMTIONOOOOOOOOOOO 00000000 0.0.00.000000000000 Principle of Operation - Gamma Spectrometer... ...... ........ Extrinsic Variables Effecting Energy Response............... NO‘O‘ Voltage............................ ......... .............. 7 Temperature............................................... 9 Photomultiplier Tube Fatigue.............................. 10 Count Rate ............................................... 10 ‘Methods of Energy Response'Measurement...................... 11 Effects of Instability Upon Composite Spectrum Resolving MethOdBOOOOOOO0.00000.000.000.00000000000.00.00.00.000000. 12 Spectrum Stripping Method........... ..... ................. 12 s 1mu1taneou' Equt ion. He chad 0 O O 0 O O O O O O O O 0 O O O O O 0 O 0 0 0 0 0 0 O 0 O 13 Least-sqmres HethOdO O O O 0 0 0 O 0 0 0 0 0 O O O O 00000 0 0 0 0 0 O 0 0 O 0 0 0 O O 0 0 13 III. DEVEIDPMENTOF BASICMEMDSseeessssaeeeeeeeasaeesasesssesea 15 Equipment and Operating Conditions.......................... 15 ‘Multichannel Gamma Spectrometer System.................... 15 Source and Counting Geometry.............................. 16 Line Voltage Control and Monitoring ........ ............... 17 Temperature Control and'Monitoring........................ l9 Detector Conditioning..................................... 20 Digital Computer..................... ..... ................ 20 Peak meation...0 ........ 0.0.0.000. 000000000000 0.0.0.0000... 22 Procedure 22 Computer Programs............. ............ ................ 24 Energy Response‘Model........... ............. ............... 27 Theory Of methadl O 0 0 0 O I I O O 0 O O O O O O O O O O O 0 O 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 28 PrxedureO 0 0 0 0 0 0 O O 0 0 0 O O O 0 0 0 0 0 O O O 0 O O O O I 0 O 0 0 0 I 0 O 0 0 0 0 0 0 0 0 0 0 0 0 29 cmuterProgr‘m000000000000 000000 000.000.00.00.000000000 3o spectrmHamlizationOoOOOOOOOOOOO 00000 000.000.00.000000000 35 iii TABLE OF CONTENTS (cont.) Chapter Page meaty Of Method. 0 0 0 0 0 0 0 0 0 0 O O O 0 0 O O O O O O O O O O 0 O O O O 0 0 0 0 0 0 0 0 0 0 0 0 0 35 Procedureoooo0.0.0.0000000000 00000000 0.000.000.0000000000000 39 cmutet Program 0 O O 0 O O O O O 0 O O O O O O O O O O O O O O O O O O O O O O O O 0 0 0 O 0 0 0 0 0 40 IV. DATA ANALYSIS FOR DEVELOPMENT OF BASIC HETHODS.. . .. 45 Pe‘k m‘tion...000.000.000.0000000000.00.000000000000000000... 45 GOOdHCSO’Of'Fit....o....o..............o..........o......o... 45 Reproducibility as a Function of Fitting Area Used........... 54 Reproducibility as a Function of Total Counts................ 67 Energy Response‘Hodel............ ........ ...................... 77 Apparent Energies O O O O O 0 0 0 O 0 0 O 0 0 O O 0 0 O 0 O O O O O O O 0 0 O 0 0 0 0 O 0 0 0 0 0 O 0 0 0 77 31‘. Paint.0..O..000000.00.00.000000000000000000.000000000000 78 Reproducibility of Energy Response Line ....... ............... 81 spectrm nomlizationa O O O O 0 0 0 0 0 0 O 0 0 O 0 0 O 0 0 O 0 0 O 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 82 SMOChRGOO'Of'P1C...coo...o.................o..o.....oo...oo. 82 Accuracy of Interpolation.................................... 85 Comparison of Normalized to Determined Energy Response....... 90 V. DETERMINATION’OF ENERGY RESPONSE STABILITY..................... 93 Prxedure00000.000 ..... 0.0.0.0000...00O...0.00.00.00.0000000.00 93 Line valtageOOOOOOOOOOOOO0.0.0.0000...0000.00.00.000000000000 93 TemeracureOOOOOOO0.00....000.0.0.00.00.00.000000000000000... 93 Photomultiplier Tube Fatigue and Count Rate.................. 94 Computer Programs.....:.............. .......... ................ 9S D‘t‘ AMlysisOOOOOOO0.00.00.00.00000.0.00.000000000000000000000 95 Line valtagebo 0.000.000.00000000000000000000.. 00000....00000. 95 reveratureOOOOOOOOOOOOOOOOOOOOOOOOOOO00.0.000000000000000000 97 Photmultiplier Tube F.t18ue. 0 0 O O O O I O O ....... O O 0 O 0 0 0 0 0 0 0 0 0 0 0 0 97 count R‘te Dependence O O 0 0 0 0 0 0 0 0 0 0 0 O O O O O 0 0 0 0 0 0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 0 0 103 VI. EVALUATION OF THE SIMULTANEOUS EQUATIONS NETHOD................. 118 Basis for Approach.............. ....... ......................... 118 Procedure....................................................... 122 Reference speCtr8000000000000 OOOOOOOOOOOOOOOOOOOOOOO 0.0.0.0... 122 variation OfAct1v1tycontentOOOOOOOOO00.00.00.000000000000000 125 iv TABLE OF CONTENTS (cont.) Chapter VII. VIII. Variation of Energy Response... ....... ...................... Simultaneous Equations Method.. ............. . ............ ... computer Program000.0.00.0..000000000.0.0.0000.000000.000..00. Data Am1ys180...000.000.000.000... ...... 0 ...... 000.000....00. Data From Simulated Milk Sample.. ............... ............ Effects Of Gain Shift80. . 0 . 00000000000000000 0.00 000.00 .000. . Effects of Bias Shifts. ....... .... .......... ................ AnAlternate ApproaCh0000000.0.0.0000... ..... 00.000.000.000... Development Of HethOdO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 . 0 0 0 0 . 0 0 . 0 0 0 0 0 0 0 Evaluation of Fractional Errors.. ............. .............. Verification For Higher n ......................... . ........ . DISCUSSION ..... .. ................ . ................. ...... ..... Control of Extrinsic Conditions.. ............... .............. Peak Location ........... ...................................... Energy Response Model............. ......... ................... Spectrum Normalization... ....... ... .......... ................. Line Voltage Effects.......................................... Temperature.. .................... . ..... . ........ .............. Fatigue. ...................................................... Count Rate Dependence................ ... ...... ................ Evaluation of the Simultaneous Equation Method................ Icmuterprograms 000000 .00000.00.000.000.0.0.0.000000000000000 SUMMARY,CONCLUSIONS AND RECOMMENDATIONS .......... ............. sumary00000.00.0.0000000000000000 00000 0.00000000000000000000. Determination of Peak Locations. ............. ............... Energy Response Model....................................... Spectrum Normalization. ............. ................... Determination of Energy Response Stability.................. Evaluation of the Simultaneous Equations Method. ........ conc1u810n8 00000000000 0 000000000000000000 00000000....00000000. RecomendationSe ....... 0000000000000. 000000000 00.00.000.000... Future Operation of the Spectrometer ......... ............... Future stUdieSOOOO0.0.0.0000000000000.000.000.00000000...... BIBLIWRAPHY...0.000000.00....00000000000. ..... 00.000.000.000. APPEmIX.0000000000..00....000.000.0000..0.00.00.00.0000000000 Appendix A: Computer Program Data Deck Descriptions.......... Appendix B: Computer Programs................................ V Page 126 126 126 128 129 131 135 143 143 144 145 147 147 147 151 152 153 153 154 154 155 156 157 157 157 157 158 159 160 160 161 161 162 163 166 167 172 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. LIST OF TABLES Nermal Control Settings of Spectrometer.................... Sums of Absolute Differences For Peak 4.................... Percent Improvement of Fit by Using Logarithmic Form - Peak 400.00000000000000000000.000000000000000000000.0000. Percent Improvement of Fit by Increasing Degree of Fitting Equation - Peak 4, 10 Kev/Channel Spectrum....... Sums of Differences For Second and Fourth Degree Mithmetic Fit-Peak 40000000000000.0000.000000000000000 Relative Sums of Differences of 10 Kev/Channel Spectra For Standard Peak Widths............. ..... ............... Peak Location Width Stability For 10 Kev/Channel Spectrum.. Peak Location Symmetry Stability For 10 Kev/Channel Spectrum... 00000000 00.000.00.00.00.0000000000000000000000 Summary of Peak Location‘nethod Selected................... Reproducibility of Peak Location Channel - Standard Deviations, Source Sodium 22, Count Rate 54,600 Cpm...... Apparent Energies (Kev) and Correction Factors Por EnergyResponseM0d610000000000000.0000000000000000000000 Energy Response Lines for Bias Pivot Point Determination... Bias Pivot Paint8000000000000000000000000000000000000...000 Comparison of Incorporated Energy Response Shift to ‘Measured Energy Response........ ..... .................... Long Term Fatigue.......................................... Rate Constants For Arbitrary Gain Change Processes......... Count Rate Variation Equation Parameters................... Calibration Data For Milk Analysis......................... Activity Concentrations Used for Manufacturing cmosite Spectra, pCi per Liter0000000000000000000.00000 Peak Areas Used for Simultaneous Equations'Hethod.......... vi Page 16 46 47 47 49 54 58 59 62 78 80 80 90 103 104 109 123 125 126 Table 21. 22. 23. 24. 25. 26. 27. 28. LIST OF TABLES (cont.) Percent Errors For Analysis of Simulated‘Milk Sample at Low Activity Concentrations ............ ................ Percent Error in Barium-Lanthanum Concentrations For Varying Amounts of Cesium and Iodine............................... Fractional Errors For Gain Shift ..................... ........ Fractional Errors For Bias Shift ..... . ....... . ....... ........ Comparison of Two'Methods of Error Ca1culation............... Data Deck Description For Programs MMODELL and MMDDELA....... Data Deck Description For Program EFFECT..................... Data Deck Description For Programs BIASPT, CNTRE, TEMP and mLT000000000000 00000 00000000000000.0000.00.00.000.000... vii Page 130 135 145 145 146 168 169 171 LIST OF FIGURES Figure Page 1. Basic Components of a Multichannel Gamma Spectrometer........ 8 2. Decay Scheme of Sodium 22.................................... 17 3. Gamma Spectrum of Sodium 22 ............... . ....... ........... 18 4. Ambient Room Temperature Cycles ................... ........... 20 5. Experimental Instrumentation......... ..... . ....... ........... 22 6. General Flow Diagram of MMODEL............................... 26 7. Energy Response Model.......... ....... . .............. ........ 29 8. General Flow Diagram of CORFAC...... ......... ................ 32 9. General Flow Diagram for BIASPT, CNTRE, TEMP, and VOLT....... 33 10. General Flow Diagram of REPROD............................... 34 11. Qualitative Effect of Smoothing Equations.................... 37 12. General Flow Diagram of Subroutine TSHIFT.................... 42 13. General Flow Diagram for CKSHIFT............................. 43 14. General Flow Diagram for CKFIT............................... 43 15. General Flow Diagram for CKERL............................... 44 16. Logarithmic Second Degree Fitting Equations For Peak 4 “81n85,9and 17Channe18000000000000000000000.00000000000 50 17. Logarithmic Fourth Degree Fitting Equation For Peak 4 UBins 21 cmme180000000000000000.000000.000.000.0000000000 51 18. Arithmetic Fitting Equations For Peak 4 Using 17 Channels.... 52 19. Arithmetic Fitting Equations For Peak 5 Using 19 Channels.... 53 20. Peak Location as a Function of Peak Width - Peaks 1, 2 and 300000000000.000000000000000000000000000000000.00000000 56 21. Peak Location as a Function of Peak Width - Peaks 4 and 5.... 57 22. Peak Location as a Function of Peak Starting Channel - Peaks1‘nd2000000000000.000.0000....0.0.0.0....0.0000000. 6o viii LIST OF FIGURES (cont.) Figure Page 23. Peak Location as a Function of Peak Starting Channel - Peaksaand 40.000000000000000...0.0000.0000000000000000... 61 24. Peak 1 - Fourth Degree Arithmetic Fitting Equation, 9 Channe180000000000000000.0000000000000000000000000....000 63 25. Peak 2 - Fourth Degree Arithmetic Fitting Equation, 11Channels.0.00000000000000000000000000000000000.000000000 64 26. Peak 3 - Fourth Degree Arithmetic Fitting Equation, 15 Channe1800000000000000000000000000000000.000000000000000 65 27. Peak 4 - Fourth Degree Arithmetic Fitting Equation, 17 Channe180000000000000000000000.0000000000000000000000... 66 28. Reproducibility of Peak 1 For Varying Total Counts........... 68 29. Reproducibility of Peak 2 For Varying Total Counts........... 69 30. Reproducibility of Peak 3 For Varying Total Counts........... 70 31. Reproducibility of Peak 4 For Varying Total Counts........... 71 32. Reproducibility of Peak 1 - Effect of Time of Count.......... 72 33. Reproducibility of Peak 4 - Effect of Time of Count.......... 73 34. Reproducibility of Peak Locations - Standard Deviations...... 74 35. Comparison of Relative Peak Location of Peaks 1 to 4......... 77 36. Determination of Bias Pivot Point............................ 79 37. Comparison of Fitted Spectrum to Observed Spectrum Points inpeakAreaa 0000000 00000000000..00.000000000000.0000000000 83 38. Comparison of Fitted Spectrum to Observed Spectrum Points in valley Area.00000000000.0000000000000000000000000000.00000. 84 39. Comparison of Fitted Spectrum to Observed Spectrum'Points in Flat Area000000000000000000000000.0000..000.....0.00..0000. 84 40. Comparison of Odd Point Interpolation Spectrum to Observed spectr‘m0000000000000.0000000000000000.00000000000000.0000. 87 41. Comparison of Odd Point Interpolation Spectrum to Observed mcount SpeCtrum0°0000000000000000000.0000000000000000... 88 ix “a! glue . 1L ‘1. 1.4- ; J- ‘4') ,l '1. LIST OF FIGURES (cont.) Figure Page 42. Comparison of Odd and Even Point Interpolation Spectra to Observed Low Count Spectrum.............................. 89 43. Comparison of Incorporated Shifts to Measured Shifts - Gainconstant0o00 000000 0000000000000000000000000.00000000000 91 44. Comparison of Incorporated Shifts to Measured Shifts - Bias constant000000000000000.0000.000000000000000000000.0000 92 45. Effect of Line Voltage Upon Bias and Channel Coefficient...... 96 46. Effect of Temperature Upon Channel Coefficient................ 98 47. Effect of Temperature Upon Bias............................... 99 48. Effect of Count Rate Change and Fatigue Upon Channel coeffic1ent0°00.0000000000000009.0.000000000000000.000000000 100 49. Determination of Count Rate Dependency Equation Constants - source Changeoto 80000000000000.00000000000000...00......0 105 50. Determination of Count Rate Dependency Equation Constants - Source ChangeAto 000.000.000.00000000000000000.0000.00.0.0 106 51. Determination of Count Rate Dependency Equation Constants - source ChangeBto A000°0000000000000000000000000000.0000000 107 52. Determination of Count Rate Dependency Equation Constants - source ChangectoA000000000000000000000000000000.000000000 108 53. Variation of CG; With Change in Count Rate.................. 110 2 54. Variation of CCo With Change in Count Rate.................. 111 55. Effect of Count Rate Change and Fatigue Upon Bias............. 113 56. Effect of Countimte Change and Fatigue Upon Bias.............. 115 57. Effect of Count Rate Change and Fatigue Upon Bias............. 116 58. Parameters for Simultaneous Equations Method.................. 119 59. Gamma Spectra of Reference Nuclides........................... 124 60. Flow Diagrm of Program EFFECT................................. 127 61. Percent Error of Iodine Concentration for Gain Shifts......... 132 LIST OF FIGURES (cont.) Figure 62. Percent Error of Barium-Lanthanum Concentrations for Gain Sh1£t80000000000000000000000.000000.000.00.00.00.0000000000 63. Percent Error of Cesium Concentration for Gain Shifts........ 64. Percent Error of Iodine Concentration for Bias Shifts........ 65. Percent Error of Barium-Lanthanum Concentration for Bias Shift80000000000000000.000000000000000000000000000000000... 66. Percent Error of Cesium Concentration for Bias Shifts........ xi Page 134 136 138 140 141 CRAP TER I mmosuouon The expanding use of radiation and nuclear power and the lowering of radiation protection standards within the last twenty years has brought ‘dth it a need for more accurate detection instrumentation. As with any instrumentation, calibration and the retention of calibration is important for efficient use. The gamma spectrometer energy calibration is particularly prone to be environmentally dependent; therefore, a quali- tative and quantitative knowledge of the calibration or energy response characteristics is necessary for accurate radioactive analysis. Statement of Problem A basic assumption of virtually. all composite gamma spectrum re- 801ving methods is that a composite spectrum is a linear combination of its component spectra. (25) This assumption is applied in the resolving Procedure by comparing standard spectra of individual nuclides to the Composite spectrum by various methods. If the energy response of the spectrometer varies from the time the standard spectra were obtained to when the composte spectrum is ob- tained, the assumption of linear combinations will be in error. It fOllows that the quantitative solution for activity content of the 8ample, which the composite spectrum represents, will also be in error. The problem, in toto, faced is to develop methods for measuring 1tlatrument energy response, to determine quantitatively and qualitatively the stability of energy response, to determine the effect of instrument 1tlstability upon the results of spectrum resolving methods, and finally to develop methods for reducing or correcting for instability. -1- ..I'hu a x _~, :" I in. .1 ”a. ‘ '10-. 0 in... \ Mug.- "0. . p,_.: '3"; I.- W.__ -2- ijectives The primary objectives of this study are to: 1) determine the energy response stability of the multichannel spectrometer operated by the Department of Civil and Sanitary Engineering, Michigan State University, and 2) evaluate the "simultaneous equation method" of spectrum resolving with regard to effect of energy response shifts. Scope of Study This study includes three separate yet related investigations dealing with the over-all problem of gamma spectrometer stability. Mathematical models to measure stability of instrument energy re- 8ponse were developed. A model for energy response was defined, and a procedure for determining peak location was developed and tested. Using these models, the effect of extrinsic variables upon energy response were determined. The variables investigated were line voltage, temperature, photomultiplier tube fatigue, and count rate. The final phase was the determination of the effects of instability uPon quantitative analysis methods. A method for mathematically intro- due ing channel shift into spectra was developed and used to measure the effects of energy response shift upon the "simultaneous equation method". The individual studies do not contain the necessary breadth of cOverage to fully depict the phenomena involved, or to even begin to eRhaust the many possible approaches to the problems involved. However, the approach used does represent a set of mathematical models and pro- cedures which are applicable to the problem at hand and are useful in a practical sense to the laboratory using the gamma spectrometer as an ahalytical tool . -3- The philosophy used in selecting an approach to meet the objectives was as follows: 1. The entire counting system was treated as an integral unit. For economic reasons, most institutions using gamma spectroscopy as an analytical tool (in contrast to those interested in the design) cannot conveniently change components in order to improve the instrument's characteristics. The possible exception to this is the photomultiplier tube, which is relatively inexpensive. 2. The study was limited to the effects of parameters external to the instrument. A user has more “ontrol over the environment in which the instrument is Operated than over the design of it. 3. A minimum of external equipment and knowledge of electronics are necessary. Elaborate test equipment and the knowledge to use it efficiently are not necessarily available to laboratories using the spectro- meter as an analytical tool. 4. Electronic digital computers are available to the majority of institutions using gamma spectroscopy. This is especially true of state and federal agencies, and universities either on machine lease basis or a time rental basis. Historical Perspe-tive of Gamma Spectroscopy Since the discovery of x-radiation by Roentgen in 1895, and of natural radioactivity by Becquerel in 1896, (9)) a need for accurate instru- mentation for qualitative and quantitative measurement of radioactivity has existed. The instrumentation has progressively become more accurate, Sensitive and efficient; but the basic principles underlying all detection methods used today, with the exception of solid state conduction, were eEisentially known as early as 1910. (15) -3- The principle of scintillation counting was first used by Becquerel in 1899, when he noticed that when radiations interacted with such sub- stances as zinc sulfide, barium platinocyanide, and diamond, lumin- escense could be observed. In 1908, Rutherford and Geiger established a direct relationship between the number of scintillations observed on a zinc sulfide screen and the number of pulses produced in an electrical counter by alpha particles thus establishing the applicability of the scintillation method for counting alpha particles. The scintillation method, though involving tedious visual observation, was the accepted procedure for alpha counting until the advent of vacuum tube amplifiers 1n the 1930's, made possible more precise electrical counters. (9) The scintillation method was essentially disregarded as a practical analytical tool until 1947. The development of the modern scintillation detector was founded on the work of Coltman and Marshall (5) in 19157, in which they success- fl111y used a photomultiplier tube for counting the scintillation pulses Produced by alpha, beta and gamma radiation. Coupling of crystal 8<=:l.ntillators to voltage discriminator-type pulse height analysers in the early 1950's, led to the gamma scintillation spectrometer. The development of computer-type analsg-to-digital converters by Wilkinson (24) in 1950, produced the instrument which is presently being used and has been commercially available since the mid 1950's, originally with V‘cuum tube circuitry and more recently with solid state circuitry. ROutine gamma spectral analysis of environmental levels of radioactivity 8‘Zarted in 1957 with the establishment of a national raw milk monitoring network by the United States Public Health Service. (34) Since then gauma la’liwsctral analysis has been applied to virtually all environmental media. LI -5- Applications of Gamma Spectroscopy Gamma spectroscopy offers many advantages over other types of radiation counting and analysis techniques. The main advantage is the simplicity of sample preparation resulting from the minimal self-absorption of gamma radiation and the property of discrete energy levels of gamma radiation, which allows specific nuclide identification and quantitative determination of each component in complex radionuclide mixtures. (12) The major disadvantages are a relatively high minimum sensitivity; high Capital expenditure for equipment; and the inability to detect pure beta emitters such as strontium 89 and 931, and phosphorous 32 when a conventional 88m: scintillation detector is used. Gamma spectroscopy has been applied to many radiation counting Problems such as determination of gamma photon energies; (27) neutron a(itiviation analysis; (6) environmental monitoring of air, food and Water; (33) whole body counting; (I?) radioactivity calibration; (21) and radioactive tracer applications. (2.5) CRAPIER H shamans ., IETGFMATIGN Principle of strstion - Gamma Spectrometer A gamma spectrometer is a cont instion of a scintillation detector which produces electrical pulses representative of the gamma photons incident upon the detector, and a pulse. height analyzer which distin- guishes electrical pulses of different amplitude and determines the number of pulses occurring at each amplitude. The detector system consists basically of a scintillation crystal, and a photomultiplier tube and its associated high voltage power supply. The function of the scintillation crystal (sodium iodide with approximately 0.1 percent thalium iodide) ($23) is to convert the energy of the incident Banana photon to light energy. The amount, or intensity, of the light) Produced is ideally directly proportional to the energy transferred to the cIfystal by the photon. For each interaction a: light pulse is produced which is representative of that interaction. The light energy produced in the crystal is then converted into electrical energy by means of a Photomultiplier tube which is composed of a photocathode and an electron multiplier. The photocathode, when struck by light, emits electrons in Proportion to the amount of light incident upon it. These electrons are multiplied by accelerating them across a series of voltage potentials OWit-.0 dynode surfaces which produce secondary emission electrons. The e:l-ectrons produced from one gamma. interaction form a pulse whose height 18 proportional to the energy trnnsferred in the original interaction between the photon and crystal. (3’) The electrical pulse then enters the pulse analyzer system which is composed of a shaping and amplifying circuit, a pulse height analyzer -6... -7- and a data storage unit. The purpose of the shaping and amplifying circuit is to convert the relatively weak pulse (microamperes) from the photomultiplier tube to a pulse of higher amplitude and of standard shape which is acceptable to the pulse height analyzer. The pulse height analyzer is basically an analog-to-digital converter. When the entering pulse (analog representation) reaches its maximum height, the pulse is 'htretched" to a corresponding constant voltage by charging a capacitpr. Simultaneously a linearly increasing sweep voltage and a time oscillator are started. The oscillator is stapped when the stretched voltage and thesweep voltage are equal. (4) The elapsed time (digital representation) is thus proportional to the incoming pulse height. Each time interval corresponds to a storage location in a ferric core memory bank which tallies the number of pulses of each pulse height. At the end of a preset counting period, the content of the memory can be read out by numerous conventional read-out devices such as an electric typewriter, oscilloscope, digital line printer, or punch paper tCpe. The entire procedure is illustratively represented in Figure l. Extrinsic Variables Effecting Energy Response The major extrinsic variables effecting energy response have been f(Dund to be voltage, temperature, photomultiplier tube fatigue and sample c=<>unt rate. Voltage. The overall gain of a photomultiplier tube operated at ‘3le inter-dynode potentials can be expressed as G - M"; where G is the tube gain, I! the single dynode gain and n the number of stages. (1) Sharpe (30) reports an almost linear response of dynode gain: to applied voltage 0"er a range of 80 to 150 volts for AgMgOCs dynodes. With an applied Voltage of 100 volts, the relationship is M - 2.50 + 0.049 xAvolts. ‘5 mm in 8.22.» a? a” ( HOME: Rhona.— _ . _ , mun—BS: :3 3E8 .533 .. mas—hm m3?“ 7 5.8 Esau; 8-".852 antes: nu ma . No mo : : nw 00 mm no an .3 : z : mm «o S «o HENQ, ES ,g mus—cam. «3551 53m: and: mam. “63m fifl q «Baa mar—5.2.... 5:53 mums—w f HS: 92 gm nova-5300mm ‘6 #083335: a mo 3:23.300. 3:: .H «a: gm Guam» 8mg 25.—oi and: qfiagu #55 .3833 Maghgam fiovozvllsz/a RE «aaflflgm _ . // / n/‘Mfi ~35? 8L m \ .3889 "8.5. >I< -9- Using this equation, the effect of applied voltage is to gain of an equal ten stage tube Operated at 1000 volts can be computed and gives the approx- imate relationship: percent gain shift?! 20 1: percent voltage drift. Crouthamel (6) reports a coefficient of 7 to 8. This probably represents the operation of the tube with the first dynode at a higher voltage than the remaining stages. This is comonly done to improve resolution. (1) As will be mentioned in the next section, the voltage from a battery high voltage supply is temperature dependent. The high voltage stability also will be a function of the age of the battery ”and the load that is being drawn. It The characteristics of the pulse height analyzer, as with any electrical equipment, will vary with the line voltage applied. The ex- tent of the variation will be a function of the design of the instrument, particularly the initial regulation of line voltage. Tmerature. Temperature effects on NaI(Tl) light output and photo- multiplier gain have been reported by a large number of investigators. (1) It is also known that components of electronic circuits change value with temperature, resulting in a change in circuit characteristics. (6', l9) Flanagan (20) reports a temperature coefficient of -0.12%/°C for 1light output of Halal) crystals over a range of 20 to 150°C. This cor- l"Qmponds to a‘ shift of -0.12 channelsl°c in channel number 100 for a channel cOefficient setting of 10 Kev/channel. Photomultiplier tubes with AngCs dynodes and 8-11 type response ‘how a temperature coefficient of approximately -O.357-/°C for light with ‘ wave length of 4200 angstroms. (1) This has been explained by Causse (3) ‘. due to the relative high vapor pressure of cesium in the photocathode and dynodes, resulting in evaporation or deposition as tenperature varies. -10- Specifications (22) of the pulse height analyzer used in this study state a bias stability of 0.06%/°C and a gain stability of 0.22%/°C. Unfortunately the direction of the changes are not given. The detector high voltage battery supply is also temperature dependent. For a range of 10 to 30°C, the temperature coefficient of a type F carbon battery is -0.012%/°C. (36) This results in a gain shift of -0.241/°C. The conversion from voltage shift to gain shift has been enumerated in the previous section. Photomultiplier tube fatigue. Fatigue has been defined by Linden (18) as the non-reversible pulse height shift caused by a decrease of 3a in in the photomultiplier tube. This is as opposed to the short term reversible pulse height shift effected by changes of the counting rate .nd usually resulting in increases of gain with increased count rate. The theory of fatigue is based on the known diffusion of free cesium .t room temperature in a vacuum. As electrons of energies greater than 100 ev (molecular binding energy of Cs38b0) strike the dynode surface, they can dissociate the surface moluecules, decreasing the cesium content ‘nd thus changing the dynode gain. (2) The last dynode, where the electron energy is greatest, is effected to the largest extent; but earlier stages are progressively affected as the current being amplified increases. The magnitude of the effect is widely variable even among tubes of the same type. (16) A short term reversible pulse height shift has been 9.222 me. This has °bOerved (18) which is a function of the change in count rate. been attributed to a temporary change in the secondary emission rate of the dynodes causing a gain change '(18), and to pulse pile up or overload in the amplifying stages of the pulse height analyser causing a bias ch-nge . (21o) -11- Methods of Energy Response Measurement The energy response of a gamma spectrometer is usually represented mathematically by a linear equation (31, 37) of the point-slope form y - a + bx 3‘10 1 where a is the energy represented by channel sero, b the channel coeffi- cient in Kev/channel, y the energy of the photon in Kev, and x the channel under representing energy y. It is assumed that a and b are mutually independent, that a is a function of the bias of the system (controlled by the analog-to-digital conversion unit), and that b is a measure of the. total gain of the system. The problem of locating at precisely (to some fraction of a channel nutter) has been treated by many different methods. The peak location c‘n be determined visually to within plus or minus 0.25 channels provided e“flush counts have been collected to give a reasonably smooth spectrum. The shape of a total-absorption photo peak is observed to be ne‘rly Gaussian in the upper portion. (14) However, Managan (20) points On: that because of the idealized nature of the derivation of this shape, the photopeak must be expected to have systematic deviations from the G.‘nsaian. The deviation becomes very noticeable when the peak is en- cumbered by the Compton continuum of higher energy peaks and by other In"Otopeaka. A graphical method for peak location makes use of the near sym- “try of the upper portion of a photopeak. (23) A horizontal line inter- '§°t1ng the peak at approximately one-half the peak height is drawn. The b1'..<=t of the portion of this line under the peak is selected as the peak loc‘tion. Ziuerman (38) proposed the use of probability paper using the f 1ft19th percentile point as the center of the distribution. This has the :5‘ wet; -12- advantage of using all the data in the peak area. The location can be found to within plus or minus 0.05 channels. Gold (10) has used a parabolic fit selecting the three upper points in a peak. The first derivative is set equal to sero and the corresponding channel location solved. Robinson (27) used an iterative least-squares fit for correcting an estimated central point of a Gaussian distribution. Data points were limited to the upper twenty percent of the photopeaks. Salmon (28) also performs a least-squares fit of a Gaussian function using the iterative Newton-Raphson method. Heath (13) has developed a very sophisticated program which allows for least-squares fitting of any combination of up to eight functions of Gaussian curves, straight lines and/or exponentials. The procedure is able to locate a peak position to 'within plus or minus 0.02 channels. Effects of Instability Upon Composite Spectrum Resolving Methods Composite gamma spectra may be resolved into their component parts by basically three methods commonly referred to as spectrum strip- ping, simultaneous equations, and least-squares methods. There are many variations of these basic procedures being used; (25) however, they all have in common the elements described below. Spectrum stripping 555929. Spectrum stripping consists of the successive subtracting of reference spectra from the composite spectrum. Normally the procedure is to start at the highest energy peak, preferably unencumbered, and to progressively work toward the lower energy peaks. The method becomes somewhat subjective when encumbered peaks are encountered. A.major disadvantage of this method is that any error intro- duced during a subtraction process is accumulative and may make the analysis of the low end of the energy spectrum impossible. (11) -13- Simultaneous equations Egghgg. The simultaneous equations method involves the solution of "n" simultaneous linear equations (n is the number of nuclides in the composite spectrum). The equations represented in metric notation are N -I x (nxl) (nxn) (nxl) Eq.2 where i is a column matrix representing the observed count rate in the energy spans selected to represent each of the nuclides, i the unknown column matrix representing the quantity of each nuclide in the composite spectrum, and E the square library matrix of reference spectra which represents the contribution of one nuclide to the energy span represent- ing another nuclide. (28) flagee and Karches (12) have applied this method to the routine analysis of fluid milk. They point out the need for instrument stabil- ity and indicate an approximate change of eight percent in the values of the i matrix.with a one channel shift in channel 73 for a channel co- efficient of 20 Kev/channel. No analysis of the effect upon the solution was made. Seefedlt (29) studied the effects of channel shift in fission pro- duct spectra by moving entire spectra plus and minus one integer channel. This corresponds to introducing a bias shift. He found errors as large as 75 percent for photopeaks located close to each other but concluded that the simultaneous equation method was less sensitive to channel shift than the least-squares method he was also examining. Least-squares method, The least-squares method is similar in con- cept to the simultaneous equation method except that all data in the spectrum may'be used. -14- The equations whose solution yields the activity content are in matrix form, ET a I 2 IT a i (In) (an) (an) (and) (run) (an) (nxl) where E is the library matrix of reference spectra, ET the transpose of matrix I, i the diagonal matrix of statistical weight factors, i the unknown matrix representing the quantity of each nuclide in the composite spectrum, 3 the column matrix representing the observed count rate in each channel of the composite spectrum, m the number of channels used, and n the number of nuclides. The advantage of the method lies princi- pally in the ability to handle spectra of low counts which have large statistical fluctuations. (28) However, the method is very sensitive to channel shift. (29) Health (18) has used a mathematical approach to correcting for gain and bias shift. The spectrum is represented by a series of poly- nomial least-squares fits to sets of three channels. These successive fits are then used to interpolate between the original points. The only modification of the original data for small shifts is to produce a slight smoothing of the data. Parr and Lucus (26) accomplish gain and bias corrections by linear interpolation between the observed channel counts which they state takes into account the fact that the only truly correct representation of that data is as a histogram. CHAPTER III DEVELOPMENT OF BASIC METHODS In order to determine energy response stability it is first necessary to define energy response precisely and then to have a method to accurately determine it. The most direct way of determining it is to observe the relationship between gamma photon energy and the position of the corresponding photopeak. In addition, to study the effects of energy response instability upon spectrum resolving methods, it is necessary to induce controlled changes of energy response. A mathematical spectrum normalisation procedure was selected as the most feasible. Thus the basic methods needed for this study deal with the deter- mination of peak locations, determination of energy response, and syn- thetic shifting of spectra to bias and channel coefficient conditions other than those from which the spectra were obtained. Equipment and Operating Conditions All equipment and operating conditions under which the data were collected were essentially constant throughout the course of the investi- gation, except when a single parameter was varied to study its effect. ‘Multichannel gaggg_spectrometer system. Detector: cylindrical 3 inch by 3 inch diameter NaI(Tl) scintillation crystal with a 1 7/8 inch by 1 1/8 inch diameter counting well, crystal canned in 0.032 inch thick aluminum. Photomultiplier tube: two inch Dumont 6292, type S-ll response, AgMgOCs dynodes, Cs3Sb photocathode, in use for three years. Background shield: cylindrical with minimum 3 inch lead shielding in all directions, shielding in nominal contact with detector except for access hole to well. -15- -l6- Pulse height analyzer: 512 channel computer type (analog-to-digital conversion) with channel integrater, 6 decade storage, Nuclear Data Inc., model 130A. Readout: oscilloscope (dead display only) and IBM electric typewriter with print speed of 70 channels per minute. Detector high voltage: battery pack composed of 36 Everyready #507 carbon batteries, 900 volts DC nominal, Nuclear Data Inc., model 202. The normal control settings of the analyzer are given in Table 1. TABLE 1. NORMAL CONTROL SETTINGS OF SPECTROMETER Channel Coefficient in Kev/channel Control 5 10 20 Sub group 0-511 0-255 0-255 Group size 512 256 256 Ramp slope 512 256 256 Course gain 8 4 2 Fine gain* 450 550 550 Bias* 650 600 600 *Nominal Source and counting geometry. Sodium 22 was used as the refer- ence source exclusively. Its selection was based on the following con- siderations. The sources retained essentially the same activity over the period of experimental work because of their relatively long half-life of 2.6 years. (35) The resultant spectrum, when counted in a well geometry, contains five readily discernible peaks. Theoretically the peaks corres- pond to the 0.511 Mev annihilation photon, 1.022'Mev annihilation summation, 1.274 Mev gamma photon, 1.785 Mev gamma and single annihilation summation, and 2.296 Mev gamma and double annihilation summation. (35) The peaks will be referred to as peaks 1 through 5. The decay scheme of sodium 22 -17- is given in Figure 2. These peaks cover a fairly wide range of energy and are evenly spaced throughout the range as seen in the gamma spectrum given in Figure 3. In addition, the first four peaks have peak-to-peak ratios (count rate in peak 1 to count rate in peak 1) of less than 9.5, resulting in counting statistics which vary by less than a factor of 3.1 from one peak to the next. .Nar22 Half life - 2.6 years \_/ 0. 544 ,HeV; 89.828 +, 109,213.06. .1.274 Mev ”1.83 Mev; 0.06284; ing-22 “iStable Figqre 2. Decay Scheme of Sodium~22“(35) Sources were prepared from a stock solution of 1.33 uc/ml plus or minus 10 percent, carrier free sodium 22. Diluted aliquots were evaporated to dryness in a 1 inch by 3 inch plastic counting vial and then sprayed with an acrylic coating to fix the sample to the vial. All counting was done in the well. (Lies voltage control and monitoring. The line voltage was held at 116 volts plus or minus 1.0 percent by use of a sole CVH model constant voltage transformer with a R-type filter. Visual observation of the wave form on an oscilloscope indicated no wave distortion and no observable line noise. -13- NN summon mo souuoonm assoc .n smouwm amass: accumnua ecu as" o- can can see can cum cam on cm as cm -a ___ __________m__e______ mouacflfi cm «06.3 muggy—.59 II Homumnu\>uu ca "umowuamwoou goddamn. u: No.o ":uwsouua ounsom «N seamom “momoom >0: oo~.N n anon >u: nm«.d a neon >otes«.~+ n seem Ill >0:_-o.~ . N seem * 5: a...” 3 ages «ad ad ~a to will {euuaqo sax OI/suunoo m C, J was -19- Instrument voltage was periodically checked with a vacuum tube voltmeter. A Digetec voltmeter was used for monitoring voltage during the voltage dependence experiment. A variable voltage transformer was used to vary instrument voltage. ngperature control and monitoring. The room temperature was controlled by a five ton water cooled air conditioning unit. Ambient room temperature was held at 23.2506 plus or minus 0.7500 with periodic temperature cycles of approximately eight minutes. Typical cyclic changes of the room temperature are shown in Figure 4. Cycle A represents a period when the outside temperature was relatively low for the preceding 48 hours, while cycle 8 was taken during a period of extended high temper- ature. The mean temperatures were computed to be 23.9°C and 23.6°C for cycles A and B respectively. These compare favorably to the observed detector temperature of 23.8°C and 23.5°C. The difference of 0.1°C in the values is due to temperature variation within different parts of the room. It should be noted that the mean temperature decreased when the out- door temperature increased. Normal changes of outdoor temperature (plus or minus 10°F) over a 24 hour period without extended periods at either temperature extreme produced less than a 0.1°C fluctuation in the detector temperature. The room.temperature was monitored with a mercury thermometer (10 to 110°C, 1°C/division) suspended twelve inches above the analyzer cabinet. The detector temperature was measured inside the shield directly below the photomultiplier tube with a mercury thermometer (30 to 180°F, 0.5°F/division). The instrument temperature was measured with a remote element thermister thermometer located inside the pulse height analyser adjacent to the amplifier and ADC circuitry. The thermister and Fahrenheit -20- thermometers were calibrated against the centigrade mercury thermometer. All equipment including room lighting was left on at all times to provide a constant internal heat load. Air conditioner cycle Lj_u_‘ off _J5§LJ“ off "— I‘jfi . Erl‘ 71 __ 0 Cycle A, outside temperature - 69° 0 Cycle B, outside temperature - 91° IN 76’ .° -- I re 75 '—' ’ Temperature - o Fahrenheit Temperature - ° centigrade N w 73l'llllllllllllliill o 2 4 6 8 1o 12 14 16 18 20 Reference time - minutes Figure 4. Ambient Room Temperature Cycles Detector conditionigg. To eliminate the effect of count rate dependence, the detector was exposed to a sample for 12 to 48 hours, depending on source strength, before counting of the sample to allow for stabilization. Source strengths were normally below that which cause detectable long term fatigue over a 48 hour period. The instrumentation used during the study is pictured in Figure 5. Digital ngputer. A Control Data Corporation 3600 digital computer was used. This is a solid state, stored program, general purpose computing system constructed in modules. The storage module is a random access magnetic core unit with a 51 bit word capacity of 32,768 words. Peripheral equipment used were an IBM 026 keypunch, IBM 56 verifier, IBM.407 lister, IBM 514 reproducing punch, IBM 557 interpreter and CDC 165-2 digital plotter. -21- A11 programing was done in FORTRAN 60. The machine program for the CDC-3600 was compiled on a CDC 160-A computer with 8,192 twelve bit word storage. T .. v Q ' (A; ‘ Pulse height analyzer Oscilloscope display Typewriter output Instrument temperature monitor Room temperature monitor Digitec line voltage monitor VTOM line voltage monitor Variable transformer Line voltage regulator and filter Detector high voltage battery pack Shield containing detector Detector temperature monitor r~71c4r4=ngauunucac>us>- Figure 5. Experimental Instrumentation -22- Peak Location The determination of peak location is complicated by two conditions not controllable in the general solution: non-systematic statistical variations and systematic non-uniformity of the observed peak distributions caused by compton continuum and overlapping photopeak contributions. In order to minimize the effects of the statistical variations, it would be desirable to use as much of the photopeak area as possible. This is in contrast to the conventional procedure of only selecting the upper 50 to 20 percent of the peak, which eliminates much of the systemmatic non-unifonmity of the peak. For purposes of this investigation, the problem of systemmatic deviations is eliminated by considering only the peaks of sodium 22 and a single counting geometry. The general method of peak location used was determining a mathe- matical expression for the peak portion of the spectrum. The peak location is then defined as the channel number where the slope of the mathematical expression of the spectrum is zero. Procedure. Spectra of sodium 22 were obtained under constant con- ditions with approximate channel coefficients of 5, 10 and 20 Kev/channel. Three spectra were obtained at each channel coefficient to verify that a gain or bias shift did not occur. The middle spectra were then used for all further analysis. The spectra contained two million or more counts collected at a rate of 54,600 counts per minute, producing a statistically smooth curve. The portions of each spectrum which correspond to the five peaks were visually selected and subjected to least-squares fitting with poly- nomial equations. -23- It should be noted that the least-squares fitting method is being used simply as a systematic computational procedure and not as a statis- tical curve fitting method. The variance of each channel count is not equal, and the fitting equation does not necessarily represent the true form of the data. The peak locations were estimated visually (usually by selecting the channel with the highest count) to within the nearest integer channel number. Polynomial equations of degree 2, 3 and 4 were generated using the count and the logarithm of the count in the estimated center channel and an even number of points on either side. An even number of points will give- an approximately symmetric portion of the peak. The total number of points used varied from one plus the degree of the fitting equation to the peak width. The general forms of the fitting equations are: n Eq. 3 y = coxo + clx1 + c2x2...... + cnx log y = coxo + clx1 + czxz...... + cnxn Eq. 4 where y is the count in channel x, n the degree of the equation, and c1 are constants. Using these fitting equations, the counts in each channel were computed along with differences and percent.differences between the original and the computed count. The sums of the absolute differences, as a function of the number of points used to generate the equation and the number of points summed, were also calculated. The peak location was then determined by taking the first derivative of the fitting equation and setting it equal to zero. The peak location was then defined as the real root closest to the estimated channel. -24- The stability of peak location as a function of area of spectra used for the curve fitting was investigated for the arithmetic second and fourth degree and logarithmic fourth degree equations. The starting channel of the peak fit was varied plus and minus two channels from a symmetric portion of the peak.while keeping the peak width constant. The peak width was next varied from three channels up to the full peak width. The fitting equation and peak width for ultimate use in the peak fitting method was selected on the basis of greatest stability. The reproduceability of peak location as a function of total counts collected and collecting period was investigated. Sets of ten spectra were obtained for counting periods of 200, 80, 40, 20, 10, 4 and 2 minutes under constant conditions. Between each set of spectra an additional 80 minute spectrum was taken to observe the long-term stability. The sample used produced 54,600 counts per minute. Two additional sets of 40 and 20 minutes were taken at a count rate of 5,400 counts per minute. The standard deviations of the peak locations about a linear regression line were calculated. ngputer programs. The programs developed for selecting a method for location of peak channels are coded MHDDELL and.MHDDELA. The last letter in each program code (L and A) refer to the form of the data used in the peak fitting. L refers to logarithmic and A to arithmetic. The necessary input data to the program are the number of spectra to be analysed, number of channels and peaks in each spectrum, peak width and the starting channel of each peak, maximum degree of the fitting poly- nomials, printing control information, and the spectral data. A complete description of the data deck is given in Appendix A. The only additional caution necessary in setting up the data deck is that the spectrum data -25- must contain a zero channel. This usually contains the time of the count although it is not used in this program. The program consists of four major DO loops" (repetitive opera- tions) as shown in the flow diagram given in Figure 6. The loops corres- pond to the spectrum being used, the degree of the fitting polynomial, the peak being analysed, and the number of channels being used in the peak fitting. All operations within a D0 loop are repeated, including inner DO loops, for a specified number of cases each time the loop is entered. The general pattern and computational methods of the program are as follows. The capital letters refer to the steps shown in Figure 6. This type of reference will also be used in the description of subsequent programs. All data is read into memory (A) and selected portions of it printed at the control of the investigator (B). The indices of the four major D0 loops are set (C,D,E,F). The spectrum index (C) goes from one to the number of spectra used; the peak index goes from one to the number of peaks (D). The degree of fitting polynomial (E) goes from two to the maximum degree set, and the number of fitting points (F) decreases from the peak.width to one plus the degree of the fitting polynomial in increments of two channels. The peak is fitted to a polynomial (equations 3 or 4) based on the conditions of the above four indices (G). The equation is computed by the method of least-squares using orthogonal polynomials as proposed by Forsythe. (8) The peak location is computed by taking the first derivative of the fitting equation and solving for the channel number where it is equal to zero (B). Newtons method is used for finding the roots of third -26- r 1 START r STOP: MAX. DEGREE #a— 810?: T 0? SPECTRA A FREAD DATfl E IT’RINT INPUT DATA! C FfiLECT SPECTRDlfl-4; .— l D SELECT DEGREE OF L FITTING 1’0me l E [SELECT Emu-e F— I F SELECT NunsER OF FITTING POINTS 1 C CALCULATE FITTING EQUATION I I— E CALCULATE PEAK L... 1%TION . N I CALCULATE COUNTS g .. FROM FITTING EQUATION SE3 _. m g I 3 u h J CALCULATE DIFFERENCES § H o a: AND PERCENT DIFFERENCES 3 + 3*- E SELECT CHANNELS -- .. g E... FOR 8111103101: E g m 3 a 5 "’ f 3 l a Q + n. I" g a: s E L l , CALCULATE SUMHATION OF ABSOLUTE DIFFERENCES I L n STORE SM I N PRINT RESULTS 0 ERIN: 3mg r Figure 6. General Flow Diagram of MODEL -27- degree equations. The new counts in each channel, based on the fitting equation, are calculated (1). No problem of round off error was found; however, register overflow did occur when working with spectra that contained six digit information. Therefore, all data is scaled down by a factor of 100 before curve fitting and returned to the original scale after calculating the new counts. Differences and percent differences between the original channel counts and the computed channel counts are calculated (J). The index for summing the absoluted differences is set, varying from the peak width to one plus the degree of the fitting polynomial (K). The summations are computed (L). Selected information is calculated or stored for a summary of results (n) and the selected portions of the results are printed out at the control of the investigator (N). At the completion of all calcu- lations, a summary is printed consisting of peak locations and the change in peak locations as the number of fitting channels is increased. The program is written in a general form so that it can be used with other curve fitting methods with a minimum of adaptation. The opera- tions which are a function of the fitting method are G, n, I and to a limited extent D. These steps correspond to the subroutines FOLTPT, RENT, and FEAR 2, 3, 4 of the actual program. The entire program is given in Appendix B. Energy Response Model The idealised energy response model given in equation 1 deviates from the actual observed response in two major respects: non-linearity and bias dependence upon gain. -23- The light output of NaI(T1) is not linear over the entire energy range with a marked deviation below 100 Rev. (6) This shows up particu- larly when pulses are summed. The analyzer amplifiers and analog-to-digital converter may also be non-linear, usually at the upper end of instrument limits. (31) The assumption that gain or bias are mutually independent is not always true, but rather the relationship is a function of the analyzer design. For them to be independent, the response line must pivot about the x-intercept when the gain is changed. In many analysers the line actually pivots about a point which has both negative x and y coordinates. The problem of non-linearity gives little trouble in qualitative analysis provided it is constant, and the analysis takes into account the non-linearity. The dependence of bias upon gain can be handled either by reinterpreting the x-intercept or by redefining bias. ‘zhgggz‘gg method. With the above factors in mind, a new energy response model is defined as Y - (B-Px - Py) + B°X, Eq. 5 where x is the channel number; 3 the slaps of the response line and will be referred to once again as the channel coefficient; Y the apparent energy represented by channel X, and equal to the true energy plus a correction factor for non-linearity; and FX and Py the coordinates of the pivot point of the response line. Figure 7 shows graphically the derivation of equation 5. The reaponse line in terms of the prime coordinate system is T' - B-X'. Transferring this to the observed coordinate system gives Y+P -B(X+Px). y Regrouping of the terms gives equation 5. The term B'Px - P corresponds Y -29- to the Y-intercept, thus the x-intercept is (B'Px - Py)]B and defines what shall be called the "observed bias". $.54 Energy Channel Observed bias point ——————_—....~ (15.13) Figure 7. Energy Reaponse M¢de1 The pOint Px,P is defined as the bias point and is a direct measure Y of the bias of the analog-to-digital converter. However, difficulty arises in using this as a measure. As there are two unknowns, FX and Py, it is necessary to determine the response lines for two widely varying gain set- tings and then solve for the point of intersection, which is the bias point. As Py approaches zero, Px approaches the x-intercept. Thus if Py is small, it can be neglected and gain and bias independence assumed. Equation 1 would then be valid if the apparent energy is substituted for the true energy. Procedure. Non-linearity correction factors were obtained by deter- mining the average peak locations in twelve 10 Kev/channel spectra taken under constant conditions. The spectra each contained 4,367,000 counts. The peak location was determined by using the arithmetic fourth degree polynomial fit. The 0.511 Mev and the 1.274 Mev peaks were used to -30- determine an energy response line. Using this response line, the ap- parent energies of the remaining three peaks were determined. The correction factors were calculated as the theoretical energy minus the apparent energy. The same procedure was appliesto single spectra taken at 5 Kev/channel and 20 Kev/channel. The characteristics of the pivot point were investigated by obtaining six spectra at permutations of two gain and three bias settings. The pivot points were found by graphically determining the points of intersection of the response lines. The reproducibility of the energy response line was determined for a single counting condition. Fourteen repetitive spectra were taken at a channel coefficient of 10 Rev, counting time of 20 minutes, and count rate of 54,600 counts per minute. Standard deviation about a linear regression line was calculated. ngputer programs. Three computer programs were used coded OORFAC, BIASPT, and REPROD. The general flow diagrams for each program are given in Figures 8, 9 and 10 respectively, while the programs are given in Appendix B. The input data for CORFAC are the number of sets of spectra, number of peaks in each spectrum, number of spectra in each set, true energies of each peak, number of channels in each spectrum, the peak width and starting channels of each peak, and the spectral data. The data common to all spectra are read into memory (A). The spectrum set index is set, varying from one to the number of sets. The data common to the spectra in a given set are read (C); and the spectra index is set, varying from one to the number of spectra in the set (D). Spectral data read (E), and the location of each peak in the spectrum is determined (F, G). when all peak locations of all spectra in a set have been determined, -31- the mean peak locations and their standard deviations are calculated (H). Using the mean peak locations of peak 1 and peak 3, an energy response line is calculated (1) based on the true energies of these peaks. The apparent energies based upon the above response line and the mean peak locations are found. The correction factors are computed (K), and the results printed out (L). The input data for BIASPT are the number of spectra, number of channels and peaks in each Spectrum, width and starting channel of each peak, gain and bias instrument settings for each peak, and the spectral data. The control data are read in and portions printed out (A, B). The spectrum index is set (C), and the spectral data read in (D). The peak locations are determined with a fourth degree arithmetic polynomial least- squares fit (E), and the locations used in determining the energy response line with a linear least-squares fit (F). The results are printed out (G). REPROD uses the initial steps of CORFAC, steps A through G being identical. At this point the peak locations have been calculated. The peak locations are used to determine the energy response line (H; I). At the completion of each set of Spectra, the mean peak locations, bias channel, and channel coefficient and their standard deviations are calcu- lated (J). The results from each Spectrum in the set are printed (K). At the completion of all spectrum sets, the mean results of each_set are printed (L) s t" Figure 8. READ MASTER CONTROL DATA L # ISELECT SPECTRAL SETI E READ SET CONTROL DA A_ [SELECT SPECTRm FREAD SPECTRAL DATAJ [SELECT PEAK CALCULATE PEAR -32- -START: I :-#’OI.PEAKS STOP l START: 'STOP: #‘OF SPECTRA 1 { .__LQQAI.IQ!__ CALCUATE'HEAN PEAR LOCATIONS AND STAIDARD DEVIATIONS [ CALCULATE RESPONSE LINE TRON PEAK 1&3 CALCULATE APPARENT gm?— CALCULATE CORRECTION FACTORS 1 1* OF SETS[ ESTART: TOP: FL {PRINT RESULTs} EL General 3.19". 91.98.14]! 015 CORP“: -33- A READ CONTROL DATA B PRINT CONTROL DATA C [ SELECT SPECTRUNF——‘ D READ SPECTRAL ‘ 3 DATA F‘s; {-3 3 E a. 5:}. E COMPUTE FEAR 3 LOCATIONS an F COMPUTE ENERGY RESPONSE LINE G PRINT RESULTS a Em] Figure 9. General Flow Diagram for BIASPT, CNTRTE, TEMP, and VOLT -34- 9 START: I ! l # OF SETS START: STOP! ”"'8TOP: #‘OF SPECTRA ‘— A [R—EAD MASTER CONTROL DAT_AJ B SELECT SPECTRAL SETj-e C lREAD SET CONTROL DAT_A] D [STZLECT SPECTij —— E [READ SPECTRAL DAM F [SELECT PEAK}: H g :13 S E m- E4 (a so as G CALCULATE PEAK J LOCATION H CALCULATE ENERGY RESPONSE LINE I CALCULATE BIAS, CHANNEL COEFFICIENT AND STANDARD DEVIATION J CALCULATE HEAN PEAK LOCATIONS, BIAS, CHANNEL COEFFICIENTS AND STANDARD DEVIATIONS x PRINT PEAK LOCATIONS, BIAS, AND CHANNEL COEFFICIENTS L PRINT MEAN PEAK LOCATIONS, BIAS, AND CHANNEL COEFFICIENTS H | END] Figure 10. General Flow diagram of REPROD -35, Spectrum Normalization Spectrum normalization is the synthetic shifting of the energy response line of a Spectrum by mathematical means. The ideal normali- sation procedure would produce a shifted Spectrum identical to one which would have been obtained if the instrument conditions had been changed. The problem is essentially one of interpolation between the ob- served channel counts of the Spectrum. Unfortunately difficulties arise when attempting to set up criteria for performing the interpolation. The energy distribution of a gamma-ray Spectrum is continuous; whereas, the observed spectrum is discontinuous, Since each channel represents an energy Span rather than a discrete energy. The observed Spectrum is further distorted by statistical variation in the instrument's ability to measure the energy transferred to the crystal. Without know- ledge of the nuclide composition of a complex Spectrum and the relative amounts of each, it is impossible to make a quantitative statement about the true energy distribution. ‘ghgggy'gf method. In lieu of information about the exact energy distribution, the criteria for interpolation were based on the following observed qualitative characteristics of gamma spectra. 1. The energy Spectrum represented is contiuous. 2. The SIOpe of the Spectrum does not change rapidly. Mere precisely, both the first and second derivative can not change Sign more than once within the span of one channel. This is based on the inherent resolution of the photomultiplier tube. 3. The area under the curve is constant for any given energy interval. The total number of nuclear decay events is assumed constant with only a statistical deviation in the distribution of these counts. -36- 4. The Spectrum must pass through the observed points. Four methods of interpolation were investigaamizl) linear interpo- lation between each point; 2) exact fitting of a data points to a n-l degree polynomial (n equals number of channels in the spectra); 3) exact fitting of n data points to a n-l degree polynomial with the additional condition that the slape of the curve at each channel be equal to the Slope between the points x-l to x+1, where x is the channel number; and 4) exact fitting of the data with the sum of n seventh degree polynomials. The last method was the only one examined in detail. Basically the normalisation procedure, coded TSBIFT, involves the sequential fitting of groups of four Channels to a seventh degree polynomial. The fitting of each set of channels is done by solving eight simultaneous equations; namely, Cj - clx; + czxfi...“ + cax‘} Eqs. 6 (C34 - cj+1)/2 - 7c1x3’ + 6c2x3...” + c7x3 Eqs. 7 C3 is the count in channel Xj, c1 through c8 are constants, and j goes from 1 to 4. The first set of equations (Eqs. 6) satisfies the requirement for the fitting equation to pass through the original data points. These alone, if reduced to third degree polynomials, would allow existence of one and only one inflection, maximum, or minimum point within one channel interval. The second set of equations is essentially a condition of smoothness of the curve and prevents large oscillations. The effects of adding this condition are shown qualitatively in Figure 11. However, the criterion of only one maximum, minimum, or inflection point within a one channel Span is now violated. This is somewhat alleviated by assuming that any non-smoothness of the curve between channels is due to a random variation -37- of the points. Thus the best estimate of any given point on the Spectrum can be taken as the average of four fitting equations generated from a group of seven channels centered on the channel nunder consideration. ° Fitting channels I CB/ __ Equations 6 ///I —-"—- Equations 6 & ? //’ Counts Channel number Figure 11. Qualitative Effect of Smoothing Equations The fitting procedure can be summarized into the mathematical model 1'4 C1 - l E Pj,1_4(x1+j_4), i = 1, 2,...n. Eq. 8 4 j-l where C1 is the count in channel 1, and Pj 1_4(X1+j_4) is the seventh degree polynomial defined by equations 6 and 7 with the addedlrestriction that Pj 1409,14.) is equal to zero when X1+J_4 is greater than X1-4 or less than X1. The sequential groups of four channels about channel 1 are referred to by j, and i-4 is the first channel used in these groups. X1 is the channel number i and n is the number of channels. This model is used to generate a new spectrum with a channel coefficient and bias channel other than the original values. A pure bias -33- Shift (SB) results in each channel being displaced by an amountAEXb; thus the new channel locations (Xi) after a bias shift are Xi-i+AXb-1+AB, 1-1, 2......n. Eq.9 A pure gain shift (aG), assuming a pivot point at the origin, results in each channel being displaced by an amountFAXf which is proportional to the channel nhmber 1. Thus x; - 1 +Axf - 1 +1; c-x Eq. 10 i where G is the fractional change in gain. For convenience,Z&G can be expressed as the Shift due to gain change in a reference channel number. For S nominal channel coefficient of 10 Kev/channel, the shift in channel 100 is equal to percent change in gain. The total channel shift can then be written x: - Ax: + Axg. Eq. 11 substituting equations 9 and 10 into equation 11 gives xf -AB + AG-Xi, or the new channel locations are x; - i + AB +AG-X1. Eq. 12 If xi is then substituted into equation 8, 01 becomes Ci, the counts in the new shifted Spectrum. However, in order to conserve the area under the Spectrum, each channel count must be multiplied by the fractional change of the energy interval,zhxi, represented by the channel 1. The condition of constant area is stated as Ci Ci AT 'AT" - constant; thus. 1 1 Ci - éK' Ci Eq. 13 AK Since the channel coefficient (B in equation 1) is Elk/channel, equation 13 can be written as ' ' ' C1 - ! Ci. B A! I 17!: ‘ctt Soc 0:, i‘» 5.3%: lg- 'PE: .03 -39- Substituting this into equation 8 gives the complete mathematical model for Shifting Spectra: 1-4 Ci ' AL; 2.11 Pj’1'4a1+j'4)’ 1 I 1, 2,... n. Eq. 14 The main assumptions are the total channel Shift is small so that linearity over similar energy intervals is unchanged, and the response line pivots about the origin. If the gain shift is expressed as the shift in channel 100, then . -4 C1 - [I - 6100 .1; f— P ,1-4(x1+3-4) Eq. 15 TM 4 17f ‘1 Procedure. The reasonableness of the preposed normalising model was investigated by three approaches. Identical analyses were done on two Spectra: one containing 7,560,000 counts representing a statistically smooth Spectrum, and one containing 10,000 counts showing marked statise tical variation. l ‘ First a visual comparison was made of the fitted curve to the original Spectrum. The complete fitting curve was generated by introduc- ing into the original spectra bias Shifts of 0.1 to 0.9 channels in incre- ments of 0.1 channels. This was done with program CKSEIFT. Next a quantitative comparison between the fitted curve and original Spectrum was made by generating a curve from every other channel of the Spectrum and calculating the predicted count in the channels of the Spectrum not used. The difference between actual and predicted count was found. These steps were done with program CKFIT. Finally a comparison was made between the amount of gain and bias shift introduced in a Shifted Spectrum and the measured amount of shift. The Spectra were shifted with gain held constant and bias variations of 0.0 to 2.0 channels in increments of 0.4 channels, and the bias held con- stant and gain variations of 0.0 to 1.0 percent in increments of 0.2 to.“ -40- percent. This analysis was done with program CKERL. Cgpputer programs. The actual spectrum shifting is accomplished with Subroutine coded TSHIFT. A general flow diagram is given in Figure 12. The subroutine is entered with the original Spectrum, the number of channels in the Spectrum, and the amount of gain and bias shift (A). The independent terms of the simultaneous equation are computed (B). The Spectrum is extended four channels in both directions by letting the counts in the added channels equal the counts in channel one and the last channel respectively (C). The channel group index is set varying from one to the number of channels plus four (D). The first channel group Consists of channel minus three to channel sero. The constant terms of the simultaneous equations are calculated for the channel group (E). The fit- ting equation for the channel group is generated by solving the simultaneous equations (F), and the eight coefficients are stored as SET - 3. The previous values for SET - 3 through SET + 2 are moved up to SET - 2 through SET + 3 (G). The total shift for each of the four channels in the channel group is calculated (R). Based on the amount of total shift, the correct SET of the 'fitting equation coefficients is selected (I). This is done by selecting the closest equation whose boundry limits include the channel in question. The channel counts are computed from the selected fitting equations (J) and stored as SET A. The previous values of SET A through SET G are moved up to SET B through SET D (K). The predicted channel count is then taken as the average of the four counts evaluated from the four equations whose boundary limits include the channel (L). A new channel group is selected. (0), and the whole process repeated until all channels have been con- sidered, at which point the subroutine is exited. The program for Sub- routine TSHIFT is given in Appendix B. -41- The program CKSBIFT and CKFIT are simply schemes to assemble the information needed to enter subroutine TSEIFT and to vary the gain and bias by the desired amounts. The flow diagrams are given in Figures 12 and 14 reapectively, and their programs in Appendix B. Program.CKERL involves a little more manipulation of data. The flow diagram is given in Figure 15. The starting channels, peak widths, apparent energies, and Spectral data are read in (A). An index controlling which energy response parameter is to be varied is set (B).. The index goes from one to three, where one indicates a bias variation, two a gain vari- ation, and three both a bias and gain variation. Next an index control- ling the amount of shift is set. It goes from one to six and corresponds to increments of 0.4 for bias and 0.2 for gain. The total Shift is calcu- lated (C), and the shift incorporated by using TSEEFT (D). The peak index is set varying from one to four (E), and the starting channel for the peak fit selected (F). To assure using approximately the same portion of the peak, it is necessary to change the starting channel from that given for the unshifted spectrum. This is done by considering the amount of shifting in the original starting channel, and then adding to this the original starting channel and truncating to an integer Channel number. The peak lo- cation is determined (G). At the completion of all the peaks in the Spectrum, the energy response line is calculated (B). After all shifts have been performed, the results are printed out (I). -42- A ENTER wITH SPECTRUM, GAIN SHIFT, AND BIAS SHIFT E COMPETE INDEPENDENT TERMS OF SIMULTANEOUS EQUATIONS C CONSTRICT EXTENDED SPECTRUM D SELECT GROUP OF FOUR CHANNELS E COHPUTE DEPENDENT TERMS OF SINULTANEOUS EQUATION F CALCULATE FITTING EQUATION .__::. G CHANGE STORED ,,*' FITTING EQUATIONS ,ig H CALCULATE CHANNEL Eg" SHIFT r—4 as m- 1 SELECT FITTING §§" EQUATION ”’gi cs J CALCULATE CHANNEL COUNT f FROM FITTING EQUATION K CHANGE STORED channel count L CALCULATE CHANNEL COUNT OF SHIFTED SPECTRUM N ExIT NITH SHIFTED J SPECTRm I Figure 12. General Flow Diagram of Subroutine TSHIFT -43- IREAD DATA] SET BIAS - 0 GAIN - O ICTLCULATE SHIFTED SPECTRUNj—e——. SET SHIFTED CHANNELS .. 9 EQUAL T0 ORIGINAL CHANNELS Ea: PLUS BIAS SHIFT (- 8 (I) m INCREASE DIAS SHIFT } BT’O.1 CHANNELS [fine RESULTs| [TE Figure 13. General Flow Diagram of CRSHIFT | iREAD DATE] CONSTRUCT ODD POINT SPECTRA [SET BIAS EQUAL 0.5l [CALCULATE SHIFTED SPECTRAJ l CALCULATE DIFFERENCESJ l PRINT RESULTS] ENE] Figure 14. General Flow Diagram of CKFIT A IREAD DATAI a SELECT TTPE'W 4 OF SHIFT C CALCULATE AMOUNT OF SHIFT D ' CALCULATE SHIFTED SPECTRUM a [SELECT—PETE..— "" d’ F DETERMINE é’ .. STARTING CHANNEL fig E4 U) (I) k G CALCULATE , PEAK LOCATION F‘" ' "' E CALCULATE ENERGY RESPONSE LINE I IPRINT RESULTSI a Eh] Figure 15. General Flow Diagram for CKERL CHAPTER IV DATA ANALTS IS FOR DEVELOPMENT OF BASIC METHODS The volume of both raw and generated data was extremely great; thus raw data (Spectral data) will not be presented. Negative generated data will be limited to examples, either in tabular or graphic form, for each general type of results; and positive generated data will be presented in a more expanded manner in tabular, graphic or both forms. A complete set of data is on file with the writer. Peak Location The ultimate choice of a fitting equation for determining peak location was based on the reproduceability of peak location using different portions of the same peak. However two other properties, goodness-of-fit and reproduceability of peak location from different spectra, were considered. Goodness-of—fit. The measure of goodness-of-fit used was the sum of absolute.differences between the calculated and the observed channel counts and will be referred to as the "sum of differences". Table 2 gives the sum of differences of peak 4 for 5, 10 and 20 Rev] channel Spectra fitted with arithmetic and logarithmic equations of degrees 2, 3 and.4. Similar results were obtained for peaks 1, 2 and 3. Several generalities are evident from this data. First, the goodness-of-fit improves when the logarithmic form is used compared to the arithmetic form. Table 3 gives the percent improvement for similar energy intervals of the three Spectra fitted with the three equations. The fit is also improved in all cases when the degree of the fitting equation is increased from 2 to 4. The improvement has an increasing trend as the area of the curve considered is increased. In contrast, increasing -45- g -46- «mama canon QNNon cum" anaca «saga Man mnowa sauna manna momma 393 £33 an mamaa awn: 5mm: 3.2a mmaue mnuuc mu momaa manna manna mandasmman nmman .nu «mama mmmaa mamaa mica momau mamuu nmou nmuma mnmma unnm 35a «o3 nw aNNOa acaOa oc~Oa ummqammhma mnmma mama «soaa mamaa mama umnNN nmnu nu mama nnmm mmmm mama «mama mamNa mmma mmou mmms ameu mmoma «mom an aomm mama mama ammo «mum mmmm aqua nmmm name mmma mmmOa mmnc ma mean amam amam moms ammo ocum mmaa mmmu mnmu mama comm mmmm ha nmam scam momm nmam asnm annm mnaa mama mesa nwwa «hum «mum cmnu amnma «mama mama cusau NcmNN ma umam ammm mnnn mmam camm cmnm maaa nmua oaua mmaa amen mmmw maaa mama ammoa amoc enema mmNma ma name mama ammm mauq mmne mafia mnw mnua NmNa mmm «mma mama mmm «mmm ammn «nma mama ohmm aa mmmm anmn «Non mmmn mmmn mmmm mms ans aou mms mmm .aaa., Nun; mmNN . cmuu mam mmme amm¢ m numu mean mmnn mama annn «man man omm och nnu «as man amn «am am» man mama amsa n a. ma- :2 c TNNN lemma 5 an .1 non o . sa mow o a: L «as no ,mea . mmu m .a q n N m m N a b m N a m N m m m a m m nu , m u. cannon acumen acumen acumen acumen cannon .3 m qua. u. maoausovu mooausovm mooauoacu acoaussvu anoaumsvu mnoaumavm m n 355233 03253.2 3:52.33 03253.2 oasfiauuwoa ”.3253: w m. I asuuuomm aooosnu\>ou n souuoomm aooomnu\>uu ea asuuooem aocuaso\>uu cm s e 3m mom mug: Edam: MO 93m .N mano swoon mammumooom ozek aoccmnu use uwanm aoncmno - muacsk nu-Z naod sa-na mood aN-na aacd na-a 39o aa-h Sod e .mou 2-: 888 2-2 Sod --- 386 --- oSé :- mac... n .83. S-a 93.8 ma-o need. 31: 03.9 na-m once a-m 306 N .moa mN-aa sac-o aa-na mood nn-Ma mood aN-n mood ma-e wood a .fiafia . nu-aa n86 S-na mood -- muod -- 25.: 321- omoé n .514. w «T: oeod ma-m Sod 3.: came na-m mood aa-n 89° u .534 no as .3 am am a .... .... u .. we a. u a. u. a. X a. u m. .... B I. I. B .... U- u T. u .l u 1 u .l u a n ..u n. a n a u ..u .4 K A A . acumen. Show, n xeee a xeom n xeom N seem a seem ooauesvo mnauuah zauaummm auzzmx ca mom MHHaHmdfim manH3.onH’,.. .//. ‘Vfll '/ 1.. ’// F." 0" I, 11, l I I I I I I 1 I — Peak 3..- __.._ 0 2nd degree arithmetic ’0 _._ El 4th degree arithmetic / —— a 4th degree logarithmic ,I’ I I I I I -1 0 1 2 Peak starting channel shift Figure 23. Peak Locationeas a Function of Peak Starting.Channe1.- Peaks 3 and 4 -62- Quite clearly the arithmetic second degree polynomial is relatively unstable; whereas, the logarithmic fourth degree polynomial gives stabil- ities for all four peaks of better than 0.022 channels per one-half channel. Combining the width and symmetry stability, the maximum error in any of the four peak locations due to non-statistical causes would be 0.047 for the fourth degree arithmetic and 0.033 for the fourth degree logarithmic equation. The fourth degree arithmetic polynomial was prematurely selected for all further peak location determinations. Because of the small difference of stability between the two forms of the fourth degree equations (0.014), data were not recalculated using the logarithmic form even though it showed the greatest stability. Likewise the peak width used could have been increased after the fitting equation was selected. Little improvement in peak location.wou1d be expected when the spectrum contained relatively high count. However, it may cause a significant improvement when working with lower count spectra which contain significant statistical variations. Table 9 summarizes the peak location method that is used for all further analysis of spectra. The "percent of peak count" value was com- puted as the peak count minus the count in the peak starting channel divided by the peak count minus the count at the highest portion of the Figures 24 through 27 show the compton continuum adjacent to the peak. resulting fitting equations for peaks 1 through 4 respectively. TABLE 9. SUMMARY OF PEAK LOCATION METl-DD SELECTED Peak 1 Peak 2 Peak 3 Peak 4 Fitting equation Arith-4 Arith-4 Arith-4 Arith-4 Peak width used 9 11 15 17 Total stability 0.038 0.018 0.047 0.022 Percent peak count used 86 83 95 83 -63- 16 15- 14— 12_— 11*- 10,000 Counts per 10 Rev channel O Observedpo ints 9 channels ‘IIIJIIIIIIIIIIJIIII aFigure 24. 44 46 48 50 52 54 56 58 60 62 Channel number Peak 1 - Fourth Degree Arithmetic Fitting Equation, 9 Channels 1000 Counts per 10 Rev channel 40 -64- 38 36 .. 34 32 30 1_ 28 .. 26 ._ 24 22 20 18 __ 16 14+— 12 .. 10 o Queueing ints . O IIIIIIIIIIIIIIIIIIII 92 94 96 98 100 102 104 106 108 110 112 114 Figure 25. 11 channels Peak 2 - Fourth Degree Arithmetic Fitting Equation, 11 Channels -55- E.“ 3 30.. A o ngerved points 2 ._ 1000 Counts per 10 Rev channel o o o 10..O ._ I. .1 15.channels----_._ 6.. “v 4 F. 2.. 0 ‘I I I. I, I I I I I I I I l l I I .I I I I I 116 118 120 122 124 126 128 130 132 134 136 138 Channel number Figure 26. Peak 3 - Fourth Degree Arithmetic Fitting Equation, 15 Channels a' aaaaaa CH! \IIVVI AV— ":Ill: 1" *‘h‘ 18 17 —- 0 Observed points 16 .. 15 —- 14.. 13-— 12-— ° 11"- e 1000 Counts, per 10 Rev channel H O I 7-b—f . 5.. 17 channels.....- 0 IfiFImIM_I If I I .I. I J .I I I I I I I 'I’ I, I 170 172 174 176 178 180 182 184 186 188 7.- Peak 4 - Fourth Degree Arithmetic Fitting Equation, ..lZWChanne18.. 190 -67 Reproducibility 23'; function of total counts. A limited investi- gation of the effect of statistical variations upon peak location was conducted for the procedure selected. It was hoped that all conditions could be held constant throughout the counting period, and that any statis- tical variation would be a function of total counts in the spectrum. How- ever, the raw data indicated that a systematic long term fatigue drift com- ponent was present. It was therefore necessary to assume that the theo-‘ retical peak location was located on a line with a slope equal to the long term fatigue rate and passing through the mean of the observed peak locations. The fatigue rate was determined independently and will be dis- cussed in Chapter 5. Figures 28 through 33 give the peak locations for repetitive counts for varying counting times where counting time is directly proportional to total count. The dotted trend lines are the theoretical peak locations based on fatigue rather than a line of best fit for the observed points. It should be noted that for the long counting times there are systemmatic deviations about the fatigue line. For short counting periods the deviation appears much more random. Disregarding systemmatic deviations, standard deviations about the fatigue line were calculated and are given in Table 10 and plotted in Figure 34. Several factors are contributing to the deviations which tend to obscure the correct interpretation of the data. First consider just the 200 minute spectra. The distribution of peak location is heavily cor- related to the spectrum number which corresponds to the thme of start of {1 count. This relationship becomes very clear when the relative peak location of each peak is plotted on the same graph as done in Figure 35. The relative peak locations were calculated on the basis of an assigned value 0.26 0.25 0.24 0.23 O O O O O 1-1 1—- H H N 0‘ \J a: ‘0 CD ,Arbitrary peak position - channel O F: U‘ 0.13 0.12 0.11 0.10 _Figure -68- ‘Fatigue line . Observed peak location 0 Mean “1959295000 counts 200 minutes -‘\;;‘ 4,368,000 counts 80 minutes § ‘CL. 2.184.000 counts———”/i7 " 40 minutes ‘‘‘‘‘ i}- - I11092,000 counts-——”’zy _ 20 minutes 546,000 counts 10 minutes 1 2 3 4 5 6 7 8 9 10 Spectrumunumber~ 28. Reproducibility of Peak 1 for Varying Total Counts ~69- 0.80 .. O .55 -— O 050 '— Arbitrary peak position - channel c: a- u- I \ ----- Fatigue line . \~. 0 Observed peak location 0 Mean 10,920, 000 counts 200 minutes 4,368,000 counts 80 minutes ~. -~ —- Illn— -.~ -- ‘- .-- ~ 2,184,000 counts -‘ ______ 40 minutes 0040 — 0.35 -— 0.30 - 1,092,000 counts 20 minutes 0.25 -— 0.20 .— 0,15 ._ 546,000 counts 10 minutes 0.10 *— I I I J I I I I I I 1 2 3 4 5 6 7 8 9 10 Spectrum number Figure 29. Reproducibility of Peak 2 for Varying Total Counts -70- 0.85.. 0.80.. 0.75.. 0.70 - 0.65 - 0.60 - 0.55 - 0.50 — Arbitrary peak position - channel 0.25.. 0.20.. 0.15.. \ -— —- ‘ Fatigue line \ ‘0 Mean 10,920,000 counts 200 minutes ,4,368,000 counts " 80 minutes" 5 ~ .‘ ~ ‘ “ ~ ‘ 2,184,000 counts ”“40 minutes 1,092,000 counts ‘ 20 minutes 546,000 counts 10 minutes I I I I I I I I I I 0.10 -Figure 30. l 2 3 4 5 6 7 8 9 10 Spectrum number Reproducibility of Peak 3 for Varying Total Counts 'Observed peak location 1.7 1.6 1.5 1.4 1.3 1.2 1.() 0.5) 0.!3 0.7 Arbitrary peak position - channel 0.6 0.5 (1.4 -~»Figure 31. '-71- rm---“ ‘~- Fatigue line “ Observed peak location #. 0 Mean 10,920,000 counts ZOO-minutes 4,368,000 counts 80 minutes 2,184,000 counts 40 minutes 1,092,000 counts 20 minutes 1546,000 counts 10 minutes I I I I I I I _- — — l 2 3 4 5 6 7 8 9 10 ,Spectrum number Reproducibility of Peak 4 for Varying Total Counts Arbitrary peak position - channe 1 0.42 . --.-.72- 0.40 0.38 0.36 0.34 0.16 (3.14 (3.12 I— 0.10 ______ Observed peak location Mean 218,400 counts 4 minutes 218,400 counts 40 minutes 109,200 counts 2 minutes .109,200 counts 20 minutes 1 2 3 4 5 6 7 8 9 10' Spectrum number FIlgure 32. Reproducibility of Peak 1 - Effect of Time of Count 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 :7 c: 5’ c: 5: 01 a: ¢~ ¢~ U! c: In :1 Ln 4: Arbitrary peak position - channel 0 h) u: 0.20 0.15 0.10 l?igure.33. -73- .... Observed peak location —--— Mean 192.290.000nts 2 minutes 109,200 counts 20 minutes -1218,400.connts1 4 minutes $218,400 counts 40 minutes l 2 3 4 5 6 Spectrum number 10 Reproducibility of Peak 4 - Effect of Time of Count -74- 25.39309 muemoeum - mooEeooa xeem mo muaaamaosmoumem .em Io-uawam Amunsou 89.3 a manna... av noun—:6 - 06a... moan—500 . QON OOa ow om om ON 0a m m a N _ _ _ . _ . _ _ HII . a w 3 m —.. . I. . . . . a , o ....... \xulo-Il.I-II.IIIX/o/1/ [WNan xIHIII-IIDIIIDII x/IIIIIn/ DI IIIIILTIIIIAYI-IIJWI/. _/ 01 III? /./ / lane a / xl I-IA.Ix / / /. / / Lag... // /n/ u. . s j? // w. // . log... 3 III //.//. n. 778 /x Inna... m / /u .... / m- / _I1cma.o u . / m / I . / mNa o m a xeom o IIIII / m... n seem E III! / I . s N Jenna x .I-I.| / 8N O .n u—Qom e IIIIIII / , Inga / L22 -75- 0f sero and ten to the lowest and highest peak locations respectively. The connon flatness of the curves between spectrum numbers 1 and 4 and between 7 and 8 is readily apparent. Only if the gain of the instrument deviated from that assumed for calculating the fatigue line would the cor- relation be present in all four peaks. The standard deviations given for the 200 minute counts in Table 10 thus are much larger than the actual ones. Calculated deviations based on a visually estimated regression line following the trend of Figure 35 indicates the standard deviation to be a factor of four below those given in Table 10. The same type of systematic variation is not detectable in any of the other spectrum sets, but it is undoubtedly present to some extent only masked by larger random variations. TABLE 10. REPRODUCIBILITY OP PEAK LOCATION CHANNEL - STANDARD DEVIATIONS, SOURCE SODIIH 22, COUNT RATE 54,600 CPM Peak Counting time in minutes 200 80 40 20 10 4 2 40* 20* 1 0.009 10.008 0.006 0.007 0.008 0.045 0.024 0.017 0.017 2 0.030 0.024 0.022 0.031 0.062 0.060 0.127 0.052 0.082 3 0.038 0.030 0.035 0.042 0.035 0.079 0.138 0.052 0.060 4 0.050 0.042 0.045 0.054 0.095 0.129 0.249 0.081 0.131 *Source strength equals 5400 cpm The general trend of the data is’ incrhased deviation with increased Pfimk nusber and decreased total count. An increasing peak number represents . broadening peak, decreasing peak channel count and decreasing total peak co\Int. Becauseuof lack of control over spectrum parameters when using only °ne nuclide, it would not be feasible to attempt to interpret the results quantitatively in terms-of the basic parameters effecting the statistical regrodu :I‘IatIo not of 1‘! I11 ~76- reproducibility of peak location. However, the data do point towards the relationship that the deviation is inversely proproportional to the square root of the total count in a peak. This is derived from log-log plots of the data which give approximately straight lines with slopes of -0.5. 11 10__ ______ ‘§\\v/” .vfiu... 9.. \\ \ 8_ \\'. \ g ‘- . I: 7.. \\x 8' -1 6_ \ ... ‘.\ u ,- 3 5— \,\ .2 , ‘ o z ,- . a. I, P . '. I: ., V ’2 3 3— -\\, :2, I, .. 2L —-—‘— Peak 1 \\°.. ----- Peak 2 he 1__ .-.-.--. Peak 3 \5 Peak 4 \g 0 -F—-a—- Fatigue line 1 I I I I 1 I I I I I 1 2 £3 4 5 6 7 8 9 10 wSpectrum«number~- Figure 35. Compairson.of.kelative Peak Location of Peaks 1 to 4 An additional random deviation is present which is related to the QOunting time, but separate from the total count. Consider the data for the four and two minute counts with the strong source, and the 40 and 20 “hinute counts with weak source,where both combinations result in the same 1:101 far I rand: 1631‘ grea -77- total count. If the deviations were only a function of total count, they should be approximately equal. However, in all peaks the shorter counting time resulted in higher deviations. The average ratios of standard devi- ations for short counting to long counting for all peaks were 1.55 and 1.86 for four and two minute counts respectively. This indicates a short term random variation in gain whose effect upon peak location is inversely pro- portional to the counting period and significant for counting periods steater than four minutes. Sun-arising the reproducibility of the peak location using the proposed method, it can be said for twenty minute counting times of sodium- 22 at a count rate of 54,600 counts per minute the peak locations can be determined within a precision of approximately plus or minus 0.01, 0.03, 0.04 and 0.05 channels at the 68 percent confidence level for the first four peaks respectively. Energy Response Model Apparent energies for nomial channel coefficients of 5, 10 and 20 Kev/channel; and pivot bias points for nomial channel coefficients of 15 and 12 Kev/channel were determined for use in the proposed energy reFlionae model given in equation 5, page 28. Apparent energies. Table 11 gives the calculated apparent energies lnd correction factors. Apparent energies for peaks 1 and 3 are identical t° the theoretical energies as these were used as the reference peaks. The v“lines for the 9.900 Kev/channel spectrum are based on the average of twelve .“ccessive spectra; whereas, the other two are based on single spectra. The Qlose agreement between the correction factors (less than 1.4 Rev) indicates -73- that the linearity of the system is sufficiently constant over a wide range of both gain and ADC ramp slope settings. Thus single values for apparent energies can be assumed over the anticipated gain shifts of plus or minus ten percent. TABLE 11. APPARENT ENERGIES (KEV) AND CORRECTION FACTORS FOR ENERGY RESPONSE MODEL Channel coefficient Peak Peak Kev/channel 2 4 Theoretical energy ---- 1022.0 1784.0 Apparent energy 5.010 1042.4 1802.4 9.900 1042.4 1803.9 20.142 1041.0 1805.2 Correction factor 5.010 20.0 18.4 9.900 20.4 19.9 20.142 19.0 21.2 The systematically higher peak locations for peaks 2 and 4 (which) are both sum peaks) can be attributed almost entirely to the non-propor- Hot-.1 response of the Hal (T1) crystal. Based on the data of Heath (6), the relative pulse height per unit energy decreases from 1.010 at 0.51 Mev to 0.985 at 1.02 Mev. This amounts to a peak shift of 25 Rev as cultured to the observed shift of approximately 20 Rev in the 1.022 anni- hillntion sum peak. The remaining 5 Rev discrepancy could be caused by loss °fF light collection efficiency when the two annihilation photons given off "51 180 degrees from each other interact on opposite sides of the counting “311. gig; 22135. Calculated x-intercepts (observed bias) and channel cOefficients for various gain and bias settings are given in Table 12. These data are plotted in Figure 36 from which bias points (intersection -79- . usaom uo>am mean no :Oaumcaauouoo .mm shaman. 00a madame weauuom mean on- can masses com masses moguuoe moan weauuoe seam ou-.J \ \ \ Homes: aocnmno \x. .. -_ ._ m A- b..- ... .. _ . a . . . \ ueaom uo>am mean 0 \ \ com woauuom name IIIII 8a meauuoa 5am I -30- of energy response lines) were determined graphically. Table 13 gives the determined bias pivot points. TABLE 12. ENERGY RESPONSE LINES FOR BIAS PIVOT POINT DETERMINATION Instrument settings Channel coefficient X-intercept Gain Bias Kev/channel Channel 100 100 15.462 3.76 100 500 15.410 -0.03 100 900 15.407 -3.59 500 100 12.282 4.09 500 500 12.267 0.30 500 900 12.260 -3.30 L..., TABLE 13. BIAS PIVOT POINTS Ins trument Px Py Bias setting Channel Rev 900 ~4.95 -20.5 500 -l.55 -22.5 100 +2.35 -21.5 The non-coincidence of the pivot point with the x-axis will result in approximately a 0.023 channel change in the x-intercept for each one percent change in the gain at a nominal channel coefficient of 10 Rev] channel. The x-intercept is clearly dependent on the gain of the system; however, the limited data does not provide evidence for determining whether the bias pivot point is independent or dependent on gain. Without a detail- ed analysis of the position of the bias pivot point as a function of both gain and bias settings it is impossible to predict a change in the pivot point based upon constant gain and bias settings and varying environmental conditions. Thus the x-intercept will be used as a psuedo measure of bias -31- with the full realisation that significant correlation between gain change and observed bias change may exist. The x-intercept hereafter will be referred to as bias. Reproducibility 2; energy response line. Originally the reproduci- bility was to be calculated from the same data.used for studying the re- producibility of peak location. The standard.-deviations about the means were calculated and printed out, but the individual values for the energy response lines were not printed out. The existence of a significant fatigue drift rate invalidated the use of these standard deviations as a measure of statistical variation. In lieu of this necessary information, the data collected for investi- gation of line voltage variation was used to determine the statistical variation for this single set of conditions. The conditions were nominal channel coefficient of 10 Kev/channel, counting time of 20 minutes and counting rate of 54,600 counts per minute. The data for channel coefficient and bias are given in Figure 45, page 96. For the channel coefficient the standard deviation about the regression line was 0.0043 Kev/channel.. The bias variation did not show a significant trend. Thus the standard deviation was computed about the mean and was 0.027 channels. The standard deviation of the channel coefficient represents an error of approximately_0.04 channels in channel 100. This corresponds to an error of 0.03 channels found in the study of peak location for the same conditions. This would indicate that the error in the slope of the energy response line is almost entirely dependent upon the error in locating the peaks used in determining the energy response line. -32- Spectrum Normalization Linear interpolation has the inherent property that any interpolated value lies between the two values used for interpolation. Very significant error could occur when working on narrow peaks of low energy.' If the true. peak would fall at say.channe1 50.5, the observed spectrum would have equal counts in channels 50 and 51 with some higher value expected in channel 50.5. Linear interpolation can not account for this situation and I was thus ruled out as a realistic method. Attempts were then made to use exact high degree fitting polynomials to represent the data points. This method proved unsuccessful because very large oscillations occurred between the fitting points. The degree of the fitting equations as varied over a range of 3 to 39, but substantial oscil- lations were present at all degrees. In an attempt to dampen out the oscillations, the criteria for poly- nomial fitting were changed to include a statement designating the slope of the curve at each data point. The results showed virtually no oscil- lation for seventh and ninth degree polynomials, but did show significant levels at higher degrees. Thus a single or several high degree polynomials could not be used to represent the entire spectrum The proposed method was then developed which represents the spectrum as the sum of "n" seventh degree polynomials where "n" is the number of Channels in the spectrum. The basic model is given in equation 15, page 39. Smoothness-gffigig. The fitting equation was first compared to the observed spectrum. Figures 37 through 39 show three narrow sections of the Spectrum. The scale has been greatly expanded to show the detail and linear interpolation lines have been added for comparison. Figure 37 is the upper 143 -33- 142 —— 141 ._ 140 —— 139 —— 13.8 P 137 _— 136 —- 1135 —— ‘l34-—— 133-—— 100 counts per 10 Kev channels 132 —- 131 _- 130-—- 129 128 - s.’ O. . . O . O O O I O O O O 0 Observed points a Polynomial interpolation spectrum Linear interpolation _,Figure 37. 52.0 53.0 54.0 Channel number Comparison of Fitted Spectrum to Observed Spectrum Points in Peak.Area -34- 36 35I 0 Observed points 3 ° Polynomial interpolation spectrum 1% 34 Linear interpolation {i r > u. “ 33.. o H In 2. 32.. Q ‘3 3 31 _ 0 § 30.- 29 l | - l l 200 201 202 203 204 205 Channel number Figure 39. Comparison of Fitted Spectrum to Observed Spectrum Points in Flat Area 1350 __ . 0 Observed points 1345 ° Polynomial interpolation spectrum . '2‘. T’ Linear interpolation E 0 1340 2; he <3 1335 N a. 8.1330- m' in.) g 1325 u c :4 1320 .— I I 95.0 96.0 97.0 Shannel and: er Figure 38. Comparison of Fitted Spectrum to Observed Spectrum Points in Valley Area -35- four channels of peak 1. The fitting curve appears reasonable in all respects including the peak location. The interpolation equation gives a peak location of 52.74 as compared to 52.735 found by methods developed previous ly . Figure 38 is the leading base of peak 2. Slight effect of oscil- lation is noted in that two minimums occur; one at channel 96.6 and another at channel 95.5. Even though the shape of the curve is not reasonable, the deviation from the most reasonable line, which probably falls between the linear interpolation and the polynomial interpolation line, may well be less than that of the linear curve. Regardless of which method is used, the deviations are on the order of 20 parts in 13,200. This particular portion of the spectrum was selected for illustration since the greatest oscillations were noted to occur in relatively flat areas of the spectrum which are adjacent to rapidly rising peaks. Figure 39 is once again a relatively flat area but is not adjacent 5° ‘1 peak. There is essentially no difference between the linear and poly- “°mial interpolation and no problem of oscillation. In all portions of both the high and low count spectra the interpolation curve appears reason- able and fits the basic criteria set up for the interpolation procedure. Accuracy _o_f_ interpolation. Two 10 Kev/channel spectra, one with. 800d counting statistics and one with poor, were next used to generate new aDectra by applying the interpolation procedure to only the odd channels a11:1 calculating a predicted count in the even channels. If the interpo- 1~ation procedure is valid, the observed count and the predicted count should agree closely for the spectrum with little statistical error. Figure 40 Shows the observed and the interpolated spectrum for peak 2. The results on this peak are typical of all peaks. In the concave portion of the peak (top) -35- the interpolated channels fall below the observed channels, while in the convex portion the interpolation slightly over estimates the counts. This is similar to the results observed early in the development of the Peak location procedure when arithmetic polynomials were used. The linear interpolation spectrum is also shown and is seen to deviate even further from the observed spectrum. The largest observed deviation of any inter- Polated channel from the observed channel was 3.7 percent. This agreement ‘ppears quite acceptable considering that part of this deviation is due to Statistical error. The percent difference over the entire spectrum is 0.065 percent. Figure 41' shows several peaks which were derived from a spectrum with poor counting statistics. The lines connecting the points represent 20 59"] channel spectra rather than 10 Kev/channel spectra since only every other channel is used. The smoothing effect of the interpolation procedure 1' very clearly seen and also the systematic under-estimation at the tops °f Peaks. The under-estimation in the peak located at channel 20 is unique 131an it is due almost entirely to statistical error. This peak is expanded in Figure 42 to illustrate this point. The connecting lines are only added to .how the points as separate groups and should not be interpreted as a continuous spectrum. The observed points show the large statistical varia- tion; particularly in channels 20, 21 and 22. If the odd channels are used for interpolation, the lowest curve results since the data in channels 20 ‘nd 22, the two highest channel counts, are lost. Conversely, if the even chaunels are used, the effect of channel 21, a low channel count, is lost "1th the result of producing a curve which is too high. The distribution °£ counts about the true spectrum is assumed to be random; thus the two h13h points would have the same probability of falling in adjacent channels flsIvIal-m'm-U >0: C! III!!! lIis-IIIIHIN Calm 37 36 35 34 33 32 - 31 30 annel N \0 ha a: h) \I a: GI Ibo Counts per 10 Rev ch BO III I th I0 0: b A! R: 121 20 19 18 17 16 -37- I O Fitting points Observed points Polynomial interpolation points Linear interpolation points Xo-E] I I, I I I ' I I I I I I 100 Figure 40. Spectrum 1 102 104 106 108 110 112 Channel number Comparison of Odd Point Interpolation Spectrum to Observed asuuuoom assoc 30a meauemao ou asuuoomm coaumaomuouoa uaaon pro mo sosaueosou non-Ha- assume”. ooa mma Nma 3a 3a 2a mma Nma mma mma .3 co mm NM mu om ON ma Na m a .a8 enemas _ _ 1 m _ J asuuooem usaom moo msuuam asuuoomm ucaoo moo mo>noamo auoaom oo>o me>uemmn O _ _ m a _ _ a _ _ can I can I GO.» I one I com 10mm .0 omm \ OON I 8 \O 1auusq0 as; 01 zed saunoo. 1 o? I on» com 870 860 850 840 830 820 810 800 790 780 N N O Counts per 10 Rev channel N as <3 750 /' , , \ / 1 \ ' 740 I’ ‘1 \ I, \ a - b\ 730 d --—— 0 Even point spectrum - ----- X Odd point spectrum \ 720 0 Observed spectrum ‘ 710 700 690 I I I I I I I I I I | 14 15 16 17 18 19 20 21 22 23 24 Channel number Figure 42. Comparison of Odd and Even Point Interpolation -39- Spectra to Observed For Low Count Spectrum -90- as in the sequence which was observed. In the non-peak areas the observed points appear to be distributed more or less randomly about the interpolated curve. ngparis n p§_normalixed £2 determined gpppgy response. The shifting procedure was next tested by determining the energy response line of an unshifted spectrum. The spectrum was then shifted mathematically by various amounts of gain and bias, and the energy response line was redetermined. The changes in gain and bias of shifted spectra were then compared to the theoretical change incorporated into the spectra. Table 14 gives results of this analysis. TABLE 14. COMPARISON OF INCORPORATED ENERGY RESPONSE SHIFT TO MEASURED ENERGY RESPONSE A— Incorporated shifts 'Measured shifts Differences Gain Bias Gain Bias Cain Bias percent channels percent channels percent channels 0.000 0.400 0.001 0.411 -0.001 -0.011 0.000 0.800 -0.027 0.851 0.027 -0.051 0.000 1.200 0.014 1.184 -0.014 0.016 0.000 1.600 -0.017 1.636 0.017 -0.036 0.000 2.000 0.000 2.000 0.000 0.000 0.2 0.000 0.214 -0.007 -0.014 0.007 0.4 0.000 0.412 0.002 -0.012 -0.002 0.6 0.000 0.594 0.027 0.006 -0.027 0.8 0.000 0.783 0.047 0.017 -0.047 1.0 0.000 0.964 0.090 0.040 -0.090 When the gain is held constant and the bias is varied, the error in the gain and bias must cycle repetitively as the bias passes through integer bias shifts since no interpolation is involved when shifting the spectrum up or down an integer channel. Thus the data which spans a range from sero to two channel bill shifts can be interpreted as if the shifts were all within zero to one channel. If only the fractional portion of the shift -91- is used, Figure 43 shows the cycling for gain and bias when gain is held constant. The two functions are almost mirror images of each other. This suggests that the error that is introduced in the bias shift is intepreted by the energy response determination procedure to be a slight gain shift. Gain Bias. \ I I I I L I I I I I ”I I I I I I I I I I 'l I‘ I 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 -Incorporated bias shift, channel Incorporated minus measured shift - channel Eigure 43. Comparison of Incorporated Shifts to Measured Shifts - Cain Constant The data for the condition of constant bias and varying gain (see Figure 44) is quite different. Here the cycle period is obviously not one channel. This can be explained if the errors are assumed to be a function of the errors in the location of each of the four peaks used to determine the energy response. The error of each peak location will cycle with a one channel period; however, the amount of gain shift required to produce a one channel shift is a function of the peak energy. Peak 4 will have the shortest period in terms of gain change, while peak 1 will have the longest. Since one percent in gain at channel coefficient of 10 Kev/channel results in a one channel shift in channel 100 (location of peak 2), a two percent change in gain would be necessary to produce a one channel shift in channel 50 (peak 1). Thus the minimum period of the energy response error would be a two percent change in gain, while the true period is probably much longer -92- since the peak energies are not simple multiples of one another. Unfortun- ately, the spectrum shifting program used was limited to only 1.4 percent gain shift, and errors for shifts were calculated only to one percent. This is not a limitation of the method, only of how it was programed. 0.04- q 0.03_. nu m 0.02 A 0.01 0.00 -0.01 .-0.02 e. “0.03I-I -0.04_ -0.05_ -0.06— -0.07— -0.os_.._ -0.09‘,_ “'7 T‘T'”1”I'“L 14 I II I‘” “”"T'T I II ‘I ‘I"I I '1‘ 0.0 0.2 0.40.0 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 .Lincorporated gain shift, channels in channel 100' It] I_J d shift - channel ncorporated minus I . reasure Lligure 44. Comparison of Incorporated Shifts to Measured Shifts - Bias Constant while the proposed method of spectrum shifting is not absolute, it appears quite satisfactory for use in incorporating small bias and gain shifts. Improvement of the basic model might be obtained by incorporating some of the features of the peak location method such as use of a logarithmic form of the channel counts. CHAPTER V DETERMINATION OF ENERGY RESPONSE STABILITY The energy response stability of the spectrometer system was measured as a function of applied line voltage, ambient room temperature, photo- multiplier tube fatigue and counting rate. The entire counting system was treated as an integral unit; thus the components causing response shift were not investigated directly. Procedure The general method used was to hold constant all parameters except the one under investigation, and to observe the effect of the varied parameter upon energy response. ‘Li2£_voltage. The line voltage was varied from 100 volts to 128 volts in increments of two volts using a variable voltage transformer. The input voltage to this transformer was regulated and filtered at a nominal 116 volts. The voltage was changed five minutes before obtaining each spectrum. The source used produced 54,600 counts per minute and was counted for twenty minutes at each voltage. The peak locations and energy response lines were then calculated using program VOLT. Temperature. The room temperature was varied from 62.9°F to 96.3°F in increments of approximately 3°F. The room air conditioner was used to obtain the lower temperatures while heat lamps, placed at the air condi- tioning outlet, were used for higher temperatures. The air conditioner fan.was kdpt on all the time to promote uniform temperatures throughout the room. The detector system was removed from the lead shield to allow for quicker temperature equilibrium between the detector and the room. Even so, two hours were necessary to reach equilibrium as the average rate -93- -94- of change was kept to 2° per hour to prevent damage to the crystal. The detector temperature was monitored within the counting vial immediately above the deposited source. The top of the vial was wrapped with 1/8 inch aluminum foil to approximate the insulation effect of the crystal°s own can and the photomultiplier tube”s magnetic shield. The temperature was left at a given level for 120 minutes or longer. When the detector and instrument temperatures were constant for 30 minutes, a set of two 20 minute spectra was taken at a nominal channel coefficient of 10 Kev/channel. The peak locations were checked visually and further spectra taken until similar energy responses were obtained. The peak locations and energy responses were then calculated using program TEMP. ghotomultiplier tube fatiggg and count rate. Three sources were used producing 5400, 54,600 and 538,000 counts per minute. They will be referred to as sources A, B and C respectively. The detector was conditioned with source B for one week. A single spectrum was taken and the source removed for 39 hours. Source B was then replaced, and spectra were ob- tained continuously for the next 71 hours. Counting times varied from 20 to 200 minutes in proportion to the rate of change of the peak locations. Source A was then immediately substituted and counted continuously for 47 hours. Counting periods were 100 and 200 minutes. Source C was sub- stituted and counted for the next 71 hours for periods of 10 to 40 minutes. Source A was returned and counted for 45 hours. The peak lo- cations and energy response lines were calculated with program CNTRATE. -95- Computer Programs The computer program coded TEMP, VOLT, and CNTRATE are identical to program BIASPT used in the development of the energy response model except for the format of the input and output data. The data deck descriptions are given in Appendix A, and programs are given in Appendix B. The flow diagram is given in Figure 9, page 33. Data Analysis The data analysis consisted of reducing the raw spectral data to energy response lines by determining the peak locations from the fourth degree arithmetic polynomial fitting equation and performing a least- squares fit of these to the energy response model (equation 1). Cor- relations between the energy response parameters and the extrinsic vari- ables were made. 'Eigg voltag . Figure 45 shows observed variations of channel coefficient and bias (x-intercept) with applied line voltage. The channel coefficient increases, thus gain decreases, with increasing line voltage. The data were fitted by least-squares to a linear function and the slope was found to be 0.0014 Kev/channel/volt or -0.014 percent gain change/ volt. As 6.7 hours were required to collect the data, long term fatigue would account for part of the slope. Based on 25 minutes per two volt reading and a fatigue rate of 0.00059 Kev/channel/hour, the equivalent fatigue rate in terms of volts would be 0.00012 Kev/channel/volt which is comparable to the statistical variation in the determination of the observed slope and an order of magnitude less than the line voltage coefficient. Thus the fatigue rate can be neglected. -96- 0.7Qe. 'nean 0.72;. .705. o o (x-intercept) - channel c: 0.66_. Bias I 10.02_. 10.01__ 10.00—- Slope = 0.0014 Kev/channel/volt Channel coefficient - Kev/channel 9. 95 I I I L I I I I I I I I I ‘ I" " ‘ I "r- 100 104 108 112 116 120 124 128 132 Line voltage .-Figure 45. Effect of Line Voltage Upon Bias and Channel Coefficient -97- The bias does not show any significant trend as the values are distributed about the mean with a possible skewness to the low side. However, the bias for 100 volts deviates from the mean by a factor of 3.8 standard deviations. The probability of this being due to random- ness is only 0.00007; therefore, it is assumed to be an observation of a true change in bias. It is quite possible that at 100 volts and below, the voltage applied to the ADC circuit dropped below that necessary for the operation of some component circuit such as a multivibrator. Temperature. Figure 46 shows the variation of channel coefficient as a function of ambient room temperature. The least-squares linear fit of the data gives a temperature coefficient (slape) of 0.055 Kev/channel/°F or -l.000 percent gain change/°C. Once again the fatigue is small com- pared to the statistical variation in the measurement. Bias (x-intercept) variation is shown in Figure 47. The resultant temperature coefficient is 0.024 channels/°F. As was previously shown, the x-intercept is partially dependent on the gain to the extent that a one per- cent gain change is approximately equal to a 0.02 channel shift in the x- intercept. Applying this to the observed gain change, the x-intercept would vary 0.010 channels/°F. The true change in bias due to temperature effect alone would be 0.024 minus 0.010 or 0.014 channels/°F. Photomultiplier tube fatigg_. 0f the three source strengths used, observable fatigue rates could be found for only the two stronger sources. The weakest source produced statistical variation much greater than that of the anticipated variation due to fatigue. The fatigue rate for source B (54,600 counts per minute) was determined from the slape of the channel coefficient versus time curve (Figure 48) between 58 and 110 hours, while 10.6 10.5 10.4 10.3-— 10.2I— 10.1 _ 10.0-— 9.9.“ , Kev/channel 9.7 — 9.6 h Channel_coefficient 9.5 — 9.4 — 9.3 — 9.2 — 9.0 — Temperature coefficient equals 0.055 Kev/channel/°F I I I | Figure 46. 70 80 90 100 .-Temperature,-degrees Fahrenheit Effect of Temperature Upon Channel Coefficient 1.6 1.5.. bserved bias change equals 0.024 channelsl°F N\7K\\H‘ 1.3.. 1.2— hias, channels 1.1_. 1.0I_ 0.9L- /<;'””Bias change due to gain dependence equals 0.010 channels/0F 0.8Lis .L l I l l 60 70 80 90 100 Temperature, degrees Fahrenheit .Eigure 47. Effect of Temperature Upon Bias washouumooo flamenco com: omwwumm one sundae cums ucnou uo uoouwm .wo «unnumfl masoc.elwuteusouououP 1suusqo]on ‘3“813195303 [anusqo o2 m2 o2 m: a: «2 c2 3 2 3 8 fl 2 3 8 an an 3 8 n o _ _ _ _ _ _ _ _ _ _ _ _ _ a _ _ _ _ _ _ _ mm.m . - . u . .I IIIIIII I 8.3- «00 VII 0 I s m scezuesaeuEuustdaIm-«sea. . I 3.2 m .. a . . I 2.2 nw .m .. m u . . I 2.2 a c, x O . 3 O I , d 2.2 a 0 a _fi 0 L _ . . , 2.2 . . . I 8.2 h...‘ Y+AI||Y._ . 4 season n Museum souaoeIoz an ca ”2.35% I 3.2 ucouuuuuooo flamenco coo: oswuumm mam mucosa oumm assoc we uuouwu .A.ucoov we ouswam .. 950: .083 oocouewoz mam OMN mNN emu 3m ouw MON 8N no.— 63 mm.— 8~ n: a: Q: as mm.— 03 n3 0.: «2 OS nu.— _ _ _ _ _ _ _ _ _ oqua .....H . g.o. . _ _ q _ . . I36 I... Hsos\~ma:m£o\>ox Nooo.o u macaw, o _ . I25 I o . _ w. u .IS.... I a ..l 0'. o I a co m w 19 J oIcn.m I u. I. . a 1 L w mu oo.o I. c /. . u. #1. I o l a .. is h 3 u. .Im . 8.... I “u. a Tl .Ic¢.m I O°~ l ~.o~ I.nN.c~ e e e . . . . I82 IllleI Ill 0 nausea" < oousou I.nn.o~ ~102~ essence coo: oswwumm one owcmco comm ucsou mo uoouum mmN owm mmm 0mm mono: new com .oewu oocouowom mam Cam mom oqm ocmfiuuuuooo .A.u:oov we seamen mmm 0mm mmm 0mm _ _ a _ _ _ _ < uouaom 1 VI ll 0 season oo.o o~.o ON.¢ om.m o¢.m Om.o oo.m 05.0 ow.a O¢.a oo.ofi o~.o~ ON.OH cm.o~ anuuq; 'I I Iouuuqs/Aon ‘3uognI3Ioua -103- that for cource 0 (538,000 counts per minute) was determined from the portion between 180 and 230 hours. The results are given in Table 15. TABLE 15. LONG TERM FATIGUE Source Count rate Fatigue rate K1, fatigue rste/ cpm Kev/channel/hr count rate B 54,600 0.00052 7.83 x 10’8 0 538,000 0.00417 9.50 x 10-8 average --- --- 8.67 x 10' With only two points it is not possible to define the relationship between count rate and fatigue rate. However, if the cause of the fatigue is due to loss of dynode material as described in the literature, then the relationship should be directly proportional to the number of electrons striking the dynode surface since the electron is the cause of dynode depletion. The number of electrons per unit time is proportional to the count rate, thus'equation 16 would be appropriate. CR -KICR Eq. 16 a proportionality constant and CR the f 1 count rate. The data support this relationship since the K1 values cal- where fCR is the fatigue rate, R culated differ by only 24 percent when the count rate was varied by a factor of ten. The difference between the values could be due to lack of sufficient control over other variables; particularly temperature. The average value of [1 (8.67 x 10'8) will be used for all further calculations. pggggg'ggtg dependence. The variation of channel coefficient with time for various count rates is also given in Figure 48. The shape of the curve is welldefined except in the area of 125 to 140 hours where the channel coefficient increased abruptly. The shape of the curve in the portion immediately after a source is changed suggests an exponential relationship. Semi-logarithmic plots (Figures 49 through 52) show that the data can be -104- reasonably represented as the sum of two exponentials. If in addition the variation due to long term fatigue is considered, the following relationship can be hypothesised. cc, -Acc;ebt +Acc§e*2t + cog + £CR(I:) Eq. 17 where t is the time after change of count rate, CC; the channel coefficient at time t, zsccl the change in channel coefficient caused by arbitrary process 1 extrapolated to time sero,z3CC§ the change in channel coefficient f o the channel caused by arbitrary process 2 extrapolated to time sero, CC coefficient at time zero extrapolated from the fatigue line, fCR the fatigue rate for count rate CR, and.A,and}g the rate constants for processes 1 and 2. The values for 2” and )q_88 evaluated from Figures 49 to 52 are given in Table 16. Considering that the rate constants are derived from the difference between extrapolated values, the agreement is sufficient to assume that;k,is independent of count rate and direction of change. Similarly ,Al is independent of count rate but is dependent on the direction of the change. Thus the average value for each direction of change will be used. TABLE 16. RATE CONSTANTS FOR.ARBITRARY GAIN CHANGE PROCESSES Initial Final Direction A I *2 source Source of change hours'1 hours“1 0 B + -Os192 --- -1098 A C + -0.178 --- -l.39 B A - -Os 192 " --- c A - -00 173 - -"' Average -0.l84 - -1.68 nnel I Kev per cha 0.60 0.50 0.40 . -105 0.90 A ccc‘;z - 0.110‘5ey/channeLM ~-~..m. .. —‘. ~—--~A---L~B kMCga Art I: 10 >12 - -l.98 hours"1 ” .35CCg‘:_Q,22§“§ey[ghannel cog.- cc: - sum -Acc:ek.t R. 7:91.192 hours‘l 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.15 I Kev per_channel o. lo 0.09 0.08 0.07 0.06 0.05 0.04 0.03 ~106- accg - 0.760 Kev/channel A \ B ~----‘- A-B“~. ACCo - 0.580 Kev/channel cc: - cc: -£c'(c) - cage *8 ° 9w- -o’. 178 hour-'1 0 O O o N ,.-.-_ aumfiakzt A2 - -l.3.9 hours'1 I I I I 3I 0 l 2 3 4 5 0 7 8 9 Reference time, hours Ffigureso Determination of Count Rate Dependency Equation Constants - mtants - Source M‘Changerx' to*C“ ““‘ mm: 1 Q 3 Rev per c -107- 0.30 0'3/A CC: .- 0.060 Kev/channel Ace; - 0.230 Kev/channel 0.20 Ct - ccfi - £011“) - chez't .10 ... .09 __ .08 _ A. - 019? hour-'1 .07 _ 000°C .06 ... 0. 01 I l l I 0 S 10 15 20 Reference time . hours Figure 51. Determination of Count Rate Dependency Equation Constants -- Source Change B to A 1.00 0.90 0.80 0.70 0.60 0.50 0.60 0.30 ilev per channel 0 p0 Ul 0.10 0.09 0.08 0.07 0.06 0.05 0.06 0.03 Figure 52. Source Change C to A -108- °3 — l .‘/////<1LCC§ - 0.140 Kev/channel AGO}; I 0.820 Kev/chennel A, - -0.173 houre'l - CC; - ”CS - fnnt‘ - CG}! )‘_'.t L O 10 15 Reference time, hours 20 25 Determinetion of Count Rate Dependency Equation.Conetento - -109- The next problem is to find relationships between count rate and Accé, Accg and 00:. Table 17 sun-arises the observed or extrapolated values for these quantities. TABLE 17. COUNT RATE VARIATION EQUATION'PARAHETERS Count rate Previous ACR f f 1 2 cpm count rate cpm 000* 000* CCD - 000* ACCo* ACC°* c m 54,600 0 54,600 10.327 9.992 0.335 0.225 0.110 538,000 5,400 532,600 10.255 8.915 1.340 0.580 0.760 5,400 538,000 -532,600 9.190 10.150 -0.960 0.820 0.140 5,400 54,600 -49,200 10.030 10.300 -0.270 0.230 0.040 * Units - Kev/channel At this point a discrepancy is apparent in the data. At time zero with source 3, the channel coefficient was 10.067 Kev/channel. The source was then removed for a period of 40 hours and then replaced. After gain equilibrium had been reached the channel coefficient was 10.000 Kev/ channel. However, the channel coefficient should have returned to the original value plus any change due to fatigue which would increase rather than decrease the channel coefficient. The most logical explanation is that the environmental conditions had changed, most likely temperature. This would indicate that the same type of gain change could have taken place at any time during the 285 hours over which the data was collected. This is substantiated by the values of CCo minus 0c: given in Table 17. When the count rate was increased by 532,600 counts per minute, the change of channel coefficient was 1.350 Kev/channel. However, when the count rate was decreased by the same amount, the channel coefficient change was only 0.96 Kev/channel. These values should have been roughly equal. When the -110- source was changed from source 0 to D and then source B to A, the changes in channel coefficient were 0.335 Kev per channel and 0.270 Kev/channel which are essentially equal in light of the accuracy of the determination. The values of A003, are plotted in Figure 53. 0.9 0.8b 0.7- Decreasedncount_rate 0.67 0.5- 0.4.. _Increased_count rate 0.3 A 0C1, Kev per channel 0.2 0.1 l I I I I O l 2 3 4 5 6 A CR - Change in countrate, 105-com“ per minute 0.0 Figure 53. Variation of ACC% With Change in Count Rate A second degree equation of the form Ace; - :2 Ace 2 + x3 ACR , Eq. 18 where R2 and K3 are empirical fitting constants and £3CR the absolute change in count rate, was selected to represent the data. _The values found for .ACCé for reversible count rate changes differ by an average of fifteen percent. This is sufficiently close to assume that the process is quanti- tatively reversible for a first approximation. Therefore, K2 and 13 were -lll- evaluated for the average from increasing and decreasing count rates and were found to be -5.7 x 10'12 and 4.5 x 10'6 respectively. The values for zxccg (see Table 17) account for the apparent discre- pancy in the total gain change. The values, when the change in count rate is negative, are smaller than the values when the count rate is increaseed by a factor greater than three. These data are plotted in Figure 54. 1.0 A003 , Kev/ channel 0 l 2 3 4 5 6 ACR - Change in count rate, 105 counts per minute IZigure 54. Variation of Axccg With Change in Count Rate However, in both the cases of increased and decreased count rate, the rela- tionship between count rate and.£>CCfi is linear. With the data at hand it is impossible to hypothesise whether this difference is real, whether it is due to lack of experimental control, or whether the difference is due to a very long term gain change component. Whatever the reason may be, it _would be prudent to limit equation 17 to only an increased count rate. For increased count rateieccg varies with count rate according to the empirical relationship A003 - KaACR, Eq. 19 where K4 is an empirical fitting constant and was found to be 1.5 x 10'6. -l12- The last value in equation 17 to be evaluated was 0C5, which is equal to the initial channel coefficient (CCO) plus the gain changes that take place; thus f - 1 2 E 20 000 CC, + ACC0 + ACC . q. Substituting equations 16, 18, 19 and 20 into equation 17 gives CC: - [12600102 + K3ACR] a)“ + MACRe’ch + CC0 + K2 (A0102 + macs. +K19CRL .3. Regrouping gives cct - cco + x103: +.scn [31 + eAvt)(K2ACR + x3) + x4(1 + e*2t) Eq. 21 The constants have been evaluated as 11 8.67 x 10'8 K2 K3 - 4.5 x 10-6 -5.7 x 10‘12 x4 - 1.5 x 10'6. The equation is entered with 000 andAACR in units of Kev/channel and counts per minute. The restrictions on the equation are that CCo is close to 10 Rev per channel and ACR is positive. The bias variation does not follow a predicted pattern as well as the gain. In addition, the statistical variation is the same order of magnitude as the systematic variation and makes interpretation of the data difficult. While source B was being counted (39 to 110 hours) the bias was constant at 0.765 plus or minus 0.051 channels. The predicted but not observed systematic variation due to bias dependence upon gain was plus 0.07 channels with a shape similar to that observed for the channel coeffi- cient. When source A replaced source B (110 to 160 hours), the bias did not change significantly. The data are presented in Figure 55. A linear -113- and cad Nag and noun comb enmquem one ease:0.0ueu assoc no uoeuwm auson .uluu ouceuuuom and and 0N“ d . «Nu w- cad .mn enough c- coueeaumem amuse; caozllll . . . J 4 _ I _ cm.o Land oo.o 86 26 £6 IAco.o Inmé stsuusqo ‘ssrg ~114- least-squares fit of the data resulted in a line with slope -0.00257 plus or minus 0.00195 channels per hour. However, the probability that the slope is sero or greater is 0.097. If the true slope is zero, the mean value is 0.690 channels which is 0.07 channels less than before the source change was made. This would indicate that an instantaneous bias shift due to pulse pile up had occurred. The placing of source C (very strong) finally did cause a dis- tinguishable trend in the bias as shown in Figure 56. A very sharp in- crease occurred within the first five minutes amounting to 0.44 channels if the sloping line of bias for source B is assumed. Part of the remain- ing increased bias could be accounted for by bias dependence upon gain; namely, about 0.23 channels. However, the observed increase of 0.38 channels is almost double this. Thus it appears that a true bias shift has occurred. In the decreasing portion of the curve the observed slope is -0.00217 channels per hour of which the gain dependence component would account for 0.00097 channels per hour. The initial increase before five minutes is unique in that it represented fifty-three percent of the maxi- mum increase, where in the same period the gain increased only twenty-two percent of the maximum increase. The replacement of source A (weak source) brought about a sharp initial decrease in the bias and a gradual long term decrease as shown in Figure 57. The initial drop was 0.48 channels. This is followed with a decreasing trend of -0.0045 channels per hour. While the pattern of true bias variation is not clearly defined by these data, a trend does appear to exist; namely, the following. 1) A very rapid change (perhaps instantaneous) in bias occurs when the source strength is changed. 2) In addition a bias component, dependent on total counts, tends to increase the bias with time until equilibrium is reached followed by a decrease of bias with slope of approximately 0.004 channeli moan coo: oamaumm one owcmno comm unnoo mo uoouuu .on shaman oou wag men was «we owe cud «ha mod «ca cod _ t a . W ,a 4 _ _ a . _ _ _ _ 4 a . a q 8..” mo.~ o~.~ U" 9-l F. o N H -115- In on ,-| stauusqo ‘ssru om.~ mm.~ o¢.~ n¢.~ om.~ 3:. uoan Sufi-m a: 9.8.6 32 058 «o 33:. .3 0.5»: mason .esau codenamed ~n~ won «as cow was «ms was . «ea cow emu «mu ‘1 fl . q . . a . u _ a omoO L _ _ mn.o o - om... no.c 05.0 ~116- nu.o o I 86 1 36 .. 85 I ma.o Oo.d stsnueqo 'ssrg -ll7- per hour after the source has been removed. This would qualitatively be similar to an "after-glow" condition that exists after a phototube has been exposed to ambient light. The photocathode continues to emit electrons for as long as 48 hours after the stimulus (light) has been removed (6). No reference in the literature to such a phenomenon has been found when the light intensity is on the order of that produced with a strong radioactive source and sodium iodide scintillators. It should be recalled that the use of the x-intercept as a measure of bias is only valid if the bias pivot point is independent of gain. This fact has not been established. CHAPTER VI BVALUATIOR’OP SIMULTANEOUS EQUATIONS METHOD In previous chapters it has been demonstrated that the spectrometer is unstable by varying degrees to several extrinsic variables. Whether the degree of instability found is significant will depend on the use of the spectra. In this investigation the effect of instability upon quanti- tative analysis results, using the simultaneous equations method for re- solving composite spectra, is studied. This will also aid in evaluating the peak location and energy response determination methods to see if they are sufficiently accurate to delineate changes which are significant from the practical point of view. Basis for Approach In order to show the factors effecting the amount of error intro- duced into the simultaneous equations method, an example using only two components is used. The same approach can‘be extended to three or more components, but the mathematics becomes more cumbersome. Figure 58 shows graphically the quantities that are being manipulated. The figure indicates the composite spectrum of nuclides X and T is simply the sum of the spectra of nuclide x and nuclide T. The count of the composite spectrum for the photopeak channel of nuclide x, denoted as Rx, can be written as Rx - Xx +Xy Eq. 22 where x, is the count contributed by nuclide x to the photopeak channel of X, and X, is the count contributed by nuclide Y to the photopeak channel of x. Likewise the count in the photopeak channel of Y, denoted Hy, can be written as Hy _ I? + 1* Eq. 23 -118- -119- Huclidex 10.. .m— Q.-—.I NucLideY 20F- .I _. 0 _ I: g __ 4:10-_ 0 H 2‘ I— 3 " ’9 3 1 IIIIIIIII III U l: 3 10, — Nuclides--x {-1 Y ,u. _ 2.9.- 14.. F I... Q--—IIIIIIJIIVIIII+IIIIiIIII 0 5 10 15 20 Channelinumber Figure 5.8. -Parameters-Fnr Simfiranenus Equations Method -120- The principle of "constant spectral shape" for a single nuclide spectrum states that the ratio of any portion of the spectrum to another is constant and independent of activity. This principle is used to define the "interference factors" which are ‘fxy ..gg zq.24 xx f - x yx '?%. Eq. 25 where fy: is the interference factor which eXpresses the fractional contri- bution of the photopeak channel of nuclide Y to the photopeak channel of nuclide x, and fxy is the reverse of this. Equations 22 and 23 can be expressed in terms of just the photopeak channels by substituting equations 24 and 25. x - Xx + fny5 Eq. 26 y - vi+ fxixx Eq. 27 Solving these simultaneous equations for Xx and Y’ gives - f N 1 - rxygy: The N‘ and fly values are determined from the composite spectra, which are in actuality the sample spectra, while the interference factors are determined from reference spectra. Now if an energy response shift occurs between the counting of the reference spectra and the sample spectra, the interference factors will be in error by an amount f which will result in an error in Xx and Ti. Equations 26 and 27 can be expressed in terms of the incorrect interference factors as follows: ~121- my - I; + (fxy +A£xy)x,} Eq. 29 where the primes refer to the incorrect quantities. If the errors in the count due to each nuclide are defined as I Ey-Yy'Yy I Ex ' xx ' xx then equations 28 and 29 can be equated to equations 26 and 27 and the error terms solved for. The result is By ' fnyy(fxy +Afxy) -Afx‘fix fyxfxy +Afyxfxy + Afxyfyx + Afxymyx - 1 Eq. 30 fyxfxy + Afyxfxy + afayf.yx + Afxy Afyx - l Eq. 31 The major conclusion that can be drawn from these equations is that, the errors and also the percentage errors, are dependent on the count of all the nuclides in the composite spectra as well as the errors in the interference factors. The difficulty of evaluating the simultaneous equations method now becomes apparent. The error will be a function of the amount of gain and bias shift as this determines f, the number of nuclides, the shape of each spectrum, and most important the relative amount of each nuclide. In essence, the amount of error is dependent on an infinite number of combi- nations, and thus the quantitative effect of instability of the method can not be expressed in a simple, direct manner. The approach selected was to limit the evaluation to the analysis of fluid milk contaminated with fallout at levels of contamination that have frequently been observed. 'Milk analysis is the most wide spread use of gamma spectroscopy in environmental radiation surveillance. -122- Procedure The general method of approach was to obtain reference spectra of four nuclides. Composite spectra were artificially manufactured by multi- plying each reference spectrum by varying constants to change the amounts of each component, and adding the components together. The reference spectra were then shifted by varying amounts of gain and bias. The shifted composite spectrum was resolved back into its component parts using the shifted reference spectra. These were compared to the amount that was originally put into the composite spectra. The detailed procedure is covered in the description of the computer program. Reference spectra. Reference spectra of iodine 131, barium- lanthanum 140, cesium 137, and potassium 40 were obtained from the South- western Radiological Health Laboratory, U.S. Public Health Service. These spectra were used rather than ones obtained on the‘nichigan State University spectrometer because they more adequately represented the counting geometry and spectrum quality that is commonly used. The'uichigsn State University spectrometer can not accommodate large volume samples (3.5 liters), and its resolution is relatively poor (approximately twelve percent for cesium 137). The reference spectra were reduced to a common activity content (nanocurie per 100 minute count) by dividing the original spectra by appropriate factors for ease of manufacturing composite spectra. Table 18 gives the calibration data. The spectra are given graphically in Figure 59. ~123- ~w~.ca Ono.cu muse» ca x nu.~ o oonaueonmoINH Oo~auooumoIN~ nnw.n cmnm name» on sumo» no.~ oumauqcuacIua Ocuuuuouauwaa aaa.n Games .aau m.~e sane wo.a~ oooeueo-~°-~e ooaeuee-~c-em onn.o owed sane mo.m sane ee.ae neoeuee-eo»~e oouauwouoauca moaned! acaxqom eon wauussou um huu>wuu< ouwdumawm osau hmuon wounsoo ouan wouauoudmu ouan ca ca ca an souaeaa..oaau maaunaou Agent» any ooa.an ca«.- occ.ooe oom.nn~ can .aua>auu< as and as” mma sauna-ooh saw-00 ssamcummqus9«ump usawoa uoxmoo «Haomuwmx.uouua m.n u huuolooufl Hoasssuxsau ca " unequummoou Among—oH mH mug 5H: «8 <93 moans—Eda .3 59: mundane: codename: no wuuuoem all-o .om «woman woolen denounu cow cam and cud cog ana cod and cad cad cod cm on as cm on as on on OH 9 _ _ e 7 a _ _ a _ _ _ _ ..., _ j _ e _ _ 3 . . .. , e. 3 // s1fs5’f “fa \ua. . a. . a \ r‘f /W~. / 1 q , . \: I332.» .TI VXJ u I 3 . r\\.{ . . I on.” \w N , \ , /I.II.HJ./) \x. . z \x /. \. .. lll/.“\..U.tW~V/\ um [I \ M . w/(u/h» \IM 1 ..\ o a . /. \ .. .. I rt /x~ mad 3.. 2: £3 IIIII ...cS Ba 2 .62: ...|.|I- 3.. S .3... ... .. .... II... ..I tauusqo as] 01 13d annurm 13d sauuug ~125- Variation of activity content. The approach of manufacturing arti- ficial spectra by addition, has the advantage that all errors from counting statistics are eliminated. Also the effect of energy response shift in the component spectra upon the composite spectrum is dependent on only the relative amount of each component and not on the absolute amount. The shape of a spectrum made up of 100 pCi of each component would be identical to one made up of 1000 pCi of each. The original intent was to vary the amount of cesium, iodine and barium-lanthanum from 0 to approximately 150 pCi/liter and to hold po- tassium at 1500 pCi/liter. The levels are representative of those found in fluid milk. Through an error in entering the spectrum multipliers, the potassium was set at 15,000 pCi/liter instead of 1500 pCi/liter. This re- sulted in unrealistic composite spectra. An alternative interpretation of the spectra is that the potassium content is 1500 pCi/liter and that the iodine, cesium and barium-lanthanum content varies from 0 to 15 pCi/liter. Unfortunately 15 pCi is approximately the practical limit of detection for these nuclides; thus the information gained is of little practical value. The activity contents used for manufacturing composite spectra are given in Table 19. Using these concentrations, 32 composite spectrum sets (4 a 4 x 4 x l - 32) were formed. TABLE 19. ACTIVITY CONCENTRATIONS USED FOR NINUEACTURINC COMPOSITE SPECTRA, pCi PER LITE! Iodine Cesium Barium-Lanthanum Potassium 0 0 0 15,000 10 20 10 50 60 50 100 150 100 -126- Variation gg_gngggz response. The gain and bias were separately varied for shifting the reference spectra. The gain shift was changed from minus 2.0 percent to plus 2.0 percent in increments of approximately 0.3 percent, while the bias shift was changed from minus 2.0 channels to plus 2.0 channels in increments of approximately 0.4 channels. The gain vari- ation represents the largest that can be incorporated with the program TSHIFT. Twenty-six different energy response shifts were used; thus the total number of composite spectra resolved was 26 x 32 - 832. I Simultaneous equations method. Rather than use a single peak channel for Ti, Kg, Ry, Ni and the interference factors as in the example, energy spans covering entire peaks were used. The usual procedure is to select a peak width of one channel less than the full width (valley to valley) of the peak. This procedure is intended to lessen the effects of instability. The peak areas used are given in Table 20. TABLE 20. PEAK AREAS USED FOR SIMULIANEOUS EQUATIONS METHOD Iodine 131 Barium-Lanthanum 140 Cesium 137 Potassium 40 Peak channel 37 49 66 146 _§umming channels 1341141 46-53 61-73 138-154 Peak width, 8 43 13’ 17 - channels Counts in peak/ 1.15 0.66 0.89 0.061 pCi/100 min. Computer Program The program developed for determining the effect of energy response shifts upon the simultaneous equations method is coded EFFECT. The flow diagram is given in Figure 60, and the data deck description and program are given in appendices A and B respectively. I. J. 0. W [Em mm: 1mg] -127- [finer srncmn mannsj: roan mosm Symon SELECT GAIN AND BIAS SHIFTS FORM SHIFTED CGIPOS ITE SPECTRIM L___A‘T*_5_ [3mm 3?:ch [gamma BEES}? SELECT INTEGRATION ‘— lst STANDARD STOP: CMOSITE TART FORE SIMULTAEEOUS EQUAT IONS camman acuvrrr or sacs worms | STORE smaTl LE l START: l START: I l STOP: # OP SHIFTS STOP: # OF SETS r1 START: 1 STOP: # or sns F [first “sums 'P PRINT SMART I E; Figure 60. General Flow Diagram of Program EFFECT -128- The input data are read in (A) consisting of 1) the number of nuclides, composite spectra, channels per spectrum, gain and bias shifts; 2) the spectrum multipliers; 3) the amounts of gain and bias shift; 4) the starting channel for integrating the peak areas; 5) the width of the integrated areas; and 6) the reference spectra. This data may be printed out for checking (B) at the control of the investigator. The first set of spectrum multipliers is selected (C) and each reference spectrum is multiplied by the appropriate factor. The adjusted spectra are then added together to form the composite spectrum (D). The bias and gain shifts are selected (E) and the subroutine TSHIFT is entered to form the shifted composite spectrum (F). The steps G through R relate to the simultaneous equations method. The integration areas are selected (G), and the required areas of the unshifted reference spectra and the composite spectrum are calculated (R, I). Using these quantities, the basic simultaneous equations are formed (J). The solution to these equations is the nuclide content (R). Selected data are stored as a summary (L) and the results are printed out (I). The program then returns to step E and selects the next set of gain and bias shift. At the completion of these, the next set of spectrum multipliers is selected and the entire process repeated. When all spectrum multipliers have been used the summary is printed out (N). Data Analysis The original intent of this portion of the study was only to ob- serve the errors introduced into the analysis of milk by energy response shifts. However, because of the error in entering the potassium.content (15,000 pCi per liter substituted for 1500 pCi per liter), this can be done only at levels which are close to the limit of detectability. The -129- remainder of the data are presented and basic properties are pointed out. However, these data are not representative of milk analysis and any inter- pretation should not be extended to milk analysis. 935; go; simulated milk sample. Percent errors are given in Table 21 for the analysis of milk samples with activity concentrations of 0 and 10 pCi/liter iodine 131, 0 and 10 pCi/liter barium-lanthanum 140, 0 and 15 pCi/liter cesium 137 and 1500 pCi/liter potassium 40, and gain shifts of plus and minus two percent and bias shifts of plus and minus two channels. The major conclusion from these data is thatthe errors are very small compared to the errors that would be introduced by counting statis- tics at these low levels. The minimum detectable activity of iodine, barium and cesium is approximately 10 pCi/liter and is defined as the amount of activity at which the error is plus or minus 100 percent at the 95 percent confidence level. It is apparent that compensating errors are present. With the exception of iodine errors, the error at all levels are due mainly to potassium since the counts due to potassium are approximately ten times greater than those from barium or cesium. In spectrum 1, the errors in the iodine, barium and cesium are all due to potassium. In spectrum 5, which has only iodine added, the error for iodine decreases indicating a negative error due to the iodine component. The same is true in spectrum 3 when barium is added: the error in barium decreases. It is also seen that some components are not significantly effected by changes in the other components. When cesium is added to spectra 1, 3, 5, and 7 to give spectra 2, 4, 6 and 8, and the gain shift is positive; the error in barium is affected very little. The error in potassium is only dependent on the amount of barium while iodine or cesium have little effect ova? sowumuusoosoo 539.33 gamma: so @023. mu mongoweml ousa-.aw..hhua#4uum 0:34.. IJIII mé- o.m no.” H6 5:7. m4 5.¢ «.5 com.— mH ca OH m a6; o.o.— n.5a ~27 5:7 «.0 o.~ N.@ can.— 9 O." o.— 5 o.¢u n.9u «.mn m.¢ N.~I ¢.o a.a A.N coma ma o ca m o.a- o.ae a.oe ¢.n ~.H- e.e a.» a.“ coma o o as m «:7 «4 N6- m.m 0.0 o; ...N m.o~ aéI 55 5.3 o.~ 5A- «A Ed o.5 82 m." ca 0 o -130- mien 0.5 no: cl.5I To m.N N.nu a.o~ «.mu o.wa a.5~ 54 5:7 n.N o.N w.5 83 o 3 o m 6| o.mI «A Hon 5A .3. n6 #5 5.5 m6- «.3 5.3 5.5 «:7 OJ 0.5 5..” 603 n." o o o.m.. o.m o.nI #6 .o a; 0...” 5.» a.a.. .o.a.:5..d.~.Imd.o- “NWT. #4 .70 a.~ Gang 9 c o H “I .0 ..n hr HI}; “ . ‘0. .ql IIH I M . .0 . .m r H . u I '0; .nl H 15:“!1818 -.IH nouaE\aoa Someone «I I n 4 ado-Essa 5+ .- n4 usouuom NI I ed I unsouom-N+IflId¢ auw>uuum.-osua enema-musoocou 532304 no woman umouuomw ilrll mZOHagnozoo :HDHHn: 33 “.4 ”as...“ 5H! Egan ho Snug mom magma Eva—mm 23 3:8 -l3l- as seen by comparing spectra 1, 2, 5 and 6 to spectra 3, 4, 7 and 8. The only other significant observation is that with the exception of the negative bias shift, the errors are in a positive direction for iodine, barium and cesium. Thus, regardless of the direction of the gain shift, the results will tend toward a value greater than the true value. It is once again cautioned that the small errors at these low levels do not necessarily indicate that the errors will be small at higher concentrations. Effects g§_gain shifts. Figure 61 shows the effect of varying gain shifts on the percent error in iodine for all the composite spectra. The shaded areas represent the changes due to varying the cesium content from 0 to 150 pCi/liter. The shape of the curves are basically the same for all four levels of iodine. However, as the iodine is increased the lobes in the negative gain shift portion move progressively to the left. The percent error decreases almost in proportion to the amount of iodine present. This indicates that a substantial portion of the error is due to nuclides other than iodine. If this were not the case, the percent error would remain approximately constant because the absolute error for a single nuclide is directly proportional to its amount. The greatest variation is caused by changing levels of barium, while cesium causes a relatively small variation (as indicated in the shaded portions). It is also seen by examining the curve representing sero iodine and sero barium, that potassium has a significant effect. The errors when barium is 50 pCi/liter are unique in that they are all in a positive direction; thus always leading to an over-estimation of the iodine activity re- gardless of the direction of the shift. Figure 62 shows similar information for the percent error in the barium-lanthanum concentrations. The shaded portions indicate the IODINE ACTIVITY I 10 pCi/LITER L Percent error sL. L I I O i 5 60' 7or so ‘- o pCi/liter “Ora-La j 50 pCi/liter 1“Ola-m 100 pCi/liter l‘ona-La ‘0: - 15,000 pCi/1. Percent Error of Iodine Concentration for Cain Shifts Figure 61a. 2.0 I ..‘. 1 .5 _ / “‘fl. ...eo e Q 1.0— |"' ,e’. / “ ’0’ 101mm acnvrrr so pCi/.LITER 0.5_ 1 " ”x I 0 ' ‘l l . I I I I I I I ‘h /f x 2 4 6 s 10 12 14 16‘ 4’ \ Percent error - -o.5 _ s N 9.. -1.o .. s $0. \ I '0. I. ‘1 '0 - , __ ' i. " -2.o _ 2.0 " I e‘ i e 1.5 _ “fl “‘ “‘ e‘...‘ e 1.0 __ / “‘ ...e I I! I" mainstay—1111400 pCi/liter 0.5 -— I x 0 1’4” L I I l I * ’ I l l : ' I I -5 -1' ’/ |\ '1 2 3 a 5 6 7 -0 5 0’ 3. PercenLerror - 0 — - ‘Q‘. "’ \\\‘ 0 pCi/liter 1l'OBa-L'a -l.0 — Q % Ml 150pCi/liter 1Sana-La \ | . ”’ I - loo-pCi! liter 140351.11 -1'5 — III 0" 401 - 15,000 pCi/liter a2.o , f ” Figure—blbr—J-ereent—irror'of'iodine‘conc‘éntration for Gain Shifts auuunm same you souusuumoomoo saswcumaAIssuuem mo uouum usouuom ..No shaman -134... o.N.. l at? [$41 1' mac... : 1 ll I/flffl/I I I [If/It I i// l/fl/ HORN” UGQOHflm // ems mma atom .3 ..cm . fl _ a q _ _ 4 ... 3.5.6.. 892 I use . .v 3-39: 33:5.— 2: g O’ «AI-moi .8323.— 3 E 3-39: 3313.. 2 a a”? . o Im.o lo; In.“ o.~ 3;;qs urns auaaasa -135- variation due to both iodine and cesium. These curves are very similar to the iodine curves. The only major difference is that both iodine and cesium levels effect the percent error in similar proportions. Table 22 shows the error when these two nuclides are varied with other conditions constant. The dotted line within the shaded portion of the 10 pCi/liter barium-lanthanum curve represents the error when all nuclide concentrations are sero except potassium 40. The small spread of the shaded portion about this line indicates that the majority of the error is due to potassium. This would be expected since its activity content is relatively high com- pared to the others. TABLE 22. PERCENT ERDR IN EARIIH-LANTEANIH CONCENTRATIONS FOR VARTINC AMOUNTS 0F CESIUM AND IODINE Ea - 100 pCi/1. R - 15,000 pCi/l G - +2 percent, 8 I 0.0 channels Iodine Activity Cesium Activity, pCi/1 pCi/1 0 20 60 150 0 55.1 58.0 63.6 76.3 10 55.7 58.6 64.2 76.9 50 58.0 60.8 66.5 79.2 100 60.9 63.7 69.4 82.0 The curves for the percent error of cesium concentrations are similar to the barium curves, except there is no indication of symmetry about the x-axis. One group of curves for a constant cesium activity is given in Figure 63. For positive gain shifts, the system is quite stable compared to negative gain shifts. Effects 2; bigg,ghi£§g, The nuclides causing the greatest percent error variations with bias shift are essentially the same as the nuclides causing the greatest percent error variations with gain shift. The similarity between gain and bias stops here. Rather than tending towards -136- H—_ —_. 33.—m case How souueuuseoooo Sumac we. noun—N E0305 .no annual anana\aoa coo.na unnsaaoa.manmm.EMEVI5).NSCIS).NUMIIoo).AALIBI70.5I.ALIa(560I DIMENSION Y(50).A(20).XBAR(100.5).SX(5)o 5x215).$D(5)oAVEX(53 DIMENSION CY(5)o CF(5)0NPU(5) READ 100‘ NTRAIL READ IOIoNSTcNP READ IO50INSPII). I I IoNST) READ lOBQIEMEV‘I).I I IoNP) PRINT 2000 PRINT IO4oNTRAIL DO 500 KS t IoNST ANSP 8 NSPIKS) NNSP I NSPIKS) READ IOOoMM READ IOZQINPU(I)0I 8 IQNP) READ IOZOINSCIIIQI 3 IoNp) DO 501 KK 8 loNNSP READ 999NUMIKK) ML 8 MM/B DO 300 I SlgNL 300 READ IO9o(AALIB(IoJ)oJ=IQB) DO 301 LxloNL DO 301 M3198 I a H+L*8-9 301 ALIB(I) = AALIB(L.M)/IOOoO DO 302 I 8 IoNP NNSC 8 NSC(I) NNPH 3 NPUII) ANSC 8 NSCCI) NEC I N$C(I) + NPU(I) - l I! I 0 DO 303 J a NNSCoNEC I! - ll+1 303 Y(III 8 ALIB(J) KDsa' CALL POLYFTIKDoNNPUoYoAcKB) CALL PEAK2(A0PK) 302 XBARIKKQI) 8 PK+ANSC-IoO CALL ENRESP‘XBARCEMEVQNPOKKOGAINQBIASOGE06E) 501 CONTINUE CALL AVER"(XBARONPONNSPOSD OAVEX) CALL AVERSIBIASoNNSPoABEoBBIAS) CALL AVERSIGAINoNNSPoAGEoGGAIN) PRINT ZOOoKS PRINT ZOIQNNSP PRINT 202 PRINT 203 00 308 I310NNSP 308 PRINT BOAoNUMtI)o(XBAR(IoJIoJ8IoNP) PRINT 205oCAVEX(J)oJ819NP) PRINT ZOGoISDIJICJ'IoNR) -189- 'c PROGRAM REPROD CONTINUED UIKS) VIKSI UCKS) ABE RIKS) AGE DO 305 I 3 IoNP SIKSOII= AVEXII) 305 TIKSOI)= SDII) 500 CONTINUE PRINT 2000 PRINT 210 PRINT 211 DO 304 KS 8 IONST 304 PRINTZIZOKSOS‘KSOl)OT(K$91’OS‘KSQE’9T‘KSQa’OS‘KSQB’OT‘KSQ3)Q ISIKSQ4)OTIKSQ4)OUIKSIORIKSIQVIKSIOUIKSI PRINT 2000 2000 FORMAT (1H1) 99 FORMAT (I4) 100 FORMAT (15) 101 FORMAT (2I5) 102 FORMAT (5I5) 103 FORMAT (5F1002) 104 FORMAT (25Xo29HREPRODUCEABILITY TRAIL NUMBER' .Is.//3 105 FORMAT (IOI5) 109 FORMAT(8F700) 200 FORMAT (3IXoI3HSPECTRUM SET 9I3) 201 FORMAT (29XoIBHNUMBER OF SPECTRA oI3ol/I 202 FORMAT (32XQI4HPEAK LOCATIONSo/) 203 FORMAT (EXCBHSPEC NUMQ4X02HPIQBXQZHPZOBXQZHpaoBXoaHPOoDXDZHPEI 204 FORMAT (3X9I505F1003l) 205 FORMAT (2X94HMEAN02X05F1003) 206 FORMAT (2X96HSTD. Do 5F1003o/l) 210 FORMAT (ZQXOZZHMEANS OF SPECTRAL SETS I 211 FORMAT(2X.3HSET.5X.2HP1.I3X02HP2oIQXoZHP3o14Xg2HP40IQXoZHCColax. I4HBIAS) 212 FORMAT (1X9I49F7o39F6030F9039F6039F9o39F6039F9O3QF6039F9049F7049 IF9030F503) END SUBROUTINE AVERMIXBARoNPvNNSPOSDoAVEX) DIMENSION SX(5)oSX2(5)o$D(5)oAVEXI5IoXBARIIOoo5) ANSP 8 NNSP DO 304 J anNP SXIJ) 3 0.0 304 SXZIJ) 3 000 DO 306 J t IONP DO 305 I ‘ 19NNSP SX(J) = SXIJI+XBAR(I9J) 305 SXZIJ) I SXZIJ) + XBARIIOJ)*XBARIIOJ) SDIJI8I(SXZIJI-SXIJ)*SX(J)/ANSP)/(ANSP-I00))**005 306 AVEXIJI 3 SXIJI/ANSP RETURN END SUBROUTINE AVERS‘XONNSPQSDQAMEAN) DIMENSION XIIOO) GGAIN BBIAS 300 -190- PROGRAM REPROD CONTINUED ANSP ' NNSP 3X 3 0.0 5x2 3 000 DO 300 I 8 loNNSP 5X3 SX + XII) 5X2 3 5X2 + XII)*X(I3 SD 8 ((SXZ - SX*SX/AN$P)/(ANSP-loo))**005 AMEAN 8 SX/ANSP RETURN END ADDITIONAL SUBROUTINES REQUIRED ENRESP POLYFT PEAKZ -191- PROGRAM CKERL DIMENSION NSC!5).NPU(5).EMEV<5).AALIBt7oo8).COMP(560).SHCONP(560) DIMENSION Y(50)QXBARI10005IOGAINIIOO)OBIASIIOOIoGEIIOOIQBEIIOO) DIMENSION AI15) DIMENSION BBSIIOOIIGGSIIOO)OTTSIIOO)QDBSIIOO’QDGSIIOO) MM 8 256 L 8 O NP 8 4 KO 8 2 PRINT 2000 READ IOOoNTRIAL READ IOIQ (NSCIIIOIII94) READ 1010 INPU‘I’OI'IG4) READ 1039(EMEVIIIQIII94) DO 300 I=Io32 300 READ IOZQCAALIBIIOJIQJ‘IQBI DO 301 L=Ic32 DO 301 M'IOB I I M+L*8-9 301 COMPII) = AALIB‘LOM) DO 501 KK 8103 BS = 0.0 GS = 000 DO 501 JJ=106 L I L+1 AK 8 JJ IFIKK-2)20092019202 200 as = AK*0.4-004 GO TO 203 201 GS 8 AK*0.2 ~0o2 GO TO 203 202 GS 8 AK*°.2-002 BS 3 AK*0.4-004 203 CALL TSHIFT (COMPOSHCOMPOMMOGSQBS) DO 502 IISIQ4 ANSC 3 NSCIII) SHIFT 8 BS+GS*ANSC/IOOoO IFISHIFTIZOAQZOSQZOS 204 NCOR 3 -SHIFT+100 NNSC = NSC(II) -NCOR+I GO TO 206 205 NCOR 3 SHIFT + 1.0 NNSC 3 NSCIII) + NCOR-I 206 NNPU 8 NPUIIII ANSC 3 NNSC NEC 3 NNSC + NNPU-I NN = 0 DO 302 I 8 NNSCQNEC NN 3 NN+I 302 YINN) ‘ SHCOMPII) CALL POLYFTIKDQNNPUQYQAQK3) CALL PEAKZIAQPK) 502 XBARILqII) 3 PK + ANSC-100 CALL ENRESPIXBAR.EMEVoNPQLvGAINoBIASoGEoBEJ 501 303 104 105 2000 100 101 102 103 600 601 602 603 604 PROGR BBS(L) GOSIL) TTSIL) CONTIN PRINT PRINT PRINT PRINT 881 a 661 2 DO 303 OAINII BIA$(I 035(1) 065(1) PRINT CONTIN PRINT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT FORMAT END SUBROU TSHIFT SLINEO ENRESP POLYFT PEAKZ -192- AM CKERL CONTINUED 3 BS 3 GS = BS+GS UE 600 601 602 603 BIAS(1) GAINII) I ' 1018 I F GAIN(I) - $61 I = BIAS(I) - BBI 3 BBSII) - BIASII) ' GGSII)-GAIN(II 6049IQGGSIIIQBBSIIIQGAIN(I)OBIASII’0 DGSIIIODBSII) UE ZOOO (1595X0F1004) (I502X0FIOO4OFIOO4) (1H1) (I5) (515) (8F7OOI (SFIOOZI (21X939HCOMPARISON OF CALCULATED TO DETERMINED 7) (30x.2OHENERGY RESPONSE LINE//’ IIXOBHSPECTRUM93X9ITHCALCULATED SHIFT$O3X017HUETERMINEU an: IFT506XoIIHDIFFERENCES ) ‘2x06HNUM8ERO7XO4HGAINO4XO4HB1A5.BXO‘HGAINO‘XO‘HBIASOEXO I4HGAIN¢4X94HBIAS I (2X0I506x.F6.392X0F60396X9F603‘2XoF60306X9F6OJOCX9F00Ji TINES REQUIRED -193- PROGRAM CKSHIFT DIMENSION ALIBISO.8). COMP(400). SNCOMPIAOOI.TCOMPIAOOOI DIMENSION TTCOMPIAOOTIOI READ IOO.NTRIAL PRINT 2000 PRINT 105 PRINT IOS.NTRIAL READ IOO.MM M8 = MM/a DO 300 I - I.M8 300 READ IOZ.(ALIB.MaI.IO) PRINT 2000 100 FORMAT (15) 102 FORMAT (8F7.O) 103 FORMAT (Ix.13.4x.IOF7.O) 105 FORMAT I30X.22HINTERPOLATION SPECTRUM.//) 106 FORMAT (35Xo13HTRIAL NUMBER . 15./I 2000 FORMAT (1H1) END C SUBROUTINES REQUIRED TSHIFT SL INEQ 300 301 302 303 304 305 100 101 102 201 202 203 301 300 -194- PROGRAM CKFIT DIMENSION ALIB(70¢B)0 OCOMPIBGOII TCOMPISOQ). DIMENSION COMPISOOI READ IOOO NTRIAL PRINT 1010 NTRAIL READ 1009 MM M8 3 MM/B DO 300 I 810MB READ 1029(ALIB(IOJIOJ‘198) DO 301 L = 10MB DO 301 M 8 198 I 8 L*8+M-9 OCOMPII) 8 ALIB(LOM) M2 8 MM/Z DO 302 I = IOMZ II = 2*Iml COMP(I) ‘ OCOMPIIII/IOOoO CALL CKSH1(COMP9TCOMP0MZI DO 303 I 8 IQMM TCOMPII) 8 TCOMPII)*IOOoO DIFII) 3 OCOMPII) - TCOMPIII DO 304 ISIQMM SUMI = SUMI + OCOMPII) SUMZ = SUMZ + TCOMP(I) SUM3 = SUM3 + DIF(I) PRINT 201 DO 305 I a loMM PRINT 2029190COMP(I)oTCOMP(I)oDIF(I) PRINT 203oSUM1o SUMZC SUM3 FORMAT (15) FORMAT (19X042HCOMPARISON 0F TSHIFT SPECTRA. FORMAT (BF700) DIFI560) TRIAL NUMBER OI59//: FORMAT (1X07HCHANNEL97XoBHORIGINAL99XI6HFITTEDIBXOIOHDIFFERENCEo/l FORMAT (3X01303F1500) FORMAT ¢3X03HSUM93F1500) END SUBROUTINE CKSHIICOMPoTCOMPoMM) DIMENSION COMPI400). SHCOMPI400). GS = 0.0 BS 8 0.0 L 8 0 DO 300 KK: 1.2 CALL TSHIFT(COMPOSHCOMPoMMoGSoBS) DO 301 I=IoMM NC 8 1*2-I+L TCOMPINC) = SHCOMP(I) BS 8 BS+005 L81+L RETURN END ADDITIONAL SUBROUTINES REQUIRED TSHIFT SLINEQ TCOMP(400) -195- SUBROUTINE POLYFT (KDoNoYoAOK3) DIMENSION Y‘4o)OV‘IOOOOIOBETAIZOIOALPHA‘20)05‘20)O RHOIZOI DIMENSION C(20920)9 AI20) TOL 8 0.0000000 DO 300 I8IIN VIIQI) 3 I VIIQZI8 YII) VII.3)= 0.0 300 V(Ic4)= 1.0 R1 8 N BETAII)=000 K1 = KD+I RHO 30.0 R3 = 0.0 DO 301 I=IoN 301 R3 8R3 + VII.2)**2 DO 500 I8IoKI R48 000 DO 302 J=IoN 302 R4 8 R4+V(Jo2)*V(J04) SCI) 8 R4/Rl R5 8 R3 -SII)**2*RI AN=N-l-I IF(AN)20092009201 201 RHO(I) 8 R5/AN GO TO 202 200 RHOII) 3 R5 202 K2 8 I-I DO 303 J=IQK2 IF (ABSF(RHO(K)-RHO(I))-TOL) 20302030303 303 CONTINUE IF (I-KI) 206.205.205 206 DO 304 K'lQN 304 - VIK95)8 VIK01)*V(KO4) T1 3000 DO 305 L810N 305 TI 8 T1 + V(L05)*V(L04) II 8 1+1 ALPHAIII) 8 TI/RI T2 3 000 DO 306 LIIvN VIL95) 8 (VILol)-ALPHA(II))*V(L04)-BETA(I)*V(L93) T2 8 T2 + VIL95)**2 VIL93)= VILQ4) 306 VIL94)8 VILQ5) BETAIII) 8 T2/R1 R3 8 R5 R1 8 T2 500 CONTINUE 205 K3 8 K1 GO TO 207 203 K3 8 I 207 DO 307 I8I¢K3 C(I91)8 1.0 307 308 309 310 311 208 312 501 -196- SUBROUTINE POLYFT CONTINUED II 8 1+1 C(IoIII 8 0.0 DO 308 I8 20KB II 8 1-1 C(Io2) 8 C(11o2) - ALPHA(1) DO 309 J8 39K3 DO 309 I8 JoK3 11 8 1-1 J1 8 J-I I2 8 1-2 J2 8 J-2 C(19J18 C(IIOJ’ - ALPHA(I)*C1110J11- BETAIIII*C(129J2) D0 310 I 8 19K3 DO 310 J8191 C(IcJ)8 C(10J)*S(I) DO 501 I8 IoK3 II 8 1-1 A11) ‘ OOO DO 311 J8 101 A11) 8 C(JoJ) + AII) IF(II)208.5010208 DO 312 JBIQII III 8 I-J A1111) 8 0.0 J1 8 J+1 DO 312 L8 JIoI L1 8 L-J A(III) 8 C(LoLI) + A(III) CONTINUE RETURN END 200 201 202 20 21 22 -197- SUBROUTINES FOR PEAK LOCATIONS SUBROUTINE PEAKZ (AAOPK) DIMENSION AAIEOI PK - -AA(2)/(2.0*AA(1)) RETURN END SUBROUTINE PEAK3 (AAoPKgNPI) DIMENSION AAIZO) A 8 (ABSFIAA(2)**2-3.O*AA(1)8AAI3)))**005 PKI 8 (-AA(2)-A)/(3.0*AA(1)) PK2 8 (-AA(2)+A)/(3.0§AA(1)) MID 8 NPI/2 + I AMID 8 MID DI 8 ABSF(PK1-AMID) DZ 8 ABSF(PK2-AMID) IF (DI-D2) 200.201.201 PK 8 PKI GO TO 202 PK 8 PKZ CONTINUE RETURN END SUBROUTINE PEAK4 (AAOPKONPI’ DIMENSION AAI4) DIMENSION 0(3) DIMENSION A14)QB(3)OYI3) A11) 8 400*AA11) A12) 3 300*AA12) A13) . 200*AA13) A(4) 8 AAI4) BIII8A12)/A(I) BIOV38B(1)/300 5(2)=A131/A(I) 8131'A14)/A11) ALF8BIZI-B(1)*BIOV3 BET8ZOO*BIOV3**3-B(Z)iBIOV3+B(3) BETOV28BET/200 ALFOV38ALF/3oO CUAOV3=ALFOV3**3 SQBOVZ:BETOV2*82 DEL8SOBOV2+CUAOV3 IF‘DELI40'20030 MTYPE‘O GAM8SORTFI'ALFOV3) IFIBETIZZQZEQZI Y(1)3'200*GAM-BIOV3 Y12)8GAM-BIOV3 Y13)‘Y(2) GO TO 50 Y(1)=200*GAM-BIOV3 Y(2)"GAM-BIOV3 Y13)‘Y(2) 30 40 41 42 43 50 302 301 -198- SUBROUTINES FOR PEAK LOCATIONS CONTINUED GO TO 50 MTYPE81 EPSISORTFIDEL) TAUI-BETOVZ R:CU8ERTF(TAU+EPS) SsCUBERTFITAu-EPS) Y(1)8R+S-810V3 Y(2)8-(R+S)/200—BIOV3 Y(3)80o86602540*(R-S) GO TO 50 MTYPE8-1 QUOT8SOBOV2/CUAOV3 ROOT8$ORTF(-OUOT) IF(8ET)42.4I¢41 PHI=(1uS7O7963+A$INFCROOT11/300 GO TO 43 PHI8ACOSFIROOT1/3o0 FACT=2.0*SORTF(-ALFOV3) Y(I)8FACTiCOSFIPHII-BIOV3 Yt2)8FACT*CO$F(PHI+2o0943951)-BIOV3 Y(3)8FACT*COSF(PHI+4¢1887902)-BIOV3 CONTINUE AMID 8 NPI/2+1 SMALL 2 100.0 00 301 I x 1.3 0(11 8 A8$F(Y(I)- AMID) IF (0(1) - SMALL) 302.301.301 SMALL 8 0(1) K 8 1 CONTINUE PK 8 Y(K) RETURN END FUNCTION CUBERTF(X) IFIX)29394 X8-X CUBERTF8-X**.333333333333 RETURN CUBERTF80. RETURN CUBERTFPXiio333333333333 RETURN END 300 301 ~199- SUBROUTINE ENRESPIXBAR. Y oNPoKKoGAINoBIASoGEoBEI DIMENSION X(5)o Y(5)oGAINIIOO)o.BIASIIOOI DIMENSION GE(100)9 BEIIOO) 211:0.0 21230.0 YI-OOO Y2=0.0 DO 300 I 8 IoNP X11) 8 XBARIKKQI) ZII8XCI)**2+211 2128 X111+212 Y18 X(1)*Y(I)+Y1 Y28 Y111+Y2 ANP - NP U 8 NP DET 8 U*211—212*212 GAINIKK) 8 (U8YI-ZIZ*Y2)/DET B 8 (211*Y2-2128Y11/DET BIASIKK) 8 ~8/GAINIKK) SX 8 0.0 5X28 0.0 SY 8 0.0 SYZ 8 OOO SXY 3 OOO DO 301 1 8 IoNP sx - sx+XIII 5x2 8 5X2 + XII)*X(I) SY 8 SY + YII) 5Y2 8 5Y2 + Y(I)*Y(1) SXY 8 SKY + Y(I)*X(I) VX 8 (5X2- tSXiSX/ANP11/‘(ANP- 1.0) VY 8 (5Y2 - (SY*$Y/ANP))/(ANP- 1.0) VXY 8 I(SXY-(SX85Y/ANP)1/(ANP-100)1§i200 VYONX 8 ((ANP'IoOIIIANP~200)1*(VY8VXY/VX) VB 8 VYONX/(IANP-100)*VX) VA 8 VYONX/ANP GEIKK) 8 VB§*O.S Bl 8 VB/IGAINIRK)*GA1N(KK)) 32 8 VA/(B*B) B3 8 (Bl+BZ)iO.5 BE1KK) 8 BIASIKK)883 RETURN END XBAR‘IOOCSI -200- SUBROUTINE TSHIFT (COMP. SHCOHP. Ho GS. 85) DIMENSION COMP(300)¢ SHCDNPI3DDIQ A(IO.ID)0 CDK3DOIQ CCIIDQID) DIMENSION STOREIIOQIOIo C(10). $HY(300) DIMENSION 5(40) DIHENSIDN 38(15015) DD 300 I 8 194 II 8 I+4 III ' I-l DD 300 J 8 1'8 JJ 8 J-I 300 A(I9J) ' I**JJ DD 306 I 8 508 A(I 9 1) 8 000 I4 8 1-4 DD 306 J 8 208 JI 8 J-l J2 8 J-Z 306 ACIQJI8 JI*I4**JZ DD 301 I 8 195 ME 8 M + 5 +I CD(I) 8 COMP(1) 301 CDIMEI 8 CDNP(M) DD 302 I 8 19M I5 8 I+5 302 CDIISI 8 CDMP(II DD 303 I 8 107 DO 303 J 8 198 CCIIQJ) 8 0.0 303 STOQEIIQJ) 8 000 M4 8 M+7 DO 500 KK: IQM4 DO 304 I 8 194 K 8 KK+I AII99) 8 CD(K) II 8 1+4 KPI 8 K+I KMI 8 K-1 A‘II99I 3 (COIKPI)-CD(KNII)/200 304 CONTINUE N8 8 DD 810 I 8 1010 DD 810 J 8 1010 810 BBIIQJ) 8 AIIOJ) CALL SLINEQ (NOBBOCOKI) DD 305 I s 207 II 8 I-I DD 305 J 8 108 305 ccuxur . ccnu) DD 437 J 8 198 “37 ccnw) :- cw) DO 307 I a 104 AI 8 I B] 8 KK + I-5 SHIFT 8 as + GS*BI/IOO.D -201- C SUBROUTINE TSHIFT CONTINUED 307 SCI) 8 AI + SHIFT DD 308 I 8 104 IFISIII'IQDIZODqZOIQZDE 202 IF‘S‘I’-‘OO) 20192010203 200 L 8 “5(II + 200 GO TO (20402050206920702070207)0L 204 J 8 3 GO TO 208 201 J 8 4 GO TO 213 205 J 8 2 GO TO 208 207 CONTINUE 206 J = 1 208 AJ 8 J StI) 8 4.0-AJ+S(I) GO TO 214 203 L 8 SII)-3o00001 GO TO (209.210.211.212.212.212)0L 209 J 8 5 GO TO 213 212 CONTINUE 210 J 8 6 GO TO 213 211 J = 7 213 AJ 8 J SII) 8 SII)-AJ+4.0 214 DO 309 K 8 1.8 309 C(K) 8 CCIJoK) 303 STORE(5oI) 8 C(1) + C(z)*5(l) + C(3)*S(I)§*2 +C(4)*S(I)**3 + 1 C(5)*S(I)**4+C(6)*S(I)§*5+C(7)*5(I18*6+C(8)*$(I)**7 DD 310 I 8 103 II 8 4-1 III 8 II+1 DO 310 J 8 1.4 310 STORE(III¢J) 8 STORE(IIoJ) DO 438 J 8 104 438 STORE¢I¢J) 8 STOREISoJ) SUM 3 000 00 311 I 8 1.4 311 SUN 8 STORE(I¢I) + SUM SHYIKK)‘ SUM/400*I10000-GSI/10000 500 CONTINUE DD 312 I 8 I." II = I + 7 312 SHCOMP (I) 8 SHY(II) RETURN END C ADDITONAL SUBROUTINE REQUIRED SLINEQ 300 42 43 10 11 12 13 14 20 30 31 32 33 34 40 41 50 75 -202- SUBROUTINE SLINEQ (NvoXoKI) DIMENSION AIISQISIO XIID) DO 300 K8 I.IO XIK) ' 00° NP" 8 N+I EP 8 0.00001 DD 34 L810N K980 28000 DO 12 K8L9N IFIZ-ABSFIAIKOLIIIII912912 28ABSFIA(K9LII KPIK CONTINUE IFIL-KPII3QZOQZO DO 14 J8L0NPM Z8ACL9J) AILCJI8AIKP9J) AIKPOJ)'Z IFIABSFIAILQL)I’EPI5005OQ3O IFIL-NI31040040 LPI8L+1 DD 34 K8LP10N IFIAIKQLII32034Q32 RATID8AIK0L)/A(L0L) DO 33 J8LP10NP" AIKQJI8AIK9JI-RATIO*A(LQJ) CONTINUE DD 43 I810N II8N+1-I S8000 IFCII‘NIQIQQ3943 IIPI8II+I DO 42 K‘IIPIQN S 8 $+AIII9K)*X(KI X(II) 8 (A(II9NPN)-S)/AIII9II’ KER'I GO TO 75 KER'Z CONTINUE RETURN END M'TITI'ITIQHILTIMJMTHMlllflfiiflflflflilllflitflITI'ES