Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-15-2007 Precise Calculation of Complex Radioactive Decay Chains Logan J. Harr Follow this and additional works at: https://scholar.afit.edu/etd Part of the Nuclear Engineering Commons Recommended Citation Harr, Logan J., "Precise Calculation of Complex Radioactive Decay Chains" (2007). Theses and Dissertations. 2924. https://scholar.afit.edu/etd/2924 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. PRECISE CALCULATION OF COMPLEX RADIOACTIVE DECAY CHAINS THESIS Logan J. Harr, Captain, USAF AFIT/GNE/ENP/07-03 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. AFIT/GNE/ENP/07-03 PRECISE CALCULATION OF COMPLEX RADIOACTIVE DECAY CHAINS THESIS Presented to the Faculty Department of Engineering Physics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science (Nuclear Engineering) Logan J. Harr, B.S. Captain, USAF March 2007 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. Abstract This thesis documents a new approach to investigate the gamma radiation activity of the fission products of three different fuels (U-235, U-238, and U-239) exposed to three different incident neutron energy spectra (thermal, fast spectrum, and high energies). An application of the exponential moments function is used with a transmutation matrix in the calculation of complex radioactive decay chains to achieve greater precision than can be attained through current methods. The result of this research is a code which can calculate the decay products from complex radioactive decay chains with a high degree of precision while quantifying the uncertainty in gamma activity due to uncertainties in the isotope properties. iv Acknowledgments Work on this thesis has been nearly a year long effort. I want to thank my wife for her patience and understanding throughout the entirety of this process. I also wish to recognize my advisor, Dr. Mathews, for the amount of time and effort he gave me in working out the inevitable bugs in my code. Special thanks also go to Dr. Gerts for his help in increasing my understanding of the Fortran language. Logan J. Harr v Table of Contents Page Abstract.............................................................................................................................. iv Acknowledgments ................................................................................................................v List of Figures.................................................................................................................. viii List of Tables........................................................................................................................x I. Introduction ...................................................................................................................1 I.A: Motivation................................................................................................................2 I.B: Statement of the Problem.........................................................................................3 I.C: Goal of the Research................................................................................................3 I.D: Scope........................................................................................................................3 I.E: Assumptions ............................................................................................................4 II. Analysis and Approach..................................................................................................5 II.A: Special Radioactive Decay Problem........................................................................5 II.B: Solution Methods of the Special Problem ...............................................................7 1. Bateman Solution Formula......................................................................................8 2. Numerical Integration of Coupled ODEs ................................................................9 3. Matrix Exponential (Transmutation Matrix) Method............................................10 4. Transmutation Matrix by Exponential Moments Function ...................................11 II.C: Generalization to the Full Problem........................................................................15 III. Implementation ............................................................................................................18 III.A: Input Data......................................................................................................18 1. Isotope Decay Information ....................................................................................18 2. Gamma Radiation Data..........................................................................................20 3. N(0) Data ..............................................................................................................21 III.B: Decay Chain Identification............................................................................22 III.C: Exponential Moments Function Implementation ..........................................23 1. One Argument .......................................................................................................24 2. Two Arguments .....................................................................................................24 3. Three Arguments ...................................................................................................25 4. N Arguments..........................................................................................................26 III.D: T-matrix Generation......................................................................................27 vi Page III.E: Calculating N(t).............................................................................................28 III.F: Calculating A(,)Et ........................................................................................29 IV. Verification Process.....................................................................................................31 IV.A: Defining the Range of Calculation Inputs .....................................................31 IV.B: Creating the Calculation Inputs .....................................................................33 IV.C: Verifying the Calculations.............................................................................33 IV.D: Verifying the Depth First Search Routine .....................................................35 V. Monte Carlo Estimation of Uncertainty ......................................................................37 VI. Performance, Results, and Analysis ............................................................................39 VI.A: Performance...................................................................................................39 VI.B: A(E,t) .............................................................................................................41 1. Total Activity.........................................................................................................43 2. 1-2 MeV.................................................................................................................54 3. 2-3 MeV.................................................................................................................58 4. 3-4 MeV.................................................................................................................64 5. 4-5 MeV.................................................................................................................68 6. 5-6 MeV.................................................................................................................73 7. 6-7 MeV.................................................................................................................79 8. 7-8 MeV.................................................................................................................83 9. >8 MeV..................................................................................................................86 VII. Conclusions and Recommendations ......................................................................87 VII.A: Conclusions....................................................................................................87 VII.B: Recommendations for Future Work ..............................................................88 Appendix A: Order Invariance of Bateman Equation .......................................................89 Appendix B: Depth-First Search Verification Test Problems ...........................................90 Appendix C: Mathematica Verification Notebook............................................................91 Bibliography ......................................................................................................................92