Theoretical analysis of the activity of radioactive isotope mixtures modeled by Laplace transform and Python simulation

Abstract

The management of nuclear waste requires a thorough understanding of the evolution of its activity over time, which is determined by the isotopic composition and decay characteristics of the various radionuclides present. In this theoretical study, the behavior of total activity in mixtures of radioactive isotopes is analyzed through the conceptual application of the Laplace transform and numerical simulation in Python. The mathematical framework of radioactive decay for individual isotopes is established and extended to the modeling of mixtures using the principle of superposition. The Laplace transform is introduced as a fundamental tool in the analysis of linear systems, and its application to radioactive decay reveals the relationship between decay constants in the time domain and the location of poles in the complex frequency domain (s). By simulating theoretical scenarios in Python, the study illustrates how different combinations of decay constants and initial quantities of isotopes influence the total activity curve of the mixture. The analysis of these simulations, interpreted through the lens of linear systems theory and the Laplace transform, provides fundamental insights into the dynamics of activity in multi-component systems, laying the groundwork for a deeper theoretical understanding in the field of nuclear waste management.