Utilizing Computational Methods to Solve the Bateman Equation for Studying the Transient Behaviours of Samarium

Abstract

Following the shutdown of a nuclear reactor, the accumulation of samarium-149, an isotope with a significant capacity to absorb thermal neutrons, will substantially decrease reactivity over a certain period. This phenomenon is commonly referred to as poisoning. Due to neutron absorption or disintegration, the amount of samarium-149 would steadily decrease over time. Control of the reactor's reactivity is necessary to forestall the reactor from achieving a critical or supercritical state. For this reason, we need to create a prediction about the relationship between the amount of time that has passed since the reactor was shut off and the toxicity of samarium. This article aims to provide a comprehensive analysis of a study that estimates the potential dangers posed by samarium in a fictitious nuclear reactor after it has been shut down. To get the forecast, solving the Bateman equations associated with the elements promethium (P_m) and samarium (S_m) is necessary. The sum of the equations shown above is referred to as an ordinary differential equation, or ODE for short. The matrix exponential method and the fourth-order Runge-Kutta technique were the two distinct methodological techniques used to solve the ODE problem. This was done successfully using MATLAB scripts. Both methodologies are evaluated for accuracy and computing efficiency in this comparison and contrast.