Algorithms and algebraic solutions of decay chain differential equations for stable and unstable nuclide fractionation

Radioactive decay processes, such as alpha decay, produce decay chains where the mass numbers of nuclides decrease as larger nuclides expel energetic particles to form smaller nuclides. Under these conditions, the coefficient matrix that describes the differential rate expressions for radioactive decay can be made lower triangular. With this special structure, formulating an algebraic solution to the decay chains can be done by first formulating the eigenvectors that make up the coefficient matrix, which can then be solved using forward substitution for a lower triangular matrix. This work details the derivation of algebraic solutions for decay chains of any number of stable and unstable nuclides with any number of branching based on this eigenvector analysis. A prototype computational code was developed to validate and compare this methodology against a number of other methods for solving similar systems. A two-phase sorting algorithm yielding the lower triangular matrix structure was established to apply the developed algebraic solutions for decay chains involving beta-emitting radionuclides transformed into daughter nuclides without change in their mass number. The methodologies produced in this work provide an efficient way to estimate nuclide fractions from natural decay processes.