Two alternative approaches to the solution of cyclic chains in transmutation and decay problems

Abstract

In several burnup and activation problems there is a recurrent issue related to the singularities in the Bateman’s solution due to its inability to solve linear transmutation schemes where there are repeated isotopes, or where there are two different isotopes with the same removal coefficients. Most of these types of transmutation schemes are called cyclic chains. In these cases, the Bateman’s solution fails due to the presence of subtractions between the mentioned coefficients in some denominators, which eventually become zero and undefined. In order to overcome this problem, two methodologies have been reported in the open literature. The first one consists in introducing small modifications in the repeated removal coefficients, preventing the presence of zeros in certain denominators. The second is to develop more general equations for the Bateman’s solution. Nevertheless, both methodologies are based on the approximation of the cyclic chains with the linear chain method, whose error has not been studied until now. The study of this error is fundamental to omit cyclic chains and to reduce the execution time of the algorithm. In the present work, a more general approach to the cyclic chains was studied, starting with the description and classification of the transmutation and decay networks that generate them. Afterward, two different approaches to solve some of these structures were proposed, which are not based on the linear chain method. One of them is based on a power series analysis, and the other one is related to a numerical analysis of the roots of a polynomial in the Laplace transform space. Additionally, computer algorithms were developed for each approach to facilitate their implementation in a burnup or activation code, and a numerical comparison with the linear chain approximation was carried out. Through the present work, it was possible to compute the actual error involved when the linear chain is used for approximating cyclic chains, and to conclude if a cyclic chain can be ignored in a burnup problem.