Nuclear Engineering and Technology

Korean Nuclear Society (한국원자력학회)
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Fractional radioactive decay law and Bateman equations

  • Cruz-Lopez, C.A. (Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa) ;
  • Espinosa-Paredes, G. (Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa)
  • Received : 2021.05.17
  • Accepted : 2021.07.16
  • Published : 2022.01.25
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Abstract

The aim of this work is to develop the fractional Bateman equations, which can model memory effects in successive isotopes transformations. Such memory effects have been previously reported in the alpha decay, which exhibits a non-Markovian behavior. Since there are radioactive decay series with consecutive alpha decays, it is convenient to include the mentioned memory effects, developing the fractional Bateman Equations, which can reproduce the standard ones when the fractional order is equal to one. The proposed fractional model preserves the mathematical shape and the symmetry of the standard equations, being the only difference the presence of the Mittag-Leffler function, instead of the exponential one. This last is a very important result, because allows the implementation of the proposed fractional model in burnup and activation codes in a straightforward way. Numerical experiments show that the proposed equations predict high decay rates for small time values, in comparison with the standard equations, which have high decay rates for large times. This work represents a novelty approach to the theory of successive transformations, and opens the possibility to study properties of the Bateman equation from a fractional approach.

Keywords

Acknowledgement

The authors acknowledge the financial support received from the Mexican Secretariat of Public Education, SEP, under the project 12513606, entitled: Study and modelling of transmutation and diffusion phenomena in nuclear reactors using fractional calculus theory, by which the present development was possible.

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