Abstract
This article develops a complete solution of the randomized nuclear radioactive decay chain model based on Bateman master equations. A multidimensional version of the random variable transformation technique is adapted to derive a full probabilistic description for this model. To present general and more realistic physical situation, the initial number of the parent radionuclides and the decay parameters are considered to be random variables. The first probability density functions for the solution processes and the time until a given number of parent radionuclides remains in its state before decaying are constructed and used to calculate the mean, the variance and the confidence intervals. To test the efficiency of the theoretical findings, some numerical results are graphically presented and found to be consistent with the observations.
Similar content being viewed by others
References
E. Segrè, Nuclei, and Particles, 2nd edn. (Benjamin/Cummings, Reading, 1977)
H. Bateman, Solution of a system of differential equations occurring in the theory of radioactive transformation. Proc. Camb. Philos. Soc. 15, 423–427 (1910)
M.C. Casabán, J.C. Cortés, J.V. Romero, M.-D. Roselló, Mediterr. J. Math. 13, 3817 (2016). https://doi.org/10.1007/s00009-016-0716-6
M.C. Casabán, J.C. Cortés, J.V. Romero, M.D. Roselló, Abstract, and Applied Analysis (Hindawi Publishing Corporation, London, 2016). https://doi.org/10.1155/2016/6372108
M.C. Casabán, J.C. Cortés, J.V. Romero, M.-D. Roselló, Abstract, and Applied Analysis, vol. 2014 (Hindawi Publishing Corporation, London, 2016), p. 248512. https://doi.org/10.1155/2014/248512
M.C. Casabán, J.C. Cortés, J.V. Romero, M.-D. Roselló, Appl. Math. Lett. 34, 27 (2014). https://doi.org/10.1016/j.aml.2014.03.010
A. Hussein, M.M. Selim, Eur. Phys. J. Plus. 130, 249 (2015). https://doi.org/10.1140/epjp/i2015-15249-3
A. Hussein, M.M. Selim, Appl. Math. Comput. 218, 7193 (2012). https://doi.org/10.1016/j.amc.2011.12.088
A. Hussein, M.M. Selim, Appl. Math. Comput. 213, 250 (2009). https://doi.org/10.1016/j.amc.2009.03.016
A. Hussein, M.M. Selim, Appl. Math. Comput. 216, 2910 (2010). https://doi.org/10.1016/j.amc.2010.04.003
H. Slama, N.A. El-Bedwhey, A. El-Depsy, M.M. Selim, Eur. Phys. J. Plus. 132, 505 (2017). https://doi.org/10.1140/epjp/i2017-11763-6
A. Hussein, M.M. Selim, J. Quant. Spect. Radiat. Transf. 232, 54 (2019). https://doi.org/10.1016/j.jqsrt.2019.04.034
M.C. Casabán, J.C. Cortés, J.V. Romero, M.-D. Roselló, Commun. Nonlinear Sci. Numer. Simulat. 32, 199 (2016). https://doi.org/10.1016/j.cnsns.2015.08.009
M.C. Casabán, J.C. Cortés, J.V. Romero, M.-D. Roselló, Commun. Nonlinear Sci. Numer. Simulat. 24, 86 (2015). https://doi.org/10.1016/j.cnsns.2014.12.016
H. Slama, A. Hussein, N.A. El-Bedwhey, M.M. Selim, Appl. Math. Comput. 361, 144 (2019). https://doi.org/10.1140/epjp/i2017-11763-6
R. Walpole, R. Myers, S. Myers, Probability and Statistics for Engineers and Scientists, vol. 7, 9th edn. (Prentice-Hall, Upper Saddle River, 2012), pp. 211–217
A. Papoulis, Probability, Random Variables and Stochastic Processes, vol. 5, 4th edn. (McGraw-Hill, Boston, 2002), pp. 123–138
W. Jack Rinka, L.M. Heamanb, Radioactive Decay Constants: A Review, Encyclopedia of Scientific Dating Methods (Springer, Dordrecht, 2014). https://doi.org/10.1007/978-94-007-6326-5_264-1
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hussein, A., Selim, M.M. A general probabilistic solution of randomized radioactive decay chain (RDC) model using RVT technique. Eur. Phys. J. Plus 135, 418 (2020). https://doi.org/10.1140/epjp/s13360-020-00389-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00389-6