A general probabilistic solution of randomized radioactive decay chain (RDC) model using RVT technique
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A general probabilistic solution of randomized radioactive decay chain (RDC) model using RVT technique A. Hussein1, M. M. Selim2,a 1 Engineering and Applied Sciences Department, Community College, Umm Al-Qura University,
Makkah 715, Saudi Arabia
2 Physics Department, Faculty of Science, Damietta University, New Damietta City 34517, Egypt
Received: 20 December 2019 / Accepted: 7 April 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction Radioactive decay is a process in which a radionuclide (which is a radioactive nuclide) loses energy by emitting ionizing radiation (e.g., alpha, beta or gamma radiations) [1, 2]. In this process, an unstable nuclide, called the parent nuclide, transforms into a different nuclide, called daughter nuclide (has a different state or different numbers of nucleons), by emitting ionizing radiation. In some decays, the parent and daughter nuclides are different elements, and thus, the decay process results in a creation of a new element. It is crucial to notice that the radioactive decay process is a stochastic process (sp) at the level of a single nuclide. In other words, from a quantum mechanical point of view, it is impossible to predict when a particular nuclide will decay. For a large number of identical nuclides, the decay rate can be predicted from the measured decay parameter, λ, of the nuclide. However, the daughter nuclide of the decay event may be unstable and it will also decay producing radiation and a second daughter nuclide. This second nuclide may also be unstable, and this can lead to a sequence of several decay events until a stable nuclide is produced. This process is called a decay chain.
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
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In this article, we study the stochastic radioactive decay chain (RDC) model for natural decay chain of three species of radionuclides, which ends with stable species. At first, the Bateman equations governing the RDC model are deterministically solved. However, since in real case the RDC is a sp, it is reasonable to consider the initial numbe
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