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European journal of applied sciences

Half-life calculation in general radioactive decay

Hasan Hüseyin Erbil

Published Dec 1, 2021• 1 citation
Publisher

Abstract

According to theories of classical physics, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by transition coefficient. Correct calculation of this coefficient is particularly important because extremely sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient was generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In the previous papers, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results were applied to calculate the half-life values of the nuclei that emit alpha particle. In this article, the half-life calculation formulas obtained in previous studies were made simpler. It has been shown that it is applicable to all radioactive decays (alpha, beta, gamma etc.). The half-life values obtained from our calculations have been compared with the experimental values, and it has been found that the new half-life values are exactly consistent with the experimental values.

DOI
10.14738/aivp.96.11235

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According to theories of classical physics, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by transition coefficient. Correct calculation of this coefficient is particularly important because extremely sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient was generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In the previous papers, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results were applied to calculate the half-life values of the nuclei that emit alpha particle. In this article, the half-life calculation formulas obtained in previous studies were made simpler. It has been shown that it is applicable to all radioactive decays (alpha, beta, gamma etc.). The half-life values obtained from our calculations have been compared with the experimental values, and it has been found that the new half-life values are exactly consistent with the experimental values.
According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is known to be. This event is called tunneling. Tunneling is a probability, and it is measured by a transition coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In this paper, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values.
According to classical physics theories, a moving particle cannot move to an environment with greater potential energy than its total energy during movement. But according to quantum theories, this event is well known. This event is called tunneling. Tunneling is a probability, and it is measured by a transmission coefficient. Correct calculation of this coefficient is very important because very sensitive and important instruments have been developed based on this event, and many events in nature can be explained by tunneling. This coefficient is generally calculated by semi-classical approaches (WKB) and the known formula is an approximate formula. In previous publications, the general transmission coefficient of a potential barrier with arbitrary form is calculated by a simple method without any approximation. The results are applied to calculate the half-life values of the nuclei that emit alpha particles. The half-life values obtained from our calculations and the classical method (WKB) have been compared, and it has been found that the new half-life values are exactly consistent with the experimental values. In this article, this general transmission coefficient is applied to calculate transmission coefficient for an arbitrary form external electrical field barrier (cold emission by the metal). The values obtained from our calculations and those of classical method (WKB) have been compared.
The halflife of αradioactive elements has already been measured out through experiments. But, classical theory's failure to explain the phenomenon of αDecay is found in this article according to the analysis of the mechanism of αDecay. And in this article, this problem is completely settled through introducing of the theory of quantum tunneling effect yet. Furthermore, the halflife of αDecay of238U is illustrated and figured out with this theory, and is approximately in accordance with the figure of experiments.
Quantum tunneling — the passage of a particle through a potential barrier that it classically lacks the energy to surmount — has been described since Gamow (1928) by the tunneling probability T ≈ exp(−2κd), where κ = √(2m(V₀−E))/ℏ and d is the barrier width. The formula is well-confirmed experimentally but has not been derived from a physical principle more fundamental than the Schrödinger equation. We derive it from the principle of information processing load minimization, using the framework established in the preceding papers of this series. We define the tunneling load as C_tunnel = ∫κ(x)dx — the integral of the spatial attenuation coefficient through the barrier — and show that T = exp(−2·C_tunnel) exactly. This is not a restatement of the WKB approximation; it is an identification of the physical meaning of the integral ∫κ(x)dx as the information processing load of the tunneling path. The tunneling path is selected by load minimization: when C_tunnel < C_classical (the load of the classical over-barrier trajectory), the system tunnels. Three physical systems are verified numerically without free parameters: (i) proton-proton nuclear fusion in the solar core — Gamow load C_Gamow = πη = 11.28, T = 1.60 × 10⁻¹⁰, consistent with solar luminosity; (ii) scanning tunneling microscope (STM) current — C_STM = κd = 5.43 at 0.5 nm, T = 1.91 × 10⁻⁵, and the factor-of-ten current change per 0.1 nm distance change follows directly; (iii) Ra-226 alpha decay — C_Gamow = 79.2, T ~ 10⁻⁶⁹, consistent with the 1,600-year half-life. Tunneling is not a mysterious violation of classical mechanics. It is the selection of the minimum-load propagation path — the same principle that explains wave-particle duality (Paper 7), wave function collapse (Paper 4), and the structure of all fundamental forces (Paper 8).
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
In the framework of Bohmian quantum mechanics supplemented with the Chetaev theorem on stable trajectories in dynamics in the presence of dissipative forces we have shown the possibility of the classical (without tunneling) universal description of radioactive decay of heavy nuclei, in which under certain conditions so called noise-induced transition is generated or, in other words, the stochastic channel of alpha decay, cluster radioactivity and spontaneous fission conditioned by the Kramers diffusion mechanism. Based on the ENSDF database we have found the parametrized solutions of the Kramers equation of Langevin type by Alexandrov dynamic auto-regularization method (FORTRAN program REGN-Dubna). These solutions describe with high-accuracy the dependence of the half-life (decay probability) of heavy radioactive nuclei on total kinetic energy of daughter decay products. The verification of inverse problem solution in the framework of the universal Kramers description of the alpha decay, cluster radioactivity and spontaneous fission, which was based on the newest experimental data of alpha-decay of even-even super heavy nuclei (Z=114, 116, 118) have shown the good coincidence of the experimental and theoretical half-life depend upon of alpha-decay energy.