Alpha decay study of Thorium isotopes using double folding model with NN interactions derived from relativistic mean field theory
Introduction
One of the crucial decay modes that can give information about the structure of nuclei is the α−decay mode [1], [2]. This decay mode has successfully been described using quantum theory [3]. α−decay of nuclei has been studied extensively using various theoretical approaches such as the preformed cluster model [4], [5], cluster formation model [6], [7], [8], [9], [10], [11], the generalised liquid drop model [12], [13], [14], the modified generalized liquid drop model [15], [16], [17], the effective liquid drop model [18], and the fission-like model [19]. These models use various interactions ranging from the phenomenological potential such as the proximity potentials [20], the Woods-Saxon potential, etc. to microscopic interactions. The first decay law to describe α−decay half-life was the Geiger-Nuttall law. This law was given theoretical explanation by Gamow in 1928. The theory explained that the α−decay was due to the quantum mechanical tunneling of a charged alpha particle through the nuclear Coulomb barrier [21]. Since the introduction of the Geiger-Nuttall law, there have been various analytical formulas introduced to compute the α-decay half-lives of many isotopes. Some of the empirical formulas are the Viola-Seaborg formula [22], the Royer formula [23], [24], [25], the universal decay law developed by Qi et al. [26], [27], the Akrawy formula [28], the Ren formula, [29], [30], the scaling law of Horoi [31], scaling law of Brown, the AKRE formula developed by Akrawy and Poenaru [32], etc.
The theoretical study on α−decay half-lives can be carried out using the semiclassical WKB approach. In this method, the effective interaction between the alpha-daughter system plays an important role in the calculations. The effective interaction consists of the nuclear potential, the Coulomb potential and the centrifugal potential. There have been various phenomenological [33], [34] and microscopic nuclear potentials [35], [36], [37], [38], [39] introduced to study the α−decay of various nuclei. In the microscopic approach, the nuclear potential is calculated using the double folding model, where the calculated nuclear densities are folded with the effective M3Y nucleon-nucleon interaction. The density-dependent double folding model has also been introduced [39], [40], [41] to study the α−decay half-lives of many nuclei. In Ref. [5], the authors introduced microscopic NN interaction derived from relativistic mean field theory Lagrangian (termed R3Y). The NN interaction derived was then used to calculate the optical potential using the double folding model. This was then successfully applied to study cluster radioactive decays.
In this work, the calculations of the α−decay half-lives of some Thorium isotopes have been carried out using both density-independent and density-dependent double folding model. The nuclear potentials are determined using various effective nucleon-nucleon interaction, from the usual M3Y-Paris and M3Y-Reid potentials to nucleon-nucleon interactions determined from relativistic mean field theory (R3Y). The different NN interaction potentials have been used to determine the most appropriate effective NN−interaction for α−decay study of the Thorium isotopes.
The article is organised as follows: in Section 2, the theoretical models, such as the density-dependent double folding model, to compute the α-decay half-lives are described. The results of the calculations are presented and discussed in Section 3 while the conclusion is given in Section 4.
Section snippets
Theory
The nuclear interaction potential between the daughter and cluster nuclei in the double folding model is given by: where the relative distance between interacting nucleon pair is , and are the ground state matter density distributions of the alpha (or cluster) and the daughter nucleus, respectively. is the kinetic energy of the α particle. The density distribution of the alpha particle is taken to be the Gaussian form:
Results and discussions
The results of the α-decay half-lives for the Thorium isotopes using the theoretical model described in the previous Section are presented and discussed here. In the calculations, the preformation factor of the alpha particle has been taken to be one. The calculations were carried out using not only the usual M3Y-Paris and M3Y-Reid NN interactions, but also the R3Y interactions derived from relativistic mean field theory with the R3Y-HS, R3Y-L1, R3Y-W, and R3Y-Z parametrizations.
Conclusion
The α-decay half-lives of isotopes have been theoretically studied within the WKB semi-classical framework and with the inclusion of the Bohr-Sommerfeld quantization factor. The nuclear potential is obtained using the double folding model. Apart from using the famous M3Y-Paris and M3Y-Reid nucleon-nucleon interactions, the microscopic nucleon-nucleon interactions derived from relativistic mean field theory Lagrangian (R3Y) were also used to calculate the α−nucleus nuclear potential.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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