Narrow Results

In this paper, we study the nuclear charge radius for 804 nuclei with $Z,N\ge 12$ in the CR2013 database (excluding the isotopic chains of Tb, Tm, Lu and Cm). By adding the Casten factor for the $Z^{1/3}$ law empirical formula to calculate the charge radii of these nuclei, the root mean square deviation (RMSD) between calculated and experimental values is 0.0530fm. A new empirical formula is then obtained by correcting using neutron–proton ratio factor $N/Z$ and the neutron factor $1/N$. The RMSD between the calculated values obtained using this new empirical formula and experimental values is 0.0197fm. The new empirical formula reproduces well not only the factors’ (i.e. the $N/Z$ and $1/N$ factors) dependence on charge radii, but also the kinks of charge radii at the neutron magic numbers N = 82 and 126 for some isotope chains. Furthermore, this paper introduces the Back propagation (BP) neural network approach using the Casten factor, the $N/Z$ factor and the $1/N$ factor as input variables to train the model. The RMSD between the calculated values obtained by the neural network model II and the experimental values is only 0.0098fm. Compared to the empirical formula, the RMSD of the neural network model is reduced by 50%. The above research results based on the improvement of the $Z^{1/3}$ law formula indicate that the relation and models proposed in this paper have accuracy and reliability.

In this paper, we get the nuclear mass-density parameter of known mass and known charge radius by using the AME2020 nuclear mass database and the CR2013 nuclear charge radius database. For the nuclei with $N\ge 21$, we obtained an empirical formula of the constant parameter equal to 0 for the difference of nuclear mass-density parameter between two neighboring isotope. That is, the nuclear mass-density parameter of the two neighboring isotopes was equal. By using this empirical formula combined with AME2020 and CR2013 databases, the calculated value of 625 nuclear charge radii was obtained. The root-mean-square deviation (RMSD) of nuclear charge radius that we have successfully obtained is $\sigma =0.011$fm. In addition, one of the corrections is the neutron factor. This correction is the key improvement which reduces the RMSD to 0.0077fm. Considering the neutron shell effect, the different shell ranges were divided into two categories for correction, and the RMSD was reduced to 0.0056fm. Based on the revised empirical formula combined with the AME2020 database, the calculated and predicted values of nuclear charge radius were obtained with high accuracy. Some of these predictions are also very consistent with experimental values measured in recent years. In addition, we use BP neural network to study the difference of nuclear mass-density parameter on the basis of two categories. The RMSDs obtained are 0.0028fm and 0.0038fm, respectively. It can be seen that such a division has a certain degree of reliability and feasibility. These results show that the new relation proposed in this paper has simplicity and reliability, which can be compared with other local nuclear charge radius relations.

In this paper, we have successfully obtained the mean mass-density parameter (${\rho }_m$) using the nuclear masses AME2020 database and nuclear charge radius (CR) CR2013 database. The empirical formula is derived based on the relationship between ${\rho }_m$ and $N∕Z$ (the ratio of the number of neutrons to protons). Subsequently, we obtained an empirical formula for the difference in ${\rho }_m$ between two neighboring isotopes. By utilizing this empirical formula along with the AME2020 and CR2013 databases, we then calculated 625 charge radii for nuclei with $N\ge 21$. The root-mean-square deviation (RMSD) between the calculated values and the experimental values in the CR2013 database is 0.0075fm. The predicted nuclear CR values are comparable to those of other researches, which some predictions closely matching the experimental values measured in recent years. Additionally, this work used the Back Propagation (BP) neural network to establish a model for describing and predicting the difference in the mean mass-density parameter between two neighboring isotopes. The RMSD between the calculated and experimental values obtained using this model is 0.0039fm. Some of our predicted values have good accuracy and compared well with experimental values. Both of the above methods indicate that the nuclear CR relationship proposed in this paper based on the difference in mean mass-density parameter has simplicity and reliability.

In this paper, we obtained the mean mass density parameter (MMDP) ${\rho }_m(Z,N)$ based on the nuclear mass AME2020 and nuclear charge radius CR2013 databases. On this basis, we study the dependence relationship between neutron number N and the two neighboring isotopes MMDP difference $\delta \rho (Z,N)$, and then obtained the empirical formula for $\delta \rho (Z,N)$. By combining this empirical formula with the AME2020 and CR2013 databases, the root-mean-square deviation (RMSD) for 606 nuclear charge radii with $Z,N\ge$ 21 between the calculated and experimental values is 0.0046fm. Considering the odd–even staggering and isospin term correction, the study found that the RMSD is reduced to 0.0040fm. This work also uses Levenberg–Marquart (L-M) neural network approach to study the nuclear charge radius. The RMSD of nuclear charge radius is only 0.0030 fm, where the accuracy is increased by about 25%. Furthermore, the predicted values of nuclear charge radius obtained using this modified empirical formula and Neural Network model are competitive with the results of other models. These results indicate that nuclear charge radius relation and Neural Network model based on $\delta \rho (Z,N)$ proposed in this work have a certain accuracy, simplicity and reliability.