Abstract
Artificial neural networks (ANN) have emerged with successful applications in nuclear physics as well as in many fields of science in recent years. In this paper, ANN have been employed on experimental nuclear charge radii. Statistical modeling of nuclear charge radii using ANN are seen to be successful. Based on the outputs of ANN we have estimated a new simple mass-dependent nuclear charge radii formula. Also, the charge radii, binding energies and two-neutron separation energies of Sn isotopes have been calculated by implementation of a new estimated formula in Hartree–Fock–Bogoliubov calculations. The results of the study show that the new estimated formula is useful for describing nuclear charge radii.
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1. Introduction
Nuclear charge radii and, more generally, distributions of charge-density give direct information on the Coulomb energy of nuclei. Because of this, they have gained attention for the nuclear mass formulae for over six decades [1]. As is well known, the nuclear charge radius is a fundamental property of atomic nuclei [2, 3]. It can be measured by various methods based on the electromagnetic interaction between the nucleus and electrons or muons. Widely used methods are measurements of transition energies in muonic atoms, elastic electron scattering experiments, Kαx-ray and optical isotope shifts. Detailed discussions of these techniques for measuring the root-mean-square (rms) charge radii of nuclei can be found in [4, 5]. The latest advances in experimental techniques have provided access to experimental nuclear charge of more nuclei far from the β-stability line.
Measurement of the nuclear charge radii related to exotic phenomena such as skin and halo has become a hot topic [6]. With its accuracy, the study of the nuclear charge radius is very important for a better understanding of not only the proton distribution in nuclei but also the skin and halo. For these reasons, if one can obtain a simple and reliable formula for nuclear charge radii, it can provide information for exotic nuclei and the effective nucleon–nucleon interaction. For this purpose, a number of nuclear charge formulae have been proposed as being based on A1/3 or Z1/3 dependence. An excellent comprehensive study on a nuclear charge formula obtained from the fitting of experimental data can be found in [6] and references therein. However, in these studies experimental data were limited. New experimental data for over 900 nuclei have been updated recently [4, 5]. Using the new data, the parameters of the nuclear charge formulae can be revised. In the present work, we used feedforward artificial neural networks (ANN) as an alternative approach for estimating a reliable and simple nuclear charge formula. As is well known, feedforward ANN may be viewed as a universal non-linear function approximator [7]. In recent years, ANN have been used in many fields in nuclear physics as in the other fields, such as the determination of one and two proton separation energies [8], developing nuclear mass systematics [9], the identification of impact parameters in heavy-ion collisions [10–12], estimating beta decay half-lives [13] and obtaining potential energy curves [14]. The fundamental task of ANN is to give outputs as a consequence of the computation of the inputs. We have estimated a new mass-dependent simple nuclear charge formula by using all the data from ANN whose inputs were taken from the latest experimental data [4, 5]. The estimated formula has been used in Hartree–Fock–Bogoliubov (HFB) calculations for Sn nuclei.
The paper is organized as follows. In section 2, ANN formalism and the HFB method are given briefly. In section 3, the results of the study and discussions are presented. Finally, a summary is given in section 4.
2. Theoretical framework
2.1. ANN
ANN are mathematical models that mimic the human brain. They consist of several neurons which are processing units. The neurons are connected to each other via adaptive synaptic weights [15]. The ANN are composed of three main layers. The first layer corresponds to the input layer, the intermediate layer is called the hidden layer and the last one is the output layer. In this study, one input layer with 2 neurons, 1 hidden layer with 24 neurons (h = 24) and 1 output layer with a neuron ANN were used. The total adjustable weights were 3h = 72.
The input layer neurons receive the data from outside and the output layer neuron gives the results. The data is transmitted via weighted connections between the neurons. The tangent hyperbolic (sigmoidlike) and linear functions were used for hidden and output units, respectively. It has been proven that one hidden layer and sigmoidlike activation function in this layer are sufficient to approximate any continuous real function [16].
The use of ANN is a two-step process: training and test stages. In this work, a back-propagation algorithm with Levenberg–Marquardt [17, 18] for the training of the ANN was used. In the training stage, ANN modify their weights until an acceptable error level between desired and predicted outputs is attained. The error function which measures this difference was mean square error. After an acceptable error level, the trained ANN are tested over the data of interest. For further and general background for ANN, the reader is referred to [15].
2.2. HFB method
In the HFB formalism, a two-body Hamiltonian of a system of fermions can be interpreted in terms of a set of annihilation and creation operators. The ground-state wave function is described as the quasi-particle vacuum and the linear Bogoliubov transformation provides connection between the quasiparticle operators and the original particle operators. The basic building blocks of the HFB method are the density matrix and the pairing tensor. The expectation value of the HFB Hamiltonian could be expressed as an energy functional (details can be found in [2, 19]). In terms of Skyrme forces, the HFB energy has the form of a local energy density functional which contains the sum of the mean field and pairing energy densities. These fields can be calculated in the coordinate space [2, 19]. In this work, HFB equations have been solved by expanding quasi-particle wave functions that conserve axial symmetry and parity on a harmonic oscillator basis expressed in coordinate space proposed by [19]. For pairing, the Lipkin–Nogami method is implemented by performing the HFB calculations with an additional term included in the HF Hamiltonian. To obtain ground-state properties of Sn isotopes the prescription of [19] was taken into account. A number of effective Skyrme forces can be found in the literature for correct prediction of the nuclear ground-state properties. In this study, the Skyrme force SLy4 [20] has been used.
3. Results and discussion
In this study, experimental nuclear charge radii of 900 nuclei [4, 5] have been used in ANN. The inputs of the ANN were neutron and proton numbers, and the output was the nuclear charge radius. The whole data set partitioned into two separate sets, 80% for the training stage and the rest for the test stage. In figure 1, the differences between the experimental rms charge radii and the results of the ANN training stage for several isotopes are shown (left panel). After the training of the network, it was tested over the test data set which had never been seen before by the network. In the right panel of the same figure, the differences between the observed rms charge radii and ANN test predictions are shown. As can be seen in figure 1, the ANN results agree exceptionally well with experimental values, especially for the charge radii of nuclei with A ⩾ 40. The rms deviations (σ) between the experimental charge radii and the results of ANN for the training and test stages are 0.036 and 0.025, respectively. This indicates that the test set ANN has consistently generalized the training set fittings. As an illustration, the nuclear rms charge radii obtained from ANN and available experimental data [4, 5] for Ca, Ni, Zr, Sn and Pb isotopes are presented in figure 2. As can be seen in the figure, ANN results are consistent with the experimental data (within 0.37%).
Figure 1. The differences between the experimental (exp.) rms charge radii and ANN training results (left panel) for several isotopes ranging from A = 6 to 240. The right panel is similar to the left panel, but for ANN test predictions.
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Figure 2. The rms charge radii versus the atomic mass number (A) for even–even Ca, Ni, Zr, Sn and Pb isotopes. The experimental rms charge radii are denoted by solid symbols while the results of ANN are represented by open symbols.
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Standard imageAfter testing of the trained network, the whole data set from ANN outputs was used for employing least squares fitting. It should be noted that the mass-dependent form of nuclear charge radii given by the formula Rc = r0Aβ has been considered for fitting. The novelly estimated formula in the present study is given by
where Rc and A are the rms charge radii and mass number of the nuclei, respectively. In figure 3, we show the differences between experimental rms charge radii and the results obtained from the estimated formula. Calculated results from this formula are in good agreement with the experimental data. The rms deviation (σ) between charge radii obtained from the formula and experimental ones is 0.081. In the case of using conventional mass-dependent charge radii formula Rc = 1.223A1/3 taken from [6], σ is 1.371. This means that our estimated formula based on ANN gives much better nuclear charge radii than the conventional formula.
Figure 3. The differences between the experimental (exp.) rms charge radii and the predictions of the formula Rcal = 1.231A0.28 derived from ANN results.
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Standard imageGenerally, the experimental charge radii of nuclei with A ⩾ 40 (825 nuclei) are taken into account for the estimation of nuclear charge formulae as in [6]. Because of this, we have estimated another formula based on ANN by considering experimental rms charge radii of nuclei with A ⩾ 40. Under this condition, the estimated formula is Rc = 1.112A0.30. The related σ is 0.070. However, obtained σ of the conventional formula, Rc = 1.223A1/3, is 1.423.
In addition, we have calculated the ground-state properties of nuclei such as binding energy per nucleon, two-neutron separation energy and the nuclear charge radius in the HFB model for Sn (Z = 50) isotopes by using the estimated formula given in equation (1) as a parameter of the harmonic oscillator basis. In order to see the success of the model, the predictions of binding energies for even–even 100 − 144Sn isotopes within the HFB method with the Skyrme force SLy4 are shown in the upper panel of figure 4. The available experimental (exp.) binding energies per nucleon taken from [21] are shown in the same figure for comparison. As can be seen in the upper panel of the figure 4, the predictions of the HFB method are in good agreement with the experimental data. The maximum difference between the calculated and experimental values is only 0.043 MeV at neutron number N = 50. In the lower panel of the figure 4, the calculated two-neutron separation energies of Sn isotopes and experimental ones are shown. Abrupt decrease of the two-neutron separation energy at neutron number N = 82 indicates that this nuclei has a shell closure. The shell effect is clearly visible in our HFB calculations.
Figure 4. Binding energies per nucleon (upper panel) and two-neutron separation energies (lower panel) for Sn Isotopes calculated with the spherical HFB code are compared with experimental data (exp.) taken from [21].
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Standard imageThe calculated rms nuclear charge radii of 108 − 132Sn using the HFB method by implementing the new formula and the available experimental charge radii of Sn isotopes taken from [4] are shown in figure 5. As can be seen in this figure, the HFB method with the SLy4 parameters reproduced rms charge radii of Sn isotopes well. It should be noted, however, that going on from neutron number N = 58 to the neutron drip line, the difference between calculated and experimental charge radii for Sn isotopes is rising. However, the maximum difference is only 0.031 fm at neutron number N = 82.
Figure 5. The calculated rms nuclear charge radii within the HFB method with the Skyrme force SLy4 for Sn isotopes obtained by implementing the new formula in the HFB code. The available experimental rms charge radii of Sn isotopes taken from [4] are also shown for comparison.
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Standard image4. Summary
ANN have been applied on new experimental nuclear charge radii data. It is clearly shown that the results obtained from ANN are in good agreement with the experimental data. Beside, ANN outputs have been used to estimate a simple mass-dependent charge radii formula. The formula is found to be very successful. Also, by using the formula in the HFB code, the ground-state properties of even–even Sn isotopes such as binding energies per nucleon, two-neutron separation energies and nuclear charge radii have been calculated.