The experimental charge densities of atomic nuclei show fluctuations in their distributions. The experimental charge densities of 40Ca, 60Ni, 100Mo, 152Sm, and 208Pb are considered in this study, representing the various shapes of density-fluctuation. The present paper investigates the limits of accuracy of two-parameter Fermi and three-parameter Fermi distributions in describing the charge density. An improved analytical function for density distribution is proposed, which allows for density-fluctuation. The proposed function reproduces the experimental charge densities with significant improvement in accuracy over other formulas.

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In the 1950’s, Hofstadter and collaborators used electron scattering to characterize the charge distribution of nucleons and nuclei 1 From the cross-section  of the electron scattering, the form factor  can be deduced. Furthermore, the charge densities  can be produced in the form of model-independent distributions, such as the Fourier–Bessel expansion 2-4, or further simplified to analytical formulae 2-5.

The two-parameter Fermi (2pF) function is always considered as an acceptable approximation of the charge and neutron distributions 6-8

The 2pF function is given by equation (1) where  is the diffuseness parameter and  is the radius parameter. The central density  is determined by the normalization to the number of protons (Z) or neutrons (N).

A common alternative to the 2pF function is the three-parameter Fermi function (3pF), given by equation (2) where a central depression parameter () is introduced.

The central depression parameter allows the central density to be depressed or raised, depending on the sign of . The 2pF function is the most widely used analytical formula in the study of nuclear structure, nuclear reactions, alpha decay and cluster decay. Although 2pF function gives acceptable results, using 3pF distribution improves the binding energy calculation especially for superheavy and ultraheavy elements, at which the ground state has depression in the central density 9. Moreover, the calculation of alpha decay half-life time and preformation probability is very sensitive to the central depression parameter10. In a recent study of 208Pb charge density, the two formulas were compared to each other 7. The study showed that the fitting of the model-independent data to 3pF distribution does not provide a significant improvement over the fitting to 2pF. The present work shows that the 3pF function could provide reasonable improvement for some nuclei more than the improvement in the case of 208Pb.

In fact, the model-independent analysis of electron scattering shows fluctuations in the charge density. The shell model and self-consistent analysis also show such fluctuations in the proton and neutron densities11-13. The 2pF function and 3pF function, as well, show smooth variation without any fluctuations at all. The fitting of the model-independent data to both functions is quite good at the tail of the distribution, but the situation is different at the core, at which the fluctuations exist.

The purpose of the present work is to propose an improved formula to describe density fluctuations at the nucleus interior. We will call the new function as double 3pF (d3pF) and it is composed of two 3pF parts, one with large radius parameter, in order to describe the tail of the distribution, and the second has a smaller radius, to describe density fluctuation at the vicinity of the center.

The d3pF function is given by equation (3) where ‘s are the weights of the two 3pF parts. In fact, this function has seven independent parameters since the density distribution should verify the normalization condition.

In the present work, the experimental charge densities of 40Ca, 60Ni, 100Mo, 152Sm, and 208Pb were used to study the ability of the aforementioned formulas to describe the nucleon density. Regardless the position of the nuclide in the chart of nuclides, this study is concerned with the quality of fitting. For this propose, this study focused on five different shapes of density distribution, which represent the common shapes.

The main results of the study are summarized in Fig. 1. The left panels show the model-independent distributions of 40Ca, 60Ni, 100Mo, 152Sm, and 208Pb, respectively from bottom to top, with the corresponding fitted 2pF, 3pF, and d3pF distributions. The right panels of Fig. 1 show the residuals of the fits of the corresponding left panels. In addition to the residuals, there are three bars on each right panel showing the difference between the highest and lowest values of residual. The bars correspond to 2pF, 3pF and d3pF distributions respectively from left to right. It is clear that 2pF and 3pF reproduce the model-independent density with appropriate accuracy, except for the interior fluctuations. As seen in fig. 1, the d3pF distribution has two advantages over the other two distributions. The first advantage is that it reproduces the model-independent density with much higher overall accuracy. The second advantage is that it allows for fluctuations in density, which can be adjusted to match the actual fluctuations.

Contrary to expectations, the difference between the highest and lowest values of residual for 3pF distribution is not always smaller than its value for 2pF distribution, moreover, it could be greater. For 60Ni and 208Pb, it is obvious that the length of the bar representing the difference between residual extremes for 3pF distribution is greater than the adjacent bar, corresponding to the 2pF distribution. In fact, the value of residual extreme is not the indicator of the goodness of fit, but it still important since it indicates how the expected value is far from the real value at the worst point. The real indicator of the goodness of fit should consider the overall deviation from the actual data. For that purpose, the residual sum of squares (RSS) is chosen to compare the goodness of fit of the three formulas considered in this study. The values of RSS are presented in Table 1 together with the parameters obtained from fitting to model-independent density.