Nuclear Charge Density Distributions Research Articles

Calibration of nuclear charge density distribution by back-propagation neural networks

Calibration of nuclear charge density distribution by back-propagation neural networks

Prediction of nuclear charge density distribution with feedback neural network

Prediction of nuclear charge density distribution with feedback neural network

Nuclear charge radii of Cr46–62 isotopes and reaction cross sections for p−Cr

The nuclear charge radii and density distributions for $^{46--62}\mathrm{Cr}$ have been calculated in the frameworks of both relativistic and nonrelativistic self-consistent mean-field models. The relativistic Hartree-Bogoliubov model with density-dependent meson-exchange functional DD-ME2 and the nonrelativistic Hartree-Fock-Bogoliubov model with the Gogny D1S interaction, hereafter referred to as DIRHB and HFB, respectively, are used to compute ground-state properties of even $^{46--62}\mathrm{Cr}$ isotopes. Both calculations for the root-mean-squared charge radii reveal the characteristic kink at the $N$ = 28 shell closure in accordance with the corresponding experimental radii. The point proton and neutron density distributions calculated from both DIRHB and HFB are used to obtain the optical potentials for $p$-Cr at an incident proton energy of 65 MeV. The elastic scattering differential and total reaction cross sections, computed from the semimicroscopic proton optical potentials, have been derived by folding each of the target matter densities with the Jeukenne-Lejeune-Mahaux-Bruye\ifmmode \acute{r}\else \'{r}\fi{}es (JLMB) energy- and density-dependent internucleon interaction. The calculated elastic scattering differential cross sections for stable even isotopes, $^{50,52,54}\mathrm{Cr}$, using the respective DIRHB and HFB densities in the folding model reproduce the corresponding cross-section data well. The correlation between root-mean-squared charge radii and nuclear reaction observables obtained from the DIRHB and HFB calculations have been used in the folding model approach to predict the reaction cross sections for $p\text{\ensuremath{-}}^{46--62}\mathrm{Cr}$ at an incident proton energy of 65 MeV.

Study of charge density distributions, elastic charge form factors and root-mean square radii for 4He, 12C and 16O nuclei using Woods- Saxon and harmonic-oscillator potentials

The nuclear charge density distributions, form factors andcorresponding proton, charge, neutron, and matter root mean squareradii for stable 4He, 12C, and 16O nuclei have been calculated usingsingle-particle radial wave functions of Woods-Saxon potential andharmonic-oscillator potential for comparison. The calculations for theground charge density distributions using the Woods-Saxon potentialshow good agreement with experimental data for 4He nucleus whilethe results for 12C and 16O nuclei are better in harmonic-oscillatorpotential. The calculated elastic charge form factors in Woods-Saxonpotential are better than the results of harmonic-oscillator potential.Finally, the calculated root mean square radii usingWoods-Saxonpotentials how overestimation in comparison with experimental dataon contrary to the results of harmonic-oscillator potential.

Elastic electron scattering form factors of deformed exotic Xe isotopes

Background: The positions of diffraction minima of Coulomb form factors $|{F}_{C}{(q)|}^{2}$ are important landmarks for electron scattering experiments, which are sensitive to the nuclear size. For the isotopic chain, previous studies show the minima of $|{F}_{C}{(q)|}^{2}$ have an outward shift as the target nucleus moves to the proton-rich side.Purpose: Based on previous studies, the Coulomb form factors of Xe isotopes are further investigated by the deformed relativistic mean-field (RMF) model and distorted wave Born approximation (DWBA) method.Method: First, the nuclear charge density distributions are calculated by the constrained calculations of deformed RMF model. Next, the axially deformed density distributions are expanded into multipole components. With the charge density multipoles, the Coulomb form factors of Xe isotopes are calculated by the DWBA method.Results: For the nucleus with deformation, there are differences on the Coulomb form factors calculated from the spherical and deformed RMF models. For the Xe isotopic chain, the changing trend of $|{F}_{C}{(q)|}^{2}$ from the deformed RMF model is different from the results from the spherical RMF model at high momentum transfers.Conclusions: The minima of $|{F}_{C}{(q)|}^{2}$ of isotopes are directly influenced by the nuclear deformation parameter ${\ensuremath{\beta}}_{2}$, not just the charge radius ${R}_{C}$. For the studies of electron scattering, the nuclear deformation should be taken into account, especially in high momentum transfers.

Effect of realistic nuclear charge distributions on isotope shifts and progress towards the extraction of higher-order nuclear radial moments

Background: Atomic spectral lines from different isotopes display a small shift in energy, commonly referred to as the line isotope shift. One of the components of the isotope shift is the field shift, which depends on the extent and the shape of the nuclear charge density distribution. Purpose: To investigate how sensitive field shifts are with respect to variations in the nuclear size and shape and what information of nuclear charge distributions that can be extracted from measured field shifts. Methods: Nuclear properties are obtained from nuclear density functional theory calculations based on the Skyrme-Hartree-Fock-Bogoliubov approach. These results are combined with multiconfiguration Dirac-Hartree-Fock methods to obtain realistic field shifts. Results: Phenomena such as nuclear deformation and variations in the diffuseness of nuclear charge distributions give measurable contributions to the field shifts. Using a novel approach, we demonstrate the possibility to extract new information concerning the nuclear charge densities from the observed field shifts. Conclusions: Combining methods used in atomic and nuclear structure theory gives an improved description of field shifts. Extracting additional nuclear information from measured field shifts is possible in the near future with improved experimental methods.

Theoretical study of the central depression of nuclear charge density distribution by electron scattering

The charge form factors of elastic electron scattering for isotones with N = 20 and N = 28 are calculated using the phase-shift analysis method, with corresponding charge density distributions from relativistic mean-field theory. The results show that there are sharp variations at the inner parts of charge distributions with the proton number decreasing. The corresponding charge form factors are divided into two groups because of the unique properties of the s-states wave functions, though the proton numbers change uniformly in two isotonic chains. Meanwhile, the shift regularities of the minima are also discussed, and we give a clear relation between the minima of the charge form factors and the corresponding charge radii. This relation is caused by the diffraction effect of the electron. Under this conclusion, we calculate the charge density distributions and the charge form factors of the A = 44 nuclei chain. The results are also useful for studying the central depression in light exotic nuclei.

Central depression of nuclear charge density distribution

The center-depressed nuclear charge distributions are investigated with the parametrized distribution and the relativistic mean-field theory, and their corresponding charge form factors are worked out with the phase shift analysis method. The central depression of nuclear charge distribution of $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ is supported by the relativistic mean-field calculation. According to the calculation, the valence protons in $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ prefer to occupy the $1{d}_{3/2}$ state rather than the $2{s}_{1/2}$ state, which is different from that in the less neutron-rich argon and sulfur isotopes. As a result, the central proton densities of $^{46}\mathrm{Ar}$ and $^{44}\mathrm{S}$ are highly depressed, and so are their central charge densities. The charge form factors of some argon and sulfur isotopes are presented, and the minima of the charge form factors shift upward and inward when the central nuclear charge distributions are more depressed. Besides, the effect of the central depression on the charge form factors is studied with a parametrized distribution, when the root-mean-square charge radii remain constant.

A comparative study of finite nucleus models for low-lying states of few-electron high- Z atoms

We studied low-lying states of lithium-like and beryllium-like ions with Z=80 and 100≤ Z≤120 for differential effects due to variation of the nuclear charge density distribution. The latter was represented by exponential, Gauss-type, and Fermi-type models, standardized to a common rms radius. Numerical Hartree–Fock and Dirac–Fock–Coulomb calculations were performed. Changes in total energies due to the use of different models become increasingly important for high Z in relativistic approaches. Different models may even lead to different sequences of atomic states, and thus energy differences between those states may vary by an order of magnitude.

Anharmonicity in thermal motion and electrostatic forces on nuclei: Pulsed neutron diffraction fromMnF2

Antiferromagnetic order in ${\mathrm{MnF}}_{2}$ induces a sharp polarization of the fluorine core electron shell. The dipolar core-deformation generates a sizable electric field at the nuclear site, thus giving rise to an apparent Coulomb force on the nucleus, which is not compensated by peripheral lattice contributions. A local source of an opposing electric field is proposed, consisting in a small skewness of the fluorine nuclear vibrational distribution. Pulsed single-crystal neutron diffraction $(T=15\mathrm{K})$ at the spallation source IPNS has been used to test the hypothesis. Data have been collected up to very high diffraction vectors, $(\mathrm{sin}\ensuremath{\Theta}/\ensuremath{\lambda}{)}_{\mathrm{max}}=2.75{\AA{}}^{\ensuremath{-}1},$ and statistically significant anharmonic parameters could be extracted. Though the predicted value for the electric field could not be confirmed quantitatively, the presence of anharmonicity at low temperature has been clearly established. The shape of the fluorine nuclear distribution function supports a subtle balance between the mean thermal electric fields caused by the electronic and the nuclear charge density distributions. The sense of the skewness around the equilibrium position is opposite to the one found for the paramagnetic state.

Mean-field description of nuclear charge density distributions

We propose a method to extract nuclear charge distributions from elastic electron-scattering data based upon a mean-field approach. The nuclear charge distributions are generated by solving the Schrödinger equation with a mean-field potential expanded in terms of Hermite functions. The coefficients of the potential are changed in order to obtain the best fit with the cross-section data. Applications to the 12 C, 16 O, 40 Ca and 208 Pb nuclei are presented.

Experimental study of shell-model occupancies employing a new sum rule

Despite of the obvious success of the nuclear shell model the occupation numbers of nucleons in the shell model orbitals could not be determined precisely. The interpretation of one-nucleon transfer and quasi-free knock-out experiments, the classical tools for nuclear structure studies, did not yield reliable absolute spectroscopic factors mainly because of the uncertain distortions. In contrast, relative spectroscopic factors can be extracted with high precision because many sources of systematic experimental errors vanish and the uncertain DWBA or DWIA predictions enter as corrections only. On the other hand, elastic electron scattering represents a high precision tool to study nuclear charge density distributions which, however, offers no direct model-independent access to occupation numbers. To exploit the advantages of the different probes a Combined Evaluation of Relative spectroscopic factors and Electron Scattering, the CERES approach, has been developed to study the occupation numbers of the 3s 1 2 proton orbital in 206Pb. New precise relative experiments on 204Hg, 205Tl and 205,206Pb provide the necessary input data for the CERES sum rule. The occupation number n 3 s (208) for the 3s 1 2 proton orbital in 206Pb is found as n 3 s (208) = 2.46(16)★ z, where z represents the absolute 3s 1 2 contribution to the charge density difference of 206Pb- 205Tl. Based on the most recent interpretation of z = 0.64(6) we obtain n 3 s (208) = 1.57(10)[17]. The resulting depletion of ≈ 20% agrees well with other recent experimental findings and constrains the mean-field predictions.

Quark exchange effects on 3He and 3H charge densities

The exchange of quarks between nucleons bound in a nucleus leads to a definite and evaluable contribution to the nuclear charge density distribution. Although this exchange contribution is a consequence of finite nucleon size, it cannot be included by the conventional procedure of folding nucleon charge distributions with the density of point nucleons. For the A = 3 nuclei investigated here, the effects of quark exchange were found to be of significance for small distances away from the nuclear centre-of-mass.

Comment on the compound-nucleus cross section from nuclear charge-density distributions

Two equations for calculating the average cross section of statistical compound nuclear reactions are recommended in order to determine the maximum value of angular momentum which contributes to the cross section. The equations are applied to several reactions. The results are in good agreement with the experimental measurements.

Elastic Electron Scattering Experiments withN15

The nucleus of ${\mathrm{N}}^{15}$ has been investigated by the electron scattering technique at incident energies of 250 and 400 MeV. The first diffraction minimum has been observed and measured. The data have been analyzed by using two nuclear charge density distributions: the Fermi model and the shell model. A phase-shift program was used for the analysis. A value for the radius and skin thickness of the ${\mathrm{N}}^{15}$ nucleus is given, although the data do not fit the diffraction minimum. For the Fermi model, the rms radius is 2.7 F; the shell model yields 2.6 F.

Electron Scattering by Calcium at 250 MeV

The doubly magic nucleus of ${\mathrm{Ca}}^{40}$ has been investigated by the electron scattering technique at an incident energy of 250 MeV. The elastic scattering behavior has been resolved from the inelastic scattering, and absolute data on cross sections have been obtained for both types of events. In the case of the elastic scattering, the first and second diffraction features have been observed and measured. It has been possible to make a comparison of the elastic data with theoretical calculations based on a phase-shift analysis for certain assumed models of the nuclear charge density distribution. The combination of such an analysis with the present data, and also with earlier data on the same nucleus, permits the determination of the radius and skin thickness of the ${\mathrm{Ca}}^{40}$ nucleus with a precision greater than has been attainable heretofore. An investigation of the dependence of the density distribution on a third parameter indicates that a Fermi distribution, or models close to this type, are required to fit the experimental data. A Fermi radial charge density distribution with radius $c=3.60$ F and skin thickness $t=2.50$ F fits the data extremely well.

Weak Collective Effects in the Nuclear Shell Model

Configuration mixing between the odd particle and the core is formulated to account for the weak collective quadrupole effects near closed shells. The configuration mixing sum is evaluated formally for the general case and applied to ${\mathrm{O}}^{17}$. An equivalent formalism in which closure is done over the configuration mixing sum is developed and applied to ${\mathrm{O}}^{17}$, giving good agreement with experiment. The closure result is found to be very insensitive to variation of parameters. The closure formalism is also applied near the lead closed shell, using a nuclear charge density distribution for the core instead of a sum over particle states. The collective effects in this region are well accounted for, and the formalism is compared with that for weak surface coupling.