Rms Charge Radius Research Articles

Nuclear Structure and Decay Data for A=231 Isobars

The present evaluation of currently known nine nuclides of A=231: ( 231 Rn, 233 Fr, 231 Ra, 231 Ac, 231 Th, 231 Pa, 231 U, 231 Np and 231 Pu) represents an update and revision of previous A=231 evaluation by 2013Br04. All the known decay and reaction data are evaluated, with recommended properties given in the Adopted Levels, Gammas datasets, for gamma-ray energies, photon branching ratios, level half-lives, spins, parities, transition probabilities, and configuration assignments. For 231 Rn, only an identification of the nuclide has been made without an experimental determination of its half-life. For 231 Fr, no excited states are known. About 30 levels in 231 Ra up to 1774 keV excitation are known only from β − decay of 231 Fr with level half-lives determined for five excited states. For 231 Ac, excited states up to 3122 keV are known from β − decay, (t, α ), and fragmentary information for high-spin levels from two heavy-ion reaction studies, with lifetimes of excited states measured for five levels. Extensive structure and rotational band information for 231 Th is available for a large number of levels up to 1714 keV from β − and α decays, (n, γ ) thermal and resonance, single-particle transfer reactions (d,p), (d,t) and ( 3 He, α ), with limited data for high-spin levels from one secondary reference, and half-lives available for five excited states. Very detailed structure data are available for 231 Pa up to 2139 keV from β − , ε and α decays, ( α ,t), (d,d′) and high-spin data from heavy-ion Coulomb excitation and (p,2n γ ) reaction, with level half-lives known for 34 excited states, mostly deduced by evaluators from transition probabilities determined in Coulomb excitation. Only eight excited states are known in 231 U up to 1268 keV from ε and α decays, with no data for half-lives of excited states. For 231 Np and 231 Pu, only the ground-state is known from α decays, with a tentative level at 324 keV in 231 Pu. Measurements of nuclear rms charge radius and isotope shifts for 231 Ra have been made by 2018Ly01, and that for rms charge radius for 231 Fr by 2014Bu06. 235 U is a well known nuclide for nuclear structure and reactor applications. There have been a large number of studies of its alpha decay to levels in 231 Th, including some recent ones such as 2018Ma03 and 2017Le03, yet evaluators' analysis of the decay scheme suggests that, while α -particle transitions are well established, but several issues still remain about intricacies of gamma-ray and conversion electron spectroscopy which need to be resolved through dedicated experiments for a complete understanding of the decay scheme characteristics, although, this sort of research project would seem a challenging spectroscopic problem. As commented in individual datasets for other decays, important spectral information is lacking for most of the decay schemes, with almost no information available for some of these, resulting in many incomplete decay schemes in A=231 mass chain.

Evolution of nuclear charge radii in copper and indium isotopes * *Supported by the Reform and Development Project of Beijing Academy of Science and Technology (13001-2110); Supported in part by the National Natural Science Foundation of China (12135004, 11635003, 11961141004, 12047513). X. J. is grateful for the support of the National Natural Science Foundation of China (11705118). L.-G. C. is

Systematic trends in nuclear charge radii are of great interest due to universal shell effects and odd-even staggering (OES). The modified root mean square (rms) charge radius formula, which phenomenologically accounts for the formation of neutron-proton (np) correlations, is here applied for the first time to the study of odd-Z copper and indium isotopes. Theoretical results obtained by the relativistic mean field (RMF) model with NL3, PK1 and NL3* parameter sets are compared with experimental data. Our results show that both OES and the abrupt changes across and 82 shell closures are clearly reproduced in nuclear charge radii. The inverted parabolic-like behaviors of rms charge radii can also be described remarkably well between two neutron magic numbers, namely to 50 for copper isotopes and to 82 for indium isotopes. This implies that the np-correlations play an indispensable role in quantitatively determining the fine structures of nuclear charge radii along odd-Z isotopic chains. Also, our conclusions have almost no dependence on the effective forces.

Calculation of Charge Density, Charge Radii and Form Factor for some Exotic Calcium Isotopes Using OXBASH Code

This work is concerned to give information of shell model calculations, limited to fp-shell with an accuracy and applicability in the work. The form factors have been calculated for J+=0+,2+,4+,6+,8+ for each nuclei depending on charge density q≤3 fm-1 using the harmonic oscillator potential based on GX1A effective interaction with ep=1.16e and en=0.7e. For charge density, the differences between proton and neutron densities increased with neutron increase while neutron radius increased as neutron number increase as an extension of neutron densities outwards on the nuclear surface (3< r <6 fm) while a slight increase appears in interior region (r < 3 fm). The values of charge radii are obtained from the rms charge radii. The value of ( ) starts with an increase from (52Ca-0.313) to reach its highest value at (58Ca-0.397) because there is no size of the energy gap between the shell in isotopes (54, 56 and 58) while orbital is specified in 52Ca doubly-magic nucleus. The results are corresponding to the experimental results of an increase in radius

Search for the α + core structure in the ground state bands of 22≤Z≤42 even-even nuclei

A systematic analysis of the $\alpha$ + core structure is performed in the ground state bands of even-even nuclei of the $22 \leq Z \leq 42$ region in terms of the Local Potential Model. The $\alpha$ + core interaction is described by the nuclear potential of (1 + Gaussian)$\times$(W.S. + W.S.$^3$) shape with 2 free parameters. Properties such as energy levels, reduced $\alpha$-widths, $B(E2)$ transition rates and rms charge radii are calculated and compared with experimental data. A good agreement with the experimental data is obtained in general, even for the nuclei without the $\alpha$ + {doubly closed shell core} configuration. The analysis of the selected nuclei in the $22 \leq Z \leq 42$ region indicates that $^{44}$Ti and $^{94}$Mo have a greater $\alpha$-clustering degree in comparison with their respective neighboring nuclei of this set. In addition, the model points to the existence of nuclei without the $\alpha$ + {doubly closed shell core} configuration with a significant $\alpha$-clustering degree compared to $^{44}$Ti, $^{60}$Zn, and $^{94}$Mo. The study shows that the $\alpha$ + core approach is satisfactorily applicable to nuclei other than those with $\alpha$-clustering above double-shell closures.

Prediction of Rms Charge Radius of Proton Using Proton–Proton Elastic Scattering Data at TeV

Using proton–proton elastic scattering data at TeV and squared four-momentum transfer 0.36 &lt; -t &lt; 0.76 (GeV/c)2 for 13 σBeam distance and 0.07 &lt; -t &lt; 0.46 (GeV/c)2 for 4.3 σBeam distance, form factor of proton is predicted. Simplest version of Chou–Yang model is employed to extract the form factor by fitting experimental data of differential cross section from TOTEM experiment (for 13σBeamand 4.3 σBeam distance) to a single Gaussian. Root mean square (rms) charge radius of proton is calculated using this form factor. It is found to be equal to 0.91 fm and 0.90 fm respectively. Which is in good agreement with experimental data and theoretically predicted values.

Prediction of rms charge radius of proton using proton-proton elastic scattering data at TeV

Using proton–proton elastic scattering data at TeV and squared four-momentum transfer 0.36 &lt; -t &lt; 0.76 (GeV/c)2 for 13 σBeam distance and 0.07 &lt; -t &lt; 0.46 (GeV/c)2 for 4.3 σBeam distance, form factor of proton is predicted. Simplest version of Chou–Yang model is employed to extract the form factor by fitting experimental data of differential cross section from TOTEM experiment (for 13σBeamand 4.3 σBeam distance) to a single Gaussian. Root mean square (rms) charge radius of proton is calculated using this form factor. It is found to be equal to 0.91 fm and 0.90 fm respectively. Which is in good agreement with experimental data and theoretically predicted values.

Static properties of skyrmions and nucleons at finite isospin chemical potential

We use the Skyrme Lagrangian with a finite pion mass term to study the stability of the skyrmion at finite isospin chemical potential [Formula: see text]. It is found that there is a critical value [Formula: see text] MeV, and there is no stable solution above the [Formula: see text]. We also explore the behavior of different skyrmion parameters as a function of [Formula: see text], in particular the isovector electric root-mean-square (rms) charge radius in this work. Then, using the Hamiltonian formulation, in terms of collective variables, we study the behavior of nucleons mass splitting at finite isospin chemical potential. We find the proton mass decreases and the neutron mass increases when [Formula: see text]. Finally, we also discuss the magnetic moments and the pion-nucleon sigma term [Formula: see text] as a function of isospin chemical potential for interest.

Nuclear shape evolution and shape coexistence in Zr and Mo isotopes

The phenomena of shape evolution and shape coexistence in even–even $$^{88-114}$$ Zr and $$^{90-116}$$ Mo isotopes are studied by employing covariant density functional theory (CDFT) with density-dependent point-coupling parameter set, DD-PCX, and with separable pairing interaction. The results for the rms deviation in binding energies, two-neutron separation energy, the differential variation of two-neutron separation energy, and rms charge radii, as a function of neutron number, are presented and compared with available experimental data. In addition to the oblate–prolate shape coexistence in $$^{96-110}$$ Zr isotopes, the correlations between shape transition and discontinuity in the observables are also examined. A smooth trend of charge radii in Mo isotopes is found to be due to the manifestation of triaxiality softness. The observed oblate and prolate minima are related to the low single-particle energy level density around the Fermi level of neutron and proton, respectively. The rapid shape transition in Zr isotopes near N $$\approx $$ 60 is identified to be caused by the evolution of the shell structure associated with massive proton excitations to 1 $$\pi g_{9/2}$$ orbit. The present calculations also predict a deformed semi-bubble structure in the $$^{100}$$ Zr isotope.

Nuclear structure and decay properties of Nd isotopes

Using the Hartree–Fock–Bogoliubov mean-field theory, the ground-state structural and decay properties of Nd isotopes are investigated from the proton-rich side up to the neutron drip-line. Quantities such as binding energies per nucleon, one and two-neutron separation energies, rms charge radii, and quadrupole deformation parameters have been calculated. Compared with the relativistic mean-field results, the present calculations are in better agreement with the available experimental data. The results show clearly the signature of a shape transition at [Formula: see text] and an abrupt increase in the deformation near the neutron drip-line. Further, the possible decay modes like alpha, cluster and [Formula: see text]-decay are analyzed in a unified fission model and phenomenological formulas. Overall, a good agreement is achieved between the calculated and experimental [Formula: see text]-values and half-lives wherever available. The most likely decay modes are thus identified throughout the isotopic chain.

Theory of the n\u2009=\u20092 levels in muonic helium-3 ions

The present knowledge of Lamb shift, fine-, and hyperfine structure of the 2S and 2P states in muonic helium-3 ions is reviewed in anticipation of the results of a first measurement of several 2S → 2P transition frequencies in the muonic helium-3 ion, μ3He+. This ion is the bound state of a single negative muon μ- and a bare helium-3 nucleus (helion), 3He++.A term-by-term comparison of all available sources, including new, updated, and so far unpublished calculations, reveals reliable values and uncertainties of the QED and nuclear structure-dependent contributions to the Lamb shift and the hyperfine splitting. These values are essential for the determination of the helion rms charge radius and the nuclear structure effects to the hyperfine splitting in μ3He+. With this review we continue our series of theory summaries in light muonic atoms [see A. Antognini et al., Ann. Phys. 331, 127 (2013); J.J. Krauth et al., Ann. Phys. 366, 168 (2016); and M. Diepold et al. arXiv:1606.05231 (2016)].Graphical abstract

Laser Spectroscopy of Muonic Hydrogen and the Puzzling Proton

Laser spectroscopy of muonic hydrogen atoms, μp, has revealed a proton root-mean-square (rms) charge radius rE that is an order of magnitude more accurate than the CODATA world average from elastic electron–proton scattering and precision spectroscopy of regular (electronic) hydrogen. Interestingly, though, the value of rE from μp is 4%, or 7 combined standard deviations smaller than the CODATA value of rE. This discrepancy has been coined “proton radius puzzle”. We summarize the experiment and give a brief overview of the theory in muonic hydrogen. Finally we discuss some possible scenarios for the resolution of the “proton radius puzzle”.

Evaluation of the strength of electron-proton scattering data for determining the proton charge radius

Precisely measured electron-proton elastic scattering cross sections [Phys. Rev. Lett. {\bf 105}, 242002 (2010)] are reanalyzed to evaluate their strength for determining the rms charge radius ($R_{\rm E}$) of the proton. More than half of the cross sections at lowest $Q^2$ are fit using two single-parameter form-factor models, with the first based on a dipole parametrization, and the second on a linear fit to a conformal-mapping variable. These low-$Q^2$ fits extrapolate the slope of the form factor to $Q^2$=0 and determine $R_{\rm E}$ values of approximately 0.84 and 0.89~fm, respectively. Fits spanning all $Q^2$, in which the single constants are replaced with cubic splines at larger $Q^2$, lead to similar results for $R_{\rm E}$. We conclude that the scattering data is consistent with $R_{\rm E}$ ranging from at least 0.84 to 0.89~fm, and therefore cannot resolve the discrepancy between determinations of $R_{\rm E}$ made using muonic and electronic hydrogen-atom spectroscopy.

Tentative study of nuclear charge radii for neutron-deficient nuclei around the Z = 82 shell from experimental α decay data

Abstract We tentatively investigate the root-mean-square (rms) nuclear charge radii of odd- A Po and Pb isotopes plus Tl isotopes, particularly concerning these difficultly-detected nuclei along with short lifetimes, via various data on α decay. Within the density-dependent cluster model, the density distributions of studied daughter nuclei are determined by exactly reproducing the corresponding experimental α decay half-lives, which leads the final results of nuclear charge radii. In addition, our recently proposed formula deducing the charge radii is extended to this study for comparison. Whether it concerns the ground or isomeric state of target nuclei, the extracted nuclear charge radii are found to be in good agreement with the measured values. Sequential predictions on the rms charge radii are subsequently made for these neutron-deficient nuclei and especially for the rarely detected Bi isotopic chain, which are expected to be useful for future measurements. Moreover, the variety of α -preformation factors is analyzed in the scheme of valence nucleon number to pursue the further improvement of the model. This may be considered as an effective effort to obtain the charge radii of ground and even low-lying excited states for exotic nuclei near the proton-dripline.

Isotope Shift Calculations of Stable and Short-Lived Beryllium Isotopes**Supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department under Grant No 14JK1402.

The isotope shifts of the 2s2 1S0 to 2s2p 1P1 and 3P1 transitions in the four-electron beryllium atom are calculated by using the multi-configuration Dirac–Hartree–Fock method and the relativistic configuration interaction approach for the stable and short-lived beryllium isotopes. The results provided herein can be employed for the consistency check with the nuclear rms charge radii from the experimental isotope shifts by using the corresponding transitions for the short-lived nuclei7,10–12Be and 14Be. The analogous isotope shift results could also be obtained for the beryllium-like ions by the methods used here.

Flavor decomposition of the nucleon electromagnetic form factors at lowQ2

Background: The spatial distribution of charge and magnetization within the proton is encoded in the elastic form factors. These have been precisely measured in elastic electron scattering, and the combination of proton and neutron form factors allows for the separation of the up- and down-quark contributions.Purpose: In this work, we extract the proton and neutron form factors from worldwide data with an emphasis on precise new data covering the low-momentum region, which is sensitive to the large-scale structure of the nucleon. From these, we separate the up- and down-quark contributions to the proton form factors.Method: We combine cross section and polarization measurements of elastic electron-proton scattering to separate the proton form factors and two-photon exchange (TPE) contributions. We combine the proton form factors with parametrization of the neutron form factor data and uncertainties to separate the up- and down-quark contributions to the proton's charge and magnetic form factors.Results: The extracted TPE corrections are compared to previous phenomenological extractions, TPE calculations, and direct measurements from the comparison of electron and positron scattering. The flavor-separated form factors are extracted and compared to models of the nucleon structure.Conclusions: With the inclusion of the precise new data, the extracted TPE contributions show a clear change of sign at low ${Q}^{2}$, which is necessary to explain the high-${Q}^{2}$ form factor discrepancy while being consistent with the known ${Q}^{2}\ensuremath{\rightarrow}0$ limit. We find that the new Mainz data yield a significantly different result for the proton magnetic form factor and its flavor-separated contributions. We also observe that the rms radius of both the up- and down-quark distributions are smaller than the rms charge radius of the proton.

The Lamb shift in muonic hydrogen and the proton radius puzzle

The recent measurement of the Lamb shift in muonic hydrogen resulted in a tenfold improved value of the rms charge radius of the proton. The value is, however, 7 standard deviations discrepant from the world average of this quantitiy which is obtained from elastic electron-proton scattering and precision spectroscopy of hydrogen and deuterium. New input from both theory and experiment is needed to resolve this so-called “proton radius puzzle”.

Tentative probe into the nuclear charge radii of superheavy odd-mass and odd-odd nuclei

The root-mean-square (rms) nuclear charge radii of superheavy odd-$A$ and odd-odd nuclei are tentatively pursued by the deduction of experimental $\ensuremath{\alpha}$ decay data. The framework of calculating $\ensuremath{\alpha}$ decay half-lives is constructed via the combination of the improved two-potential approach with the density-dependent cluster model. In this procedure, the charge distribution of daughter nuclei is determined to exactly reproduce the measured $\ensuremath{\alpha}$ decay half-lives. Next, the rms charge radius of daughter nuclei is obtained by using the corresponding charge distribution. For comparison, the previously proposed formula of our group is employed to estimate the rms charge radii as well. Besides the reasonable agreement between the extracted nuclear charge radii and the available experimental values, the nuclear radii of heaviest odd-$A$ and odd-odd nuclei are extracted from the $\ensuremath{\alpha}$ decay energies and half-lives. This can be considered as an effective attempt in terms of the nuclear size in the superheavy mass region.

Nuclear Polarization Corrections to theμHe+4Lamb Shift

Stimulated by the proton radius conundrum, measurements of the Lamb shift in various light muonic atoms are planned at PSI. The aim is to extract the rms charge radius with high precision, limited by the uncertainty in the nuclear polarization corrections. We present an ab-initio calculation of the nuclear polarization for mu-4He+ leading to an energy correction in the 2S-2P transitions of $\delta_{pol}=-2.47$ meV $\pm$ 6%. We use two different state-of-the-art nuclear Hamiltonians and utilize the Lorentz integral transform with hyperspherical harmonics expansion as few-body methods. We take into account the leading multipole contributions, plus Coulomb, relativistic and finite-nucleon-size corrections. Our main source of uncertainty is the nuclear Hamiltonian, which currently limits the attainable accuracy. Our predictions considerably reduce the uncertainty with respect to previous estimates and should timely serve the mu-4He+ experiment planned for 2013.

Coefficients of different macro–microscopic mass formulae from the AME2012 atomic mass evaluation

Abstract The coefficients of different possible macro–microscopic mass formulae previously proposed have been adjusted on 2264 experimental atomic masses extracted from the AME2012 atomic mass evaluation [1] assuming N , Z ⩾ 8 and the one standard deviation uncertainty on the mass lower than 150 keV. All the formulae include the volume and surface energies, the Coulomb energy, the diffuseness correction to the sharp radius Coulomb energy, the shell and pairing energies and take into account or not the curvature energy, different forms of the Wigner term, a free charge radius, the experimental equivalent rms charge radius or a fixed short central radius. Masses of 976 more exotic nuclei are extrapolated from the most accurate formula.

MICROSCOPIC ANALYSIS OF SHAPE EVOLUTION IN Mo ISOTOPES

The shape evolution of even–even Mo isotopes from the line of stability up to the two-neutron drip-line is investigated within the self-consistent Hartree–Fock–Bogoliubov theory in both the axial and triaxial symmetries. The Skyrme energy density functional SLy4 has been considered in the particle-hole channel, while the zero range delta-interaction has been employed in the particle–particle channel. In order to correctly treat the pairing correlations, a particle-number projection was carried out by the Lipkin–Nogami (LN) method. The two-neutron separation energies and root-mean-square (rms) charge radii are investigated and compared with available experimental data. The evolution of the potential energy surfaces in the (β, γ) deformation plane is presented and discussed. In addition, the obtained ground state deformation parameters are compared to those obtained by other models.