Charge Form Factor Research Articles

Emergence of Simple Patterns in Complex Atomic Nuclei from First Principles

We study the structure of low-lying states in 6Li, 6He, 8Be, 8B, 12C, and 16O, using ab initio symmetry-adapted no-core shell model. The results of our study demonstrate that collective modes in light nuclei emerge from first principles. We investigate the impact of the symmetry-adapted model space on spectroscopic properties and, in the case of the ground state of 6Li, on elastic electron scattering charge form factor. The results confirm that only a small symmetry-adapted subspace of the complete model space is needed to accurately reproduce complete-space observables and the form factor momentum dependence.

Neutron density distributions analyzed in terms of relativistic impulse approximation for nickel isotopes

Observables of proton elastic scattering from nickel isotopes (48–82 Ni ) are calculated based on relativistic impulse approximation (RIA), and nuclear density distributions are provided by relativistic mean-field (RMF) calculations. Contributions of a medium effect and multiple scattering to observables are evaluated and shown to be small at incident proton energies from 200 MeV through 500 MeV so that it is confirmed to perform a model analysis based on the fundamental RIA calculation. For 58,60,62,64 Ni isotopes, are considered proton distributions which are obtained by means of unfolding the charge form factor of proton from charge densities determined by the experiments of electron scattering. Through comparisons between results for the different proton densities, contributions of proton form factor to proton distributions and to elastic scattering observables at 300 MeV are discussed. It is shown that the neutron distribution is determined from the restricted observables, reaction cross-section and the first dip of differential cross-section, based on a model analysis of Woods–Saxon distribution in the case of 64 Ni target at 300 MeV. Contributions of tensor density and empirical proton density are shown to obtaining the neutron distribution with the model analysis. Compared with the similar studies for 40,60 Ca and 208 Pb , problems of the model analysis, which arise out of errors in observables, are discussed.

Electron-scattering form factors forLi6in theab initiosymmetry-guided framework

We present an ab initio symmetry-adapted no-core shell-model description for $^{6}$Li. We study the structure of the ground state of $^{6}$Li and the impact of the symmetry-guided space selection on the charge density components for this state in momentum space, including the effect of higher shells. We accomplish this by investigating the electron scattering charge form factor for momentum transfers up to $q \sim 4$ fm$^{-1}$. We demonstrate that this symmetry-adapted framework can achieve significantly reduced dimensions for equivalent large shell-model spaces while retaining the accuracy of the form factor for any momentum transfer. These new results confirm the previous outcomes for selected spectroscopy observables in light nuclei, such as binding energies, excitation energies, electromagnetic moments, E2 and M1 reduced transition probabilities, as well as point-nucleon matter rms radii.

Meson cloud contributions to baryon axial form factors

The axial form factor as well as the axial charges and radii of octet $N$, $\mathrm{\ensuremath{\Sigma}}$, and $\mathrm{\ensuremath{\Xi}}$ baryons are studied in the perturbative chiral quark model with the quark wave functions predetermined by fitting the theoretical results of the proton charge form factor to experimental data. The theoretical results are found, based on the predetermined quark wave functions, in good agreement with experimental data and lattice values. This may indicate that the electric charge and axial charge distributions of the constituent quarks are the same. The study reveals that the meson cloud plays an important role in the axial charge of octet baryons, contributing 30%--40% to the total values, and strange sea quarks have a considerable contribution to the axial charges of the $\mathrm{\ensuremath{\Sigma}}$ and $\mathrm{\ensuremath{\Xi}}$.

Error estimates for the Skyrme–Hartree–Fock model

There are many complementary strategies to estimate the extrapolation errors of a model calibrated in least-squares fits. We consider the Skyrme–Hartree–Fock model for nuclear structure and dynamics and exemplify the following five strategies: uncertainties from statistical analysis, covariances between observables, trends of residuals, variation of fit data, and dedicated variation of model parameters. This gives useful insight into the impact of the key fit data as they consist of binding energies, charge rms radii, and charge formfactor. Amongst others, we check in particular the predictive value for observables in the stable nucleus 208Pb, the super-heavy element 266Hs, r-process nuclei, and neutron stars.

The Higgs oscillator on the hyperbolic plane and light-front holography

The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schr\"odinger equation with the second hyperbolic P\"oschl-Teller potential, whose flat-space (small-angle) limit reduces to the conformally invariant inverse square distance plus harmonic oscillator interaction, on which LFH is based. In consequence, energies and wave functions of the LFH spectrum can be approached by the solutions of the Higgs oscillator on the hyperbolic plane in employing its curvature and the potential strength as fitting parameters. Also the proton electric charge form factor is well reproduced within this scheme by means of a Fourier-Helgason hyperbolic wave transform of the charge density. In conclusion, in the small angle approximation, the Higgs oscillator on the hyperbolic plane is demonstrated to satisfactory parallel essential outcomes of the Light Front Holographic QCD. The findings are suggestive of associating the hyperboloid curvature of the with a second scale in LFH, which then could be employed in the definition of a chemical potential.

Electromagnetic production ofKΣon the nucleon near threshold

Photo- and electroproduction of $K\Sigma$ have been investigated near their production thresholds by using an effective Lagrangian approach. For this purpose, the background amplitude is constructed from suitable Feynman diagrams, whereas the resonance terms are calculated by means of the Breit-Wigner form of multipoles. Experimental data available in the proton channels ($K^+\Sigma^0$ and $K^0\Sigma^+$) with energies up to 50 MeV above the thresholds have been utilized to extract the unknown parameters. In these proton channels the calculated observables fit nicely the experimental data, whereas in the neutron channels ($K^+\Sigma^-$ and $K^0\Sigma^0$) the predicted observables contain some uncertainties due to the the uncertainties in the values of helicity photon couplings. To this end, new $K^0\Sigma^+$ photoproduction data are urgently required. The present analysis indicates the validity of the $P_\Lambda = -\frac{1}{3} P_\Sigma$ relation derived a long time ago. In the electroproduction sector the present analysis confirms the smooth transition between photoproduction and low $Q^2$ electroproduction data. The effect of new Crystal Ball data is shown to be mild at the backward angles. It is also found that the electroproductions of $K^0\Sigma^+$ and $K^0\Sigma^0$ are practically not the suitable reactions for investigating the $K^0$ charge form factor, since the effect is small.

Form factors and charge radii in a quantum chromodynamics-inspired potential model using variationally improved perturbation theory

We use variationally improved perturbation theory (VIPT) for calculating the elastic form factors and charge radii of D,D s,B,B s and B c mesons in a quantum chromodynamics (QCD)-inspired potential model. For that, we use linear-cum-Coulombic potential and opt the Coulombic part first as parent and then the linear part as parent. The results show that charge radii and form factors are quite small for the Coulombic parent compared to the linear parent. Also, the analysis leads to a lower as well as upper bounds on the four-momentum transfer Q 2, hinting at a workable range of Q 2 within this approach, which may be useful in future experimental analyses. Comparison of both the options shows that the linear parent is the better option.

Comparison between shell model and self-consistent mean field calculations for ground charge density distributions and elastic form factors of 12C and 16O nuclei

The ground charge density distributions, elastic form factors and root mean square radii for 12C and 16O nuclei in 1p-shell nuclei are calculated in shell model using core plus valence and no-core–shell model calculations. The single-particle wave functions of harmonic-oscillators potential are used. For such potential, two harmonic-oscillators size parameters are used: one for neutron and the other for proton. Calculations are compared with the results of self-consistent mean field using SKX Skyrme parameterization and experimental data. The calculated charge density distributions in shell model for 12C and 16O nuclei show good agreement with experimental data. The calculated charge form factors at low q in shell model for 12C and 16O show good agreement with experimental data. At high q, the result for 16O shows no prediction for the position of second diffraction minimum. In general, it is found that enlarging the model space in shell model leads to suppress slightly the behavior of density distribution around central region and also the behavior of form factor at high q, which is good to approach experimental data. On the other side, such increase in model space leads to increase the calculated root mean square radii, which is the same behaviour of the results of Hartree–Fock calculations. The calculated charge form factors in Hartree–Fock calculations show the existence of many diffraction minima, which do not agree with experimental data in contrary to shell model, which predicts less diffraction minima.

Investigation of nuclear structure of 30–44S isotopes using spherical and deformed Skyrme–Hartree–Fock method

Nuclear many-body system is usually described by a mean-field built upon a nucleon–nucleon effective interaction. In this work, we investigate ground state properties of the sulfur isotopes covering a wide range from the line of stability up to the dripline region (30–44S). For this purpose the Hartree–Fock mean field theory in coordinate space with a Skyrme parameterization SkM* has been utilized. In particular, we calculate the nuclear charge, neutrons, protons, mass densities, the associated radii, neutron skin thickness and binding energy. The charge form factors have been also investigated using SkM*, SkO, SkE, SLy4 and Skxs15 Skyrme parameterizations and the results obtained using the theoretical approach are compared with the available experimental data. To investigate the potential energy surface as a function of the quadrupole deformation for isotopic sulfur chains, Skyrme–Hartree–Fock–Bogoliubov theory has been adopted with SLy4 parameterization.

A Discussion of Deuteron Transverse Charge Densities

The deuteron transverse charge density ρC(b) is the two-dimensional Fourier transform of its charge form factor in the impact space. We show that different parameterizations of the charge form factors provide different ρC(b), in particular at the central value of impact parameter (b = 0), although all the parameterizations can well reproduce the form factors in the region of small Q2. In addition, we also check the explicit contributions from the different coordinate intervals of the deuteron wave function to its root-mean-square radius.

Separated response function ratios in exclusive, forward π(±) electroproduction.

The study of exclusive π(±) electroproduction on the nucleon, including separation of the various structure functions, is of interest for a number of reasons. The ratio RL=σL(π-)/σL(π+) is sensitive to isoscalar contamination to the dominant isovector pion exchange amplitude, which is the basis for the determination of the charged pion form factor from electroproduction data. A change in the value of RT=σT(π-)/σT(π+) from unity at small -t, to 1/4 at large -t, would suggest a transition from coupling to a (virtual) pion to coupling to individual quarks. Furthermore, the mentioned ratios may show an earlier approach to perturbative QCD than the individual cross sections. We have performed the first complete separation of the four unpolarized electromagnetic structure functions above the dominant resonances in forward, exclusive π(±) electroproduction on the deuteron at central Q(2) values of 0.6, 1.0, 1.6 GeV(2) at W=1.95 GeV, and Q(2)=2.45 GeV(2) at W=2.22 GeV. Here, we present the L and T cross sections, with emphasis on RL and RT, and compare them with theoretical calculations. Results for the separated ratio RL indicate dominance of the pion-pole diagram at low -t, while results for RT are consistent with a transition between pion knockout and quark knockout mechanisms.

JLab measurement of the 4He charge form factor at large momentum transfers.

The charge form factor of 4He has been extracted in the range 29 fm(-2) ≤ Q2 ≤ 77 fm(-2) from elastic electron scattering, detecting 4He recoil nuclei and electrons in coincidence with the high resolution spectrometers of the Hall A Facility of Jefferson Lab. The measurements have uncovered a second diffraction minimum for the form factor, which was predicted in the Q2 range of this experiment. The data are in qualitative agreement with theoretical calculations based on realistic interactions and accurate methods to solve the few-body problem.

Study of baryon octet electromagnetic form factors in perturbative chiral quark model

The electromagnetic properties of baryon octet are studied in the perturbative chiral quark model (PCQM). The relativistic quark wave function is extracted by fitting the theoretical results of the proton charge form factor to experimental data and the predetermined quark wave function is applied to study the electromagnetic form factors of other octet baryons as well as magnetic moments, charge and magnetic radii. The PCQM results are found, based on the predetermined quark wave function, in good agreement with experimental data.

Elastic and quasi-elastic electron scattering on theN=14, 20, and 28 isotonic chains

We present theoretical predictions for electron scattering on the N = 14, 20, and 28 isotonic chains from proton-deficient to proton-rich nuclei. The calculations are performed within the framework of the distorted-wave Born approximation and the proton and neutron density distributions are evaluated adopting a Relativistic Hartree-Bogoliubov (RHB) approach with a density dependent meson-exchange interaction. We present results for the elastic and quasi-elastic cross sections and for the parity-violating asymmetry parameter. Owing to the correlations between the evolution of the electric charge form factors along each chain with the underlying proton shell structure of the isotones, elastic electron scattering experiments on isotones can provide useful informations about the occupation and filling of the single-particle levels of protons.

Translationally Invariant Calculations of Form Factors, Densities and Momentum Distributions for Finite Nuclei with Short–Range Correlations Included: A Fresh Look

The approach proposed in the 70s (Dementiji et al. in Sov J Nucl Phys 22:6–9, 1976), when describing the elastic and inelastic electron scattering off 4He, and elaborated in (Shebeko et al.in Eur Phys J A27:143–155, 2006) for calculations of the one-body, two-body and more complex density matrices of finite bound systems has been applied (Shebeko and Grigorov in Ukr J Phys 52:830–842, 2007; Shebeko et al. in Eur. Phys. J. A48:153–172, 2012) in studying a combined effect of the center-of-mass motion and nucleon–nucleon short-range correlations on the nucleon density and momentum distributions in light nuclei beyond the independent particle model. Unlike a common practice, suitable for infinite bound systems, these distributions are determined as expectation values of appropriate intrinsic operators that depend upon the relative coordinates and momenta (Jacobi variables) and act on the intrinsic ground–state wave functions (WFs). The latter are constructed in the so-called fixed center-of-mass approximation, starting with a mean–field Slater determinant modified by some correlator (e.g., after Jastrow or Villars). Our numerical calculations of the charge form factors (FCH(q)), densities and momentum distributions have been carried out for nuclei 4He and 16O choosing, respectively, the 1s and 1s−1p Slater determinants of the harmonic oscillator model as trial, nontranslationally invariant WFs.

Effective field theory for proton halo nuclei

We use halo effective field theory to analyze the universal features of proton halo nuclei bound due to a large S-wave scattering length. With a Lagrangian built from effective core and valence-proton fields, we derive a leading-order expression for the charge form factor. Within the same framework we also calculate the radiative proton capture cross section. Our general results at leading order are applied to study the excited 1/2+ state of Fluorine-17, and we give results for the charge radius and the astrophysical S-factor.

Study of baryon octet charge form factors in perturbative chiral quark model

The charge form factors of baryon octet are studied in the perturbative chiral quark model (PCQM). The relativistic quark wave function is extracted by fitting the theoretical results of the nucleon charge form factors to the experimental data and the predetermined quark wave function is applied to study the charge form factors of other octet baryons. The PCQM results are found, based on the predetermined quark wave function, in good agreement with the experimental data.

Muon Elastic Scattering with MUSE at PSI

The proton radius puzzle is the disagreement between the much more precise radius determined from muonic hydrogen spectroscopy and the numerous atomic hydrogen and electron scattering determinations. The puzzle has several possible resolutions, including physics beyond the Standard Model, missing conventional physics, and errors or underestimated uncertainties in the extraction of the radius from the data. New experiments are needed to confirm and / or resolve the puzzle. The MU on S cattering E xperiment (MUSE) recently approved at PSI has been designed to help resolve the puzzle by measuring the radius in a way not yet done. Similar to electron scattering, the radius will be extracted from the observed change of the charge form factor with momentum transfer. The experiment uses the π M1 beamline to provide a mixed secondary muon and electron (and pion) beam of either positive or negative charge. The comparison of muon and electron scattering measured simultaneously determines the consistency of the form factors in the two cases with high precision. Comparison of yields from both charge signs will at the same time disentangle the effect of two-photon exchange. The proton charge radius can be extracted from each set of scattering data. The physics case and status of MUSE will be discussed.

From nuclear structure to neutron stars

Recent progress in quantum Monte Carlo with modern nucleon-nucleon interactions have enabled the successful description of properties of light nuclei and neutron-rich matter. As a demonstration, we show that the agreement between theoretical calculations of the charge form factor of 12C and the experimental data is excellent. Applying similar methods to isospin-asymmetric systems allows one to describe neutrons confined in an external potential and homogeneous neutron-rich matter. Of particular interest is the nuclear symmetry energy, the energy cost of creating an isospin asymmetry. Combining these advances with recent observations of neutron star masses and radii gives insight into the equation of state of neutron-rich matter near and above the saturation density. In particular, neutron star radius measurements constrain the derivative of the symmetry energy.