Elastic Charge Form Factors Research Articles

Quark counting, Drell-Yan West, and the pion wave function

The relation between the pion's quark distribution function, q(x), its light-front wave function, and the elastic charge form factor, F(Δ2), is explored. The square of the leading-twist pion wave function at a special probe scale, ζH, is determined using models and Poincaré covariance from realistic results for q(x). This wave function is then used to compute form factors with the result that the Drell-Yan-West and quark counting relationships are not satisfied. A new relationship between q(x) and F(Δ2) is proposed. Published by the American Physical Society 2024

Imprints of α clustering in the density profiles of C12 and O16

The $^{4}$He nucleus is a well bound and highly correlated four-nucleon system that is also found in form of $\alpha$-clusters as substructure in nuclear many-body systems. The standard single-particle shell model cannot represent these four-body correlations. It is highly desirable to find a simple way to identify and distinguish these fundamental structures without large-scale computations. In this paper, we investigate with intrinsic Slater determinants as trial states in how far these two competing pictures prevail in the ground states of $^{12}$C and $^{16}$O. The trial states can describe both the $j$-$j$ coupling shell-model and $\alpha$-cluster configurations in a unified way. One-body density distributions of the trial states are calculated and compared to elastic scattering cross section data. The parameters of the trial state are chosen to reproduce the experimental charge radii. A well developed cluster structure is characterized by an enhancement of the differential elastic scattering cross sections, as well as the elastic charge form factors around the first peak positions. The density profiles of the internal regions can also be probed at higher momentum transfer regions beyond the second minimum. With these trial states one can visualize the competition between cluster and shell structure in an intuitive and simple way.

Two-photon exchange in (muonic) deuterium at N3LO in pionless effective field theory

We present a study of the two-photon-exchange (2upgamma -exchange) corrections to the S-levels in muonic (mu D) and ordinary (D) deuterium within the pionless effective field theory (EFT). Our calculation proceeds up to next-to-next-to-next-to-leading order (N3LO) in the EFT expansion. The only unknown low-energy constant entering the calculation at this order corresponds to the coupling of a longitudinal photon to the nucleon–nucleon system. To minimise its correlation with the deuteron charge radius, it is extracted using the information about the hydrogen–deuterium isotope shift. We find the elastic 2upgamma -exchange contribution in mu D larger by several standard deviations than obtained in other recent calculations. This discrepancy ameliorates the mismatch between theory and experiment on the size of 2upgamma -exchange effects, and is attributed to the properties of the deuteron elastic charge form factor parametrisation used to evaluate the elastic contribution. We identify a correlation between the deuteron charge and Friar radii, which can help one to judge how well a form factor parametrisation describes the low-virtuality properties of the deuteron. We also evaluate the higher-order 2upgamma -exchange contributions in mu D, generated by the single-nucleon structure and expected to be the most important terms beyond N3LO. The uncertainty of the theoretical result is dominated by the truncation of the EFT series and is quantified using a Bayesian approach. The resulting extractions of the deuteron charge radius from the mu D Lamb shift, the 2S{-}1S transition in D, and the 2S{-}1S hydrogen–deuterium isotope shift, with the respective 2upgamma -exchange effects evaluated in a unified EFT approach, are in perfect agreement.

The ground state properties of some exotic nuclei studied by the two-body model

The ground state properties including the density distributions of the neutrons, protons and matter as well as the corresponding root mean square (rms) radii of proton-rich halo candidates 8B, [Formula: see text]N, [Formula: see text]Al and [Formula: see text]P have been studied by the single particle Bear–Hodgson (BH) wave functions with the two-body model of [Formula: see text]. It is found that the rms radii of these proton-rich nuclei are reproduced well by this model and the radial wave functions describe the long tail of the proton and matter density distributions. These results indicate that this model achieves a suitable description of the possible halo structure. The plane wave Born approximation (PWBA) has been used to compute the elastic charge form factors.

Nuclear matter distributions of neutron rich 6He, 11Li, 14Be and 17B halo nuclei studied by the Bear–Hodgson potential

The radial wave functions of the Bear–Hodgson potential have been used to study the ground state features such as the proton, neutron and matter densities and the associated rms radii of two neutrons halo 6He, [Formula: see text]Li, [Formula: see text]Be and [Formula: see text]B nuclei. These halo nuclei are treated as a three-body system composed of core and outer two-neutron [Formula: see text]. The radial wave functions of the Bear–Hodgson potential are used to describe the core and halo density distributions. The interaction of core-neutron takes the Bear–Hodgson potential form. The outer two neutrons of 6He and [Formula: see text]Li interact by the realistic interaction REWIL whereas those of [Formula: see text]Be and [Formula: see text]B interact by the realistic interaction of HASP. The obtained results show that this model succeeds in reproducing the neutron halo in these nuclei. From the calculated densities, it is found that 6He, [Formula: see text]Li, [Formula: see text]Be and [Formula: see text]B have a long tail in neutron and matter densities which is consistent with the experimental data. Elastic charge form factors for these halo nuclei are analyzed via the plane wave Born approximation.

The dependence of the nuclear charge form factor on short range correlations and surface fluctuation effects

An investigation is carried out in order to study effects, originating from fluctuations of the nuclear surface, to the elastic charge form factor of light nuclei in which the two-body part of the short range correlation factor is also included through a Jastrow-type correlation function. It is found that if these effects are taken into account in the uncorrected (harmonic oscillator) part of such a form-factor for the 4He, l6O and 40Ca nuclei, the quality of the fitting is improved. In addition, they lead to a drastic change in the asymptotic behaviour of the point-proton form factor which now drops off for large values of the momentum transfer q quite slowly that is as const.•q^-4.

Systematic Study of the Nuclear Structure for Some Exotic Nuclei Using Skyrme–Hartree–Fock Method

The ground state charge, neutron, proton and matter densities, the associated nuclear radii and the binding energy per nucleon of 8B, 17Ne, 23Al and 27P halo nuclei have been investigated using the Skyrme–Hartree–Fock (SHF) model with the new SKxs25 parameters. According to the calculated results, it is found that the SHF model with these Skyrme parameters provides a good description on the nuclear structure of above proton-rich halo nuclei. The elastic charge form factors of 8B and 17Ne halo nuclei and those of their stable isotopes 10B and 20Ne are calculated using plane-wave Born approximation with the charge density distributions obtained by SHF model to investigate the effect of the extended charge distributions of proton-rich nuclei on the elastic electron scattering. Calculations show that the major difference between the elastic charge form factors of halo nuclei and those of their stable isotopes is caused by the variation in the charge density distributions, especially the details of the outer parts.

Electron scattering from some even fp- and N50-shell nuclei with implanting the influence of short-range correlation functions

The influence of two body short range correlations on elastic electron scattering charge form factors, charge densities as well as root mean square charge radii of some fp-shell nuclei (for example, 48Ca, 50Cr, 54Fe, 58Ni, 70Ge and 74Se) and some N50-shell nuclei (for example, 88Sr and 90Zr) is analyzed using the one- and two-body terms in the cluster expansion together with the single particle harmonic oscillator wave functions. The Jastrow-type correlation function is utilized to embed the effect of short range correlations into elastic charge form factors F(q) and charge densities ρ(r). Both F(q) and ρ(r) depend upon the harmonic oscillator parameter b and the correlation parameter β (which initiates from the Jastrow correlation function). Here, the parameters b and β are determined via the fitting to the measured charge form factors. The embedding of short range correlations imitates the measured charge form factors at the high momentum transfers (q ≥ 2 fm−1). It is noticed that the implanting of short range correlations is required for obtaining a remarkable alteration in the computed elastic charge form factors which in turn leads to explain the data of electron scattering astonishingly throughout the entire range of considered momentum transfers.

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.

Study of matter density distributions, elastic charge form factors and size radii for halo 11Be, 19C and 11Li nuclei

In this work, the calculation of matter density distributions, elastic charge form factors and size radii for halo 11Be, 19C and 11Li nuclei are calculated. Each nuclide under study are divided into two parts; one for core part and the second for halo part. The core part are studied using harmonic-oscillator radial wave functions, while the halo part are studied using the radial wave functions of Woods-Saxon potential. A very good agreement are obtained with experimental data for matter density distributions and available size radii. Besides, the quadrupole moment for 11Li are generated.

Elastic electron scattering from 6He and 11Li halo nuclei

The binary cluster model (BCM) and the two-frequency shell model (TFSM) have been used to study the ground state matter densities of neutron-rich 6 He and 11 Li halo nuclei. Calculations show that both models provide a good description on the matter density distribution of above nuclei. The root-mean square (rms) proton, neutron and matter radii of these halo nuclei obtained by TFSM have been successfully obtained. The elastic charge form factors for these halo nuclei are studied through combining the charge density distribution obtained by TFSM with the plane wave Born approximation (PWBA).

Study of Charge Density Distributions and Elastic Charge Form Factors for 40Ca and 48Ca

Study of Charge Density Distributions and Elastic Charge Form Factors for 40Ca and 48Ca

Elastic Electron Scattering from <sup>88</sup>Sr and <sup>89</sup>Y Nuclei

The ground state proton momentum distributions and elastic charge form factors for ⁸⁸Sr and ⁸⁹Y nuclei have been derived and studied using the Coherent Density Fluctuation Model and formulated by means of the fluctuation function (weight function) |<i>f(x)</i>|² The fluctuation function has been connected to the charge density distribution of the nuclei and determined from the theory and experiment. The feature of the long-tail behavior at high momentum region of the PMD has been calculated by both the theoretical and experimental fluctuation functions. The calculated form factors F (q) of all nuclei under study are in good agreement with those of experimental data throughout all values of momentum transfer q.

Spherical Hartree-Fock calculations with linear-momentum projection before the variation

Spherical Hartree--Fock calculations with projection onto zero total linear momentum before the variation are performed for the nuclei 4He, 12C, 16O, 28Si, 32S and 40Ca using a density--independent effective nucleon--nucleon interaction. The results are compared to those of usual spherical Hartree--Fock calculations subtracting the kinetic energy of the center of mass motion either before or after the variation and to the results obtained analytically with oscillator occupations. Total energies, hole--energies, elastic charge form factors and charge densities and the mathematical Coulomb sum rules are discussed.

Large-basisab initiono-core shell model and its application to12C

We present the framework for the ab initio no-core nuclear shell model and apply it to obtain properties of ${}^{12}\mathrm{C}.$ We derive two-body effective interactions microscopically for specific model spaces from the realistic CD-Bonn and the Argonne V8' nucleon-nucleon $(\mathrm{NN})$ potentials. We then evaluate binding energies, excitation spectra, radii, and electromagnetic transitions in the $0\ensuremath{\Elzxh}\ensuremath{\Omega},$ $2\ensuremath{\Elzxh}\ensuremath{\Omega},$ and $4\ensuremath{\Elzxh}\ensuremath{\Omega}$ model spaces for the positive-parity states and the $1\ensuremath{\Elzxh}\ensuremath{\Omega},$ $3\ensuremath{\Elzxh}\ensuremath{\Omega},$ and $5\ensuremath{\Elzxh}\ensuremath{\Omega}$ model spaces for the negative-parity states. Dependence on the model-space size, on the harmonic-oscillator frequency, and on the type of the $\mathrm{NN}$ potential, used for the effective interaction derivation, are studied. In addition, electromagnetic and weak neutral elastic charge form factors are calculated in the impulse approximation. Sensitivity of the form-factor ratios to the strangeness one-body form-factor parameters and to the influence of isospin-symmetry violation is evaluated and discussed. Agreement between theory and experiment is favorable for many observables, while others require yet larger model spaces and/or three-body forces. The limitations of the present results are easily understood by virtue of the trends established and previous phenomenological results.

Elastic charge form factors of and K mesons

An approach to the electromagnetic structure of composite systems is developed. The method is based on the instant form of relativistic Hamiltonian dynamics and modified impulse approximation. Soft elastic charge form factors of and K mesons as systems are calculated. The results are in good agreement with the available data. Relativistic effects are shown to be large even at low values of momentum transfer.

The dependence of the nuclear charge form factor on short-range correlations and surface fluctuation effects

We investigate the effects of fluctuations of the nuclear surface on the harmonic oscillator elastic charge form factor of light nuclei, while simultaneously approximating the short-range correlations through a Jastrow correlation ~factor. Inclusion of surface-fluctuation effects within this description, by truncating the cluster expansion at the two-body part, is found to improve somewhat the fit to the elastic charge form-factor of $^{16}O$ and $^{40}Ca$. However, the convergence of the cluster expansion is expected to deteriorate. An additional finding is that the surface-fluctuation correlations produce a drastic change in the asymptotic behavior of the point-proton form factor, which now falls off quite slowly (i.e. as $const. \cdot q^{-4}$) at large values of the momentum transfer $q$.

Relationship Between the Proton Charge Form Factor and the Quark-Quark Interaction

We demonstrate how the parameterized proton elastic charge form factor may be used to determine the quark-quark interaction assuming a constituent-quark model of the nucleon and non-relativistic kinematics. The relation between the hypercentral component of the three-quark bound-state wave function and the charge density is obtained by using an Abel-type integral equation. To a high degree of accuracy, this component fixes the hypercentral potential, from which the quark-quark interaction may be determined exactly via another Abel transform. We find that the presence of the meson cloud makes it impossible to determine the long-range behaviour of the quark-quark potential. However, our results indicate that the Gari-Krupelmann and dipole parametrizations of the form factor both generate potentials that are singular at short distances, indicating a degree of sensitivity of the form factor to the quark-quark interaction at small distances. We also generalize our technique to systems of arbitrarily large number of particles.

Effect of relativistic kinematics on the quark-quark interaction obtained from the proton form factor.

We have recently shown, assuming nonrelativistic kinematics, that the parametrized elastic charge form factor data for an {ital N}-particle system may be used to determine the underlying two-particle interaction. We now extend the validity of our formalism, using the prescription of Mitra and Kumari, to include relativistic kinematics, and apply it to a constituent quark model of the proton. We show analytically that, for a dipole fit to the form factor, the essential features of the quark-quark potential do not depend on the form of the kinematics. This provides an explanation of our earlier somewhat paradoxical finding that the small {ital r} singularity in the potential, which is predicted by first order QCD, is reproduced by a totally nonrelativistic inversion of the form factor data.

Elastic charge form factors for K mesons

The elastic charge form factors for the charged K + and neutral K 0 are presented. The model employed makes use of solutions of the coupled Bethe-Salpeter and Schwinger-Dyson equations. This covariant model provides good predictions for the pion's elastic charge and transition form factor, and here we extend the model to include strange quarks. The results for the kaon are in good agreement with the available data. This model is partially motivated by the expectation of precision data from CEBAF approved experiment E-93-018. The non vanishing K 0 form factor is due to the mass difference between the strange and down quarks.