Diffraction Radius Research Articles

Nuclear charge densities in spherical and deformed nuclei: Toward precise calculations of charge radii

Background: Precise measurements of atomic transitions affected by electron-nucleus hyperfine interactions offer sensitivity to explore basic properties of the atomic nucleus and study fundamental symmetries, including the search for new physics beyond the standard model of particle physics. In particular, such measurements, augmented by atomic and nuclear calculations, will enable extraction of the higher-order radial moments of the charge-density distribution in spherical and deformed nuclei. The new data impose higher precision requirements on a theoretical description.Purpose: The nuclear charge density is composed of the proton point distribution folded with the nucleonic charge distributions. The latter induce subtle relativistic corrections due to the coupling of nucleon magnetic moments with the nuclear spin-orbit density. Additional corrections come from the effect of center-of-mass projection. We assess the precision of nuclear charge density calculations by studying the behavior of relativistic and center-of-mass motion corrections to the second and fourth charge radial moments. Special attention has been paid to the magnetic spin-orbit density associated with the local variations of the spin-orbit current.Methods: The calculations for semimagic and open-shell nuclei are performed in the framework of self-consistent mean-field theory using quantified energy density functionals and density-dependent pairing forces. We used the general expression for the spin-orbit form factor that is valid for spherical and deformed nuclei.Results: We studied the impact of various correction terms on the charge radii, fourth radial moments, diffraction radii, and surface thickness of spherical and deformed nuclei. The spin-orbit corrections to charge radial moments and surface thickness show strong shell fluctuations which can make an appreciable effect when aiming at high-precision predictions of isotopic shifts. The inclusion of relativistic and center-of-mass corrections impacts the quality of energy density functionals optimized to charge radii data.Conclusions: To establish reliable constraints on the existence of new forces from isotope shift measurements, precise calculations of nuclear charge densities of deformed nuclei are needed. The proper inclusion of the spin-orbit charge density and other correction terms is essential when aiming at extraction of subtle effects which become particularly visible in isotopic trends. It is also important when developing high-quality nuclear energy density functionals optimized using heterogeneous datasets involving absolute charge radii, differential charge radii, and charge form factor properties deduced from electron-scattering data.

Nonlinear propagation dynamics of Gaussian beams in fractional Schrödinger equation

We investigated theoretically the nonlinear propagation dynamics of Gaussian beams in the fractional Schrödinger equation (FSE). When the nonlinearity is introduced into FSE without invoking an external potential, the evolution behaviors of incident Gaussian beams are modulated regularly and some novel phenomena arise. In the one-dimensional case, by changing the values of Kerr or saturated nonlinear coefficient, specific localized or diffracted phenomena appear in the corresponding intensity region, where the splitting angle of one-dimensional incident beam will be modulated flexibly to become larger or smaller in weak nonlinear region, besides, when the self-focusing strength is moderate, the energy of the beam is highly concentrated to form a breathing soliton structure. For two-dimensional case, Kerr or saturated nonlinearity will modulate the energy to the middle or edge in a certain nonlinear region, corresponding to the decrease or increase of the conical diffraction radius, it should be noted that there are two evolution periods under the saturated self-focusing nonlinearity. The work may provide more possibilities for beam modulation in FSE from a nonlinear perspective.

Parametrisations of relativistic energy density functionals with tensor couplings

The relativistic density functional with minimal density dependent nucleon–meson couplings for nuclei and nuclear matter is extended to include tensor couplings of the nucleons to the vector mesons. The dependence of the minimal couplings on either vector or scalar densities is explored. New parametrisations are obtained by a fit to nuclear observables with uncertainties that are determined self-consistently. The corresponding nuclear matter parameters at saturation are determined including their uncertainties. An improvement in the description of nuclear observables, in particular for binding energies and diffraction radii, is found when tensor couplings are considered, accompanied by an increase of the Dirac effective mass. The equations of state for symmetric nuclear matter and pure neutron matter are studied for all models. The density dependence of the nuclear symmetry energy, the Dirac effective masses and scalar densities is explored. Problems at high densities for parametrisations using a scalar density dependence of the couplings are identified due to the rearrangement contributions in the scalar self-energies that lead to vanishing Dirac effective masses.

Rotational band in12C based on the Hoyle state

α + 12 C inelastic differential cross-sections were measured at the energies 65 and 110 MeV. A new broad state at 13.75 MeV was observed. Its spin-parity has been determined as 4 + and the diffraction radius of the corresponding L = 4 transition is ~ 0.8 fm larger than that of the excitation of the 4+, 14.8 MeV level. The 13.75 MeV state was considered to be the third member of the rotational band based on the Hoyle state.

Energy density functional for nuclei and neutron stars

We aim to develop a nuclear energy density functional that can be simultaneously applied to finite nuclei and neutron stars. We use the self-consistent nuclear density functional theory (DFT) with Skyrme energy density functionals and covariance analysis to assess correlations between observables for finite nuclei and neutron stars. In a first step two energy functionals -- a high density energy functional giving reasonable neutron properties, and a low density functional fitted to nuclear properties -- are matched. In a second step, we optimize a new functional using exactly the same protocol as in earlier studies pertaining to nuclei but now including neutron star data. This allows direct comparisons of performance of the new functional relative to the standard one. The new functional TOV-min yields results for nuclear bulk properties (energy, r.m.s. radius, diffraction radius, surface thickness) that are of the same quality as those obtained with the established Skyrme functionals, including SV-min. When comparing SV-min and TOV-min, isoscalar nuclear matter indicators vary slightly while isovector properties are changed considerably. We discuss neutron skins, dipole polarizability, separation energies of the heaviest elements, and proton and neutron drip lines. We confirm a correlation between the neutron skin of $^{208}$Pb and the neutron star radius. We demonstrate that standard energy density functionals optimized to nuclear data do not carry information on the expected maximum neutron star mass, and that predictions can only be made within an extremely broad uncertainty band. For atomic nuclei, the new functional TOV-min performs at least as well as the standard nuclear functionals, but it also reproduces expected neutron star data within assumed error bands.

Simulation of optical interstellar scintillation

Stars twinkle because their light propagates through the atmosphere. The same phenomenon is expected on a longer time scale when the light of remote stars crosses an interstellar turbulent molecular cloud, but it has never been observed at optical wavelengths. The aim of the study described in this paper is to fully simulate the scintillation process, starting from the molecular cloud description as a fractal object, ending with the simulations of fluctuating stellar light curves. Fast Fourier transforms are first used to simulate fractal clouds. Then, the illumination pattern resulting from the crossing of background star light through these refractive clouds is calculated from a Fresnel integral that also uses fast Fourier transform techniques. Regularisation procedure and computing limitations are discussed, along with the effect of spatial and temporal coherency (source size and wavelength passband). We quantify the expected modulation index of stellar light curves as a function of the turbulence strength --characterised by the diffraction radius $R_{diff}$-- and the projected source size, introduce the timing aspects, and establish connections between the light curve observables and the refractive cloud. We extend our discussion to clouds with different structure functions from Kolmogorov-type turbulence. Our study confirms that current telescopes of ~4m with fast-readout, wide-field detectors have the capability of discovering the first interstellar optical scintillation effects. We also show that this effect should be unambiguously distinguished from any other type of variability through the observation of desynchronised light curves, simultaneously measured by two distant telescopes.

Radius of 12C in the excited 2 2 + Hoyle state

The differential cross sections of the inelastic α + 12C scattering leading to the excitation of the short-lived 2 2 + (E x = 9.84 or 9.6 MeV) state in 12C have been analysed within a modified diffraction model. We determined the diffraction radii of 12C in this excited state at E α = 386 and 240 MeV and compared them with the diffraction radius for the 0 2 + (E x = 7.65 MeV) Hoyle state. We found that the rms radii for the 2 2 + state is 〈R(2 2 + )〉 = 3.07 ± 0.13fm, which agrees well with the rms radius of the 0 2 + , 7.65 MeV Hoyle state and is a factor of 1.3 larger than the rms radius for the ground state of 12C. The similarity between the rms radii of the 0 2 + and 2 2 + states provides a strong argument in favor of a hypothesis that the 2 2 + state is the first member of a rotational band based on the 0 2 + Hoyle state.

Effect of neutron halos on excited states of nuclei

The differential cross sections of the inelastic scattering leading to the excitation of short-lived states in the stable ${}^{13}$C and ${}^{9}$Be nuclei as well as the radioactive ${}^{11}$Be nucleus have been analyzed. Signatures of neutron halos in the excited states located close to the neutron emission thresholds have been investigated by applying a recently developed modified diffraction model. The abnormally large rms radius was identified for the 3.089-MeV $1/{2}^{+}$ state of ${}^{13}$C. Significantly enlarged diffraction radii were found for the 1.68-MeV $1/{2}^{+}$ and the 3.05-MeV $5/{2}^{+}$ states of ${}^{9}$Be. The analysis of the diffraction radii of the weakly bound radioactive ${}^{11}$Be nuclei have shown that the 1.78-MeV $5/{2}^{+}$ and 3.41-MeV $3/{2}^{+}$ states in ${}^{11}$Be have the same diffraction radii as the $1/{2}^{+}$ ground state, which is known as the best example of a state with one-neutron halo.

Study of the diffraction scattering 12C + 12C with the excitation of the 12C exotic state 0 2 + (the Hoyle state)

New results from a series of experiments dedicated to the study of the 12C exotic state (the so-called Hoyle state) are presented. In spite of the many investigations that have been carried out, the structure of this state (which lies above the threshold for breaking up into three alpha particles) is still unknown. The different models assume that the nucleus has an abnormally large size in this excited state. However, until recently, methods for measuring the radii of unbound states have not been suggested. The best way to solve this problem seems to be by measuring the angular distributions of elastic and inelastic scattering of 12C on different target nuclei, and the determination of the radii is based on the fact that, at small scattering angles, the cross sections for direct reactions at high enough energies behave like Frauenhofer diffraction on a black ball. Accordingly, an experiment was performed aimed at measuring the elastic and inelastic angular distributions of 12C with an energy of (121.5 ± 0.5) MeV on a 12C target. The elastic scattering was measured in the angular range from 18° to 50° in the c.m. system with uncertainty in the angle of measurement equal to Δθ = ± 0.6°. The inelastic cross section was measured for the 12C excited state 2+ (4.44 MeV) and 0+ (7.65 MeV). Estimates were made for the diffraction radii for the ground and excited states. An increase was observed in the radius of the state at 7.65 MeV compared to those of the ground and first excited states.

α + 12C scattering at 110 MeV: Has exotic 7.65 MeV Hoyle’s state signatures “alpha-condensate” structure?

We present new measurements of the α + 12C elastic and inelastic (to the states 4.44, 7.65, and 9.64 MeV) scattering at Elab = 110 MeV in the wide angular range from ∼10° to 175°, which enable us to examine the condensate and cluster properties of the low-lying excited states in 12C. We present the diffraction-radius analysis of our data together with a considerable amount of the existing data. The magnitudes of the diffraction radii for the ground and the first excited (4.44 MeV) states are found to be equal, whereas they appear to be enhanced by ∼0.6 fm both for 7.65 and 9.64 MeV states. This result shows that the radius of the Hoyle’s 02+, 7.65 MeV state in 12C is by a factor of ∼1.2–1.3 larger than that of the ground state. It is demonstrated that the direct transfer mechanism of 8Be dominates at the largest angles in all four reactions reported here. The configuration corresponding to the transfer of 8Be in its ground state (Iπ = 0+) with L = 0 turns out to be the most important for the 7.65 MeV state of 12C. Evidence of existence of some features of α-condensed structure of the Hoyle’s 02+ state in 12C was obtained: its enhanced radius and large contribution of α-particle configuration with L = 0.

Nuclear mass form factors from coherent photoproduction of πmesons

Data for coherent photoproduction of $\pi^0$ mesons from nuclei ($^{12}$C, $^{40}$Ca, $^{93}$Nb, $^{nat}$Pb), recently measured with the TAPS detector at the Mainz MAMI accelerator, have been analyzed in view of the mass form factors of the nuclei. The form factors have been extracted in plane wave approximation of the $A(\gamma ,\pi^0)A$ reaction and corrected for final state interaction effects with the help of distorted wave impulse approximations. Nuclear mass rms-radii have been calculated from the slope of the form factors for $q^2\to 0$. Furthermore, the Helm model (hard sphere form factor folded with Gaussian) was used to extract diffraction radii from the zeroes of the form factor and skin thicknesses from the position and height of its first maximum. The diffraction radii from the Helm model agree with the corresponding charge radii obtained from electron scattering experiments within their uncertainties of a few per cent. The rms-radii from the slope of the form factors are systematically lower by up to 5% for PWIA and up to 10% for DWIA. Also the skin thicknesses extracted from the Helm model are systematically smaller than their charge counter parts.

Study on Application of the cos .ALPHA. Method to Neutron Stress Measurement

In order to study on a method of neutron stress measurement using the cos a method, a numerical simulation study was performed comparing with experimental data. The resultant diffraction profiles, which are defined as the distribution of the intensity of diffracted beams in the radial direction from the center of the diffraction ring, were calculated by summing all diffraction profiles which were diffracted at entire depth in the material. The locations of the peak points which were determined using the half value breadth method (or the centroid method) showed dent distribution with respect to the central angle of the diffraction ring even in case of stress free state. Though the stresses calculated by the cos α method showed different values from the initial stresses, the difference between the calculated and initial stresses almost kept constant for different stress level. These misfit stresses can be vanished after correcting original diffraction radius using that in stress free. The results obtained by the present simulation study agreed with those obtained in the previous experiment, which suggests the validity of the method of neutron stress measurement combining both an area detector and the cos α method.

Unified model for partially coherent solitons in logarithmically nonlinear media

We investigate the propagation of a partially coherent beam in a nonlinear medium with logarithmic nonlinearity. We show that all information about the properties of the beam, as well as the condition for formation of incoherent solitons, can be obtained from the evolution equation for the mutual coherence function. The key parameter is the detuning $\ensuremath{\Delta}$ between the effective diffraction radius and the strength of the nonlinearity. Stationary partially coherent solitons exist when $\ensuremath{\Delta}=0$ and the nonlinearity exactly compensates for the spreading due to both diffraction and incoherence. For nonzero detunings the solitons are oscillating in nature, and we find approximate solutions in terms of elliptic functions. Our results establish an elegant equivalence among several different approaches to partially coherent beams in nonlinear media.

Sputtered tungsten films on polyimide, an application for X-ray masks

Among processes of lithography for submicron technologies, X-ray lithography shows the great advantage of high resolution. In the fabrication of X-ray masks, tungsten (W) is one of the alternative absorber materials that can be used instead of gold, but it requires a low stress film with high density. Tungsten films were deposited on Kapton and Upilex polyimides using a triode discharge system. The dependence of thin W films properties on the working argon pressure is reported. X-ray diffraction, AFM and curvature radius measurements were used to characterize the coatings in terms of microstructure and mechanical properties. Average stress and density were found to be significantly dependent on the argon pressure. Microstructural analysis indicated that the grain size and the proportion of metastable phase, β-W, could vary in a wide range. The adhesion of W to polyimide verified with the scratch testing technique seemed very acceptable.

- optical potential at 1016 MeV by inversion: effect of non-eikonal correction[Brief report

We study the effect of the non-eikonal corrections on the determination of heavy-ion optical potential from the diffraction model phase shifts at intermediate energies. Considering the case of a - system at 1016 MeV and the McIntyre parametrization for the diffraction model phase shifts, we find that the effect is significant for the imaginary potential around the diffraction radius.

Three‐dimensional time‐slice migration

Three‐dimensional migration of zero‐offset data using a velocity varying with depth can be performed in one pass using Fourier transforms of time slices. The migration process is carried out entirely in the two‐dimensional spatial Fourier domain. The algorithm consecutively filters and adds time slices of the 3-D data volume in a way that is equivalent to summing energy over the diffraction surface of a point scatterer. The partial energy being distributed along a circle in a time slice is properly added in each summation step. Time‐slice migration is based on an integral solution of the acoustic wave equation known as the “Kirchhoff integral.” The wavelet shape in a 3-D data volume is preserved throughout the entire migration process. The frequency characteristics are maintained by summing weighted differences between time slices instead of summing the time slices themselves. Automatic weighting is achieved by time slicing at equal increments of diffraction radius. Tapering the summation operator reduces effects introduced by limiting the summation window. Time‐slice migration preserves the frequency content of a 3-D data volume during summation in a natural way. Since the migration scheme assumes a constant velocity within the entire time slice, it is a local process in time which migrates a 3-D data volume with a constant velocity or with a velocity which varies with depth. The migration algorithm is applied to numerical and physical model data. This method is especially suitable for a migration of a targeted subset of the 3-D data volume.

Skyrme-force parametrization: Least-squares fit to nuclear ground-state properties.

We investigate systematically the possibilities and the limits of the Skyrme force for reproducing nuclear ground-state properties in a spherical Hartree-Fock calculation. This investigation is performed by means of least-squares fits of the force parameters to the measured binding energy, diffraction radius, and surface width of eight selected nuclei. Particular emphasis is put on the density dependence of the interaction, which turns out to be determined mainly by the surface width. The least-squares fitting procedure yields the best-fit parameters together with uncertainties on them, and it also allows one to estimate the uncertainties of an extrapolation to other fields, e.g., nuclear matter properties. We also study the contribution of random-phase-approximation correlations to the ground-state properties and their influence on the parameters of the effective interaction. Here, we also compare to giant dipole resonance energies.

Dynamics of theO16+O16→S32fusion process

The collective mass and potential, the quantum corrections, and the inertia parameters of the /sup 16/O+ /sup 16/O..-->.. /sup 32/S system are evaluated by means of the quantized adiabatic time dependent Hartree-Fock theory in a three-dimensional coordinate and momentum lattice. The interaction used, consisting of a direct finite range Yukawa force and a density-dependent term, is fitted by static Hartree-Fock calculations (including center-of-mass corrections) to binding energies and elastic electron scattering form factors in the mass region of interest here. The result of the fit is a new interaction, still without exchange terms, which for light nuclei reproduces the binding energies, the diffraction radii, and the surface widths. The sub-barrier fusion cross section for /sup 16/O+ /sup 16/O..-->.. /sup 32/S calculated with this force in quantized adiabatic time dependent Hartree-Fock theory is in excellent agreement with experimental data in contrast to calculations using other interactions. For the evaluation of the fusion cross section above the barrier classical trajectory calculations are performed using a suitable phenomenological friction force. This yields agreement with experimental data up to energies of 40 MeV above the barrier. The quantum corrections and the dependence of the mass parameter on the fragment distance are shown to havemore » an important impact only below the barrier.« less

Possible origin of oscillations and exponential fall-off in high-energy large-angle pp scattering

Recently SCHR~PP and SCHI~I~PP (1) have observed that, contrary to popular belief, high-energy large-angle pp scattering is well described by exponential fall-off in transverse momentum. In fact, the exponential fall-off fits the data over a wider range of energy than the power fall-off based on the consti tuent models, i.e. the quarkpar ton models (3). Furthermore, they found oscillations in the large-angle differential cross-sections with a period 2g/R, where R is roughly the diffraction radius. The appearance of the diffraction radius in the large-angle behavior is somewhat of a surprise, since one usually expects the diffraction effects to be important only for the small-angle scattering. That oscillations governed by the diffraction radius R appear in large-angle pp scattering has also been independently noted by H~DRY (a). SCHR~MPr and SC~REMPP point out that one may conclude from the exponential fall-off and oscillations that a geometrical picture is emerging, and there is no evidence of constituents, i.e. quarks. In this note we, however, take a different view of their analysis. We first suggest that the oscillations are due to a sudden change in absorption from b R seen by the large-angle scattering process in the impact-parameter plane (R = diffraction radius). We then suggest that the exponential fall-off of the large-angle amplitude is due to the existence of a nucleon core, which can be identified as a quark bag. Thus, we conclude that the exponential fall-off can be regarded as evidence for the coherent scattering of quarks in one nucleon off those of the other. We begin with the eikonal representation of the elastic-scattering amplitude