Nuclear Matter Distributions Research Articles

Quadrupole moments and nucleon excitation strengths in even-A Sm isotopes

We present a coherent coupled-channel analysis of 7 MeV neutron and 16 MeV proton elastic and inelastic scattering from 148, 152, 154Sm. The optical potential and nuclear deformation parameters are determined so as to fit not only these elastic and inelastic scattering data but also the low-energy neutron scattering properties and the total cross sections over a wide energy range. This analysis provides evidence of the same excitation strengths for both projectiles in the case of 152, 154Sm, and of a smaller excitation strength for the proton than for the neutron in case of 148Sm. Moreover the quadrupole moments of these deformed optical potentials are in good agreement with those extracted from Coulomb excitation measurements and from nuclear matter distribution calculations.

Proton and neutron density distributions for A=16-58 nuclei

A new method of calculating nuclear charge and matter distributions is described. It uses the sum of the squares of single-particle wavefunctions to give a first approximation and then iterates the proton and neutron densities in a formalism that takes into account the interaction between these densities and between the valence and core nucleons. The single-particle potentials are adjusted to give the experimental centroid separation energies and configuration mixing is included using shell-model occupation probabilities. The method is applied to the core nuclei 16O and 40Ca, to their isotopes and to other nuclei in the sd and pf shells. Detailed comparisons are made with all available experimental data on nuclear charge and matter distributions.

Optical potential for nucleus-nucleus systems

The electrostatic and nuclear parts of the nucleus-nucleus optical potential are constructed using the double folding model by taking into account nuclear charge and matter distributions, along with a suitably chosen effective nucleon-nucleon interaction. For nuclei with uniform charge distributions, a closed form expression for the electrostatic potential has been obtained earlier. This is found to be a very good approximation to the electrostatic potential corresponding to commonly known realistic charge distributions. Optical model calculations for the $^{16}\mathrm{O}$-$^{16}\mathrm{O}$ system using this improved electrostatic potential are reported which provide a slightly better fit to elastic scattering data. The six dimensional integral for the nuclear part of the optical potential is reduced to a three dimensional integral for the case of spherically sysmmetric density distributions and this leads to substantial reduction in computations. In contrast with the electrostatic part, it is found that the nuclear part of the optical potential is quite sensitive to the shape of the density distributions. The optical potential constructed by the "double folding" method with only a single adjustable parameter corresponding to the strength of the imaginary part, is used to fit successfully $^{12}\mathrm{C}$-$^{12}\mathrm{C}$ scattering data at ${E}_{\mathrm{lab}}=102.1$ MeV.NUCLEAR REACTIONS Optical model for nucleus-nucleus system, double folding model, realistic electrostatic potentials.

Critical radii for alpha-particle elastic scattering

The critical radii obtained from analyses of the elastic scattering of alpha-particles by a range of nuclei are found to be closely connected with the nuclear matter distributions obtained by summing the squares of single-particle wavefunctions. This provides a method of calculating the details of the optical potentials from the structure of the target nucleus.

Nuclear Size Comparison of Even Titanium Isotopes Using 140-MeVα-Particle Scattering

Systematic variations in nuclear-matter distributions have been determined by analyzing the measured elastic scattering of 140-MeV alpha particles from $^{46,48,50}\mathrm{Ti}$. The "unique" optical potentials obtained ($\frac{{J}_{\mathrm{R}}}{4A}\ensuremath{\simeq}300$ MeV ${\mathrm{fm}}^{3}$, $\frac{{J}_{\mathrm{I}}}{4A}\ensuremath{\simeq}100$ MeV ${\mathrm{fm}}^{3}$) indicate that isotopic differences occur primarily in the strength of the imaginary potential. The rms matter radii increase with $A$, in contrast to those of the charge distributions. The matter-radius results are in agreement with Hartree-Fock calculations.

The systematics of nuclear bound states

A single physically realistic potential that gives a good overall account of the bound single-particle states of a range of nuclei is described. This potential may be used to calculate nuclear charge and matter distributions either on its own or supplemented by other data.

Density oscillations due to pion condensation in pionic atoms

The assumption of the existence of pion condensation leads to the presence of oscillations in the density distribution of nuclear matter. The energy levels of pionic atoms are calculated by using an optical potential which takes into account these density oscillations. In this sense, the electromagnetic interaction plays an insignificant role; however, the energy shift due to the strong interaction depends mainly upon the parameters which we have introduced for the oscillations.

Yields of projectile fragments in hadron-nucleus interactions and the quark structure of hadrons

Multiplicities of secondary hadrons in the fragmentation region of a projectile are computed as a function of the atomic number of the target, A. The calculation is based on the notion of a nucleon (meson) composed of three (two) spatially discrete constituent quarks. For an incident proton, yields of secondary nucleons near x = 2 3 (as well as π − or K + mesons near x = 1 3 ) are determined solely by the magnitude of the quark-nucleon cross section, assumed to be ∼ 1 3 of the inelastic proton-proton cross section, and the distribution of nuclear matter. Our results agree well with the data on A dependence of the p, π − and K + yields in proton-nucleus collisions. Predictions are given for the A dependence of the projectile fragment yields in the cases of π, K and hyperon beams.

Studies of the nuclear inertia in fission and heavy-ion reactions

On the basis of the non-self-consistent cranking model we study some aspects of the nuclear inertia of interest in fission and heavy-ion reactions. First, we consider in the adiabatic limit the inertia for a doubly closed-shell nucleus in a deformed spheroidal harmonic-oscillator single-particle potential plus a small perturbation. When expressed in terms of a coordinate that describes the deformation of the nuclear matter distribution, the inertia for small oscillations about a spherical shape is exactly equal to the incompressible, irrotational value. For large distortions it deviates from the incompressible, irrotational value by up to about ±1 % away from level crossings. Second, in order to study the dependence of the inertia upon a level crossing, we consider in detail two levels of the above system. This is done both in the adiabatic limit and for large collective velocities. At level crossings the adiabatic inertia relative to the deformation of the matter distribution diverges as 1/| ΔV|, where | ΔV| is the magnitude of the perturbation. However, for large collective velocities the contribution to the inertia from a level crossing is less than 4| ΔV| r 2 m, where r m is the collective velocity of the matter distribution. Although we have not considered the effect of large velocities on the remaining levels of the many-body system or the effect of a statistical ensemble of states, some of our results suggest that for high excitation energies and moderately large collective velocities the nuclear inertia approaches approximately the irrotational value.

Density matrix and momentum distribution of helium liquids and nuclear matter

The one-body density matrix $n(r)$ and the associated momentum distribution $n(k)$ (for given spin-isospin projection) are calculated for the ground states of liquid $^{4}\mathrm{He}$, liquid $^{3}\mathrm{He}$, and a simple model of nuclear matter, based on suitable Jastrow wave functions $\ensuremath{\psi}={\ensuremath{\Pi}}_{ilj}f({r}_{0})\ensuremath{\Phi}$. The model function $\ensuremath{\Phi}$ is taken as the ground-state wave function of a system of noninteracting bosons or fermions, as appropriate. The evaluations are performed within the framework of the diagrammatic formalism developed recently by Ristig and Clark. For the most part, the correlations incorporated via the two-body factor $f(r)$ are assumed to be of short range. In the case of $^{4}\mathrm{He}$, the asymptotic value ${\ensuremath{\rho}}^{\ensuremath{-}1}n(r\ensuremath{\rightarrow}\ensuremath{\infty})$, which may be identified with the condensate fraction, is found to be 0.119 at the experimental equilibrium value of the particle density $\ensuremath{\rho}$. In the cases of $^{3}\mathrm{He}$ and nuclear matter, there is no long-range order, i.e., $n(r\ensuremath{\rightarrow}\ensuremath{\infty})=0$. On the other hand, for these Fermi systems the momentum distribution $n(k)$ exhibits a discontinuity ${z}_{{k}_{F}}$ at the Fermi wave number ${k}_{F}$, which measures the strength of the quasiparticle pole. The calculation yields ${z}_{{k}_{F}}=0.350$ for liquid $^{3}\mathrm{He}$ and ${z}_{{k}_{F}}=0.747$ for nuclear matter, at the respective equilibrium densities. The results presented for the helium liquids are of considerable interest in connection with proposed scattering experiments using intense neutron beams.

Nuclear matter distribution inSn112,Sn116, andSn124from the (p,n) reaction

Angular distributions have been obtained for the quasielastic (p, n) reaction on /sup 112/Sn, /sup 116/Sn, and /sup 124/Sn at a proton energy of 22.8 MeV. Data are of sufficient precision to reveal differences in both the shape and magnitude of the angular distributions. Analysis in terms of a nuclear matter model indicates that the data are consistent with an increase in the ratio of the neutron to proton root-mean-square radius with increase in neutron excess. (AIP)

Application of the Strutinsky method to the rotational motion of 20Ne

The cranking model rotational behavior of 20Ne obtained from the Strutinsky method is compared with that obtained from the analytical solution for the cranked deformed harmonic oscillator potential. It is found that the agreement is excellent except in the vicinity of the gs-band cut-off where the Strutinsky results are somewhat sensitive to the behavior of the liquid drop under rotation. The selfconsistency condition on the shapes of the equipotentials and the nuclear matter distribution is also found to have little influence except in the cut-off region.

Isobaric analog states of the tin and tellurium isotopes studied with the ( 3He, t) reaction at θ L = 0°

The Sn( 3He, t)Sb and Te( 3He, t)I reactions to the g.s. isobaric analog states (IAS) of 112, 114, 116, 117, 118, 119, 120, 122, 124Sn and 124, 125, 126, 128, 130Te have been studied at θ = 0° and E( 3 He) = 45.9 MeV . The Coulomb displacement energies decrease more slowly than A − 1 3 indicating that the rms radius of the nuclear matter distribution increases more slowly than A 1 3 . The total widths of the IAS are 10 to 60 keV and exhibit a pronounced odd- A even- A variation correlated with the excitation and decay energies of the IAS. Estimates of the decay and spreading widths have been obtained, with the latter yielding an isospin-mixing matrix element of about 1 keV. The ( 3He, t) cross sections at 0° appear to be directly related to (p, n) total cross sections for the 0 + IAS. The cross sections for the Sn-Sb and Te-I isotopes, in contrast to accepted expressions, depend in a complicated manner on the neutron excess. A microscopic analysis gives an effective interaction ( μ = 1 fm −1) of V t· t = 11 ± 4 MeV. This is smaller than values obtained from analyses of ( 3He, t) reactions on nuclei with A ⪅ 90, if one assumes a simple direct charge-exchange reaction.

Theory of kaonic atoms

A nonlocal energy-dependent self-consistent kaon-nucleus optical potential is derived for kaonic atoms. Energy level shifts and widths are calculated for several light nuclei, and the results are compared with experiment. The sensitivity of the results to changes in parameters of the nuclear matter distribution is studied. Nonlocality and off-energy-shell effects are examined. The optical potential is derived by means of a Brueckner-type many-body theory with the independent pair approximation for the kaon and the nucleon. The two-body interaction on which the optical potential depends is represented by separable potentials of the Yamaguchi form. Coupled channels ( KN and Σπ) are used for the I = 0 states, which are dominated by the Y 0 ∗ resonance, and only a single channel ( KN ) is used for the I = 1 state. Calculations are carried out in three levels of approximation of the nonlocal energy-dependent optical potential. In no approximation is the potential found to be proportional to the nuclear density. Indeed, the real part of the potential changes sign in the nuclear surface. Sensitivity of the results to variations in the nuclear matter distribution is investigated and found to be on the order of experimental error. Nonlocality and off-energy-shell effects are estimated to be at least as large as this error, so that these effects must be included if one wishes to extract information about the nuclear surface from the existing experimental data. The use of correct nucleon wavefunctions and binding energies is similarly found to be essential in the calculation.

Deformation of56Fe from 104 MeV α-particle scattering

Measured differential cross sections for elastic and inelastic scattering of 104 MeV alpha -particles from 56Fe are analyzed in terms of coupled channels on the basis of symmetric and asymmetric rotator models. The analyses use a semi-microscopic folding procedure and in some cases the conventional extended optical model is also used. Deformation parameters, RMS radii, multipole moments of the nuclear matter distribution, and transition probabilities have been extracted.

Intensities and strong interaction attenuation of kaonic X-rays

Relative intensities of numerous kaonic X-ray transitions have been measured for the elements C,P,S, and C1, from which level widths due to the strong K-nucleus absorption have been determined. From these and earlier published data, optical potential parameters have been derived and possible consequences on the nuclear matter distribution are discussed.

Real Potential in Alpha-Nucleus Scattering

Measurements of $\ensuremath{\alpha}$-nucleus elastic differential-cross-section data at 27 MeV from $^{45}\mathrm{Sc}$, $^{65}\mathrm{Cu}$, $^{90}\mathrm{Zr}$, and $^{122}\mathrm{Sn}$, over the angular range 15 (\ifmmode\times\else\texttimes\fi{}2.5) to 175\ifmmode^\circ\else\textdegree\fi{} are reported. The data are analyzed using both a standard six-parameter optical model and a folding model. The latter uses an effective nucleon-$\ensuremath{\alpha}$ interaction obtained from a global analysis of free nucleon-$\ensuremath{\alpha}$ scattering data below 12 MeV incident nucleon energy which is folded with a nuclear matter distribution to obtain the real part of the $\ensuremath{\alpha}$-nucleus potential. The fits with the folding model are improved appreciably over those obtained using the standard optical model and are sensitive to the choice of matter parameters. The matter mean-square radii deduced are in good agreement with other estimates.

An analysis of 104 MeV α-scattering from 24Mg using a folded collective model

Using a folding method in which the spherical and non-spherical components of the optical potential are calculated consistently, it is shown that it is possible to get a simultaneous fit to the elastic and inelastic scattering of 104 MeV α-particles from 24Mg. The deformation of the underlying nuclear matter distribution is much greater than that of the phenomenological optical potential.

Nuclear matter distributions for 16,17,18O from a microscopic analysis of elastic proton scattering

An analysis of 65 MeV elastic proton scattering by 16,17,18O has been made in terms of a reformulated optical model. Matter distributions for 17O and 18O have been obtained relative to 16O. The results for the rms matter radii are R 17− R 16 = 0.04±0.03 fm and R 18− R 16 = 0.35± 0.07 fm.

Study of Some Properties of Tin Isotopes Through High-Energy Alpha-Particle Scattering

Angular distributions of 166-MeV $\ensuremath{\alpha}$ particles scattered from the Sn isotopes 116, 118, 120, and 124 have been measured. An analysis of the elastic scattering data was performed using an optical potential calculated from an effective interaction between the $\ensuremath{\alpha}$ particle and a nuclear-matter distribution. The variation of nuclear sizes among the isotopes is considered. Assuming charge densities derived from the results of elastic electron scattering and/or mesonic atom data, values of neutron-matter radii for the tin isotopes are deduced which agree well with theoretical values. An analysis of the inelastic $\ensuremath{\alpha}$-particle scattering to the first ${2}^{+}$ and ${3}^{\ensuremath{-}}$ levels for each isotope was effected using the optical potentials calculated from the elastic scattering, the distorted-wave Born-approximation, and quasiparticle wave functions for the excited states. Electromagnetic transition rates and form factors for electron scattering were calculated with the same wave functions, and a comparison of the calculations to the experimental results provides information about the proton and neutron configurations in these isotopes.