Nuclear compression moduli and density-dependent forces

Density-dependent zero-range forces of the form of the modified delta interaction (MDI) are generalized (MDI3, MDI4) in order to yield reasonable values of the compression modulus in nuclear matter (K_N = 200 MeV). This low value can be fitted by introducing two terms with different density dependence in the force. The four free parameters of MDI3 are adjusted to reproduce the nuclear matter values of the binding energy, density and compression modulus, and to fulfil the condition that the total energy of ¹⁶O in harmonic oscillator wave functions has a minimum at the oscillator length b = 1.75 fm, corresponding to the correct rms radius. MDI4 contains in addition a two-body spin-orbit interaction. The five parameters of MDI4 are fitted to the above three nuclear matter data and by requiring that Hartree-Fock (HF) calculations in ²⁰⁸Pb yield the experimental charge rms radius and reasonable values of certain single-particle spin-orbit splittings. The quality of MDI4 is checked by comparing calculated rms radii, binding energies, and elastic electron scattering cross sections with available experimental data for doubly closed shell nuclei. As a test the energy levels and the nuclear monopole polarization of muonic ²⁰⁸Pb are calculated self-consistently yielding impressive agreement with experiment.     

Thick skin in neutron/proton-rich sodium isotopes

Nucleon (both neutron and proton) density distributions of the chain of sodium isotopes are calculated using a semi-phenomenological model of nuclear density which incorporates correctly the asymptotic behaviour and the behaviour near the centre. The experimental charge root-mean-square radii and the single neutron and proton separation energies, required as input, are used. The calculated interaction cross-sections using these densities in the Glauber model agree well with the experiment. The calculated neutron rms radii r _n and the nuclear skin thickness ( r _n - r _p) closely agree with the corresponding experimental values and also are consistent with the Relativistic Hartree-Bogoliubov (RHB) calculations. Received: 24 April 2001 / Accepted: 28 June 2001     

Symmetry and surface energy coefficients with an effective interaction

Symmetry energy coefficients in the first and second order have been estimated by using an effective interaction developed in a previous paper. The rearrangement contribution to the first-order symmetry energy coefficient has also been estimated. The effect of neutron excess on the first- and second-order single particle energies of nuclear matter have been studied by determining the isotopic spin dependent part of the single particle potential. The 1s state single particle energies of²⁰⁸Pb and¹⁶O have been estimated from the single particle energy of nuclear matter by incorporating the values of the isotopic spin dependent part of the single particle potential. The surface energy coefficient has been determined by using the semi-infinite model in the Thomas Fermi approach.Financial assistance from University Grants Commission, New Delhi, is gratefully acknowledged. The authors are also grateful to Professor K. C. Pande and Dr. S. P. Pandya for helpful suggestions.     

Nuclear dissipation,fission probability and neutron multiplicity prior to fission

We discuss the effects of nuclear dissipation on fission probabilities that are characteristic of a diffusion model of the fission process. Reproducing the experimental fission probabilities at low excitation energies fixes the ratioa _f/a_n of the level density parameters for a given strength of the reduced dissipation coefficientβ. These low energy constraints ona _f/a_n andβ balance the effects of transients on neutron multiplicities prior to fission at higher excitation energies. For the competitive decay of¹⁵⁸Er formed in the reaction¹⁶O+¹⁴²Nd at 207 MeV we show that dueto transients only the multiplicity of pre-fission neutrons is enhanced with respect to the prediction of the statistical model in a manner consistent with our earlier general analysis.     

Off-shell effects in O

We calculate the binding energy of ¹⁶O for a set of phase-shift-equivalent potentials previously studied in nuclear matter. Off-shell variations of up to 2.8 MeV per particle occur compared with about a 10 MeV per particle variation in nuclear matter. As in nuclear matter calculations a nearly linear relation exists between the variations in the binding-energy results and the wound integral k. We compare the ¹⁶O results with a nuclear matter calculation at the “equivalent” nuclear matter density of k_F = 1.13 fm⁻¹. This “equivalent” density reflects the fact that ¹⁶O has a surface and hence a lower average density than nuclear matter. The ¹⁶O and nuclear matter off-shell variations are comparable once one takes into account the lower average density of ¹⁶O and the suppression of the relative D-wave interaction — also a surface effect. We present a method of computing the correlated wave functions of finite nuclear systems and display such wave functions for ¹⁶O. The correlated wave functions of ¹⁶O and of nuclear matter are strikingly similar for all of the potentials studied.     

Nuclear-rainbow scattering and nucleus-nucleus potentials at short distances

The elastic scattering of strongly bound nuclei at energies of 10 to 70 MeV per nucleon shows the phenomenon of “rainbow scattering.” A nuclear rainbow appears because of deflection to negative angles. This process involves a strong overlap of nuclear densities, with values of up to twice the saturation density of nuclear matter. The ¹⁶O+¹⁶O system is studied with a high precision over a wide energy range from 7 to 70 MeV per nucleon in several laboratories. Primary Airy maxima and higher order Airy structures are observed. At all energies, excellent fits are obtained with deep potentials as deduced from the double-folding model involving a nucleon-nucleon interaction weakly dependent on the density. It is shown that Pauli blocking expected at low energies is strongly reduced if the local momenta are calculated self-consistently. Systematics confirms a refractive origin of large-angle scattering, at low energies inclusive. Thus, nuclear-rainbow scattering yields unique information about the properties of cold nuclear matter at higher densities.     

Coulomb energies and nuclear radii in the energy density formalism

The relation between Coulomb displacement energies,ΔE _c , andΔr=r _n -r _p , the difference between the rms radii of neutrons and protons in nuclei, is investigated within the energy density formalism (EDF). The variational equation, obtained by minimizing the Coulomb plus symmetry energies, is solved assuming the symmetry interaction is a simple functional of the local nuclear matter density. Varying parameters of the model, rather unique relation betweenΔE _c andΔr is obtained (within ±50 keV).ΔE _c isindependent ofr _ex, the rms radius of the excess neutrons distribution. Using the experimental values ofr _p and adjusting the model to reproduce the recent data onΔr (Δr∽~0.05 fm for⁴⁸Ca and²⁰⁸Pb), which are significantly smaller than those obtained from current Hartree-Fock calculations, the calculatedΔE _c agree with the experimental results. Using the value ofΔr～0.05 fm and the experimental values ofr _ex, a small compression (<0.02 fm) of the proton core in the analogue state relative to its parent state emerges, thus contributing an additional electrostatic term to the Coulomb displacement energy. The size of this relative core-compression effect depends on the values assumed forΔr andr _ex, it increases with the decreasing ofΔr and the increasing ofr _ex. IfΔr～0.05 fm the effect is large enough to remove the long standing Coulomb energy anomaly. The main result of the present work is the correlation betweenΔE _c andΔr, suggesting that the difficulties of current Hartree-Fock calculations in reproducing isotope shifts ofr _p , the small value ofr _n ?r _p and the values ofΔE _c may all be different manifestations of some missing residualp n effective interaction.     

Coulomb displacement energies and decay widths of isobaric analog states from Sm(3He,t) at θ = 0°

The ^{144,147,148,149,150,154}Sm(³He, t)^{144,147,148,149,150,152,154}Eu reactions leading to ground-state isobaric analog states (IAS) have been studied at θ = 0° and E(3He) = 45.9 MeV. The Coulomb displacement energies decrease more rapidly than $\text{A}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$. Approximately 110 keV of the total decrease of about 470 keV from A = 144 (spherical) to A = 154 (deformed) can be ascribed to deformation. No discontinuity is apparent at the transition from spherical to deformed shapes at N = 88–90. This is attributed to two effects: (i) rms radii increase with static deformations and with dynamical vibrations; (ii) Coulomb displacement energies depend on rms charge radii and on the rms radii of the neutron excess. The data suggest neutron deformations greater than proton deformations for A = 148 and 150 but smaller for A = 152 and 154. The IAS widths increase from ～30 keV to ～90 keV and can be attributed to mixing with the (T₀?1) component of a proposed isovector giant monopole resonance.     

Proton radii determinations from the ratio of π+ elastic scattering from 11B and 12C

A new method using low energy π⁺ elastic scattering is employed to determine the proton matter distribution of a nucleus relative to an adjacent nucleus differing by one proton. The difference in the rms charge radii of ¹²C and ¹¹B was determined to be 0.072 ± 0.021 fm with π⁺ beam energies of 38.6 and 47.7 MeV.     

Magnetic moments of <Superscript>33</Superscript>Mg in the time-odd relativistic mean field approach

The configuration-fixed deformation constrained relativistic mean field approach with time-odd component has been applied to investigate the ground state properties of 33Mg with effective interaction PK1.The ground state of 33Mg has been found to be prolate deformed,β2=0.23,with the odd neutron in 1/2[330] orbital and the energy -251.85 MeV which is close to the data -252.06 MeV.The magnetic moment -0.9134 μN is obtained with the effective electromagnetic current which well reproduces the data -0.7456 μN se...     

Some characteristics of nuclear densities

We propose a simple expression for nuclear densities, which brings out the following important nuclear properties: (i) shell effects in proton and neutron central densities, (ii) approximate global constancy of neutron central densities, (iii) approximate constancy ofRN ^?1/3 and R_pZ^?1/3 whereR is the nuclear half-density radius andR _p is the rms radius of the proton density, (iv) larger surface thickness and rms radius for neutron density as compared to those for proton density.     

Elastische Elektronenstreuung an14N,15N,16O und18O bei kleiner Impulsübertragung

Elastic electron scattering cross sections of¹⁴N and¹⁶O have been measured relative to the proton and of¹⁵N and¹⁸O relative to the lighter isotope (¹⁴N,¹⁶O resp.) using gas targets. The momentum transfer ranged from 0.22 to 0.48 fm^?1. The data were analyzed by phase shift calculations assuming harmonic oscillator shell model charge distributions. The following rms charge radii have been deduced: R_m(¹⁴N)=2.540±0.020 fm R_m(¹⁵N)=2.580±0.026 fm R_m(¹⁶O)=2.718±0.021 fm R_m(¹⁸O)=2.789±0.027 fm. The errors include statistical and systematic uncertainties and an estimate of effects due to the choice of the model. The radius differences of the isotopes are smaller than the values predicted by anA ^1/3 relation     

Untersuchung von Kernniveaus des16O bis 14 MeV Anregungsenergie durch unelastische Elektronenstreuung

Using electrons with up to 60 MeV energy, ten transitions in¹⁶O have been studied: twoE0 (6.05 and 12.05 MeV), oneE1 (13.10 MeV), fourE2 (6.92, 9.85, 11.52 and 13.1 MeV), twoM2 (12.53 and 12.96 MeV) and oneE3 (6.13 MeV). The cross sections measured as a function of momentum transfer have been analyzed to yield transition probabilities to the ground state and transition radii. The results are compared with the theory ofBrown andGreen, and with the particle-hole calculations ofLewis andDeForest. For levels at 11.08 and 13.67 MeV, upper limits for the transition probabilities are given.     

Neutron Skin Thickness of Nuclei and Effective Nucleon-Nucleon Interactions

The Skyrme energy density functional is applied to study the ground state properties of a series of finite nuclei. The charge rms radii, neutron rms radii, and the neutron skin thickness for some nuclei are calculated and compared with the experimental data. The constraint on the effective interactions, especially, the density dependence of the isospin-dependent. part of Skyrme interactions is extracted by the data of neutron skin thicknesses of ^208Pb and isotopes of Sn.     

Prompt mass distributions in fission

A previously developed structure-sensitive statistical model, incorporating the nuclear structure and deformation effects, has been utilised to study the prompt mass distributions in fision and other related characteristics. The principle of detailed balance has been applied to calculate the yield probabilities. The fusion barriers and the penetrabilities of the fragments were estimated from a nuclear plus Coulomb interaction potential. The prompt fragment mass yields were then obtained as a product of three quantities: the product level densities of the conjugates at their most probable excitation energies, the barrier penetrabilities and the isobaric charge yields. The calculations are free from any arbitrary parameters. Comparison with experimental data for²²⁶Ra(p, f) atE _p =11.1MeV,²³²Th(n, f) atE _n =1.0 MeV,²³⁵U(n, f) and²³⁹Pu(n, f) at thermal neutron energies and for the spontaneous fission of²⁵²Cf shows fairly good agreement.     

Messung des totalen Wirkungsquerschnitts von27Al für schnelle Neutronen mit Si(Li)-Detektoren

The total fast neutron cross section of²⁷Al has been determined by transmission measurements using Si(Li)-solid state detectors for neutron spectroscopy. The energies of the bombarding neutrons have been in the range betweenE _n=5,24 andE _n=7,26 MeV. A comparision has been made between the total cross sections obtained in this experiment and values obtained via the time-of-flight technique. At most neutron energies the agreement was within experimental errors quoted for the works being compared.     

Neutron and proton densities in nuclei

A simple expression for nuclear densities which incorporates the correct asymptotic and central behaviour, gives quite accurate values for rms radii and surface thicknesses. It also brings out some important nuclear properties, (i) global constancy of neutron densities, (ii) equality of half-density radiiR for neutrons and protons, (iii) near constancy ofRN ^?1/3, (iv) larger surface thickness and rms radius for neutrons (v) shell effects on charge density.     

On the nuclear interaction potential in the reactions with heavy ions

The interaction potential of heavy ions⁴He,⁶Li,¹²C and¹⁶O is constructed in the folding model. The density distribution of nuclear matter for these nuclei is calculated in the framework of the hyperspherical function method. For the calculation of the folding potentials we have employed the Skyrme nucleon-nucleon forces. The influence of several effects on the results of calculations is studied: the role of the three-body forces of the nucleon-nucleon interaction, dependence of the folding potential on the mass numbers of the colliding nuclei and the possibility of observing the monopole resonance in the ion inelastic scattering. Using our folding potential as a real part of the optical potential we have calculated the differential cross section of elastic scattering of⁶Li from¹²C at laboratory energy of lithium ionsT _L =90.0 MeV. Reasonable agreement with experiment is obtained.     

Skyrme-Hartree-Fock Description of the Nuclear Structure in Ca Isotopes (Ⅰ)Binding Energy and Shell Effects,Radii and Density Distributions

The ground state properties of Ca isotopes far from stability line were systematically studied using the Skyrme Hartree Fock model.The shell effects on the binding energy and two neutron separation energy are discussed.The isospin dependency of the unclear radii and nucleon density distributions and the shell effects on these properties are also studied.It is shown that the neutron magic number affests the width of nuclear surface and the nucleon density distributions beyond the nuclear surface.The change of proton rms radii R_rms with neutron number excess I=(N-Z)/A follows R_rms=3/5(1+αI+βI²)r_pZ^1/3.The effect of the centrifugal potential on the nuclear density in the outer trach of nuclear surface is clearly shown.