Coulomb contributions to nuclear radii and energies

The effect of including the isospin non-scalar Coulomb interaction (also taking into account the finite proton size) in binding energy and rms radius calculations for the closed shell nuclei ⁴He, ¹⁶O and ⁴⁰Ca is discussed in detail. Using the saturating Sussex matrix elements and including 0 + 2?ω excitations it is found that the major Coulomb contribution is in the first order, and that the pure second-order Coulomb contribution to the energy, near the saturation value of the size parameter for each nucleus, is more important than the mixed second-order contribution. Finally, for third order and above one can safely neglect the pure Coulomb contributions compared to the mixed ones which are in turn small compared to the pure nuclear contribution, in that order.     

Coulomb energies and charge asymmetry of nuclear forces

Charge-symmetry-breaking potentials suggested in the literature to resolve the discrepancy between calculated Coulomb energy differences of analog states and the experimental values, are considered in detail. We calculate the contributions of these potentials to the ground state energy differences of the mirror nuclei ³He³H, ¹⁵O¹⁵N, ¹⁷F¹⁷O, ³⁹Ca³⁹K and ⁴¹Sc⁴¹Ca. It turns out, due to the short range character of these symmetry-breaking potentials, that their inclusion may resolve the ³He³H difficulty but not the ⁴¹Sc⁴¹Ca discrepancy.     

Role of excited nuclear states in photoabsorption shadowing at medium energies

The effect of intranuclear motion of nucleons on the photoabsorption shadowing due to the conversion of a γ-ray photon into a vector meson in nuclear matter is analyzed. It is shown within the method of time-correlation functions that, in comparison with the results of the standard approach, the correction to the total photoabsorption cross section due to the nuclear shadowing is determined by two additional factors: the Fermi motion of nucleons in a nucleus and nucleon recoil.     

Heavy-ion induced fission at near-barrier energies

The study of heavy-ion induced fission fragment angular distributions continues to be a source of rich information as regards fission process in general and fission dynamics in particular. Considerable progress has been made towards understanding many features of the fission phenomenon. While some of the new sets of data measured in the last few years have confirmed the theoretical expectations, the others have provided surprises not quite anticipated. In the present review article the emphasis will be mainly on the recent experimental results of heavy-ion-induced fission fragment angular distributions at energies near the fusion barrier, their implications and new puzzles in this area which require not only suitable explanation but also additional measurements.     

Basic results of investigations of scission neutrons in nuclear fission at low excitation energies

To estimate the main characteristics of neutrons emitted shortly before the scission of a fissioning nucleus, various experiments sensitive to the presence of these scission neutrons in thermal-neutron-induced fission of ²³⁵U and spontaneous fission of ²⁵²Cf were performed. The results of the experiments were analyzed within theoretical calculations allowing for various possible neutron-emission mechanisms, including the possibility of the emergence of neutrons from the scission of a nucleus.     

A folded diagram microscopic calculation of nuclear Coulomb displacement energies

We have performed a rather extensive microscopic calculation of the ¹⁷F-¹⁷O Coulomb energy differences ΔE_C. Our main purpose has been to study the effects of (i) folded diagrams, (ii) core polarization and (iii) the effects due to different nucleon-nucleon potentials, different single-particle spectra and different radial wave functions. Using a proton-neutron representation we have included higher-order Coulomb corrections like e.g. the coupling of valence particles to collective vibrations of the core.The inclusion of folded diagrams is very important; it is equivalent to a self-consistent treatment of the Q-box starting energies. It causes a significant suppression of the effect due to core polarization, its contribution to ΔE_C becoming small, about 0.03 MeV. As a consequence of this “self-correcting” behavior our results for ΔE_C are quite stable with respect to the choice of different single-particle spectra. Two rather different nucleon-nucleon potentials, the Reid soft-core potential and a meson exchange potential of the Bonn-Jülich group also lead to quite similar results for ΔE_C. In the case of $\text{J =}\text{5}\text{2}{}^{\mathrm{+}}$, for example, Δ_C calculated with the Reid potential ranges, depending on other assumptions, from 3.36 to 3.43 MeV and from 3.36 to 3.40 MeV with the Bonn-Jülich potential: Both are significantly smaller than the experimental value of 3.54 MeV. Optimizing the radial wave functions in the spirit of a Brueckner-Hartree-Fock theory improves the calculated values of ΔE_C for the $\text{J =}\text{1}\text{2}{}^{\mathrm{+}}$ state but not for $\text{J =}\text{5}\text{2}{}^{\mathrm{+}}\text{or}\text{3}\text{2}{}^{\mathrm{+}}$.We feel that other processes such as the explicit inclusion of core deformation or mesonic degrees of freedom or both are needed to explain the Coulomb displacement energies.     

The determination of the Coulomb energy of a nuclear system at the scission point upon fission

The dependence of the Coulomb energy of deformed nuclei on the shape and the nuclear matter density distribution is considered. For an ellipsoidal shape of the nucleus and a homogeneous distribution of nuclear matter, the result was obtained in a simple analytical form convenient for further use. Separate attention was paid to the consideration of a Fermi-like distribution of nuclear matter using different shapes of the nucleus that reflect several different kinds of collective motions. After these considerations, the dependence of the Coulomb energy of the fission products at the scission point of binary fission reactions on configurations, shapes and nuclear matter distributions was investigated. Calculation of the dependence of the shape of the nuclei deformation parameters at the scission point from their total kinetic energies was made in detail. Finally, the influence of shell effects on the mass yield of fission products is discussed. The text was submitted by the authors in English.     

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.     

Shell-model Coulomb energies

Analytical expressions have been derived for the total Coulomb energies, for isoscalar, isovector and isotensor Coulomb energies, and for Coulomb energy differences of nuclei with protons and neutrons in equivalent and in different major shell regions. The approach is related to earlier theoretical treatments of Carlson and Talmi and of Hecht. A simple A-dependence is assumed to account for the increase in radius. Introducing average matrix elements for the Coulomb interactions between valence-valence, valence-core, and core-core protons in each of the six major shell regions as adjustable parameters, a global fit with 21 parameters to 288 experimental Coulomb displacement energies with 4 < A < 240 gives good agreement with a standard deviation of σ = 45 keV. The interaction energies display discontinuities particularly at major proton shell crossings, and they generally decrease within shells less rapidly than $\text{A}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$. The average decrease, however, follows approximately an $\text{A}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}$ dependence which explains the success of Coulomb energy equations based on liquid-drop models.     

Puzzling features of heavy-ion fission at sub-barrier energies

Elastic and inelastic scatterings cross-sections of¹²C+¹²C in the energy range 11–24 MeV/n have been analyzed within the coupled channels framework including appropriate symmetrization. The set obtained withr ₀=1.01 fm,r _I =0.86 fm for seven incident energies gave the best agreement with the data and could be, therefore, considered as really significant when compared with other sets. Coupled channels calculations have been made to fit the data and it has been found that they require a reduction of the imaginary potential. The total reaction cross-sectionσ _R (E) determined from this analysis is in good agreement with a direct and recent measurement of σ _R . Quadrupole deformation parameterβ ₂ has been extracted and compared with those obtained for proton and alpha scatterings. Some small energy dependence of β₂ over this energy range has been found, ¦β ₂¦ ～0.39–0.47. Finally, these calculations further confirm that β₂ depends on the probe used in the various scattering experiments. For composite projectile such asα or¹²C,β ₂ is clearly smaller than that one for nucleon as projectile.     

Puzzling features of heavy-ion fission at sub-barrier energies

The heavy-ion induced fission fragment angular distributions measured for systems with Th, U and Np as targets have revealed “anomalous” values of anisotropies at energiesE≤V _B (fusion barrier) and this feature is observed to be independent of the entrance channel mass-asymmetry. While this puzzling feature is exhibited by the deformed targets like Th, U and Np, most of the fission data measured for the spherical targets like Pb and Bi can be satisfactorily explained using the standard saddle point statistical model with moderate correction for pre-fission neutron emission. Plausible reasons for this anomalous behaviour are explored.     

Pion scattering from a bound nucleon at medium energies

In order to examine the validity of the impulse approximation for pion-nucleus scattering in the 33-resonance energy region, we consider pion-scattering from a “nucleus” which consists of a single nucleon bound in a harmonic oscillator potential. A separable πN interaction is assumed. The oscillator parameter is chosen such that the nuclear sizes are fitted for ⁴He ～ ¹⁶O. The binding effect is found to result in a downward shift of the resonance energy (by about 20 MeV), and an increase (by 50 ～ 70%) of the total cross section near the resonance. The angular distribution is also strongly modified. In connection with the binding effect, the importance of a careful treatment of nucleon recoil is emphasized. It is pointed out that the closure approximation which is often used to sum over intermediate nuclear states leads to very misleading results. The effect of the Pauli principle is also examined by excluding some intermediate states.     

Coulomb energies from quark exchanges between nucleons

The Coulomb exchange kernel between two-nucleon clusters of quarks is calculated by using the resonating-group method. The resulting Coulomb quark-exchange energy increases the Coulomb energy difference in the A = 3 mirror nuclei (by as much as 4% for large nucleons), but decreases it (by ?2 to ?4%) in p-shell nuclei. Other quark effects in charge-dependent (and asymmetric) interactions are briefly discussed.     

Nuclide yields of light fission products from thermal-neutron induced fission of 233U at different kinetic energies

The yields of light fission products from thermal-neutron induced fission of ²³³U are measured as a function of their mass A, their nuclear charge Z, their kinetic energy E and their ionic charge state q at the recoil spectrometer Lohengrin of the Institut Laue-Langevin in Grenoble. The mass yields are determined by intercepting the fragments with an ionization chamber of high energy resolution positioned at the focal plane of the spectrometer. The nuclear charges and their yields are determined with the same ionization chamber by measuring the residual energy of fission products, selected monoenergetically by Lohengrin, behind a passive absorber made of parylene-C. The nuclear charge resolution enabled by this detector device is considerably improved to Z/dZ = 58. The nuclear charge and mass distributions summed over all ionic charge states are listed within the mass range 79 ⩽ A ⩽ 106 at 6 energies: E = 85.34, 90.41, 95.46, 100.50, 105.55 and 110.55 MeV. The energy-integrated nuclear charge and mass yields are also given. The isotonic and isotopic yields are shown. An odd-even effect in the yields is found for the protons as well as for the neutrons at all kinetic energies. The yield weighted total odd-even effect for the protons is found to be (22.1 ± 2.1)%, for the neutrons (5.4 ± 1.7)%. An odd-even effect for the protons in the mean kinetic energy is also observed. The displacement of the mean isobaric nuclear charges from the unchanged charge-density values and the variances of the isobaric nuclear-charge distributions reveal fine structures in their mass dependences.     

Manifestation of the nuclear viscosity effects in induced fission reactions at excitation energies below 30 MeV

A large set of experimental observables for the ²³²Th(α, xnf)reaction was analyzed theoretically within the dynamic-statistical approach, making it possible to interconsistently consider the manifestation of nuclear viscosity, the double-humped structure of the fission barrier, and the phenomenon of shell effect damping with temperature. Analyses were performed for the energy dependence of the finite lifetime effect in the investigated reaction, obtained using the crystal blocking technique; the fission probability isotopes produced in this reaction during the development of a neutron emission cascade; and the anisotropy of angular distributions of fission fragments. It is shown that this analysis allows us to obtain information regarding nuclear viscosity and its energy dependence at relatively low excitation energies (<30 MeV).     

Nuclear structure studies at medium energies

Reactions initiated by electrons, nucleons, pions and kaons in the energy region 100–600 mev are discussed. It is shown that valuable information on nuclear structure can be obtained from these reactions, although precise determination of the nucleon momentum distribution in nuclei requires a combined study of a range of reactions and elastic scattering. It is suggested that attempts to determine the nucelon pair correlation function may be premature.     

Semiclassical description of reactions at energies near the Coulomb barrier

The modified semiclassical approximation of Coulomb matrix elements is extended to include effects of distorting nuclear potentials in the scattering wave functions. The applicability and efficiency of the proposed semiclassical method are discussed. The advantages of this approximation are shown for a typical heavy-ion transfer reaction.     

Scattering of α-particles at medium energies

Elastic scattering of α-particles is treated within the context of high energy multiple diffraction theory. Terms up to fourth order in the expansion of the optical phase-shift function are retained (when necessary). The theoretical predictions are compared with the recent 5.05 and 4.30 GeV/c α-α measurements. The effects of short-range dynamical correlations are discussed.