S.A. Moszkowski

Nuclear forces and the properties of nuclear matter

Abstract The application of the Brueckner theory to the nuclear many-body problem can be greatly simplified if one separates the two-nucleon interaction (for any given state and relative momentum) into a short range part vs and a long range part vl. The cut is made such that vs alone gives no phase shift for free particle scattering. If the separation is made in this way, then the presence of nuclear matter, as manifested by the Pauli principle, has only a small effect on the two-particle wave function at short distances. On the other hand, the Pauli principle drastically reduces the induced correlations due to vl. In fact, vl is the first approximation to the effective interaction in nuclear matter. This procedure permits one to systematically expand the effective interaction in terms of ts, the reaction operator for free particles caused by vs alone. The contribution of all first- and second-order terms to the binding energy of extended nuclear matter has been calculated numerically. Most of the work was done using a central spin-independent two-body interaction containing a hard core of radius 0.4 fermi followed by an attractive exponential well with parameters chosen to give infinite scattering length and intrinsic range 2.5 fermis. If the potential is assumed to act only in S-states, we obtain a minimum energy of −12 Mev per particle at a density corresponding to a nuclear radius 1.0 A 1 3 fermis. The largest of the second-order terms contributes only 5.4 Mev to the energy per particle at this density (as compared to −42 Mev for the first-order term) and the other second-order terms much less. Thus our expansion appears to converge fairly rapidly. The effect of changes in the well parameters has been investigated. Also, our method was applied to interactions used by Gomes et al., Brueckner and Gammel, and Tagami, and our results compared with some of those obtained by the above authors.

Nuclear matter calculations using the separation method

Abstract The convergence of the separation method for calculating the two-body reaction matrix is established for central interactions by comparing the separation method results to those obtained by solving the integral equation numerically. A spin-dependent central well which approximately satisfies two-body data as well as giving reasonable nuclear saturation properties is presented.

The tensor interaction and nuclear matter

Abstract The separation method of Moszkowski-Scott has been applied to the calculation of the properties of nuclear matter using two different nucleon-nucleon potentials, both in reasonable agreement with two-body data. Calculations with the potential of Brueckner-Gammel gave A E = −14.2 Mev at an equilibrium densitty corresponding to k f = 1.5 f −1 . The difference from the results of B and G may be caused by slow convergence of the series (especially in the triplet-even state where the tensor interaction has a large second order contribution). An important factor in obtaining nuclear saturation is shown to be the weakening of tensor interaction effects by the Fermi sea. Evidence for this may also be seen from the results obtained using a different two nucleon potential which, however, still gives good fits to two body data. The potential chosen has a much weaker tensor component and shows no sign of saturation at normal densities (at k f = 1.5 f −1 , A E = −23.4 Mev ). The difference in the two results appears to be much larger than can be accounted for either by higher order terms or by differences in the phase shift approximation to the reaction matrix.

Incompressibility in finite nuclei and nuclear matter

The incompressibility (compression modulus) K0 of infinite symmetric nuclear matter at saturation density has become one of the major constraints on mean-field models of nuclear many-body systems as well as of models of high density matter in astrophysical objects and heavy-ion collisions. It is usually extracted from data on the Giant Monopole Resonance (GMR) or calculated using theoretical models. We present a comprehensive re-analysis of recent data on GMR energies in even-even Sn and Cd and earlier data on 58 ≤ A ≤ 208 nuclei. The incompressibility of finite nuclei KA is calculated from experimental GMR energies and expressed in terms of A −1/3 and the asymmetry parameter β = (N-Z)/A as a leptodermous expansion with volume, surface, isospin and Coulomb coefficients Kvol, Ksurf , Kτ and Kcoul. Only data consistent with the scaling approximation, leading to a fast converging leptodermous expansion, with negligible higher-order-term contributions to KA, were used in the present analysis. Assuming that the volume coefficient Kvol is identified with K0, the Kcoul = -(5.2 ± 0.7) MeV and the contribution from the curvature term KcurvA −2/3 in the expansion is neglected, compelling evidence is found for K0 to be in the range 250 < K0 < 315 MeV, the ratio of the surface and volume coefficients c = Ksurf/Kvol to be between -2.4 and -1.6 and Kτ between -840 and -350 MeV. In addition, estimation of the volume and surface parts of the isospin coefficient Kτ , Kτ,v and Kτ,s, is presented. We show that the generally accepted value of K0 = (240 ± 20) MeV can be obtained from the fits provided c ∼ -1, as predicted by the majority of mean-field models. However, the fits are significantly improved if c is allowed to vary, leading to a range of K0, extended to higher values. The results demonstrate the importance of nuclear surface properties in determination of K0 from fits to the leptodermous expansion of KA . A self-consistent simple (toy) model has been developed, which shows that the density dependence of the surface diffuseness of a vibrating nucleus plays a major role in determination of the ratio Ksurf/Kvol and yields predictions consistent with our findings.

Models of Nuclear Structure

Nuclear physics is a young science dating back only to the discovery of radioactivity in 1896. The development of nuclear physics can be divided into two stages. During the first stage (1896–1932) many fundamental facts about atomic nuclei were discovered. Thus Lord Rutherford’s experiments showed that the nucleus occupies only a very small fraction of the atomic volume. Also, isotopes were discovered for the first time, the field of mass spectroscopy was developed, and a few nuclear reactions were induced in the laboratory.

Skyrme-Landau parameterization of effective interactions (I). Hartree-Fock ground states

Abstract An extended Skyrme-Landau interaction - SL1, which includes velocity-dependent three-body forces and a tensor force is developed. Unlike the effective interactions with density-dependent two-body forces, this form of the interaction yields, in finite nuclei, an anti-symmetric particle-particle interaction from the particle-hole interaction with the phonon-induced interaction included. The interaction parameters are determined by the better known Landau-Migdal parameters in nuclear matter and other physical quantities like the surface energy and the dipole sum rule. Due to the fact that sufficient degrees of freedom are introduced, previous problems with the high compression modulus K ∞ and spin instability, which plagued the earlier Skyrme interactions are thus removed. We present results on the Hartree-Fock ground states of spherical nuclei : 16 O, 40 Ca, 48 Ca, 90 Zr and 208 Pb. The fitted binding energies, the radii and the single-particle energies are all comparable to those of the earlier Skyrme interactions. Comparison with experiments is also made. The self-consistent RPA calculation of the electric and magnetic resonances, Fermi and Gamow-Teller transitions will be presented in the sequel of the present paper.

Nonlinear mean field theory for nuclear matter and surface properties

Abstract Nuclear matter properties are studied in a nonlinear relativistic mean field theory. We determine the parameters of the model from bulk properties of symmetric nuclear matter and a reasonable value of the effective mass. In this work, we stress the nonrelativistic limit of the theory which is essentially equivalent to a Skyrme hamiltonian, and we show that most of the results can be obtained, to a good approximation, analytically. The strength of the required parameters is determined from the binding energy and density of nuclear matter and the effective nucleon mass. For realistic values of the parameters, the nonrelativistic approximation turns out to be quite satisfactory. Using reasonable values of the parameters, we can account for other key properties of nuclei, such as the spin-orbit coupling, surface energy, and diffuseness of the nuclear surface. Also the energy dependence of the nucleon-nucleus optical model is accounted for reasonably well except near the Fermi surface. It is found, in agreement with empirical results, that the Landau parameter F 0 is quite small in normal nuclear matter. Both density dependence and momentum dependence of the NN interaction, but especially the former, are important for nuclear saturation. The required scalar and vector coupling constants agree fairly well with those obtained from analyses of NN scattering phase shifts with one-boson-exchange models. The mean field theory provides a semiquantitative justification for the weak Skyrme interaction in odd states. The strength of the required nonlinear term is roughly consistent with that derived using a new version of the chiral mean field theory in which the vector mass as well as the nucleon mass is generated by the σ-field.

Binding three or four bosons without bound subsystems

We estimate the ratio R = g3/g2 of the critical coupling constants g2 andg3 which are required to achieve binding of 2 or 3 bosons, respectivel y, with a short-range interaction, and examine how this ratio depends on the shape of th potential. Simple monotonous potentials give R ≃ 0.8. A wide repulsive core pushes this ratio close to R = 1. On the other hand, for an attractive well protected by an ext ernal repulsive barrier, the ratio approaches the rigorous lower bound R = 2/3. We also present results forN = 4 bosons, sketch the extension to N > 4, and discuss various consequences.

Three-alpha structures in 12C

Abstract We search for three-alpha resonances in 12 C by using the complex scaling method in a microscopic cluster model. All experimentally known low-lying natural-parity states of 12 are localized. For the first time we unambiguously show in a microscopic model that the 0 2 + state in 12 C, which plays an important role in stellar nucleosynthesis, is a genuine three-alpha resonance.

The nucleus-nucleus interaction potential using density-dependent delta interaction

Abstract The interaction potential between two nuclei has been calculated by using a generalized folding model. The direct and the exchange terms are computed by folding in Skyrme interaction with the nuclear density distributions and the density matrices of the two nuclei, respectively. A new definition of the one-body optical potential is also suggested. The results agree quite well with standard phenomenology.