Reid Soft-core Interaction Research Articles

Singlet pairing gaps of neutrons and protons in hyperonic neutron stars**Supported by the National Natural Science Foundation of China.

The 1S0 nucleonic superfluids are investigated within the relativistic mean-field model and Bardeen-Cooper-Schrieffer theory in hyperonic neutron stars. The 1S0 pairing gaps of neutrons and protons are calculated based on the Reid soft-core interaction as the nucleon-nucleon interaction. In particular, we have studied the influence of degrees of freedom for hyperons on the 1S0 nucleonic pairing gap in neutron star matter. It is found that the appearance of hyperons has little impact on the baryonic density range and the size of the 1S0 neutronic pairing gap; the 1S0 protonic pairing gap also decreases slightly in this region where ρB = 0.0–0.393 fm−3. However, if baryonic density becomes greater than 0.393 fm−3, the 1S0 protonic pairing gap obviously increases. In addition, the possible range for a protonic superfluid is obviously enlarged due to the presence of hyperons. In our results, the hyperons change the 1S0 protonic pairing gap, which must change the cooling properties of neutron stars.

The LOCV nuclear matter calculation and the magnetic susceptibility of neutron matter

The lowest order constrained variational (LOCV) calculation for asymmetrical nuclear matter is revisited by using Reid93, Av 18 , etc., potentials. It is shown that our previous two-body cluster energy formulations need some modifications when they switch from the old Reid soft core interaction to the Argonne type potentials which have the operator basis structure. The modified equations increase the saturation energy of nuclear matter by ∼ 5 MeV for Av 18 interaction and it is in agreement with that of Reid68 potential. Similar to our 1979 work, the new formulation is used to calculate the magnetic susceptibility of neutron matter (MSNM). The predicted MSNM is in agreement with the others methods such as the Brueckner–Hartree–Fock (BHF) for the Av 14 and Av 18 potentials, but as before, it is considerably larger than the similar calculation (BHF) performed by Haensel with Reid68 interaction. While no evidence of ferromagnetic phase is found with these potentials, the magnetic phase transition is observed for Reid93 interaction.

Relative wavefunctions, vertex functions and coupling constants for the virtual decay of 4He

The d + d, t + p and h + n relative wavefunctions and their asymptotic normalizations are considered in the framework of the generator coordinate method (GCM) and compared with ATMS (amalgamation of two-body correlation into multiple scattering processes) method which used the realistic Reid soft core interaction. The asymptotic normalization of relative wavefunctions provide various coupling constants, the cluster probability amplitude (the so-called Z 1/2-factor) and matter RMS radii. These wavefunctions are also used to obtain 4He − d − d, 4He − t − p and 4He − h − n vertex functions in the virtual decay of 4He. The extrapolation of vertex functions for negative values of q 2 upto the corresponding poles provide the vertex constants which are comparable with other estimates. It is noticed that in GCM the coupling constants C 2 for 4He − d − d vertex is less than 2 as has been obtained in the forward dispersion relation technique.

Magnetic dipole properties in deformed nuclei

The magnetic dipole response in deformed nuclei is investigated. Spurious states due to the rotation of the whole nucleus are removed by restricting the residual interaction in such a way that it commutes with the total angular momentum as a ground state expectation value in the quasi-particle random phase approximation. By this one guarantees that the spurious state due to rotations lies at energy zero and that it is a solution of the quasi-particle random phase equation (QRPA) and can easily be removed. The Hamiltonian is chosen to be a deformed Saxon Woods potential with a pairing force and an isoscalar and isovector generalized quadrupole-quadrupole and spin-spin interaction. The generalized isoscalar quadrupole operator is defined as the commutator of the independent quasi-particle Hamiltonian with the total angular momentum. The restoration of rotational symmetry mentioned above determines the isoscalar coupling constant. The generalized isovector quadrupole-quadrupole force does not commute with the total angular momentum. But one can find a linear “isovector” combination of the proton and the neutron total angular momentum which produces by a commutator with the independent quasiparticle Hamiltonian an “isovector” quadrupole operator which yields an isovector quadrupole-quadrupole force which always commutes with the total angular momentum. Thus one gets no restriction of the isovector quadrupole-quadrupole force constant. We fit it to the isovector giant quadrupole resonance ( E(E2) = 130 A 1 3 MeV ). The Hamiltonian contains in addition a spin-spin force, which always commutes with the total angular momentum and therefore does not spoil the restoration of rotational symmetry. The proton-proton, neutron-neutron and proton-neutron spin force constant is determined by solving the nuclear matter problem with the Reid soft core interaction within the Brückner theory and translating the result with the help of the Migdal force into finite nuclei. This yields an almost pure repulsive isovector spin-spin force. This Hamiltonian predicts a high lying ΔN = 2 scissors mode which lies between 17 and 26 MeV. It is spread over very many 1 + states with an average spacing of 15 keV. This highlying scissors mode consists of several hundred states and the overlap with an artificially constructed scissors mode is of the order of 38%, while the low lying scissors mode which shows up as 1 + states around 3 MeV has an overlap with the scissors mode of more than 50 %. In the same energy range as the high lying scissors mode lies also the isovector giant quadrupole resonance. Even at 180°, that means at completely backward angles, the E2 is comparable with the M1 excitation. This is due to the transversal quadrupole transition probability which is not disappearing in electron scattering at 180°. The same Hamiltonian is nicely describing the low lying scissors mode around 3 MeV in the rare earth and the transuranic nuclei. It gives the energies and the reduced magnetic dipole transition probabilities. It also can reproduce the formfactors measured by inelastic electron scattering. The Hamiltonian also reproduces the spin-flip states between 5 and 9 MeV excitation energy, which can be measured by inelastic proton scattering. One obtains in this region two maxima. The lowest one around 6 MeV is mainly of isoscalar nature and is weaker than the spin flip maximum at 7 to 8 MeV, which is mainly of isovector nature. The detailed form of this two-spin flip maxima depend very sensitively on the ratio of the isoscalar and isovector strength of the spin-spin force. In conclusion one can say: There exist a high lying scissors mode, but it is strongly fragmented over several hundred 1 + states between 17 and 26 MeV and even in electron scattering at backward angles the cross section for the excitation of the isovector giant quadrupole resonance is of comparable size. Thus it will be difficult to establish experimentally the existence of this high lying scissors mode.

Nuclear shell model with delta excitations

Working in an oscillator basis, we define the nuclear Hamiltonian in a no-core model space to consist of an effective nucleon-nucleon interaction obtained with Brueckner theory from the Reid Soft Core interaction, a Coulomb term, nucleondelza transition potentials, and delta-delta interaction terms. This Hamiltonian is suitable for direct diagonalization and will account for interaction effects usually attributed to three-body, four-body, etc., forces arising from the virtual excitation of deltas. A natural outcome of this approach is a nuclear shell-model involving single particle states which are primarily nucleon in character and others, at higher energies, which are primarily delta in character. By performing spherical Hartree Fock calculations with this “realistic baryon Hamiltonian” we have found that the ground states of 16O, 40Ca and 56Ni are nearly pure nucleonic states while 2 Sn has about 10% of one delta in the ground state. For an estimate of how the delta degree of freedom is excited as one goes away from the ground state, we have performed spherical Hartree-Fock calculations with a radial constraint to compress the nucleus. The delta degree of freedom is gradually populated as the nucleus is compressed. Our results suggest that inclusion of the delta in the nuclear dynamics could lead to a significant softening of the nuclear equation of state.

Influence of tensor and short-range correlations on nucleon properties in the nuclear medium

The properties of nucleons in the nuclear medium at normal density are studied including the effects of short-range and tensor correlations induced by the realistic Reid soft-core interaction. A complete characterization of the energy dependence of the self-energy and the nucleon spectral functions is obtained. These results clarify the limitations of the mean-field picture of nucleon motion in matter and provide an explanation of a number of problems associated with theoretical calculations in nuclear physics which assume the validity of the mean-field picture.

Pairing gaps in nucleonic superfluids

Singlet S-wave nucleonic superfluids are studied within a microscopic many-body theory that incorporates explicit spatial correlations due to strong short-range repulsive forces as well as the momentum-space pairing correlations of BCS theory. The theory is formulated within the method of correlated basis functions (CBF). Within this scheme, there results a nonlinear problem for the superfluid energy gap that is identical in form to the gap problem of conventional BCS theory. However, the input single-particle energies and pairing matrix elements are dressed by the short-range spatial correlations and accordingly incorporate an important class of medium corrections. The effective pairing force of the theory is finite even if the bare two-nucleon potential contains an infinitely hard core; both the pairing matrix elements and single-particle energies are to be constructed from normal-state CBF matrix elements and may be evaluated by cluster-expansion techniques. The theory is explicated and applied at a variational level that is equivalent to the leading order of a CBF superstate perturbation theory. New results are presented for the 1s 0 pairing gap Δ k F in pure neutron matter at densities relevant to the inner crust of a neutron star, based on a simplified version of the Reid soft-core interaction and spin-dependent spatial correlations optimized in the correlated normal state. Careful consideration is given to the treatment of the gap equation at large intermediate-state momenta, correcting quantitative defects of earlier calculations. The variational gap function evaluated at the Fermi surface, Δ F, which serves as a critical input to models of the cooling and internal dynamics of neutron stars, is found to be larger than predicted in earlier work. Estimates of the suppression of the gap due to polarization processes (and other particle-particle and hole-hole irreducible medium effects of higher order within CBF superstate perturbation theory) yield values for Δ k F that are generally compatible with results from the semi-empirical polarization-potential approach, except at higher densities where the gap is closing. Updated solutions of the gap problem at the variational-CBF level are also given for 1S 0 proton pairing in the quantum fluid interior of a neutron star and for 1S 0 like-nucleon pairing in symmetrical nuclear matter at densities typical of the nuclear surface.

Single-particle properties and short-range correlations in nuclear matter

Ladder correlations are studied with inclusion of hole-hole propagation to all orders. The resulting effective interaction is separated into a forward-going and a backward-going contribution with the numerical use of dispersion relations. This procedure allows the correct calculation of the corresponding self-energy terms. Self-consistency between the ladders and the self-energy is established for the quasi-particle energy. Calculations are performed with semi-realistic interactions derived from Reid's soft-core interaction to avoid the appearance of pairing instabilities. A careful study of the complete momentum and energy dependence of the resulting self-energy is made for various densities. Effective mass contributions are studied coming from both the k- and ω-dependence of the self-energy. Accurate calculation of all relevant spectral functions is performed with proper fulfillment of the sum rule and quasi-particle properties are discussed. An important fraction of the single-particle strength is found at very high energy due to the realistic short-range repulsion in the interaction. From the hole spectral function the momentum distribution is calculated at various densities. The depletion due to the influence of short-range correlations around normal nuclear matter density amounts to about 13%.

Excitation of the Delta (3,3) resonance in compressed finite nuclei from a constrained mean-field method.

In this paper we present a nonrelativistic microscopic mean-field framework for finite nuclei in which the nucleus is described as a system of A baryons where each baryon is a dynamically determined superposition of a nucleon (N) and a ..delta..(3,3) resonance. The N-N interaction is a Brueckner G matrix based on the Reid soft-core interaction while the N-..delta.. transition potentials and ..delta..-..delta.. interactions are one-boson-exchange potentials. The ground-state properties of the nuclei /sup 16/O and /sup 40/Ca are determined within the spherical Hartree-Fock approximation in a no-core basis consisting of up to six major oscillator shells. A detailed study of the zero-temperature properties of these nuclei under compression and dilation is presented in order to gain insight into the nuclear equation of state. Under certain conditions, a transition to a mixture of nucleons and deltas is found in /sup 40/Ca but not in /sup 16/O. The transition is reminiscent of a first-order phase transition.

Analysis of the proton momentum distribution in 3He

The proton momentum distribution in 3 He, calculated variationally from the Reid soft core interaction, is compared with momentum distributions for other nuclei and for nuclear matter.

Unified Skyrme approach to the real and imaginary parts of the heavy-ion optical potential

We derive a set of energy-dependent Skyrme interaction parameters to be used in the calculation of the complex heavy-ion optical potential. They are determined by fitting the binding energy of 16O as well as nuclear matter constants and the real volume part of the 16O+ 16O optical potential previously obtained from the Reid soft-core interaction. For the binding energy of heavier nuclei and the single-particle levels the present parameters lead to predictions comparable with those of other Skyrme interactions. The charge radii turn out to be somewhat smaller than the experimental values but a better fit cannot be achieved within the present constraints.

Nuclear matter approach to the heavy-ion optical potential: (II). Imaginary part

The heavy-ion optical potentials are constructed in a nuclear matter approach, for the 16O + 16O, 40Ca + 16O and 40Ca + 40Ca elastic scattering at the incident energies per nucleon E lab/ A ⩽ 45 MeV. The energy density formalism is employed assuming that the complex energy density of colliding heavy ions is a functional of the nucleon density ϱ( r), the intrinsic kinetic energy density τ (2)( r) and the average momentum of relative motion per nucleon K r(≦ 1.5 fm −1). The complex energy density is numerically evaluated for the two units of colliding nuclear matter with the same values of ρ, τ (2) and K r. The Bethe-Goldstone equation is solved for the corresponding Fermi distribution in momentum space using the Reid soft-core interaction. The “self-consistent” single-particle potential for unoccupied states which is continuous at the Fermi surface plays a crucial role to produce the imaginary part. It is found that the calculated optical potentials become more attractive and absorptive with increasing incident energy. The elastic scattering and the reaction cross sections are in fair agreement with the experimental data.

A complex effective force for heavy-ion collisions

A local, complex, density and kinetic energy density-dependent effective force is derived from the G-matrix corresponding to two colliding nuclear matters. The Bethe-Goldstone equation uses the Reid soft-core interaction as input. The auxiliary single-particle potential is chosen self-consistently for all momenta. The results are presented in parametrized form.

Contributions from high-momentum intermediate states to effective nucleon-nucleon interactions

We reexamine the intermediate state convergence problem for the core polarization diagram (${G}_{3\mathrm{p}1\mathrm{h}}$) in the expansion of an effective shell-model interaction using a modern meson exchange Bonn-J\"ulich potential in comparison to the phenomenological Reid soft-core interaction. The Bonn-J\"ulich $N\ensuremath{-}N$ potential contains a much weaker tensor force as compared with the Reid soft-core interaction. As a consequence its repulsive contributions to ${G}_{3\mathrm{p}1\mathrm{h}}$ from high momentum intermediate particle states are much reduced and ${G}_{3\mathrm{p}1\mathrm{h}}$ can be calculated in a good approximation with $2\ensuremath{\hbar}\ensuremath{\omega}$ particle-hole excitations alone. This opens up the possibility of calculating higher order processes in a reliable and expendient way. We have further determined the spectra of $^{18}\mathrm{O}$ and $^{18}\mathrm{F}$ using the folded diagram perturbation theory to determine the effective interaction. The energy levels are best reproduced by the Bonn-J\"ulich $N\ensuremath{-}N$ potential.NUCLEAR STRUCTURE $^{18}\mathrm{O}$, $^{18}\mathrm{F}$; dependence of effective interaction on $N\ensuremath{-}N$ tensor force. Convergence of intermediate state summation in core polarization.

Lowest-order brueckner theory with nonstandard dispersion

Lowest-order Brueckner energies of nuclear matter for thev1 andv2 versions of the Reid soft-core interaction, Baker-Hind-Kahane and Hamada-Johnstonv6 potentials are calculated by a method recently proposed by the author. The method is designed to incorporate many-body effects into the lowest-order Brueckner energy. The results are in close agreement with those of variational calculations. The relative merits of the present method and a similar method of the Liege group are discussed. For the full Hamada-Johnston potential, our method predicts saturation atkF=1.42 fm−1 with the average energy per particle of −12.6 MeV.

The potential energy of an infinite system of nucleons and delta resonances

Investigates the nature and properties of the delta resonance, Delta (1236), in infinite nuclear and neutron matter, performing calculations of the potential energy of a system of separate Fermi seas of nucleons and delta resonances, using Jastrow lowest-order constrained variational techniques The author uses the Reid soft-core nucleon-nucleon interaction and a model, consistent, energy-dependent, static one-pion- and one-rho-meson-exchange nucleon-delta potential, and find a significant reduction in the potential energy of the system at densities above nuclear matter density (0.17 fm-3) when deltas are present. The author estimates the density at which the formation of a separate Fermi sea of deltas is favourable, and discuss the consequences for the possible formation of a pion condensate and the properties of neutron star matter.

Nuclear matter approach to the heavy-ion optical potential

An energy-dependent local potential for heavy-ion (HI) scattering is derived from Reid's softcore interaction using the Brueckner theory. The Bethe-Goldstone equation in momentum space is first solved with the outgoing boundary condition for two colliding systems of nuclear matter with the relative momentum K r per nucleon. The Fermi distribution is assumed to consist of two spheres without double counting of their intersection separated by the relative momentum K r. The angle-averaged Pauli projection function is given in the form of a one-dimensional integral. Secondly the optical potential for HI scattering is evaluated using the energy-density formalism. The energy density is calculated for two limiting cases: (i) the sudden approximation in which the spatial distribution of the two HI is described by an antisymmetrized cluster wave function, and (ii) the adiabatic limit represented by an antisymmetrized two-centre wave function. The complex HI potential is defined in terms of the energy density from nuclear matter so that both volume elements in the finite and the infinite systems have the same nucleon and kinetic energy density. This method is applied to the 16O + 16O, 40Ca + 16O, and 40Ca + 40Ca potentials. The calculated results are compared with phenomenological potentials. Though in principle our approach can generate an imaginary part for the HI potential, the magnitude is too small. Reasons and possible improvements of this point are discussed.

Realistic nucleon-nucleon interactions and the three-body electrodisintegration ofH3

The three-body variational wave functions resulting from two realistic nucleon-nucleon interactions featuring different deuteron $D$-wave probabilities have been used in the calculation of the three-body electrodisintegration of triton in the quasi-elastic region. The angular distributions of the coincidence cross section $^{3}\mathrm{H}(e,{e}^{\ensuremath{'}}p)2n$ are found to depend sensitively upon the $D$-wave probability in the triton. In the case of the Reid soft core interaction a comparison of the $T=1$ spectral functions corresponding to the Faddeev and variational wave functions reveals an appreciably larger high momentum content in the latter.NUCLEAR REACTIONS $^{3}\mathrm{H}(e,{e}^{\ensuremath{'}}p) 2n$, calculated spectral function $P(k,{E}_{2})$, angular distributions of coincidence process, quasi-elastic peak, energy-weighted sum rule. Three-body variational wave functions; Reid and RHEL-1 nucleon-nucleon interactions.

Nuclear properties with realistic Hamiltonians through spectral distribution theory

Motivated by the need of non-perturbative methods for utilizing realistic nuclear Hamiltonians H, we use spectral distribution theory, based on calculated moments of H, to obtain specific bulk and valence properties of finite nuclei. The primary emphasis here is to present results for the binding energies of nuclei obtained with and without an assumed core. We first concentrate on results for the binding energy of sd shell nuclei calculated with respect to an 16O core since a number of meaningful tests of the spectral distribution theory can be performed through comparisons with exact diagonalizations. We also present some initial results for the binding energy of 16O in no-core calculations. Specifically, we treat 16O as 16 nucleons interacting in model spaces which increase in size up to the s-p-sd-pf oscillator shells. The Hamiltonian consists of a sum of relative kinetic energy operators and a G-matrix obtained from the Reid soft-core interaction with an exact treatment of the Pauli operator. In the largest space studied, this approach yields a binding energy which is significantly independent of the gap between the active space and the excluded space and appears to be converging on the binding energy obtained in the exp( S) method. It is felt that larger space calculations and further tests must be performed in order to confirm convergence and establish convincingly that working, as we do, with a few leading moments of H is adequate in the very large spaces of the no-core studies.

The convergence of the binding energy expansion in the Brueckner-Bethe-Goldstone theory of nuclear matter

Two-, three- and four-body contributions to the binding energy of nuclear matter are evaluated in the framework of the Bethe-Brueckner expansion. Special attention is devoted to the choice of the auxiliary single particle field and to the potential diagrams at the level of three- and four-hole lines present when such a field is different from zero. Two nucleon-nucleon interactions are used: a model interaction V 1 and the Reid soft-core interaction. For V 1 our results are compared with those obtained from variational calculations; this comparison supports the reliability of the perturbative expansion.