Microscopic calculation of a pairing gap in semi-infinite nuclear matter

The pairing gap in semi-infinite nuclear matter has been calculated microscopically by solving the gap equation for a nonlocal interaction with the aid of the method proposed by V.A. Khodel, A.V. Khodel, and J.W. Clark [Nucl. Phys. A 598, 390 (1996)] for the case of infinite nuclear matter. The calculation employs the effective pairing interaction obtained previously for semi-infinite geometry on the basis of the separable 3×3 representation of the Paris nucleon-nucleon potential. The gap found in this way changes sharply in the surface region, where it has a pronounced maximum. The dependence of the surface effect on the chemical potential of nuclear matter has been investigated.     

Chemical-potential dependence of the pairing gap in a nuclear-matter slab

The equation for the pairing gap Δ in a slab of nuclear matter governed by the Paris nucleon-nucleon potential is solved for various values of the chemical potential μ in the range from −8 MeV, which corresponds to stable nuclei, to −0.1 MeV, which corresponds to nuclei in the vicinity of the nucleon drip line. The slab is placed in a one-dimensional Woods-Saxon potential whose parameters are set to values typical of nuclei. Two models are considered. In the first, the potential-well depth is fixed at U ₀ = −50 MeV, the density within the slab growing as |μ| is reduced. In the second model, the density is fixed at the center of the slab, |U ₀| decreasing as |μ| is reduced. The behavior of the gap Δ as a function of μ is model-dependent. In the first model, Δ decreases with decreasing |μ|, while, in the second, it increases. At the same time, the effect of the surface enhancement of Δ becomes more pronounced with decreasing |μ| in both models. Original Russian Text ? S.S. Pankratov, E.E. Saperstein, M.V. Zverev, 2006, published in Yadernaya Fizika, 2006, Vol. 69, No. 12, pp. 2052–2063.     

Collective modes in a slab of interacting nuclear matter

We study the properties of a slab of nuclear matter. The behaviour with the slab thickness of the particle density, kinetic energy density and surface tension are given in the non-interacting case, together with the slab free response to an external field. Next we introduce a zero-range isovector interaction among the nucleons and analyze the slab collective excitations. For moderate momenta hard and soft modes are found, which exhaust most of the excitation strength. Their position and splitting in energy favourably compares with the splitted giant dipole resonance experimentally seen in deformed nuclei.     

Collective modes in a slab of interacting nuclear matter

We consider a slab of nuclear matter and investigate the collective excitations, which develop in the response function of the system. We introduce a finiterange realistic interaction among the nucleons, which reproduces the full G-matrix by a linear combination of gaussian potentials in the various spin-isospin channels. We then analyze the collective modes of the slab in theS=T=1 channel: for moderate momenta hard and soft zero-sound modes are found, which exhaust most of the excitation strength. At variance with the results obtained with a zero range force, new “massive” excitations are found for the vector-isovector channel.     

Medium polarization and pairing in asymmetric nuclear matter

The many-body theory of asymmetric nuclear matter is developed beyond the Brueckner–Hartree–Fock approximation to incorporate the medium polarization effects. The extension is performed within the Babu–Brown induced interaction theory. After deriving the particle–hole interaction in the form of Landau–Migdal parameters, the effects of the induced component on the symmetry energy are investigated along with the screening of ¹ S ₀ proton–proton and ³ PF ₂ neutron–neutron pairing, which are relevant for the neutron-star cooling. The crossover from repulsive (screening) to attractive (anti-screening) interaction going from pure neutron matter to symmetric nuclear matter is discussed.     

Brueckner G matrix for a planar slab of nuclear matter

The equation for the Brueckner G matrix is investigated for planar-slab geometry. A method for calculating the G matrix for a planar slab of nuclear matter is developed for a separable form of NN interaction. Actually, the separable version of the Paris NN potential is used. The singlet ¹ S ₀ and the triplet ³ S ₁–³ D ₁ channel are considered. The present analysis relies on the mixed momentum-coordinate representation, where use is made of the momentum representation in the slab plane and of the coordinate representation in the orthogonal direction. The full two-particle Hilbert space is broken down into the model subspace, where the two-particle propagator is considered exactly, and the complementary subspace, where the local-potential approximation is used, which was proposed previously for calculating the effective pairing potential. Specific calculations are performed for the case where the model subspace is constructed on the basis of negative-energy single-particle states. The G matrix is parametrically dependent on the total two-particle energy E and the total momentum P _⊥ in the slab plane. Since the G matrix is assumed to be further used to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at the value E=2μ, where μ is the chemical potential of the system under investigation. The calculations are performed predominantly for P _⊥=0. The role of nonzero values of P _⊥ is assessed. The resulting G matrix is found to depend greatly on μ in the surface region.     

Surface properties of nuclear pairing with the Gogny force in a simplified model

Surface properties of neutron-neutron (T=1) pairing in semi-infinite nuclear matter in a hard wall potential are investigated in BCS approximation using the Gogny force. Surface enhancement of the gap function, pairing tensor and correlation energy density is put into evidence.     

Isotope shifts in finite nuclei and the pairing properties of nuclear matter

A uniform nuclear matter with s-wave pairing is studied within the local energy-density functional approach, incorporating a few parameter sets extracted from the analysis of isotope shifts in finite nuclei. The dilute limit, in which the regime changes from weak to strong pairing, is considered in detail, and, for strong coupling, the ground state properties of that system are found to be completely determined in leading order by the singlet scattering length a _nn . The combination of a density-dependent contact pairing interaction and an energy cutoff adjusted to produce a realistic value of a _nn is shown to be the preferred choice among the deduced parameter sets. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 4, 235–241 (25 August 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Triplet cooper pairing in nuclear matter and rotational states of superdeformed nuclei

One hundred and sixty-one rotational bands of superdeformed states in nuclei are considered on the basis of a model that admits triplet Cooper pairing in superfluid nuclear matter. The behavior of the dynamical moment of inertia for such states is investigated within this model, which is shown to comply well with available experimental data and to describe successfully the rotational spectra of superdeformed states.     

Fission of a slab of charged nuclear matter in the time-dependent hartree fock (TDHF) theory

The time evolution of a slab of charged nuclear matter is studied within the TDHF method as a function of the initial conditions. The effective interaction is of Skyrme type plus a repulsive Yukawa interaction representing the Coulomb repulsion. The implications on the time evolution of specific collective as well as intrinsic excitations are investigated. Fission is found to occur at large enough excitation energies for a variety of initial conditions after timest _fi ≈(1–3)·10^?21 s. The final disintegration of the slab (“snatching”) is characterized by arapid rise of a potential barrier between the nascent fragments, the rise occurring int _sn≈0.2·10^?21 s. The importance for the fission process of the reflection symmetry of the occupied single particle states is demonstrated.     

Equation of state of superfluid neutron matter and the calculation of the 1S0 pairing gap

We present a quantum Monte Carlo study of the zero-temperature equation of state of neutron matter and the computation of the 1S0 pairing gap in the low-density regime with rho < 0.04 fm(-3). The system is described by a nonrelativistic nuclear Hamiltonian including both two- and three-nucleon interactions of the Argonne and Urbana type. This model interaction provides very accurate results in the calculation of the binding energy of light nuclei. A suppression of the gap with respect to the pure BCS theory is found, but sensibly weaker than in other works that attempt to include polarization effects in an approximate way.     

Superfluid pairing gap in strong coupling

The zero-temperature pairing gap is a fundamental property of interacting Fermions, providing a crucial test of many-body theories in strong coupling. We analyze recent cold-atom experiments on imbalanced Fermi systems using Quantum Monte Carlo results for the superfluid and normal phases. Through this analysis we extract, for the first time, the experimental zero-temperature pairing gap in the fully paired superfluid state at unitarity where the two-body scattering length is infinite. We find that the zero-temperature pairing gap is greater than 0.4 times the Fermi energy E(F), with a preferred value of (0.45+/-0.05) E(F). The ratio of the pairing gap to the Fermi Energy is larger here than in any other Fermi system measured to date.     

On the ab initio calculation of a pairing gap in atomic nuclei

The role of an effective mass in the ab initio BCS equation for a pairing gap in atomic nuclei has been analyzed.     

Role of the surface in nuclear pairing

Recent results obtained by solving the ab initio equation for the pairing gap in a slab of nuclear matter are reviewed. They conclusively prove a surface nature of nucleon pairing in nuclei. First, the effective pairing interaction in the vicinity of the slab surface makes a dominant contribution to the gap. Second, there is a sizable surface enhancement of the pairing gap Δ(X) itself. Finally, an analysis of spatial correlation features of the anomalous density matrix shows that the local correlation length at the slab surface is very small, about 1.5 fm, but, in the interior of the slab, it is rather large, about the slab thickness. These results are in qualitative agreement with those obtained recently by Pillet and his coauthors for spherical nuclei by using the phenomenological Gogny force. An increase in the concentration of Cooper pairs at the surface of nuclei was yet another result of their study, and our present analysis confirms this indirectly.     

Electrostatic mapping of nuclear pairing

The traditional nuclear pairing problem is shown to be in one-to-one correspondence with a classical electrostatic problem in two dimensions. We make use of this analogy in a series of calculations in the tin region, showing that the extremely rich phenomenology that appears in this classical problem can provide interesting new insights into nuclear superconductivity.     

Variational methods in nuclear pairing theory

Particle number conserving groundstate wave functions are constructed by use of the Thouless-Peierls formalism. A comparison with PBCS- and FBCS-solutions shows an encouraging advantage of this method: 1. the groundstate is lower in energy than the FBCS groundstate, 2. the computer time necessary to obtain the groundstate wave function as well as its energy is smaller than with the FBCS method.     

Surface incompressibility of semi-infinite nuclear matter via constrained Hartree-Fock calculations

With a view to calculating the incompressibility \(\ddot \sigma \) of the nuclear surface, we develop a constrained Hartree-Fock treatment of semi-infinite nuclear matter. Our approach leads first to a proof of the “ \(\dot \sigma = 0\) ” theorem of Myers and Swiatecki, and then to a corroboration of a simple expression for \(\ddot \sigma \) , previously obtained in an intuitive way by Stocker. This expression is used to calculate \(\ddot \sigma \) for theS3, SkM and Gogny forces. The results are significantly different from those of a scaled HF calculation, but in very good agreement with the values of \(\ddot \sigma \) that are implied by RPA calculations of the breathing mode in various finite nuclei.