Pairing Correlations In Nuclei Research Articles

Neutron-proton pairing in the unstable N=Z nuclei of the f-shell through two-nucleon transfer reactions

Pair transfer is a unique tool to study pairing correlations in nuclei. Neutron-proton pairing is investigated in the N=Z nuclei of the f-shell, through the reaction (p,3He) in inverse kinematics, that allows to populate at the same time the lowest J=0+, T=1 (isovector pairing) state and J=1+, T=0 (isoscalar pairing) state. Radioactive beams of 56Ni and 52Fe produced by fragmentation at the GANIL/LISE facility combined with particle and gamma-ray detection make it possible to carry out this study from 48Cr (mid-shell nucleus) to 56Ni (doubly-magic nucleus). The cross-sections were extracted and compared with second-order distorted-wave born approximation (DWBA) calculations performed with neutron-proton amplitudes obtained from shell model calculations with GXPF1 interaction. Very low cross-sections for the J=1+,T=0 state (isoscalar channel) were observed. The cross-section for 56Ni is one of order of magnitude lower than for 40Ca showing a strong reduction of the isoscalar channel in the f-shell as compared to the sd-shell. On the other hand, the increase of the cross-section towards the middle of the shell for the isovector channel points towards a possible superfluid phase.

Higgs response and pair condensation energy in superfluid nuclei

Abstract The pairing correlation in nuclei causes a characteristic excitation, known as the pair vibration, which is populated by the pair transfer reactions. Here we introduce a new method of characterizing the pair vibration by employing an analogy to the Higgs mode, which emerges in infinite superconducting/superfluid systems as a collective vibrational mode associated with the amplitude oscillation of the Cooper pair condensate. The idea is formulated by defining a pair-transfer probe, the Higgs operator, and then describing the linear response and the strength function to this probe. We will show that the pair condensation energy in nuclei can be extracted with use of the strength sum and the static polarizability of the Higgs response. In order to demonstrate and validate the method, we perform numerical analysis for Sn isotopes based on the quasi-particle random phase approximation to the Skyrme–Hartree–Fock–Bogoliubov model. We discuss the possibility of applying this new scheme to pair-transfer experiments.

Two-neutron transfer as a tool to study pairing correlations in nuclei

A feasibility study of two-neutron transfer reactions, using 12C and 19F targets, was performed. In this experiment, a 6He beam at 20 MeV incident energy was delivered by the RIBRAS facility. The main goal of this work was to test the RIBRAS system, operated in a single-solenoid mode, to investigate (6He,α) reactions in light nuclei. The excitation function of the 14C nucleus, populated by the 12C(6He,α) reaction, was extracted from the data. Evidence of nn-transfer reactions populating the ground state and the first state in 14C were observed. However, the present spectrum might have contribution from other reaction channels such as the 6He breakup. Future experiments with the RIBRAS dual-solenoid mode will improve the quality of the data by reducing a significant amount of background.

Exact solutions of mean-field plus various pairing interactions and shape phase transitions in nuclei

The exact solutions of either spherical or deformed mean-field plus various types of pairing models are briefly reviewed. It is shown that, besides the standard pairing model, there are several special types of pairing interaction that can be solved exactly. In comparison to the standard pairing, the results of several pairing-driven quantities, such as pairing excitation energies, even–odd mass differences, the moment of inertia, etc., show that these types of pairing interaction can indeed be used to describe pairing correlations in nuclei either more accurately or efficiently. Moreover, the shape phase transitional behaviors of nuclei described by the consistent-Q formalism of the interacting boson model are summarized.

Isovector pair correlations in analytically solvable models

The eigensolutions of the collective Hamiltonian with different potentials suggested for description of the isovector pair correlations are obtained, analyzed and compared with the experimental energies. It is shown that the isovector pair correlations in nuclei around [Formula: see text]Ni can be described as anharmonic pairing vibrations. The results obtained indicate the presence of the [Formula: see text]-particle type correlations in these nuclei and the existence of the interaction different from isovector pairing which also influences on the isospin dependence of the energies.

Exact isovector pairing in a shell-model framework: Role of proton-neutron correlations in isobaric analog states

We utilize a nuclear shell model Hamiltonian with only two adjustable parameters to generate, for the first time, exact solutions for pairing correlations for light to medium-mass nuclei, including the challenging proton-neutron pairs, while also identifying the primary physics involved. In addition to single-particle energy and Coulomb potential terms, the shell model Hamiltonian consists of an isovector $T=1$ pairing interaction and an average proton-neutron isoscalar $T=0$ interaction, where the $T=0$ term describes the average interaction between non-paired protons and neutrons. This Hamiltonian is exactly solvable, where, utilizing 3 to 7 single-particle energy levels, we reproduce experimental data for 0$^+$ state energies for isotopes with mass $A=10$ through $A=62$ exceptionally well including isotopes from He to Ge. Additionally, we isolate effects due to like-particle and proton-neutron pairing, provide estimates for the total and proton-neutron pairing gaps, and reproduce $N$ (neutron) = $Z$ (proton) irregularity. These results provide a further understanding for the key role of proton-neutron pairing correlations in nuclei, which is especially important for waiting-point nuclei on the rp-path of nucleosynthesis.

Localization of pairing correlations in nuclei within relativistic mean field models

We analyze the localization properties of two-body correlations induced by pairing in the framework of relativistic mean field (RMF) models. The spatial properties of two-body correlations are studied for the pairing tensor in coordinate space and for the Cooper pair wave function. The calculations are performed both with Relativistic-Hatree-Bogoliubov (RHB) and RMF+Projected-BCS (PBCS) models and taking as examples the nuclei $^{66}$Ni, $^{124}$Sn and $^{200}$Pb. It is shown that the coherence length have the same pattern as in previous non-relativistic HFB calculations, i.e., it is maximum in the interior of the nucleus and drops to a minimum in the surface region. In the framework of RMF+PBCS we have also analysed, for the particular case of $^{120}$Sn, the dependence of the coherence length on the intensity of the pairing force. This analysis indicates that pairing is reducing the coherence length by about 25-30 $\%$ compared to the RMF limit.

Spatial properties of pairing and quarteting correlations in nuclei

Interest in the study of the spatial properties of pairing correlations in nuclei has recently been revived. The main interesting physical quantity is the coherence length (CL). We investigate the CL as obtained from schematic pairing potentials. We then generalize to quarteting correlations by considering the CL build out of proton and neutron pairs. We point out the main differences between the pairing and quarteting CLs. Finally we comment on the important role of proton-neutron correlations as they strongly modify the behaviour of the CL when taken into account. The coherence properties of the spatial distribution of the two-particle density are known to give information about nuclear correlations [1, 2]. The main physical quantity that describes the coherence property is the CL, defined as the root mean square relative distance averaged over the pairing density, in the case of superfluid nuclei. It has been found [3, 4] that this quantity is of the order of the nuclear radius inside the nucleus and smaller around the nuclear surface. In contrast, the correlations between proton and neutron pairs by the analogous quarteting CL.

Probing the pairing interaction through two-neutron transfer reactions

We analyze two-neutron transfers in nuclei within a mean-field approach. We first show that the response associated with pairing excitations can be affected by the surface/volume localization of the employed effective pairing interaction. Then, ground-state-to-ground-state transitions are studied and some improvements on the currently used theoretical formulas are discussed. Finally, Cr isotopes at the drip line are analyzed. The analysis of pair-transfer reactions in these nuclei may improve our understanding of two aspects: the spatial distibution of pairing correlations in nuclei and the debated question of the persistence of pairing at the drip lines.

Two-neutron transfer probabilities and spatial-localization effects at the drip line

We examine Cr isotopes at the drip line, where surface effects related to the existence of a weakly bound ${s}_{1/2}$ state are known to be important and tightly connected with the pairing phenomenon (antihalo effect). For these weakly bound isotopes, we evaluate the ground state to ground state two-neutron transfer probabilities within a mean-field-based approach. An important part of the discussion is devoted to the analysis of several procedures that can be employed to constrain the parameters of a phenomenological pairing interaction. The parameters are first adjusted to reproduce the experimental gaps evaluated with the five-point formula. This choice has, however, some consequences on the evolution of the pairing correlations along the isotopic chains and, in particular, at shell closures. Other procedures are then followed (adjustment on a theoretical pairing gap at mid-shell and on the two-neutron separation energies). For the transfer probabilities, we discuss the effects associated with different choices of the spatial localization of the pairing interaction. We indicate that the analysis of pair-transfer reactions for such cases (where the last bound state is a low-$l$ state in a weakly bound nucleus) may improve our understanding of two aspects: the spatial distribution of pairing correlations in nuclei and the general problem of the persistence of pairing at the drip lines.

Pairing correlations in exotic nuclei

The BCS and HFB theories which can accommodate the pairing correlations in the ground states of atomic nuclei are presented. As an application of the pairing theories, we investigate the spatial extension of weakly bound Ne and C isotopes by taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method and a 3-body model, respectively. We show that the odd-even staggering in the reaction cross sections of 30,31,32Ne and 14,15,16C are successfully reproduced, and thus the staggering can be attributed to the unique role of pairing correlations in nuclei far from the stability line. A correlation between a one-neutron separation energy and the anti-halo effect is demonstrated for s- and p-waves using the HFB wave functions. We also propose effective density-dependent pairing interactions which reproduce both the neutron-neutron (nn) scattering length at zero density and the neutron pairing gap in uniform matter. Then, we apply these interactions to study pairing gaps in semi-magic finite nuclei, such as Ca, Ni, Sn and Pb isotopic chains.

Axially deformed solution of the Skyrme-Hartree–Fock–Bogoliubov equations using the transformed harmonic oscillator basis (II) hfbtho v2.00d: A new version of the program

We describe the new version 2.00d of the code hfbtho that solves the nuclear Skyrme-Hartree–Fock (HF) or Skyrme-Hartree–Fock–Bogoliubov (HFB) problem by using the cylindrical transformed deformed harmonic oscillator basis. In the new version, we have implemented the following features: (i) the modified Broyden method for non-linear problems, (ii) optional breaking of reflection symmetry, (iii) calculation of axial multipole moments, (iv) finite temperature formalism for the HFB method, (v) linear constraint method based on the approximation of the Random Phase Approximation (RPA) matrix for multi-constraint calculations, (vi) blocking of quasi-particles in the Equal Filling Approximation (EFA), (vii) framework for generalized energy density with arbitrary density-dependences, and (viii) shared memory parallelism via OpenMP pragmas. Program summaryProgram title: HFBTHO v2.00dCatalog identifier: ADUI_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUI_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License version 3No. of lines in distributed program, including test data, etc.: 167228No. of bytes in distributed program, including test data, etc.: 2672156Distribution format: tar.gzProgramming language: FORTRAN-95.Computer: Intel Pentium-III, Intel Xeon, AMD-Athlon, AMD-Opteron, Cray XT5, Cray XE6.Operating system: UNIX, LINUX, WindowsXP.RAM: 200 MwordsWord size: 8 bitsClassification: 17.22.Does the new version supercede the previous version?: YesCatalog identifier of previous version: ADUI_v1_0Journal reference of previous version: Comput. Phys. Comm. 167 (2005) 43Nature of problem:The solution of self-consistent mean-field equations for weakly-bound paired nuclei requires a correct description of the asymptotic properties of nuclear quasi-particle wave functions. In the present implementation, this is achieved by using the single-particle wave functions of the transformed harmonic oscillator, which allows for an accurate description of deformation effects and pairing correlations in nuclei arbitrarily close to the particle drip lines.Solution method:The program uses the axial Transformed Harmonic Oscillator (THO) single- particle basis to expand quasi-particle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogoliubov Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions until a self-consistent solution is found. A previous version of the program was presented in: M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring, Comput. Phys. Commun. 167 (2005) 43–63.Reasons for new version:Version 2.00d of HFBTHO provides a number of new options such as the optional breaking of reflection symmetry, the calculation of axial multipole moments, the finite temperature formalism for the HFB method, optimized multi-constraint calculations, the treatment of odd–even and odd–odd nuclei in the blocking approximation, and the framework for generalized energy density with arbitrary density-dependences. It is also the first version of HFBTHO to contain threading capabilities.Summary of revisions:1.The modified Broyden method has been implemented,2.Optional breaking of reflection symmetry has been implemented,3.The calculation of all axial multipole moments up to λ=8 has been implemented,4.The finite temperature formalism for the HFB method has been implemented,5.The linear constraint method based on the approximation of the Random Phase Approximation (RPA) matrix for multi-constraint calculations has been implemented,6.The blocking of quasi-particles in the Equal Filling Approximation (EFA) has been implemented,7.The framework for generalized energy density functionals with arbitrary density-dependence has been implemented,8.Shared memory parallelism via OpenMP pragmas has been implemented.Restrictions:Axial- and time-reversal symmetries are assumed.Unusual features:The user must have access to (i)the LAPACK subroutines DSYEVD, DSYTRF and DSYTRI, and their dependences, which compute eigenvalues and eigenfunctions of real symmetric matrices,(ii)the LAPACK subroutines DGETRI and DGETRF, which invert arbitrary real matrices, and(iii)the BLAS routines DCOPY, DSCAL, DGEMM and DGEMV for double-precision linear algebra (or provide another set of subroutines that can perform such tasks). The BLAS and LAPACK subroutines can be obtained from the Netlib Repository at the University of Tennessee, Knoxville: http://netlib2.cs.utk.edu/.Running time:Highly variable, as it depends on the nucleus, size of the basis, requested accuracy, requested configuration, compiler and libraries, and hardware architecture. An order of magnitude would be a few seconds for ground-state configurations in small bases Nmax≈8−12, to a few minutes in very deformed configuration of a heavy nucleus with a large basis Nmax>20.

Effects which will not blur the message of the 1H (11Li, 9Li) 3H reaction: observation of phonon–exchange pairing correlations in nuclei

The results of the 1H (11Li, 9Li) 3H experiment carried out recently at TRIUMF have given detailed information concerning the structure and the reaction mechanism at work in the case of halo, exotic nuclei, providing important stepping stones for an eventual quantitative description of superfluidity in finite nuclei. In fact, the central role played by renormalization processes as testified by the excitation of the 1/2– (2.69 MeV) member of the 9Li (2+ ⊗p3/2(π))j− multiplet (nuclear structure) together with the post-prior symmetry (reaction), testify to the fact that a detailed knowledge of the "bare" quantities, like e.g. the NN-interaction is likely to be less important, in a unified Nuclear Field Theory description of the structure and the reaction aspects of the experiment, than previously thought.

Persistent contribution of unbound quasiparticles to the pair correlation in the continuum Skyrme-Hartree-Fock-Bogoliubov approach

The neutron pair correlation in nuclei near the neutron drip-line is investigated using the selfconsistent continuum Skyrme-Hartree-Fock-Bogoliubov theory formulated with the coordinate-space Green's function technique. Numerical analysis is performed for even-even N = 86 isotones in the Mo-Sn region, where the 3p3/2 and 3p1/2 orbits lying near the Fermi energy are either weakly bound or unbound. The quasiparticle states originating from the l = 1 orbits form resonances with large widths, which are due to the low barrier height and the strong continuum coupling caused by the pair potential. Analyzing in detail the pairing properties and roles of the quasiparticle resonances, we found that the l = 1 broad quasiparticle resonances persist to feel the pair potential and contribute to the pair correlation even when their widths are comparable with the resonance energy.

Microscopic evaluation of the pairing gap

We discuss the relevant progress that has been made in the last few years on the microscopic theory of the pairing correlation in nuclei and the open problems that still must be solved in order to reach a satisfactory description and understanding of the nuclear pairing. The similarities and differences with the nuclear matter case are emphasized and described by a few illustrative examples. The comparison of calculations of different groups on the same set of nuclei, besides agreements, also shows discrepancies that remain to be clarified. The role of the many-body correlations, like screening, that go beyond the BCS scheme, is still uncertain and requires further investigation.

Microscopic calculation and local approximation of the spatial dependence of the pairing field with bare and induced interactions

The bare nucleon-nucleon interaction is essential for the production of pair correlations in nuclei, but an important contribution also arises from the induced interaction resulting from the exchange of collective vibrations between nucleons moving in time reversal states close to the Fermi energy. The pairing field resulting from the summed interaction is strongly peaked at the nuclear surface. It is possible to reproduce the detailed spatial dependence of this field by using either a local approximation, which fully takes into account finite size effects, or a contact interaction, with parameters that are quite different from those commonly used in more phenomenological approaches.

Variational multiparticle-multihole configuration mixing method applied to pairing correlations in nuclei

Applying a variational multiparticle-multihole configuration mixing method whose purpose is to include correlations beyond the mean field in a unified way without particle number and Pauli principle violations, we investigate pairing-like correlations in the ground states of $^{116}\mathrm{Sn}$, $^{106}\mathrm{Sn}$, and $^{100}\mathrm{Sn}$. The same effective nucleon-nucleon interaction, namely, the D1S parametrization of the Gogny force, is used to derive both the mean field and correlation components of nuclear wave functions. Calculations are performed using an axially symmetric representation. The structure of correlated wave functions, their convergence with respect to the number of particle-hole excitations, and the influence of correlations on single-particle level spectra and occupation probabilities are analyzed and compared with results obtained with the same two-body effective interaction from BCS, Hartree-Fock-Bogoliubov, and particle number projected after variation BCS approaches. Calculations of nuclear radii and the first theoretical excited ${0}^{+}$ states are compared with experimental data.

Thermal Signatures of Pairing Correlations in Nuclei and Nanoparticles

The Bardeen-Cooper-Schrieffer (BCS) mean-field theory of the pairing interaction breaks down for nuclei and ultra-small metallic grains (nanoparticles). Finite-temperature pairing correlations in such finite-size systems can be calculated beyond the BCS theory in an auxiliary-field Monte Carlo approach. We identify thermal signatures of pairing correlations in both nuclei and nanoparticles that depend on the particle-number parity.

Pairing correlations in nuclei: bare, induced and effective interactions

The bare nucleon–nucleon interaction is essential for the production of pair correlations in nuclei, but the induced interaction due to phonon exchange also contributes. In this paper, we shall present examples of the interplay between these two sources of pairing interaction in the case of finite nuclei and of the inner crust of neutron stars.

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis. The program HFBTHO (v1.66p)

We describe the program HFBTHO for axially deformed configurational Hartree–Fock–Bogolyubov calculations with Skyrme-forces and zero-range pairing interaction using Harmonic-Oscillator and/or Transformed Harmonic-Oscillator states. The particle-number symmetry is approximately restored using the Lipkin–Nogami prescription, followed by an exact particle number projection after the variation. The program can be used in a variety of applications, including systematic studies of wide ranges of nuclei, both spherical and axially deformed, extending all the way out to nucleon drip lines. Program summary Title of the program: HFBTHO (v1.66p) Catalogue number: ADUI Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUI Licensing provisions: none Computers on which the program has been tested: Pentium-III, Pentium-IV, AMD-Athlon, IBM Power 3, IBM Power 4, Intel Xeon Operating systems: LINUX, Windows Programming language used: FORTRAN-95 Memory required to execute with typical data: 59 MB when using N sh = 20 No. of bits in a word: 64 No. of processors used: 1 Has the code been vectorized?: No No. of bytes in distributed program, including test data, etc.: 195 285 No. of lines in distributed program: 12 058 Distribution format: tar.gz Nature of physical problem: The solution of self-consistent mean-field equations for weakly bound paired nuclei requires a correct description of the asymptotic properties of nuclear quasiparticle wave functions. In the present implementation, this is achieved by using the single-particle wave functions of the Transformed Harmonic Oscillator, which allows for an accurate description of deformation effects and pairing correlations in nuclei arbitrarily close to the particle drip lines. Method of solution: The program uses the axially Transformed Harmonic Oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the Hartree–Fock–Bogolyubov Hamiltonian based on the Skyrme-forces and zero-range pairing interaction until a self-consistent solution is found. Restrictions on the complexity of the problem: Axial-, time-reversal-, and space-inversion symmetries are assumed. Only quasiparticle vacua of even–even nuclei can be calculated. Typical running time: 4 s per iteration on an Intel Xeon 2.8 GHz processor when using N sh = 20 Unusual features of the program: none