Nuclear Pairing Correlations Research Articles

Coordinate-space solution of the Skyrme–Hartree–Fock– Bogolyubov equations within spherical symmetry. The program HFBRAD (v1.00)

We describe the first version (v1.00) of the code hfbrad which solves the Skyrme–Hartree–Fock or Skyrme–Hartree–Fock–Bogolyubov equations in the coordinate representation with spherical symmetry. A realistic representation of the quasiparticle wave functions on the space lattice allows calculations to be performed up to the particle drip lines. Zero-range density-dependent interactions are used in the pairing channel. The pairing energy is calculated by either using a cut-off energy in the quasiparticle spectrum or the regularization scheme proposed by A. Bulgac and Y. Yu. Program summary Title of the program: hfbrad (v1.00) Catalogue indentifier:ADVM Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVM Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: none Computers on which the program has been tested: Pentium-III, Pentium-IV Operating systems: LINUX, Windows Programming language used:FORTRAN-95 Memory required to execute with typical data: 30 MBytes No. of bits in a word: The code is written with a type real and uses the intrinsic function selected_real_kind at the beginning of the code to ask for at least 12 significant digits. This can be easily modified by asking for more significant digits if the architecture of the computer can handle it. No. of processors used:1 Has the code been vectorized?:No No. of bytes in distributed program, including test data, etc.: 40 308 No. of lines in distributed program, including test data, etc.: 5370 Distribution format:tar.gz Nature of physical problem: For a self-consistent description of nuclear pair correlations, both the particle–hole (field) and particle–particle (pairing) channels of the nuclear mean field must be treated within a common approach, which is the Hartree–Fock–Bogolyubov theory. By expressing these fields in spatial coordinates one can obtain the best possible solutions of the problem; however, without assuming specific symmetries the numerical task is often too difficult. This is not the case when the spherical symmetry is assumed, because then the one-dimensional differential equations can be solved very efficiently. Although the spherically symmetric solutions are physically meaningful only for magic and semi-magic nuclei, the possibility of obtaining them within tens of seconds of the CPU makes them a valuable element for studying nuclei across the nuclear chart, including those near or at the drip lines. Method of solution: The program determines the two-component Hartree–Fock–Bogolyubov quasiparticle wave functions on the lattice of equidistant points in the radial coordinate. This is done by solving the eigensystem of two second-order differential equations using the Numerov method. A standard iterative procedure is then used to find self-consistent solutions for the nuclear product wave functions and densities. Restrictions on the complexity of the problem: The main restriction is related to the assumed spherical symmetry. Typical running time: One Hartree–Fock iteration takes about 0.4 s for a medium mass nucleus, convergence is achieved in about 40 s. Unusual features of the program: none

Nuclear pairing: New perspectives

Nuclear pairing correlations are known to play an important role in various single-particle and collective aspects of nuclear structure. After the first idea by A. Bohr, B. Mottelson and D. Pines on similarity of nuclear pairing to electron superconductivity, S.T. Belyaev gave a thorough analysis of the manifestations of pairing in complex nuclei. The current revival of interest in nuclear pairing is connected to the shift of modern nuclear physics towards nuclei far from stability; many loosely bound nuclei are particle-stable only due to the pairing. The theoretical methods borrowed from macroscopic superconductivity turn out to be insufficient for finite systems as nuclei, in particular for the cases of weak pairing and proximity of continuum states. We suggest a simple numerical procedure of exact solution of the nuclear pairing problem and discuss the physical features of this complete solution. We show also how the continuum states can be naturally included in the consideration bridging the gap between the structure and reactions. The path from coherent pairing to chaos and thermalization and perspectives of new theoretical approaches based on the full solution of pairing are discussed.

Pairing gaps from nuclear mean-fieldmo dels

We discuss the pairing gap, a measure for nuclear pairing correlations, in chains of spherical, semi-magic nuclei in the framework of self-consistent nuclear mean-field models. The equations for the conventional BCS model and the approximate projection-before-variation Lipkin-Nogami method are formulated in terms of local density functionals for the effective interaction. We calculate the Lipkin-Nogami corrections of both the mean-field energy and the pairing energy. Various definitions of the pairing gap are discussed as three-point, four-point and five-point mass-difference formulae, averaged matrix elements of the pairing potential, and single-quasiparticle energies. Experimental values for the pairing gap are compared with calculations employing both a delta pairing force and a density-dependent delta interaction in the BCS and Lipkin-Nogami model. Odd-mass nuclei are calculated in the spherical blocking approximation which neglects part of the the core polarization in the odd nucleus. We find that the five-point mass difference formula gives a very robust description of the odd-even staggering, other approximations for the gap may differ from that up to 30% for certain nuclei.

Nuclear pairing correlations by a quantum Monte Carlo method

A quantum Monte Carlo approach is applied to the treatment of the pairing force in nuclear systems. The method makes possible the calculation of the pairing energy and the pair occupation probabilities for the ground state. It is computationally efficient and also very simple to implement. The reliability of the method is checked with the use of two exactly solvable schematic shell models. The computed pairing energy is also compared with the predictions of several approximation methods of common use. Numerical results for some nuclei with a realistic interaction are also provided.

Low-spin identical bands in neighboring odd-A and even-even nuclei

A comprehensive study of odd-A rotational bands in normally deformed rare-earth nuclei indicates that a large number of seniority-one configurations (21% for odd-Z nuclei) at low spin have moments of inertia nearly identical to that of the seniority-zero configuration of the neighboring even-even nucleus with one less nucleon. It is difficult to reconcile these results with conventional models of nuclear pair correlation, which predict variations of about 15% in the moments of inertia of configurations differing by one unit in seniority.

Low-spin indentical bands in neighboring odd-A and even-even nuclei: A challenge to mean-field theories.

A comprehensive study of odd-A rotational bands in normally deformed rare-earth nuclei indicates that a large number of seniority-one configurations (30% for odd-Z nuclei) at low spin have moments of inertia nearly identical to that of the seniority-zero configuration of the neighboring even-even nucleus with one less nucleon. It is difficult to reconcile these results with conventional models of nuclear pair correlation, which predict variations of about 15% in the moments of inertia of configurations differing by one unit in seniority.

One possible mechanism for coriolis attenuation

The blocking effect on the nuclear pairing correlation in combination with some small change of the single-particle level scheme near the Fermi surface may lead to a large Coriolis attenuation.

High-spin states in 157,158Er and 158,159Tm.

High-spin states in /sup 157,158/Er and /sup 158,159/Tm have been studied via the /sup 122/Sn(/sup 40/Ar,5,4n) and /sup 141/Pr(/sup 22/Ne,5,4n) reactions, respectively, using in-beam gamma-ray spectroscopy techniques which included gamma - gamma coincidence, angular correlation, angular distribution, excitation function, and multiplicity measurements. The favored ( pi =+, alpha =+1/2) /b i//sub 13/2/ neutron band in /sup 157/Er has been established up to the 53/2/sup +/ level ( h omega =0.417 MeV), and the favored ( pi =-, alpha =-1/2) and unfavored ( pi =-, alpha =+/sup 1///sub 2/)/b h//sub 11/2 / proton bands in /sup 159/Tm, up to the 59/2/sup -/ ( h omega =0.454 MeV) and 61/2/sup -/( h omega =0.480 MeV) levels, respectively, i.e. well above the range of rotational frequency h omega where a second backbending has been observed in the neighboring even-even nuclei. In /sup 158/Er, the yrast levels have been observed up to 38/sup +/, and in /sup 158/Tm, up to 25/sup -/. The favored /b i//sub 13/2/ band of the odd-/b N/ nucleus /sup 157/Er displays a strong upbending at h omega =0.40 MeV, whereas, in the two branches of the /b h/ /sub 11/2/ yrast band of the odd-/b Z/ nucleus /sup 159/Tm, no alignment effect has been observed between h omega =0.27 and 0.48 MeV. The experimental results have been analyzed in the framework of the cranked shell model. On the basis of blocking arguments and cranked shell model calculations, the obtained results show that the second backbending in the yrast band of even-even nuclei near /b A/=158 is due to an /b h//sub 11/2/ proton alignment. The signature splitting of the intrinsic excitation energies (Routhians) and aligned angular momenta in /sup 159/Tm has been reproduced in cranked shell model calculations by introducing a triaxial deformation. The observed steep rise of the /b B/(/b M/ 1)//b Q//sub 0/ /sup 2/ values in the favored and unfavored bands of /sup 159/Tm are consistent with an increase of |/b g//sub K/-/b g//sub R/| at the firsts backbending due to the alignment of a neutron pair around h omega =0.27 MeV. The evolution of the kinematical and dynamical moments of inertia at very high rotational frequencies ( h omega >0.50 MeV) yields information on the nuclear pairing correlations in this frequency range.

Residual quasiparticle interaction due to blocking

The residual quasiparticle interaction due to the blocking effect is analyzed with the particlenumber-conserving (PNC) method for treating the nuclear pairing correlation. The calculation shows that this kind of residual interaction depends sensitively on the distribution of the single-particle levels near the Fermi surface and the location of the blocked levels. If both the blocked levels are above the Fermi surface the residual interaction due to blocking turns out to be small in general. However, if one of the blocked levels is just below the Fermi surface the residual interaction due to blocking may be rather large.

Particle-number-conserving method for treating the nuclear pairing correlation

A particle-number-conserving (PNC) method for treating the nuclear pairing correlation is presented. The difficulties encountered in the BCS method (particle-number nonconservation, spurious states, blocking effect and orthogonality, etc.) disappear in the PNC treatment. A detailed comparison between the BCS and PNC methods is made. It is found that the shape and the state density of the low-lying spectra obtained are quite different from those of the quasiparticle spectra obtained in the BCS method. Large differences between the two methods are found in the occupation probabilities in the excited states, the branching ratios for two-particle transfer reactions, etc. Blocking effects in the energy and the moment of inertia are analyzed. The residual quasiparticle interaction and the relation between the even-odd mass difference and quasiparticle energy are investigated. Pair vibration and pair rotation are investigated in the framework of the PNC formalism.

The structure factor for liquid bromine by neutron diffraction

Thermal neutrons from steady-state (reactor) and pulsed (linac) neutron facilities have been used to study the structure factor for liquid bromine at 20°C, covering a wide range of values (0·6-35 A-1) of the momentum transfer (ħQ). The diffraction pattern at high Q-values (>10 A-1) gives information on the structural properties of the individual molecules but detailed interpretation is complicated by the vibrational motion which causes a systematic variation in the periodic oscillations of the molecular form-factor. An internuclear distance (bond length) of 2·28 ± 0·01 A is found to be suitable for data at lower Q-values and has been used in the analysis of the liquid structure. The results show that some form of orientational correlation between molecules must be present and the nuclear (atom) pair correlation function is split into two peaks for the coordination shell corresponding to nearest-neighbour molecules. The results are compared with other studies of liquid bromine and similar neutron experimen...

Glauber's multiple scattering series for nuclei in the presence of short range correlations

Using the eikonal expansion of Sugar and Blankenbecker, we investigate the conditions under which Glauber's multiple scattering formalism may be applied to nuclei. If no information about nuclear pair correlations is to be extracted from the experiment, the momentum k of the projectile is restricted by kR ⪢ 1, where R is the nuclear radius. The angular range is found to include the domain of single and double scattering and thus is considerably larger than the usually stated limit of θ 2 ⪡ ( kR) −1. Pair correlations of dimensions ν c can be determined only if kν c ⪢ 1. Furthermore, only those correlations can be extracted reliably for which ν c larger than the range of the projectile-nucleon interaction.

On scattering by nuclei at high energies

We study how nucleon-nucleon correlations influence high-energy scattering by nuclei. Using the multiple scattering theory of Kerman, McManus, and Thaler, we formulate the scattering problem in terms of an infinite system of coupled equations. For the calculation of elastic scattering, we propose replacing the infinite system by a pair of coupled equations, containing the elastic channel, and another effective one which carries the inelastic strength. The coupling potential is proportional to the nuclear pair correlation function, and a first approximation to the potential is obtained. These equations are accurate to the extent that the pair correlations and not higher-order correlations are important. There are no restrictions to forward scattering, nor are any “on the energy shell” approximations made. In order to obtain insight into the structure of the many-channel S-matrix for high energies, the infinite system of coupled equations is solved by semi-classical methods, and explicit formulae for the S-matrix elements as a function of various potentials and the nuclear pair correlation function are obtained. We show how properties of excited target states may complicate a reliable extraction of correlations in high-energy scattering by nuclei. A close relation between our semi-classical solution and Glauber's multiple scattering is established. A numerical study of high-energy nucleon-nucleus scattering using the methods developed in this paper is under way.

Collective modes associated with nuclear pair correlations

The general properties of the pair addition and the pair removal modes are investigated within the framework of the microscopic model with the pairing forces.

Über die Anwendung von Greenfunktionen zur Beschreibung von Paarungseffekten bei deformierten Atomkernen

The theory of nuclear pairing correlations, recently developed byMigdal, introduces only one constant, which should be nearly the same for all nuclei. This Green's function method is compared with the well known BCS-theory. The constant is fitted and the problem has been solved by numerical methods. We found that the constant varies likeA−1/2. The individual magnitude of the pairing energy depends strongly on the single particle level density which determines the value of the renormalized coupling constant of the pairinteraction. This enables us to reproduce the deviations of the pairing energy from the curveδn=11,2A−1/2. The results are independent from the choice of the cut-off level.

Nuclear Relaxation and Pair Correlation in Paramagnets

Nuclear Relaxation and Pair Correlation in Paramagnets

Two-nucleon correlation function and final-state interactions

A triple coincidence experiment in which an inelastically scattered projectile sends off two closely correlated, high-momentum nucleons from a nucleus is proposed as a direct means of measuring the short range nuclear pair correlation function. An analysis based on potential theory shows that the Fourier transform of the correlation function is amenable to the relative momentum spectrum of the ejected nucleon pair. Final state interactions modify the low momentum components of this spectrum. However, marked characteristics of the short range effects that are not present in a pure independent particle model are expected to persist. These characteristics manifest themselves in the high Fourier components of the relative momentum spectrum.

Optical properties of low-energy pions in nuclei

A simple nonlocal potential for low-energy pions in finite nuclei is calculated from the amplitudes for πN scattering and for π production in NN collisions. The potential includes absorption and has no free parameters. The appropriate multiple scattering equations are derived in the coordinate representation with nuclear pair correlations included. Owing to the large mass of the scatterers the pion field behaves nearly classically. It is shown that short-range pair correlations are important in the multiple scattering owing to the dominant dipole component in the πN scattering. To a good approximation this produces a Lorentz-Lorenz effect analogous to the one occurring in the scattering of electromagnetic waves in dense media. The contributions of Fermi motion to the potential are shown to be small. The isospin part of the potential is derived and is shown to have a tensor component in addition to the ordinary vector one. The former gives rise to direct double charge exchange of pions; its strength becomes important for high momentum pions. The potential is further found to have a term which gives a small but possibly observable hyperfine coupling. A comparison is made between predictions of the potential and experimental data on level shifts and widths for π mesic atoms. Satisfactory agreement is found within experimental uncertainties which is quantitative evidence for nuclear pair-absorption of pions. There is also some indication of short range anticorrelations between nucleons.

The disappearance of nuclear pairing correlations as influenced by rotation

The disappearance of nuclear pairing correlations as influenced by rotation

Nuclear structure and pairing correlations for the heavy elements

The advent of BCS calculations for deformed nuclei has brought extensive progress in understanding many nuclear properties. With the aid of fast and versatile computer facilities a number of previously-made simplifying assumptions have been tested. (1) By use of delta- force matrix elements between Nilsson eigenfunctions, instead of a constant- G pairing force, some feeling for the nature of fluctuations in δ ν values and their variation as the systems are filled is gained. (2) By carrying the usually-neglected G vv′ matrix elements of Belyaev a generally uniform lowering of Nilsson levels is observed as the system is filled. (3) The systematic effects on variational energies and average configuration mixing caused by projecting fixed-particle terms from BCS wave functions (PBCS) is studied. (4) The variation has been performed on the fixed-particle expression (FBCS). The FBCS method gives the lowest possible energies for a variational solution of the projected BCS form. The FBCS method is found always to lead to non-trivial solutions no matter how weak the residual force strength; the abrupt transition between superfluid and normal states of the nuclear fluid, predicted by the BCS method, is thus found to be a spurious result. The FBCS method always gives more configuration mixing than PBCS. The PBCS energies are close to FBCS in the region of high pairing correlation but are worse where pairing correlation is lower. This tendency is reflected strikingly in the odd-even mass differences, where the PBCS method gives differences much too large in the region of low pairing correlation. Most of the simpler methods are found to predict too-high level densities for odd-proton systems, relative to the FBCS method.