Publications Related to the Project

1. Axially Deformed Solution of the Skyrme-Hartree-Fock-Bogoliubov Equations Using the Transformed Harmonic Oscillator Basis. The program HFBTHO (v1.66p)
   M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, and P. Ring,
   Comp. Phys. Commun. 167, 43­63 (2005)
   In this work, we describe the program HFBTHO for axially deformed configurational Hartree-Fock-Bogoliubov calculations with Skyrme-forces and a 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.

2. Systematic Study of Deformed Nuclei at the Drip Lines and Beyond
   M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, S. Pittel, and D. J. Dean
   Phys. Rev. C 68, 054312 (2003)
   This work contains the first large-scale application of the code HFBTHO (v1.66p) described above. The HFB+THO framework that follows accurately reproduces the results of coordinate-space HFB calculations for spherical nuclei, including those that are weakly bound. Furthermore, it is fully automated, facilitating its use in systematic investigations of large sets of nuclei throughout the periodic table. As a first application, we have carried out calculations using the Skyrme Force SLy4 and volume pairing, with exact particle number projection following application of the Lipkin-Nogami prescription. Calculations were performed for all even-even nuclei from the proton drip line to the neutron drip line having proton numbers Z=2,4,...,108 and neutron numbers N=2,4,...,188.

3. Local Density Approximation for Proton-Neutron Pairing Correlations. Formalism
   E. Perlińska, S. G. Rohoziński, J. Dobaczewski, and W. Nazarewicz
   Phys. Rev. C 69, 014316 (2004)
   In this study we generalize the self-consistent Hartree-Fock-Bogoliubov (HFB) theory formulated in the coordinate space to the case which incorporates an arbitrary mixing between protons and neutrons in the particle-hole (p-h) and particle-particle (p-p or pairing) channels. We define the HFB density matrices, discuss their spin-isospin structure, and construct the most general energy density functional that is quadratic in local densities. The complete list of expressions required to calculate total energy is presented.

4. Particle-Number-Projected HFB method with Skyrme Forces and Delta Pairing
   M. Stoitsov, J. Dobaczewski, and W. Nazarewicz Proc. 8th International Sping Seminar on Nuclear Physics, Key Topics in Nuclear Structure, Paestum, Italy, 23-27 May 2004. Ed. by Aldo Covello (World Scientific, Singapore); pp. 167-176.
   Particle-number restoration before variation is implemented in the HFB method employing the Skyrme force and zero-range delta pairing. Results are compared with those obtained within the Lipkin-Nogami method, with or without the particle-number projection after variation. Shift invariance property is proven to be valid also in the case of density functional calculations which allows the well known singularity in PNP HFB calculations to be safely avoided.

5. Shell Energy in the Heaviest Nuclei Using the Green's Function Oscillator Expansion Method
   S. Ćwiok, W. Dudek, P. Kaszyński, and W. Nazarewicz
   Eur. Phys. J. A 23, 387-393 (2005)
   In this work, the Green's function oscillator expansion method and the generalized Strutinsky smoothing procedure are applied to shell corrections in the heaviest elements. A microscopic-macroscopic method with a finite, deformed Woods-Saxon potential is used. The stability condition for the shell correction is discussed in detail and the parameters defining the smoothing procedure are carefully determined. It is demonstrated that the spurious contribution to the total binding energy due to the unphysical particle gas that appears in the standard method can be as large as 1.5 MeV for weakly bound neutron-rich superheavy nuclei, but the effect on energy differences (e.g., alpha decay values) is fairly small.

6. Skyrme Hartree Fock Calculations of Fission Barriers of The Heaviest and Superheavy Nuclei
   A. Staszczak, J. Dobaczewski, and W. Nazarewicz
   Int. J. Mod. Phys. E 14 (2005) 395-402
   In this work, self-consistent Skyrme-Hartree-Fock (SHF) calculations of static fission barriers are presented for even-even Fermium isotopes as well as for superheavy even-even N=184 isotones. In the particle-hole channel, we use the SLy4 Skyrme parametrization, while in the particle-particle channel we take a T=1 seniority pairing force treated in the BCS approximation. The influence of reflection-asymmetric and triaxial degrees of freedom on the static fission paths are investigated.

7. Self-Consistent Study of Fission Barriers of Even-Even Superheavy Nuclei
   A. Staszczak, J. Dobaczewski, and W. Nazarewicz
   Proc. "3rd International Workshop on Nuclear Fission and Fission-Product Spectroscopy", 11-14 May 2005 Cadarache, France, AIP Conf. Proc. 798, 93 (2005).
   In this work, static fission barriers of even-even nuclei with 100≤Z≤110 are investigated using the Skyrme-Hartree-Fock model with particular attention paid to symmetry-breaking effects along the fission path. Effects of reflection-asymmetric and triaxial degrees of freedom on the fission barriers are discussed.

8. Shape coexistence and triaxiality in the superheavy nuclei
   S. Ćwiok, P.-H. Heenen, and W. Nazarewicz
   Nature 433, 705-709 (2005).
   Superheavy nuclei represent the limit of nuclear mass and charge; they inhabit the remote corner of the nuclear landscape, whose extent is unknown. The discovery of new elements with atomic numbers Z≥110 has brought much excitement to the atomic and nuclear physics communities. The existence of such heavy nuclei hangs on a subtle balance between the attractive nuclear force and the disruptive Coulomb repulsion between protons that favours fission. Here we model the interplay between these forces using self-consistent energy density functional theory; our approach accounts for spontaneous breaking of spherical symmetry through the nuclear Jahn­Teller effect. We predict that the long-lived superheavy elements can exist in a variety of shapes, including spherical, axial and triaxial configurations. In some cases, we anticipate the existence of metastable states and shape isomers that can affect decay properties and hence nuclear half-lives.

9. Large-Scale HFB Calculations for Deformed Nuclei with the Exact Particle-Number Projection
   M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, and J. Terasaki
   Eur. Phys. J. A 25, s1.567-s1.568 (2005).
   Recent theoretical advances in the large-scale HFBTHO calculations of nuclear ground-state properties are presented with the emphasis on the exact particle number projection. The applicability of the widely used Lipkin-Nogami procedure is discussed together with the analysis of the particle number projection after variation.

10. On the non-Unitarity of the Bogoliubov Transformation due to the Quasiparticle Space Truncation
    J. Dobaczewski, P.J. Borycki, W. Nazarewicz, and M. Stoitsov
    Eur. Phys. J. A 25, s1.541-s1.542 (2005).
    We show that due to the energy cutoff in the Hartree-Fock-Bogoliubov quasiparticle space, the Bogoliubov transformation becomes non-unitary. We propose a method of restoring the unitarity by introducing a truncated single-particle Hilbert space, in which the HFB equations are to be solved.

11. Nuclear Fission with Mean-Field Instantons
    J. Skalski
    Proc. Int. Workshop on "New Developments in Nuclear Self-Consistent Mean-Field Theories", Yukawa Institute for Theoretical Physics, Kyoto, Japan May 30 - June 1, 2005; YITP-W-05-01 (Soryushi-ron Kenkyu) p. B62.
    Quantum tunneling process in spontaneous nuclear fission may be analyzed in terms of imaginary-time solutions to selfconsistent mean-field theory. An advantage of such formulation is that it identifies a unique semiclassical prediction for the dominant (exponential) part of the fission half-life as instanton action, and that this prediction is based solely on the underlying mean-field theory (there is no arbitrariness in the selection of mass parameters). We try to advance a variational approach to finding instanton action, hoping that a good estimate of action may be easier to find than that of the instanton itself.

12. From Finite Nuclei to the Nuclear Liquid Drop: Leptodermous Expansion Based on the Self-consistent Mean-Field Theory
    P.-G. Reinhard, M. Bender, W. Nazarewicz, and T. Vertse
    Phys. Rev. C 73, 014309 (2006)
    The parameters of the nuclear liquid drop model, such as the volume, surface, symmetry, and curvature constants, as well as bulk radii, are extracted from the non-relativistic and relativistic energy density functionals used in microscopic calculations for finite nuclei. The microscopic liquid drop energy, obtained self-consistently for a large sample of finite, spherical nuclei, has been expanded in terms of powers of inverse nuclear radius and the isospin excess (or neutron-to-proton asymmetry). In order to perform a reliable extrapolation in the inverse radius, the calculations have been carried out for nuclei with huge numbers of nucleons. The Coulomb interaction has been ignored to be able to approach nuclei of arbitrary sizes and to avoid radial instabilities characteristic of systems with very large atomic numbers. The main contribution to the fluctuating part of the binding energy has been removed using the Green's function method to calculate the shell correction. The limitations of applying the leptodermous expansion to actual nuclei are discussed. While the leading terms in the macroscopic energy expansion can be extracted very precisely, the higher-order, isospin-dependent terms are prone to large uncertainties due to finite-size effects.

13. Fission Barriers of Superheavy Nuclei in the Skyrme-Hartree-Fock Model
    A. Staszczak, J. Dobaczewski, and W. Nazarewicz
    Int. J. Mod. Phys. E15, 302 (2006)
    Constrained Skyrme-Hartree-Fock calculations of static fission barriers are performed for even-even elements with Z=100-110 as well as for superheavy N=184 isotones. In our study we use the Skyrme parametrization SLy4 and a seniority pairing force treated in the BCS approximation. The computations are carried out applying a code that makes it possible to break all self-consistent symmetries of the nuclear mean field, including axial symmetry and reflection symmetry. The influence of reflection-asymmetric and triaxial degrees of freedom on fission barriers are discussed.

14. Large-Scale Self-Consistent Nuclear Mass Calculations
    M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz and P. Borycki
    Int.J. Mass Spectrometry 251, 243 (2006).
    The program of systematic large-scale self-consistent nuclear mass calculations that is based on the nuclear density functional theory represents a rich scientific agenda that is closely aligned with the main research directions in modern nuclear structure and astrophysics, especially the radioactive nuclear beam physics. The quest for the microscopic understanding of the phenomenon of nuclear binding represents, in fact, a number of fundamental and crucial questions of the quantum many-body problem, including the proper treatment of correlations and dynamics in the presence of symmetry breaking. Recent advances and open problems in the field of nuclear mass calculations are presented and discussed.

15. Relative kinetic energy correction to self-consistent fission barriers
    J. Skalski
    Phys. Rev. C 74, 051601(R) (2006)
    The effect of spurious relative kinetic energy removal on the fission barriers is discussed within the Skyrme Hartree-Fock method. Calculations for medium-heavy nuclei show that this correction is large and in the right direction.

16. Pairing renormalization and regularization within the local density approximation
    P.J. Borycki, J. Dobaczewski, W. Nazarewicz and M. V. Stoitsov
    Phys. Rev. C73, 044319 (2006)
    We discuss methods used in mean-field theories to treat pairing correlations within the local density approximation. Pairing renormalization and regularization procedures are compared in spherical and deformed nuclei. Both prescriptions give fairly similar results, although the theoretical motivation, simplicity, and stability of the regularization procedure makes it a method of choice for future applications.

17. Collective inertia and fission barriers within the Skyrme-Hartree-Fock theory
    A. Baran, A. Staszczak, J. Dobaczewski, and W. Nazarewicz
    Int. J. Mod. Phys. E16, 443 (2007)
    Spontaneous fission barriers, quadrupole inertia, and zero-point quadrupole-energy corrections are calculated for 252,256,258Fm in the framework of the self-consistent Skyrme-Hartree-Fock+BCS theory. Two ways of computing dynamical inertia are employed: the Gaussian Overlap Approximation to the Generator Coordinate Method and cranking ansatz. The Skyrme results are compared with those of the Gogny-Hartree-Fock-Bogolyubov model.

18. Pairing properties of superheavy nuclei
    A. Staszczak, J. Dobaczewski, and W. Nazarewicz
    Int. J. Mod. Phys. E 16, 310 (2007)
    Pairing properties of even-even superheavy N=184 isotones are studied within the Skyrme-Hartree-Fock+BCS approach. In the particle-hole channel we take the Skyrme energy density functional SLy4, while in the particle-particle channel we employ the seniority pairing force and zero-range delta-interactions with different forms of density dependence. We conclude that the calculated static fission trajectories weakly depend on the specific form of the delta-pairing interaction. We also investigate the impact of triaxiality on the inner fission barrier and find a rather strong Z dependence of the effect.

19. Bimodal fission in the Skyrme-Hartree-Fock approach
    A. Staszczak, J. Dobaczewski, and W. Nazarewicz
    Acta. Phys. Pol. B, 38, 1589 (2007)
    Spontaneous-fission properties of 256Fm, 258Fm, and 260Fm isotopes are studied within the Skyrme-Hartree-Fock+BCS framework. In the particle-hole channel we take the Skyrme SkM* effective force, while in the particle-particle channel we employ the seniority pairing interaction. Three static fission paths for all investigated heavy fermium isotopes are found. The analysis of these fission modes allows to describe observed asymmetric fission of 256Fm, as well as bimodal fission of 258Fm and symmetric fission in 260Fm.

20. Variation after particle-number projection for the Hartree-Fock-Bogoliubov method with the Skyrme energy density functional
    M. V. Stoitsov, J. Dobaczewski, R. Kirchner, W. Nazarewicz, and J. Terasaki
    Phys. Rev. C 76, 014308 (2007)
    Variation after particle-number restoration was incorporated for the first time into the Hartree-Fock-Bogoliubov framework employing the Skyrme energy density functional with zero-range pairing. The resulting projected HFB equations can be expressed in terms of the local gauge-angle-dependent densities. Results of projected calculations are compared with those obtained within the Lipkin-Nogami method in the standard version and with the Lipkin-Nogami method followed by exact particle-number projection

21. Particle-Number Projection and the Density Functional Theory
    J. Dobaczewski, M. V. Stoitsov, W. Nazarewicz, and P. -G. Reinhard
    Phys. Rev. C 76, 054315 (2007)
    In the framework of the Density Functional Theory for superconductors, we studied the restoration of the particle number symmetry by means of the projection technique. Conceptual problems were outlined and numerical difficulties discussed. Both are related to the fact that neither the many-body Hamiltonian nor the wave function of the system appear explicitly in the Density Functional Theory. Similar obstacles are encountered in self-consistent theories utilizing density-dependent effective interactions.

22. Adiabatic fusion barriers from selfconsistent calculations
    J. Skalski
    Phys. Rev. C 76, 044603 (2007)
    We studied adiabatic fusion barriers within the static Hartree-Fock method with the effective Skyrme interactions SkM* and SLy6. We discussed the problem of kinetic energy of the relative motion becoming spurious for separate fragments, relevant for fusion and fission barriers. We also discussed specific assumptions necessary to compensate for the non-uniqueness of the static method. Barriers obtained with two forces agree to within 2 MeV and seem nearly decoupled from errors in binding energies, specific to each force. For a number of reactions, comparisons are made with experimental estimates of barriers and barriers calculated with the frozen densities. The adiabatic barriers are generally lower than the experimental estimates. The offset amounts to less than 3 MeV in lighter systems and varies between zero and ~10 MeV in heavy ones. We also calculated HF energy surfaces for three heavy systems looking for a relation between adiabatic potential and the fusion hindrance at large ZTZP. One can see a link between quasi-fission and the force opposing fusion, acting inside the Coulomb barrier. One surface illustrates the identity of the adiabatic fusion barrier with the fission saddle of a compound nucleus.

23. Empirical proton-neutron interactions and nuclear density functional theory: global, regional, and local comparisons
    M. Stoitsov, R.B. Cakirli, R.F. Casten, W. Nazarewicz, and W. Satula
    Phys. Rev. Lett. 98, 132502 (2007)
    Calculations of nuclear masses, using nuclear density functional theory, are presented for even-even nuclei spanning the nuclear chart. The resulting binding energy differences can be interpreted in terms of valence proton-neutron interactions. These are compared globally, regionally, and locally with empirical values. Overall, excellent agreement is obtained. Discrepancies highlight neglected degrees of freedom and can point to improved density functionals.

24. Universal Nuclear Energy Density Functional: Computing Atomic Nuclei
    G.F. Bertsch, D.J. Dean, and W. Nazarewicz
    SciDAC Review, Issue 6, 42 (2007)
    A popular article for the SciDAC Review that also features our fission research.

25. Nuclear fission with mean-field instantons
    J. Skalski
    Phys. Rev. C 77, 064610 (2008)
    We present a description of nuclear spontaneous fission, and generally of quantum tunneling, in terms of instantons - periodic imaginary-time solutions to time-dependent mean-field equations - that allows for a comparison with more familiar and used generator coordinate (GCM) and adiabatic time-dependent Hartree-Fock (ATDHF) methods. It is shown that the action functional whose value for the instanton is the quasiclassical estimate of the decay exponent fulfils the minimum principle when additional constraints are imposed on trial fission paths. In analogy with mechanics, these are conditions of energy conservation and the velocity-momentum relations. In the adiabatic limit the instanton method reduces to the time-odd ATDHF equation, with collective mass including the time-odd Thouless-Valatin term, while the GCM mass completely ignores velocity-momentum relations. This implies that GCM inertia generally overestimates instanton-related decay rate. The very existence of the minimum principle offers a hope for a variational search for instantons, and sharply contrasts with absence of a suitable functional for real-time mean-field dynamics. After the inclusion of pairing, the instanton equations and the variational principle can be expressed in terms of the imaginary-time-dependent Hartree-Fock- Bogolyubov (TDHFB) theory. The adiabatic limit of this theory reproduces ATDHFB inertia.

26. Relative motion correction to fission barriers
    J. Skalski
    Int. J. Mod. Phys. E, in press.
    We discuss the effect of kinetic energy of the relative motion becoming spurious for separate fragments on the selfconsistent mean-field fission barriers. The treatment of the relative motion in the cluster model is contrasted with the necessity of a simpler and approximate approach in the mean-field theory. A scheme of the energy correction to the Hartree-Fock is proposed. The results obtained with the effective Skyrme interaction SLy6 show that the correction, previously estimated as ~8 MeV in A=70-100 nuclei, amounts to 4 MeV in the medium heavy nucleus 198Hg and to null in 238U. However, the corrected barrier implies a shorter fission half-life of the latter nucleus. The same effect is expected to lower barriers for multipartition (i.e. ternary fission, etc) and make hyperdeformed minima less stable.

27. Broyden's Method in Nuclear Structure Calculations
    A. Baran, A. Bulgac, M.M. Forbes, G. Hagen, W. Nazarewicz, N. Schunck and M.V. Stoitsov
    Phys. Rev. C 78, 014318 (2008).
    Broyden's method, widely used in quantum chemistry electronic-structure calculations for the numerical solution of nonlinear equations in many variables, is applied in the context of the nuclear many-body problem. Examples include the unitary gas problem, the nuclear density functional theory with Skyrme functionals, and the nuclear coupled-cluster theory. The stability of the method, its ease of use, and its rapid convergence rates make Broyden's method a tool of choice for large-scale nuclear structure calculations.

28. Deformed Coordinate-Space Hartree-Fock-Bogoliubov Approach to Weakly Bound Nuclei and Large Deformations
    J.C. Pei, M.V. Stoitsov, G.I. Fann, W. Nazarewicz, N. Schunck and F.R. Xu
    submitted to Phys. Rev. C
    The coordinate space formulation of the Hartree-Fock-Bogoliubov (HFB) method enables self-consistent treatment of mean-field and pairing in weakly bound systems whose properties are affected by the particle continuum space. Of particular interest are neutron-rich, deformed drip-line nuclei which can exhibit novel properties associated with neutron skin. To describe such systems theoretically, we developed an accurate 2D lattice Skyrme-HFB solver HFB-AX based on B-splines. Compared to previous implementations, we made a number of improvements aimed at boosting the solver's performance. These include: explicit imposition of axiality and space inversion, use of the modified Broyden's method to solve self-consistent equations, and a partial parallelization of the code. {hfbax} has been benchmarked against other HFB solvers, both spherical and deformed, and the accuracy of the B-spline expansion was tested by employing the multiresolution wavelet method. Illustrative calculations are carried out for stable and weakly bound nuclei at spherical and very deformed shapes, including constrained fission pathways. In addition to providing new physics insights, {hfbax} can serve as a useful tool to assess the reliability and applicability of coordinate-space and configuration-space HFB solvers, both existing and in development.