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DIRHB -- a relativistic self-consistent mean-field framework for atomic nuclei
T. Niksic, N. Paar, D. Vretenar, P. Ring
TL;DR
Nuclear energy density functionals offer a self-consistent description of nuclei, but open-shell systems require a unified treatment of mean-field and pairing correlations. DIRHB provides relativistic Hartree-Bogoliubov codes for spherical and deformed even-even nuclei, using efficient basis-space and coordinate-space calculations. The package implements established relativistic functionals and supports constrained quadrupole-deformation studies across these symmetries.
Problem
Open-shell nuclei require a unified and self-consistent treatment of mean-field and pairing correlations because phenomenological BCS pairing can be a poor approximation, especially for weakly bound nuclei.
Method
DIRHB solves stationary relativistic Hartree-Bogoliubov equations with spherical, axial, or triaxial harmonic-oscillator bases and combines configurational- and coordinate-space representations.
Results
DIRHB implements relativistic energy functionals including DD-ME2 and DD-PC1 for calculations of binding energies, radii, deformations, neutron skins, and excitation energies.
Takeaways & Limitations
The package supports self-consistent ground-state calculations for even-even nuclei with spherical, axial quadrupole, and triaxial quadrupole shapes.
Abstract
from arXiv · showhide
The DIRHB package consists of three Fortran computer codes for the calculation of the ground-state properties of even-even atomic nuclei using the framework of relativistic self-consistent mean-field models. Each code corresponds to a particular choice of spatial symmetry: the DIRHBS, DIRHBZ and DIRHBT codes are used to calculate nuclei with spherical symmetry, axially symmetric quadrupole deformation, and triaxial quadrupole shapes, respectively. Reflection symmetry is assumed in all three cases. The latest relativistic nuclear energy density functionals are implemented in the codes, thus enabling efficient and accurate calculations over the entire nuclide chart.
PROGRAM SUMMARY
DIRHB is a Fortran package for calculating ground-state properties of even-even open-shell nuclei with spherical, axial, or triaxial spatial symmetries. Its relativistic Hartree-Bogoliubov calculations use self-consistent iterations and harmonic-oscillator bases, subject to time-reversal and reflection symmetries.
- DIRHB comprises DIRHBS, DIRHBZ, and DIRHBT for spherical, axially symmetric, and triaxial quadrupole nuclear shapes, respectively.
- The codes solve stationary relativistic Hartree-Bogoliubov equations for even-even open-shell nuclei using a self-consistent iteration scheme.Matrix elements are updated with modified Broyden or linear mixing methods.
- Single-nucleon wave functions are expanded in spherical, axially symmetric, or triaxial harmonic-oscillator bases according to the imposed symmetry.
- The calculations assume time-reversal and reflection symmetries and cover open-shell even-even spherical and quadrupole-deformed nuclei.
- Running time depends on symmetry and oscillator-shell number, ranging from a few seconds for spherical cases to a few hours for triaxial test cases.The triaxial test case requires 300 Mb of RAM.
1. Introduction
Nuclear energy density functionals provide a self-consistent framework for describing ground-state and collective properties across a broad range of nuclei. DIRHB implements relativistic covariant functionals for spherical, axially deformed, and triaxial even-even open-shell nuclei.
- Energy density functionals describe nuclear ground-state properties and collective excitations from relatively light to superheavy nuclei and from stability to the drip-lines.
- Self-consistent mean-field models construct an energy density functional from one-body nucleon density matrices associated with a single product state.The approach maps the nuclear many-body problem onto a one-body problem.
- Relativistic covariant energy density functionals have achieved accuracy comparable to non-relativistic Hartree-Fock-Bogoliubov approaches based on Skyrme or Gogny interactions.
- DIRHB solves stationary relativistic Hartree-Bogoliubov equations for even-even open-shell nuclei with spherical, axial quadrupole, or triaxial quadrupole shapes.
2. Covariant density functional theory
The covariant density functional framework represents nuclear interactions through meson-exchange or point-coupling models and derives self-consistent Dirac equations and field equations from an energy density functional. DIRHB implements density-dependent functionals, including DD-ME2 and DD-PC1, for nuclear structure calculations.
- Meson-exchange models: Meson-exchange models describe Dirac nucleons coupled through σ, ω, and ρ meson fields, with electromagnetic interactions included.Heavy-meson exchange represents short-distance dynamics in the effective interaction.
- Meson-exchange models: The Hamiltonian density and total energy are obtained from the effective Lagrangian, with densities and currents built from occupied positive-energy states in the no-sea approximation.Dirac-sea effects are incorporated through adjustment of model parameters to experimental data.
- Meson-exchange models: Density-dependent meson-nucleon couplings generate a rearrangement contribution to the vector self-energy and are parameterized using nuclear-matter and finite-nucleus data.DIRHB includes the DD-ME2 density-dependent meson-exchange functional.
- Point-coupling models: Point-coupling models replace meson fields with isoscalar-scalar, isoscalar-vector, and isovector-vector contact interactions, while retaining electromagnetic proton coupling.A derivative term accounts for leading finite-range effects important for nuclear density distributions such as radii.
- Point-coupling models: Variation of the energy density functional with respect to Dirac spinors produces the Dirac equation, while density-dependent couplings contribute rearrangement terms to the vector potential.
3. Numerical implementation of the RHB equations
DIRHB solves the relativistic Hartree-Bogoliubov problem efficiently by combining configurational and coordinate-space representations. It supports constrained calculations of axial and triaxial quadrupole deformation while addressing convergence limitations through augmented Lagrangian methods.
- For deformed nuclei, DIRHB combines configurational-space and coordinate-space representations to avoid the intensive direct solution of coordinate-space integro-differential equations.The method is also applicable to spherical nuclei.
- The RHB eigenvalue problem is solved by diagonalizing a matrix in configurational space, yielding quasiparticle wave functions and a resulting density matrix.
- The density matrix is transformed to coordinate space, where scalar and vector densities are used to calculate the potentials.
- Energy surfaces are mapped by solving constrained RHB equations for axial and triaxial mass quadrupole moments.Quadratic constraints impose target multipole moments through stiffness constants and associated force terms.
- Smaller stiffness constants increase deviations from constrained quadrupole moments, whereas larger constants can disrupt self-consistent convergence.The augmented Lagrangian method is implemented to resolve this deficiency.
3.1. The spherically symmetric case
The spherical implementation uses radial Dirac-spinor equations and spherical harmonic-oscillator expansions, with analytic angular reduction and numerical radial integration. Pairing matrix elements are simplified through angular-momentum restrictions, coordinate transformations, and selection rules.
- Radial formulation: Spherical nucleon densities and meson fields depend only on radius, reducing the Dirac problem to coupled radial equations for large and small spinor components.The spinor is labeled by angular momentum, projection, parity, and isospin.
- Basis expansion: The large and small Dirac-spinor components are expanded separately in spherical harmonic-oscillator radial functions.The small components extend to Nmax + 1 to avoid spurious contributions.
- Field calculation: The angular dependence is integrated analytically, leaving radial factors that are evaluated numerically in the spherical Coulomb treatment.The direct proton potential includes the Coulomb field, whose logarithmic integrand singularity is removed analytically.
- Pairing matrix elements: Pairing matrix elements separate into spin and coordinate-space factors, with the spin-singlet projector and J = 0 restricting the relevant quantum numbers.The spatial basis transformation uses Talmi-Moshinsky brackets and harmonic-oscillator selection rules.
- Pairing matrix elements: The center-of-mass integration imposes equal quantum numbers, while angular selection rules reduce the pairing expression to a single sum over separable terms.Additional radial integrals can be computed analytically using generating functions for associated Laguerre polynomials.
3.2. Nuclei with axially symmetric quadrupole deformation
The axial implementation expands Dirac spinors and meson fields in an axially deformed oscillator basis labeled by conserved Ω and deformation parameters. Pairing and field equations are reduced using symmetry, selection rules, and basis transformations.
- Axial basis: Axial symmetry conserves Jz, whose eigenvalue Ωi labels each nucleon spinor.Parity and the third component of isospin are also conserved.
- Axial basis: The nucleon basis is an axially deformed harmonic oscillator with frequencies determined by β0 under volume conservation.Its states use cylindrical coordinates and quantum numbers for radial, axial, orbital, and spin projections.
- Dirac expansion: Large and small Dirac components are expanded independently in oscillator eigenfunctions, with their truncations offset by one major shell to avoid spurious states.The corresponding major quantum numbers are bounded by Nmax and Nmax + 1.
- Meson fields: Meson fields are expanded in the same deformed basis, producing an inhomogeneous linear system from the Klein-Gordon equations that is solved by inversion.The basis uses the same β0 and oscillator frequency as the nucleon wave functions.
- Pairing matrix elements: In the axial pairing channel, Ωtot = 0 and the spin-singlet projector restrict the matrix elements, while Talmi-Moshinsky transformations and selection rules reduce the sums.The remaining matrix elements are expressed through separable terms and generating-function integrals.
3.3. Nuclei with triaxial quadrupole shapes
The triaxial implementation uses a three-dimensional Cartesian harmonic-oscillator basis controlled by β0 and γ0, with parity, simplex, and reflection symmetries reducing the problem. Coulomb and meson fields are solved through basis and Green’s-function methods, while pairing becomes separable in Cartesian coordinates.
- Triaxial basis: Triaxial Dirac spinors are expanded in three-dimensional Cartesian harmonic-oscillator eigenfunctions labeled by α = {nx, ny, nz}.The basis is determined by ℏω0, β0, and γ0 under volume conservation.
- Symmetries and truncation: Parity and x-simplex symmetry organize the basis into four blocks, while large and small components use truncations differing by one major shell.Reflection symmetry with respect to the coordinate planes is also imposed on densities and fields.
- Coulomb interaction: The triaxial Coulomb problem avoids prohibitively large three-dimensional meshes by using a Coulomb Green’s-function method with Dirichlet boundary conditions.The Green’s function is expressed in separable form, and the surface potential uses a multipole expansion retaining monopole, quadrupole, and hexadecapole terms.
- Meson fields: Meson fields are expanded in a deformed basis and inserted into the Klein-Gordon equations, yielding an inhomogeneous linear system solved by inversion.The same deformation parameter used for nucleon wave functions is used for the field basis.
- Pairing matrix elements: The pairing interaction reduces to separable Cartesian matrix elements through spin-singlet projection, one-dimensional Talmi-Moshinsky transformations, and Hermite-polynomial generating functions.Reflection symmetry eliminates matrix elements with disallowed parity, including odd relevant quantum-number contributions.
4. Structure of the DIRHB program package
Each DIRHB code combines parameter and nucleus input with iterative self-consistent computation and post-convergence observables. The package supports symmetry-specific deformation constraints, mixing procedures, and outputs for canonical states, quasiparticles, observables, and densities.
- Program structure: Each DIRHB code consists of a Fortran program plus DIRHB.PAR and DIRHB.DAT files.DIRHB.PAR controls array dimensions and mesh-related sizes, while DIRHB.DAT contains data for the selected nucleus.
- Self-consistent computation: The self-consistent computation iterates intermediate solutions using either linear mixing or Broyden mixing.The iteration stops when changes between consecutive steps fall below epsi, while center-of-mass correction is computed after convergence.
- Post-processing: After convergence, the codes print quasiparticle and canonical-basis states, evaluate the center-of-mass correction, and compute expectation values.The output includes single-particle energies, pairing gaps, occupation probabilities, quasiparticle energies, and U/V norms.
- Constraints: DIRHBZ and DIRHBT accept quadrupole constraints, with β constrained in axial calculations and β–γ values available for the triaxial case.The stiffness constant cqad controls the constraint strength.
- Observables: The codes report nuclear observables including charge radii, quadrupole moments, deformation parameters, hexadecapole moments, and corrected binding energies.The charge-radius expression includes a 0.64 fm2 finite-proton-size term, and the center-of-mass correction is subtracted after solving the RHB equations.
- Density output: Coordinate-space density output depends on symmetry: radial data for DIRHBS, an xz-plane representation for DIRHBZ, and xy, xz, and yz planes for DIRHBT.The axial output is limited to x > 0 because of axial symmetry.