Density Functional Energy Calculations Research Articles

Fermi operator expansion method for nuclei and inhomogeneous matter with a nuclear energy density functional

The nuclear energy density functional method at finite temperature is a useful tool for studies of nuclear structure at high excitation, and also for researches of nuclear matter involved in explosive stellar phenomena and neutron stars. However, its unrestricted calculation requires large computational costs for the three-dimensional coordinate-space solvers, especially for the Hamiltonian matrix diagonalization and (or) the Gram-Schmidt orthonormalization of the single-particle wave functions. We test numerical performance of the Fermi operator expansion method, that requires neither the diagonalization nor the Gram-Schmidt orthonormalization, for finite nuclei and inhomogeneous nuclear matter. The method is applied to isolated finite N=Z nuclei and to non-uniform symmetric nuclear matter at finite temperature, which turns out be very effective with the three-dimensional coordinate-space representation, especially at high temperature. The Fermi operator expansion method is a useful tool for studies of various nuclear phases at finite temperature with the energy density functional calculations. The method is suitable for massively parallel computing with distributed memory. Furthermore, when the space size is large, the calculation may benefit from its order-N scaling property.

Structure of $$^{128,129,130}$$Xe through multi-reference energy density functional calculations

Recently, values for the Kumar quadrupole deformation parameters of the nucleus $^{130}$Xe have been computed from the results of a Coulomb excitation experiment, indicating that this xenon isotope has a prominent triaxial ground state. Within a different context, it was recently argued that the analysis of particle correlations in the final states of ultra-relativistic heavy-ion collisions performed at the Large Hadron Collider (LHC) points to a similar structure for the adjacent isotope, $^{129}$Xe. In the present work, we report on state-of-the-art multi-reference energy density functional calculations that combine projection on proton and neutron number as well as angular momentum with shape mixing for the three isotopes $^{128,129,130}$Xe using the Skyrme-type pseudo-potential SLyMR1. Exploring the triaxial degree of freedom, we demonstrate that the ground states of all three isotopes display a very pronounced triaxial structure. Moreover, comparison with experimental results shows that the calculations reproduce fairly well the low-energy excitation spectrum of the two even-mass isotopes. By contrast, the calculation of $^{129}$Xe reveals some deficiencies of the effective interaction.

Precision Determination of the Neutral Weak Form Factor of ^{48}Ca.

We report a precise measurement of the parity-violating (PV) asymmetry A_{PV} in the elastic scattering of longitudinally polarized electrons from ^{48}Ca. We measure A_{PV}=2668±106(stat)±40(syst) parts per billion, leading to an extraction of the neutral weak form factor F_{W}(q=0.8733 fm^{-1})=0.1304±0.0052(stat)±0.0020(syst) and the charge minus the weak form factor F_{ch}-F_{W}=0.0277±0.0055. The resulting neutron skin thickness R_{n}-R_{p}=0.121±0.026(exp)±0.024(model) fm is relatively thin yet consistent with many model calculations. The combined CREX and PREX results will have implications for future energy density functional calculations and on the density dependence of the symmetry energy of nuclear matter.

Experimental and theoretical characterization of Al3Sc precipitates in Al–Mg–Si–Cu–Sc–Zr alloys

Abstract The structure of Al3Sc precipitates in Al –Mg– Si–Cu – Sc –Zr alloys annealed at 450 °C was investigated using transmission electron microscopy and energy-dispersive X-ray spectroscopy. The Al3Sc particles contain mainly Sc and small amounts of Zr, Si, and Cu. The addition of Zr limits the size of the Al3Sc precipitates to about 10– 30 nm, and these precipitates are coherent with the matrix. Density functional energy calculations showed that exceptionally small energies are required to dissolve Zr and Cu in Sc and Al sublattices, respectively. On the contrary, Cu, Mg, and Si are difficult to be dissolved in the Sc sublattice.

Reduced spin-orbit splitting in 35Si: Weak binding or density-depletion effect?

The reduction of the neutron spin-orbit (SO) splitting 2p3/2−2p1/2 between the 41Ca and 35Si isotones is a unique feature throughout the chart of nuclides, as the SO splitting usually increases with decreasing A. The way it is reduced, either gradually between 41Ca and 35Si or abruptly between 37S and 35Si, as well as the origin of its reduction, whether from the weak binding energy of the p states or from the sudden depletion in the central proton density moving to 35Si, are subject of debate. The results reported here using the self-consistent Covariant Energy Density Functional calculations with the DD-ME2 parametrization rather point to an abrupt, local decrease in 35Si, and to the large dominance of the central density depletion effect. It is concluded that weak binding, central density depletion as well as correlations must be taken into account to fully evaluate the amplitude and causes of this SO splitting reduction. To broaden the scope of the present work, discussions also include the recent experimental results of proton and neutron SO splittings around 132Sn and 56Ni.

Macroscopic and microscopic description of low-energy collective states in Se86,88

Algebraic collective model (ACM) calculations for the even-even selenium isotopes $^{86,88}\mathrm{Se}$ have been performed to describe collective properties of these two transitional nuclei with only two and four valence neutrons, respectively. From the perspective of the ACM the nuclei turn out to be only weakly deformed, described by a very $\ensuremath{\gamma}$-soft potential. This is confirmed by Skyrme and Gogny energy density functional calculations. Additionally, the fully microscopic symmetry conserving configuration mixing method with the Gogny interaction has been applied to the study of these isotopes. The results show the transitional character of $^{86}\mathrm{Se}$ and the relevant role of the triaxial degree of freedom in the structure of these nuclei.

Norm overlap between many-body states: Uncorrelated overlap between arbitrary Bogoliubov product states

State-of-the-art multi-reference energy density functional calculations require the computation of norm overlaps between different Bogoliubov quasiparticle many-body states. It is only recently that the efficient and unambiguous calculation of such norm kernels has become available under the form of Pfaffians~[L. M. Robledo, Phys. Rev. C79, 021302 (2009)]. The goals of this work is (i) to propose and implement an alternative to the Pfaffian method to compute unambiguously the norm overlap between arbitrary Bogoliubov quasiparticle states and (ii) to extend the first point to explicitly correlated norm kernels at play in recently developped particle-number-restored Bogoliubov coupled-cluster (PNR-BCC) and particle-number-restored many-body perturbation (PNR-BMBPT) ab initio theories~[T. Duguet and A. Signoracci, J. Phys. G44, 015103 (2017)]. Point (i) constitutes the purpose of the present paper while point (ii) is addressed in a forthcoming companion paper. We generalize the method used in~[T. Duguet and A. Signoracci, J. Phys. G44, 015103 (2017)] to obtain the norm overlap between arbitrary Bogoliubov product states under a closed-form expression. The formula is physically intuitive, accurate, versatile and relies on elementary linear algebra operations. It equally applies to norm overlaps between Bogoliubov states of even or odd number parity. Numerical applications illustrate these features and provide a transparent representation of the content of the norm overlaps. Furthermore, the closed-form expression extends naturally to correlated overlaps at play in PNR-BCC and PNR-BMBPT. As such, the straight overlap between Bogoliubov states is the zeroth-order reduction of more involved norm kernels to be studied in the forthcoming paper.

Fission fragment charge and mass distributions inPu239(n,f)in the adiabatic nuclear energy density functional theory

Background: Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the $r$ process to fuel cycle optimization for nuclear energy. The need for a predictive theory applicable where no data are available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics.Purpose: In this work, we calculate the pre-neutron emission charge and mass distributions of the fission fragments formed in the neutron-induced fission of $^{239}\mathrm{Pu}$ using a microscopic method based on nuclear density functional theory (DFT).Methods: Our theoretical framework is the nuclear energy density functional (EDF) method, where large-amplitude collective motion is treated adiabatically by using the time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). In practice, the TDGCM is implemented in two steps. First, a series of constrained EDF calculations map the configuration and potential-energy landscape of the fissioning system for a small set of collective variables (in this work, the axial quadrupole and octupole moments of the nucleus). Then, nuclear dynamics is modeled by propagating a collective wave packet on the potential-energy surface. Fission fragment distributions are extracted from the flux of the collective wave packet through the scission line.Results: We find that the main characteristics of the fission charge and mass distributions can be well reproduced by existing energy functionals even in two-dimensional collective spaces. Theory and experiment agree typically within two mass units for the position of the asymmetric peak. As expected, calculations are sensitive to the structure of the initial state and the prescription for the collective inertia. We emphasize that results are also sensitive to the continuity of the collective landscape near scission.Conclusions: Our analysis confirms that the adiabatic approximation provides an effective scheme to compute fission fragment yields. It also suggests that, at least in the framework of nuclear DFT, three-dimensional collective spaces may be a prerequisite to reach 10% accuracy in predicting pre-neutron emission fission fragment yields.

Skyrme functional from a three-body pseudopotential of second order in gradients: Formalism for central terms

In one way or the other, all modern parametrizations of the nuclear energy density functional (EDF) do not respect the exchange symmetry associated with Pauli's principle. It has been recently shown that this practice jeopardizes multi-reference (MR) EDF calculations by contaminating the energy with spurious self-interactions that, for example, lead to finite steps or even divergences when plotting it as a function of collective coordinates. As of today, the only viable option to bypass these pathologies is to rely on EDF kernels that enforce Pauli's principle from the outset by strictly and exactly deriving from a genuine, i.e. density-independent, Hamilton operator. We wish to develop the most general Skyrme-like EDF parametrization containing linear, bilinear and trilinear terms in the density matrices with up to two gradients, under the key constraint that it derives strictly from an effective Hamilton operator. The most general three-body Skyrme-like pseudo-potential containing up to two gradient operators is constructed to generate the trilinear part. The present study is limited to central terms. Spin-orbit and tensor will be addressed in a forthcoming paper. (See paper for full abstract)

New parametrization of Skyrme's interaction for regularized multireference energy density functional calculations

Background: Symmetry restoration and configuration mixing in the spirit of the generator coordinate method based on energy density functionals have become widely used techniques in low-energy nuclear structure physics. Recently, it has been pointed out that these techniques are ill defined for standard Skyrme functionals, and a regularization procedure has been proposed to remove the resulting spuriosities from such calculations. This procedure imposes an integer power of the density for the density-dependent terms of the functional. At present, only dated parametrizations of the Skyrme interaction fulfill this condition.Purpose: To construct a set of parametrizations of the Skyrme energy density functional for multireference energy density functional calculations with regularization using the state-of-the-art fitting protocols.Method: The parametrizations were adjusted to reproduce ground-state properties of a selected set of doubly magic nuclei and properties of nuclear matter. Subsequently, these parameter sets were validated against properties of spherical and deformed nuclei.Results: Our parameter sets successfully reproduce the experimental binding energies and charge radii for a wide range of singly magic nuclei. Compared to the widely used SLy5 and to the SIII parametrization that has integer powers of the density, a significant improvement of the reproduction of the data is observed. Similarly, a good description of the deformation properties at $A\ensuremath{\sim}80$ was obtained.Conclusions: We have constructed new Skyrme parametrizations with integer powers of the density and validated them against a broad set of experimental data for spherical and deformed nuclei. These parametrizations are tailor-made for regularized multireference energy density functional calculations and can be used to study correlations beyond the mean field in atomic nuclei.

Qvalue and half-life of double-electron capture in184Os

Os-184 has been excluded as a promising candidate for the search of neutrinoless double-electron capture. High-precision mass measurements with the Penning-trap mass spectrometer TRIGA-TRAP resulted in a marginal resonant enhancement with = -8.89(58) keV excess energy to the 1322.152(22) keV 0+ excited state in W-184. State-of-the-art energy density functional calculations are applied for the evaluation of the nuclear matrix elements to the excited states predicting a strong suppression due to the large deformation of mother and daughter states. The half-life of the transition in Os-184 exceeds T_{1/2} > 1.3 10^{29} years for an effective neutrino mass of 1 eV.

Infrared spectra and density functional calculations of the M←NCCCH3, η2–M(NC)–CH3, CH3–MNC, CH2[dbnd]M(H)NC, and CH[tbnd]M(H)2NC complexes produced by reactions of Group 6 metal atoms with acetonitrile

The M←NCCH3, η2–M(NC)–CH3, CH3–MNC, CH2M(H)NC, and HCM(H)2NC complexes are produced by reactions of laser-ablated Group 6 metal atoms with acetonitrile and identified from the matrix IR spectra through isotopic substitution and comparison to frequencies calculated by density functional theory in the harmonic approximation (B3LYP/6-311++(3df,3pd) and BPW91/6-311++G(3df,3pd), this all-electron set for Cr, and SDD for Mo and W). Good general agreement is shown in tables for all observed frequencies and these calculations. The products follow the reaction path proposed earlier for Group 4 and 5 metal reactions, which are supported by density functional energy calculations. The M←NCCH3 complex strengthens the assumption that the lone electron pair on the N-end first attracts the metal atom. The π-complexes show that the back-donation of the Group 6 metal is weaker than those of Zr and Group 5 metals. The observed products clearly show the increasing preference for the higher oxidation-state complex with increasing atomic number of the Group 6 metal, and subsequent photolysis increases the yields for the higher oxidation-state complexes. The methylidyne isocyanide HCW(H)2NC has a Cs structure similar to the HCW(H)2X complexes prepared earlier.

New analysis method of the halo phenomenon in finite many-fermion systems: First applications to medium-mass atomic nuclei

A new analysis method to investigate halos in finite many-fermion systems is designed, as existing characterization methods are proven to be incomplete/inaccurate. A decomposition of the internal wave-function of the $N$-body system in terms of overlap functions allows a model-independent analysis of medium-range and asymptotic properties of the internal one-body density. The existence of a spatially decorrelated region in the density profile is related to the existence of three typical energy scales in the excitation spectrum of the $(N\ensuremath{-}1)$-body system. A series of model-independent measures, taking the internal density as the only input, are introduced. The new measures allow a quantification of the potential halo in terms of the average number of fermions participating to it and of its impact on the system extension. Those new ``halo factors'' are validated through simulations and applied to results obtained through energy density functional calculations of medium-mass nuclei. Performing spherical Hartree-Fock-Bogoliubov calculations with state-of-the-art Skyrme plus pairing functionals, a collective halo is predicted in drip-line Cr isotopes, whereas no such effect is seen in Sn isotopes.

Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

Multi reference calculations along the lines of the generator coordinate method or the restoration of broken symmetries within the nuclear energy density functional (EDF) framework are becoming a standard tool in nuclear structure physics. These calculations rely on the extension of a single-reference energy functional, of the Gogny or the Skyrme types, to nondiagonal energy kernels. There is no rigorous constructive framework for this extension so far. The commonly accepted way proceeds by formal analogy with the expressions obtained when applying the generalized Wick theorem to the nondiagonal matrix element of a Hamilton operator between two product states. It is pointed out that this procedure is ill defined when extended to EDF calculations as the generalized Wick theorem is taken outside of its range of applicability. In particular, such a procedure is responsible for the appearance of spurious divergences and steps in multi reference EDF energies, as was recently observed in calculations restoring particle number or angular momentum. In the present work, we give a formal analysis of the origin of this problem for calculations with and without pairing, i.e., constructing the density matrices from either Slater determinants or quasiparticle vacua. We propose a method to regularize nondiagonal energy kernels such that divergences and steps are removed from multi reference EDF energies. Such a removal is a priori quasiparticle-basis dependent. A special feature of the method we use to proceed to the actual regularization is that it singles out one basis among all possible ones. The regularization method is applicable to calculations based on any symmetry restoration or generator coordinate but is limited to EDFs depending only on integer powers of the normal and anomalous density matrices. Eventually, the method is formally illustrated for particle-number restoration and is specified to configuration mixing calculations based on Slater determinants.

Superheavy Elements from r-Process Calculations with an Energy-Density Mass Formula

Astrophysical superheavy-element synthesis by rapid neutron capture is studied by utilizing nuclear masses from energy-density functional calculations and a new $\ensuremath{\beta}$-decay formula in a modified r-process code. We find reduced abundances for elements with large atomic numbers relative to those predicted by conventional mass formulas. Calculations have been performed with fission barriers corresponding to both large and small values of the surface-symmetry contribution to the deformation energy. Our results, based on a different mass-formula parametrization, show that it is possible to synthesize superheavy elements in the r process even for very large surface-symmetry contributions if certain mass-formula terms are modified. Also, a more realistic formula for $\ensuremath{\beta}$-decay rates yields r-process fission-cycle times which are about 1 order of magnitude shorter than previous values.