Energy Density Functional Formalism Research Articles

Finite-temperature calculation of the isovector giant dipole resonance in Sn120 using self-consistent inputs

Background: Although microscopic models are frequently used to investigate giant resonances built on the ground state of a nucleus, similar calculations at nonzero temperature are limited. Approaches based on the classical thermal shape-fluctuation model (TSFM) are often utilized to analyze the properties of giant dipole resonance at finite temperature. However, TSFM fails to predict the correct trend of resonance width over a wide range of temperature starting from zero.Purpose: To study the isovector giant dipole resonance (IVGDR) at finite temperature, we present an improved version of TSFM, where driving potential and IVGDR centroid energies are obtained self-consistently using the nuclear energy density functional formalism. The temperature dependence of IVGDR width in $^{120}\mathrm{Sn}$ is calculated to benchmark the proposed method.Method: Nuclear free energy surfaces and entropy surfaces are simulated at different temperatures for two different parametrizations of Skyrme energy density functionals. Moreover, IVGDR centroid energies are calculated by incorporating nuclear deformations extracted from the corresponding self-consistent densities. Subsequently, IVGDR widths are obtained within TSFM. Also, we demonstrate the role of pairing fluctuation in improving the low temperature behavior of IVGDR width.Result: We found good overall agreement of our results with the measured IVGDR widths. Specifically, except at a very low temperature, unprecedented accuracy has been achieved in TSFM. Calculated IVGDR widths are shown to be robust against the choices of driving potential and Skyrme parametrization.Conclusion: The present paper provides guidance on using TSFM based calculations in predicting the IVGDR width over a broad range of temperature. For a more consistent calculation, pairing fluctuations are required to be included as additional degrees of freedom.

Primary rainbow and Airy minima in 12C + 12C elastic scattering with families of non-monotonic potentials*

The experimental differential cross section of the 12C + 12C elastic scattering over a wide laboratory energy scale 25 ⩽ E lab ⩽ 360 MeV are analyzed within the framework of the optical model (OM) using non-monotonic (NM) nucleus–nucleus potential. The real parts of the NM potentials are derived from the energy density functional (EDF) formalism with the sudden approximation embodying the Pauli exclusion principle and the imaginary parts are taken as phenomenology. The diffractive and the refractive scattering with Airy structures in the whole angular region of the 12C + 12C elastic scattering in the above mentioned energy range are found profoundly successful in OM with the NM nucleus–nucleus potentials in five of different potential families. All the families at lower energies converge at 240 MeV to the unique potential family starting with the EDF potential. This, in conjunction with our recent findings for the 16O + 16O system in Islam et al (2021 J. Phys. G 48 075109), not only verifies the Goldberg criterion for elimination of discrete potential ambiguities for NM potentials, but also establishes that this leads uniquely to the EDF family. The 90° excitation function is also successfully reproduced with extension using NM potentials. The near- and far-side (N/F) decomposition of total elastic scattering amplitude has also been studied using our NM potentials.

Non-monotonic potential description of the cross-section, vector and tensor analyzing powers of the 6Li+12C elastic scattering at 30 and 50 MeV†

This work illustrates, for the first time, the analysis of tensor analyzing powers (T 20, T 21, T 22) along with the differential cross-section (CS) and the vector analyzing power iT 11 for the 6Li+12C elastic scattering at 30 and 50 MeVwithin the framework of an optical model (OM) using microscopic shallow non-monotonic (NM) potentials. The NM potential is generated from the energy density functional formalism (EDF) [Brueckner et al., Phys. Rev., 168 (1968) 1184] using a realistic two-nucleon interaction incorporating Pauli Exclusion principle. The shallow NM potential can describe the experimental angular distributions of CS and analyzing powers of the elastic scattering data. The OM analysis of the data at this energy does not indicate their sensitivity on the nuclear matter incompressibility K.

Cross-section, vector and tensor analyzing powers of the 6Li + 12C elastic scattering at 30 and 50 MeV: a non-monotonic potential description

This work reports, for the first time, the analysis of tensor analyzing powers (T20, T21, T22) along with the differential cross-section (CS) and the vector analyzing power iT11 for the 6Li + 12C elastic scattering at 30 and 50 MeV within the framework of an optical model (OM) using microscopic shallow non-monotonic (NM) potentials. The NM potential is generated from the energy density functional formalism (EDF) (Brueckner et al (1968) Phys. Rev. 168 1184) using a realistic two-nucleon interaction incorporating the Pauli exclusion principle. The shallow NM potential can describe the experimental angular distributions of CS and analyzing powers of the elastic scattering data. The OM analysis of the data at 30 and 50 MeV does not indicate their sensitivity on the nuclear matter incompressibility K.

Variation of nuclear matter properties in fusion reaction of the 64Ni+64Ni

In this paper, the variation of the nuclear matter properties during the fusion reaction and also its relationship with the hindrance phenomenon are examined in the 64Ni+64Ni reaction. For this purpose, the inter-nuclear potential is calculated by using the Skyrme energy density functional formalism in which the used forces are in a wide range of the incompressibility values. The obtained results indicate that by increasing bombarding energy the nuclear matter incompressibility is increasing. Also, this variation shows that nuclear matter exhibits a very soft behavior when moving from the sub-barrier to the deep sub-barrier region in this reaction, which can cause a large overlapping between the interacting nuclei. Since the repulsion arising from the Pauli exclusion principle affects this large overlapping, so it can lead to the fusion hindrance and the fall-off of the cross-sections in the 64Ni+64Ni reaction.

Clustering at the nuclear surface and symmetry energy

With in the energy density functional formalism a phenomenological theory of nuclei is developed which incorporates clustering at the nuclear surface in a general form. This explains the large values of symmetry energy extracted recently at low values of nuclear matter density. It is shown that the nuclear matter binding energy per nucleon (B/A), in the neighbourhood of zero density, must approach its value at the saturation density. The parameters of the theory are mainly constrained from the binding energies and root mean square radii of 376 spherical nuclei as well as the large values of the recently extracted symmetry energy at low densities. Importance of quartic term in symmetry energy is demonstrated. It is shown that it originates due to clustering as well as due to contribution of three-nucleon interaction in the state-of-the-art equation of state of neutron matter at and below the saturation densities. It is found that clustering significantly reduces the neutron skin thickness in nuclei.

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.

Particle-number restoration within the energy density functional formalism

We give a detailed analysis of the origin of spurious divergences and finite steps that have been recently identified in particle-number restoration calculations within the nuclear energy density functional framework. We isolate two distinct levels of spurious contributions to the energy. The first one is encoded in the definition of the basic energy density functional itself whereas the second one relates to the canonical procedure followed to extend the use of the energy density functional to multi-reference calculations. The first level of spuriosity relates to the long-known self-interaction problem and to the newly discussed self-pairing interaction process which might appear when describing paired systems with energy functional methods using auxiliary reference states of Bogoliubov or BCS type. A minimal correction to the second level of spuriosity to the multi-reference nuclear energy density functional proposed in [D. Lacroix, T. Duguet, M. Bender, arXiv:0809.2041] is shown to remove completely the anomalies encountered in particle-number restored calculations. In particular, it restores sum-rules over (positive) particle numbers that are to be fulfilled by the particle-number-restored formalism. The correction is found to be on the order of several hundreds of keVs up to about 1 MeV in realistic calculations, which is small compared to the total binding energy, but often accounts for a substantial percentage of the energy gain from particle-number restoration and is on the same energy scale as the excitations one addresses with multi-reference energy density functional methods.

Particle-number restoration within the energy density functional formalism: Nonviability of terms depending on noninteger powers of the density matrices

We discuss the origin of pathological behaviors that have been recently identified in particle-number-restoration calculations performed within the nuclear energy density functional framework. A regularization method that removes the problematic terms from the multi-reference energy density functional and which applies (i) to any symmetry restoration- and/or generator-coordinate-method-based configuration mixing calculation and (ii) to energy density functionals depending only on integer powers of the density matrices, was proposed in [D. Lacroix, T. Duguet, M. Bender, arXiv:0809.2041] and implemented for particle-number restoration calculations in [M. Bender, T. Duguet, D. Lacroix, arXiv:0809.2045]. In the present paper, we address the viability of non-integer powers of the density matrices in the nuclear energy density functional. Our discussion builds upon the analysis already carried out in [J. Dobaczewski \emph{et al.}, Phys. Rev. C \textbf{76}, 054315 (2007)]. First, we propose to reduce the pathological nature of terms depending on a non-integer power of the density matrices by regularizing the fraction that relates to the integer part of the exponent using the method proposed in [D. Lacroix, T. Duguet, M. Bender, arXiv:0809.2041]. Then, we discuss the spurious features brought about by the remaining fractional power. Finally, we conclude that non-integer powers of the density matrices are not viable and should be avoided in the first place when constructing nuclear energy density functionals that are eventually meant to be used in multi-reference calculations.

NON-MONOTONIC ALPHA- AND6Li-POTENTIALS FROM ENERGY DENSITY FUNCTIONAL FORMALISM

The present status of the α-nucleus potential, generated from the energy density functional (EDF) formalism using a realistic two-nucleon potential, which incorporates the Pauli principle, is discussed. The EDF potentials, calculated using a density distribution of α-particle that yields a binding energy of 20 MeV with a reasonable root-mean-squared radius and observed density distributions of6Li and various target nuclei, are found to be shallow and non-monotonic in character. This non-monotonic EDF potential reproduces satisfactorily the experimental elastic scattering data, particularly at energies above the Coulomb barrier. Since the elastic scattering data and the binding energies of all nuclei considered herein are well reproduced using the mean field generated from a realistic two-nucleon potential for nuclear and nucleonic matter, one may conclude to have reasonable information on the equation of states of nuclear and nucleonic matter from a very low to the saturation density from the present investigation.

Extraction of Nuclear Matter Properties from Nuclear Masses by a Model of Equation of State

The extraction of nuclear matter properties from measured nuclear masses is investigated in the energy density functional formalism of nuclei.It is shown that the volume energy and the nuclear incompressibility depend essentially on ,whereas the symmetry energy and the density symmetry coefficient as well as symmetry incompressibility depend essentially on ,where , and are the neutron and proton chemical potentials respectively, the nuclear energy,and the Coulomb energy.The obtained symmetry energy is , while other coefficients are uncertain within ranges depending on the model of nuclear equation of state.

A simple analysis of the classical hard-sphere one-component plasma. II. Density functional theory

The generalized van der Waals free energy density functional formalism of inhomogeneous fluids has been applied to study the thermodynamics and structural properties of a system of identical charged hard spheres neutralized by a uniform penetrating background. Nonlocal entropic effects are included through a simple density independent coarse-graining kernel and the hard-sphere truncation of the Coulomb interaction in the ionic atmosphere is accounted for within a mean-field approximation. In the first instance a parametric charge density of known form is introduced yielding essentially analytic results. Second, the functional is optimized with full variational flexibility to produce a theory directly analogous with the Poisson–Boltzmann approach to Coulomb fluids. A linearized version corresponding to a Debye–Hückel approximation is also discussed. The results compare favorably with Monte Carlo simulation in the regime of low to intermediate bulk density. At high volume fractions (>0.1) the functional becomes ill conditioned as the excluded volume effect is exaggerated and the mean-field hole correction fails to account for the hard-sphere structure.

Gradient expansion correction to the Dirac exchange term in statistical models for the Na atom with shell structure

AbstractIn the study of the ground‐state binding energy of atoms, relatively more attention has been paid to the gradient expansion of the kinetic energy term of the Thomas–Fermi–Dirac model than to the gradient expansion of the exchange term of this model. Recently Shih and Shih, Murphy, and Wang have focused on this problem and calculated the first gradient expansion correction to the Dirac exchange term by making use of electron densities constructed from Hartree–Fock wave functions. In previous work, aimed to introduce the shell structure of atoms via a variational procedure using the energy density functional formalism, electron densities have been obtained for the Na atom within the Thomas–Fermi–Dirac model with and without the Weizsäcker and Hodges gradient expansion corrections to the kinetic energy term. In this paper we make use of the respective electron densities and calculate the first gradient expansion corrections to the Dirac exchange term. The results show that, for the Na atom, the magnitude of this correction is about 1% of the magnitude of the total binding energy.

On the nuclear curvature energy

We present a leptodermous expansion of the nuclear binding energy which allows its decomposition into volume, surface and curvature parts. The two contributions to the curvature energy are thus identified and estimated in the framework of an energy density functional formalism. A general procedure for the calculation of the curvature energy with Skyrme like forces is outlined.

Density Functional theory of the atomic electronegativity

The atomic electronegativity, identified with the electronic chemical potential of the Energy Density Functional formalism has been studied by using an approximate functional containing gradient corrections. Only the valence electrons are taken into account and the effect of the ionic core is simulated by an empty core pseudopotential. The results are in close agreement with the experiments.

A theory of order-Disorder and antiphase domain boundary energies

A simiempirical theory of the energy of formation of binary alloys that is based on the Energy Density Functional formalism is extended to cases of incomplete order. This is used to derive a theory of order-disorder alloys, from which the ordering energy and transition temperature of several CsCl type compounds are calculated. The theory is also applied to compute the antiphase domain boundary energies for these alloys, as well as the separation distance between superpartial dislocations in CuZn.

The energy density functional formalism for excited states

It is shown that the density can be used as the basic variable for calculating the properties of excited states. The correspondence is not between an eigenstate and its density, as is the case with the ground state, but between the subspace spanned by the number of lowest-energy eigenstates and the sum of their densities. An extension of the Hohenberg-Kohn-Sham theory (1964-5) for excited states has also been developed. The equations derived are similar in form to those for the ground-state density but the interpretation is different. The lowest-order approximation of the present theory coincides with Slater's 'transition-state' theory (1974).

Calculation of ground state surface properties of simple metals including non-local effects of the exchange and correlation energy

The energy density functional formalism is used to calculate the ground state surface energies and work functions for simple metals. Non-local corrections to the exchange and correlation energies were included by means of a gradient expansion in the density. Good agreement is obtained with experiment.