Random Phase Approximation Calculations Research Articles

Microscopic mean field approximation and beyond with the Gogny force

Fully consistent axially-symmetric-deformed quasiparticle random phase approximation calculations have been performed with the D1S Gogny force. A brief review on the main results obtained in this approach is presented. After a reminder on the method and on the first results concerning giant resonances in deformed Mg and Si isotopes, the multipole responses up to octupole in the deformed and heavy nucleus 238U are discussed. In order to analyse soft dipole modes in exotic nuclei, the dipole responses have been studied in Ne isotopes and in N=16 isotopes, for which results are presented. In these nuclei, the QRPA results on the low lying 2+ states are compared to the 5-Dimensional Collective Hamiltonian ones. © Owned by the authors, published by EDP Sciences, 2014.

Measurement of the Dipole Polarizability of the Unstable Neutron-Rich NucleusNi68

The E1 strength distribution in 68Ni has been investigated using Coulomb excitation in inverse kinematics at the R3B-LAND setup and by measuring the invariant mass in the one- and two-neutron decay channels. The giant dipole resonance and a low-lying peak (pygmy dipole resonance) have been observed at 17.1(2) and 9.55(17) MeV, respectively. The measured dipole polarizability is compared to relativistic random phase approximation calculations yielding a neutron-skin thickness of 0.17(2) fm. A method and analysis applicable to neutron-rich nuclei has been developed, allowing for a precise determination of neutron skins in nuclei as a function of neutron excess.

Beyond the random phase approximation: Improved description of short-range correlation by a renormalized adiabatic local density approximation

We assess the performance of a recently proposed renormalized adiabatic local density approximation (rALDA) for \textit{ab initio} calculations of electronic correlation energies in solids and molecules. The method is an extension of the random phase approximation (RPA) derived from time-dependent density functional theory and the adiabatic connection fluctuation-dissipation theorem and contains no fitted parameters. The new kernel is shown to preserve the accurate description of dispersive interactions from RPA while significantly improving the description of short range correlation in molecules, insulators, and metals. For molecular atomization energies the rALDA is a factor of 7(4) better than RPA(PBE) when compared to experiments, and a factor of 3(1.5) better than RPA(PBE) for cohesive energies of solids. For transition metals the inclusion of full shell semi-core states is found to be crucial for both RPA and rALDA calculations and can improve the cohesive energies by up to 0.4 eV. Finally we discuss straightforward generalizations of the method, which might improve results even further.

Giant monopole energies from a constrained relativistic mean-field approach

Background: Average energies of nuclear collective modes may be efficiently and accurately computed using a nonrelativistic constrained approach without reliance on a random phase approximation (RPA).Purpose: To extend the constrained approach to the relativistic domain and to establish its impact on the calibration of energy density functionals.Methods: Relativistic RPA calculations of the giant monopole resonance (GMR) are compared against the predictions of the corresponding constrained approach using two accurately calibrated energy density functionals.Results: We find excellent agreement---at the 2% level or better---between the predictions of the relativistic RPA and the corresponding constrained approach for magic (or semimagic) nuclei ranging from ${}^{16}$O to ${}^{208}$Pb.Conclusions: An efficient and accurate method is proposed for incorporating nuclear collective excitations into the calibration of future energy density functionals.

Deformation of Sr and Rb isotopes close to theN=Zline viaβ-decay studies using the total absorption technique

A study of the Gamow-Teller strength distributions $B$(GT) in the beta decay of ${}^{78}$Sr and ${}^{76,78}$Rb has been made using a total absorption spectrometer (TAS). Following the success in deducing the sign of the deformation for ${}^{76}$Sr, a similar approach is adopted for ${}^{78}$Sr based on a comparison of the measured $B$(GT) with quasiparticle random-phase approximation calculations. This work confirms its previously expected prolate deformation in the ground state. Conclusions about the structure of the odd-odd ${}^{76,78}$Rb isotopes have been drawn based on their measured $B$(GT) distributions.

Graphics Processing Unit acceleration of the Random Phase Approximation in the projector augmented wave method

The Random Phase Approximation (RPA) for correlation energy in the grid-based projector augmented wave (gpaw) code is accelerated by porting to the Graphics Processing Unit (GPU) architecture. The acceleration is achieved by grouping independent vectors/matrices and transforming the implementation from being memory bound to being computation/latency bound. With this approach, both the CPU and GPU implementations have been enhanced. We tested the GPU implementation on a few representative systems: molecules (O2), bulk solids (Li2O and MoO3) and molecules adsorbed on metal surfaces (N2/Ru(0001) and CO/Ni(111)). Improvements from 10× to 40× have been achieved (8-GPUs versus 8-CPUs). A realistic RPA calculation for CO/Ni(111) surface can be finished in 5.5 h using 8 GPUs. It is thus promising to employ the non-self-consistent RPA for routine surface chemistry simulations.

Giant Resonances in 40Ca and 48Ca

We present results of fully self-consistent Hartree-Fock based random phase approximation calculations of the centroid energies ECEN of isoscalar (T = 0) and isovector (T = 1) giant resonances of multipolarities L = 0 – 3 in 40Ca and 48Ca, using a wide range of commonly employed Skyrme type nucleon-nucleon effective interactions. We determined the sensitivities of ECEN and of the isotopic differences ECEN(48Ca) – ECEN(40Ca) to physical quantities, such as nuclear matter incompressibility coefficient, symmetry energy density and effective mass, associated with the Skyrme interactions. We compare the results with the available experimental data, in particular, with the recent data indicating that the ISGMR centroid energy in 48Ca is somewhat higher than in 40Ca.

Random phase approximation applied to solids, molecules, and graphene-metal interfaces: From van der Waals to covalent bonding

The random phase approximation (RPA) is attracting renewed interest as a universal and accurate method for first-principles total energy calculations. The RPA naturally accounts for long-range dispersive forces without compromising accuracy for short range interactions making the RPA superior to semi-local and hybrid functionals in systems dominated by weak van der Waals or mixed covalent-dispersive interactions. In this work we present plane wave-based RPA calculations for a broad collection of systems with bond types ranging from strong covalent to van der Waals. Our main result is the RPA potential energy surfaces of graphene on the Cu(111), Ni(111), Co(0001), Pd(111), Pt(111), Ag(111), Au(111), and Al(111) metal surfaces, which represent archetypical examples of metal-organic interfaces. Comparison with semi-local density approximations and a non-local van der Waals functional show that only the RPA captures both the weak covalent and dispersive forces which are equally important for these systems. We benchmark our implementation in the GPAW electronic structure code by calculating cohesive energies of graphite and a range of covalently bonded solids and molecules as well as the dissociation curves of H2 and H2+. These results show that RPA with orbitals from the local density approximation suffers from delocalization errors and systematically underestimates covalent bond energies yielding similar or lower accuracy than the Perdew-Burke-Ernzerhof (PBE) functional for molecules and solids, respectively.

Giant resonances in40Ca and48Ca

We present results of fully self-consistent Hartree-Fock-based random phase approximation calculations of the strength functions $S$($E$) and centroid energies ${E}_{\mathrm{CEN}}$ of isoscalar ($T=0$) and isovector ($T=1$) giant resonances of multipolarities $L=$ 0--3 in ${}^{40}$Ca and ${}^{48}$Ca, using a wide range of commonly employed 18 Skyrme-type nucleon-nucleon effective interactions. We determined the sensitivities of ${E}_{\mathrm{CEN}}$ and of the isotopic differences ${E}_{\mathrm{CEN}}$(${}^{48}$Ca) -- ${E}_{\mathrm{CEN}}$(${}^{40}$Ca) to physical quantities, such as nuclear matter incompressibility coefficient, symmetry energy density, and effective mass, associated with the Skyrme interactions and compare the results with the available experimental data.

Dipole strength distribution in56Fe

Electromagnetic dipole transitions in ${}^{56}$Fe were measured in photon-scattering experiments with a linearly polarized photon beam. The parity quantum numbers of the excited dipole states were determined by the intensity asymmetry of resonantly scattered $\ensuremath{\gamma}$ rays with respect to the polarization plane of the incident photon beam. While the summed magnetic dipole ($M1$) strength was determined as $\ensuremath{\Sigma}B(M1)\ensuremath{\uparrow}=3.52(17)$ ${\ensuremath{\mu}}_{N}^{2}$ at excitation energies between 7 and 10 MeV, the summed electric dipole ($E1$) strength below 10 MeV was obtained as $\ensuremath{\Sigma}B(E1)\ensuremath{\uparrow}=78.0(15)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ ${e}^{2}$ fm${}^{2}$. The observed $M1$ strength was compared with shell-model predictions in the $pf$ shell using the GXPF1J and KB3G effective interactions. In addition, the $E1$ strength was compared with random-phase approximation calculations with the Skyrme interaction.

Particle–particle Tamm–Dancoff approximation and particle–particle random phase approximation calculations for 18O and 18F nuclei

The nuclear structures of 18O and 18F nuclei are studied using particle–particle Tamm–Dancoff approximation (pp TDA) and particle–particle random phase approximation (pp RPA). All possible single-particle states of the allowed angular momenta are considered in the 0p and 1s–0d shells. The Hamiltonian is diagonalized in the presence of Warburton and Brown interactions. The results containing energy-level schemes and transition strength B(E2) are compared with the available experimental data.

Van der Waals interactions between hydrocarbon molecules and zeolites: Periodic calculations at different levels of theory, from density functional theory to the random phase approximation and Møller-Plesset perturbation theory

The adsorption of small alkane molecules in purely siliceous and protonated chabazite has been investigated at different levels of theory: (i) density-functional (DFT) calculations with a gradient-corrected exchange-correlation functional; DFT calculations using the Perdew-Burke-Ernzerhof (PBE) functional with corrections for the missing dispersion forces in the form of C(6)∕R(6) pair potentials with (ii) C(6) parameters and vdW radii determined by fitting accurate energies for a large molecular data base (PBE-d) or (iii) derived from "atoms in a solid" calculations; (iv) DFT calculations using a non-local correlation functional constructed such as to account for dispersion forces (vdW-DF); (v) calculations based on the random phase approximation (RPA) combined with the adiabatic-coupling fluctuation-dissipation theorem; and (vi) using Hartree-Fock (HF) calculations together with correlation energies calculated using second-order Møller-Plesset (MP2) perturbation theory. All calculations have been performed for periodic models of the zeolite and using a plane-wave basis and the projector-augmented wave method. The simpler and computationally less demanding approaches (i)-(iv) permit a calculation of the forces acting on the atoms using the Hellmann-Feynman theorem and further a structural optimization of the adsorbate-zeolite complex, while RPA and MP2 calculations can be performed only for a fixed geometry optimized at a lower level of theory. The influence of elevated temperature has been taken into account by averaging the adsorption energies calculated for purely siliceous and protonated chabazite, with weighting factors determined by molecular dynamics calculations with dispersion-corrected forces from DFT. Compared to experiment, the RPA underestimates the adsorption energies by about 5 kJ/mol while MP2 leads to an overestimation by about 6 kJ/Mol (averaged over methane, ethane, and propane). The most accurate results have been found for the "hybrid" RPA-HF method with an average error of less than 2 kJ/mol only, while RPA underestimates the adsorption energies by about 8 kJ/mol on average. MP2 overestimates the adsorption energies slightly, with an average error of 5 kJ/mol. The more approximate and computationally less demanding methods such as the vdW-DF density functional or the C(6)∕R(6) pair potentials with C(6) parameters from "atoms in a solid" calculations overestimate the adsorption energies quite strongly. Relatively good agreement with experiment is achieved with the empirical PBE+d method with an average error of about 5 kJ/mol.

Large-scale continuum random-phase approximation predictions of dipole strength for astrophysical applications

Large-scale calculations of the E1 strength are performed within the random phase approximation (RPA) based on the relativistic point-coupling mean field approach in order to derive the radiative neutron capture cross sections for all nuclei of astrophysical interest. While the coupling to the single-particle continuum is taken into account in an explicit and self-consistent way, additional corrections like the coupling to complex configurations and the temperature and deformation effects are included in a phenomenological way to account for a complete description of the nuclear dynamical problem. It is shown that the resulting E1-strength function based on the PCF1 force is in close agreement with photoabsorption data as well as the available experimental E1 strength data at low energies. For neutron-rich nuclei, as well as light neutron-deficient nuclei, a low-lying so-called pygmy resonance is found systematically in the 5-10 MeV region. The corresponding strength can reach 10% of the giant dipole strength in the neutron-rich region and about 5% in the neutron-deficient region, and is found to be reduced in the vicinity of the shell closures. Finally, the neutron capture reaction rates of neutron-rich nuclei is found to be about 2-5 times larger than those predicted on the basis of the nonrelativistic RPA calculation and about a factor 50 larger than obtained with traditional Lorentzian-type approaches.

Structural phase transitions in Si and SiO2crystals via the random phase approximation

We have assessed the performance of the non-self-consistent random phase approximation (RPA) on two pressure-induced structural phase transitions, diamond to \ensuremath{\beta}-Sn Si in Si and \ensuremath{\alpha}-quartz to stishovite in SiO${}_{2}$. The calculated equilibrium lattice properties of the four structures are in better agreement with experimental results than are those from several semilocal functionals. The energy differences between the high- and low-pressure phases are found to be 0.37 eV/Si and 0.39 eV/SiO${}_{2}$, respectively. The transition pressure obtained from our RPA calculations for diamond to \ensuremath{\beta}-Sn in Si is 12.2 GPa, in excellent agreement with the experimental value 11.3--12.6 GPa. However, the \ensuremath{\alpha}-quartz to stishovite phase-transition pressure in SiO${}_{2}$ is found to be 5.6 GPa, lower than the experimental 7.46 GPa; the Perdew-Burke-Ernzerhof (PBE) semilocal functional gives the transition pressure closest to experiment in this case. We conclude that the non-self-consistent, nonlocal RPA accurately describes the insulator-to-metal transition in Si, where semilocal density functionals tend to fail. But the RPA error cancellation that is nearly perfect in many solids, including Si, may be less perfect in solid SiO${}_{2}$, as it is in many molecules.

Evolution of the neutron resonances inAFe2Se2

Recent experiments on the alkali-intercalated iron selenides have raised questions about the symmetry of the superconducting phase. Random phase approximation calculations of the leading pairing eigenstate for a tight-binding 5-orbital Hubbard-Hund model of $A$Fe${}_{2}$Se${}_{2}$ find that a $d$-wave (${B}_{1g}$) state evolves into an extended ${s}^{\ifmmode\pm\else\textpm\fi{}}$ (${A}_{1g}$) state as the system is hole-doped. However, over a range of doping these two states are nearly degenerate. Here, we calculate the imaginary part of the magnetic spin susceptibility ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(q,\ensuremath{\omega})$ for these gaps and discuss how the evolution of neutron scattering resonances can distinguish between them.

Self-consistent RPA calculations with Skyrme-type interactions: The skyrme_rpa program

Random Phase Approximation (RPA) calculations are nowadays an indispensable tool in nuclear physics studies. We present here a complete version implemented with Skyrme-type interactions, with the spherical symmetry assumption, that can be used in cases where the effects of pairing correlations and of deformation can be ignored. The full self-consistency between the Hartree–Fock mean field and the RPA excitations is enforced, and it is numerically controlled by comparison with energy-weighted sum rules. The main limitations are that charge-exchange excitations and transitions involving spin operators are not included in this version. Program summaryProgram title: skyrme_rpa (v 1.00)Catalogue identifier: AENF_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENF_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5531No. of bytes in distributed program, including test data, etc.: 39435Distribution format: tar.gzProgramming language: FORTRAN-90/95; easily downgradable to FORTRAN-77.Computer: PC with Intel Celeron, Intel Pentium, AMD Athlon and Intel Core Duo processors.Operating system: Linux, Windows.RAM: From 4 MBytes to 150 MBytes, depending on the size of the nucleus and of the model space for RPA.Word size: The code is written with a prevalent use of double precision orREAL(8) variables; this assures 15 significant digits.Classification: 17.24.Nature of problem: Systematic observations of excitation properties in finite nuclear systems can lead to improved knowledge of the nuclear matter equation of state as well as a better understanding of the effective interaction in the medium. This is the case of the nuclear giant resonances and low-lying collective excitations, which can be described as small amplitude collective motions in the framework of the Random Phase Approximation (RPA). This work provides a tool where one starts from an assumed form of nuclear effective interaction (the Skyrme forces) and builds the self-consistent Hartree–Fock mean field of a given nucleus, and then the RPA multipole excitations of that nucleus.Solution method: The Hartree–Fock (HF) equations are solved in a radial mesh, using a Numerov algorithm. The solutions are iterated until self-consistency is achieved (in practice, when the energy eigenvalues are stable within a desired accuracy). In the obtained mean field, unoccupied states necessary for the RPA calculations are found. For all single-particle states, box boundary conditions are assumed. To solve the RPA problem for a given value of total angular momentum and parity Jπ a coupled basis is constructed and the RPA matrix is diagonalized (protons and neutrons are treated explicitly, and no approximation related to the use of isospin formalism is introduced). The transition amplitudes and transition strengths associated to given external operators are calculated. The HF densities and RPA transition densities are also evaluated.Restrictions: The main restrictions are related to the assumed spherical symmetry and absence of pairing correlations.Running time: The typical running time depends strongly on the nucleus, on the multipolarity, on the choice of the model space and of course on the computer. It can vary from a few minutes to several hours.

Electron doping evolution of the anisotropic spin excitations in BaFe2−xNixAs2

We use inelastic neutron scattering to systematically investigate the Ni-doping evolution of the low-energy spin excitations in BaFe${}_{2\ensuremath{-}x}$Ni${}_{x}$As${}_{2}$ spanning from underdoped antiferromagnet to overdoped superconductor ($0.03\ensuremath{\le}x\ensuremath{\le}0.18$). In the undoped state, BaFe${}_{2}$As${}_{2}$ changes from paramagnetic tetragonal phase to orthorhombic antiferromagnetic (AF) phase below about 138 K, where the low-energy ($\ensuremath{\le}\ensuremath{\sim}80$ meV) spin waves form transversely elongated ellipses in the $[H,K]$ plane of the reciprocal space. Upon Ni doping to suppress the static AF order and induce superconductivity, the $c$-axis magnetic exchange coupling is rapidly suppressed and the momentum distribution of spin excitations in the $[H,K]$ plane is enlarged in both the transverse and longitudinal directions with respect to the in-plane AF ordering wave vector of the parent compound. As a function of increasing Ni-doping $x$, the spin excitation widths increase linearly but with a larger rate along the transverse direction. These results are in general agreement with calculations of dynamic susceptibility based on the random phase approximation (RPA) in an itinerant electron picture. For samples near optimal superconductivity at $x\ensuremath{\approx}0.1$, a neutron spin resonance appears in the superconducting state. Upon further increasing the electron doping to decrease the superconducting transition temperature ${T}_{c}$, the intensity of the low-energy magnetic scattering decreases and vanishes concurrently with vanishing superconductivity in the overdoped side of the superconducting dome. Comparing with the low-energy spin excitations centered at commensurate AF positions for underdoped and optimally doped materials ($x\ensuremath{\le}0.1$), spin excitations in the overdoped side ($x=0.15$) form transversely incommensurate spin excitations, consistent with the RPA calculation. Therefore, the itinerant electron approach provides a reasonable description to the low-energy AF spin excitations in BaFe${}_{2\ensuremath{-}x}$Ni${}_{x}$As${}_{2}$.

Complete sets of polarization transfer observables for the208Pb(p⃗,n⃗)reaction at 296 MeV and Gamow-Teller and spin-dipole strengths for208Pb

We report measurements of cross sections and analyzing powers for reaction angles of ${\ensuremath{\theta}}_{\mathrm{lab}}={0.0}^{\ensuremath{\circ}}$--${10.0}^{\ensuremath{\circ}}$ and complete sets of polarization transfer observables and induced polarizations at ${\ensuremath{\theta}}_{\mathrm{lab}}={0.0}^{\ensuremath{\circ}}$--${7.0}^{\ensuremath{\circ}}$ for the ${}^{208}\mathrm{Pb}(\stackrel{P\vec}{p},\stackrel{P\vec}{n})$ reaction at a bombarding energy of ${T}_{p}=296$ MeV. A multipole decomposition technique was applied to the data to extract the Gamow-Teller (GT) and spin-dipole (SD) components from the continuum. The polarization observables were also used, for the first time, to separate the SD components into different spin-parity transfer ($\ensuremath{\Delta}{J}^{\ensuremath{\pi}}={0}^{\ensuremath{-}}$, ${1}^{\ensuremath{-}}$, and ${2}^{\ensuremath{-}}$) contributions. A significant strength with $\ensuremath{\Delta}{J}^{\ensuremath{\pi}}={1}^{+}$ is identified in the continuum beyond the GT giant resonance, which is due to configuration mixing effects and the isovector spin monopole (IVSM) contribution. The SD strength distributions were clearly dependent on $\ensuremath{\Delta}{J}^{\ensuremath{\pi}}$, which is reasonably reproduced by Hartree-Fock (HF) plus random phase approximation (RPA) calculations that include the Skyrme interaction with tensor components [Phys. Rev. C 83, 054316 (2011)].

Two-nucleon spectral function of the16O nucleus using the lowest-order constrained variational state-dependent correlation functions of the Reid and Av18interactions

In this work, the two-nucleon spectral functions (TNSFs) are defined in terms of the state- and the density-dependent correlation functions in the framework of the lowest-order constrained variational (LOCV) method to calculate the TNSF of the ${}^{16}\text{O}$ nucleus in the ${}^{16}\text{O}$$(e,{e}^{\ensuremath{'}}NN)$${}^{14}\text{C}$ reaction. The Reid soft-core (Reid68) and the ${\text{Av}}_{18}$ potentials are used as the internucleon interactions. Since, the short-range correlation effects are imposed on the wave functions for the individual channels (e.g., the ${}^{1}{S}_{0}$ and ${}^{3}{P}_{J}$ channels); therefore, the defect wave functions are obtained for various channels such that the high relative momenta ($p>4\phantom{\rule{0.28em}{0ex}}{\text{fm}}^{\ensuremath{-}1}$) are ignored. The resulting TNSFs for the ${}^{16}\text{O}$ nucleus are compared with those of the dressed random phase approximation (DRPA) calculations of Geurts et al. and the experimental predictions, especially those of Onderwater et al., (NIKHEF group), where reasonable agreement is found. It is shown that the optimized state-dependent defect wave functions have substantial effects on the TNSF and it is not justified to use the simplified parametrized two-body correlation functions in all of the channels. In agreement with the experimental data of Onderwater et al., the knockout of a ${}^{1}{S}_{0}$ pair proton dominates the above reaction cross section. Finally, it is demonstrated that the ${0}^{+}$ and ${2}^{+}$ peaks, which are expected to be observed in the above reaction cross section, are moved to the lower momenta of out-going protons when the state-dependent correlation functions are imposed.

Magnetic excitations in nuclei with neutron excess

The excitation of the $1^+$, $2^-$ and $3^+$ modes in $^{16}$O, $^{22}$O, $^{24}$O, $^{28}$O, $^{40}$Ca, $^{48}$Ca, $^{52}$Ca and $^{60}$Ca nuclei is studied with self-consistent random phase approximation calculations. Finite-range interactions of Gogny type, containing also tensor-isospin terms, are used. We analyze the evolution of the magnetic resonances with the increasing number of neutrons, the relevance of collective effects, the need of a correct treatment of the continuum and the role of the tensor force.