RPA Formalism Research Articles

Excitations of pygmy dipole resonances in exotic and stable nuclei via Coulomb and nuclear fields

We study heavy-ion inelastic scattering processes in neutron-rich nuclei including the full response to the different multipolarities. Among these we focus in particular on the excitation of low-lying dipole states commonly associated to the pygmy dipole resonance. The multipole response is described within the Hartree-Fock plus RPA formalism with Skyrme interaction. We show how the combined information from reactions processes involving the Coulomb and different mixtures of isoscalar and isovector nuclear interactions can provide a clue to reveal the characteristic features of these states. We have performed calculation for the excitation of ${}^{132}$Sn generated in the reactions with ${}^{4}$He, ${}^{40}$Ca, and ${}^{48}$Ca at several incident energies, as well as for the system ${}^{17}$O $+{\phantom{\rule{0.28em}{0ex}}}^{208}$Pb. Our results suggest that the investigation of the PDR states can be better carried out at low incident energies (below 50 MeV/nucleon). In fact, at these energies the PDR peak is relatively stronger than the giant dipole resonance (GDR) one and the narrow width of the low-lying quadrupole and octupole states should not blur its presence.

Giant and Pygmy Dipole Resonances in neutron-rich nuclei: their excitation via Coulomb and nuclear fields

We study the excitation of low-lying dipole states in very-neutron rich nuclei (so called Pygmy Dipole Resonances). The states are described within the Hartree-Fock plus RPA formalism, using different parametrizations of the Skyrme interaction. We show how the combined information from reaction processes involving the Coulomb and different mixtures of isoscalar and isovector nuclear interactions can provide a clue to reveal the characteristic features of these states.

Excitation of pygmy dipole resonance in neutron-rich nuclei via Coulomb and nuclear fields

We study the nature of the low-lying dipole strength in neutron-rich nuclei, often associated with the pygmy dipole resonance. The states are described within the Hartree-Fock plus RPA formalism, using different parametrizations of the Skyrme interaction. We show how the information from combined reaction processes involving the Coulomb and different mixtures of isoscalar and isovector nuclear interactions can provide a clue to reveal the characteristic features of these states.

Exploring the random phase approximation: Application to CO adsorbed on Cu(111)

The adsorption of CO on the Cu111 surface is investigated in the random phase approximation RPA as formulated within the adiabatic connection fluctuation-dissipation theorem. The RPA adsorption energy is obtained by adding a “local exchange-correlation correction” that is extrapolated from cluster calculations of increasing size, to the Perdew-Burke-Ernzerhof PBE value for the extended system. In comparison to density-functional theory calculations with the generalized gradient functionals PBE and AM05 and the hybrid functionals PBE0 and HSE03, we find a hierarchy of improved performance from AM05/PBE to PBE0/HSE03, and from PBE0/HSE03 to RPA, both in terms of the absolute adsorption energy as well as the adsorptionenergy difference between the atop and the hollow fcc sites. In particular, the very weak atop site preference at the PBE0/HSE03 level is further stabilized by about 0.2 eV in the RPA. The mechanism behind this improvement is analyzed in terms of the GW density of states that gives a spectral representation en par with the RPA formalism for the total energy.

Quantum interference terms in non-mesonic weak decay of Λ-hypernuclei within an RPA formalism

Single and double coincidence nucleon spectra in the Λ-hypernuclei weak decay are evaluated and discussed using a microscopic formalism. Nuclear matter is employed together with the local density approximation which allows us to analyze the 12 Λ C hypernucleus non-mesonic weak decay. Final state interactions (FSI) are included via the first order (in the nuclear residual interaction) terms to the RPA, where the strong residual interaction is modelled by a Bonn potential. At this level of approximation, these FSI are pure quantum interference terms between the primary decay ( Λ N → N N ) and ( Λ N → N N → N N ) , where the strong interaction is responsible for the last piece in the second reaction. Also the Pauli exchange contributions are explicitly evaluated. We show that the inclusion of Pauli exchange terms is important. A comparison with data is made. We conclude that the limitations in phase space in the RPA make this approximation inadequate to reproduce the nucleon spectra. This fact does not allow us to draw a definite conclusion about the importance of the interference terms.

The role of the Δ(1232) on the longitudinal response in the inclusive electron scattering reaction

We have performed a many-body calculation of the longitudinal nuclear ( e, e′) reaction employing a Second RPA (SRPA) formalism which contains the Δ(1232). More explicitly, our scheme contains RPA correlations as well as Hartree–Fock and second order self-energies, where an accurate evaluation of exchange terms is achieved. Using this formalism we have evaluated the longitudinal response function for 40Ca. We give final results at momentum transfers ranging from 300 up to 500 MeV/ c, obtaining a good agreement with data.

A New Description of Nuclear Rotational Motion in Terms of the Intrinsic Pair Mode

A new method describing nuclear rotational motion microscopically is proposed. We extract the rotational Hamiltonian by introducing the intrinsic pair modes which commute with the rotational mode. Thereby the rotational mode is not treated as zero energy mode in contrast with the conventional RPA formalism so that we circumvent the difficulty related with infrared divergence. The wave function is constructed by angular momentum projection on each intrinsic state. Without numerical integration for projection we calculate the matrix elements analytically under a certain approximation. The numerical calculations are carried out to illustrate the applicability of our method and they show that our method works well.

Inter-subband relaxation due to electron–electron scattering in quantum well structures

We present a full RPA formalism, and an efficient computational scheme for the inter-subband relaxation processes due to the electron–electron interactions in quantum well structures. Based on a full self-energy calculation, our formalism includes both single particle (e.g. Auger) and collective (plasmon) processes, and is applicable to strongly inhomogeneous, non-equilibrium steady-state systems. To show the efficacy of our formalism, we analyze two recent experiments which report different scattering rates for similar structures and similar carrier densities. We obtain quantitative agreement with the experimental results, showing that the observed difference is due to different excitation levels of the electron population. We have also shown, that the electron–plasmon scattering dominates inter-subband electron–electron scattering at low temperatures.

Computer Simulations of the Static Scattering from Model Polymer Blends

To understand the unusual composition and wave vector, q, dependent Flory interaction parameters, χ, obtained from small angle neutron scattering (SANS), we have calculated the three partial structure factors for symmetric binary polymer blends with relatively strong interactions between dissimilar monomers using molecular dynamics simulations. In agreement with past work we find that the radii of gyration of the chains are altered on blending. The following new results emerged from our simulations. The single-chain form factors in the blends follow Gaussian statistics at all distances larger than a monomer diameter, confirming the Flory−de Gennes conjecture that correlations in condensed polymer phases are screened over short length scales. The Rg's deduced from these form factors are in agreement with simulation-determined values and illustrate conclusively that the single chain term in the incompressible random-phase approximation (i-RPA) is altered on blending. The three partial structure factors obtained from the simulations, each of which follows an Ornstein−Zernike behavior, have different q dependences and extrapolate to different values in the thermodynamic limit. The different q dependences can be explained qualitatively by a compressible RPA formalism proposed by Tang and Freed. We then show that any form chosen to combine these partial structure factors into a single total scattering function will, in general, not have a q dependence that is describable by the incompressible RPA. This explains the experimentally observed q dependence of the χ parameter derived from the i-RPA. Finally, a thermodynamic analysis on the zero wave-vector limit of the partial structure factors shows that their absolute values are different from each other due to the effects of density fluctuations, as well as the different partial molar volumes of the two species in the blend. These results serve to emphasize the important role played by compressibility on the scattering obtained from simple polymer mixtures with large excess volumes on mixing.

Quenching of 1 + excitations in the double giant resonance

The electromagnetic excitation of the two-phonon isovector giant dipole resonance in relativistic projectiles incident on heavy targets can proceed via several intermediate 1 − one-phonon giant resonance states. In two-step electric dipole transitions the population of 0 +, 1 +, and 2 + two-phonon states are possible. We calculate the amplitude distribution of 1 − excitations with an RPA formalism, and use it to calculate the electromagnetic excitation of two-phonon states in second order perturbation theory and coupled-channels. We show that a conspiracy between angular momentum coupling and the strength of the electromagnetic fields suppresses contributions of 1 + states to the total cross sections.

Collective excitations and ground state correlations

A generalized RPA formalism is presented which treats pp and ph correlations on an equal footing. The effect of these correlations on the single-particle Green function is discussed and it is demonstrated that a self-consistent treatment of the single-particle Green function is required to obtain stable solutions. A simple approximation scheme is presented which incorporates for this self-consistency requirement and conserves the number of particles. Results of numerical calculations are given for 16O using a G-matrix interaction derived from a realistic one-boson-exchange potential.

Giant dipole resonance in nuclei with exotic shape

The giant dipole resonance is studied in nuclei with exotic shapes utilizing the Woods-Saxon potential for the single-particle states, and the RPA formalism for the phonon states. Requirements on translational invariance is used to determine the residual interaction including coupling constants. The low-lying resonance component in superdeformed nuclei, which corresponds to vibrations along the longer axis, almost disappears if a strong necking is formed.

Promotion of the scissors motion by consistent interactions

A relationship is found between the quantized isovector rotor and the microscopic RPA formalism. The scissors restoring force is obtained from a sum-rule approach. It is not influenced significantly by symmetry-restoring interactions, derived consistently from the deformed potential. The resulting neutron-proton interaction generates a scissors restoring force, which almost coincides with that extracted by the relative angular momentum from the deformed mean field alone. Such consistent interactions do not favour much orbital M1 strength at high energy and the scissors mode fragments mainly over the low-lying orbital 1 + excitations.

Super-RPA ground-state correlations

A new RPA formalism is developed which treats pp and ph correlations on an equal footing. The resulting comprehensive expressions for closed-shell ground-state correlation energies, for the corresponding wavefunctions, and for single-particle (and single-hole) occupation probabilities, are given. Additionally various specializations of these expressions are detailed, e.g., standard pp or ph RPA. This formalism is then applied to a pilot 16O calculation using as the realistic two-body interaction three one-boson exchange potentials which differ in their tensor strenghts. The full super-RPA results for this case and seven specializations are studied and compared. Finally, some conclusions are drawn from our numerical results.

Pairing correlations in rotating nuclei and the frequency-deformation scaling

The dynamical pairing correlations are discussed using the rotating harmonic oscillator model and the RPA formalism. The dependence of the pairing strength distribution on rotational frequency and deformation is given special attention. In particular, the influence of coupling between different oscillator shells is analyzed in detail. Explicit formulae are given for the overlap matrix elements between the rotating and nonrotating states of the cranked harmonic oscillator.

Nuclear response in an extended RPA formalism; an application to 48Ca 0103

An extension of the standard (1p1h) RPA is derived, by considering the Feynman diagram expansion of the polarization propagator and the relationship between the self-energy and the particle-hole interaction that must be fulfilled in order to obey conservation laws. The resulting Extended RPA (ERPA) equations include the dynamic coupling of 1p1h states to 2p2h states, which leads to a fragmentation of single-particle and single-hole strength and screening of the interaction by the medium. The method has been applied to 48Ca using a realistic G-matrix interaction based on meson-exchange. The results show an improved description of the response over the whole energy range upto 100 MeV. Remaining discrepancies point in the direction of further strength reduction due to short-range correlations as well as a stronger coupling to 2p2h states at low energy.

The continuum rpa with the exchange term explicitly included and its applications to the spin-isospin and longitudinal response functions

The continuum RPA formalism, taking into account the exchange term of the particle-hole interaction explicitly, is developed and applied to the spin-isospin and longitudinal response functions in the quasielastic scattering region. It is shown that the ring approximation, in which the exchange term is approximated to be of zero-range, works well in the spin-isospin channel. On the other hand, the appropriate treatment of the finite range effect in the exchange term is necessary for investigating the role of the particle-hole correlation in the longitudinal channel. The experimentally observed quenching phenomena in the longitudinal response function cannot be explained in the RPA framework with the G-matrix which is calculated in order to examine the nuclear ground-state properties.

On the description of the giant resonances within an RPA formalism with good isospin

The method developed previously for the treatment of a charge-independent hamiltonian with wave functions having isospin as a good quantum number has been carried further, giving explicit results for the matrix elements and transition probabilities. As an example of application of the formalism we performed a calculation of the well-known Gamow-Teller giant resonances.

Microscopic structure of collective 1 + states in deformed nuclei

A self-consistent cranked HFB plus RPA formalism is used to calculate the collective M1 mode, which is adjusted to explain the backbending curve and g-factors along the yrast line of 156Gd. The isovector M1 mode is fragmented into several 1 + states, depending on the relative strength of the quadrupole pairing force to the quadrupole-quadrupole force.

The isovector spin response in a finite nucleus

A polarization propagator is evaluated in a finite nucleus by using the continuum RPA formalism and applied to the isovector spin response in a quasielastic region for a large momentum transfer.