Dielectric function of dense plasmas and sum rules

Optical properties can be determined by either the transverse dielectric function ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\perp}}}$(q,\ensuremath{\omega}) or the longitudinal dielectric function ${\mathrm{\ensuremath{\epsilon}}}_{\mathcal{n}}$(q,\ensuremath{\omega}) in the long wavelength limit, i.e., q\ensuremath{\rightarrow}0. By expressing ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\perp}}}$(q\ensuremath{\rightarrow}0,\ensuremath{\omega}) in terms of density response functions, we calculate the third, fifth, and seventh frequency moments of Im${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\perp}}}$(q\ensuremath{\rightarrow}0,\ensuremath{\omega}) in addition to the first frequency moment, the well-known conductivity sum rule. While the third and fifth moments reflect the inhomogeneity of the system, correlation effects contribute explicitly to the seventh moment. In addition, the frequency moments of Im[-1/${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\parallel}}}$(q\ensuremath{\rightarrow}0,\ensuremath{\omega})] can be constructed from these moments. The dielectric functions evaluated in a self-consistent field (SCF) approximation, such as the random phase approximation (RPA) or the adiabatic local density approximation (ALDA), satisfy the first three odd frequency moments provided the single particle states on which the SCF calculation is based are calculated in the corresponding SCF approximation. The seventh frequency moment constitutes a severe requirement that ${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{\ensuremath{\perp}}}$(q\ensuremath{\rightarrow}0,\ensuremath{\omega}) evaluated in the RPA or the ALDA cannot satisfy. This is demonstrated for nearly free electron systems.

The collective excitation phenomena in atomic nuclei are studied in two different formulations of the Random Phase Approximation (RPA): (i) RPA based on correlated realistic nucleon-nucleon interactions constructed within the Unitary Correlation Operator Method (UCOM), and (ii) relativistic RPA (RRPA) derived from effective Lagrangians with density-dependent meson-exchange interactions. The former includes the dominant interaction-induced short-range central and tensor correlations by means of an unitary transformation. It is shown that UCOM-RPA correlations induced by collective nuclear vibrations recover a part of the residual long-range correlations that are not explicitly included in the UCOM Hartree-Fock ground state. Both RPA models are employed in studies of the isoscalar monopole resonance (ISGMR) in closed-shell nuclei across the nuclide chart, with an emphasis on the sensitivity of its properties on the constraints for the range of the UCOM correlation functions. Within the Relativistic Quasiparticle RPA (RQRPA) based on Relativistic Hartree-Bogoliubov model, the occurrence of pronounced low-lying dipole excitations is predicted in nuclei towards the proton drip-line. From the analysis of the transition densities and the structure of the RQRPA amplitudes, it is shown that these states correspond to the proton pygmy dipole resonance.

The implementation of dielectric functions based on the Random Phase Approximation (RPA), including those by Lindhard, Kaneko, and Levine-Louie, is presented systematically, incorporating the effects of electron relaxation (finite width) and lattice interactions. A straightforward approach is proposed to introduce electron relaxation time and binding effects without needing Mermin corrections or Kramers–Kronig (KK) relations. This method yields the same dielectric function as the Mermin-corrected Lindhard and Kaneko models in several limiting cases (optical, high momentum transfer, and static limits). Moreover, this method adheres to the Bethe and F sum rule and the real and imaginary part are Kramers–Kronig pairs. Still, it shows some variation at intermediate energy and momentum transfer. Additionally, the description of dispersion at high momentum transfer is refined to account for relativistic effects. A small library containing the implementation of these dielectric functions is provided as supplementary material.

Our understanding of nuclear structure is built upon mean-field theories such as Hartree-Fock and time-dependent Hartree-Fock. The small-amplitude limit of the latter is the random phase approximation (RPA), which is widely used to model giant resonances in nuclei. Despite this popularity, RPA has been mostly validated against toy models; tests against complex models are scarce in the literature. We perform a thorough test of the RPA against full 0hw shell model (SM) calculations, including in our investigation binding energies, scalar ground-state observables, for which we develop a new method, and transition strengths. We allow deformed Hartree-Fock solutions and compare results for spherical and deformed nuclei. We obtain reasonable agreement between RPA and SM, albeit with some significant failures. Particularly, we found that the low-lying collectivity is poorly described for deformed mean-field solutions, which we interpret as incomplete symmetry restoration in RPA. Results for observables, and in particular for J2, also point out toward the same conclusion regarding the symmetries of the ground state. We also prove, both analytically and numerically, that a long-standing theorem regarding RPA is violated in the case of deformation. The worse violation appears for low-lying transitions, such as isoscalar E2, which we consider as a third argument for an incomplete symmetry restoration.

Infrared spectroscopy of solids

A random phase approximation (RPA) calculation and a direct sum over states is used to calculate second−order optical properties and van der Waals coefficients. A basis set expansion technique is used and no continuumlike functions are included in the basis. However, unlike other methods we do not force the basis functions to satisfy any sum−rule constraints but rather the formalism (RPA) is such that the Thomas−Reiche−Kuhn sum rule is satisfied exactly. Central attention is paid to the dynamic polarizability from which most of the other properties are derived. Application is made to helium and molecular hydrogen. In addition to the polarizability and van der Waals coefficients, results are given for the molecular anisotropy of H2, Rayleigh scattering cross sections, and Verdet constants as a function of frequency. Agreement with experiment and other theories is good. Other energy weighted sum rules are calculated and compare very well with previous estimates. The practicality of our method suggests its applications to larger molecular systems and other properties.

Perspectives on study of unstable nuclei near drip lines

Dynamic Response at Metal Surfaces: Violation of Sum Rules by Density-Functional-Based Random Phase Approximation

The effect of particle correlations in a classical system of charged particles moving in a static uniform background is investigated by means of a dielectric theory developed by Singwi et al. for a completely degenerate electron gas. In this theory, the short-range correlations arising from the Coulomb interaction is taken into account through a local-field correction depending on the pair correlation function. In a first approximation (I), the pair correlation function is chosen to be time independent. The extension of allowing also the pair correlation function to adjust itself to the external field results in a screening of the Coulomb potential entering the local field. The effect of this screening has been investigated for the cases of a static random-phase-approximation (RPA) dielectric function (II) and a static fully self-consistent dielectric function (III). Numerical self-consistent calculations have been carried out for the cases I-III in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. From $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ thus obtained, the Helmholtz free energy, the correlation energy, the pair correlation functions, and the isothermal compressibility have been calculated numerically, The plasma dispersion and the compressibility sum rule have also been investigated. The present method can be regarded as a rather natural extension of the RPA. Considerable improvement upon the simple Debye-H\"uckel (or, equivalently, classical RPA) is also found. The results of the present calculations are also compared with the results of other elaborate methods.

The most fundamental approach to an understanding of electronic, optical, and transport phenomena which the condensed matter physics (of conventional as well as nonconventional systems) offers is generally founded on two experiments: the inelastic electron scattering and the inelastic light scattering. This work embarks on providing a systematic framework for the theory of inelastic electron scattering and of inelastic light scattering from the electronic excitations in GaAs/Ga1−xAlxAs quantum wells. To this end, we start with the Kubo's correlation function to derive the generalized nonlocal, dynamic dielectric function, and the inverse dielectric function within the framework of Bohm-Pines’ random-phase approximation. This is followed by a thorough development of the theory of inelastic electron scattering and of inelastic light scattering. The methodological part is then subjected to the analytical diagnoses which allow us to sense the subtlety of the analytical results and the importance of their applications. The general analytical results, which know no bounds regarding, e.g., the subband occupancy, are then specified so as to make them applicable to practicality. After trying and testing the eigenfunctions, we compute the density of states, the Fermi energy, the full excitation spectrum made up of intrasubband and intersubband – single-particle and collective (plasmon) – excitations, the loss functions for all the principal geometries envisioned for the inelastic electron scattering, and the Raman intensity, which provides a measure of the real transitions induced by the (laser) probe, for the inelastic light scattering. It is found that the dominant contribution to both the loss peaks and the Raman peaks comes from the collective (plasmon) excitations. As to the single-particle peaks, the analysis indicates a long-lasting lack of quantitative comparison between theory and experiments. It is inferred that the inelastic electron scattering can be a potential alternative of the inelastic light scattering for investigating elementary electronic excitations in quantum wells.

Phase separation in dispersions of weakly-interacting particles in solutions of non-adsorbing polymer

The Coulomb excitation of the neutron rich 68Ni was measured using it as as secondary radioactive beam at 600 MeV/nucleon impinging on a 197Au target. The dipole strength in the region at around the nucleon binding energy was extracted from the gamma decay measurement. A sizable fraction of "pygmy" dipole strength, energetically located below the giant dipole resonance, is observed in this nucleus. The experimental data is discussed in connection with predictions obtained using different Random Phase Approximation (RPA) models for the dipole response, based on a representative set of Skyrme effective forces plus meson-exchange effective Lagrangians. A comparison with the experimental data for both 68Ni and 132Sn has allowed us to constrain the value of the derivative of the symmetry energy at saturation. The neutron skin radius is deduced under this constraint.

Gamow–Teller excitations attract the constant interest of physicists not only within nuclear structure but also in astrophysics. In this paper, the random phase approximation (RPA) plus particle-vibration coupling model is applied to the investigation of Gamow–Teller excitations in the doubly magic nuclei 208Pb and 132Sn. It is found that the Gamow–Teller resonance develops a large spreading width and that the excitation energies are shifted downward by phonon coupling. Good agreement with experiment is obtained for the line shape of the Gamow–Teller distribution for 208Pb using the Skyrme interactions SkM* and SGII. It is also found that the collective isovector giant dipole phonons play an important role in the width of both nuclei, and the related microscopic mechanism is analyzed in detail.

We present a comprehensive study of plasmon dispersions in simple metals and Heusler compounds based on an accurate ab-initio evaluation of the momentum and frequency dependent dielectric function in the random-phase approximation. Using a momentum-dependent tetrahedron method for the computation of the dielectric function, we extract and analyze "full" and "intraband" plasmon dispersions: The "full" plasma dispersion is obtained by including all bands, the intraband plasma dispersion by including only intraband transitions. For the simple metals silver and alu- minum, we show that the intraband plasmon dispersion has an unexpected downward slope and is therefore markedly different from the results of an effective-mass electron-gas model and the full plasmon dispersion. For the two Heusler compounds Co2FeSi and Co2MnSi, we present spectra for the dielectric function, their loss functions and plasmon dispersions. The latter exhibit the same negative intraband plasmon dispersion as found in the simple metals. We also discuss the influence of spin-mixing on the plasmon dispersion.

We derive the zero-temperature second random phase approximation using Rowe's double-commutator equation. Subsequently, we show that the zero-temperature second random phase approximation exhibits formal properties in analogy with the usual 1p-1h random phase approximation. In particular, the second random phase approximation preserves the energy-weighted sum rule.