Inverse Energy-weighted Sum Rule Research Articles

Direct ab initio calculation of the 4He nuclear electric dipole polarizability

The calculation of nuclear electromagnetic sum rules by directly solving for numerous eigenstates in a large basis is numerically challenging and has not been performed for A>2 nuclei. With the significant progress of high performance computing, we show that calculating sum rules using numerous discretized continuum states obtained with the ab initio no-core shell model in the harmonic oscillator basis is achievable numerically. Specifically, we calculate the 4He electric dipole (E1) polarizability, that is an inverse energy weighted sum rule, employing the Daejeon16 NN interaction. We demonstrate that the calculations are numerically tractable as the dimension of the basis increases and are convergent. Our results for the 4He electric dipole polarizability are consistent with the most recent experimental data and are compared with those of other theoretical studies employing different techniques and various interactions.

Benchmark calculations of electromagnetic sum rules with a symmetry-adapted basis and hyperspherical harmonics

We demonstrate the ability to calculate electromagnetic sum rules with the \textit{ab initio} symmetry-adapted no-core shell model. By implementing the Lanczos algorithm, we compute non-energy weighted, energy weighted, and inverse energy weighted sum rules for electric monopole, dipole, and quadrupole transitions in $^4$He using realistic interactions. We benchmark the results with the hyperspherical harmonics method and show agreement within $2\sigma$, where the uncertainties are estimated from the use of the many-body technique. We investigate the dependence of the results on three different interactions, including chiral potentials, and we report on the $^4$He electric dipole polarizability calculated in the SA-NCSM that reproduces the experimental data and earlier theoretical outcomes. We also detail a novel use of the Lawson procedure to remove the spurious center-of-mass contribution to the sum rules that arises from using laboratory-frame coordinates. We further show that this same technique can be applied in the Lorentz integral transform method, with a view toward studies of electromagnetic reactions for light through medium-mass nuclei.

Complex-energy approach to sum rules within nuclear density functional theory

The linear response of the nucleus to an external field contains unique information about the effective interaction, correlations, and properties of its excited states. To characterize the response, it is useful to use its energy-weighted moments, or sum rules. By comparing computed sum rules with experimental values, the information content of the response can be utilized in the optimization process of the nuclear Hamiltonian or EDF. But the additional information comes at a price: compared to the ground state, computation of excited states is more demanding. To establish an efficient framework to compute sum rules of the response that is adaptable to the optimization of the nuclear EDF and large-scale surveys of collective strength, we have developed a new technique within the complex-energy FAM based on the QRPA. To compute sum rules, we carry out contour integration of the response function in the complex-energy plane. We benchmark our results against the conventional matrix formulation of the QRPA theory, the Thouless theorem for the energy-weighted sum rule, and the dielectric theorem for the inverse energy-weighted sum rule. We demonstrate that calculated sum-rule values agree with those obtained from the matrix formulation of the QRPA. We also discuss the applicability of both the Thouless theorem about the energy-weighted sum rule and the dielectric theorem for the inverse energy-weighted sum rule to nuclear density functional theory in cases when the EDF is not based on a Hamiltonian. The proposed sum-rule technique based on the complex-energy FAM is a tool of choice when optimizing effective interactions or energy functionals. The method is very efficient and well-adaptable to parallel computing. The FAM formulation is especially useful when standard theorems based on commutation relations involving the nuclear Hamiltonian and external field cannot be used.

Electric dipole response of 208Pb from proton inelastic scattering: Constraints on neutron skin thickness and symmetry energy

The electric dipole (E1) response of 208Pb has been precisely determined by measuring Coulomb excitation induced by proton scattering at very forward angles. The electric dipole polarizability, defined as inverse energy-weighted sum rule of the E1 strength, has been extracted as 20.1+-0.6 fm^3. The data can be used to constrain the neutron skin thickness of 208Pb to 0.168(+-0.009)_expt(+-0.013)_theo(+-0.021)_est fm, where the subscript "expt" refers to the experimental uncertainty, "theor" to the theoretical confidence band and "est" to the uncertainty associated with the estimation of the symmetry energy at the saturation density. In addition, a constraint band has been extracted in the plane of the symmetry energy (J) and its slope parameter (L) at the saturation density.

Nuclear response for the Skyrme effective interaction with zero-range tensor terms. III. Neutron matter and neutrino propagation

The formalism of the linear response for the Skyrme energy density functional including tensor terms derived in articles [1,2] for nuclear matter is applied here to the case of pure neutron matter. As in article [2] we present analytical results for the response function in all channels, the Landau parameters and the odd-power sum rules. Special emphasis is given to the inverse energy weighted sum rule because it can be used to detect non physical instabilities. Typical examples are discussed and numerical results shown. Moreover, as a direct application, neutrino propagation in neutron matter is investigated through its neutrino mean free path at zero temperature. This quantity turns out to be very sensitive to the tensor terms of the Skyrme energy density functional.

Nuclear response for the Skyrme effective interaction with zero-range tensor terms. II. Sum rules and instabilities

The formalism of linear response theory for Skyrme forces including tensor terms presented in article [1] is generalized for the case of a Skyrme energy density functional in infinite matter. We also present analytical results for the odd-power sum rules, with particular attention to the inverse energy weighted sum rule, $M_{-1}$, as a tool to detect instabilities in Skyrme functionals.

Low-energy isovector and isoscalar dipole response in neutron-rich nuclei

The self-consistent random phase approximation (RPA), based on the framework of relativistic energy density functionals, is employed in the study of isovector and isoscalar dipole response in $^{68}$Ni, $^{132}$Sn, and $^{208}$Pb. The evolution of pygmy dipole states (PDS) in the region of low excitation energies is analyzed as a function of the density-dependence of the symmetry energy for a set of relativistic effective interactions. The occurrence of PDS is predicted in the response to both the isovector and isoscalar dipole operators, and its strength is enhanced with the increase of the symmetry energy at saturation and the slope of the symmetry energy. In both channels the PDS exhausts a relatively small fraction of the energy-weighted sum rule but a much larger percentage of the inverse energy-weighted sum rule. For the isovector dipole operator the reduced transition probability $B(E1)$ of the PDS is generally small because of pronounced cancellation of neutron and proton partial contributions. The isoscalar reduced transition amplitude is predominantly determined by neutron particle-hole configurations, most of which add coherently, and this results in a collective response of the PDS to the isoscalar dipole operator.

Relativistic Continuum Random Phase Approximation and Applications II. Applications

The fully consistent relativistic continuum random phase approximation (RCRPA) has been constructed in the momentum representation in the first part of this paper. In this part we describe the numerical details for solving the Bethe–Salpeter equation. The numerical results are checked by the inverse energy weighted sum rules in the isoscalar giant monopole resonance, which are obtained from the constraint relativistic mean field theory and also calculated with the integration of the RCRPA strengths. Good agreement between them is achieved. We study the effects of the self-consistency violation, particularly the currents and Coulomb interaction to various collective multipole excitations. Using the fully consistent RCRPA method, we investigate the properties of isoscalar and isovector collective multipole excitations for some stable and exotic from light to heavy nuclei. The properties of the resonances, such as the centroid energies and strength distributions are compared with the experimental data as well as with results calculated in other models.

Isoscalar Giant Monopole Resonance in Relativistic Continuum Random Phase Approximation

The isoscalar giant monopole resonance (ISGMR) in nuclei is studied in the framework of a fully consistent relativistic continuum random phase approximation (RCRPA). In this method the contribution of the continuum spectrum to nuclear excitations is treated exactly by the single particle Green's function technique. The negative energy states in the Dirac sea are also included in the single particle Green's function in the no-sea approximation. The single particle Green's function is calculated numerically by a proper product of the regular and irregular solutions of the Dirac equation. The strength distributions in the RCRPA calculations, the inverse energy-weighted sum rule m−1 and the centroid energy of the ISGMR in 120Sn and 208Pb are analysed. Numerical results of the RCRPA are checked with the constrained relativistic mean field model and relativistic random phase approximation with a discretized spectrum in the continuum. Good agreement between them is achieved.

Sum Rule in a Consistent Relativistic Random-Phase Approximation**The project supported by National Natural Science Foundation of China with Grant Nos 19675070 and 19835010 and the Science Foundation for Nuclear Industry of China

A fully consistent relativistic random-phase approximation (RRPA) is studied in the sense that the relativistic mean-field (RMF) wavefunction of nucleus and the particlehole residual interactions in the RRPA are calculated from the same effective Lagrangian. A consistent treatment of RRPA within the RMF approximation, i.e., no sea approximation, has to include also the pairs formed from the Dirac states and occupied Fermi states, which is essential for satisfying the current conservation. The inverse energy-weighted sum rule for the isoscalar giant monopole mode is investigated in the constrained RMF. It is found that the sum rule is fulfilled only in the case where the Dirac state contributions are included.

The effect of surface diffusibility on the collective modes of metal clusters

We calculate the eigenmode spectrum for collective multipole vibrations of the electrons of alkali clusters including coupling between surface and volume plasmons. We formulate the equations of motion for the collective variables (density and velocity potential) starting from the hydrodynamic approximation. We investigate the effect of the diffusibility of the valence electrons on their collective modes, considering an equilibrium density of the valence electrons with a smooth surface profile. This effect was not considered in previous work carried out by the first author, who assumed the equilibrium density of the valence electrons to be a constant that has the bulk value. The results indicate that, within the hydrodynamic model, surface spill-out effects lead to a tendency for a red-shift from the Mie frequency and a mixing of resonance modes when the size of the cluster decreases. The eigenmodes fulfil the linear energy-weighted sum rule, the inverse energy-weighted sum rule and orthogonality relations.

Surface modes in metal clusters and cavities

Solving the hydrodynamical equations by means of a variational principle, we obtain a simultaneous description of the surface and volume modes of the valence electrons in a metal. This variational scheme, which has been previously used in the context of nuclear fluid dynamics, is applied to describe the dynamics of the valence electrons in a spherical metal cluster and of the valence electrons in the metal surrounding a spherical cavity (void). The eigenmodes fulfil the linear energy-weighted sum rule , the inverse energy-weighted sum rule and orthogonality relations. The surface modes predicted by Mie (in clusters) and by Natta (in cavities) appear in this model as natural solutions of the equations of motion and boundary conditions. We have considered a stabilized spherical jellium model. The parameters of the effective interaction are obtained by means of a variational method taking into account the experimental values of the density, compressibility and bulk energy. In the present model we have ignored the diffuseness of the equilibrium electron density, taking into account, however, the surface degrees of freedom of the valence electrons and thus allowing them to penetrate into the vacuum (outside of the jellium) when undergoing collective oscillations. The spectra of the excitation energies, and the electronic transition currents and transition densities are obtained for spherical clusters and voids.

Small-amplitude vibrations at a finite temperature in the liquid drop model

Abstract The ground state of a hot nucleus is studied in the classical limit. The equations of motion and boundary conditions of the liquid drop model are derived from the variational principle. The effect of the surface tension is taken into account. The temperature dependence of small-amplitude vibrations in the liquid drop model is investigated. It is shown that the breathing mode suffers a 6.3% decrease in energy when the temperature increases from 0 to 5 MeV. The present model allows for a description of surface modes with an A −1 2 dependence of the energy. It is also found that the surface modes will show an appreciable temperature dependence if a reasonable temperature dependence of the surface tension is postulated. It is shown that the model satisfies the energy-weighted sum rule and the inverse energy-weighted sum rule.

Inverse energy weighted sum-rules

A new derivation of the inverse energy-weighted sum-rules is given by applying the spectral distribution methods to the Rayleigh-Schrodinger perturbation theory. The scalar space result is then extended to the configurations. This is applied to obtain corrections to the ground-state energy estimates when the effective interaction is approximated by a model Hamiltonian obtained by taking linear combinations of various parts of the pairing and the Q.Q operators.

RPA correlation effects in radiative pion capture

A sum rule approach to radiative pion capture is given in terms of the energy-weighted and inverse-energy weighted sum rules. They are evaluated with RPA accuracy for Skyrme-like interactions. The static polarizabilities of the dipole, quadrupole and octupole isovector modes turn out to be the relevant quantities to determine the total branching ratios and mean photon energies.

Fluid-dynamical description of nuclear collective excitations

Fluid-dynamic models are derived in the framework of the time-dependent Hartree-Fock theory. The generalized scaling (GS) and the hydrodynamic (HD) models are discussed in detail and their applicability to nuclear giant resonances is explicitly investigated. The response function of the system (dynamic polarizability) and its connections to sum rules are analyzed in the framework of the fluid-dynamic approach. It is shown that the GS model correctly reproduces the high-frequency limit of the response function, while the HD model is more suited to investigate its static limit. In terms of sum rules, this means that the two models are expected to reproduce the cubic energy-weighted and inverse energy-weighted sum rules, respectively. Arguments based on sum rules and on the form of the boundary conditions suggest that the HD model provides a useful tool to investigate isoscalar compression excitations and, in general, isovector modes, while it dramatically fails in the description of divergency free motions, where the GS model results are much more successful. The role of surface effects on the nuclear motion is also discussed. It is shown that the effect of the surface symmetry energy term on isovector excitations can be taken into account by means of a suitable choice of the boundary conditions.

Sum rule approach to the nuclear response in the isovector spin channel

We study the global features of the response of infinite nuclear matter in the spin-isospin channel through the energy-weighted sum rules S1 and S−1. In particular we compare the outcome of the ring approximation with the exact RPA evaluation of the sum rules. We also investigate the influence of the collective character of the response, induced by the particle-hole force for longitudinal and transverse spin couplings. We show that S1 is insensitive to the collectivity of the response, as long as the Δ-degree of freedom is ignored. The inverse energy-weighted sum rule on the other hand, which is linked to the paramagnetic susceptibility, always reflects the hardening or softening of the nuclear response, due to the repulsive or attractive character of the p-h force. This quantity is well suited to the comparison with the experiments, which we perform for 12C and 56Fe.

A statistical approach to perturbation theory and inverse‐energy‐weighted sum rules

A distribution function formalism is developed for perturbation theory and inverse‐energy‐weighted sum‐rules (IEWSR). The second‐order perturbation is linear in the CLT (central limit theorem) limit. A polynomial expansion is also developed for the same. Finally, some applications are indicated.

Magnetic susceptibility and M1 transitions in 208Pb

M1 transitions in 208Pb are studied by evaluating energy-weighted and inverse energy-weighted sum-rules. The role of the nuclear interaction is widely discussed. It is shown that the nuclear potential increases the energy-weighted sum rule and lowers the inverse energy-weighted sum rule, with respect to the prediction of the pure shell model. Values of strengths and excitation energies are compared with experimental results and other theoretical calculations.

A relation between adiabatic inertial parameters and the cubic inverse energy-weighted sum rule

In the case of a collective path generated by a constrained Hartree-Fock calculation, we derive a simple expression for the adiabatic inertial parameter. This parameter is given by the cubic inverse energy-weighted sum rule calculated in the random phase approximation around each point of the path.