Isoscalar Giant Monopole Resonance Research Articles

Low-energy monopole strength in exotic nickel isotopes

Low-energy strength is predicted for the isoscalar monopole response of neutron-rich Ni isotopes, in calculations performed using the microscopic Skyrme HF + RPA and relativistic RHB + RQRPA models. Both models, although based on different energy density functionals, predict the occurrence of pronounced monopole states in the energy region between 10 and 15 MeV, well separated from the isoscalar giant monopole resonance. The analysis of transition densities and corresponding particle-hole configurations shows that these states represent almost-pure neutron single hole-particle excitations. Even though their location is not modified with respect to the corresponding unperturbed states, their strength is considerably enhanced by the residual interaction. The theoretical analysis predicts the gradual enhancement of low-energy monopole strength with neutron excess.

Finite temperature effects on monopole and dipole excitations

The relativistic random phase approximation based on effective Lagrangian with density dependent meson-nucleon couplings has been extended to finite temperature and employed in studies of multipole excitations within the temperature range T = 1 − 2 MeV. The model calculations showed that isoscalar giant monopole and isovector giant dipole resonances are only slightly modified with temperature, but additional transition strength appears at low energies because of thermal unblocking of single-particle orbitals close to the Fermi level. The analysis of low-lying states shows that isoscalar monopole response in 132Sn results from single particle transitions, while the isovector dipole strength for 60Ni, located around 10 MeV, is composed of several single particle transitions, accumulating a small degree of collectivity.

Isoscalar giant resonances inCa48

The giant resonance region from 9.5 MeV ${E}_{x}$ 40 MeV in $^{48}\mathrm{Ca}$ has been studied with inelastic scattering of 240-MeV $\ensuremath{\alpha}$ particles at small angles, including 0\ifmmode^\circ\else\textdegree\fi{}. ${95}_{\ensuremath{-}15}^{+11}$% of $E$0 energy-weighted sum rule (EWSR), ${83}_{\ensuremath{-}16}^{+10}%$ of $E$2 EWSR, and 137 \ifmmode\pm\else\textpm\fi{} 20% of $E$1 EWSR were located below ${E}_{x}=40$ MeV. A comparison of the experimental data with calculated results for the isoscalar giant monopole resonance, obtained within the mean-field-based random-phase approximation, is also given.

Effect of pairing correlations on incompressibility and symmetry energy in nuclear matter and finite nuclei

The role of superfluidity in the incompressibility and in the symmetry energy is studied in nuclear matter and finite nuclei. Several pairing interactions are used: surface, mixed and isovector dependent. Pairing has a small effect on the nuclear matter incompressibility at saturation density, but the effects are significant at lower densities. The pairing effect on the centroid energy of the isoscalar Giant Monopole Resonance (GMR) is also evaluated for Pb and Sn isotopes by using a microscopic constrained-HFB approach, and found to change at most by 10% the nucleus incompressibility $K_A$. It is shown by using the Local Density Approximation (LDA) that most of the pairing effect on the GMR centroid come from the low-density nuclear surface.

The influence of the symmetry energy on the giant monopole resonance of neutron-rich nuclei analyzed in Thomas–Fermi theory

We analyze the influence of the density dependence of the symmetry energy on the average excitation energy of the isoscalar giant monopole resonance (GMR) in stable and exotic neutron-rich nuclei by applying the relativistic extended Thomas–Fermi method in scaling and constrained calculations. For the effective nuclear interaction, we employ the relativistic mean field model supplemented by an isoscalar–isovector meson coupling that allows one to modify the density dependence of the symmetry energy without compromising the success of the model for binding energies and charge radii. The semiclassical estimates of the average energy of the GMR are known to be in good agreement with the results obtained in full RPA calculations. The present analysis is performed along the Pb and Zr isotopic chains. In the scaling calculations, the excitation energy is larger when the symmetry energy is softer. The same happens in the constrained calculations for nuclei with small and moderate neutron excess. However, for nuclei of large isospin the constrained excitation energy becomes smaller in models having a soft symmetry energy. This effect is mainly due to the presence of loosely bound outer neutrons in these isotopes. A sharp increase of the estimated width of the resonance is found in largely neutron-rich isotopes, even for heavy nuclei, which is enhanced when the symmetry energy of the model is soft. The results indicate that at large neutron numbers the structure of the low-energy region of the GMR strength distribution changes considerably with the density dependence of the nuclear symmetry energy, which may be worthy of further characterization in RPA calculations of the response function.

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 resonances in the Sn nuclei and implications for the asymmetry term in the nuclear-matter incompressibility

We have investigated the isoscalar giant resonances in the Sn isotopes using inelastic scattering of 386-MeV alpha-particles at extremely forward angles, including 0deg. We have obtained completely "background-free inelastic-scattering spectra for the Sn isotopes over the angular range 0deg--9deg and up to an excitation energy of 31.5 MeV. The strength distributions for various multipoles were extracted by a multipole decomposition analysis based on the expected angular distributions of the respective multipoles. We find that the centroid energies of the isoscalar giant monopole resonance (ISGMR) in the Sn isotopes are significantly lower than the theoretical predictions. In addition, based on the ISGMR results, a value of K_{tau} = -550 \pm 100 MeV is obtained for the asymmetry term in the nuclear incompressibility. Constraints on interactions employed in nuclear structure calculations are discussed on the basis of the experimentally-obtained values for K_(infty) and K_(tau).

10.1007/s11496-008-2005-2

Accurate assessment of the value of the incompressibility coefficient, K ∞, of symmetric nuclear matter, which is directly related to the curvature of the equation of state (EOS), is needed to extend our knowledge of the EOS in the vicinity of the saturation point. We review the current status of K ∞ as determined from experimental data on isoscalar giant monopole and dipole resonances (compression modes) in nuclei by employing the microscopic theory based on the Random Phase Approximation (RPA). The importance of full self-consistent calculations is emphasized. In recent years, a comparision between RPA calculations based on either non-relativistic effective interactions or relativistic Lagrangians has been pursued in great detail. It has been pointed out that these two types of models embed different ansatz for the density dependence of the symmetry energy. This fact has consequences on the extraction of the nuclear incompressibility, as it is discussed. The comparison with other ways of extracting K ∞ from experimental data is highlighted.

Giant monopole resonance in Pb isotopes

The extraction of the nuclear incompressibility from isoscalar giant monopole resonance (GMR) measurements is analyzed. Pairing may play a role in the shift of the GMR energy between the doubly closed shell $^{208}\mathrm{Pb}$ nucleus and other Pb isotopes. Pairing effects are predicted microscopically using the constrained Hartree-Fock-Bogoliubov method. Accurate measurements of the GMR in open-shell Pb isotopes are necessary.

Description of the giant monopole resonance in the even-ASn112−124isotopes within a microscopic model including quasiparticle-phonon coupling

We have calculated the strength distributions of the isoscalar giant monopole resonance (ISGMR) in the even-$A$ tin isotopes ($A=112\text{\ensuremath{-}}124$) that were recently measured in inelastic $\ensuremath{\alpha}$ scattering. The calculations were performed within two microscopic models: the quasiparticle random phase approximation (QRPA) and the quasiparticle time blocking approximation (QTBA), which is an extension of the QRPA including quasiparticle-phonon coupling. We used a self-consistent calculational scheme based on the Hartree-Fock$+$Bardeen-Cooper-Schrieffer approximation. Within the RPA the self-consistency is full. The single-particle continuum is also exactly included at the RPA level. The self-consistent mean field and the effective interaction are derived from the Skyrme energy functional. In the calculations, two Skyrme force parametrizations were used: T5 with a comparatively low value of the incompressibility modulus of infinite nuclear matter (${K}_{\ensuremath{\infty}}=202$ MeV) and T6 with ${K}_{\ensuremath{\infty}}=236$ MeV. The T5 parametrization gives theoretical results for tin isotopes in good agreement with the experimental data including the resonance widths. The results of the ISGMR calculations in $^{90}\mathrm{Zr}$, $^{144}\mathrm{Sm}$, and $^{208}\mathrm{Pb}$ performed with these Skyrme forces are discussed and compared with the experiment.

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.

Microscopic linear response calculations based on the Skyrme functional plus the pairing contribution

A self-consistent quasiparticle random-phase approximation (QRPA) model that employs the canonical Hartree-Fock-Bogoliubov (HFB) basis and an energy-density functional with a Skyrme mean-field part and a density-dependent pairing is used to study the monopole collective excitations of spherical even-even nuclei. The influence of the spurious state on the strength function of the isoscalar monopole excitations is clearly assessed. We compare the effect of different kinds of pairing forces (volume pairing, surface pairing, and mixed pairing) on the monopole excitation strength function. The energy of the isoscalar giant monopole resonance (ISGMR), which is related to the nuclear incompressibility ${K}_{\ensuremath{\infty}}$, is calculated for tin isotopes and the results are discussed.

The breathing-mode giant monopole resonance and the surface compressibility in the relativistic mean-field theory

The breathing-mode isoscalar giant monopole resonance (GMR) is investigated using the generator coordinate method within the relativistic mean-field (RMF) theory. Employing the Lagrangian models of the nonlinear- σ model (NL σ), the scalar–vector interaction model (SVI) and the σ– ω coupling model (SIGO), we show that each Lagrangian model exhibits a distinctly different GMR response. Consequently, Lagrangian models yield a different value of the GMR energy for a given value of the nuclear matter incompressibility K ∞ . It is shown that this effect arises largely from a different value of the surface incompressibility K surf inherent to each Lagrangian model, thus giving rise to the ratio K surf / K ∞ which depends upon the Lagrangian model used. This is attributed to a difference in the density dependence of the meson masses and hence to the density dependence of the nuclear interaction amongst various Lagrangian models. The sensitivity of the GMR energy to the Lagrangian model used and thus emergence of a multitude of GMR energies for a given value of K ∞ renders the method of extracting K ∞ on the basis of interpolation amongst forces as inappropriate. As a remedy, the need to ‘calibrate’ the density dependence of the nuclear interaction in the RMF theory is proposed.

Nonlinear classical model for the decay widths of isoscalar giant monopole resonances

The decay of the isoscalar giant monopole resonance (ISGMR) in nuclei is studied by means of a nonlinear classical model consisting of several noninteracting nucleons (particles) moving in a potential well with an oscillating nuclear surface (wall). The motion of the nuclear surface is described by means of a collective variable that appears explicitly in the Hamiltonian as an additional degree of freedom. The total energy of the system is therefore conserved. Although the particles do not directly interact with each other, their motions are indirectly coupled by means of their interaction with the moving nuclear surface. We consider as free parameters in this model the degree of collectivity and the fraction of nucleons that participate to the decay of the collective excitation. Specifically, we have calculated the decay width of the ISGMR in the spherical nuclei {sup 208}Pb, {sup 144}Sm, {sup 116}Sn, and {sup 90}Zr. Despite its simplicity and its purely classical nature, the model reproduces the trend of the experimental data that show that with increasing mass number the decay width decreases. Moreover the experimental results (with the exception of {sup 90}Zr) can be well fitted using appropriate values for the free parameters mentioned above. It is alsomore » found that these values allow for a good description of the experimentally measured {sup 112}Sn and {sup 124}Sn decay widths. In addition, we give a prediction for the decay width of the exotic isotope {sup 132}Sn for which there is experimental interest. The agreement of our results with the corresponding experimental data for medium-heavy nuclei is dictated by the underlying classical mechanics, i.e., the behavior of the maximum Lyapunov exponent as a function of the system size.« less

Erratum: Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations [Phys. Rev. C 76, 034327 (2007)

The isospin dependence of incompressibility is investigated in the Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin dependent term of the finite nucleus incompressibility is elucidated by using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. The symmetry energy coefficient of incompressibility is extracted to be K_{\tau}=-(500\pm50) MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes. Microscopic HF+random phase approximation (RPA) calculations are also performed with Skyrme interactions for ^{208}Pb and Sn isotopes to study the strength distributions of ISGMR. .

Giant Resonance Overtones: Compression Modes of the Nucleus

Giant resonances are high-frequency collective excitations of atomic nuclei, the occurrence of which is a typical phenomenon in many-body quantum systems confined in a potential well [1]. In hydrodynamic models, giant resonances can be described as density and shape oscillations around the ground-state density distribution and shape. Such collective excitations of a nucleus may, similarly to other media with given boundary conditions, exhibit standing wave oscillations with a discrete series of frequencies as solutions of the applied wave equations. In this simple picture, most of the giant resonances studied extensively so far can be associated with the lowest energy, leading-order solutions (main tone). Giant resonances corresponding to higher-order solutions (overtones) have not been identified, except for the special cases of the isoscalar giant monopole (ISGMR) and isoscalar giant dipole (ISGDR) resonances, where excitation of the main-tone components is physically forbidden. Strong evidence for the overtone mode of the isoscalar giant quadrupole resonance, which is denoted as ISGQR2, has been found in our experiments [2,3], which is highly supported by predictions of recent RPA calculations [4]. Moreover, the microscopic structures of high-energy isoscalar giant resonances have been studied through measurements of direct particle decay, and in the case of the ISGDR for the first time in our measurements of such branching ratios.

Sigma–omega meson coupling and properties of nuclei and nuclear matter

We have constructed a Lagrangian model with a coupling of σ and ω mesons in the relativistic mean-field theory. Properties of finite nuclei and nuclear matter are explored with the new Lagrangian model SIG–OM. The study shows that an excellent description of binding energies and charge radii of nuclei over a large range of isospin is achieved with SIG–OM. With an incompressibility of nuclear matter K = 265 MeV , it is also able to describe the breathing-mode isoscalar giant monopole resonance energies appropriately. It is shown that the high-density behaviour of the equation of state of nuclear and neutron matter with the σ – ω coupling is much softer than that of the non-linear scalar coupling model.

The compression modes in atomic nuclei and their relevance for the nuclear equation of state

Accurate assessment of the value of the incompressibility coefficient, K∞, of symmetric nuclear matter, which is directly related to the curvature of the equation of state (EOS), is needed to extend our knowledge of the EOS in the vicinity of the saturation point. We review the current status of K∞ as determined from experimental data on isoscalar giant monopole and dipole resonances (compression modes) in nuclei by employing the microscopic theory based on the Random Phase Approximation (RPA). The importance of full self-consistent calculations is emphasized. In recent years, a comparision between RPA calculations based on either non-relativistic effective interactions or relativistic Lagrangians has been pursued in great detail. It has been pointed out that these two types of models embed different ansatz for the density dependence of the symmetry energy. This fact has consequences on the extraction of the nuclear incompressibility, as it is discussed. The comparison with other ways of extracting K∞ from experimental data is highlighted.

First Measurement of the Giant Monopole and Quadrupole Resonances in a Short-Lived Nucleus:Ni56

The isoscalar giant monopole resonance (GMR) and giant quadrupole resonance (GQR) have been measured in the 56Ni unstable nucleus by inducing the 56Ni(d,d') reaction at 50A MeV in the Maya active target at the GANIL facility. The GMR and GQR centroids are measured at 19.3+/-0.5 MeV and 16.2+/-0.5 MeV, respectively. The corresponding angular distributions are extracted from 3 degrees to 7 degrees . A multipole decomposition analysis using distorted wave Born approximation with random phase approximation transition densities shows that both the GMR and the GQR exhaust a large fraction of the energy-weighted sum rule. The demonstration of this new method opens a broad range of giant resonance studies at intermediate-energy radioactive beam facilities.

Isospin dependence of incompressibility in relativistic and nonrelativistic mean field calculations

The isospin dependence of incompressibility is investigated in Skyrme Hartree-Fock (SHF) and relativistic mean field (RMF) models. The correlations between the nuclear matter incompressibility and the isospin-dependent term of the finite nucleus incompressibility is elucidated using the Thomas-Fermi approximation. The Coulomb term is also studied by using various different Skyrme Hamiltonians and RMF Lagrangians. Microscopic $\mathrm{HF}+\mathrm{random}$ phase approximation (RPA) calculations are performed with Skyrme interactions for $^{208}\mathrm{Pb}$ and Sn isotopes to study the strength distributions of isoscalar giant monopole resonances. The symmetry term of nuclear incompressibility is extracted to be ${K}_{\ensuremath{\tau}}=\ensuremath{-}(500\ifmmode\pm\else\textpm\fi{}50)$ MeV from the recent experimental data of isoscalar giant monopole resonances (ISGMR) in Sn isotopes.