Compression modes and the nuclear matter incompressibility coefficient

We review the current status of the nuclear matter (N=Z and no Coulomb interaction) incompressibility coefficient, K _nm , and describe the theoretical and the experimental methods used to determine K _nm from properties of compression modes in nuclei. In particular we consider the long standing problem of the conflicting results obtained for K _nm , deduced from experimental data on excitation cross sections for the isoscalar giant monopole resonance (ISGMR) and data for the isoscalar giant dipole resonance (ISGDR).     

Relativistic and nonrelativistic calculations of the isoscalar monopole and dipole states

The study of the isoscalar giant monopole resonance (ISGMR) should allow extracting a value for the nuclear incompressibility coefficient K_∞. In this contribution, we review the most recent attempts along this line. While the nonrelativistic (Skyrme, Gogny) models predict K_∞ to be around 220–235 MeV, the values obtained from the relativistic calculations are significantly larger (250–270 MeV). We argue that the most plausible reason for this discrepancy lies in the different behavior of the symmetry energy in the two classes of models. We also discuss the role of the isoscalar giant dipole resonance (ISGDR). We conclude that a number of experimental ambiguities still prevent us from deducing K_∞ from the ISGDR with a comparable accuracy as from the ISGMR.     

巨单极共振和对称能

基于带有混合同位旋标量-矢量非线性耦合的有效拉格朗日量, 在完全自洽的相对论无规位相近似的框架内, 通过单极压缩模式讨论了核物质的不可压缩性. 比较核²⁰⁸Pb,¹⁴⁴Sm,¹¹⁶Sn和⁹⁰Zr实验和计算的巨单极共振能量, 给出了核物质不可压缩系数的取值范围. 新的同位旋标量-矢量非线性耦合软化核物质的对称能, 但无损于基态性质与实验的一致性. 讨论了对称能的软化对巨单极共振的影响.     

201 MeV proton excitation of giant resonances in 208Pb: Macroscopic and microscopic analysis

The region of the giant resonances in ²⁰⁸Pb has been investigated by inelastic scattering of 201 MeV protons. To test the analysis, angular distributions were measured for the low-lying 3^?, 5^?, 2⁺ and 4⁺ collective states. The giant isoscalar quadrupole resonance (ISGQR) is split into two structures, one at 9.0 MeV with a full width at half-maximum Γ = 1.0 MeV, the other one at 10.6 MeV (Γ = 2.0 MeV), with fine structures at 8.9, 9.3, 10.1, 10.6 and 11 MeV. A macroscopic analysis using the distorted-wave Born approximation (DWBA) leads for the low-lying collective levels, as well as for the ISGQR, to transition probabilities too small by a factor of two, compared with those obtained in other reactions. Microscopic analysis using the distorted-wave impulse approximation (DWIA), with three different sets of random phase approximation (RPA) transition densities, is in very good agreement with the data. At forward angles, in the 12 to 16 MeV excitation energy region, a strong resonance at 13.5 MeV (Γ = 3.6 MeV) is accounted for by the Coulomb excitation of the isovector giant dipole resonance (IVGDR); at larger angles the results are compatible with the excitation of the isoscalar monopole resonance (ISGMR) located at 13.9 MeV (Γ = 2.6 MeV).A resonance located at 21.5 MeV (Γ = 5.7 MeV) appears as the superposition of an isovector quadrupole resonance (IVGQR) excited by Coulomb interaction and a resonance of multipolarity L = 1 ΔT = 0 (ISGDR “squeezing mode”).     

Excitation of the giant dipole resonance in the scattering of isoscalar projectiles

The excitation of the giant dipole resonance in nuclei with N > Z by isoscalar projectiles α and d is discussed within a simple collective model for isoscalar dipole excitations. Calculations have been performed for ²⁰⁸Pb; they are compared to recent data on the excitation of the new giant resonance at E_x = 13.8 MeV. For α scattering the effect of dipole excitation is quite weak but significant contributions are obtained for d scattering.     

Giant monopole resonance in even-A Cd isotopes,the asymmetry term in nuclear incompressibility,and the “softness” of Sn and Cd nuclei

The isoscalar giant monopole resonance (ISGMR) in even-A Cd isotopes has been studied by inelastic α  -scattering at 100 MeV/u and at extremely forward angles, including 0°. The asymmetry term in the nuclear incompressibility extracted from the ISGMR in Cd isotopes is found to be _Kτ=−555±75 MeV$K_{\tau }=-555\pm 75\text{MeV}$, confirming the value previously obtained from the Sn isotopes. ISGMR strength has been computed in relativistic RPA using NL3 and FSUGold effective interactions. Both models significantly overestimate the centroids of the ISGMR strength in the Cd isotopes. Combined with other recent theoretical effort, the question of the “softness” of the open-shell nuclei in the tin region remains open still.     

A Study of Giant Monopole an. d. Quadrupole States in the Relativistic Thomas-Fermi Approximation

The isoscalar giant monopole and quadrupole states in finite nuclei are studied in a relativistic δ-ω model by making use of a local Lorentz boost and scaling method, and the nuclear surface and the density distribution are treated in the relativistic Thomas-Fekmi approximation.The excitation energies of the giant resonances are self-consistently calculated. It is found that the excitation energies of giant monopole arehbasically in agreement with experimental data for all nuclei and those of giant quadrupole for light nuclei. Coupling constants and δ-meson mass in the theory are chosen to fit static properties of nuclear matter.     

Deducing the nuclear-matter incompressibility coefficient from data on isoscalar compression modes

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).     

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 energyweighted 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.     

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. The text was submitted by the author in English.     

Study of the 208Pb(n, γ0)209Pb reaction between 0.8 and 7.7 MeV

The reaction ²⁰⁸Pb(n, γ₀)²⁰⁹Pb was studied from 0.8 to 7.7 MeV to investigate relative contributions of the compound-nucleus and direct and semidirect processes in this energy range. Compound-nucleus reactions dominate below about 5 MeV and semidirect processes above 6 MeV. The direct-semidirect (DSD) model with a complex particle-vibration coupling describes the experimental data in the giant resonance region. A relatively large imaginary term is necessary to obtain a good fit to the data indicating either that the reactions proceed to a large extent in more complicated ways than the simple two-step semidirect reaction or that the model has a serious defect in its present formulation. A second objective was to search for a possible excitation of the isoscalar E2 and the M1 giant resonances by measuring asymmetries around 90° in the angular distribution of the γ-rays. The results indicate no (or very weak) asymmetry effects.     

Core polarization effects and giant quadropole resonances in the A ≈ 90 region

The existence, in A ≈ 90 nuclei, of large E2 core polarization effects evident from experimental and shell model fits to decay rates is examined in two independent model calculations: a linearized Hartree-Fock calculation studying the nuclear response function and a macroscopic model involving excitations of giant quadrupole resonances. Both investigations confirm the need for large renormalization of proton and neutron E2 effective charges consistent with the recently discovered isoscalar and isovector giant quadrupole resonances.     

Effect of large neutron excess on the dipole response in the region of the giant dipole resonance

The evolution of the dipole response in nuclei with strong neutron excess is studied in the Hartree-Fock plus random phase approximation with Skyrme forces. We find that the neutron excess increases the fragmentation of the isovector giant dipole resonance, while pushing the centroid of the distribution to lower energies beyond the mass dependence predicted by the collective models. The radial separation of proton and neutron densities associated with a large neutron excess leads to non-vanishing isoscalar transition densities to the GDR states, which are therefore predicted to be excited also by isoscalar nuclear probes. The evolution of the isoscalar compression dipole mode as a function of the neutron excess is finally studied. We find that the large neutron excess leads to a strong concentration of the strength associated with the isoscalar dipole operator ∑_ir_i³Y₁₀, that mainly originates from uncorrelated excitations of the neutrons of the skin.     

Semimicroscopic description of giant octupole resonances in deformed nuclei

The strength functions b(E3, ω) are calculated, and the positions and widths of giant octupole resonances in deformed nuclei are found. It is shown that the giant octupole isoscalar resonances have energies (19–20) MeV for the rare-earth nuclei and (17–18) MeV for the actinides and the widths (5–7) MeV. The energies of the giant octupole isovector resonances are defined by the value of the isovector constant κ⁽³⁾₁.     

Nuclear viscosity and widths of giant resonances

We write down a set of coupled hydrodynamic equations of the Navier-Stokes type which describe the motion of two compressible, viscous nuclear fluids. The solutions of these equations give rise to giant resonances of both isoscalar and isovector type. The viscosity terms in the equations are responsible for the damping of these resonances. Within this framework we obtain expressions for the width of the resonances as a function of the mass number A, and relations between the widths and the excitation energies for various multipolarities (J = 0⁺, 1^?, 2⁺, 3^?, 4⁺), and isospins (T = 0,1). The A dependence of the calculated widths exhibit the experimental trends of the giant dipole and isoscalar quadrupole widths. Also, as a result of the calculation we obtain estimates of the values of viscosity coefficients in nuclei.     

Microscopic description of the E0, E2 and E1 giant resonances in 40Ca, 48Ca and 56Ni

The isovector E1 as well as the isoscalar, isovector and full electromagnetic E2 and E0 strength distributions for ⁴⁰Ca, ⁴⁸Ca and ⁵⁶Ni have been calculated in a large energy range up to 50 MeV of excitation. The microscopic model used includes the continuum RPA, 1p1h⊗phonon configurations and ground state correlations induced by these configurations. It is shown that the latter effect gives an increase of the energy weighted sum rules of 4–7%. In all these nuclei the isoscalar E0 and E2 resonances are spread out over the broad energy region. We obtained a reasonable good agreement with the available experimental data including the recent ones for the isoscalar E0 resonance in ⁴⁰Ca. The theoretical transition densities show a rather strong dependence on the excitation energy.     

Microscopic analysis of the breathing mode in 40Ca and 58Ni

Recent experimental studies of the giant electric resonance region in ⁵⁸Ni and ⁴⁰Ca with inelastically scattered α-particles of energy E_α= 240 MeV are analyzed within a microscopic nuclear structure model. The model includes the continuum RPA and more complex 1p1h⊗phonon configurations. By superimposing the contributions of different multipoles up to L = 4 we obtain good agreement with the newest (reanalyzed) data for the isoscalar monopole strength and for the total (α,α′) cross section in ⁵⁸Ni. Agreement with experiment for the isoscalar monopole resonance in ⁴⁰Ca is obtained too. We emphasize the necessity of using microscopic transition densities and discuss consequences for the analyses of such experiments in light and medium mass nuclei. It is shown that the gross structure of the isoscalar monopole resonance in ⁴⁰Ca is caused by the 1p1h⊗phonon configurations. Received: 23 December 1999 / Revised version: 28 February 2000     

Observation of the isoscalar giant resonance in the reaction 10B(d, α)8Be(g.s.)

The excitation functions of the reaction ¹⁰B(d, α)⁸Be show a wide resonant structure near E_x = 27 MeV which is interpreted as an isoscalar giant resonance in the ¹²C nucleus.     

Compression Properties of Finite Nuclear Matter at Zero and Finite Temperatures with the Skyrme Force SKM

By taking into account the finite-size effect rationally in the framework of the Hartree-Fock theory, compression properties at zero and finite temperatures of finite nuclear matter are investigated with the Skyrme force SKM^*. Energies of the isoscalar giant monopole resonance of nuclei at zero temperature calculated are found to be in agreement with experimental and empirical values.     

Evaluating emission widths of giant dipole resonances in a Fermi liquid model

It is shown in the theory of finite Fermi systems that giant resonances can be considered as zero-sound excitations in concrete nuclei. Solutions to the zero-sound dispersion equation in symmetric nuclear matter, ωs(k), are obtained. Hydrodynamic models determine the wave vector k _A corresponding to giant resonance in a nucleus with atomic number A. The real and imaginary parts of ωs(k _A ) compare to the energy and escape width of giant resonance. Calculations are performed for giant dipole resonances.