Application of an extended random-phase approximation to giant resonances in light-, medium-, and heavy-mass nuclei

We present results of the time blocking approximation (TBA) for giant resonances in light-, medium-, and heavy-mass nuclei. The TBA is an extension of the widely used random-phase approximation (RPA) adding complex configurations by coupling to phonon excitations. A new method for handling the single-particle continuum is developed and applied in the present calculations. We investigate in detail the dependence of the numerical results on the size of the single-particle space and the number of phonons as well as on nuclear matter properties. Our approach is self-consistent, based on an energy-density functional of Skyrme type where we used seven different parameter sets. The numerical results are compared with experimental data.

Figure 1

Dipole strength (lower panel) and quadrupole strength (upper panel) for four parameter sets as indicated. The smooth spectra are obtained from folding with Gaussians of linearly increasing width $\mathrm{\Gamma }=\text{max}(0.2,(E-8)/5)$ MeV.

Figure 2

ISGMR in ${}^{16}\mathrm{O}$ calculated within fully self-consistent RPA based on the Skyrme energy density functional with the parameter set T6. [36]. Fractions of the $E0$ EWSR are shown. Upper panel: the ${\mathrm{CRPA}}_{\text{c.r.}}$ function obtained by making use of the method of Ref. [37] is presented by the red solid line. The ${\mathrm{CRPA}}_{\text{100}}$ function obtained in the discrete basis with ${\varepsilon }_{\text{max}}=100$ MeV is presented by the black dashed line. Lower panel: the ${\mathrm{CRPA}}_{\text{c.r.}}$ function (red solid line) is compared with the ${\mathrm{DRPA}}_{\text{c.r.}}$ function (black dashed line) obtained by the same method as in Ref. [37]. Smearing parameter $\mathrm{\Delta }=200$ keV was used in all the calculations.

Figure 3

Photoabsorption cross sections in the nuclei ${}^{16}\mathrm{O},{}^{48}\mathrm{Ca},{}^{132}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ calculated within the CRPA (red solid lines) and the DRPA (black dashed lines) with different smearing parameters $\mathrm{\Delta }$: ${\mathrm{\Delta }}_1=200$ keV, ${\mathrm{\Delta }}_2=400$ keV, ${\mathrm{\Delta }}_3=1$ MeV, and ${\mathrm{\Delta }}_4=2$ MeV. The discrete basis representation with $E_{\text{cut}}=100$ MeV is used both in the CRPA and the DRPA. The calculated cross sections have been normalized to the classical values ${\sigma }_{\text{class.}}=\frac{5}{3}\pi \langle r^2\rangle$ (see text for more details). The results are obtained with the Skyrme force parameter set SV-bas [40].

Figure 4

Discrete and continuum TBA results for ${}^{16}\mathrm{O}$ which were obtained with the parameter set SV-m64k6. The fractions EWSR for GMR and GQR and photoabsorption cross section for GDR are presented in the upper, middle, and lower panels, respectively. The DTBA for smearing parameters $\mathrm{\Delta }=400$ and 700 keV are given by blue dashed and red dashed-doted lines, respectively. Thick green and thin brown full lines represent CTBA with $\mathrm{\Delta }=400$ keV and experimental data, respectively. The data are taken from Refs. [43, 44].

Figure 5

Same as in Fig. 4 but for ${}^{40}\mathrm{Ca}$. The corresponding data are taken from Refs. [45, 46].

Figure 6

Energy (lower part) and width (upper part) of the GDR in ${}^{16}\mathrm{O}$ obtained from TBA calculations. The energy $E_0$ and the width $\mathrm{\Gamma }$ are the corresponding parameters of a Lorentzian fit to the theoretical results. In the left corner of each figure the RPA result is given. In the left column we present $E_0$ and $\mathrm{\Gamma }$ as a function of the maximal phonon energy $E_{\text{max}}^{\text{phon}}$ used in complex configurations. In the right column we present $E_0$ and $\mathrm{\Gamma }$ as function of the minimal collectivity $B_{\text{cut}}$ of the phonons but with fixed value $E_{\text{max}}^{\text{phon}}=80$ MeV. Table 2 gives the relation between values $E_{\text{max}}^{\text{phon}},B_{\mathrm{cut}}$, and the number of phonons.

Figure 7

Same as in Fig. 6 but for the GMR and GQR in ${}^{16}\mathrm{O}$.

Figure 8

Spectral strength distributions for ${}^{208}\mathrm{Pb}$ and the three modes under consideration: isoscalar monopole (left panels), isoscalar quadrupole (middle panels), and isovector dipole (right panels). Photoabsorption strength is shown in case of the dipole mode. Results are obtained with the seven Skyrme parameter sets discussed in Sec. 2b. Compared are strengths derived from RPA (blue dashed) and TBA (full red) with experimental data (full brown) from [48] for the GDR and [49] for the GMR and the GQR.

Figure 9

Same results as in the previous figure but for ${}^{48}\mathrm{Ca}$. The data are taken from [46] for the GDR and from [45] for the GMR and the GQR.

Figure 10

Comparison of giant resonance energies in ${}^{208}\mathrm{Pb}$ for a variety of Skyrme parameter sets as indicated. The energy centroids $E_0=m_1/m_0$ are computed in the window $E_0\pm 2\delta$, where $\delta$ is a dispersion (with the condition $\delta \ge 2\mathrm{MeV}$).Open and filled symbols show the values calculated in the frameworks of RPA and TBA, respectively. The experimental data are taken from Refs. [48] for the GDR, [49] for the GMR, and the GQR, and [19] for ${\alpha }_D$.

Figure 11

Same as in Fig. 10, but for ${}^{16}\mathrm{O}$.

Figure 12

Difference between TBA and RPA for the giant resonance energies in ${}^{208}\mathrm{Pb}$ and ${}^{16}\mathrm{O}$ for a variety of Skyrme parameter sets as indicated.