Systematic study of isovector and isoscalar giant quadrupole resonances in normal and superfluid deformed nuclei

The systematic study of isoscalar (IS) and isovector (IV) giant quadrupole responses (GQRs) in normal and superfluid nuclei presented in Scamps and Lacroix [Phys. Rev. C 88, 044310 (2013)] is extended to the case of axially deformed and triaxial nuclei. The static and dynamical energy density functionals based on Skyrme effective interaction are used to study static properties and dynamical response functions over the whole nuclear chart. Among the 749 nuclei that are considered, 301 and 65 are respectively found to be prolate and oblate, while 54 do not present any symmetry axis. For these nuclei, the IS- and IV-GQR response functions are systematically obtained. In these nuclei, different aspects related to the interplay between deformation and collective motion are studied. We show that some aspects like the fragmentation of the response induced by deformation effects in axially symmetric and triaxial nuclei can be rather well understood using simple arguments. Besides this simplicity, more complex effects show up like the appearance of nontrivial deformation effects on the collective motion damping or the influence of hexadecapole or higher-order effects. A specific study is made on the triaxial nuclei where the absence of symmetry axis adds further complexity to the nuclear response. The relative importance of geometric deformation effects and coupling to other vibrational modes are discussed.

Figure 1

Nuclei retained in the present study. The different markers refer to the deformation type found in the ground state using the ev8 code with SkM* functional. The black open squares indicate nuclei for which an insufficient convergence has been obtained in ev8 (see text).

Figure 2

Illustration of the energy splitting of the different $\left|K\right|$ components in the IS-GQR strength function owing to deformation. The cases of (a) ${}^{254}$No, (b) ${}^{232}$Th, and (c) ${}^{200}$Hg are considered. In each case, the $\left|K\right|=0$ (green solid line), $\left|K\right|=1$ (orange short dashed line), and $\left|K\right|=2$ (red long dashed line) are shown, as well as the total strength (black solid line) obtained by summing up the different contributions. Note that the two first nuclei are prolate while the last one is oblate. A smoothing parameter $\gamma =1$ MeV is used in this figure.

Figure 3

Same as Fig. 2 for the IV-GQR.

Figure 4

Comparison between the $\mathrm{TDHF}+\mathrm{BCS}$ IS-GQR strength distributions obtained in ${}^{154}$Sm for $\left|K\right|=0$ (green open squares), $\left|K\right|=1$ (orange triangles), and $\left|K\right|=2$ (red circles). The different curves are the corresponding QRPA results from Ref. [25] for $\left|K\right|=0$ (green solid line), $\left|K\right|=1$ (orange short dashed line), and $\left|K\right|=2$ (red long dashed line). Note that the $\mathrm{TDHF}+\mathrm{BCS}$ strengths have been smoothed with a width of 2 MeV consistently with the QRPA results.

Figure 5

Comparison between the mean GQR energy obtained by fitting the strength distribution with a Lorentzian shape around the collective energy. Mean energies of the IVGQR and ISGQR for the Sm isotopic chains are presented in panels (a) and (b), respectively. The IVGQR and ISGQR for the Nd isotopic chains are presented in panels (c) and (d). The fit has been performed on the strength distribution smoothed by a Lorentzian distribution of 2 MeV width. The crosses correspond to the results given in Ref. [25]. The different symbols correspond to the spherical symmetric (green solid circles), prolate (blue solid squares), and triaxial (red solid triangles) nuclei.

Figure 6

Widths of the IS and IV-GQR resonances in the Sm and Nd isotopic chains. The conventions are the same as in Fig. 5. The total width is obtain by fitting the GQR response with a single Lorentzian around the collective energy. Note that the smoothing parameter has been removed from the width as in Ref. [16] to focus on the physical width.

Figure 7

Mean value of the principal peak as a function of the mass $A$ for the IS excitation (red open circles) and the IV excitation (black crosses). The two lines correspond to the formula given by Eq. (17) of Ref. [16] fitted on spherical nuclei.

Figure 8

Main peak energy of the IS-GQR as a function of $\delta$ for $\left|K\right|=0$ (green open triangles), $\left|K\right|=1$ (orange crosses), and $\left|K\right|=2$ (red open squares). The fluid-dynamical model (directly taken from Fig. 1(a) of Ref. [34]) is shown with thick black solid lines, while the simple scaling model is shown by dashed blue lines.

Figure 9

Same as Fig. 8 for the IV-GQR. Note that we also show the scaling model result of the IS-GQR as a reference (blue dashed line).

Figure 10

Heights of the main collective IS-GQR peak for the components (a) $\left|K\right|=0$, (b) $\left|K\right|=1$, and (c) $\left|K\right|=2$. The solid lines are the average trends obtained using Eqs. (15) together with the fitted mean collective energies [Eq. (9)]. Only nuclei with $A>100$ are shown here.

Figure 11

Same as Fig. 10 for the IV-GQR.

Figure 12

(Top) Total damping width ${\Gamma }_{\mathrm{tot}}$ of the IS-GQR is plotted a function of the deformation parameter $\delta$ in axially symmetric nuclei (red crosses). The black solid line is the result of the fit [Eq. (16)]. The widths for nuclei with triaxial deformation are also reported with black open squares. (Bottom) The quantity ${\Gamma }_{\mathrm{tot}}^C/{\Gamma }_{\mathrm{tot}}$, computed from formula (17) is plotted as a function of $\delta$ (red open circles). For comparison, the quantity ${\Gamma }_E^C/{\Gamma }_{\mathrm{tot}}$ that only account for the energy splitting of the different $\left|K\right|$ components is also shown (blue crosses). The corresponding quantities for triaxial nuclei are displayed by black open squares and black open triangles. Only nuclei with $A>100$ are shown here.

Figure 13

Widths of the main collective IS-GQR peak for the components (a) $\left|K\right|=0$, (b) $\left|K\right|=1$, and (c) $\left|K\right|=2$. Note that the smoothing parameter has been removed from the width as in Ref. [16] to focus on the physical width. Only nuclei with $A>100$ are shown here.

Figure 14

Same as Fig. 12 for the IV-GQR.

Figure 15

Same as Fig. 13 for the IV-GQR.

Figure 16

Deformation parameters as a function of the mass for the SkM* functional. The ${\beta }_2$ (blue crosses), ${\beta }_4$ (green circles), and ${\beta }_6$ (red triangles) parameters are shown.

Figure 17

Deviation of different collective peaks obtained with $\mathrm{TDHF}+\mathrm{BCS}$ and the reference results, denoted by ${\mathcal{E}}_{\left|K\right|}$, given by the fluid-dynamical model as a function of ${\beta }_4$ (top) and ${\beta }_6$ (bottom) for the IS-GQR resonance. The different markers correspond to $\left|K\right|=0$ (green open triangles), $\left|K\right|=1$ (orange crosses), and $\left|K\right|=2$ (red open squares).

Figure 18

Position of the triaxial nuclei considered in the present work in the $(\delta ,\gamma )$ plane. Each point corresponds to one nucleus.

Figure 19

Examples of IS-GQR (left) and IV-GQR (right) strength distributions obtained for the different $K$ projection for the triaxial nuclei selected in the isotopic Pt chain. The $\left|K\right|=0$ response is shown by the thick green solid line, the two $\left|K\right|=1$ and the two $\left|K\right|=2$ projections are shown, respectively, by the thin orange lines and red dashed lines. Note that a smoothing width of 1 MeV has been used here to better see the peak separation.

Figure 20

Energy difference between the two $\left|K\right|=1$ (green crosses) and the two $\left|K\right|=2$ (red open triangles) main collective peaks in triaxial nuclei as a function of the deformation angle $\gamma$. The two lines correspond to the prediction of the adiabatic model when only geometric effects in the diagonal part of the mass matrix are accounted for. For the $\left|K\right|=2$ case (horizontal line) no splitting is predicted, while for the $\left|K\right|=1$, Eq. (25) is used where $E_{\mathrm{sph}}$ is replaced by $\overline{E_{2+}}$.

Figure 21

Restoring force of the IS-GQR computed with the formula (23) as a function of the mass.

Figure 22

Same as Fig. 20 for the IV-GQR. For the $\left|K\right|=2$ case (horizontal line) no splitting is predicted, while for the $\left|K\right|=1$, Eq. (25) is used.

Figure 23

Evolution of the energies of different modes—(a) monopole, (b) $K=0$, (c) $K=\pm 1$ and $K=\pm 2$—as a function of the $\gamma$ parameter. In all cases, the calculation neglecting the couplings (black solid line) are compared to the case with coupling (red solid circles). In panel (c), the two components without coupling for $\left|K\right|=1$ and 2 are shown with solid lines. When coupling is included, the negative projection is displayed by red solid circles while the positives components are shown by blue crosses. Note that the $\left|K\right|=1$ components are not affected by the coupling.