Linear response of light deformed nuclei investigated by self-consistent quasiparticle random-phase approximation

We present a calculation of the properties of vibrational states in deformed, axially-symmetric even-even nuclei, within the framework of a fully self-consistent quasiparticle random phase approximation (QRPA). The same Skyrme energy density and density-dependent pairing functionals are used to calculate the mean field and the residual interaction in the particle-hole and particle-particle channels. We have tested our software in the case of spherical nuclei against fully self-consistent calculations published in the literature, finding excellent agreement. We investigate the consequences of neglecting the spin-orbit and Coulomb residual interactions in QRPA. Furthermore we discuss the improvement obtained in the QRPA result associated with the removal of spurious modes. Isoscalar and isovector responses in the deformed ${}^{24–26}\mathrm{Mg}$, ${}^{34}\mathrm{Mg}$ isotopes are presented and compared to experimental findings.

Figure 1

HFB potential energy curves $E_{\mathrm{HFB}}$ (left hand panel); neutron (thick lines) and proton (thin lines) pairing energies $E_{\mathrm{pair}}$ (right hand panel). The quantities are calculated in HO (dashed curves) and THO (solid curves) basis as functions of axial deformation parameter $\beta$ in the nucleus ${}^{24}\mathrm{Mg}$.

Figure 2

Isoscalar (IS) and isovector (IV) $2^{+}$ strength functions in ${}^{48}\mathrm{Ca}$ in the spherical basis of J. Terasaki (JT, dashed-dotted curve), in the HO (dashed curve) and THO (solid curve) basis.

Figure 3

Energies on the complex plane of the spurious state for the $1^{-}$ mode with ${\Omega }^{\pi }=0$${}^{-}$ (circles) and ${\Omega }^{\pi }=1$${}^{-}$ (crosses) projections in ${}^{24}\mathrm{Mg}$ in HO basis. The figure illustrates the change in energy of spurious modes when specific terms of the interaction are omitted.

Figure 4

The same as Fig. 3 but for the $2^{+}$ mode with ${\Omega }^{\pi }=1$${}^{+}$ projection.

Figure 5

IS $2^{+}$ strength functions in ${}^{20}\mathrm{O}$, in HO (upper panel) and THO (lower panel) basis without the residual spin-orbit (SO) and Coulomb (C) interaction (dashed curve) and with all the terms included (solid curve).

Figure 6

The same as Fig. 5 but for IV $1^{-}$ strength functions.

Figure 7

IS $2^{+}$ strength functions in ${}^{24}\mathrm{Mg}$, in HO (upper panel), and THO (lower panel) basis for the ${\Omega }^{\pi }=0^{+}$ (solid curve), $1^{+}$ (dashed curve), and $2^{+}$ (dashed-dotted curve) excitations.

Figure 8

IS $2^{+}$ strength functions in ${}^{26}\mathrm{Mg}$, in HO (upper panel), and THO (lower panel) basis for the ${\Omega }^{\pi }=0^{+}$ (solid curve), $1^{+}$ (dashed curve), and $2^{+}$ (dashed-dotted curve) excitations.

Figure 9

Fractions of IS quadrupole EWSR in ${}^{24}\mathrm{Mg}$, in HO (upper panel) and THO (lower panel) basis. Dashed and dot-dashed curves are obtained from folding of QRPA spectra with a Lorentian distribution having $\Gamma =1$ MeV and $\Gamma =3$ MeV, respectively. These are compared with the experimental data by Youngblood $\mathit{et}$ $\mathit{al}.$ (solid curve).

Figure 10

IS $2^{+}$ strength functions in ${}^{24}\mathrm{Mg}$, in HO (upper panel) and THO (lower panel) basis without the residual spin-orbit (SO) and Coulomb (C) interaction (dashed curve) and with all the terms included (solid curve).

Figure 11

IV $1^{-}$ strength functions in ${}^{24}\mathrm{Mg}$, in HO (upper panel), and in THO (lower panel) basis for the ${\Omega }^{\pi }=0^{-}$ (solid curve), $1^{-}$ (dashed curve) excitations.

Figure 12

IV $1^{-}$ strength functions in ${}^{26}\mathrm{Mg}$, in HO (upper panel), and in THO (lower panel) basis for the ${\Omega }^{\pi }=0^{-}$ (solid curve), $1^{-}$ (dashed curve) excitations.

Figure 13

Fractions of IV dipole EWSR in ${}^{24}\mathrm{Mg}$, in HO (upper panel), and in THO (lower panel) basis. Dashed and dot-dashed curves are obtained from folding of QRPA spectra with a Lorentian distribution having $\Gamma =1$ MeV and $\Gamma =3$ MeV, respectively. These are compared with the experimental data by Irgashev $\mathit{et}$ $\mathit{al}.$ (solid curve).

Figure 14

Fractions of IV dipole EWSR in ${}^{26}\mathrm{Mg}$, in HO (upper panel), and in THO (lower panel) basis. Dashed and dot-dashed curves are obtained from folding of QRPA spectra with a Lorentian distribution having $\Gamma =1$ MeV and $\Gamma =3$ MeV, respectively. These are compared with the experimental data by Fultz $\mathit{et}$ $\mathit{al}.$ (solid curve).

Figure 15

The same as Fig. 10 but for IV $1^{-}$ strength functions.

Figure 16

The same as Fig. 9 but for fractions of IS monopole EWSR.

Figure 17

Fractions of IS monopole EWSR in ${}^{26}\mathrm{Mg}$, in HO basis (dashed curve) and THO (solid curve) basis.

Figure 18

IS $2^{+}$ strength functions in ${}^{34}\mathrm{Mg}$, in the HO (upper panel) and in the THO (lower panel) basis for the ${\Omega }^{\pi }=0^{+}$ (solid curve), $1^{+}$ (dashed curve), and $2^{+}$ (dashed-dotted curve) excitations.

Figure 19

The same as Fig. 10 but for ${}^{34}\mathrm{Mg}$.