Hybrid model for the damped transient response of giant dipole resonances

We suggest the hybrid model for the description of the dipole spreading widths, in which the strength function is built on the one-phonon strength distribution folded with the energy-dependent Lorentzian. Its energy dependence is brought about by (i) the energy-dependent smearing parameter and (ii) the energy-dependent shift of the one-phonon states. These variables are estimated by means of the two-phonon density, provided by a modified Fermi gas model with uniformly spaced states, and with the aid of the weak constant interaction between the one-phonon states and the two-phonon states, considered as background states. Based on calculated one-phonon states, the model is analytically solvable. To explore the validity of the model we analyze the properties of the $1^{-}$ spectrum in ${}^{206}\mathrm{Hg}$, ${}^{206}\mathrm{Pb}$, and ${}^{210}\mathrm{Po}$ nuclei. Our results demonstrate that the description of the decay widths within the latter approach is in good agreement with that obtained in the quasiparticle random phase approximation, by means of the random distribution of the coupling between microscopic one-phonon states and two-phonon states, generated by the Gaussian orthogonal ensemble distribution.

Figure 1

Comparison of the results for the dipole strength distribution for ${}^{206}\mathrm{Pb}$, obtained by means of (a) the one-phonon approximation, (b) the PPC approach, and (c) the random coupling matrix elements between the one- and two-phonon configurations (the doorway model, solid line). The strength distributions are described by the Lorentzian function with the smoothing parameter of 100 keV. The results, based on the one-phonon approximation (the hybrid model), are connected by dotted and dashed lines [see panel (c)]. The dashed curve is associated with the results where the perturbation correction to the one-phonon energy, $\mathrm{\Delta }E$ [Eq. (30)], is taken into account (see Sec. 3b). The dotted line is associated with the results where $\mathrm{\Delta }E=0$. The experimental centroid and width of the GDR are 13.59 and 3.85 MeV [28], which are denoted by arrows and lines in panels (a)–(c), respectively.

Figure 2

Cumulative level densities of one-phonon, $N_1\left(E\right)$ (top and bottom, left-hand scale), and two-phonon, $N_2\left(E\right)$, states (top and bottom, right-hand scale) of ${}^{206}\mathrm{Hg}$ (top) and ${}^{210}\mathrm{Po}$ (bottom). All one-phonon states with $0^{+}$, $1^{-}$, $2^{+}$, $3^{-}$, and $4^{+}$ are included. The two-phonon states are coupled to the angular momentum $1^{-}$. Results from the QRPA calculations are shown by solid black lines, and those from the Fermi gas model are shown by dashed red lines.

Figure 3

Energy shift $\mathrm{\Delta }E\left(E\right)$ [Eq. (30)] vs energy, $E$, for ${}^{210}\mathrm{Po}$ with $E_1=8$ MeV and $E_{\max }=20.5$ MeV.

Figure 4

Similar to Fig. 1, for ${}^{206}\mathrm{Hg}$ (left panels) and ${}^{210}\mathrm{Po}$ (right panels). The empirical systematics of the centroid energy and width are denoted by arrows and lines in panels (a)–(f), respectively.