Role of exact treatment of thermal pairing in radiative strength functions of ${}^{161–163}\mathrm{Dy}$ nuclei

The enhancement of radiative strength function (RSF) in the region of low $\gamma$-rays energy ($E_{\gamma }\le 12\mathrm{MeV}$), which is caused by the pygmy dipole resonance (PDR), is treated within the phonon damping model (PDM) plus exact thermal pairing (EP) without adding any extra PDR strength function. The numerical calculations performed for ${}^{161–163}\mathrm{Dy}$ show that, because of the effect of EP, the $\mathrm{EP}+\mathrm{PDM}$ can describe reasonably well the total RSF data in both low- and high-energy regions of $\gamma$ rays. Consequently, as compared to the conventional description within the phenomenological generalized Lorentzian and standard Lorentzian models, the $\mathrm{EP}+\mathrm{PDM}$ calculations can eliminate, at least, eight free parameters. This indicates the important role of microscopic approaches towards the precise description of the RSF. In particular, temperature is found to have significant contributions to the RSF below the neutron separation energy, questioning again the validity of the Brink-Axel hypothesis in this energy region.

Figure 1

Exact proton (dashed lines) and neutron (solid lines) pairing gaps as functions of $T$ for ${}^{161–163}\mathrm{Dy}$.

Figure 2

(a)–(c) Total RSFs obtained within the PDM (thin solid lines), $\mathrm{EP}+\mathrm{PDM}$ (thick solid lines) versus the QRPA RSFs for the $E1$ and $E1+M1$ excitations and the experimental data taken from Ref. [18] for ${}^{161–163}\mathrm{Dy}$. The dashed, dash-dotted, and dotted lines stand for the RSFs obtained within the phenomenological GLO-SLO models with two PDRs, one PDR, and without a PDR, respectively. The QRPA $E1$ and $E1+M1$ RSFs are taken from Fig. 10 of Ref. [18]. (d)–(f) GDR widths $\mathrm{\Gamma }\left(E_{\gamma }\right)$ for the $E_1^I$ and $E_1^{II}$ excitations obtained within the PDM and $\mathrm{EP}+\mathrm{PDM}$ for ${}^{161–163}\mathrm{Dy}$.

Figure 3

Total RSFs obtained within the $\mathrm{EP}+\mathrm{PDM}$ for the $E1$ (thin solid line) and the $E1+\mathrm{SR}$ (thick solid line) excitations versus the QRPA RSFs for the lower (${\mathrm{QRPA}+0\lim }^{-}$) (dash-dotted line) and upper (${\mathrm{QRPA}+0\lim }^{+}$) (dashed line) limits taken from Refs. [30, 31] for ${}^{162}\mathrm{Dy}$. The RSF obtained for the $E1$ excitation only within the PDM (dotted line) is also shown for comparison. Experimental RSF data are the same as those in Fig. 2.