Effects of thermal shape fluctuations and pairing fluctuations on the giant dipole resonance in warm nuclei

Apart from the higher limits of isospin and temperature, the properties of atomic nuclei are intriguing and less explored at the limits of lowest but finite temperatures. At very low temperatures there is a strong interplay between the shell (quantal fluctuations), statistical (thermal fluctuations), and residual pairing effects as evidenced from the studies on giant dipole resonance (GDR). In our recent work [Phys. Rev. C 90, 044308 (2014)], we have outlined some of our results from a theoretical approach for such warm nuclei where all these effects are incorporated along within the thermal shape fluctuation model (TSFM) extended to include the fluctuations in the pairing field. In this article, we present the complete formalism based on the microscopic-macroscopic approach for determining the deformation energies and a macroscopic approach which links the deformation to GDR observables. We discuss our results for the nuclei ${}^{97}\mathrm{Tc},{}^{120}\mathrm{Sn},{}^{179}\mathrm{Au}$, and ${}^{208}\mathrm{Pb}$, and corroborate with the experimental data available. The TSFM could explain the data successfully at low temperature only with a proper treatment of pairing and its fluctuations. More measurements with better precision could yield rich information about several phase transitions that can happen in warm nuclei.

Figure 1

Role of pairing (within the simple BCS approach) in calculations with thermal shape fluctuations in the case of ${}^{120}\mathrm{Sn}$. (a) Average pairing gap and (b) average quadrupole deformation parameter, as a function of temperature calculated without pairing in a canonical ensemble approach (CE) and in a grand canonical ensemble approach (GCE) using the pairing strengths obtained from Strutinsky calculations ($\tilde{G}$) and those quoted in Ref. [49] ($G$).

Figure 2

The variation of pairing gap for protons (solid line) and neutrons (dashed line), as a function of pairing strength in the case of ${}^{120}\mathrm{Sn}$. The prescriptions by Strutinsky calculations ($\tilde{G}$) and those quoted in Ref. [49] ($G$) are marked in the bottom axis.

Figure 3

The probability distribution $P\left(\Delta \right)$ for gap parameter $\Delta$ of protons and neutrons in the case of ${}^{120}\mathrm{Sn}$ with $\beta =0$ at different temperatures. $P\left(\Delta \right)\propto exp(-F_{\mathrm{TOT}}/T)$ where $F_{\mathrm{TOT}}$ corresponds to Eq. (5) and we have normalized the peak of $P\left(\Delta \right)$. The simple BCS calculations yielding $\Delta =0$ translate (in the calculations with pairing fluctuations) into the scenario where $\Delta =0$ is the most probable value and the occurrence of $\Delta >0$ has a small but finite probability.

Figure 4

Variation of different probabilities ($P$) with respect to the order parameters in the nucleus ${}^{120}\mathrm{Sn}$ for a fixed triaxiality parameter $\gamma =-{180}^{\circ }$. Solid and dashed lines represent the probability distributions for the gap parameter $\Delta$ of protons and neutrons, respectively, where we fix $\beta =0$. Dotted lines represent the probability distributions for deformation parameter $\beta$ where the pairing gaps are fixed at their BCS values (${\Delta }_P=0$ and ${\Delta }_N=1.39$). Different temperatures are assumed in different panels as in the inset. $P\left(\Delta \text{or}\beta \right)\propto exp(-F_{\mathrm{TOT}}/T)$ where $F_{\mathrm{TOT}}$ corresponds to Eq. (5) and we have normalized the peak. In the top axis, the maximum value of $\beta$ is chosen such that $F_{\mathrm{TOT}}\left({\beta }_{\max }\right)-F_{\mathrm{TOT}}(\beta =0)\approx F_{\mathrm{TOT}}({\Delta }_P=2.0)-F_{\mathrm{TOT}}({\Delta }_P=0.0)$.

Figure 5

(a) Average pairing gap, (b) average quadrupole deformation parameter, and (c) GDR width in the case of ${}^{120}\mathrm{Sn}$, as a function of temperature. The calculations done without pairing utilize a CE approach (CE), whereas the calculations with simple BCS pairing (BCS) and with pairing fluctuations (PF) utilize a GCE approach (GCE). The results obtained by using the liquid drop model (LDM) are also presented. Experimental values for ${}^{120}\mathrm{Sn}$, taken from Refs. [30, 67], are shown by solid squares. For comparison, data for ${}^{119}\mathrm{Sb}$, taken from Ref. [9], are also shown with open circles.

Figure 6

The free energy surfaces for ${}^{120}\mathrm{Sn}$ at different temperatures are plotted without pairing (CE) and with pairing (GCE). The contour line spacing is 0.5 MeV, the most probable shape is marked by a solid red circle, and the first two minima are shaded.

Figure 7

Similar to Fig. 5 but for the nucleus ${}^{179}\mathrm{Au}$. Experimental values for the spherical width (${\Gamma }_0$) are taken from Ref. [34] and are shown with solid circles. The corresponding deformed widths ($\Gamma$) are estimated using the relation $\Gamma ={\Gamma }_0{(E_{\mathrm{GDR}}/14.2)}^{1.6}$ and are shown with solid squares.

Figure 8

Similar to Fig. 6 but for the nucleus ${}^{179}\mathrm{Au}$.

Figure 9

Similar to Fig. 5 but for the nucleus ${}^{208}\mathrm{Pb}$. Experimental values shown with solid squares correspond to those given in Refs. [27, 67].

Figure 10

Similar to Fig. 5 but for the nucleus ${}^{97}\mathrm{Tc}$. Experimental values shown with solid squares correspond to those given in Ref. [11]. The legend PF (GCE)* denotes the results obtained in the calculations by using the parameter $\delta =1.9$. In all other calculations $\delta =1.8$.

Figure 11

Similar to Fig. 6 but for the nucleus ${}^{97}\mathrm{Tc}$.

Figure 12

The GDR strength functions ${}^{97}\mathrm{Tc}$ at different temperatures ($T$) are compared with the results obtained using the pairing fluctuations (PF) within a grand canonical ensemble approach (GCE). Experimental data are taken from Ref. [11] and are shown by solid circles. The legend PF (GCE)* denotes that the calculations are with the parameter $\delta =1.9$; in all other calculations $\delta =1.8$.