Instantaneous shape sampling: A model for the $\gamma$-absorption cross section of transitional nuclei

The influence of the quadrupole shape fluctuations on the dipole vibrations in transitional nuclei is investigated in the framework of the instantaneous shape sampling model, which combines the interacting boson model for the slow collective quadrupole motion with the random phase approximation for the rapid dipole vibrations. Coupling to the complex background configurations is taken into account by folding the results with a Lorentzian with an energy-dependent width. The low-energy portion of the $\gamma$-absorption cross section, which is important for photonuclear processes, is studied for the isotopic series of Kr, Xe, Ba, and Sm. The experimental cross sections are well reproduced. The low-energy cross section is determined by the Landau fragmentation of the dipole strength and its redistribution caused by the shape fluctuations. Collisional damping only wipes out fluctuations of the absorption cross section, generating the smooth energy dependence observed in experiment. In the case of semimagic nuclei, shallow pygmy resonances are found, in agreement with experiment.

Figure 1

Absorption cross section of ${}^{130}\mathrm{Ba}$ calculated in the QRPA with the equilibrium deformation $\beta =0.171$ (cf. Table ). Solid black curve, width$\Gamma =3$ MeV; thin blue curve, width $\Gamma =0.1$ MeV.

Figure 2

IBA symmetry triangles for the even-$A$ isotopic chains ${}^{78–86}$Kr, ${}^{124–136}$Xe, ${}^{130–138}$Ba, and ${}^{144–-154}$Sm.

Figure 3

Probability distributions for ${}^{78–86}$Kr. The instantaneous nuclear shapes over the $\beta –\gamma$ plane were found by means of the IBA with the parameters given in Table .

Figure 4

Photoabsorption cross section of ${}^{138}\mathrm{Ba}$ calculated with the ISS-QRPA and energy-dependent width $\Gamma (E_x)=2.5(E_x/15){}^2$ MeV. (b) $E_x$ $=$ 6–10 MeV. Red solid curve, $E1$ contribution; red dotted curve, $M1$ contribution; solid green curve, summed $E1+M1$ cross section. (a) $E_x$ $=$ 10–20 MeV. Black curve, QRPA with the equilibrium deformation in Table ; green curve, ISS-QRPA.

Figure 5

Cross sections for the isotopes ${}^{78–86}$Kr. Left panel, $E_x$ $=$ 7–12.5 MeV; right panel, $E_x$ $=$ 12–20 MeV. Red curve (ISS): calculated with ISS-QRPA ($\Gamma =0.1$ MeV). Dotted blue curve (MM): QRPA ($\Gamma$ $=$ 0.1 MeV) with the equilibrium deformations ${\beta }_{\mathrm{MM}},{\gamma }_{\mathrm{MM}}$ in Table . Green curve ($\mathrm{ISS}+\mathrm{CD}$): ISS-QRPA averaged with energy-dependent width $\Gamma (E_x)=2.5(E_x/15){}^2$ MeV. Magenta curve (mean ISS): QRPA ($\Gamma$ $=$ 0.1 MeV) with the average ISS deformation parameters ${\beta }_{\mathrm{ISS}},{\gamma }_{\mathrm{ISS}}$ in Table . The arrows labeled by $n$ mark the position of the neutron-emission threshold of the respective isotope.

Figure 6

Cross sections for the isotopes ${}^{124–134}$Xe (left panel) and ${}^{130–138}$Ba (right panel) in the low-energy region 7–11 MeV. Notations are as in Fig. 5. The arrows labeled by $n$ mark the positions of the neutron-emission threshold of the isotopes. The black dots with error bars in ${}^{138}\mathrm{Ba}$ display the measured $(\gamma ,n)$ cross section from Ref. [23]. The blue dots below the neutron threshold are $(\gamma ,\gamma {}^{\text{'}})$ data from the recent measurement by Tonchev et al. [50].

Figure 7

Cross sections for the isotopes ${}^{144–154}$Sm. Left panel, $E_x$ $=$ 7–11 MeV; right panel, $E_x$ $=$ 7–20 MeV. Red curves (ISS): calculated with ISS-QRPA ($\Gamma =0.1$ MeV). Dotted blue curves (MM): QRPA ($\Gamma =0.1$ MeV) with the equilibrium deformations in Table . Green line ($\mathrm{ISS}+\mathrm{CD}$): QRPA with a energy-dependent width $\Gamma (E_x)=2.5(E_x/15){}^2$ MeV. The black dots are the measured $(\gamma ,n)$ cross sections and the arrows labeled by $n$ mark the position of the neutron-emission threshold of the respective isotope.

Figure 8

Ground-state eigenvectors for the harmonic oscillator (left) and square well (right).

Figure 9

Strength functions of a RPA pole coupled to the collective quadrupole mode. Red circles, exact; red dashed curve,exact $+$ CD ($\Gamma$ $=$ 0.8 MeV); green squares, ISS; green curve, $\mathrm{ISS}+\mathrm{CD}$ ($\Gamma$ $=$ 0.8 MeV). Change of the pole energy between $\beta =0.1$ and 0.3: upper panels, 1 MeV; lower panels, 3 MeV. Potentials: left panels, harmonic oscillator; right panels, square well.