Derivation of an optical potential for statically deformed rare-earth nuclei from a global spherical potential

The coupled-channel theory is a natural way of treating nonelastic channels, in particular those arising from collective excitations characterized by nuclear deformations. A proper treatment of such excitations is often essential to the accurate description of experimental nuclear-reaction data and to the prediction of a wide variety of scattering observables. Stimulated by recent work substantiating the near validity of the adiabatic approximation in coupled-channel calculations for scattering on statically deformed nuclei, we explore the possibility of generalizing a global spherical optical model potential to make it usable in coupled-channel calculations on this class of nuclei. To do this, we have deformed the Koning-Delaroche global spherical potential for neutrons, coupling a sufficient number of states of the ground-state band to ensure convergence. We present an extensive study of the effects of collective couplings and nuclear deformations on integrated cross sections as well as on angular distributions for neutron-induced reactions on statically deformed nuclei in the rare-earth region. We choose isotopes of three rare-earth elements (Gd, Ho, W), which are known to be nearly perfect rotors, to exemplify the results of the proposed method. Predictions from our model for total, elastic, and inelastic cross sections, as well as for elastic and inelastic angular distributions, are in reasonable agreement with measured experimental data. These results suggest that the deformed Koning-Delaroche potential provides a useful regional neutron optical potential for the statically deformed rare-earth nuclei.

Figure 1

Total cross sections for neutrons scattered by a ${}^{165}\mathrm{Ho}$ and ${}^{182,184,186}\mathrm{W}$ targets for incident energies ranging from as low as $\approx 3$ keV to as high as 200 MeV, which is the upper limit of validity for the KD optical potential [2]. The solid black curves correspond to the predictions of our CC model, while the dashed red curves are the results of calculations within the spherical model. The experimental data were taken from the EXFOR nuclear data library [39].

Figure 2

Angular distributions for elastic (top panels) and first and second inelastic (middle and bottom panels, respectively) channels for the neutron-induced reaction on ${}^{158}\mathrm{Gd}$ and ${}^{160}\mathrm{Gd}$. The curves correspond to predictions by our CC model. Numbers on the left of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right side correspond to the multiplicative factor applied to facilitate plotting data from different incident energies in the same graph. Experimental data taken from Bauge et al. [9].

Figure 3

Elastic angular distributions for neutron-induced reactions on ${}^{\mathrm{Nat}}\mathrm{Gd}$. The curves correspond to predictions by our CC model. Numbers on the left of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Smith [40].

Figure 4

Sum of elastic and inelastic differential cross sections for neutrons scattered by ${}^{\mathrm{Nat}}\mathrm{Gd}$. The curves correspond to predictions by our CC model. Numbers on the left of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right side correspond to a multiplicative offset. Experimental data taken from Bauge et al. [9].

Figure 5

Elastic angular distribution for the neutron-induced reaction on ${}^{165}\mathrm{Ho}$ at the incident energy of 0.350 MeV. The curve corresponds to the elastic-channel results obtained within our CC model. Experimental data taken from Ref. [41].

Figure 6

Comparison of calculations using our CC model with measured angular distributions for the neutron-induced reaction on ${}^{165}\mathrm{Ho}$. The black curves correspond to the elastic-channel results, while the blue dashed curve corresponds to the elastic differential cross sections with the added contributions from inelastic channels. Even though the measurements were described in Ref. [42], the experimental data were actually taken directly from Ref. [35, Fig. 3].

Figure 7

Comparison between the ${}^{165}\mathrm{Ho}(n,n+n^{\prime })$ angular distributions calculated using our CC model and the data set from Ref. [35] for two different values of incident energies, $E_{\mathrm{inc}}=4.51$ and 9.99 MeV. Experimental data contain contributions from the elastic and inelastic channels.

Figure 8

Angular distribution for the neutron-induced reaction on ${}^{165}\mathrm{Ho}$, at $E_{\mathrm{inc}}=11.0$ MeV. The solid black curve corresponds to the results for the elastic channel summed with the inelastic channels. The red dashed curve corresponds to the same calculation but with reduced imaginary potential. Experimental data taken from Ref. [43].

Figure 9

Elastic angular distributions for neutron-induced reactions on ${}^{182}\mathrm{W}$. The curves correspond to predictions by our CC model. Numbers on the left-hand side of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right-hand side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Refs. [36, 37, 38] and their correspondence to each data set is indicated in the legends.

Figure 10

Inelastic angular distributions for neutron-induced reactions on ${}^{182}\mathrm{W}$. The curves correspond to predictions by our CC model. Numbers on the left-hand side of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right-hand side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Refs. [36, 37, 38] and their correspondence to each data set is indicated in the legends.

Figure 11

Elastic angular distributions for neutron-induced reactions on ${}^{184}\mathrm{W}$. The curves correspond to predictions by our CC model. Numbers on the left-hand side of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right-hand side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Refs. [36, 37, 38] and their correspondence to each data set is indicated in the legends.

Figure 12

Inelastic angular distributions for neutron-induced reactions on ${}^{184}\mathrm{W}$. The curves correspond to predictions by our CC model. Numbers on the left-hand side of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right-hand side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Refs. [36, 37, 38] and their correspondence to each data set is indicated in the legends.

Figure 13

Inelastic angular distributions for neutron-induced reactions on ${}^{182}\mathrm{W}$. The curves correspond to predictions by our CC model. Numbers on the left-hand side of each plot indicate, in MeV, the values of incident energy at which the cross sections were measured, while the numbers on the right-hand side correspond to the multiplicative factor applied to be able to plot data from different incident energies in the same graph. Experimental data taken from Refs. [36, 37, 38] and their correspondence to each data set is indicated in the legends.