Nucleon scattering on actinides using a dispersive optical model with extended couplings

The Tamura coupling model [Rev. Mod. Phys. 37, 679 (1965)] has been extended to consider the coupling of additional low-lying rotational bands to the ground-state band. Rotational bands are built on vibrational bandheads (even-even targets) or single-particle bandheads (odd-$A$ targets) including both axial and nonaxial deformations. These additional excitations are introduced as a perturbation to the underlying axially symmetric rigid-rotor structure of the ground-state rotational band. Coupling matrix elements of the generalized optical model are derived for extended multiband transitions in even-even and odd-$A$ nuclei. Isospin symmetric formulation of the optical model is employed. A coupled-channels optical-model potential (OMP) containing a dispersive contribution is used to fit simultaneously all available optical experimental databases including neutron strength functions for nucleon scattering on ${}^{232}\mathrm{Th},{}^{233,235,238}\mathrm{U}$, and ${}^{239}\mathrm{Pu}$ nuclei. Quasielastic $\left(p,n\right)$ scattering data on ${}^{232}\mathrm{Th}$ and ${}^{238}\mathrm{U}$ to the isobaric analog states of the target nucleus are also used to constrain the isovector part of the optical potential. Lane consistent OMP is derived for all actinides if corresponding multiband coupling schemes are defined. For even-even (odd-$A$) actinides almost all low-lying collective levels below 1 MeV (0.5 MeV) of excitation energy are coupled. OMP parameters show a smooth energy dependence and energy-independent geometry. A phenomenological optical-model potential that couples multiple bands in odd-$A$ actinides is published for a first time. Calculations using the derived OMP potential reproduce measured total cross-section differences between several actinide pairs within experimental uncertainty for incident neutron energies from 50 keV up to 150 MeV. The importance of extended coupling is studied. Multiband coupling is stronger in even-even targets owing to the collective nature of the coupling; the impact of extended coupling on predicted compound-nucleus formation cross section reaches 5% below 3 MeV of incident neutron energy. Excitation of multiple bands in odd-$A$ targets is weaker owing to the single-particle nature of the coupling. Coupling of ground-state rotational band levels in odd-$A$ nuclei is sufficient for a good description of the compound-nucleus formation cross sections as long as the coupling is saturated (a minimum of seven coupled levels are typically needed).

Figure 1

Multiband coupling scheme (not to scale) is shown for a generalized optical-model description of $n+{}^{238}\mathrm{U}$ reaction. Transitions between excited bands and the ground-state band are indicated; those between excited vibrational bands are forbidden within the vibrational phonon model. Numbers in parentheses next to the level spin are corresponding level excitation energies in keV. The trivial in-band quadrupolar transitions with $\mathrm{\Delta }K=0$ are omitted. The selection rules are discussed in Appendix pp2. Eighteen levels are coupled for neutron-induced reactions including the ground-state rotational band $\left(K=0^{+}\right)$ and four excited vibrational bands ($K=0^{+} \beta$ band, $K=0^{+} \gamma$ band, $K=0^{-}$ octupole band, and anomalous $K=2^{+}$ nonaxial band). For proton-induced reactions the coupling from the ground-state band to isobar-analog states (IASs) should be added.

Figure 2

Calculated total cross sections using the dispersive potential with multiple-band coupling for neutron-induced reactions on ${}^{233,235,238}\mathrm{U}$ and ${}^{232}\mathrm{Th}$ targets are compared with calculations using the RIPL 2408 potential [20, 21]. Experimental data are taken from Refs. [80, 81, 82, 83, 84, 85, 86, 87].

Figure 3

Energy dependence of the measured cross-section ratio vs calculated values using RIPL potential 2408 and the potential derived in the current work with parameters shown in Table 1 combined with deformation parameters tabulated in Appendix pp3.

Figure 4

Energy dependence of the calculated compound-nucleus formation cross section on ${}^{235}\mathrm{U}$ and ${}^{238}\mathrm{U}$ targets using the rigid-rotor RIPL 2408 potential [20, 21, 22] (red dashed line) and the potential derived in the current work (solid green line). Calculations using the present work potential with coupling limited to the ground-state rotational band are represented by a solid black line.