Determination of nuclear deformations with an emulator for sub-barrier fusion reactions

TL;DR

The paper develops an emulator for heavy-ion fusion CC calculations using eigenvector continuation (EC) integrated with a discrete basis formulation. This emulator accelerates parameter exploration and enables extraction of intrinsic nuclear shapes from subbarrier fusion data, demonstrated on $^{16}$O+$^{144,154}$Sm and $^{186}$W, including octupole vibrations and static deformations. It shows that EC reproduces exact CC results with substantial speed-ups (up to several hundred-fold) and accurately recovers deformation parameters consistent with independent probes, supporting EC as a practical tool for nuclear-structure-informed reaction studies. The approach promises systematic mapping of nuclear shapes and can be extended to more complex deformations and heavier systems.

Abstract

Based on the eigenvector continuation, we construct an emulator for coupled-channels calculations for heavy-ion fusion reactions at energies around the Coulomb barrier. We apply this to the $^{16}$O+$^{144,154}$Sm, $^{186}$W reactions and examine whether the emulator can be used to extract the deformation parameters of the target nuclei. We show that the emulator not only accelerates the calculations but also has an ability to accurately extract the nuclear shapes. This indicates that the emulator provides a powerful tool to systematically explore intrinsic shapes of atomic nuclei, enhancing our understanding of the fundamental properties of nuclear systems.

Figures (6)

• Figure 1: The workflow of the emulator for determining the nuclear shape. High-fidelity eigen-functions $\Psi$ of the coupled-channels equations are first obtained for a selected set of parameters (Snapshot). The full Hamiltonian $H$ is projected onto the subspace formed by these eigen-functions (Projection) to build an emulator, which facilitates rapid predictions of fusion cross sections (Emulation). The results of the emulator are then compared with experimental data to perform a $\chi^2$ analysis, with which optimum values of the parameters are extracted (Constraint).
• Figure 2: (a) The $\chi^{2}$ function for the $^{16}$O+$^{144}$Sm fusion reaction. It is plotted as a function of the octupole deformation $\beta_3$ of the $^{144}\mathrm{Sm}$ nucleus. The black dots are obtained with the exact fusion cross sections, while the red solid line with the fusion cross sections with the emulator. The blue dashed line indicates the minimum point of the $\chi^{2}$ function. (b) The fusion cross sections $\sigma_{\mathrm{fus}}$ for the $\mathrm{^{16}O}+\mathrm{^{144}Sm}$ reaction as a function of the center-of-mass energy $E$. The black dashed line shows the exact result with the optimum deformation parameter, while the red solid line is obtained with the emulator. The experimental data are taken from Ref. PhysRevC.52.3151. (c) The corresponding fusion barrier distributions.
• Figure 3: The relative error for the penertrability $P_J$ at the center-of-mass energy $E$=60 MeV between the exact calculations and the EC method as a function of the relative computational consumption. Here, the relative error is defined by $|P^{\mathrm{EC}}_{J}(E)-P^{\mathrm{Exact}}_{J}(E)| / P^{\mathrm{Exact}}_{J}(E)$, where $P^{\mathrm{Exact}}_{J}(E)$ and $P^{\mathrm{EC}}_{J}(E)$ are the exact penetration probability and that obtained with the eigenvector continuation, respectively. On the other hand, the relative time is defined by $T^{\mathrm{Exact}}_{J}(E) / T^{\mathrm{EC}}_{J}(E)$, where $T^{\mathrm{Exact}}_{J}(E)$ and $T^{\mathrm{EC}}_{J}(E)$ are the exact comsuming time and that obtained with the eigenvector continuation, respectively. Each point is for several combinations of the number of basis state for the EC, $N_{\rm EC}$, and the number of channels in the coupled-channels calculations, $N_{\rm Ch}$, and for various values of the angular momentum, $J$.
• Figure 4: (a) The two-dimensional $\chi^{2}$ function in the ($\beta_2,\beta_4$) plane for the $^{16}$O+$^{154}$Sm fusion reactions, obtained with the exact coupled-channels calculations with the quadrupole deformation parameter $\beta_2$ and the hexadecapole deformation parameter $\beta_4$ of the target nucleus, $^{154}$Sm. The minimum point is denoted by the star. (b) Same as (a), but obtained with the emulator. The black points indicate the training points used for a construction of the emulator. (c) The fusion cross sections, $\sigma_{\mathrm{fus}}$, as a function of the energy $E$ in the center of mass frame for the $\mathrm{^{16}O}+\mathrm{^{154}Sm}$ reaction, calculated with the optimum values of the deformation parameters. The black dashed line shows the exact result, while the red solid line is obtained with the emulator. The experimental data are taken from Ref. PhysRevC.52.3151. (d) The corresponding barrier distribution.
• Figure 5: Sames as Fig. \ref{['Pic_4']}, but for the $^{16}$O+$^{186}\mathrm{W}$ reaction.