Investigation of the determination of nuclear deformation using high-energy heavy-ion scattering

Background: Nuclear deformation provides a crucial characteristic of nuclear structure. Conventionally, the quadrupole deformation length of a nucleus, $\delta_{2}$, has often been determined based on a macroscopic model through a deformed nuclear potential with the deformation length $\delta^{\rm (pot)}_{2}$, which is determined to reproduce the nuclear scattering data. This approach assumes $\delta_{2}=\delta^{\rm (pot)}_{2}$ although there is no theoretical foundation. Purpose: We clarify the relationship between $\delta_{2}$ and $\delta^{\rm (pot)}_{2}$ for high-energy heavy-ion scattering systematically to evaluate the validity of the conventional approach to determine the nuclear deformation. Method: The deformation lengths for the $^{12}$C inelastic scattering by $^{12}$C, $^{16}$O, $^{40}$Ca, and $^{208}$Pb targets at $E/A$ = 50--400 MeV are examined. First, we perform microscopic coupled-channel (CC) calculations to relate $\delta_{2}$ of the deformed density into the inelastic scattering cross section. Second, we use the deformed potential model to determine $\delta^{\rm (pot)}_{2}$ so as to reproduce the microscopic CC result. We then compare $\delta^{\rm (pot)}_{2}$ with $\delta_{2}$. Results: We find that $\delta^{\rm (pot)}_{2}$ is about 20--40 \% smaller than presumed $\delta_{2}$, showing strong energy and target dependence. Further analysis, which considers higher-order deformation effects beyond the derivative model, reveals that $\delta^{\rm (pot)}_{2}$ is still about 15--35 \% smaller than $\delta_{2}$. Conclusion: Our results suggest that one needs to be careful when the deformed potential model for the high-energy heavy-ion scattering is used to extract the nuclear deformation. The conventional approach may underestimate the deformation length $\delta_2$ systematically.