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, δ₂, has often been determined based on a macroscopic model through a deformed nuclear potential with the deformation length δ⁽ᵖᵒᵗ⁾₂, which is determined to reproduce the nuclear scattering data. This approach assumes δ₂=δ⁽ᵖᵒᵗ⁾₂ although there is no theoretical foundation. Purpose: We clarify the relationship between δ₂ and δ⁽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 ¹²C inelastic scattering by ¹²C, ¹⁶O, ⁴⁰Ca, and ²⁰⁸Pb targets at $E/A$ = 50--400 MeV are examined. First, we perform microscopic coupled-channel (CC) calculations to relate δ₂ of the deformed density into the inelastic scattering cross section. Second, we use the deformed potential model to determine δ⁽ᵖᵒᵗ⁾₂ so as to reproduce the microscopic CC result. We then compare δ⁽ᵖᵒᵗ⁾₂ with δ₂. Results: We find that δ⁽ᵖᵒᵗ⁾₂ is about 20--40 % smaller than presumed δ₂, showing strong energy and target dependence. Further analysis, which considers higher-order deformation effects beyond the derivative model, reveals that δ⁽ᵖᵒᵗ⁾₂ is still about 15--35 % smaller than δ₂. 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 δ₂ systematically.