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 δ₂⁽ᵖᵒᵗ⁾, which is determined to reproduce the nuclear scattering data. This approach assumes δ₂=δ₂⁽ᵖᵒᵗ⁾ although there is no theoretical foundation.Purpose: We clarify the relationship between δ₂ and δ₂⁽ᵖᵒᵗ⁾ 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 δ₂⁽ᵖᵒᵗ⁾ so as to reproduce the microscopic CC result. We then compare δ₂⁽ᵖᵒᵗ⁾ with δ₂.Results: We find that δ₂⁽ᵖᵒᵗ⁾ 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 δ₂⁽ᵖᵒᵗ⁾ 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.