Understanding the intrinsic framework of the Hall-Petch relationship of metals from the view of the electronic-structure level

The Hall-Petch relationship in the form of σ = σ₀ + kd^−0.5 has been widely used to study the relationship between grain size and yield strength of metals; however, the specific physical factors that affect the coefficients σ₀ and k remain unclear. Here we propose two intrinsic descriptors to determine the coefficients of the Hall-Petch relation across different metals and alloys. Inspired by tight-binding theory, we find that σ₀ strongly depends on the group and period number, valence-electron number and electronegativity, while k is determined by the cohesive energy. Our framework establishes a predictive structure-property relationship for the size-dependent yield strength of various metals, and unravels that both coefficients of the Hall-Petch relationship physically originate from the d-band properties. This novel correlation provides a new perspective for understanding the mechanical strength of metals, which is useful for the design of high-performance alloys.