Universal binding-energy relation for crystals that accounts for surface relaxation

We present a universal relation for crack surface cohesion including surface relaxation. Specifically, we analyze how N atomic planes respond to an opening displacement at its boundary, producing structurally relaxed surfaces. Via density-functional theory, we verify universality for metals (Al), ceramics (α-Al₂O₃), and semiconductors (Si). When the energy and opening displacement are scaled appropriately with respect to N, the uniaxial elastic constant, the relaxed surface energy, and the equilibrium interlayer spacing, all energy-displacement curves collapse onto a single universal curve.