Lambda-Theory: A Unified Energy-Density Criterion for Thermal Softening, Lindemann Melting and Gibbs–Thomson Size Effects

Melting, thermal softening, and size-dependent fusion have been described by separate empirical laws for over a century—Lindemann's geometric threshold, temperature-dependent elastic moduli, and the Gibbs–Thomson relation—yet their common physical origin has remained obscure. Here we show that all three phenomena emerge from a single criterion: the dimensionless energy-density ratio Λ ≡ K/|V|, where K is the kinetic (disruptive) energy density and |V| the cohesive (binding) energy density, crosses a universal threshold Λ ≈ 1. Starting from the mean-square atomic displacement ⟨u²⟩ in Debye–Waller theory, we derive a thermal-softening model (Λ³) whose exponential-with-acceleration functional form captures elastic modulus evolution E(T) across seven metals (Fe, W, Cu, Al, Ni, Ti, Mg) spanning BCC, FCC, and HCP structures, with material-specific parameters reflecting each metal's cohesive energy and anharmonic phonon coupling. The model reproduces experimental data with mean residual < 3% across temperatures from 300 K to 0.9Tm. We further demonstrate that experimental Lindemann ratios are reproduced with 5.4% mean absolute error using structure-dependent Born-type shear collapse—without per-element fitting. For nanoparticles, coordination reduction at surfaces naturally yields the Gibbs–Thomson 1/r law (R² > 0.77 for six materials, r ≥ 3 nm). The Λ framework thus replaces three disconnected rules with a unified energy-balance principle, providing predictive power for high-temperature alloy design, nanoscale stability, and localized failure phenomena (fatigue, machining, additive manufacturing).