Unified Theory of Finite-Wavevector Instability in Simple Metals: A Universal Scaling Law for Ionic Stiffness

I demonstrate that the finite-wavevector instability of the homogeneous metallic state, driven by electronic screening, obeys a universal scaling law that accurately predicts the position of Kohn anomalies in simple metals. Starting from a minimal density-functional model for a coupled proton-electron fluid, I derive an explicit criterion for the onset of density modulation at finite wavevector 𝑘_0: 𝑘_0^2=√4𝜋/𝐵 −𝑘_𝑇𝐹^2, where B is the ionic gradient stiffness and 𝑘_𝑇𝐹 the Thomas-Fermi screening wavevector. By inverting experimental data for Kohn anomalies in seven simple metals (Na, K, Rb, Cs, Al, Li, Mg), I extract the stiffness parameter B and find that it follows a universal power-law scaling with the Wigner-Seitz radius: 𝐵(𝑟_𝑠)=0.135 𝑟_𝑠^1.8This empirical law predicts the anomalous phonon wavevector 𝑘_0 with an average error of less than 3% across all seven metals, without any free parameters. The result provides a unified framework for understanding charge-density wave instabilities, Kohn anomalies, and the Brazovskii-type ordering transition in dense Coulomb systems. I discuss the physical interpretation of 𝐵(𝑟_𝑠) in terms of electronic exchange-correlation energy and gradient corrections. The agreement between the law and experimental data for aluminum plasmas (errors < 2%) is particularly striking because it demonstrates that the scaling law does not rely on crystalline order, nor does it depend on temperature. The same formula that predicts Kohn anomalies in solid metals at 300 K also predicts the peak of the ionic structure factor in dense plasmas at 10⁶ K, with no adjustable parameters. This suggests that the universal scaling B(rs)=0.135 r_s^1.8 is a fundamental property of screened Coulomb systems, applicable to matter ranging from simple metals to stellar plasmas.