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The highest superconducting temperature T$_c$ observed in any elemental metal(Li with T$_c$ ~ 20 K at pressure P ~ 40 GPa) is shown to arise from critical(formally divergent) electron-phonon coupling to the transverse T$_1$ phononbranch along intersections of Kohn anomaly surfaces with the Fermi surface.First principles linear response calculations of the phonon spectrum andspectral function $\alpha^2 F(\omega)$ reveal (harmonic) instability already at25 GPa. Our results imply that the fcc phase is anharmonically stabilized inthe 25-38 GPa range.Comment: 4 pages, 3 embedded figure

Superconductivity and Lattice Instability in Compressed Lithium from Fermi Surface Hot Spots