First Analytic Treatment of keV Electronic Screening via Finite-Length Nanomeshes: Implications for D + D Fusion Studies

Low-energy Deuterium–Deuterium fusion in condensed matter remains a sensitive probe of electronic screening. We present an analytic framework for finite-length metallic wire meshes, outlining how wire radius, lattice spacing, and length can influence the collective electronic response. Within our model, finite-length confinement can logarithmically enhance an effective (Drude-like) electronic inertia, shorten the Thomas–Fermi screening length, and increase the static screening energy while preserving a transparent separation between static and dynamic dielectric regimes. We formulate four complementary static closures—(M1) classical Thomas–Fermi, (M2) D-ion fast response, (M3) momentum-integrated TF, and (M4) Lindhard with a constant local-field factor—and combine them with bound-electron background permittivity, image-charge, and cohesive (lattice) corrections in a way that aims to reduce double counting. For fabrication-realistic geometries and standard metals (Cu, Ag, Au, Pd), the model indicates that 1–2.5 keV screening energies are possible within selected parameter ranges, with geometry acting as the primary driver; copper tends to yield comparatively larger values in this framework, whereas palladium offers practical advantages for deuterium uptake and thermal stability. The corresponding increase in sub-Coulomb D+D fusion cross sections may reach multiple orders of magnitude, owing to the exponential tunneling dependence, suggesting that electronic screening can, at least within this model, be approached as an potentially engineerable design parameter for compact fusion studies. While the present analysis is fully analytic, its predictions are intended as testable upper bounds rather than definitive values.