Casimir-Bragg Coincidence in Transition Metal Lattices: A Route to Cooperative X-ray Emission Without Population Inversion

We report a mathematical identity between the Casimir characteristic energy of a planar cavity and the Bragg backscattering energy at the same spacing: both equal hc/(2d). In transition metal lattices with interplanar spacings d ≈ 2.0–2.9 Å, this places the Casimir- Bragg energy in the 2–3 keV range, coinciding with M-shell absorption edges of heavy dopant atoms. We perform a systematic survey of 805 host-dopant combinations (23 metallic hosts × 35 heavy-element dopants) and identify 42 pairs for which the host Casimir energy matches a dopant M-shell edge to within 1%, 201 within 5%, and 297 within 10%. The best match is Ti:Pt (Δ = 0.03%), where the Ti Casimir energy of 2.646 keV falls on the Pt M₃ edge at 2.645 keV. We analyze the conditions for Bragg-enhanced cooperative emission driven by the Casimir vacuum field rather than an external source, propose specific experiments at third-generation synchrotron facilities, and discuss ap- plications to compact X-ray sources, spectroscopy, imaging, and lithography.