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Whether this trade-off is a law of nature or a limitation of current materials is unknown.","posed_since":"","precise":"Near threshold, MTE decreases as the photon energy approaches the work function while QE falls steeply, so brightness per unit laser energy has an optimum. Determine whether a lower bound on MTE as a function of QE exists for photoemission from any crystalline surface, and identify materials that approach it (for example cryogenic single-crystal Cu(100), which reached $\\mathrm{MTE}\\sim 5\\,\\mathrm{meV}$ at 35 K, or alkali antimonides).","problem_ref":"c5ef3fbe2a8115d2489d6a660f71a5ce785dd095e6a538d7e3ec9c19a2a95faf","references":"","settled_by":"A derived bound $\\mathrm{MTE}_{\\min}(\\mathrm{QE})$ from photoemission theory checked against measurements on candidate materials.","status_note":"","title":"Fundamental trade-off between quantum efficiency and mean transverse 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