Fundamental trade-off between quantum efficiency and mean transverse energy
In plain words
Cathodes that emit many electrons per photon tend to emit them with more sideways energy. Whether this trade-off is a law of nature or a limitation of current materials is unknown.
Precise statement
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).
What would settle it
A derived bound $\mathrm{MTE}_{\min}(\mathrm{QE})$ from photoemission theory checked against measurements on candidate materials.
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