Einstein photoelectric equation ($K_{\max }=h\nu -\Phi$)

For the ideal single-photon metal model, conservation of energy givesHere $h$ is the Planck constant, $\nu$ the incident frequency, $\Phi$ the work function, and $K_{\max }$ the largest emitted kinetic energy relative to the nearby vacuum level. An Electron at the highest occupied energy can reach this edge if it escapes without an additional loss. An Electron starting lower or losing energy within the metal emerges below it. The material takes recoil momentum; its recoil energy is neglected in this model.

For an original chosen example, take $\Phi =2.30\mathrm{eV}$ and $\nu =8.00\times 10^{14}\mathrm{Hz}$. Using $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$ and the elementary charge $e=1.602176634\times 10^{-19}\mathrm{C}$ gives $h\nu =3.308534\mathrm{eV}$ and $K_{\max }=1.008534\mathrm{eV}$. The ideal stopping potential is therefore $1.008534\mathrm{V}$. At $4.00\times 10^{14}\mathrm{Hz}$ the same model supplies only $1.654267\mathrm{eV}$ per photon, so this single-photon channel is closed. A negative result from $h\nu -\Phi$ means no emission through that channel, not an emitted Electron with negative kinetic energy.

The photoelectric threshold frequency is the lower frequency boundary allowed by the ideal single-photon energy balance. With vacuum wavelength $\lambda =c/\nu$ and speed of light $c$, its corresponding longest wavelength is ${\lambda }_0=hc/\Phi$. The chosen $2.30\mathrm{eV}$ work function gives ${\nu }_0\simeq 5.561375\times 10^{14}\mathrm{Hz}$ and ${\lambda }_0\simeq 539.062\mathrm{nm}$. These are model parameters, not a measured threshold for a named metal. Detectable photocurrent need not begin abruptly at the ideal boundary: collection, finite spectral width, thermal broadening and surface inhomogeneity matter.