Table 1.

The work functions of heterogeneous solid metal surfaces for different emission phenomena.

Emission Phenomena	Work Function Expression
Photoemission	${\phi }_a\left(E\right)={\int }_0^{+\infty }{k}_B^2{T}^2\frac{{\partial }^3}{\partial E^3}F\left(\frac{E-\mathrm{\Phi }}{k_{B}T}\right)\phi \left(\mathrm{\Phi }\right)d\mathrm{\Phi }$
Thermionic emission	$\phi \left(\mathrm{\Phi }\right)=\frac{1}{2\pi i}{\int }_{\epsilon -i\infty }^{\epsilon +i\infty }\exp \left(\beta \mathrm{\Phi }\right)\tilde{y}\left(\beta \right)d\beta$
Field emission	$\psi \left(\xi \right)={\int }_{\epsilon -i\infty }^{\epsilon +i\infty }\exp \left(\zeta \xi \right)\chi \left(\zeta \right)d\zeta$ where, $\psi \left(\xi \right)=2\phi \left({\xi }^{2/3}\right)/3\xi$, ζ = b/F and ξ = Φ3/2