\(E_s ={ m }_{ e }c^{ 2 } \{C_o-f({ r}_{ ph })\}\)
where, \(f(r)=ln(r)-C.r\) and
\(C_o=f(r_s)\).
\(r_s\) is the normal radius of the electron around the proton that the electron return to after the passing of an incident photon; \(r_{ph}\) is the minimum radius of the orbit as the electron is pushed towards the proton when an incident photon passes through.
When \(r_s=r_{ph}\), ie that the electron is not pushed further towards the proton.
\(E_s =0\)
\(E_P=\cfrac{1}{2}E_b\) which is a constant.
We move \(r_{ph}\) towards \(r_s\) by decreasing the incident photon frequency.
\(E_P=constant\)
suggests that there is a minimum frequency of the incident photon that must provide for \(E_P=\frac{1}{2}E_b\), below which there is no emitted photon. \(E_P\) does not decrease continuously to zero with decreasing incident photon frequency.
\(E_P=\cfrac{1}{2}E_b=\Phi_w\)
From this we deduce that the work function, \(\Phi_w\) is half the bond energy.
