What We Have Always Suspected...Fodo

From the previous post "Energy Shadow Of A Dipole" dated 12 Apr 2016,

$E_s ={ m }_{ e }c^{ 2 } \{C_o-f({ r}_{ ph })\}$

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.