It is important to understand that collision between photon and electron is a random process. More importantly, the KE measured of the ejected electrons is a random variable that is the aggregated result of many random collisions. This KE is the arithmetic mean of KEs of many independent collisions. By the Central Limit Theorem, KE is a random variable described by the Normal Distribution.
From the previous post, we know that photons and electrons are interacting when the raito \(\cfrac{{r}_{e}}{r}\) is small (\(\cfrac{1}{300}\)). \({r}_{e}\) is atomic radius, and \(r\) is the radius of the incident photons' helix path. And it is certain that when \(\cfrac{{r}_{e}}{r}\) = 1 that collisions is a certainty, and photon frequency \(f\) no longer effect \(E_{max}\). This corresponds to the plateau part of the proposed \(E_{max}\) vs \(f\) curve in the posting "Fodo Electric Effect". Therefore the range of concern on the probability graph, given in the post "Why two Ejection Rates? Fido", governing each individual photon-electron collision narrows to \(0\lt x \lt 1\).
If we assume that all energy from the photon is transferred to the ejected electron on collision, the expected KE of these electrons at an incident photon frequency of \(f\) is,
\(E[KE(x)]=\cfrac{2}{\pi}x.KE_{max}\) , \(x=\cfrac{{r}_{e}}{r}\)
where \(KE_{max}={m}_{p}c^2-\Phi\),
\({m}_{p}c^2\) KE of a photon,
\(\Phi\) the potential energy holding the electron in orbit.
since, \(r=\cfrac{c}{2\pi f}\) and \(x=\cfrac{{r}_{e}}{r}\)
\(x=\cfrac { { r }_{ e }2\pi. f }{ c } \)
\(\therefore E[KE(f)]=\cfrac { 4{ r }_{ e } }{ c } .KE_{ max }.f\)
It is a property of the Normal Distribution that its mean is also the maximum,
\(E_{max}(f)=E[KE(f)]=\cfrac { 4{ r }_{ e } }{ c } .KE_{ max }.f\)
If we compare this with Planck's relation,
\(E=h.f =\cfrac { 4{ r }_{ e } }{ c } .KE_{ max }.f\)
then,
\(h=\cfrac { 4{ r }_{ e } }{ c } .KE_{ max }\)
\(h=\cfrac { 4{ r }_{ e } }{ c }({m}_{p}c^2-\Phi)\)
If \(\Phi\propto\cfrac{1}{{r}_{e}}\) then,
\(h=4{ r }_{ e } {m}_{p}c-\cfrac { 4 }{ c }A\)
where is \(A={r}_{e}.\Phi\) is independent of \({r}_{e}\)
Even if the term \(\cfrac { 4 }{ c }{r}_{e}\Phi\) is small, \(4{ r }_{ e } {m}_{p}c\) still varies with \({r}_{e}\). The Planck Constant is not a constant.
