From the previous post "Like Wave, Like Particle, Not Attracted to Electrons", the energy in required in moving from \(r\) at \(\infty\) to \({r}_{eo}\) is
\(PE_e={ m }_{ e }c^{ 2 }\lim _{ r\rightarrow \infty }{ \{ln{( \cfrac{{r}}{{r_{eo} }})}+C({ r }_{ eo}-{ r })\}}\)
Energy required in moving from \(\infty\) to \({r}_{ef}\) is
\(PE_e={ m }_{ e }c^{ 2 }\lim _{ r\rightarrow \infty }{ \{ln{( \cfrac{{r}}{{r_{ef} }})}+C({ r }_{ ef}-{ r })\}}\)
So, the energy required in moving the electron from from \({r}_{eo}\) to \({r}_{ef}\) is
\(E_s =PE_e({r}_{ef})-PE_e({r}_{eo})\)
\(E_s={ m }_{ e }c^{ 2 }\lim _{ r\rightarrow \infty }{ \{ln{( \cfrac{{r }}{{r}_{ef}})}-ln{( \cfrac{{r }}{{r}_{eo}})}+C({ r }_{ ef}-{ r })-C({ r }_{ eo}-{ r })\}}\)
\({E}_{s}={ m }_{ e }c^{ 2 } \{ln{( \cfrac{{r }_{ e o}}{{r}_{ef}})}+C({ r }_{ ef }-{ r }_{ e o})\}\)
which is the same expression as \({E}_{s}\) as before.
From previous calculation of atomic radius we find that, the centripetal force is of the form,
\(\cfrac{{m}_{e}c^2}{{r}_{eo}}=\cfrac{q^2}{4\pi\varepsilon_o {r}^2_{eo}}.(4-\cfrac{3}{\frac{8}{3}}).\cfrac{\sqrt{3}}{2\sqrt{2}}\cfrac{3}{2}\)
and
\(\cfrac{m_ec^2}{{r}_{eo}}= \cfrac{q^2}{4\pi\varepsilon_o{r}^2_{eo}}(3-\cfrac{\sqrt{3}}{3}) \)
In general,
\(\cfrac{m_ec^2}{{r}_{eo}}= \cfrac{q^2}{4\pi\varepsilon_o{r}^2_{eo}}.A\)
where \(A\) is a numerical constant dependent on the configuration of the electrons around the positive charge. Using this expression to bring a charge from infinity to its final configuration position along the line joining it and the positive center.
\(PE=-A\int^{{r}_{eo}}_{r\rightarrow\infty}{ \cfrac{-q^2}{4\pi\varepsilon_o{r}^2}d r}\)
\(PE= -\cfrac{q^2A}{4\pi\varepsilon_o}\{ \cfrac{1}{r}\}|^{{r}_{eo}}_{r\rightarrow\infty}\)
\(PE= -\cfrac{q^2A}{4\pi\varepsilon_o} \cfrac{1}{{r}_{eo}}\)
The negative sign indicates that this potential is lower than that at \(\infty\) which is zero. Numerically, the energy stored when the electron is pushed to a lower orbit, (photon and electron repel) is equal to this PE when the electron is ejected (ie the atom is ionized) after the photon has passed.
\(E_s ={ m }_{ e }c^{ 2 }\{ ln{( \cfrac{{r }_{ e o}}{{r}_{ef}})}+C({ r }_{ ef}-{ r }_{ eo})\}=\cfrac{q^2A}{4\pi\varepsilon_o} \frac{1}{{r}_{eo}}\)
\(ln{ ( \cfrac{{r }_{ e o}} {{r}_{ef}}e^{ C({ r }_{ ef }-{ r }_{ eo }) } })=\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } } \)
\( \cfrac{{r }_{ e o}}{{r}_{ef}} e^{ C({ r }_{ ef }-{ r }_{ eo }) } =e^{\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } }}\)
\({ r }_{ ef }e^{ -C.{ r }_{ ef } }={ r }_{ eo }e^{ -C.{ r }_{ eo } }e^{ \cfrac { -q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } } }\)
because of the \(e^{ -\cfrac { q^{ 2 }A }{ 4\pi \varepsilon _{ o }{ m }_{ e }c^{ 2 } } \cfrac { 1 }{ { r }_{ eo } } }\lt 1\) factor, and \({x}{e^{-Cx}}\) is monotonous increasing for \(x \lt 1\) for all values of \(C\). \({r}_{ef}\lt {r}_{eo}\) for \({r}\lt 1\). Which is an acceptable result.
Given an element the constant \(A\) is determined from distribution of electrons around the positive center. The factor \(C\)....
