From the post "Energy Band Gap....Gap...Gap.Gap",
\(\cfrac{d\,PE_{re}}{d\,t}=\cfrac{1}{3}m_e\dot r_{ e }^3\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }\)
It is possible to rearrange the terms to give,
\(\cfrac{d\,PE_{re}}{d\,t}=\cfrac{1}{2}m_e\dot r_{ e }^2\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }\cfrac{2}{3}\dot r_{ e }\)
\(\cfrac{d\,PE_{re}}{d\,t}=KE_{re}\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }\cfrac{2}{3}\dot r_{ e }\)
\(\cfrac{d\,PE_{re}}{d\,t}=\cfrac{2}{3}KE_{re}\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }\cfrac{d\, r_e}{d\,t}\)
We will formulate the change of total energy,
\(U=T+V=KE+(-PE)\)
\(\cfrac{d\,U}{d\,t}=\cfrac{d\,KE_{re}}{d\,t}-\cfrac{d\,PE_{re}}{d\,t}\)
So the total change in energy is,
\(\Delta U=\int{1.}d\,KE_{re}-\cfrac{2}{3}\int{KE_{re}\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }d\,r_e}\)
\(\Delta U= KE_{re}-\cfrac{2}{3}\int{KE_{re}\cfrac{ d\, r_e } { d\, T }\cfrac { d^2 T }{ d\, r^2_e }d\,r_e}\)
\(\Delta U= KE_{re}-\cfrac{2}{3}\int{KE_{re}\cfrac{ d\, r_e } { d\, T }\cfrac{d(\cfrac { d T }{ d\, r_e })}{d\,r_e}d\,r_e}\)
\(\Delta U= KE_{re}-\cfrac{2}{3}\int{KE_{re}\cfrac{ d\, r_e } { d\, T }d(\cfrac { d T }{ d\, r_e })}\)
\(KE_{re}\) is an expression in \(\dot r_e\) which in the use of the Lagrangian is treated as an independent variable.
\(\Delta U= KE_{re}\left\{1-\cfrac{2}{3}ln(\cfrac { d T }{ d\, r_e })\right \}+C\)
If we focus on the kink point \(r_{e1}\) to \(r_{e1}+\varepsilon\) where \(\varepsilon\) is very small, and
let \(r_{e2}=r_{e1}+\varepsilon\). \(r_{e1}\) is just before the kink and \(r_{e2}\) is just after the kink in the \(r_e\) vs \(T\) profile.
\(\Delta U= \left\{KE_{re}\left\{1+\cfrac{2}{3}ln(\cfrac { d\, r_e }{ d T })\right \}\right\}^{b}_{a}\)
where \(a=\cfrac { d\, r_e }{ d T }|_{re2}\) and \(b=\cfrac { d\, r_e }{ d T }|_{re1}\)
Note that the direction of integration is from higher orbit to lower orbit.
\(\Delta U= \left\{KE_{re}+\cfrac{2}{3}KE_{re}ln(\cfrac { d\, r_e }{ d T })\right\}^{b}_{a}\)
\(\Delta U= KE_{re}|_{r{e1}}-KE_{re}|_{r{e2}}+\cfrac{2}{3}KE_{re}|_{r{e1}}ln(\cfrac{ d\, r_e } { d T }|_{r{e1}})-\cfrac{2}{3}KE_{re}|_{r{e2}}ln(\cfrac { d\, r_e } { d T }|_{r{e2}})\)
Since, \(r_e\) did not change, \(PE_{re}\) does not change. \(PE\) after all is defined as energy stored by virtue of position. All the energy of the band gap is then due to a change in \(KE_{re}\) alone.
\(\Delta U=\Delta KE_{re}=KE_{re}|_{re1}-KE_{re}|_{re2}\)
So,
\(-\cfrac{2}{3}KE_{re}|_{r{e2}}ln(\cfrac { d\, r_e }{ d T }|_{r{e2}})+\cfrac{2}{3}KE_{re}|_{r{e1}}ln(\cfrac { d\, r_e }{ d T }|_{r{e1}})=0\)
and
\(KE_{re}|_{r{e2}}=KE_{re}|_{r{e1}}{ln(\cfrac{ d\, r_e }{ d T }|_{r{e1}})\over ln(\cfrac { d\, r_e }{ d T }|_{r{e2}})}\)
Moving from lower to higher orbit,
\(\Delta U=KE_{re}|_{re2}-KE_{re}|_{re1}=KE_{re}|_{r{e1}}\left\{{ln(\cfrac { d\, r_e }{ d T }|_{r{e1}})\over ln(\cfrac { d\, r_e }{ d T }|_{r{e2}})}-1\right\}\)
\(E_{BG}=\Delta U=KE_{re}|_{r{e1}}\left\{{ln(\cfrac{ d\, r_e } { d T }|_{r{e1}})\over ln(\cfrac { d\, r_e }{ d T }|_{r{e2}})}-1\right\}\)
This is a much more involved quantification of Energy Band Gap. We know that,
\(\cfrac{ d\, r_e } { d T }<0\)
But, \(\int{\cfrac{1}{-x}}d(-x)=ln(-x)+c\)
So, \(\int{\cfrac{1}{x}}d(x)=ln(|x|)+c\)
and so, knowing that we have negative gradients,
\(E_{BG}=KE_{re}|_{r{e1}}\left\{{ln(|(\cfrac{ d\, r_e } { d T }|_{r{e1}})|)/ ln(|(\cfrac { d\, r_e }{ d T }|_{r{e2}})|)}-1\right\}\)
we use their absolute values only. And since,
\(|(\cfrac{ d\, r_e } { d T }|_{r{e1}})|\) is large
\(E_{BG}<0\), when
\(|(\cfrac { d\, r_e }{ d T }|_{r{e2}})|<1\)
This is the condition for quantum emission on transit to a higher orbit. Very strange indeed. For most material \(r_e\) is small, the change of \(r_e\) with \(T\) is even smaller. So this is a common phenomenon, the top part of the \(r_e\) vs \(T\) graph has a very gentle slope less then \(1\).
Losing energy going from a lower orbit to a higher one, is as if the electron is repulsed by the nucleus.
