\(r_e=\cfrac{m_e}{2A_of(lnT)}\left\{{1- \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}\)
\(\cfrac{d\,r_e}{d\,T}=\cfrac{d}{d\,T}\left\{\cfrac{m_e}{2A_of(lnT)}\right\}.\left\{{1- \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}+\cfrac{m_e}{2A_of(lnT)}.\cfrac{d}{d\,T}\left\{{1- \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}\)
The gradient just before the kink at \(r_e=2r_{ec}\) is,
\(\cfrac{d\,r_e}{d\,T}=\cfrac{d}{d\,T}\left\{\cfrac{m_e}{2A_of(lnT)}\right\}\)
\(\cfrac{d\,r_e}{d\,T}=\cfrac{m_e}{2A_o}\cfrac{d}{d\,T}\left\{\cfrac{1}{f(lnT)}\right\}\)
\(\cfrac{d\,r_e}{d\,T}=\cfrac{m_e}{2A_o}\cfrac{d}{d\,ln(T)}\left\{\cfrac{1}{f(lnT)}\right\}\cfrac{d\,ln(T)}{d\,T}\)
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{m_e}{2A_o}\cfrac{1}{f(lnT)^2}.f^{'}(ln(T)).\cfrac{1}{T}\)
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{m_e}{2A_o}\cfrac{1}{T}\cfrac{1}{f(lnT)^2}.f^{'}(ln(T))\)
From the post "Band Gap? Just A Kink",
\(A_of(lnT)=\cfrac{m_e}{4r_{ec}}\)
\(f(lnT)^2=\cfrac{{m_e}^2}{16A^2_or^2_{ec}}\)
So,
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{m_e}{2A_o}\cfrac{1}{T}.\cfrac{16A^2_or^2_{ec}}{{m_e}^2}f^{'}(ln(T))\)
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{8A_or^2_{ec}}{m_e}\cfrac{1}{T}.f^{'}(ln(T))\)
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{8A_or^2_{ec}}{m_e}\cfrac{1}{T}.\cfrac{d\,f(ln(T))}{d\,T}\cfrac{d\,T}{d\,ln(T)}\)
\(\cfrac{d\,r_e}{d\,T}=-\cfrac{8A_or^2_{ec}}{m_e}.\cfrac{d\,f(ln(T))}{d\,T}\)
And for the case of a single electron orbiting about a single positive charge, \(r_{ec}\) the orbiting radius without considering drag is,
\( r_{ ec }=\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }v^{ 2 } } \)
and \(v^2=2c^2\)
\(\cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { 8A_{ o } }{ m_{ e } } .\cfrac { q^{ 4 } }{ 64\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 2 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T } \)
\( \cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { A_{ o }q^{ 4 } }{ 8\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 3 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T }\)
For a nucleus of atomic number \(n\),
\( \cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { A_{ o }n^2q^{ 4 } }{ 8\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 3 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T }\)
and \(v^2=2c^2\)
\(\cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { 8A_{ o } }{ m_{ e } } .\cfrac { q^{ 4 } }{ 64\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 2 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T } \)
\( \cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { A_{ o }q^{ 4 } }{ 8\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 3 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T }\)
For a nucleus of atomic number \(n\),
\( \cfrac { d\, r_{ e } }{ d\, T } =-\cfrac { A_{ o }n^2q^{ 4 } }{ 8\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 3 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T }\)
The gradient just after the kink is, theoretically,
\(\cfrac{d\,r_e}{d\,T}\rightarrow-\infty\)
From the post "Gap, Gap, Gap and Quack",
\(\cfrac{d(r_e)}{dt}=\cfrac{d(r_e)}{dT}\cfrac{dT}{dx}\)
where \(\cfrac{d\,T}{d\,x}\) is the temperature profile around the nucleus. Thus, the velocity before the kink at \(r_e=2r_{ec}\),
\(\cfrac{d(r_e)}{dt}=-\cfrac{8A_or^2_{ec}}{m_e}.\cfrac{d\,f(ln(T))}{d\,T}\cfrac{dT}{dx}\)
the negative sign indicates that the electron is moving towards the nucleus, decreasing \(r_e\). In general for a nucleus of \(n\) positive charge.
\(\cfrac{d(r_e)}{dt}=-\cfrac { A_{ o }n^2q^{ 4 } }{ 8\pi ^{ 2 }\varepsilon ^{ 2 }_{ o }m^{ 3 }_{ e }c^{ 4 } } .\cfrac { d\, f(ln(T)) }{ d\, T }\cfrac{dT}{dx}\)
And the velocity just after the kink is,
\(\cfrac{d\,r_e}{d\,t}\rightarrow-\infty\)
theoretically.
