\(\cfrac { \partial ^{ 2 }\, B_{ o } }{ \partial \, r_{ e }^{ 2 } } =2\cfrac { \mu _{ o }\varepsilon _{ o }\omega E_{ o } }{ a_{ e } } f(\phi )cot(\theta _{ o })\left\{ Ar^{ 2 }_{ e }-Br_{ e }+C \right\} \)
where,
\( C=\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) }\)
\( B=\cfrac { 4sin(\phi /2) }{ a_{ e } } \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 5 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) }\)
\( A=2\left( \cfrac { sin(\phi /2) }{ a_{ e } } \right) ^{ 2 }\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 5 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) }\)
\(A\gt0\) and \(C\gt0\) for \(0\lt\theta\lt\pi/2\) but \(B\) depends on \(\phi/2\).
\(B^{ 2 }-4AC=16\left( \cfrac { sin(\phi /2) }{ a_{ e } } \right) ^{ 2 }\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 5 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) }\\\left[ \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 5 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) }-\cfrac { 1 }{ 2 } cot(\theta _{ o })\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta )\, } { d\, (cos(\theta )) } \right] \)
From the graph of \(Ar^{ 2 }_{ e }-Br_{ e }+C\) which determines the sign of the second derivative,
The second derivative is negative over a range of \(r_e\), beyond this range it is positive. Under normal circumstances at,
\(\cfrac { \partial \, B_{ o } }{ \partial \, r_{ e } } =0\)
a small increase in \(r_e\) result in a negative change in \(\cfrac { \partial \, B_{ o } }{ \partial \, r_{ e } }\) as
\(\cfrac { \partial ^{ 2 }\, B_{ o } }{ \partial \, r_{ e }^{ 2 } }\lt0\).
Since we started with zero, a negative change means that the derivative itself is negative,
\(\cfrac { \partial \, B_{ o } }{ \partial \, r_{ e } }\lt0\)
and the system is stable.
However, if there is energy input that move \(r_e\) beyond this range, where the second derivative becomes positive, an increase in \(r_e\) will result in a positive \(\cfrac { \partial \, B_{ o } }{ \partial \, r_{ e } }\). \(B_o\) increases and the repulsion between the spinning nucleus and the revolving electron increases causing \(r_e\) to increase further. The electron breaks away and ionization occurs.
Similarly, if \(r_e\) is moved below the stable region, an decrease in \(r_e\) results in a positive \(\cfrac { \partial \, B_{ o } }{ \partial \, r_{ e } } \) that decreases \(B\) correspondingly. A reduction in repulsion between the spinning nucleus and the revolving electron decreases \(r_e\) further and the electron collapses into the nucleus. We have plasma.
So, \(r_e\) has a range of stable values beyond which the particle breaks away at the increasing end or collapses at the decreasing end.
Energy input is required to move \(r_e\). This can be achieved by decreasing drag through increasing temperature, by pushing the orbiting electron inwards/outwards using photons at resonance, decreasing/increasing the \(B\) field between the nucleus and electron by applying an opposing/parallel \(B\) field, etc.
