Just Rolling Along...Conducting Electricity

We continue from the previous post "Hot And Electrifying" dated 2 May 2016.

When,

$1-AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } \lt 0$

we have a set of complex roots.

$r_{ e }=\cfrac { m_{ e }c^{ 2 } }{ 2AT } \{ 1\pm i\sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } }  -1} \}$

If we interpret the complex part of  $r_e$ to be perpendicular to $r_e$ then, the electron now perform circular motion with radius $r_c$,

$r_c=complex(r_e)=\cfrac { m_{ e }c^{ 2 } }{ 2AT }\sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } }  -1}$

and the distance from the proton is,

$r_d=real(r_e)=\cfrac { m_{ e }c^{ 2 } }{ 2AT }$

the electron is no longer in a helical orbit.  It is rolling along with the proton in a circle of radius $r_r$, outside the proton orbit by a distance of $r_d$ along the line joining the proton and the center of the paired orbit where the weak field originates.

Both complex roots correspond to the same configuration.

Both radii decrease with increasing temperature.  It is possible then, that $real(r_e)\rightarrow0$ as $T\rightarrow\infty$ and the electron clashes into the proton and we have matter/anti-matter total annihilation.

As $real(r_e)$ changes continuously, there is no bandgap.  A conductive metal!  And it is the rolling electrons along proton orbits that generates the proverbial $B$ fields that enables electric conduction.