When A Metal Is Not Electriclally Conductive

We continue from the last post "Just Rolling Along...Conducting Electricity" dated 3 May 2016.

As $T$ increases, $r_c$ decreases, since the $B$ field generated by the spinning electron increases.

$B=\cfrac{\mu_o}{4\pi}\cfrac{qv\times \hat{r}}{r_c^2}$

Why should this decrease electric conductivity?  The Lorentz force that acts on passing charges due to this $B$ field is perpendicular to both the $B$ field and the velocity of the charge.  A high $B$ field sends the charge clashing into the structure lattice and reduces its drift velocity.

For high conductivity, $B$ field should be minimum but non zero.  For a given material, this occurs when,

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

is a maximum, ie

$\cfrac{d\,r_c}{d\,T}=0$

or

$\cfrac { q^{ 2 } }{ 4\pi T\varepsilon _{ o }m_{ e }c^{ 2 } } \frac { 1 }{ \sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 }  } -\cfrac { m_{ e }c^{ 2 } }{ 2AT^{ 2 } } \sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 } =0$

$\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } } =\cfrac { m_{ e }c^{ 2 } }{ 2AT } \left( { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 } \right)$

$\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } } =\cfrac { q^{ 2 } }{ 2\pi \varepsilon _{ o }m_{ e }c^{ 2 } } -\cfrac { m_{ e }c^{ 2 } }{ 2AT }$

$\cfrac { m_{ e }c^{ 2 } }{ 2AT } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } }$

$T=\cfrac { Aq^{ 2 } }{ 2\pi \varepsilon _{ o }(m_{ e }c^{ 2 })^{ 2 } }$

where

$A=\cfrac{\tau_o}{2}\cfrac{qv_{\small{T}}}{\{r^2_{\small{T}}+r_{or}^2\}^{3/2}}.{r_{or}}$

A particular temperature exists for which the material is most conductive.  Beyond the maximum value, $r_c$ decreases monotonously, $B$ increases monotonously, drift velocity decreases and conductivity decreases.  As temperature decreases, the complex roots changes to real roots.  What was a conductor can change rapidly into a semiconductor after attaining maximum conductivity with decreasing temperature.

And superconductors are all stuck at minimum $B$, at a specific temperature.

Good night.