Gap, Gap, Gap and Quack

From the post "Band Gap? Just A Kink",

$r_e=\cfrac{m_e}{2A_of(lnT)}\left\{{1\pm \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}$

and

This is a graphical explanation for bandgap, as a result of drag of free space near terminal velocity, $2c^2$.

Band gap energy is given by,

$\Delta v^2_{BG}={\cfrac{d(r_e)}{d\,t}|_{r_{e1}}}^2-{\cfrac{d(r_e)}{d\,t}|_{r_{e2}}}^2$

$E_{BG}=\cfrac{1}{2}m_e\Delta v^2_{BG}$   this energy is released, moving from  $r_{e1}$  to  $r_{e2}$.

Band gap can be calculated theoretically from the difference of the square of the gradient,

$\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{d\,t}\cfrac{d\,t}{dx}\cfrac{dx}{dT}$

If we assume a conservative field where the energy changes are independent of the path taken, the time taken, and where only the starting and ending points mattered, we let,

$\cfrac{d x}{d\,t}=1$    --- (*)

when we move from  $r_{e1}$  to  $r_{e2}$,  this path is also constrained by Hamilton’s principle of stationary action.   (This principle and the differential equation (*) above fully describe the path from  $r_{e1}$  to  $r_{e2}$.)  And so we have,

$\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{dt}\cfrac{dx}{dT}$

$\cfrac{d(r_e)}{dt}={\cfrac{d(r_e)}{dT}\over\cfrac{dx}{dT}}=\cfrac{d(r_e)}{dT}\cfrac{dT}{dx}$

where  $\cfrac{dT}{dx}$  is the postulated temperature gradient around the nucleus.  Obviously  $x$  is in the direction of  increasing $r_e$ and the unit dimension of the last expression is consistent given that it is multiplied by 1 ms^-1 from  $\cfrac{d x}{d\,t}=1$.

Both  $\cfrac{d(r_e)}{dT}$  and  $\cfrac{dT}{dx}$  are negative which makes  $\cfrac{d(r_e)}{dt}$  positve.  If  $T$  is not defined as energy (Joules) then a multiplicative constant is needed for a consistent unit dimension on both sides of the equation.

A derivation using the Lagrangian is presented in the posts "Energy Band Gap....Gap...Gap.Gap"  and "Pag.Pag...Pag....Pag Dnab Ygrene".