\(r_e=\cfrac{m_e}{2A_of(lnT)}\left\{{1\pm \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}<r_{n}\)
where a electron crashes into the nucleus.
As \(A=A_of(lnT)\) decreases with increasing \(T\), \(4\cfrac{A_of(lnT)}{m_e}{r_{ec}}\) decreases and so \(\sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}\) tends towards \(1\).
As such for decreasing orbital radius \(r_e\), we admit only the decreasing expression,
\(r_e=\cfrac{m_e}{2A_of(lnT)}\left\{{1- \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}<r_{n}\)
such that \(\left\{{1- \sqrt{1-4\cfrac{A_of(lnT)}{m_e}{r_{ec}}}}\right\}\) decreases to zero as \(T\) increases.
A plot of 1/x*(1-(1-1/x)^(1/2)),
A special point is observed in the plot where the gradient has two values, one of which is infinity. The corresponding point in the expression for \(r_e\) is,
\(1-4\cfrac{A_of(lnT)}{m_e}r_{ec}=0\)
where
\(A_of(lnT)=\cfrac{m_e}{4.r_{ec}}\)
and
\(r_e=\cfrac{m_e}{2A_of(lnT)}=2.r_{ec}\) independent of \(A\), the drag factor of space.
As \(T\) increases drag factor decreases, electron speed increases and so increasing kinetic energy, and orbital radius decreases with decreasing potential energy (negative electron around positive nucleus).
A sudden change in gradient, \(\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{dt}\cfrac{dt}{dT}\) suggests a discontinuity in kinetic energy of the electron as temperature increases. Notice that \(r_e\) remains continuous at this special point, so the electrostatic potential energy does not change suddenly.
In moving from lower to higher \(r_e\) over the kink in the \(r_e\) vs \(T\) profile there is an decrease in kinetic energy. This disparity is released as a packet of energy without a change in potential energy. Conversely, a packet of energy equivalent to the change in kinetic energy necessary is needed to drop from a higher to a lower \(r_e\). This behavior is as if the interaction between the nucleus and the orbiting electron is repulsive.
A sudden change in gradient, \(\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{dt}\cfrac{dt}{dT}\) suggests a discontinuity in kinetic energy of the electron as temperature increases. Notice that \(r_e\) remains continuous at this special point, so the electrostatic potential energy does not change suddenly.
In moving from lower to higher \(r_e\) over the kink in the \(r_e\) vs \(T\) profile there is an decrease in kinetic energy. This disparity is released as a packet of energy without a change in potential energy. Conversely, a packet of energy equivalent to the change in kinetic energy necessary is needed to drop from a higher to a lower \(r_e\). This behavior is as if the interaction between the nucleus and the orbiting electron is repulsive.
But the electron must first gain enough kinetic energy to climb the steep curve towards the kink. In the reverse direction, when an electron drops from high orbit, at the steep region of the curve, once again the electron requires high kinetic energy.
So, the steep region of the graph present a forbidding region that requires high kinetic energy, to both electron dropping from high orbits and electron climbing to high orbits. A sort of band gap...Does the drag at terminal velocity of space explains the band gap phenomenon? Since band gap is an observed phenomenon under normal circumstances, (without increasing \(T\)),
\(\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{dt}\cfrac{dt}{dx}\cfrac{dx}{dT}\)
that travelling at \(\cfrac{dx}{dt}\) away from the nucleus, there is a change in \(T\), \(\cfrac{dT}{dx}\) and a jump in \(\cfrac{d(r_e)}{dT}\) suggests a sudden change in \(\cfrac{d(r_e)}{dt}\) is necessary to satisfy the expression for \(r_e\) considering drag, \(A\).
Since the band gap phenomenon is seen without increasing temperature, a temperature gradient exists along the radial path towards the nucleus. \(T\) increases towards the nucleus.
In summary, three points can be drawn from the \(r_e\) vs \(T\) profile. Firstly, a steep region requiring high kinetic energy to surmount for both outward direction to higher orbit and inward direction from higher orbit. Secondly, a kink in the profile that represent a discontinuity in the gradient of the profile. \(r_e\) remain continuous suggest that potential energy remains continuous but the is a discontinuity in kinetic energy. The disparity is release as a quantum of energy on going from lower to higher orbit and a similar quantum is needed to return from higher orbit. And thirdly, a temperature gradient exist around the nucleus. \(T\) increases on approach to the nucleus.
What?? d(r_e) /dt is not kinetic energy?? Think again.
So, the steep region of the graph present a forbidding region that requires high kinetic energy, to both electron dropping from high orbits and electron climbing to high orbits. A sort of band gap...Does the drag at terminal velocity of space explains the band gap phenomenon? Since band gap is an observed phenomenon under normal circumstances, (without increasing \(T\)),
\(\cfrac{d(r_e)}{dT}=\cfrac{d(r_e)}{dt}\cfrac{dt}{dx}\cfrac{dx}{dT}\)
that travelling at \(\cfrac{dx}{dt}\) away from the nucleus, there is a change in \(T\), \(\cfrac{dT}{dx}\) and a jump in \(\cfrac{d(r_e)}{dT}\) suggests a sudden change in \(\cfrac{d(r_e)}{dt}\) is necessary to satisfy the expression for \(r_e\) considering drag, \(A\).
Since the band gap phenomenon is seen without increasing temperature, a temperature gradient exists along the radial path towards the nucleus. \(T\) increases towards the nucleus.
In summary, three points can be drawn from the \(r_e\) vs \(T\) profile. Firstly, a steep region requiring high kinetic energy to surmount for both outward direction to higher orbit and inward direction from higher orbit. Secondly, a kink in the profile that represent a discontinuity in the gradient of the profile. \(r_e\) remain continuous suggest that potential energy remains continuous but the is a discontinuity in kinetic energy. The disparity is release as a quantum of energy on going from lower to higher orbit and a similar quantum is needed to return from higher orbit. And thirdly, a temperature gradient exist around the nucleus. \(T\) increases on approach to the nucleus.
What?? d(r_e) /dt is not kinetic energy?? Think again.
