If a temperature field and an electric field are orthogonal, why should a change in temperature effect electric conductivity? From these posts,
"Drag and A Sense of Lightness" dated 28 Jul 2014
"Science Fantasy, My Very Own..." dated 14 Sep 2014
"Band Gap? Just A Kink" dated 15 Sep 2014
"KaBoom " dated 10 Sep 2014
"Oops! Smooth Operator" dated 5 Sep 2014
"Stress When The Heat Is On, Thermal Stress" dated 27 Aug 2014
"Gravity Exponential Form Again" dated 8 Aug 2014
Where a drag factor \(A\) was introduced at electron speed of \(v^2=2c^2\) and later the dependence of \(A\) on temperature \(T\) was formulated based on \(A\) dependence on space density, \(d_s\) and finally, space density dependence on \(T\), temperature.
What a load of rubbish!
Based on the new model where a positive temperature particle in spin generates an \(E\) field and pulls the electron away from the proton. The electric field from the spinning temperature particle plays a role immediately. From the post "Drag and A Sense of Lightness" dated 28 Jul 2014,
\(\cfrac { m_{ e }v^{ 2 } }{ r_{ e } } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }r^{ 2 }_{ e } } +Av^{ 2 }\) --- (*)
we have instead...
Consider the magnetic field generated by a moving charge,
\(B=\cfrac{\mu_o}{4\pi}\cfrac{qv\times \hat{r}}{r^2}\)
In an analogous way,
\(E=\cfrac{\tau_o}{4\pi}\cfrac{Tv_{\small{T}}\times \hat{r}}{r^2}\)
where \(E\) is the \(E\) field generated by a temperature particle, \(T\) and \(\tau_o\) is a constant analogous to \(\mu_o\) and \(v_{\small{T}}\), the speed of the spinning temperature particle. Then we wrap this wire into a circle and concentrate the \(E\) field along a line through the center of the circle perpendicular to the plane of the circle, in which case \(\hat{r}\) is always perpendicular to \(v_{\small{T}}\) and \(r\) is the radius of the circle.
At a distance \(r_{or}\) long the line through the center,
\(E_{or}=2\pi Ecos(\theta)=\cfrac{\tau_o}{2}\cfrac{Tv_{\small{T}}}{(r^2_{\small{T}}+r^2_{or})}.\cfrac{r_{or}}{\sqrt{r^2_{\small{T}}+r^2_{or}}}\)
as the radial component of \(E\) cancels around the circle. \(r_{\small{oT}}\) is the orbital radius of the spinning positive temperature particle.
If we assume that \(d\lt\lt r_{or}\), then
\(r^{'}=r_{or}\)
The \(E\) field due to the spinning positive temperature particle along \(r^{'}\) is approximately the same as the \(E\) field along \(r_{or}\), the line through the center of the temperature particle orbit.
Expression (*) becomes,
\(\cfrac { m_{ e }v^{ 2 } }{ r_{ e } } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }r^{ 2 }_{ e } } +qE^{'}_{or}\)
where \(r_{h}=r_{e}\) is just a change in notation, and \(E^{'}_{or}\) is,
\(E_{or}=\cfrac{\tau_o}{2}\cfrac{Tv_{\small{T}}}{\{r^2_{\small{T}}+(r_{or}-r_{e})^2\}^{3/2}}.({r_{or}-r_{e}})\)
The attraction due to the weak field adds to the centrifugal force on the outside of the proton orbit away from the proton orbit center. Another situation arise when the electron is inside the proton orbit closer the proton orbit center. This scenario gives rise to other possible energy levels as the electron travel along its helical path in and out of the proton orbit.
If we again assume that \(r_{e}\lt\lt r_{or}\)
\(qE_{or}=A.T\)
where
\(A=\cfrac{\tau_o}{2}\cfrac{qv_{\small{T}}}{\{r^2_{\small{T}}+r_{or}^2\}^{3/2}}.{r_{or}}\) is a constant, then
\(\cfrac { m_{ e }v^{ 2 } }{ r_{ e } } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }r^{ 2 }_{ e } }+AT\)
\(m_{ e }v^{ 2 } .r_e= \cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } }+AT.r^2_{e} \)
and we obtain the quadratic,
\(AT.r^2_{e}-m_{ e }v^{ 2 } .r_e+ \cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } }=0 \)
from which we once again obtain a kink in the solution of \(r_e\) over temperature,
\(r_{ e }=\cfrac { 1 }{ 2AT } \{ m_{ e }v^{ 2 }\pm \sqrt { m^{ 2 }_{ e }v^{ 4 }-4AT\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o } } } \} \)
\( r_{ e }=\cfrac { m_{ e }v^{ 2 } }{ 2AT } \{ 1\pm \sqrt { 1-AT\cfrac { q^{ 2 } }{ \pi m^{ 2 }_{ e }v^{ 4 }\varepsilon _{ o } } } \} \)
\( r_{ e }=\cfrac { m_{ e }v^{ 2 } }{ 2AT } \{ 1\pm \sqrt { 1-AT\cfrac { q^{ 2 } }{ \pi (m_{ e }v^{ 2 })^{ 2 }\varepsilon _{ o } } } \} \)
Since, \( v=c\), ie at light speed,
\( r_{ e }=\cfrac { m_{ e }c^{ 2 } }{ 2AT } \{ 1\pm \sqrt { 1-AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } } \} \)
Unlike the treatment in the post "Band Gap? Just A Kink" dated 15 Sep 2014, we admit both roots of the quadratic. An illustrative plot is shown below,
We see that the rate of change of \(r_e\) with \(T\) tends towards infinity at the kink point,
\(\cfrac{\partial\,r_e}{\partial\,T}\rightarrow\infty\)
where,
\(1-AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } =0\)
There is no discontinuity (not considering infinite gradient) in the \(r_e\) verses \(T\) graph, but the argument for a high kinetic energy barrier that must be surmounted in both directions of increasing \(r_e\) and decreasing \(r_e\) corresponding to high to low energy transition and low to high energy transition of the electron respectively, is the same. This kinetic energy barrier was presented as the bandgap.
High values of \(r_e\) are achieved with low values of \(T\). But \(T\) is an independent variable, what happen to \(r_e\) when \(T\) is high?
\(T\) does not get higher than one positive temperature particle. It is possible to have a \(2T\) particle as the result of a radioactive decay, just as a \(2p\) proton is created after nucleus collapses inward after the ejection of \(g^{+}\) and \(T^{+}\) particles from the nucleus.
When we apply heat, the negative temperature particles around the nucleus is driven away. The remaining \(T^{+}\) particle makes the material hot. In a cloud of negative temperature particles, the effective magnitude of the positive temperature particle is reduced. As heat is applied, with the departure of negative temperature particles, the effective magnitude of the positive temperature particle increases. This is how \(T\) changes with ambient temperature.
\(T\) is quantized, \(T\) changes by integer multiples of a particle \(T\). It is possible that given \(T\), as in the case of insulators and semiconductors,
\(1-AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } \gt 0\)
in which case two distinct solutions to \(r_e\) exist separated by a forbidden zone. It is possible to inject energy via a photon and propel an electron across the forbidden zone, but heat alone will bring the electron only up to \(r_v\). As \(T\) increases, the vertical line on the graph above moves to the right, towards the kink point; the bandgap narrows as temperature is increased.
