\(\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=\cfrac{cosh^{-1}(e)}{cosh^{-1}(e^{1/4})}\)
\(\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=2.24921\)
From the post "Deep Blue Deeper" dated 01 Jun 2016,
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi } } =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi}}\)
where,
\(\cfrac{\partial\psi}{\partial x}=\cfrac{\partial V}{\partial x}+\cfrac{\partial T}{\partial x}=k\)
a constant and, \(x\le a_{\psi\,\pi}\).
and the post "Why A Positron And Deep Blue..." dated 01 Jun 2016,
\(\cfrac { dq }{ dx } =\left[3 \cfrac { \partial V\, }{ \partial \, x } +\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi}}\)
But we know that \(a_{\psi\,c}\) is at a higher frequency than \(a_{\psi\,\pi}\). And intuitively,
\(\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}\lt \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}\)
\(\because a_{\psi\,c}\lt a_{\psi\,\pi} \) and both particles are waves wrap around a center at light speed.
If \(\cfrac { \partial \, T }{ \partial \, x }\propto \cfrac{1}{a_{\psi}}\), as in the case of gravitational force under earth's surface (a force being the rate of change of energy with distance), that,
\(\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}=\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}} \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}\)
\(\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}=2.24912 \left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}\)
So,
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}\)
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2*2.24912\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,\pi}}\)
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3\left.\cfrac{\partial V}{\partial x}\right|_{a_{\psi\,\pi}}-1.49824\left.\cfrac{\partial T}{\partial x}\right|_{a_{\psi\,\pi}}\)
where \(\left.\cfrac{\partial\psi}{\partial x}\right|_{a_{\psi\,\pi}}=\left.\cfrac{\partial V}{\partial x}\right|_{a_{\psi\,\pi}}+\left.\cfrac{\partial T}{\partial x}\right|_{a_{\psi\,\pi}}=k\)
since,
\(\cfrac{\partial\psi}{\partial x}=\cfrac{\partial V}{\partial x}+\cfrac{\partial T}{\partial x}=k\)
is true for values of \(x\le a_{\psi\,\pi}\). So,
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } }=\left[3 \cfrac { \partial V\, }{ \partial \, x } -1.49824\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi\,\pi}}\)
but,
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,\pi } }=\left[3 \cfrac { \partial V\, }{ \partial \, x } +\cfrac { \partial \, T }{ \partial \, x } \right]_{x=a_{\psi\,\pi}}\)
\(\cfrac { dq }{ dx }\) is also a force; from the two expressions, the resultant interaction of \(a_{\psi\,c}\) and \(a_{\psi\,\pi}\) is attractive, as the common component in the two sums,
\(3 \cfrac { \partial V\, }{ \partial \, x } \)
cancels and the leading coefficients of the term
\(\cfrac { \partial \, T }{ \partial \, x }\)
sums to a negative value (\(-0.49824\)) and results in a decrease in \(T\) (kinetic energy \(KE\)) along \(x\).
In the case when \(a_{\psi\,\pi}\) interacts with \(a_{\psi\,\pi}\) both particle experience positive change in kinetic energy in the direction of \(x\) and they move further apart. The potential component still cancels, as both particles change \(PE\) in the opposite directions. The interaction of two \(a_{\psi\,c}\) particles however, is still attractive due to the negative coefficient to the \(KE\) term,
\(\cfrac { \partial \, T }{ \partial \, x }\)
So, \(a_{\psi\,c}\) particles will coalesce to size \(n=n_e\), where
\(\cfrac{a_{\psi\,\pi}}{a_{\psi\,ne}}=ratio_e\)
until the substitution into the expression,
\(\left.\cfrac { dq }{ dx }\right|_{ a_{ \psi \,c } } =3k-2\left.\cfrac { \partial \, T }{ \partial \, x }\right|_{a_{\psi\,c}}\)
does not result in a negative coefficient to the term,
\(\cfrac { \partial \, T }{ \partial \, x }\)
ie,
\(2*ratio_e\lt 3\)
\(ratio_e\lt 1.5\)
or
\(\cfrac{a_{\psi\,\pi}}{a_{\psi\,ne}}\lt 1.5\)
Only after attaining the size \(n_e\), \(a_{\psi\,ne}\), do the particles repel each other. There is not just one opposite charge particle but \(a_{\psi\,c}\) till \(a_{\psi\,ne}\) are all opposite charge to \(a_{\psi\,\pi}\).
If \(a_{\psi\,c}\) is the complement negative particle, from the post "Sizing Them Up" dated 3 Dec 2014,
| \(a_{\psi_c}\) (nm) | \(f_c\) (GHz) | \(\lambda_c\)(nm) | \(a_{\psi\,\pi}\)(nm) | \(f_{\pi}\)(GHz) | \(\lambda_{\pi}\)(nm) | |||
| 19.34 | 2466067.5 | 121.57 |
|
|
| |||
| 16.32 | 2922728.6 | 102.57 |
|
|
| |||
| 15.48 | 3082568.8 | 97.25 |
|
|
| |||
| 14.77 | 3230699.3 | 92.79 |
|
|
| |||
where the spectra lines are due to basic particles \(a_{\psi\,c}\), \(n=1\). And the size of big particles \({a_{\psi\,\pi}}\) are given by,
\(\cfrac{a_{\psi\,\pi}}{a_{\psi\,c}}=2.24921\)
and
\(2\pi a_{\psi}=\lambda\)
It was expected that \(a_{\psi\,c}\) are negative particles. But the value of \(2466067\,Hz\) suggests integer reduced micro wave frequency that agitates \(T^{+}\) particles. Based on this, \(a_{\psi}=19.34\,nm\) is a \(T^{+}\) particle. What gives?
Simple; an input of energy at reduce resonance frequency (by an integer divisor) imparts energy onto the particle; \(a_{\psi\,c}\), a negative temperature particle, grows into \(a_{\psi\,\pi}\), a positive temperature particle; the matter heats up.
Goodnight...
