$T^{+}$ Oscillations, Brownian Motion

From the plot of $U_B$ due to the presences of two electrons in orbit, $T^{+}$ can oscillate about $\phi=\cfrac{\pi}{2}$, between the two electrons.

As $T^{+}$ oscillates, $\theta_0$ changes in the expression for $U_B$,

$\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2 } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 2 }_{ e } }{ a^{ 2 }_{ e } }  \right\} ^{ 2 }\\ \left( { \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) }  }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\ +\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) }  }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\ -\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) }  } \right| ^{ \theta _{ 0 } }_{ 0 } } \right) ^{ 2 }$

and

$x=2r_esin(\phi/2)$

$\cfrac{r_{\small{T^{+}}}}{x}=tan(\theta_0)$

where $r_{\small{T^{+}}}$ is the $T^{+}$ particle orbit.

$\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2 } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 2 }_{ e } }{ a^{ 2 }_{ e } }  \right\} ^{ 2 }\left\{ \left( { \cfrac { \sqrt { 2 }  }{ 2cos^{ 2 }(\theta _{ 0 }) } +\cfrac { \sqrt { 2 } a_{ e } }{ r_{ e }cos(\theta _{ 0 }) } -\cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } }  } \right) e^{ -\cfrac { r_{ e }\sqrt { 2 }  }{ a_{ e }cos(\theta _{ 0 }) }  }+\left( -\cfrac { \sqrt { 2 }  }{ 2 } -\sqrt { 2 } \cfrac { a_{ e } }{ r } +\cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } }  \right) e^{ -\cfrac { r_{ e }\sqrt { 2 }  }{ a_{ e } }  } \right\} ^{ 2 }$

at $\phi=\cfrac{\pi}{2}$.

The change of $U_B$ at $\phi=\cfrac{\pi}{2}$ with $\theta_0$ due to the electrons is obtained illustratively from a plot of

(((2)^(1/2)/(2*cos(x)*cos(x))+(2)^(1/2)*a/(cos(x))-a^2)*e^(-(2^(1/2))/(a*cos(x)))+(-(2^(1/2)/2)-2^(1/2)*a+a^2)*e^(-(2^(1/2)/a)))^2,

where $1.5\ge a\ge 0.7$

At $\theta_0=\cfrac{\pi}{2}$, the $T^{+}$ particle is in an orbit just over the electron.  At $\theta_0=0$, the $T^{+}$ particle is far from the electron.  $theta_0$ can attain zero only if $r_e$ is sufficiently large.  Otherwise, $\theta_0$ is limited by,

$tan(\theta_{0\,min})=\cfrac{r_{\small{T^{+}}}}{\sqrt{2}.r_e}$

as $r_e\rightarrow \infty$,   $\theta_{0\,min}\rightarrow 0$

It is possible that $a\gt 1$, the electron is then spinning about an axis passing through it.  In this case, the orbit that captures $T^{+}$ particles has only one electron.

Since $cos(\cfrac{\pi}{2})=cos(-\cfrac{\pi}{2}$, to account for $U_B$ in the presence of another electron, we simply add two graphs and obtain $2U_B$.  The same plot scaled by two along $U_B$ is obtained.

As $T^{+}$ oscillates about $\phi=0$, $\theta$ varies from $\theta_{0\,max}$ to $\theta_{0\,min}$.  As the amplitude of the oscillation increases $\theta_{0\,max}$ increases.  The energy states available for the $T^{+}$ particles in oscillation are between $\theta_{0\,max}$ and $\theta_{0\,min}$.

Energy at specific $\theta_0$ within this range can be absorbed by the $T^{+}$ particles and free itself from the orbit.

If $U_B$ is directly proportional to the number of $T^{+}$ particles captured in orbit,

$U_B\propto n_{\small{T^{+}}}$,

$n_{\small{T^{+}}}$ being the number of $T^{+}$

If intensity is the number of $T^{+}$ emitting through an unit area per unit time and the distribution of $\theta_0$ among that $T^{+}$ particles is uniform, then the zoomed graph above also presents the emission spectrum of $T^{+}$ over the range of permissible $\theta_0$.

When broad spectrum illumination is passed through a material with $T^{+}$ particles, this is the emission spectrum recorded after parts of the illumination has been absorbed by the material.

As $r_e$ increases with increasing temperature, so $a$ increases with increasing temperature.  At a specific value of $U_B$ across the graphs of varying $a$,

$\theta_0$ can decrease as temperature increases.  This means for that range of temperature, the amplitude of oscillation decreases with increasing temperature.

This type of oscillation is characteristic of Brownian motion where the amplitude of oscillation can decrease with higher temperature.  Oscillating $T^{+}$ particles can be the reason for Brownian motion.