Mood Stone

From the previously post "Color of Material",  The color present by a material is determined by the discontinuity in gradient at the kink point in the  $r_e$  vs  $T$ profile.  A material that changes color noticeably with temperature means that the discontinuity in gradient,  ie. the band gap, changes widely with changing temperature.

In the case of a mood stone,  a few degrees change in temperature from ambient to body temperature shift the emitted frequencies from dark blue to red.  A decrease in emitted energy.

As temperature,  $T$  is increased,  temperature of the nucleus increases, and the temperature profile extending radially outwards shift upwards to increasing  $T$  and reaches further out to increasing  $r_e$.

The electron's  $r_e$  vs  $T$  profile stretches upwards and outwards correspondingly.

Since,  $T$  increases more than the small value of  $r_e$,  there is an decrease in the gradient,  $\cfrac{d\,r_e}{d\,T}$  atop of the kink point.  The gradient below the kink point is due to a square-root term and does not change. The result is a decrease in band gap and the emitted quantum has a lower energy corresponding to a lower frequency.

$grad T_L=\cfrac { d\, r_e }{ d T }|_{TL}$,    $grad T_H=\cfrac { d\, r_e }{ d T }|_{TH}$

The gradient atop of the kink is steeper when the temperature is higher.

$grad T_L>grad T_H\implies$

$0>ln(|\cfrac { d\, r_e }{ d T }|_{TL}|)>ln(|\cfrac { d\, r_e }{ d T }|_{TH}|)$

since,  $|\cfrac { d\, r_e }{ d T }|_{TL}|, |\cfrac { d\, r_e }{ d T }|_{TH}|<1$

$\left|ln(|\cfrac { d\, r_e }{ d T }|_{TL}|)\right|<\left|ln(\cfrac { d\, r_e }{ d T }|_{TH}|)\right|$

and From the post "Pag.Pag...Pag....Pag Dnab Ygrene", the band gap factor,

$E_{BG}=KE_{re}|_{r{e1}}\left\{{ln(\cfrac{ d\, r_e } { d T }|_{r{e1}})\over ln(\cfrac { d\, r_e }{ d T }|_{r{e2}})}-1\right\}$

where  $re1$  and  $re2$  are points just before and after the kink point.  We have,

$\left|{ln(|\cfrac{ d\, r_e } { d T }|_{kink}|)\over ln(|\cfrac { d\, r_e }{ d T }|_{TL}|)}-1\right|=\left|{ln(|\cfrac{ d\, r_e } { d T }|_{kink}|)\over| ln(|\cfrac { d\, r_e }{ d T }|_{TL}|)|}+1\right|$

and so,

$\left|{ln(|\cfrac{ d\, r_e } { d T }|_{kink}|)\over |ln(|\cfrac { d\, r_e }{ d T }|_{TL}|)|}+1\right|>\left|{ln(|\cfrac{ d\, r_e } { d T }|_{kink}|)\over |ln(|\cfrac { d\, r_e }{ d T }|_{TH}|)|}+1\right|$

$KE_{kink}$ also increases with temperature, but overall the band gap decreases.

The material changes to red color.

It is important that the electron move freely outwards,  corresponding to  $r_e$  stretching upwards.  If the valence electrons from neighboring atoms pushes against the profile as in the case of a close lattice structure, the change in gradient on top of the kink will be limited.

Moreover, given the fact that valence electrons are in SHM along the radial line, the band gap of the electron in the neighborhood will vary as the top of  the $r_e$  vs  $T$  profile is pushed by moving valence electron from adjacent atom(s).  The band gap will vary in a SHM matter and the result is the spread of the quantum over a range of energies.  So, the emitted energies spreads over a spectrum and is not a single spectra line.

The bandwidth of this spectrum, is twice the SHM frequency, as both electrons are moving.