= Band-gap extraction from intrinsic carrier activation
For fixed <band effective masses> and <band gap> $E_g$, the <intrinsic carrier concentration> obeys $n_i=C T^{3/2}e^{-E_g/(2k_BT)}$. Thus
$$
\frac{d\ln(n_i/T^{3/2})}{d(1/T)}=-\frac{E_g}{2k_B}.
$$
A base-ten plot against $x=1000\,\mathrm K/T$ instead has slope $-E_g/(2000\,k_B\mathrm K\ln10)$ after the prefactor correction. The intrinsic branch must be used, rather than the extrinsic plateau or <dopant freeze-out> branch. A changing gap or effective mass makes this an activation-energy estimate, and a small gap uncertainty can produce a much larger multiplicative uncertainty in a room-<temperature> minority density.
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