Solution
= Solution
At $500\ \mathrm K$, the same approximation gives
$$
\boxed{\frac{n_{i,\beta}}{n_{i,\alpha}}\simeq e^{0.1/(k_B500)}=10.18}.
$$
As <temperature> rises, the relative advantage from a fixed gap difference becomes smaller, even while the intrinsic carrier densities of both phases increase.