= Solution
For an intrinsic, nondegenerate <semiconductor>, charge neutrality and the <Boltzmann approximation> to the <Fermi-Dirac distribution> give $n_i=\sqrt{N_CN_V}\exp[-E_g/(2kT)]$. Assume the two polymorphs have equal effective-density-of-states prefactors and retain the quoted <band gaps> at the <temperature> being estimated. Their carrier-density ratio is then
$$
\frac{n_{i,\beta}}{n_{i,\alpha}}\simeq
\exp\left(\frac{3.4-3.2}{2k_BT}\right),
$$
with $k_B$ expressed in eV per kelvin. At $77\ \mathrm K$,
$$
\boxed{n_{i,\beta}/n_{i,\alpha}\simeq3.51\times10^6}.
$$
Both absolute intrinsic densities are extremely small at this <temperature>; a large ratio does not imply that either material is a good intrinsic conductor. Defects or unintended dopants can dominate real “undoped” specimens.
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