= Solution
For the quoted cubic-phase gap,
$$
\boxed{\lambda_{\beta}=\frac{1239.84\ \mathrm{eV\,nm}}{3.2\ \mathrm{eV}}
=387.45\ \mathrm{nm}}.
$$
The same <exciton> binding argument can shift the actual emission to longer wavelength; impurity-bound transitions can also lie below the fundamental gap. The comparison must use $E_g$ at the operating <temperature>. Ordinary temperature-dependent gap changes can shift the band edge in the opposite direction, so the numerical values based on fixed supplied gaps are not universal lower bounds for every cooled device.
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