= Solution
Warming thermally ionizes neutral donors, depleting the occupied bound ground states from which these absorption lines start. Increased scattering can also broaden the lines. Once the donors are predominantly ionized, their bound-to-bound absorption becomes weak or disappears.
The inferred binding provides the natural thermal scale:
$$
\boxed{T_b\sim\frac{R^*}{k_{\rm B}}
=\frac{10.63\times10^{-3}\ \mathrm{eV}}
{8.61733\times10^{-5}\ \mathrm{eV\,K^{-1}}}
\simeq123\ \mathrm K.}
$$
The first excitation <energy> similarly corresponds to about $92.5\ \mathrm K$. \b[An order-of-magnitude estimate is therefore a <temperature> of order $10^2\ \mathrm K$.] This is not a sharp onset: significant ionization can start below that scale because free carriers have a large translational <density of states>. The precise loss of line strength requires donor density and compensation, as in part (ii), rather than just equating one <energy> with $k_{\rm B}T$.
Back to article page
