= Solution
The sharp solid curve is the $4.2$ K spectrum; the broadened dashed curve is the $77$ K spectrum. At low temperature, a strong discrete <exciton> transition appears just below the continuum edge, with weaker higher members converging toward the <band gap>. Heating broadens those resonances through scattering, reduces their spectral resolution, and shifts the band edge to lower photon energy. The lower-energy broad dashed maximum is therefore not evidence for a second unrelated low-temperature <band gap>.
Read approximate energies from the native figure: the main low-temperature peak is near $1.610$ eV and a weaker feature is near $1.617$–$1.618$ eV, with the continuum edge around $1.620$ eV. A three-dimensional hydrogenic series $E_n=E_g-R^*/n^2$ then gives
$$
\boxed{E_g\simeq1.62\ \mathrm{eV},\qquad R^*\simeq10\ \mathrm{meV}.}
$$
For example, $E_2-E_1=3R^*/4$ predicts about $7.5$ meV, consistent with those feature spacings. These are graphical estimates at roughly meV precision, not extra digits extracted from the line widths. The strong direct interband onset and subgap series support a direct-gap Wannier-Mott interpretation. The continuum absorption scale is of order $10^6$ m$^{-1}$, corresponding to an <optical absorption> length of order a micrometre. At $77$ K, $k_BT\simeq6.6$ meV is already comparable with the inferred binding scale, explaining why a sharp Rydberg series is poorly resolved; at $4.2$ K, $k_BT\simeq0.36$ meV.
With $m_e^*=m_h^*=0.2m_e$, the <reduced mass> is $\mu^*=0.1m_e$. In the screened hydrogen model,
$$
R^*=13.6057\ \mathrm{eV}\frac{\mu^*/m_e}{\epsilon_r^2},\qquad
a_B^*=a_0\frac{\epsilon_r}{\mu^*/m_e}.
$$
It follows that
$$
\boxed{\epsilon_r\simeq11.7,\qquad a_B^*\simeq6.2\ \mathrm{nm}.}
$$
A radius of several nanometres is consistent with an extended <exciton> in an ordinary semiconductor lattice. The <relative permittivity> inferred here is the effective <dielectric screening> parameter of that model; identifying the material chemically or determining a full dielectric spectrum is not possible from these data alone.
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