Plasmon Excitation Spectra for an Electron Gas with an Energy Gap

The plasmon excitation spectrum of an electron gas in a real crystal is analyzed in the random-phase approximation without exchange for the weak-binding case. The features of this spectrum are studied as a function of energy gap and electron density. Several plasmon branches, which do not always go down to $q=0\mathrm{}$, appear with increasing energy gap and with the filling of the valence band. The contribution of normal and umklapp processes to the behavior of the real part of the dielectric constant is analyzed and correlated with a phenomenological model.