= Solution
{c}
The broad feature has the scale expected for a carrier <plasmon>, a longitudinal collective oscillation of the mobile electron density. Using the parameters just estimated and $N=10^{24}\,\mathrm m^{-3}$ gives
$$
\widetilde\nu_p=\frac{\omega_p}{2\pi c}=\frac1{2\pi c}\sqrt{\frac{Ne^2}{\epsilon_0\epsilon_\infty m_e^*}}\simeq\boxed{316\,\mathrm{cm}^{-1}}.
$$
Conversely a bare screened <plasma oscillation> at $340\,\mathrm{cm}^{-1}$ would require $N\simeq1.16\times10^{18}\,\mathrm{cm}^{-3}$ for the same parameters, consistent with the source's approximate density. Its broadness is also consistent with carrier damping, unlike the much sharper lattice line near $270\,\mathrm{cm}^{-1}$.
This identifies a \b[plasmon-related] feature, not a uniquely determined undamped pure electronic mode. <Gallium arsenide> is polar, and its longitudinal <optical phonon> couples to the carrier <plasmon>. A more complete dielectric model has longitudinal eigenfrequencies
$$
\omega_\pm^2=\frac12\left[\omega_{\mathrm{LO}}^2+\omega_p^2\pm\sqrt{(\omega_{\mathrm{LO}}^2+\omega_p^2)^2-4\omega_p^2\omega_{\mathrm{TO}}^2}\right].
$$
The measured <Raman scattering> maximum can also shift with damping and the scattering matrix elements. Experimental <Raman scattering> from these coupled modes and carrier-collision broadening is established independently.<Raman Scattering by Coupled Optical-Phonon-Plasmon Modes in GaAs (1968)>[Abstract: differently doped samples, coupled modes, and linewidths dominated by electron collisions]{ref} The supplied approximate density and one broad spectrum do not determine a unique coupled-mode fit. Treating $340\,\mathrm{cm}^{-1}$ as an exact pure-<plasmon> eigenfrequency would overstate the available information.
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