= Solution
{c}
Use the long-<wavelength> linear <dispersion relation> $\omega\simeq v_L|\mathbf k|$ as an estimate. The <phonon> <energy> is then
$$
E=\hbar\omega\simeq\hbar v_L\frac{2\pi}{a}
=\frac{h(8433\,\mathrm{m\,s^{-1}})}{0.5431\times10^{-9}\,\mathrm m},
\qquad
\boxed{E\simeq0.0642\,\mathrm{eV}.}
$$
This extrapolates the measured low-<wavevector> <speed of sound> to the $X$ point, the first <Brillouin zone> boundary along $[100]$. It is not an exact <phonon dispersion relation> there: the discrete <crystal structure> generally bends the branch, and the initial slope alone cannot determine its boundary <energy>. In particular, $(2\pi/a,0,0)$ is not a <reciprocal lattice vector> of the <face-centered cubic lattice>; the smallest such axial vector is $(4\pi/a,0,0)$. There is therefore no reason for this <acoustic phonon> <energy> to vanish.
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