= Solution
{c}
Subtracting the three <acoustic phonon branches> from the six displacement <normal modes> leaves
$$
\boxed{3\text{ optical phonon branches}.}
$$
At small <wavevector>, these involve relative motion of the two <atomic basis> sites rather than a rigid <translation> of the <primitive unit cell>. Their <angular frequencies> remain nonzero at the zone centre. The count uses the <primitive unit cell>; using all eight <chemical atoms> in the conventional <unit cell> would spuriously multiply the number of distinct branches in the first <Brillouin zone>.
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