= Solution
{c}
In the <diamond cubic structure> of <silicon>, the longitudinal <acoustic phonon branch> and longitudinal <optical phonon branch> meet at $X=(2\pi/a,0,0)$. The two <atomic basis> sites are identical, and the associated <crystal symmetry> protects their boundary <degeneracy of normal modes>. In <aluminium phosphide>, the two sites contain different <chemical atoms>. The corresponding <zinc blende structure> lacks the site-exchange symmetry, and these two longitudinal branches generally split at $X$.
A simple way to see the splitting is a <diatomic-chain dispersion relation with equal springs>, representing successive $[100]$ atomic planes by alternating <masses> $M_1,M_2$ at spacing $d=a/4$. With effective <spring constant> $K$, its <angular frequencies> obey
$$
\omega_\pm^2=K\left(\frac1{M_1}+\frac1{M_2}\right)
\pm K\sqrt{\left(\frac1{M_1}+\frac1{M_2}\right)^2-\frac4{M_1M_2}\sin^2(kd)}.
$$
At $X$, $kd=\pi/2$, so the inter-sublattice coupling cancels and the two squared <angular frequencies> are $2K/M_1$ and $2K/M_2$. Equal <masses> give a contact; unequal <masses> give
$$
\boxed{\omega_{\rm acoustic}(X)\simeq\sqrt{\frac{2K}{M_{\rm heavy}}},\qquad
\omega_{\rm optical}(X)\simeq\sqrt{\frac{2K}{M_{\rm light}}}.}
$$
In <aluminium phosphide>, the heavier <phosphorus> plane moves in the lower-frequency edge mode, and the lighter <aluminium> plane in the higher-frequency mode in this model. Their similar <atomic masses> explain why the splitting is relatively small. Different electronic environments can also change the <force constant matrix>. The chain calculation illustrates the mechanism; a quantitative three-dimensional <phonon spectrum> requires its full <dynamical matrix>. It does not assert that all transverse acoustic and optical branches meet in <silicon>.
\Image[/past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2021-longitudinal-zone-edge-splitting.png]
{title=Equal-mass and alternating-mass chain models illustrating longitudinal branch contact and splitting at X}
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