= Solution
{c}
For $\mathbf B\parallel[111]$, its squared projection onto the long axis of every valley is $1/3$. All six valleys therefore have the same <cyclotron effective mass>:
$$
\boxed{m_{111}=\sqrt{\frac{3m_1^2m_2}{2m_1+m_2}}\simeq0.28m_e.}
$$
At the fixed photon energy,
$$
\boxed{B_{111}=\frac{m_{111}\omega}{e}\simeq1.4\,\mathrm T.}
$$
Thus the two-peak spectrum is replaced by one peak near $1.4\,\mathrm T$. The source does not determine the linewidth or absolute oscillator strength for the new direction; the sketch below shows only a normalized illustrative line at the computed position.
\Image[/past-exam-of-the-condensed-matter-physics-major-option-course-of-the-university-of-oxford/2019-silicon-valleys-and-resonance.png]
{title=Silicon valley geometry and the single resonance for a $[111]$ field}
{description=Original schematic. Left: six electron <Fermi surface> pockets near the X faces of the face-centred-cubic <first Brillouin zone>. Middle: a schematic hole surface about $\Gamma$ after lowering the <chemical potential> into the <valence band>; its radius and warping are not inferred from the damaged source. Right: a unit-height illustrative absorption line at the calculated $[111]$ resonance; the width is chosen for readability and is not predicted.}
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