= Solution
Write the lattice reciprocal vector as $G=2\pi/a$ in one dimension. The <nearly-free electron model> starts from $\varepsilon_k=\hbar^2k^2/(2m_e)$ and couples plane waves separated by G. Near a zone boundary, the two-wave <Hamiltonian> is
$$
\begin{pmatrix}\varepsilon_k&V_G\\V_G^*&\varepsilon_{k-G}\end{pmatrix},\qquad
E_\pm=\frac{\varepsilon_k+\varepsilon_{k-G}}2
\pm\sqrt{\left(\frac{\varepsilon_k-\varepsilon_{k-G}}2\right)^2+|V_G|^2}.
$$
At $k=G/2$ the free waves are degenerate and the periodic potential opens a gap $2|V_G|$. A reduced-zone drawing folds k modulo G into $[-G/2,G/2]$, displaying multiple bands at each reduced wavevector. The original plot shows free folded parabolas, avoided crossings and an allowed vertical optical transition. <Solid State Physics by David Tong>[Chapter2, nearly-free plane-wave mixing and Brillouin-zone boundary gaps]{ref}
Including two spin states, filling a free <Fermi sphere> gives $n=2(4\pi k_F^3/3)/(2\pi)^3=k_F^3/(3\pi^2)$. A monovalent BCC cell contributes two electrons, so
$$
\boxed{k_F=(3\pi^2n)^{1/3}=\frac{(6\pi^2)^{1/3}}a,\qquad
\frac{k_F}{\sqrt2\pi/a}=0.877<1.}
$$
The sphere therefore lies inside every first-zone face, since the supplied distance is the nearest boundary.
For the shortest BCC reciprocal vector, $G=2\sqrt2\pi/a$. In the zero-potential folded-band limit, the energy of a vertical transition from k along G to the neighbouring free branch is $\hbar^2(G^2-2Gk)/(2m_e)$. It decreases over the occupied interval and is smallest at $k=k_F$. Thus the <monovalent BCC direct optical threshold> is
$$
\boxed{\frac{E_0}{E_F}=\left(\frac G{k_F}\right)^2-2\frac G{k_F}=0.6376\simeq0.64.}
$$
The final state is empty because $G-k_F>k_F$. Finite weak $V_G$ changes the energies and makes optical matrix elements nonzero; an exactly free electron gas by itself has no interband absorption. The expression is a leading weak-potential threshold, not a claim that every peak equals the onset.
Using the supplied lattice constants and bare-electron free bands gives $E_F=(3.24,2.12,1.85)\,\mathrm{eV}$ and thresholds $(2.06,1.35,1.18)\,\mathrm{eV}$ for Na,K,Rb. The source plot's broad higher-frequency features shift toward lower energy in the same order as a increases, qualitatively consistent with the predicted $a^{-2}$ scale. Their peak shapes, heights and exact positions need matrix elements, band corrections and scattering; low-frequency intraband conductivity is a separate contribution. No measured trace is reproduced.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1-pa2010-nearly-free-bands.png]
{title=Nearly-free folded bands}
{description=Illustrative weak periodic potential; gray curves are free bands and the arrow is a vertical transition.}
{source=paper-1-pa2010-nearly-free-bands.py}
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