= Solution
A partly filled one-dimensional <energy band> is especially sensitive to a lattice modulation at $2k_F$. At half filling, $2k_F=\pi/a$, so the <Peierls distortion> alternates the bond lengths and doubles the primitive unit cell. Mixing states at the Fermi points opens a <band gap>, lowering the energy of occupied states. In the ideal weak-coupling, zero-temperature chain, the electronic gain behaves as $-u^2\ln(1/|u|)$ while the elastic cost is proportional to $u^2$, favouring a small distortion. The <dimerized tight-binding chain> gives an explicit band calculation.
The structural signature is extra diffraction intensity at the modulation wavevector; optical absorption can detect the gap and transport can detect loss of metallic conduction. A dynamic precursor is a <Kohn anomaly>, with phonon softening near $2k_F$. <Inelastic neutron scattering of TTF-TCNQ>, using a deuterated sample over $60$–$300\,\mathrm K$, found an anomaly near $2k_F$ developing below about $150\,\mathrm K$ as the $54\,\mathrm K$ transition was approached. This probes a lattice mode coupled to the electronic instability rather than inferring distortion from resistance alone.<Phonon dispersion and Kohn anomaly in TTF-TCNQ by Gen Shirane and others>[Abstract: inelastic neutron scattering, temperature range, 2kF anomaly and transition temperature]{ref}
Away from half filling, the modulation need not simply double the lattice. Finite-temperature ordering in real quasi-one-dimensional solids also involves interchain coupling and fluctuations; the ideal isolated-chain argument alone does not establish a finite-temperature transition.
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