 Solution

A charge density wave is a spatial modulation of the electron charge density, often accompanied by a periodic lattice distortion. Write a weak modulation as $\rho (x)={\rho }_0+{\rho }_1\cos (Qx+\phi )$. In a one-dimensional metal, the Peierls distortion couples states near opposite Fermi points most strongly for $Q=2k_F$. Opening a gap near the Fermi energy lowers occupied electronic energies, while the lattice deformation has a positive elastic energy cost. Their competition can make a periodic state favorable below a transition temperature.

The modulation can be commensurate or incommensurate with the underlying Bravais lattice. Diffraction detects additional satellite peaks, while transport can show reduced low-energy carrier density. Impurities can pin its phase; above a depinning field a collectively sliding modulation can contribute nonlinear current. These effects distinguish collective density modulation from simply assigning independent static charges to isolated lattice sites.