Microscopic theory of phase and amplitude instabilities of an incommensurate charge-density wave

We show that an incommensurate charge-density wave, having wave vector $\overrightarrow{\mathrm{Q}}$, will always exhibit a phason instability of wave vector $n(\overrightarrow{\mathrm{Q}}-\overrightarrow{\mathrm{G}})$ ($\overrightarrow{\mathrm{G}}$ is a reciprocal-lattice vector and $n$ is an integer). The microscopic theory presented is based on the dielectric formulation of lattice dynamics, frequently employed for simple metals. However, special attention is given to the structure components at $\pm \overrightarrow{\mathrm{Q}}$, which are inherently nonlinear. For a monatomic metal with $|\overrightarrow{\mathrm{Q}}-\overrightarrow{\mathrm{G}}|\ll \left|\overrightarrow{\mathrm{G}}\right|$ we find that the dominant instability has $n=2\mathrm{}$, which causes third harmonics of $\overrightarrow{\mathrm{Q}}$ in the diffraction pattern. For a polyatomic metal (lacking inversion symmetry) an $n=1\mathrm{}$ instability is also possible. Amplitude-modulation instabilities also occur but are expected to be smaller in magnitude on account of their higher intrinsic frequencies, compared to those for phasons.