Phonon peaks in the dynamic structure factor of a two-dimensional harmonic crystal

The static structure factor $S\left(\overrightarrow{\mathrm{Q}}\right)$ of a two-dimensional (2D) harmonic crystal exhibits Bragg peaks with power-law singularities $\sim {\left|\overrightarrow{\mathrm{q}}\right|}^{-2+\eta }$ with $\overrightarrow{\mathrm{q}}=\overrightarrow{\mathrm{Q}}-\overrightarrow{\mathrm{K}}$, where $\overrightarrow{\mathrm{K}}$ is the closest 2D reciprocal-lattice vector to $\overrightarrow{\mathrm{Q}}$. We have evaluated $S(\overrightarrow{\mathrm{Q}},\omega )$ and have shown that both longitudinal- and transverse-phonon resonances are described by power-law singularities of the kind ${|{\omega }^2-v_{\lambda }^2{q}^2|}^{-1+\eta }$. This agrees with the work of Mikeska, who only considered $v_{\mathrm{L}}=v_{\mathrm{T}}$. We also show that a one-phonon expansion is possible for the time-dependent part. Finite-size effects are included by a Warren-type approximation.