Coulomb crystals in the harmonic lattice approximation

The dynamic structure factor $\tilde{S}(\mathbf{k},\omega )$ and the two-particle distribution function $g(\mathbf{r},t)$ of ions in a Coulomb crystal are obtained in a closed analytic form using the harmonic lattice (HL) approximation which takes into account all processes of multiphonon excitation and absorption. The static radial two-particle distribution function $g\left(r\right)\mathrm{}$ is calculated for classical $(T\gtrsim ħ{\omega }_p,$ where ${\omega }_p$ is the ion plasma frequency) and quantum $(T\ll ħ{\omega }_p)$ body-centered-cubic (bcc) crystals. The results for the classical crystal are in a very good agreement with extensive Monte Carlo (MC) calculations at $1.5\lesssim r/a\lesssim 7,$ where a is the ion-sphere radius. The HL Coulomb energy is calculated for classical and quantum bcc and face-centered-cubic crystals, and anharmonic corrections are discussed. The inelastic part of the HL static structure factor $S^{″}\mathrm{}\left(k\right),$ averaged over orientations of wave vector k, is shown to contain pronounced singularities at Bragg diffraction positions. The HL method can serve as a useful tool complementary to MC and other numerical methods.