Coulomb crystals in the harmonic lattice approximation

The dynamic structure factor ${\tilde S}({\bf k},ω)$ and the two-particle distribution function $g({\bf r},t)$ 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 multi-phonon excitation and absorption. The static radial two-particle distribution function $g(r)$ is calculated for classical ($T \gtrsim \hbar ω_p$, where $ω_p$ is the ion plasma frequency) and quantum ($T \ll \hbar ω_p$) 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 $S''(k)$, averaged over orientations of wave-vector {\bf k}, is shown to contain pronounced singularities at Bragg diffraction positions. The type of the singularities is different in classical and quantum cases. The HL method can serve as a useful tool complementary to MC and other numerical methods.