Coulomb crystals in the harmonic lattice approximation

Abstract

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 multi-phonon excitation and absorption. The static radial two-particle distribution function $g(r)$ is calculated for classical ($T\gtrsim \hslash {\omega }_p$, where ${\omega }_p$ is the ion plasma frequency) and quantum ($T\ll \hslash {\omega }_p$) 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 $S^{\prime \prime }(k)$, 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.

8 pages, 3 figures; accepted to Phys. Rev. E

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