Decay of pair correlations in three-dimensional crystals

The long-range behavior of spatial correlations in three-dimensional crystals is analyzed in the context of a Landau model. A scaling argument is used to show that the two-particle density distribution ${\rho }_2({\overrightarrow{\mathrm{r}}}_1,{\overrightarrow{\mathrm{r}}}_2)$ decays to its asymptotic value ${\rho }_1\left({\overrightarrow{\mathrm{r}}}_1\right){\rho }_1\left({\overrightarrow{\mathrm{r}}}_2\right)$ as $\frac{1}{r_{12}}$ when the distance $r_{12}$ between the positions ${\overrightarrow{\mathrm{r}}}_1$ and ${\overrightarrow{\mathrm{r}}}_2$ in the crystal becomes large. An elastic analogy is developed whereby this asymptotic behavior may also be interpreted in terms of displacement fields induced by the action of point forces. The slow $\frac{1}{r_{12}}$ decay of the density correlations is seen to be entirely consistent with expressions for the elastic moduli and the thermal diffuse scattering intensity.