Direct Correlation Function of a Crystalline Solid

Direct correlation functions (DCFs), linked to the second functional derivative of the free energy with respect to the one-particle density, play a fundamental role in a statistical mechanics description of matter. This holds, in particular, for the ordered phases: DCFs contain information about the local structure including defects and encode the thermodynamic properties of crystalline solids; they open a route to the elastic constants beyond low temperature expansions. Via a demanding numerical approach, we have explicitly calculated for the first time the DCF of a solid: based on the fundamental measure concept, we provide results for the DCF of a hard sphere crystal. We demonstrate that this function differs at coexistence significantly from its liquid counterpart—both in shape as well as in its order of magnitude—because it is dominated by vacancies. We provide evidence that the traditional use of liquid DCFs in functional Taylor expansions of the free energy is conceptually wrong and show that the emergent elastic constants are in good agreement with simulation-based results.

Figure 1

Direct correlation function $c^{\mathrm{cr}}({\mathbf{r}}_1,{\mathbf{r}}_2)$ from FMT as function of distance (with $\sigma$ the sphere diameter) in a hard sphere fcc crystal at the melting density ($\eta =0.545$, $n_{\mathrm{vac}}=2.18\times {10}^{-5}$). (a) $c^{\mathrm{cr}}$ in the RLV expansion up to ${10}^4$ shells; $\mathrm{\Delta }r$ is started from a lattice site. The black solid line is from the brute force FT and the red dotted line is the fluid DCF [26] at $\eta =0.545$ with scale on the right side. (b) $c^{\mathrm{cr}}(0,\mathrm{\Delta }\mathbf{r})$ in three directions by the brute force FT (solid line) and RLV expansion up to ${10}^4$ shells (dots). (c) $c^{\mathrm{cr}}({\mathbf{r}}_1,{\mathbf{r}}_1+\mathrm{\Delta }\mathbf{r})$ with distance measured from the interstitial point ${\mathbf{r}}_1=[(a/4),(a/4),(a/4)]$ and $a$ the side length of the cubic unit cell, lines are as in (b).

Figure 2

DCF route to elastic constants: (a) generalized elastic constants and (b) Voigt elastic constants at coexistence ($\eta =0.545$) as a function of vacancy concentration. Symbols connected with full lines are results from Eqs. (8) and (9), and dotted lines use Eqs. (8) and (9) with $c^{\mathrm{cr}}({\mathbf{r}}_1,{\mathbf{r}}_2)\to c^{\mathrm{liq}}(|{\mathbf{r}}_1-{\mathbf{r}}_2|;{\eta }_0)$, the fluid DCF [26] at ${\eta }_0=0.545$; “eq” with the arrow indicates the equilibrium $n_{\mathrm{vac}}$.