  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 numerical tour de force 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.
