  The five independent elastic constants of hexagonal $\beta$-eucryptite have
been determined using density functional theory (DFT) total energy
calculations. The calculated values agree well, to within 15%, with the
experimental data. Using the calculated elastic constants, the linear
compressibility of $\beta$-eucryptite parallel to the c-axis, $\chi_c$, and
perpendicular to it, $\chi_a$, have been evaluated. These values are in close
agreement to those obtained from experimentally known elastic constants, but
are in contradiction to the direct measurements based on a three-terminal
technique. The calculated compressibility parallel to the c-axis was found to
positive as opposed to the negative value obtained by direct measurements. We
have demonstrated that $\chi_c$ must be positive and discussed the implications
of a positive $\chi_c$ in the context of explaining the negative bulk thermal
expansion of $\beta$-eucryptite.
