  We calculate the Casimir interaction between parallel planar crystals of Au
and the anisotropic cuprate superconductor Bi2Sr2CaCu2O8+delta (BSCCO), with
BSCCO's optical axis either parallel or perpendicular to the crystal surface,
using suitable generalizations of the Lifshitz theory. We find that the strong
anisotropy of the BSCCO permittivity gives rise to a difference in the Casimir
force between the two orientations of the optical axis, which depends on
distance and is of order 10-20% at the experimentally accessible separations 10
to 5000 nm.
