  In this paper, we show how to incorporate cubic and hexagonal anisotropies in
interfacial energies in phase field models; this incorporation is achieved by
including upto sixth rank tensor terms in the free energy expansion, assuming
that the free energy is only a function of coarse grained composition, its
gradient, curvature and aberration. We derive the number of non-zero and
independent components of these tensors. Further, by demanding that the
resultant interfacial energy is positive definite for inclusion of each of the
tensor terms individually, we identify the constraints imposed on the
independent components of these tensors. The existing results in the invariant
group theory literature can be used to simplify the process of construction of
some (but not all) of the higher order tensors. Finally, we derive the relevant
phase field evolution equations.
