Symmetry of flexoelectric response in ferroics

Using direct matrix method we establish the structure, including the number of nonzero independent elements, of the static flexoelectric coupling tensor f_ijkl for all 32 point groups. We use the point symmetry of elementary cell, previously known evident index-permutation symmetry (f_ijkl= f_jikl) and recently established "hidden" index-permutation symmetry (f_ijkl= f_ilkj). We compare these results and demonstrated that the hidden symmetry of f_ijkl significantly reduces the number of its nonzero independent elements. Using group theory, we find out the explicit form of the tensor f_ijkl and the flexoelectric coupling energy in the form of Lifshitz invariant for several point symmetries most important for applications. For these symmetries we visualize the effective flexoelectric response of the bended plate allowing for both evident and hidden index-permutation symmetries, analyze its anisotropy, and angular dependences. We discuss how the hidden symmetry can significantly simplify the problem of the flexoelectric constants determination from inelastic neutron and Raman scattering experiments due to the minimization of f_ijkl independent components, opening the way for its unambiguous determination from the scattering experiments.