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Fcc lattices are Bravais lattices and so are invariant under a set of discrete translations plus inversions over the 3 axis ($x\rightarrow -x$,$y\rightarrow -y$,$z\rightarrow -z$). When one superposes two of these fcc, the discrete translation invariances are still there (supposing that the lattice spaces are the same), but what happens to the inversions over the axis? My poor drawings suggest me that two of them are still valid and one is broken. It seems to agree with the fact that piezolectricity can't appear in lattices with centers of symmetry (transformations on its tensor show that all of its coordinates vanish), when it can have one component in a superposition of two fcc lattices, agreeing with the spoiling of on of its inversions.

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