Fundamentals of crystallography

The calculation of crystal properties often makes use of a spherical modeling of atoms on their crystal lattice sites. But the question is: are atoms always as round as a sphere? Strictly speaking, the approach is only justified for atoms and ions on positions of high symmetry like in the fcc lattice or the NaCl structure. In most crystal structures, however, atoms reside on positions of lower symmetry, where a significant deviation of electron density from the shape of a sphere may arise. This holds for many crystals that serve in high technology applications like piezo- or ferroelectrics, high-k dielectrics, photovoltaics and many others. For crystal constituents on polar positions (point group symmetry C1, Cs, Cn, Cnv, n = 2, 3, 4, 6) induced dipole moments may be elicited [13] that are of importance for the crystal energy [9, 15], the calculation of pyro- and piezoelectric constants [14] or the optical absorption coefficient [11]. It may generally be stated that atoms on these positions represent a key issue in understanding anisotropic crystal properties. It has recently been shown how the concept of ionic radii has to be extended and that anions should appropriately be modelled by ellipsoids rather than by spheres. For the case studies of pyrite-type compounds, ellipsoidal deformations were shown to occur for S ions in disulfides MS2 [48] and, more generally, for chalcogen ions X in dichalcogenides MX2 [62].

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