The thermal expansion of interatomic bonds, illustrated by experimental evidence from certain niobates

The thermal expansion of structures with frameworks of linked octahedra may be broken up into two parts: the changes in shape and size of individual octahedra, and their changes of tilt relative to one another. Expansion due to the second cause, when allowed by symmetry, is generally much greater than expansion of the ectahedron itself, which is always small and may even be negative. Typical values of the average macroscopic expansion are 16 × 10^-6 deg^-1 in NaNbO₃ where change of tilt can occur, 4 × 10^-6 in KNbO₃, where it cannot. At a first approximation, the octahedron may be treated as regular at all temperatures. At a second approximation, differences of shape can be associated with off-centre displacement of the Nb atom, of such a kind that the O-O edges associated with a `relaxed' O-Nb-O link are on the average longer (typically by about 0.02 Å) than the other O-O edges. This extension e is independent of the tilt of the octahedron, its exact symmetry, the nature of the second cation, or the existence of ferroelectricity rather than antiferroelectricity, but is directly proportional to the Nb displacement. This empirical rule applies to all octahedral niobates examined (including LiNbO₃), and when symmetry allows deduction of e from lattice parameters it can be used to predict the value of the Nb displacement, and does so correctly for KNbO₃ (the only symmetrical structure about which sufficient information is available). A general consequence of the rule is that O-O edges parallel to a temperature-dependent Nb displacement (or having a component parallel to it) will tend to contract as the temperature rises. Geometrical relations are developed allowing easy calculation of O-O edge lengths and their thermal changes, both individual and mean values, from macroscopically measured data, for structures of different symmetries.