Structure of composites A1+x(A′xB1−x)O3 related to the 2H hexagonal perovskite: relation between composition and modulation

An elementary introduction to superspace crystallography

“…Other special functions include the block-wave, a combination of segments with saw-tooth shape and segments with block-wave shape, and the soliton wave [23]. An interesting class of compounds with large, non-harmonic modulations are the hexagonal perovskites, which can be described as the stacking of two or three types of atomic layers, where the stacking sequences range from short and long periods to incommensurate order [24,25]. An alternative to refinements of parameters of model functions is provided by the Maximum Entropy Method (MEM) in superspace, with which a model-independent shape can be determined for the modulation functions [26,27].…”

Structure Determination of Ba8CoRh6O21, a New Member of the 2H-Perovskite Related Oxides

“…The fractional atomic average coordinates and thermal parameters are given in Table I [22,27] From the value of γ = 0.57143(1) (or 4/7), the value of x can be calculated according to the following relationship γ = (1+x)/2, thereby x = 1/7. For this compound, x is a rational fraction indicating a commensurate structure.…”

“…As shown previously, a better structural formulation of such composites is A 1+x (A' x B 1-x )O 3 , where x = n/(3m+2n) and ranges continuously between 0 and 1/2, corresponding to chains containing all face-shared octahedra and alternating face-sharing octahedra and trigonal prisms, respectively. [27][28][29] For simple fractional values of x such as 1/5, 2/7, or 1/3, the structure is commensurate and the endmembers, the 2H perovskite (BaNiO 3 , x = 0) and the K 4 CdCl 6 (x = 1/2) structure type, are well known.…”

“…An early general structural classification of these materials based on the filling of interstitial sites generated by the stacking of [A 3 O 9 ] and [A 3 A'O 6 ] layers was developed by Darriet and Subramanian. [26][27] This approach easily describes the structural composition of all the commensurate members of this family of structures and can be extended to encompass members that form incommensurately modulated (aperiodic) structures.…”

Stability of aperiodic crystals

“…The incommensurability of a composite is determined by the ratio of the lattice constants of the subsystems, although the subsystems do not exist necessarily as such. In so called flexible composition compounds [3] one subsystem consists of columns of octahedra and triangular prisms, the other of rows of atoms. The building blocks in the first system are characterised by a pseudo-spin (s n ¼ þ1 for an octahedron, ¼ À1 for a prism).…”