The modulated structure of γ-Na2CO3 in a harmonic approximation

Abstract

Starting from the known average structure (Dubbeldam & de Wolff, Acta Cryst. (1969), B25, 2665-2667) and four-dimensional space group, the actual modulated structure has been determined by taking into account 758 newly measured main reflexions and 1152 first-order satellites. The analysis is based on a harmonic displacive modulation model, with 14 modulation parameters as well as 9 positional and 26 thermal parameters (individual anisotropic, in Debye-Waller factors of the usual type).The final value of YA/Y~Fo is 0.108 excluding, and 0.238 including non-observed reflexions. The main features of the modulation are (a) an overall modulation with an amplitude of close to 0.33 A, for all atoms except oxygen; (b) an orientational modulation of the COa ion which is out of phase with the overall modulation. Calculation of the interatomic distances yields the result that the Dab symmetry of the CO3 ions is hardly distorted by the modulation. Certain second-order effects point to anharmonic components in the modulation, with amplitudes up to 0"03 A,. The structure turns out to consist of chains of NaO6 octahedra sharing faces and with Na-O bond lengths under 2"45 /~, showing little variation. The chains run in the direction of c and are mutually coupled in other directions by CO3 ions. The remaining Na atoms occupy the remaining spacious voids; their distances (averages > 2.60/~) to eight out of the nine surrounding oxygen atoms vary strongly, and for each with a different phase.

An elementary introduction to superspace crystallography

“…The use of additional information is required to select appropriate models for modulation functions, that are based on a finite number of parameters. One such type of information is that in first approximation the intensities of the satellite reflections up to order m are determined by the Fourier components of the modulation functions up to the same order [20]. Because scattering information is always limited to main reflections and satellites up to a particular order m, the standard method of analysis of modulated crystals employs truncated Fourier series for the modulated functions, Despite the successes of structure refinements based on Eqs.…”

The aperiodic nature of incommensurately modulated structures

“…Our first example is the structure of γ-Na 2 CO 3 which is incommensurate over a large temperature do- main, including room temperature. This is the first structure which was solved in the superspace formalism by de Wolff and his team [17]. The structure is represented in fig.…”

Statistical Approach to Diffraction of Periodic and Non-Periodic Crystals—Review

“…As discussed in Section 2.3 calculations within statistical method require slightly more advanced mathematics. The method allows, however, for fully analytical derivation of the structure factor formula with no need of complex expansion methods used in classical approach to the subject [39][40][41]. General formula (8) can be again reduced to 1D integration over distribution P(u) given in (4) in 3 intervals defined in (5).…”