Abstract
A combination of structure refinements, analysis of the superspace MEM density and interpretation of difference-Fourier maps has been used to characterize the incommensurate modulation of rubidium tetrachlorozincate, Rb(2)ZnCl(4), at a temperature of T = 196 K, close to the lock-in transition at T(lock-in) = 192 K. The modulation is found to consist of a combination of displacement modulation functions, modulated atomic displacement parameters (ADPs) and modulated third-order anharmonic ADPs. Up to fifth-order Fourier coefficients could be refined against diffraction data containing up to fifth-order satellite reflections. The center-of-charge of the atomic basins of the MEM density and the displacive modulation functions of the structure model provide equivalent descriptions of the displacive modulation. Modulations of the ADPs and anharmonic ADPs are visible in the MEM density, but extracting quantitative information about these modulations appears to be difficult. In the structure refinements the modulation parameters of the ADPs form a dependent set, and ad hoc restrictions had to be introduced in the refinements. It is suggested that modulated harmonic ADPs and modulated third-order anharmonic ADPs form an intrinsic part, however small, of incommensurately modulated structures in general. Refinements of alternate models with and without parameters for modulated ADPs lead to significant differences between the parameters of the displacement modulation in these two types of models, thus showing the modulation of ADPs to be important for a correct description of the displacive modulation. The resulting functions do not provide evidence for an interpretation of the modulation by a soliton model.
Highlights
The construction of a model-independent electron-density map from phased structure factors is an important application of the maximum entropy method (MEM) in crystallography (Takata, 2008; van Smaalen & Netzel, 2009)
A combination of structure refinements, analysis of the superspace MEM density and interpretation of difference-Fourier maps has been used to characterize the incommensurate modulation of Rb2ZnCl4 at a temperature of T = 196 K, close to the lock-in transition at TlockÀin = 192 K
The harmonic atomic displacement parameters (ADPs) are modulated with up to second-order Fourier coefficients and the third-order anharmonic ADPs are modulated with Fourier coefficients up to fifth order, while the basic structure or average components of the third-order anharmonic ADPs are zero
Summary
The construction of a model-independent electron-density map from phased structure factors is an important application of the maximum entropy method (MEM) in crystallography (Takata, 2008; van Smaalen & Netzel, 2009). The MEM has been used to determine the locations of the metal atoms in endohedral fullerenes (Nishibori et al, 2006), to obtain information about disorder (multiple positions) in crystal structures (Dinnebier et al, 1999; Wang et al, 2001; Samy et al, 2010) and about anharmonic atomic displacements (Kumazawa et al, 1995; Bagautdinov et al, 1998). The MEM has been generalized towards the determination of the generalized electron density in ð3 þ dÞ-dimensional superspace The MEM provides information about the locations of the atoms, which result in a description of the doi:10.1107/S0108768111013814 205 research papers modulation functions of incommensurately modulated crystals or incommensurate composite crystals (Palatinus & van Smaalen, 2004; van Smaalen & Li, 2009). We will use the MEM to obtain information about the modulation functions of incommensurately modulated Rb2ZnCl4
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