Reconstruction of electron density by the Maximum Entropy Method from X-ray powder diffraction data based on incomplete crystal structure models: a case study of apatites with different intercalated metal atoms

Conventional Fourier analysis is one of the most commonly used methods for localization of missing atoms in crystal structures from powder diffraction data. A “perfect” Fourier map would require a complete set of structure factors up to a resolution of at least sin/ = 5.0 Å. In the case of powder diffraction, accessible information is limited, as compared to single crystal diffraction, mainly due to the projection of three-dimensional reciprocal space onto the one dimensional 2θ axis, resulting in intrinsic and accidental peak overlap. In addition, the resolution of powder diffraction data measured on laboratory instruments is generally limited to sin/  0.6Å. As a result, the Fourier transform is affected by series termination errors (e.g., spurious peaks of positive electron densities that do not correspond to atoms in the structure and unphysical local minima with negative densities). The concept of informational entropy was introduced in the field of crystallography to handle series termination effects in Fourier maps. The Maximum Entropy Method (MEM) allows to maximize the information extracted from the intrinsically limited experimental X–ray powder diffraction data. It has been demonstrated that MEM can be successfully used with powder diffraction data for localization of missing atoms with high occupancies in incomplete crystal structures [1, 2], for revealing the true nature of structural disorder [3, 4] and for determination of integrated atomic charges [5]. On the other hand, the capability of the MEM to locate missing atoms with low occupancies and to reconstruct their accurate electron density distribution is not fully investigated. This holds in particular true in case of commonly used high-resolution laboratory X-ray powder diffraction data. In the present work, the possibilities and limitations of the MEM for localization of missing intercalated metal atoms in apatites (general formula is Ae5(PO4)3MxOH1-x, Ae = Sr or Ca, M = Cu, Ni or Zn) with intercalated copper, nickel or zinc metal atoms were evaluated.