The crystal structure of SiAl5O2N5 was characterized by laboratory X-ray powder diffraction (CuKα1). The title compound is hexagonal with space group P63/mmc (Z = 2). The unit-cell dimensions are a = 0.303153(3) nm, c = 3.28153(3) nm, and V = 0.261178(5)  nm3. The initial structural model was successfully derived by the direct methods and further refined by the Rietveld method. The final structural model showed the positional disordering of two of the four (Si,Al) sites. The maximum-entropy method-based pattern fitting (MPF) method was used to confirm the validity of the split-atom model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The reliability indices calculated from the MPF were R wp = 5.00%, S (=R wp/R e) = 1.25, R p = 3.76%, R B = 1.26%, and R F  = 0.90%. The disordered crystal structure was successfully described by overlapping four types of domains with ordered atom arrangements. The distribution of atomic positions in each of the domains can be achieved in the space group P63 mc. Two of the four types of domains are related by a pseudo-symmetry inversion, and the two remaining domains also have each other the inversion pseudo-symmetry.

A computer program, Dysnomia, for the maximum-entropy method (MEM) has been tested for the evaluation and advancement of MEM-based pattern fitting (MPF). Dysnomia is a successor to PRIMA, which was the only program integrated with RIETAN-FP for MPF. Two types of MEM algorithms, i.e., 0th-order single-pixel approximation and a variant of the Cambridge algorithm, were implemented in Dysnomia in combination with a linear combination of the “generalized F constraints” and arbitrary weighting factors for them. Dysnomia excels PRIMA in computation speed, memory efficiency, and scalability owing to parallel processing and automatic switching of discrete Fourier transform and fast Fourier transform depending on sizes of grids and observed reflections. These features of Dysnomia were evaluated for MPF analyses from X-ray powder diffraction data of three different types of compounds: taurine, Cu2CO3(OH)2 (malachite), and Sr9In(PO4)7. Reliability indices in MPF analyses proved to have been improved by using multiple F constraints and weighting factors based on lattice-plane spacings, d, in comparison with those obtained with PRIMA.

Detailed procedures for solving small crystal structures ab initio with the maximum-entropy (ME) methods using X-ray powder diffraction data are described by determining the structure of the low-pressure phase of magnesium boron nitride, Mg3BN3(L), which was previously solved by one of the authors and co-workers using the Patterson method and the direct methods. The simple ME method devised by Gull, Livesey, and Sivia failed to correctly phase the structure-factor data, leading to noninterpretable electron-density maps. This method, maximizing the entropy under the constraints of the observed structure factors with subsequent incorporation of strong extrapolates in the basis set, trapped the solution in a local entropy maximum, from which there is no way to move. The multisolution method of phase determination by entropy maximization and likelihood evaluation, developed by Bricogne and Gilmore, successfully located all the Mg, B, and N atoms in one cycle of phase extension. The correct solution had a highest log-likelihood gain, but a minimum entropy, among the multisolutions generated by phase permutation.

Crystal structure of Ca1-x/2AlSi(N3-xOx):Eu2+ (x ∼ 0.11) has been characterized using an X-ray powder diffractometer and a transmission electron microscope equipped with an energy dispersive X-ray analyzer (EDX) and an electron energy loss spectrometer (EELS). The title compound is orthorhombic with space group Cmc21, Z = 4, unit-cell dimensions a = 0.979780(7) nm, b = 0.565197(4) nm, c = 0.506356(3) nm, and V = 0.280404(3) nm3. The atom ratio Al:Si was determined to be 1:1 by EDX, and the presence of O atoms in the crystal structure was confirmed by EELS. The x-value and the atomic coordinates of the final structural model were determined by the Rietveld method. The maximum-entropy methods-based pattern fitting (MPF) method was used to confirm the validity of the structural model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The reliability indices calculated from MPF are Rwp = 9.18%, S = 1.17, Rp = 6.77%, RB = 1.91%, and RF = 0.86%. Atomic arrangements of the final structural model are in an excellent agreement with the three dimensional electron-density distributions determined by MPF.

Crystal structure and structural disorder of Ba3MgSi2O8 were reinvestigated by laboratory X-ray powder diffraction. The title compound was found to be trigonal with space group P3m1, Z=1, and unit-cell dimensions a=0.561 453(4) nm, c=0.727 629(4) nm, and V=0.198 641(2) nm3. The initial structural model used for structure refinement was taken from that of glaserite (K3NaS2O8) and modified by a split-atom model. In the split-atom model, one of the two types of Ba sites and that of SiO4 tetrahedra were, respectively, positionally and orientationally disordered. The new crystal structure and structural disorder were refined by the Rietveld method. The maximum-entropy-method-based pattern fitting (MPF) method was used to confirm the validity of the split-atom model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The final reliability indices calculated from MPF were Rwp=6.52%, S=1.36, Rp=4.84%, RB=0.97%, and RF=0.52%. Details of the disorder structure of Ba3MgSi2O8 are shown in the three-dimensional and two-dimensional electron-density distribution maps determined by MPF.

Crystal structure of Li2BaSiO4 was reinvestigated by laboratory X-ray powder diffraction. The title compound was hexagonal with space group P63cm, Z=6, unit-cell dimensions a=0.810 408(2) nm, c=1.060 829(4) nm, and V=0.603 370(3) nm3. The initial structural model was successfully derived by the direct methods and further refined by the Rietveld method, with the anisotropic atomic displacement parameters being assigned for all atoms. The reliability indices calculated from the Rietveld refinement were Rwp=6.72%, S=1.17, Rp=5.06%, RB=1.86%, and RF=0.98%. The maximum-entropy method-based pattern fitting (MPF) method was used to confirm the validity of the structural model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The final reliability indices calculated from MPF were Rwp=6.74%, S=1.17, Rp=5.10%, RB=1.49%, and RF=0.69%. Atomic arrangements of the final structural model were in excellent agreement with the three-dimensional electron-density distributions determined by MPF.

Crystal structure of Li2SrSiO4 was reinvestigated by laboratory X-ray powder diffraction. The title compound was trigonal with space group P3121, Z=3, unit-cell dimensions a=0.502 281 (4) nm and c=1.245 520(8) nm, and V=0.272 129(3) nm3. The initial structural model was derived by the direct methods and further refined by the Rietveld method. The maximum-entropy method-based pattern fitting (MPF) method was used to confirm the validity of the structural model, in which conventional structure bias caused by assuming intensity partitioning was minimized. The final reliability indices calculated from MPF were Rwp=8.04%, S=1.22, Rp=6.01%, RB=1.50%, and RF=0.66%. Atomic arrangements of the final structural model were in excellent agreement with the three-dimensional electron-density distributions determined by MPF.