P. T. Beurskens

Direct Methods for Incommensurate Intergrowth Compounds. I. Determination of the Modulation

A direct method is proposed for the determination of the modulation in incommensurate intergrowth compounds. The method is based on a new type of Sayre equation that relates the phase of a satellite reflection to the sum of structure-factor products of pairs of main reflections. Phases of satellite reflections are thus uniquely determined by the phases of main reflections. This reflects the fact that the modulation in intergrowth structures is the result of the interaction of all of the subsystems that form the basic structure. Test calculations were done with experimental data of two known composite structures of the inorganic misfit layer compounds (LaS)1.14NbS2 and (PbS)1.18TiS2. The results showed that the method is accurate and efficient and is fully independent of any preliminary assumption of the model of modulation.

Direct Methods for Incommensurate Intergrowth Compounds. II. Determination of the Modulation using only Main Reflections

A modified Sayre equation for incommensurate intergrowth compounds is presented. With this equation, both magnitude and phase for structure factors of satellite reflections can be estimated quantitatively through the observed intensities of main reflections, provided their phases are already known. Modulation functions can then be revealed by the Fourier synthesis calculated using the observed main reflections and the estimated satellites. The method has been tested with the known structures of two inorganic misfit layer compounds, (LaS)1.14NbS2 and (PbS)1.18TiS2. Satisfactory results were obtained.

DETERMINATION OF THE INCOMMENSURATELY MODULATED STRUCTURE OF (PERYLENE)CO(MNT)(2)-(CH2CL2)(0.5) BY DIRECT-METHODS

For the organic conductor (perylene)Co(mnt)(2)-(CH2Cl2)(0.5), where mnt is maleonitriledithiolate, the incommensurate displacive modulation is determined using X-ray diffraction data for main reflections and first- and second-order satellites, collected at a temperature of 283 K. The lattice parameters of the unit cell of the average structure are: a = 6.5441 (13), b = 11.7173 (15), c = 16.4251 (17) Angstrom, alpha = 92.092 (11), beta = 95.343 (16), gamma = 94.67 (2)degrees, with V = 1248.6(3)Angstrom(3) and Z = 2. The components of the modulation wavevector are given by: q(1) = 0.211 (13), q(2) = -0.1374(5), q(3) = -0.368(2). The symmetry of the modulated structure is given by the (3+1)-dimensional superspace group P (

INTENSITY STATISTICS AND NORMALIZED STRUCTURE FACTORS FOR CRYSTALS WITH AN INCOMMENSURATE ONE-DIMENSIONAL MODULATION

) over bar 1 (q(1), q(2), q(3)). Direct methods were used to obtain a starting model for the modulation. The subsequent refinement converged to R = 0.126 for 2835 observed (I/sigma > 2.5) reflections. Partial R factors are 0.111 for 1450 main reflections, 0.143 for 1188 first-order satellites and 0.263 for 197 second-order satellites. The modulation is described by sawtooth-shaped functions for the Co and S atoms and by rigid-body modulations, up to third-order harmonics, for the perylene units and parts of the mnt fragments. The largest amplitudes were found for the Co (0.77 Angstrom) and S atoms (0.48-0.63 Angstrom) and were mainly directed along the a axis. The four equatorial Co-S distances are only slightly affected by the modulation, but the two apical Co-S distances show large variations with distances ranging from 2.05 to 3.86 Angstrom. These variations are out of phase. This causes the coordination of the Co atom to vary from a distorted octahedral coordination by six S atoms to a region with fivefold coordination and vice versa. The valence of the Co atom, as calculated by the bond-valence method, varies between 2.92 and 3.57. The stacking of the Co(mnt)(2) units can be described by oligomeric packages of four or five dimerized Co(mnt)(2) units.

SCALING OF X-RAY-DIFFRACTION INTENSITIES FOR CRYSTALS WITH A ONE-DIMENSIONAL, INCOMMENSURATE, DISPLACIVE MODULATION

An analytical expression is derived for the probability density function (p.d.f.) of X-ray structure-factor amplitudes of a crystal with an incommensurate one-dimensional modulation. The influence of the (3+1)-dimensional superspace symmetry is taken into account. It is shown that, in first-order approximation, this p.d.f. has the same functional form as the p.d.f. for a nonmodulated crystal, with a suitable modification of the atomic form factor. For main reflections and satellite reflections, an expression for the average intensity is derived. This leads to a definition of normalized structure factors for a crystal with an incommensurate one-dimensional modulation. In the same first-order approximation, the p.d.f. for the amplitudes of these normalized structure factors is identical to the p.d.f. for a nonmodulated crystal and does not distinguish between main reflections and satellite reflections. The theoretical p.d.fs are compared to p.d.fs obtained from X-ray diffraction data of some incommensurate one-dimensionally modulated crystals.

A more general expression for the average X-ray-diffraction intensity of crystals with an incommensurate one-dimensional modulation

A statistical method is presented for the determination of the scale factor, an overall isotropic temperature factor and an overall modulation amplitude from the X-ray diffraction intensities of crystals with a one-dimensional, incommensurate, displacive modulation. Application to several compounds with a known modulation illustrates the accuracy of our method. The results may provide a starting point for a structure determination. A preliminary definition is given of normalized structure factors which can be used in direct methods for the solution of the phase problem.

AB-INITIO SOLUTION OF MISFIT LAYER STRUCTURES BY AUTOMATIC PATTERSON AND DIRECT-METHODS

Statistical methods are used to derive an expression for the average X-ray diffraction intensity, as a function of (sin theta)/lambda, of crystals with an incommensurate one-dimensional modulation. Displacive and density modulations are considered, as well as a combination of these two. The atomic modulation functions are given by truncated Fourier series that may contain higher-order harmonics. The resulting expression for the average X-ray diffraction intensity is valid for main reflections and low-order satellite reflections. The modulation of individual atoms is taken into account by the introduction of overall modulation amplitudes. The accuracy of this expression for the average X-ray diffraction intensity is illustrated by comparison with model structures. A definition is presented for normalized structure factors of crystals with an incommensurate one-dimensional modulation that can be used in direct-methods procedures for solving the phase problem in X-ray crystallography. A numerical fitting procedure is described that can extract a scale factor, an overall temperature parameter and overall modulation amplitudes from experimental reflection intensities.

Ab initio solution of misfit layer structures by automatic Patterson and direct methods

A procedure is presented for the automatic solution of composite (misfit) layer compounds, for the case when the composite crystal structure consists of two types of layer, each of which can be approximately described as a three-dimensional periodic structure with, however, mutually incommensurate lattices and hence mutually induced incommensurate modulations. The composite structure can be described as a periodic structure in four-dimensional superspace [van Smaalen (1992), Mater. Sci. Forum, 100&101, 173-2221. From reflection data indexed with four integer indices HKLM, the phase problem is solved as follows. The basic structures of layers 1 and 2 are solved by routine application of automated Patterson interpretation and Fourier recycling using the main reflections only and ignoring the modulation effects. The two layers are brought to a common origin by a shift function based on correlating F(obs) and F(calc) using the main reflections common to both layers. All other reflections, which are the (usually weaker) satellite reflections, are phased from the known phases of the main reflections of either layer by application of the Sayre equation in four-dimensional superspace [Hao, Liu & Fan (1987), Acta Cryst. A43, 820-824; Fan, van Smaalen, Lam & Beurskens (1993), Acta Cryst. A49, 704-7081. The procedure is performed by the program MISFIT, which is embedded in the DIRDIF system.

Crystal structures of di-tin-hexa(seleno)hypodiphosphate, Sn2P2Se6, in the ferroelectric and paraelectric phase

A procedure is presented for the automatic solution of composite (misfit) layer compounds, for the case when the composite crystal structure consists of two types of layer, each of which can be approximately described as a three-dimensional periodic structure with, however, mutually incommensurate lattices and hence mutually induced incommensurate modulations. The composite structure can be described as a periodic structure in fourdimensional superspace [van Smaalen (1992), Mater. Sci. Forum, 100&101, 173-222]. From reflection data indexed with four integer indices HKLM, the phase problem is solved as follows. The basic structures of layers I and 2 are solved by routine application of automated Patterson interpretation and Fourier recycling using the main reflections only and ignoring the modulation effects. The two layers are brought to a common origin by a shift function based on correlating F,,b.~ and F,.~v using the main reflections common to both layers. All other reflections, which are the (usually weaker) satellite reflections, are phased from the known phases of the main reflections of either layer by application of the Sayre equation in four-dimensional superspace [Hao, Liu & Fan (1987), Acta Cryst. A43, 820-824; Fan, van Smaalen, Lam & Beurskens (1993), Acta Cryst. A49, 704-708]. The procedure is performed by the program MISFIT, which is embedded in the DIRDIF system.