ALBERT

Notes: A direct method for the determination of polytype structures of SiC, ZnS and similar substances from X-ray data is described. It is based on the values of a Patterson-like function (the `Pattersonian') which only depends on the stacking of the translationally equivalent layers of the structure. The way of obtaining the Pattersonian function from the experimental intensities is described and an algorithm given by which the sequence of layers may be deduced. This sequence is conveniently characterized by the sequence of the digits of a binary number. The influence of experimental errors in the intensities on the possibilities of determining the real sequence is discussed.

Notes: Seventeen sets of measurements of structure factors of D(+)-tartaric acid, within the range (sin θ)/λ 〈 0.5 Å−1, were provided by the participants in the International Union of Crystallography Single Crystal Intensity Measurement Project. Each participant used a different crystal, all being derived from a single crystallization batch. The results in the Project are representative of those from a wide variety of currently used diffractometers and techniques. The instruments included four-circle, normal-beam and equi-inclination diffractometers. Cu and Mo radiations were used – unfiltered, with single and balanced filters, and with crystal monochromators. The aims of the project were twofold: (a) to provide an estimate of the spread of F values associated with the range of variables involved in the project and (b) to locate, if possible, the sources of error. A number of agreement indices were used to measure the spread of F values both for equivalent reflections within any one experiment and for comparisons between experiments. In an attempt to allocate errors to certain plausible sources, an analysis-of-variance was applied to the weighted deviations of individual values of F from the set of mean values. The variables specified were intensity I, a θ angle factor d* and the Miller indices h,k,1. From the values of the agreement indices and the interaction curves from the analysis-of-variance, it was possible to recognize outlier sets that differ considerably from the mean and to isolate these, where necessary, before arriving at an estimate of the error spread of the main group. In this project, there is no one simple figure of merit which provides a ready assessment of the accuracy of measurement of structure factors. Rather, there are several ways of indicating the probable accuracy. One way is to present the spread of values of Rij(Σ(|Fi| − |Fj|)/Σ ½(|Fi| + |Fj|). This shows that two scaled experimental sets of structure factors, measured under circumstances similar to those of the project, will most probably differ by 6%, agree no better than 3%, and usually no worse than 10% except in cases of extreme systematic error where it may rise to 50% or more. From the analysis-of-variance, inferences are drawn concerning the concordance of results derived from the different types of diffractometer, on features of technique associated with the diffractometers and on other aspects, including 2 dependence, monochromaticity, count rates and extinction in the crystals. It is concluded that other sources of error may be present and that future projects should be designed to reveal these.

Notes: Any crystal structure may be described in terms of a sublattice of points, each of which represents a certain fraction of the electron density. Multiplying this sublattice by a density function f(x) and applying a shift function s(x), which brings the atoms into the right positions, the correct crystal structure can be given in many different ways. It is shown that the shift function s(x) yields phase relations between the structure factors F(h), which may be evaluated directly, if the coefficients of the Fourier representation of s(x) converge rapidly. This behaviour is demonstrated for the case of a one-dimensional acentric model structure consisting of 50 atoms. Complete information on the structure may be obtained by routine methods with the aid of 5 given phases of the structure factor. This procedure may also be applied to three-dimensional structures, if the corresponding computer programs are available.

Notes: The analysis in part I is generalized to any crystal containing rigid molecules which undergo anisotropic translational and librational motion about a site fixed by symmetry. The treatment is correct to terms in (ui2)2 and (ωi2)2, where (ui2) is the mean-square translational displacement of the molecule along the ith axis and (ωi2) is the mean-square angular libration about the same axis. The first-order treatment to terms in (ui2) and (ωi2) is shown to be equivalent to the rigid-body theory in current use.

Notes: It is shown that the concept of the molecular centre of libration in the description of the average rigid-body thermal behaviour of molecules is a useful approximation. A restriction is made on the full theory, making the centre of libration model a constrained version of the complete model. The relationship between the models is discussed. Some examples are chosen to demonstrate the closeness of the approximation. These results show that for data reaching a reliability factor of no better than 7% an analysis using the full theory might yield a meaningless result. Some data of much higher accuracy does, however, show the need for the full theory.