It is shown that a general method of approach, based on statistical considerations, can be formulated to deal with the problem of devising weighting schemes for different situations in crystal-structure analysis. This approach leads readily to the well known method of applying a weighting function for the improvement of a Fourier synthesis when a part of the structure is known. The method is applied to a new case, namely that of defining the “best Patterson” function to locate the heavy-atom vectors from a pair of isomorphous crystals. The possible application of this method of approach for the treatment of unobserved reflections is pointed out.
The theoretical results concerning the normalized discrepancy index and other statistical parameters such as correlation functions are tested on hypothetical models in projection. Some practical aspects of using these parameters, in particular, the normalized discrepancy index are discussed. Graphical methods have been developed for the estimation of errors in the coordinates of atoms which form only a part of the structure.
A statistical analysis of different numbers in the Zhdanov sequence of polytypes of SiC, CdI2 and ZnS is made. A simple binomial model is used for SiC and CdI2. It is shown that in SiC the various polytypes could conform to the hypothesis that they are built up of random combination of blocks of 3's and 2's with finite probabilities which have been estimated from experimental observations. Relative probabilities of various polytypes are worked out. A similar analysis is made for CdI2 with 2's and 1's in the Zhdanov symbol. The observed distribution of different numbers for ZnS shows a systematic larger occurrence of odd numbers over even.
An attempt is made to correlate some of the observed features in the layer stacking sequences of the polytypes of SiC, ZnS and CdI2 based on energetics at the atomic and molecular level. Semiempirical potential functions have been used for the atomic interactions and the difference in energy of the cubic and hexagonal packing has been estimated for the different cases. The observed relative proportions of these two types of packings is shown to be governed by the above difference in energy.
The results indicate that the occurrence of numbers such as 2, 3 in the Zhdanov symbol of SiC and larger numbers in the Zhdanov symbol of ZnS are a consequence of the slightly lower energy favouring the cubic packing. In the case of CdI2 the occurrence of numbers 1 and 2 in the Zhdanov symbol is explained by the almost equal energy of packing of the sandwiched layers. The possibility of observing polytypism in the structure of diamond is pointed out.
These results indicate that among various possible factors governing the phenomenon of polytypism, the energetics at the atomic and molecular level are important to be considered.