THE ROLE OF CRYSTALLOGRAPHIC SYMMETRY IN THE DIRECT METHODS OF X-RAY CRYSTALLOGRAPHY

Abstract

The number of equations relating the intensities of X-ray diffraction maxima with the crystal structure is in general far greater than the number of parameters needed to define the structure. Since the elucidation of the crystal structure requires also a knowledge of the complex numbers EH = |EH| exp(iϕH), the normalized structure factors, of which only the magnitudes |EH| are obtainable from the measured intensities while the needed phases ϕH are lost in the diffraction experiment, it is necessary first to determine the phases. Owing to the redundancy of observed magnitudes |EH|, relationships exist among the complex numbers EH means of which the desired phases ϕ may be expressed in terms of the known magnitudes |E|. Thus, the lost phase information is to be found among the observed intensities. The direct methods are those which exploit relationships among the normalized structure factors EH in order to extract the values of the individual phases ϕH from the measured magnitudes |EH|. When elements of crystallographic symmetry are present then there exist corresponding, space group dependent, relationships among the structure factors EH which may also be employed. The major goal of the present article is to show how these additional relationships, arising from the space group symmetries, may be effectively exploited.