The Interpretation of Harker Syntheses

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Abstract
An elegant way of studying the geometrical meaning of a Harker synthesis is to transform it into a diagram here termed the implication diagram. This is done by transformation of polar coordinates, the characteristics of the transformation depending on the number of operations in the cyclical group of the axial symmetry for which the Harker synthesis is prepared. The implication diagram has the important property that it is a map of the location of atoms in the crystal structure plus additional locations (ambiguities) which would give rise to the same Harker synthesis, plus satellitic locations. Fortunately the satellitic locations can be identified as such. Satellitic peaks are caused by powers of rotation operations and by reflections. Satellitic peaks occur in specialized locations which are discussed for the several cases. On the other hand, it is not generally possible to decide between ambiguities. An implication diagram which is a map of the crystal structure, with ambiguities, but without satellitic locations, exists for the following 18 of the 21 possible parallel‐axial symmetries: P2, P21, C2, H3, H31, H32, R3, P4̄, I4̄, P41, P42, P43, I41, H61, H65, H62, H64, and H63. This is also true of certain axial symmetries combined with reflections, such as 2c and 3c. Only for the three axial symmetries P4, I4, and H6 do no implications exist without satellitic peaks. The implication diagrams of the Harker syntheses P(xy⅙) for the space groups H61, H65, H612, and H652 and for Harker synthesis P(xy0) for R3̄ lack both ambiguities and satellitic peaks. For these space groups the implication diagrams are true maps of the crystal structure. Harker syntheses greatly exaggerate the electron densities of atoms in the structure, and for this reason their greatest usefulness lies in the location of the more compact atoms. Diffuse atoms, and therefore especially anions, provide Harker peaks of such low magnitude as to be often lost in the background. Background due to non‐Harker interactions is discussed. An important characteristic of Harker syntheses, which is independent of any interpretation of their meaning with regard to the location of atoms in the crystal structure, is that they provide criteria for distinguishing space groups which cannot be distinguished in ordinary qualitative x‐ray crystallography. This is because the Harker synthesis provides definite criteria for identifying reflection planes and rotation axes. It fails only to distinguish pairs of space groups which differ by a group of inversions alone. Harker synthesis thus realizes the ultimate possibilities of x‐ray crystallography because it is quantitative.