# The |E|'s of H and 2H: The Relationship

The relation is the first phase relationship which will be considered here; it estimates in centrosymmetric space groups the phase of reflection 2H on the basis of the magnitudes |EH| and |E2H|. To start with, geometrical considerations will be applied to reflections with simple indices.

In a centrosymmetric crystal only phases of 0 and occur; provided that the phase of the 110 reflection is 0 the maxima of the associated electron density wave are found at the lines I of Fig. 2 and the minima at the lines II. If the phase of 110 is , the maxima and minima are interchanged. The lines where the electron density wave has 0 value are marked with III. Thus in the event |E110| is large and = 0, the electron density is mainly concentrated in the shaded areas of Fig. 3. For the electron density wave associated with the 220 reflection the maxima are found at both lines I and II in Fig. 2 in the case its phase is 0 and the minima at the lines III. Thus, when |E220| is large and = 0 the atoms must lie in shaded areas in Fig. 4. A similar drawing can be made for = .

The combination of the two electron density waves associated with the reflections 110 and 220 leads to Fig. 5, in which in the areas I maxima are found of both density waves. In the areas II the maximum of 220 coincides with the minimum of 110, resulting in a low density. In the event that both reflections have a large |E| value it is likely that the atoms are concentrated in the double shaded area.

In case the phase = , the vertically shaded areas shift to the blank regions of Fig. 5 and then there is no overlap between the horizontally (110) and vertically (220) shaded areas; this implies that no position for the atoms can be found in which they contribute strongly to both structure factors. As a result for = and = 0 it is not likely that both structure factor magnitudes |E110| and |E220| are large.

In conclusion, for large structure factors |E110| and |E220|, it is likely that = 0; this relationship is known as the relation.

Up to now no attention is paid to the situation = , the reader is invited to show that this gives no change in the formulation of the relation.

The comparison of H and 2H can be considered as a one-dimensional problem which can be understood by looking along line A in Fig. 2. In Fig. 6 the situation along this line is sketched with = = 0 while in Fig. 7 = 0 and = . Areas labelled P in Fig. 6 denote regions of considerable positive overlap, whereas in Fig. 7 only regions of minor positive overlap are seen. The implication is that for large |EH| and |E2H| the situation depicted in Fig. 6 is more probably true and thus = 0. When = , as denoted by the dotted line in Fig. 6 the overlap areas marked Q show that is still zero.

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