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Next: The -Relation from a Harker-Kasper Inequality Up: An Introduction to Direct Methods. The Previous: Normalized Structure Factors EH

The |E|'s of H and 2H: The $\sum_1$ Relationship

The $\sum_1$ 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 $\pi$ 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 $\pi$, 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 $\phi_{110}$ = 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 $\phi_{220}$= 0 the atoms must lie in shaded areas in Fig. 4. A similar drawing can be made for $\phi_{220}$ = $\pi$.


Figure 2: Lines of equal contribution to the electron density of the reflections hh0. E.g. for reflections 110 with $\phi_{110}$ = 0 the contribution to the electron density is maximum at lines I, minimum at lines II and zero at lines III.
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\includegraphics {fig2.ps}
\end{figure}


Figure 3: In case |E110| is large and $\phi_{110}$ = 0 the atoms are likely to be found in the shaded areas.
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\includegraphics {fig3.ps}
\end{figure}


Figure 4: For |E220| large and $\phi_{220}$ = 0 the electron density is more likely to be present in the vertically shaded areas.
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\includegraphics {fig4.ps}
\end{figure}


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 $\phi_{220}$ = $\pi$, 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 $\phi_{220}$ = $\pi$ and $\phi_{110}$ = 0 it is not likely that both structure factor magnitudes |E110| and |E220| are large.


Figure 5: Superposition of Figs. 3 and 4. In the areas I the shaded areas from the 110 and 220 reflections coincide. In case both reflections are large this is a rather likely situation.
\begin{figure}
\includegraphics {fig5.ps}
\end{figure}

In conclusion, for large structure factors |E110| and |E220|, it is likely that $\phi_{220}$ = 0; this relationship is known as the $\sum_1$relation.

Up to now no attention is paid to the situation $\phi_{110}$ = $\pi$, the reader is invited to show that this gives no change in the formulation of the $\sum_1$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 $\phi_H$ = $\phi_{2H}$ = 0 while in Fig. 7 $\phi_H$ = 0 and $\phi_{2H}$ = $\pi$. 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 $\phi_{2H}$ = 0. When $\phi_H$ = $\pi$, as denoted by the dotted line in Fig. 6 the overlap areas marked Q show that $\phi_{2H}$ is still zero.


Figure 6: The drawn line H gives the electron density wave for $\phi_H$ = 0, and its dotted mirror image the wave for $\phi_H$ = $\pi$. The maximum of the dashed line 2H coincides with the maxima of the drawn line H in P and with the maxima of the dotted one in Q. Thus if |EH| and |E2H| are large, it is likely that $\phi_{2H}$ = 0 whatever the phase of H.
\begin{figure}
\includegraphics {fig6.ps}
\end{figure}


Figure 7: Here the unlikely situation is depicted that for strong reflections H and 2H $\phi_H$ = 0 and $\phi_{2H}$ = $\pi$. There is no positive overlap and therefore if |EH| and |E2H| are both large this situation is much more unlikely to exist than the situation of Fig. 7.
\begin{figure}
\includegraphics {fig7.ps}
\end{figure}



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Next: The -Relation from a Harker-Kasper Inequality Up: An Introduction to Direct Methods. The Previous: Normalized Structure Factors EH

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