[IUCr Home Page] [Commission Home Page]

next up previous
Next: How Numerous are the Reliable Triplets Up: An Introduction to Direct Methods. The Previous: The Positive Quartet Relation

The Negative Quartet Relation

In the event that the sum of the four phases is equal to $\pi$:

\begin{displaymath}
\phi_H + \phi_K + \phi_L + \phi_{-H-K-L} = \pi\end{displaymath} (34)
the resulting relationship is referred to as the negative quartet relation and such a relationship exists for reasonably strong intensities for H, K, L and -H- K-L.

The planes of maximum for the 4 reflection involved in relation (34) are indicated in Fig. 16. It can be seen that for all indicated positions three out of four planes intersect. If atoms are located at these points the resulting unitary structure factors of H, K, L and -H-K-L will be 0.5, because three atoms lie in the planes and one lies halfway between. Thus for a negative quartet relation the reflections H, K, L and -H-K-L will in general not be found amongst the very strongest. The next question to be answered is: what is the intensity of reflection H + K, if the electron density is located near the marked points of Fig. 16? From Fig. 17 it can be easily seen that H + K will have a small |E|-magnitude: equal numbers of points of electron density concentration lie on the H + K planes and halfway in between. As the same holds for the other cross terms it can be stated that the negative quartet relation (34) is likely to be true for reasonably large values of E4 and small |EH+K|, |EH+L| and |EK+L|.


Figure 16: In the case of the negative quartet relation the planes of maximum electron density of H, K, L and -H-K-L run as indicated here. At distinct positions three planes intersect and there most likely the electron density will be found.
\begin{figure}
\includegraphics {fig16.ps}
\end{figure}


Figure 17: As Fig. 16 with the reflection H+K indicated by means of dotted lines. It is easily checked that this reflection is weak, because the electron density is distributed in equal amounts on and between the planes.
\begin{figure}
\includegraphics {fig17.ps}
\end{figure}



next up previous
Next: How Numerous are the Reliable Triplets Up: An Introduction to Direct Methods. The Previous: The Positive Quartet Relation

Copyright © 1984, 1998 International Union of Crystallography

IUCr Webmaster