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The triplet relation, although a twodimensional phase relation, is very
successful in solving threedimensional crystal structures. Nevertheless, it
may be more appropriate to try to solve structures with threedimensional phase
relationships, the quartet relations.
The positive quartet relation is formulated as:
 
(32) 
for large E_{4} = N^{1}E_{H}E_{K}E_{L}E_{HKL}.
Analogous to the treatment of the triplet relation now three strong reflections
H, K and L are combined and the electron density must be found in the sets
of planes of Fig. 12. As a result the electron density will be found near the
points of intersection of the three planes which are indicated for only a few
planes from the sets in Fig. 13. For a strong reflection H K L it is
much more likely that its plane of maximum electron density will run through the
points of intersection (Fig. 14a) than that it will clear these points (Fig.
14b). From Fig. 14a the quartet relation (32) follows as straightforward as the
triplet relation from Fig. 10a (Schenk, 1981).
Figure 12:
If the reflections H, K and L are strong, the electron
density will probably lie in the neighbourhood of the three sets of
equidistant planes defined by H, K and L.

Figure 13:
The electron density will be found near the points of intersection
of the three sets of planes H, K and L.

Figure 14:
If reflection HKL has also a large intensity, it is more
likely that its planes of maximum intensity run through the points of
intersection of H, K and L (a) than that they run clear of them (b).

This quartet relation, however, is not as strong as the triplet relation because
of the factor N^{1} in E_{4}. It will be recalled that in E_{3} a term
N^{1/2} appears. The reliability is improved by combining the quartet
with an identical one constructed from two triplets:
 
(33) 
which holds for large E_{4} and large E_{H+K}. This can be understood by
drawing the H+K reflection in Fig. 13, as indicated in Fig. 15a. Reflection
H+K strong indicates that the electron density will be found near the
intersection of H and K and thus a large E_{H+K} is an additional
indication that quartet (32) is true. In Figs. 15b and 15c similar situations
are sketched for the other two cross terms H + L and K + L, which both
leads to sums of triplets analogous to (33) involving the phases and respectively. So in conclusion a large value of E_{4} and
large E_{H+K}, E_{H+L} and E_{K+L} are indications that the positive
quartet relation (32) is likely to be true, and thus positive quartets are
controlled by the magnitudes of 7 structure factors.
Figure 15:
If H, K, L and HKL are strong and the electron density
will be found near their points of intersecting, the H + K reflection,
indicated in (a) with a dotted line, is expected to be strong, conversely a
large E_{H+K} supports the quartet relation (11). Figs (b) and (c)
give the analogous situations with respect to the H + L and K + L
reflections.

Next: The Negative Quartet Relation
Up: An Introduction to Direct Methods. The
Previous: The Triplet Relation from Sayre's Equation
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