How Numerous are the Reliable Triplets and Quartets?

In the following table numbers of relations are given together with their percentage of correct indications for triplets, quartets and negative quartets above variable thresholds of respectively the triplet product E₃ and a quartet product E^*₄ (Schenk, 1973). The numbers are given for an aza-steroid with N = 40, in space group $P\overline{1}$.

 

		Triplets		Positive quartets		Negative quartets
E3	no. relations	% correct relations	E4	no.	%	no.	%
6.0	21	100	6.0	185	100
4.0	143	100	4.0	1213	100
3.0	353	100	3.0	3295	100	1	100
2.5	583	99.8	2.5	5813	99.8	2	100
2.0	980	99.7	2.0	10,006	99.5	17	100
1.5	1823	99.2	1.5	13,114	98.8	38	100
1.0	3395	96.9

As can be seen many relations are available to solve this small N = 40 structure. As a rule the number of useful triplets and quartets diminishes as N increases; this effect is quite noticeable for quartets.

One comment regarding the use of negative quartets. If phase relationships such as the triplet relation

$$\phi_H + \phi_K + \phi_{-H-K} \approx 0.$$

are used exclusively and there is no translational symmetry, the trivial solution with all phases $\phi_H$ = 0 is the most consistent one. To find phases equal to $\pi$ (e.g. in space group $P\overline{1}$) it is necessary to use relations of the type

$$\phi_H + \phi_K + \cdots \approx \pi.$$

Thus relations such as negative quartets (34), although few in number, play an important role in these structure determinations.

Copyright © 1984, 1998 International Union of Crystallography