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In the preceding chapters the main object was to clarify the basis of the present direct methods. In this chapter a brief guide to additional literature is given.

This triplet relation originates from the early fifties and was implicitly present in the important papers by Harker and Kasper (1948), Karle and Hauptman (1950) and Sayre (1952). For the cenytrosymmetric case it was explicitly formulated by Sayre (1952), Cochran (1952), Zachariasen (1952) and Hauptman and Karle (1953). The latter authors gave it its probability basis, which was independently derived by Kitaigorodsky (1954) as well. The noncentrosymmetric case was formulated first by Cochran (1955). Another useful expression related to the relation is the tangent formula (31) derived by Karle and Hauptman (1956).

A very important development was the use of symbols for tackling the set of triplet relations (1) in order to find the phases. Symbols are assigned to unknown phases such that a successful phase extension can be carried out. Later in the process in most cases the numerical values of the symbols can be determined. The use of symbols was first introduced by Gillis (1948) and later successfully applied by Zachariasen (1952) and Rumanova (1954), but due to the work of Karle and Karle (1963, 1966) the method could develop to a standard technique in crystallography. In particular the first structure determination of a non-centrosymmetric structure (Karle and Karle, 1964) proved the value of direct methods. The method has recently been described in detail by J. Karle (1974) and Schenk (1980a). The latter gives also some exercises. For centrosymmetric structures the symbolic addition procedure has been automized amongst others by Beurskens (1965), Germain and Woolfson (1968), Schenk (1969), Ahmed (1970), Dewar (1970), and Stewart (1970).

In noncentrosymmetric structures the programming problems are much greater and therefore the number of successful automatic program systems is smaller, examples are the systems of Dewar (1970) and the interactive system SIMPEL (Overbeek and Schenk, 1978).

Nowadays most of the structures are solved by multisolution tangent refinement procedures, which use many sets of numerical phases to start with and the tangent refinement (31) to extend and refine the phases. The most widely used procedure of this sort is the computer package MULTAN (Germain and Woolfson, 1968; Main, 1978; Main, 1980).

The positive seven-magnitude quartet relation (32) was first formulated by Schenk (1973) and at the same time a two-dimensional analogue of the negative quartet relationship proved to be useful (Schenk and de Jong, 1973; Schenk, 1973b). The negative quartet in theory and practice was then published by Hauptman (1974) and Schenk (1974). In the latter paper the first Figure of Merit based on negative quartets was successfully formulated and tested. Theories concerning 7 magnitude-quartets were developed later, among which the one of Hauptman (1975) is best established. Applications of quartets include their use in starting set procedures and figures of merit, further brief details of which can be found in a recent review article (Schenk, 1980b).

**Copyright © 1984, 1998 International Union of
Crystallography**