Direct methods in crystallography

The basic problem of structure determination in x ray crystallography is described together with the methods available up to 1948 - primarily the Patterson, isomorphous replacement and heavy-atom methods. In 1948 inequality relationships were derived by Harker and Kasper but these could only give a solution for very simple structures. A breakthrough came in 1952 with the introduction of the triple-product sign relationship, which could only be applied to centrosymmetric structures. In 1955 Cochran showed that the general-valued phases for non-centrosymmetric structures were also related. After several years, during which a number of more-or-less involved methods were proposed for applying sign relationships, it was shown by Karle and Karle that a comparatively simple approach, the symbolic addition method, could be applied even to complex structures. A variant of the method is also applicable to non-centrosymmetric structures. A different technique of applying phase relationships, developed by Germain, Main and Woolfson has been fully automated and quite complex structures can now be solved with computer programmes which require as input only unprocessed observed data. Some future trends in the development of direct methods may be seen in present work in the use of inequalities involving determinants of high rank and also in methods which attempt to derive values of certain types of structure-invariant quantities.