The Sayre Equation

Abstract

A landmark in direct methods was the simultaneous publication of three papers by Sayre (1952), Cochran (1952) and Zachariasen (1952) . They all dealt with the same ultimate topic, the sign relationship

$$ {\text{s}}\left( {\text{h}} \right){\text{ s}}\left( {\text{k}} \right){\text{ s}}\left( {{\text{h}} + {\text{k}}} \right) \approx + 1 $$

((1))

, from different points of view but in the paper by Sayre there was derived a powerful and elegant equation which gave exact relationships between structure factors. While exact equations, in particular cases, also resulted from the determinantal inequalities of Karle and Hauptman (1950) the Sayre equation had the advantage that it linked the structure factors in a very simple way.