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Powder auto-indexing is the crystallographic problem of lattice determinationfrom an average theta series. There, in addition to all the multiplicities, thelengths of part of lattice vectors cannot be obtained owing to systematicabsences. As a consequence, solutions are not always unique. We develop a newalgorithm to enumerate powder auto-indexing solutions. This is a novelapplication of the reduction theory of positive-definite quadratic forms to aproblem of crystallography. Our algorithm is proved to be effective for alltypes of systematic absences, using their newly obtained common properties. Theproperties are stated as distribution rules for lattice vectors correspondingto systematic absences on a topograph. Conway defined topographs for2-dimensional lattices as graphs whose edges are associated with $|l_1|^2$,$|l_2|^2$, $|l_1+l_2|^2$, $|l_1-l_2|^2$. In our enumeration algorithm,topographs are utilized as a network of lattice vector lengths. As a crystalstructure is a lattice of rank 3, the definition of topographs is generalizedto any higher dimensional lattices using Voronoi's second reduction theory. Theuse of topographs allows us to speed up the algorithm. The computation time isreduced to 1/250--1/32, when it is applied to real powder diffraction patterns.Another advantage of our algorithm is its robustness to missing or falseelements in the set of lengths extracted from a powder diffraction pattern.Conograph is the powder indexing software which implements the algorithm. Wepresent results of Conograph for 30 diffraction patterns, including some verydifficult cases.

Distribution rules of crystallographic systematic absences on the Conway topograph and their application to powder auto-indexing