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Characterization of Crystal Lattices

Tony Santoro (NIST Center for Neutron Research)

Crystal lattices are classified by means of the 44 reduced forms, derived by Niggli (1928) with the reduction procedure of positive definite quadratic forms, and tabulated in the International Tables for Crystallography (vol. A). This classification is far more detailed than that provided by the 14 Bravais types, and it is often used to find the relationship between different and unconventional descriptions of crystal structures, and to derive the conventional cell of a lattice from any three non-coplanar primitive vectors determined experimentally under limiting conditions such as, for example, a compound analyzed under pressure.

Reduced cells are primitive and are unique (except, in certain cases, in orientation), for any given lattice. We have recently found, however, that in many cases, a reduced cell does not define uniquely a corresponding reciprocal cell. A clear example of this occurs in the orthorhombic system where there are two face-centered (oF#16 and oF#26) and three body-centered lattices (oI#8, oI#19 and oI#42). This 'anomaly' shows that there is no one-to-one relationship between direct and reciprocal lattice types, and can be explained only if one of the two oF direct lattices can originate two different oI reciprocal types and, inversely, only if two direct oI lattices can generate the same reciprocal oF type.

A detailed study of the 44 reduced forms in direct and reciprocal space has shown that the same type of ambiguities found in the orthorhombic face- and body-centered lattices is present in all crystal systems and for almost all the lattice types. These results will be discussed together with possible practical applications.

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