STRUCTURE FACTORS WITH RATIONAL INDICES: THE STATISTICS AND THE PHASE RELATIONSHIPS
1Carmelo Giacovazzo, 1,2Dritan Siliqi, 3Cristina Fernandez-Castano and 4G. Comunale
1IRMEC c/o Dipartimento
Geomineralogico, Universita di Bari, Campus Universitario, Via
Orabona 4, 70125 Bari , Italy
2Laboratory of X-ray Diffraction,
Department of Inorganic Chemistry, Faculty of Natural Sciences,
Tirana, Albania 3Departamento de
Cristalografia, Instituto de Quimica-Fisica
"Rocasolano", CSIC, Madrid, Spain.
4Centro di Geodinamica, Universita della
Basilicata, Potenza, Italy
Wilson distributions of standard reflections are the first step in direct methods procedures. The second step is the search and the application of the phase relationships via the joint probability distribution functions for the crystal structure solution. Structure factors with rational indices play a marginal role in the present scenario, but they can find numerous applications in the modern crystallography, macromolecular crystallography included.
This communication describes the statistics of the structure
factors with rational indices.1,2,3,4 It will be shown
that: a) the Wilson statistics for standard reflections is a
particular case of such a more general statistics; b) the
statistical properties of the structure factors with rational
indices can be quite different from the standard ones; c) phase
relationships among structure factors with rational indices can
be found. Our method encompasses previous results obtains via the
Hilbert transform techniques and the sampling Shannon theorem;5,6
d) the concept of structure invariant, even if useful, is not
necessary for the existence of phase relationships; e) some
applications have been initiated.