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.

  1. C. Giacovazzo, & D. Siliqi, Acta Cryst. A54, (1998), 000-000
  2. C. Giacovazzo, D. Siliqi, B. Carrozzini, A. Guagliardi, & A.G.G. Moliterni, Acta Cryst. A54, (1998), 000-000
  3. C. Giacovazzo, D. Siliqi, A. Altomare, G. Cascarano, R. Rizzi, & R. Spagna, submitted
  4. C. Giacovazzo, D. Siliqi & C. Fernandez-Castano, submitted
  5. A. F. Mishnev, Acta Cryst. A49, (1993), 159-161; Acta Cryst. A52, (1996), 629-633
  6. G. Zanotti, F. Fogale, & G. Capitani, Acta Cryst. A52, (1996), 757-765