Exact Conditional Probability Density Function of Three-Phase Invariant in P1: Some Theoretical and Practical Considerations

  • Uri Shmueli
  • George H. Weiss
Part of the NATO ASI Series book series (NSSB, volume 274)


The applicability of probabilistic approaches to crystallography was first realized by Wilson (1949) in his study of intensity statistics, and found its place as the fundamental mathematical avenue to direct methods of phase determination (e.g. Hauptman & Karle, 1953; Cochran & Woolfson, 1955; Cochran, 1955; see also the Chapters by M. M. Woolfson on “The Cochran Distribution” and by S. Fortier and I. R. Castleden on “Some Applications of Probability Theory in Direct Methods”, in this book). The actual probabilistic formalisms, used until 1984, have been based either on low-order approximations afforded by the central limit theorem, as invoked, e.g., by Wilson (1949) and Cochran (1955), or constitute higher approximations (e.g. Hauptman & Karle, 1953; Klug, 1958; Naya, Nitta & Oda, 1964,1965 - to mention a few early ones) which can be cast into the formalism of Gram-Charlier or Edgeworth expansions. These expansions are rather lengthy and have poor convergence properties. In fact, most structure-solving packages are still based on the central limit theorem approximations, which perform well for nearly equal-atom structures, with a not too small number of atoms in the asymmetric unit and with no significant population of special positions or other sites imparting rational dependence (Karle & Hauptman, 1959). There remain, of course, small- and intermediate-molecule cases in which these approximations are not adequate. This presentation includes a description of some of our work on the development of exact probabilistic approaches, which properly account for the space-group symmetry and (arbitrarily heterogeneous) atomic composition.


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  1. Cochran, W., 1955, Relations Between Phases of Structure Factors, Acta Cryst., 8 : 473.CrossRefGoogle Scholar
  2. Cochran, W. & Woolfson, M.M., 1955, The Theory of Sign Relations Between Structure Factors, Acta Cryst., 8:1.CrossRefGoogle Scholar
  3. Hauptman, H. & Karle, J., 1953, “Solution of the Phase Problem I. The The Centrosymmetric Crystal”. A.C.A. Monograph No. 3. Pittsburgh: Polycrystal Book Service.Google Scholar
  4. Hauptman, H. & Karle, J., 1959, Rational Dependence and Renormalization of Structure Factors for Pase Determination, Acta Cryst., 12 : 846.CrossRefGoogle Scholar
  5. Klug, A., 1958, Joint Probability Distributions of Structure Factors and the Phase Problem, Acta Cryst., 11 : 515.CrossRefGoogle Scholar
  6. Naya, S., Nitta, I. & Oda, T. 1964, A Study on the Statistical Method for Determination of Signs of Structure Factors, Acta Cryst., 17 : 421.CrossRefGoogle Scholar
  7. Naya, S., Nitta, I. & Oda, T. 1965, A Theory of the Joint Probability Distribution of Complex-Valued Structure Factors, Acta Cryst., 19 : 734.CrossRefGoogle Scholar
  8. Shmueli, U., Weiss, G.H., Kiefer, J.E. & Wilson, A.J.C., 1984, Exact Random-Walk Models in Crystallographic Statistics. I. Space Groups P1̄ and P1, Acta Cryst., 40 : 559.CrossRefGoogle Scholar
  9. Shmueli, U. & Weiss, G.H., 1985, Exact Joint Probability Distribution for Centrosymmetric Structure Factors. Derivation and Application to the ∑1 Relationship in the Space Group P1̄, Acta Cryst., 41 : 401.CrossRefGoogle Scholar
  10. Shmueli, U. & Weiss, G.H., 1986, Joint Distribution of E h , E k and E h+k, and the Probability for the Positive Sign of the Triple Product in the Space Group P1̄, Acta Cryst., 42 : 240.CrossRefGoogle Scholar
  11. Shmueli, U., Rabinovich, S. & Weiss, G.H., 1989a, Exact Conditional Distribution of a Three-Phase Structure Invariant in the Space Group P1. I. Derivation and Simplification of the Fourier Series, Acta Cryst., 45 : 361.CrossRefGoogle Scholar
  12. Shmueli, U., Rabinovich, S. & Weiss, G.H., 1989b, Exact Conditional Distribution of a Three-Phase Structure Invariant in the Space Group P1. II. Calculation and Comparison with the Cochran Approximation, Acta Cryst., 45 : 367.CrossRefGoogle Scholar
  13. Wilson, A.J.C., 1949, The Probability Distribution of X-ray Intensities, Acta Cryst., 2 : 318.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Uri Shmueli
    • 1
  • George H. Weiss
    • 2
  1. 1.School of ChemistryTel Aviv UniversityTel AvivIsrael
  2. 2.Physical Sciences Laboratory Division of Computer Sciences and TechnologyNational Institutes of HealthBethesdaUSA

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