Some Applications of Probability Theory in Direct Methods

  • Suzanne Fortier
  • Ian R. Castleden
Part of the NATO ASI Series book series (NSSB, volume 274)


It may seem somewhat surprising at first that probability theory has played such an important role in the solution of the crystal structure determination problem. Crystals, which are usually defined in terms of the long range ordering they display, might appear to be poor samples on which to apply a mathematical model devised to deal with random experiments. It is the realization that the periodically repeating motif could itself be depicted as consisting of atoms randomly distributed (Wilson, 1949) that allowed the phase problem to be phrased and solved in a probabilistic framework. Since then, there have been numerous applications of probability theory to the problem of crystal structure determination. They have essentially provided a solution for the case of small molecule crystal structures.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bricogne, G., 1988, A Bayesian Statistical Theory of the Phase Problem. I. A Multichannel Maximum-Entropy Formalism for Constructing Generalized Joint Probability Distributions of Structure Factors, Acta Cryst., A44:517.CrossRefGoogle Scholar
  2. Castleden, I.R., 1987, A Joint Probability Distribution of Invariants for all Space Groups, Acta Cryst., A43:384.CrossRefGoogle Scholar
  3. Cochran, W. and Woolfson, M.M., 1955, The Theory of Sign Relations Between Structure Factors, Acta Cryst., 8:1.CrossRefGoogle Scholar
  4. Cramér, H., 1962, “The Elements of Probability Theory and some of its Applications”, John Wiley & Sons, New York.Google Scholar
  5. Fortier, S. and Nigam, G.D., 1989, On the Probabilistic Theory of Isomorphous Data Sets: General Joint Distributions for the SIR, SAS, and Partial/Complete Structure Cases, Acta Cryst., A45:247.CrossRefGoogle Scholar
  6. French, S. and Wilson, K., 1978, On the Treatment of Negative Intensity Observations, Acta Cryst., A34:517.CrossRefGoogle Scholar
  7. Giacovazzo, C., 1976, A Probabilistic Theory of the Cosine Invariant cos (ϕ h + ϕ k + ϕ 1ϕ h+k+1), Acta Cryst., A32:91.CrossRefGoogle Scholar
  8. Giacovazzo, C., 1977, A General Approach to Phase Relationships: The Method of Representations, Acta Cryst., A33:933.CrossRefGoogle Scholar
  9. Giacovazzo, C., 1980, “Direct Methods in Crystallography”, Academic Press, London.Google Scholar
  10. Giacovazzo, C., 1983a, The Estimation of Two-Phase and Three-Phase Invariants in P1 when Anomalous Scatterers are Present, Acta Cryst., A39:585.CrossRefGoogle Scholar
  11. Giacovazzo, C., 1983b, From a Partial to the Complete Crystal Structure, Acta Cryst., A39:685.CrossRefGoogle Scholar
  12. Hauptman, H.A., 1972, “Crystal Structure Determination. The Role of the Cosine Semi-Invariants”, Plenum Press, New York.CrossRefGoogle Scholar
  13. Hauptman, H., 1975a, A Joint Probability Distribution of Seven Structure Factors, Acta Cryst., A31.–671.CrossRefGoogle Scholar
  14. Hauptman, H., 1975b, A New Method in the Probabilistic Theory of the Structure Invariants, Acta Cryst., A31:680.CrossRefGoogle Scholar
  15. Hauptman, H., 1982a, On Integrating the Techniques of Direct Methods and Isomorphour Replacement: I. The Theoretical Basis, Acta Cryst., A38:289.CrossRefGoogle Scholar
  16. Hauptman, H., 1982b, On Integrating the Techniques of Direct Methods with Anomalous Dispersion. I. The Theoretical Basis, Acta Cryst., A38:632.CrossRefGoogle Scholar
  17. Hauptman, H. and Karle, J., 1953a, “Solution of the Phase Problem. I. The Centrosymmetric Crystal”, A.C.A. Monograph No. 3, Polycrystal Book Service, Brooklyn.Google Scholar
  18. Hauptman, H. and Karle, J., 1953b, The Probability Distribution of the Magnitude of a Structure Factor. II. The Non-Centrosymmetric Crystal, Acta Cryst., 6:136.CrossRefGoogle Scholar
  19. Heinerman, J.J.L., 1977, Some Contributions to the Theory of Triplet and Quartet Structure Invariants, in “Direct Methods in Crystallography”, H. Hauptman, ed., Proceedings of the 1976 Intercongress Symposium.Google Scholar
  20. Heinerman, J.J.L., Krabbendam, H. and Kroon, J., 1977, The von Mises Distribution of the Phase of a Structure Invariant, Acta Cryst., A33:873.CrossRefGoogle Scholar
  21. Oatley, S. and French, S., 1982, A Profile-Fitting Method for the Analysis of Diffractometer Intensity Data, Acta Cryst., A38:537.CrossRefGoogle Scholar
  22. Papoulis, A., 1984, “Probability, Random Variables, and Stochastic Processes”, McGraw-Hill Book Company, New York.Google Scholar
  23. Peschar, R. and Schenk, H., Computer-Aided Derivation of Theoretical Joint Probability Distributions of Normalized Structure Factors, Acta Cryst.Google Scholar
  24. Schwarzenbach, D., Abrahams, S.C., Flack, H.D., Gonschorek, W., Hahn, T., Huml, K., Marsh, R.E., Prince, E., Robertson, B.E., Rollett, J.S. and Wilson, A.J.C., 1989, Statistical Descriptors in Crystallography, Acta Cryst., A45:63.CrossRefGoogle Scholar
  25. Shmueli, U., Weiss, G.H., Keifer, J.E. and Wilson, A.J.C., 1984, Exact Random-Walk Models in Crystallographic Statistics. I. Space Groups PI and P1̄, Acta Cryst., A40:651.CrossRefGoogle Scholar
  26. Shmueli, U. and Weiss, G.H., 1987, Exact Random-Walk Models in Crystallographic Statistics. III. Distributions of |E| for Space Groups of Low Symmetry, Acta Cryst., A43:93.CrossRefGoogle Scholar
  27. Wilson, A.J.C., 1942, Determination of Absolute from Relative X-Ray Intensity Data, Nature, 150:152.CrossRefGoogle Scholar
  28. Wilson, A.J.C., 1949, The Probability Distribution of X-Ray Intensities, Acta Cryst., 2:318.CrossRefGoogle Scholar
  29. Woolfson, M.M., 1954, The Statistical Theory of Sign Relationships, Acta Cryst., 7:61.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Suzanne Fortier
    • 1
  • Ian R. Castleden
    • 1
  1. 1.Dept. of ChemistryQueen’s University-KingstonCanada

Personalised recommendations