Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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.
Castleden, I.R., 1987, A Joint Probability Distribution of Invariants for all Space Groups, Acta Cryst., A43:384.
Cochran, W. and Woolfson, M.M., 1955, The Theory of Sign Relations Between Structure Factors, Acta Cryst., 8:1.
Cramér, H., 1962, “The Elements of Probability Theory and some of its Applications”, John Wiley & Sons, New York.
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.
French, S. and Wilson, K., 1978, On the Treatment of Negative Intensity Observations, Acta Cryst., A34:517.
Giacovazzo, C., 1976, A Probabilistic Theory of the Cosine Invariant cos (ϕ h + ϕ k + ϕ 1 — ϕ h+k+1), Acta Cryst., A32:91.
Giacovazzo, C., 1977, A General Approach to Phase Relationships: The Method of Representations, Acta Cryst., A33:933.
Giacovazzo, C., 1980, “Direct Methods in Crystallography”, Academic Press, London.
Giacovazzo, C., 1983a, The Estimation of Two-Phase and Three-Phase Invariants in P1 when Anomalous Scatterers are Present, Acta Cryst., A39:585.
Giacovazzo, C., 1983b, From a Partial to the Complete Crystal Structure, Acta Cryst., A39:685.
Hauptman, H.A., 1972, “Crystal Structure Determination. The Role of the Cosine Semi-Invariants”, Plenum Press, New York.
Hauptman, H., 1975a, A Joint Probability Distribution of Seven Structure Factors, Acta Cryst., A31.–671.
Hauptman, H., 1975b, A New Method in the Probabilistic Theory of the Structure Invariants, Acta Cryst., A31:680.
Hauptman, H., 1982a, On Integrating the Techniques of Direct Methods and Isomorphour Replacement: I. The Theoretical Basis, Acta Cryst., A38:289.
Hauptman, H., 1982b, On Integrating the Techniques of Direct Methods with Anomalous Dispersion. I. The Theoretical Basis, Acta Cryst., A38:632.
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.
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.
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.
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.
Oatley, S. and French, S., 1982, A Profile-Fitting Method for the Analysis of Diffractometer Intensity Data, Acta Cryst., A38:537.
Papoulis, A., 1984, “Probability, Random Variables, and Stochastic Processes”, McGraw-Hill Book Company, New York.
Peschar, R. and Schenk, H., Computer-Aided Derivation of Theoretical Joint Probability Distributions of Normalized Structure Factors, Acta Cryst.
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.
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.
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.
Wilson, A.J.C., 1942, Determination of Absolute from Relative X-Ray Intensity Data, Nature, 150:152.
Wilson, A.J.C., 1949, The Probability Distribution of X-Ray Intensities, Acta Cryst., 2:318.
Woolfson, M.M., 1954, The Statistical Theory of Sign Relationships, Acta Cryst., 7:61.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Fortier, S., Castleden, I.R. (1991). Some Applications of Probability Theory in Direct Methods. In: Schenk, H. (eds) Direct Methods of Solving Crystal Structures. NATO ASI Series, vol 274. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3692-9_13
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3692-9_13
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-3694-3
Online ISBN: 978-1-4899-3692-9
eBook Packages: Springer Book Archive