Estimation Problems in Crystallography

  • Robert Schaback
Part of the The IBM Research Symposia Series book series (IRSS)


One of the most important problems in crystallography is the determination of the unit cell of a crystalline substance and the relative positions of atoms within that cell. This contribution is mainly concerned with the estimation of unit cell dimensions from data obtained by x-ray diffraction of a polycrystalline substance. Compared to the number of parameters to be estimated and to the desired accuracy, the given information turns out to be rather limited and relatively noisy. Therefore some deeper insight into the underlying problem is necessary. This reveals a nonlinear mixed-integer approximation problem and gives some hints for the numerical solution of the problem. By combination of common approximation methods and combinatorial strategies of branch- and bound-type a numerical estimation method for a high-speed computer can be developed. In practice this method provides the user with a set of estimates which fit into the noise limits of the data, as is shown by a series of test examples. A reduction of the number of possible estimates can be made by introducing additional restrictions involving more crystallographic information. The purpose of this contribution is not to give a purely crystallographical or a purely mathematical description of the topic but to try to arouse some interest for crystallographical questions among mathematicians working with high-speed computers.


Estimation Problem Phase Relationship Phase Problem Phenyl Ester Polycrystalline Substance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Robert Schaback
    • 1
  1. 1.Lehrstühle für Numerische und Angewandte MathematikGöttingenGermany

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