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Distribution Fitting Methods

  • Jindrich Hašek
  • Henk Schenk
Chapter
Part of the NATO ASI Series book series (NSSB, volume 274)

Abstract

The measured intensities alone are not sufficient to determine the structure. The solubility of the phase problem of a structure determination is dependent on the Amount of a Priori Structure Information (APSI) - i.e. atomicity, the known number of atoms, known molecular fragment, etc. Therefore the optimal procedure for the structure determination can be formulated as an optimal transmission of APSI among all phases. The mathematical condition for “best” utilization op APSI can be found in [1] as:
$$ s = {V^{ - 1}}\int\limits_V {{{\left( {{\rho _{calc}} - {\rho _{exaxt}}} \right)}^{ \to 2}}dV = {V^{ - 3}}\sum {\left( {{{\left| {{E_H}} \right|}^2}\operatorname{var} {\varphi _H}} \right) = \min imum}} $$
where variances of phases can be derived from the recurent formula:
$$ \operatorname{var} {\varphi _H} = {\left( {\sum {{{\left( {{b_j}\quad {V_j} + \sum {{a_{ij}}\quad \operatorname{var} {\varphi _{ij}}} } \right)}^{ - 1}}} } \right)^{ - 1}} $$
following from the Graph of Phase Relations (GPR). The Vi is the variance of the i-th seminvariant used for calculation of the phase φH in the (n+1)-th level, varφH is a variance of the phase φH and varφij are variances of known phases in preceding levels. The Theory of Seminvariant Graphs described in [2] gives a procedure how to find the optimal choice of seminvariants, their optimal utilizations, and how to find the best starting set supposing given set of seminvariants [3,4,5].

References

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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Jindrich Hašek
    • 1
  • Henk Schenk
    • 1
  1. 1.Institute of Macromolecular Chemistry, Academy of Sciences, Prague Tsjechoslowakia and Laboratory for CrystallographyUniversity of AmsterdamThe Netherlands

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