Harker-Kasper inequalities

The methods used to ensure that the Fourier summation does not give a negative electron-density map are mathematical in nature. David Harker and John Kasper in 1948 used the inequality relationships of Augustin Louis Cauchy, Hermann Amandus Schwarz, and Victor Buniakowsky [Buniakovski] (generalized to the Cauchy—Schwarz inequality) to derive relationships between the structure factors (the Harker—Kasper inequalities). These were used by David Harker. John Kasper, and Charlys Lucht to determine the structure of decab-orane, BioH, which was unknown at that time. For this study they... 

Harker-Kasper inequalities Inequalities among structure factors, dependent on the space group, that lead to equations that allow the determination of the relative phases of certain intense Bragg reflections. 

Woolfson, M. M. Some thoughts on Harker-Kasper inequalities. Acta Crysi. A44, 222-225 (1988). 

Lavine, L. R. Corrections to Grison s paper on the Harker-Kasper inequalities and to Zachariasen s paper on the statistical method. Acta Cryst. 5, 846-847 (1952). 

After the set of determinantal inequalities (18) had been obtained on the basis of the non-negativity of the electron density distribution in a crystal, it was of interest to investigate their relationship to the inequalities derived by Marker and Kasper from use of the Schwarz and Cauchy inequalities, It was shown that, when the appropriate symmetry was introduced into the third-order determinant inequality by means of certain relationships among the structure factors, the Marker-Kasper inequalities could be derived. Examination of the derivation of the Harker-Kasper inequalities shows, as would be expected, that the non-negativity of the electron density distribution is a requirement for their validity,... 

The first important step in this direction was taken by Harker and Kasper (1948), who derived relations between pairs or small groups of reflections in a centrosymmetric structure in the form of inequality expressions. The simplest of these says that if Uhkl is the unitary structure amplitude —the structure amplitude expressed as a fraction of what it would be if the waves from all atoms were exactly in phase with each other f—then... 

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