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Symmetric functions and the phase problem in crystallography

Joe Buhler·Zinovy Reichstein
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Abstract

The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.

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Joe Buhler, Zinovy Reichstein. 2003-12-18. Symmetric functions and the phase problem in crystallography. https://arxiv.org/abs/math/0312368

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