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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symmetric functions and the phase problem in crystallography


Authors: J. Buhler and Z. Reichstein
Journal: Trans. Amer. Math. Soc. 357 (2005), 2353-2377
MSC (2000): Primary 05E05, 13A50, 13P99, 20C10
Published electronically: August 11, 2004
MathSciNet review: 2140442
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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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Additional Information

J. Buhler
Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202
Email: jpb@reed.edu

Z. Reichstein
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: reichst@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03550-0
PII: S 0002-9947(04)03550-0
Keywords: Crystallography, structure factor, phase problem, symmetric function, group action, field of invariants, SAGBI basis, algorithmic computation, multiplicative invariant, rational invariant field
Received by editor(s): January 2, 2003
Received by editor(s) in revised form: October 15, 2003
Published electronically: August 11, 2004
Additional Notes: The second author was partially supported by NSF grant DMS-901675 and by an NSERC research grant
Article copyright: © Copyright 2004 American Mathematical Society