Symmetric functions and the phase problem in crystallography
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- by J. Buhler and Z. Reichstein PDF
- Trans. Amer. Math. Soc. 357 (2005), 2353-2377 Request permission
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.References
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Additional Information
- J. Buhler
- Affiliation: Department of Mathematics, Reed College, Portland, Oregon 97202
- MR Author ID: 43035
- Email: jpb@reed.edu
- Z. Reichstein
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
- MR Author ID: 268803
- Email: reichst@math.ubc.ca
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2353-2377
- MSC (2000): Primary 05E05, 13A50, 13P99, 20C10
- DOI: https://doi.org/10.1090/S0002-9947-04-03550-0
- MathSciNet review: 2140442