Symmetric functions and the phase problem in crystallography
Authors:
J. Buhler and Z. Reichstein
Journal:
Trans. Amer. Math. Soc. 357 (2005), 23532377
MSC (2000):
Primary 05E05, 13A50, 13P99, 20C10
Published electronically:
August 11, 2004
MathSciNet review:
2140442
Fulltext PDF Free Access
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Abstract: The calculation of crystal structure from Xray 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 equal atoms in the unit cell, with 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/S0002994704035500
PII:
S 00029947(04)035500
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 DMS901675 and by an NSERC research grant
Article copyright:
© Copyright 2004
American Mathematical Society
