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Symmetric functions and the phase problem in crystallography
Author(s):
J.
Buhler;
Z.
Reichstein
Journal:
Trans. Amer. Math. Soc.
357
(2005),
2353-2377.
MSC (2000):
Primary 05E05, 13A50, 13P99, 20C10
Posted:
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  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:
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
Posted:
August 11, 2004
Additional Notes:
The second author was partially supported by NSF grant DMS-901675 and by an NSERC research grant
Copyright of article:
Copyright
2004,
American Mathematical Society
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