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

DOI:
https://doi.org/10.1090/S0002-9947-04-03550-0

Published electronically:
August 11, 2004

MathSciNet review:
2140442

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-04-03550-0

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