Fourier-Space Crystallography as Group Cohomology

We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation, and identified by gauge transformation, the cohomological description becomes natura...

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Bibliographic Details
Published inarXiv.org
Main Authors Rabson, David A, Fisher, Benji
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.05.2001
Subjects
Crystallography
Homology
Linear functions
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Summary:We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation, and identified by gauge transformation, the cohomological description becomes natural. We review Fourier-space crystallography and group cohomology, quote the fact that cohomology is dual to homology, and exhibit several results, previously established for special cases or by intricate calculation, that fall immediately out of the formalism. In particular, we prove that {\it two phase functions are gauge equivalent if and only if they agree on all their gauge-invariant integral linear combinations} and show how to find all these linear combinations systematically.
ISSN:2331-8422
DOI:10.48550/arxiv.0105036