Fourier-Space Crystallography as Group Cohomology

Details Fourier-Space Crystallography as Group Cohomology Fourier-Space Crystallography as Group Cohomology

2001-05-02

arxiv.org/abs/cond-mat/0105036v2

Physics

Condensed Matter

plain tex, 14 pages (replaced 5/8/01 to include archive preprint number for reference 22)

Scientific paper

10.1103/PhysRevB.65.024201

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.

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