A Unified Crystallography for Periodic or Quasiperiodic Materials

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Research will be done to complete and extend a broader, simpler, and more physically based reformulation of the crystallographic description of periodic and quasiperiodic materials. There are now great simplifications to be made in both the derivation and the description of the so-called superspace-group categories of quasiperiodic materials. In the case of the (3+1) non-cubic and the (3+3) cubic superspace groups this reformulation will clarify conceptually, condense in form, and extend in scope the existing scheme, while demonstrating explicitly that its present restriction to weakly modulated crystals is entirely unnecessary. Rules will be extracted for this new crystallography (including rules for the compatibility of point group and lattice that are obscure in the quasiperiodic case) by deriving the generalized crystallography from the general stability conditions for materials with diffraction patterns consisting of Bragg peaks. The extension of this approach to magnetic and color groups will be investigated in both the periodic and quasiperiodic case and the question of commensurate-incommensurate transitions will be reexamined from this new perspective. %%% Research will be done on a new formulation of crystallography. The work is motivated by the discovery of quasiperiodic materials which are not adequately described by conventional crystallography. In reformulating crystallography to account for these quasiperiodic materials, a new scheme has been devised to cover periodic materials as well. This unique approach could have a major impact in crystallography.