A Unified Crystallography for Periodic or Quasiperiodic Materials

研究将完成并扩展一个更广泛、更简单、更基于物理的周期和准周期材料晶体学描述的重新表述。现在准周期物质的所谓超空间群范畴的推导和描述都有很大的简化要做。在（3+1）非立方和（3+3）立方超空间群的情况下，这种重新表述将在概念上澄清，在形式上浓缩，并在范围上扩展现有方案，同时明确表明其目前对弱调制晶体的限制是完全不必要的。从具有布拉格峰衍射图样的材料的一般稳定性条件推导出广义晶体学，从而提取出这种新晶体学的规则（包括在准周期情况下模糊的点群和晶格的相容性规则）。我们将在周期和准周期情况下研究这种方法对磁性和色群的扩展，并从这个新的角度重新研究公度-不相称跃迁的问题。将研究一种新的晶体学公式。这项工作的动机是发现了传统晶体学不能充分描述的准周期材料。在重新制定晶体学以解释这些准周期材料时，也设计了一种新的方案来涵盖周期材料。这种独特的方法可能对晶体学产生重大影响。

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.