论文标题
二维晶格的数学
Mathematics of 2-dimensional lattices
论文作者
论文摘要
欧几里得空间中的周期性晶格是基矢量的所有整数线性组合的无限集。任何晶格都可以通过无限的许多不同的基础产生。这种歧义仅是部分解决的，但是在对晶体振动建模的扰动下，标准降低仍然不连续。本文完成了二维晶格的连续分类，直到欧几里得等轴测图（或一致性），刚性运动（无反射）和相似性（具有均匀的缩放）。新的均匀不变符允许在上面的等效范围内易于计算指标。刚性运动之前的指标尤其是不平凡的，并解决了有关晶格基础（DIS）连续性的所有剩余问题。这些指标导致了实值的手性距离，该距离连续测量与高对称性邻居的晶格偏差。
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard reductions remained discontinuous under perturbations modelling crystal vibrations. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure a lattice deviation from a higher-symmetry neighbour.
