Stróż Kazimierz
Institute of Material Science, University of Silesia, Poland.
Acta Crystallogr A. 2011 Sep;67(Pt 5):421-9. doi: 10.1107/S0108767311020113. Epub 2011 Jul 6.
A fixed set, that is the set of all lattice metrics corresponding to the arithmetic holohedry of a primitive lattice, is a natural tool for keeping track of the symmetry changes that may occur in a deformable lattice [Ericksen (1979). Arch. Rat. Mech. Anal. 72, 1-13; Michel (1995). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek & S. Walcerz. Singapore: Academic Press; Pitteri & Zanzotto (1996). Acta Cryst. A52, 830-838; and references quoted therein]. For practical applications it is desirable to limit the infinite number of arithmetic holohedries, and simplify their classification and construction of the fixed sets. A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and geometric description were analyzed and offered as the framework for dealing with the symmetry of reduced lattices. This matrix space covers all arithmetic holohedries of primitive lattice descriptions related to the three shortest lattice translations in direct or reciprocal spaces, and corresponds to the unique list of 39 fixed points with integer coordinates in six-dimensional space of lattice metrics. Matrices are presented by the introduced dual symbol, which sheds some light on the lattice and its symmetry-related properties, without further digging into matrices. By the orthogonal lattice distortion the lattice group-subgroup relations are easily predicted. It was proven and exemplified that new symbols enable classification of lattice groups on an absolute basis, without metric considerations. In contrast to long established but sophisticated methods for assessing the metric symmetry of a lattice, simple filtering of the symmetry operations from the predefined set is proposed. It is concluded that the space of symmetry matrices with elements from {0, ±1} is the natural environment of lattice symmetries related to the reduced cells and that complete geometric characterization of matrices in the arithmetic holohedry provides a useful tool for solving practical lattice-related problems, especially in the context of lattice deformation.
一个固定集，即与原始晶格的算术全形相对应的所有晶格度量的集合，是追踪可变形晶格中可能发生的对称性变化的自然工具[埃里克森（1979年）。《理性力学与分析档案》72卷，第1 - 13页；米歇尔（1995年）。《凝聚态物质的对称性和结构性质》，由T. 卢莱克、W. 弗洛雷克和S. 瓦尔采尔编辑。新加坡：学术出版社；皮特里和赞佐托（1996年）。《晶体学报》A52卷，第830 - 838页；以及其中引用的参考文献]。对于实际应用，希望限制算术全形的无限数量，并简化它们的分类以及固定集的构建。分析了一个由具有循环连续幂、行列式为1、元素来自{0, ±1}且有几何描述的480个矩阵组成的空间，并将其作为处理约化晶格对称性的框架。这个矩阵空间涵盖了与直接或倒易空间中三个最短晶格平移相关的原始晶格描述的所有算术全形，并且对应于晶格度量六维空间中具有整数坐标 的39个固定点的唯一列表。矩阵由引入的对偶符号表示，这为晶格及其与对称性相关的性质提供了一些启示，而无需进一步深入研究矩阵。通过正交晶格畸变，可以很容易地预测晶格群 - 子群关系。已证明并举例说明，新符号能够在不考虑度量的情况下，绝对地对晶格群进行分类。与长期以来用于评估晶格度量对称性的复杂方法不同，提出了从预定义集合中简单过滤对称性操作的方法。得出的结论是，元素来自{0, ±1}的对称矩阵空间是与约化晶胞相关的晶格对称性的自然环境，并且算术全形中矩阵的完整几何表征为解决实际的晶格相关问题提供了一个有用的工具，特别是在晶格变形的背景下。
