Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals
Representation of orientation and disorientation data for cubic, hexagonal, tetragonal and orthorhombic crystals
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DOI：
10.1107/s0108767391006864
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发表时间：
1991-11
期刊：
影响因子：
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通讯作者：
A. Heinz;P. Neumann
中科院分区：
文献类型：
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作者：
A. Heinz;P. Neumann
给出了一种描述立方、六边形、四边形和正交晶体取向数据的简便方法。该方法也可用于定向失向数据的表示，其中任意两个特定对称晶格的晶体之间的定向失向被考虑。它是基于grimer[晶体学报]引入的四元数形式论来讨论取向和失取向的。（1974）， （30）， 685-688 [j]。Proc, Int。《材料的纹理8》（INCOTOM 8），圣达菲，美国，第3-13页)等。由于方向和失向可以解释为旋转，而旋转又可以只用三个参数表示，因此使用矢量描述。这些向量张成一个旋转空间对应于通常的欧拉角空间。它被称为Rodrigues向量空间[Rodrigues(1840)]。j .数学。纯苹果，5,380 - 440；Becker & Panchanadeeswaran（1989）。文本。[j].中国机械工程，2016,27。罗德里格斯矢量的方向平行于旋转轴，其长度为tan (θ/2)，其中θ表示旋转角度。给出了从众多对称等价Rodrigues向量中选取唯一代表向量的方法。由于这些选择规则取决于所考虑的晶体晶格的对称性，因此它们在Rodrigues向量空间中产生紧致域，这是每种晶格或晶格对的典型特征。这些域总是以平面为界。Frank（1987）称它们为基本带，并描述了立方体、六边形和正交晶体的取向。
A convenient method for the description of orientation data for cubic, hexagonal, tetragonal and orthorhombic crystals is given. The method can also be used for the representation of disorientation data, where disorientations between any two crystals of the specified symmetry lattices are considered. It is based on the quaternion formalism introduced into the discussion of orientations and disorientations by Grimmer [Acta Cryst. (1974), A30, 685–688], Frank [(1987). Proc. Int. Conf. on Texture of Materials 8 (INCOTOM 8), Santa Fe, USA, pp. 3–13] and others. Since orientations and disorientations can be interpreted as rotations which in turn can be represented by only three parameters a vector description is used. These vectors span a rotation space corresponding to the usual space of Eulerian angles. It is called Rodrigues vector space [Rodrigues (1840). J. Math. Pure Appl. 5, 380–440; Becker & Panchanadeeswaran (1989). Text. Microstruct. 10, 167]. The direction of a Rodrigues vector is parallel to the rotation axis and its length is tan (θ/2), where θ describes the rotation angle. A method for selecting a unique representative out of the numerous symmetrically equivalent Rodrigues vectors is given. Since these selection rules depend on the symmetry of the crystal lattices considered they yield compact domains in the Rodrigues vector space which are typical for each type of lattice or lattice pair. These domains are always bounded by planes. Frank (1987) called them fundamental zones and described them for the orientations of cubic, hexagonal and orthorhombic crystals.