Nespolo Massimo, Ferraris Giovanni
LCM3B UMR-CNRS 7036, Université Henri Poincaré Nancy I, BP 239, F-54506 Vandoeuvre-lès-Nancy CEDEX, France.
Acta Crystallogr A. 2007 May;63(Pt 3):278-86. doi: 10.1107/S0108767307012135. Epub 2007 Apr 13.
It is shown that the twin index n calculated, according to Friedel, as a function of the indices (hkl) and [uvw] of the lattice plane and lattice direction defining the cell of the twin lattice applies only to twofold twins, i.e. twins where the twin element is of order 2. For manifold twins, where the twin operation is a three-, four- or sixfold (direct or inverse) rotation, it is shown that the generalized formula becomes n=NXi/xi, where N is the number of lattice planes of the (hkl) family passing within the cell of the twin lattice, Xi the two-dimensional coincidence index for a plane of the (hkl) family and xi the number of planes out of N of that family that are partially restored by the twin operation. The existence of twin lattice quasi-symmetry (TLQS) twins with zero-obliquity in manifold twins leads to the introduction of a new parameter as a general measure of the pseudo-symmetry of TLQS rotation twins: the twin misfit delta, defined as the distance between the first nodes along the two shortest directions in the plane of LT (quasi-)perpendicular to the twin axes that are quasi-restored by the twin operation. Taking the example of staurolite twins, several inconsistencies in the treatment of manifold twins are pointed out.
结果表明，根据弗里德尔（Friedel）计算的孪晶指数n，作为定义孪晶晶格晶胞的晶格平面和晶格方向的指数（hkl）和[uvw]的函数，仅适用于二次孪晶，即孪晶要素为2次的孪晶。对于多重孪晶，其中孪晶操作是三次、四次或六次（正或逆）旋转，结果表明广义公式变为n = NXi / xi，其中N是在孪晶晶格晶胞内通过的（hkl）族晶格平面的数量，Xi是（hkl）族平面的二维重合指数，xi是该族N个平面中通过孪晶操作部分恢复的平面数量。多重孪晶中具有零倾角的孪晶晶格准对称（TLQS）孪晶的存在导致引入一个新参数，作为TLQS旋转孪晶伪对称性的一般度量：孪晶失配δ，定义为在LT平面中沿着与孪晶轴准垂直且通过孪晶操作准恢复的两个最短方向上的第一个节点之间的距离。以十字石孪晶为例，指出了多重孪晶处理中的几个不一致之处。
