准晶体中十二方晶系点群的对称性与矩阵表示

Symmetry and matrix representation of dodecagonal system's point groups in quasicrystal

从理论上对准晶体中十二方晶系各点群进行了研究,绘出了十二方晶系7个点群的极赤投影图;列出了各点群的所有对称操作及生成操作;填写出了其中最大固有点群12 22的群乘表;在自定义的十二方坐标系中,导出了十二方晶系各点群所有对称操作的矩阵,这48个3×3矩阵的矩阵元有7种可能取值:0,±1,±2,±3,而3恰恰是反映十二方晶系准晶体所具有的准周期平移序的无理数.

The point groups of dodecagonal system in quasicrystal were studied. By using stereographic projection of each point group, all symmetry operation and generating operation of the point groups were listed, and group multiplication table of maximal proper point group 822 was filled in. In dodecagonal coordinate system defined by us, the matrixes of the point groups symmetry operation were derived. There were fortyeight 3 × 3 matrixes. The matrixes forms were compact, their matrix element possible values were： 0, ± 1, ± 2,±√3. Thereinto, √3 was an especial irrational number,which indicated the quasi-periodicity of dodecagonal system quasicrystal.