介绍一种关于四重映轴S_(4)及四重反轴I_(4)独立性的简洁推导

Introduction to a Concise Deduction for the Independence of the Rotation-Reflection Axis S_(4) and the Rotation-Inversion Axis I_(4)

分子点群与晶体学点群是结构化学教学重要内容之一,同时也是教学的一个难点。学生经常提出“为何只有S_(4)轴及I_(4)轴是独立的,而其余映轴及反轴均是非独立的;在描述分子对称性时为何选用映轴,而描述晶体对称性时则选用反轴”等问题。本文从循环群的生成元出发,得到映轴群S_(n)及反轴群I_(n)的对称操作群元素。根据映轴是否可以用其他对称元素代替,或对称操作群是否可以表示为两个子群的直积的性质,推导出当n≤6时只有S_(4)及I_(4)是独立的结论。这种推导方式避免了图形说明繁琐和抽象,有利于学生从定义推导上理解对称元素的独立性,便于教学中推广。

The point groups of molecular and crystal symmetry are among the important contents in structural chemistry,and they constitute a challenge in teaching and learning.Students are often confused regarding why only the rotation-reflection axis S_(4) and the rotation-inversion axis I_(4) are independent,while other rotation-reflection and rotation-inversion axes are not,or why S_(4) is used in describing molecular symmetry,while I_(4) is used in describing crystal symmetry.In this article,starting from the generators of the cyclic group,we obtained the symmetrical operation elements of the point groups Sn and In,respectively.According to whether Sn(I_(n))can be replaced by another symmetric element,or whether the symmetric operating group can be represented as the direct product of two subgroups,we deducted that only S_(4) and I_(4) are independent when n≤6.This deduction avoids the tediousness and abstraction of graphical illustrations.It is helpful for students to understand the independence of symmetrical elements from the definition derivation,and convenient to promote in teaching.