大学化学 >> 2023, Vol. 38 >> Issue (5): 249-253.doi: 10.3866/PKU.DXHX202207007

师生笔谈 上一篇    下一篇

介绍一种关于四重映轴S4及四重反轴I4独立性的简洁推导

王渭娜, 刘峰毅, 王文亮()   

  • 收稿日期:2022-07-01 录用日期:2022-07-31 发布日期:2022-08-12
  • 通讯作者: 王文亮 E-mail:wlwang@snnu.edu.cn

Introduction to a Concise Deduction for the Independence of the Rotation-Reflection Axis S4 and the Rotation-Inversion Axis I4

Weina Wang, Fengyi Liu, Wenliang Wang()   

  • Received:2022-07-01 Accepted:2022-07-31 Published:2022-08-12
  • Contact: Wenliang Wang E-mail:wlwang@snnu.edu.cn

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摘要：

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S4轴及I4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Sn及反轴群In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S4及I4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。

关键词: 对称操作群, 四重映轴, 四重反轴, 独立性

Abstract:

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 S4 and the rotation-inversion axis I4 are independent, while other rotation-reflection and rotation-inversion axes are not, or why S4 is used in describing molecular symmetry, while I4 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 (In) 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 S4and I4 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.

Key words: Symmetry operation group, Rotation-reflection axis S4, Rotation-inversion axis I4, Independence