Matrix Derivation of the Crystallographic Point Groups and Associated Computer Programs

• Main Authors: David-King Chai, 翟大鈞
• Other Authors: 陳卓
• Format: Others
• Language: zh-TW
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/07648081647011509487

碩士 === 國立臺灣大學 === 物理研究所 === 98 === The aim of this thesis is to propose a method of matrix algebra to discuss the symmetry of crystals. From the results, we can explain why nature exist only seven crystal systems. The crystal structures both have translational symmetry and rotational symmetry. However the operation of the two symmetries must be compatible with each other thus restricted the angle of rotation. As we know that any successive rotation will produce another rotation, therefore the angle of this rotation also must be limited in an allowed angle of rotation, and this will lead to a specified angle between the rotational axes, that is the real reason why nature exist only seven crystal systems. In the paper we first constructed the matrix representation of three dimensional symmetry transformations, and use of these formulas to discuss the possible combinations of two rotation axes, and classify the rotation axes according the allowed angle of rotation, to find all the possible combinations of them. As the result, full completely the conclusion as the discussion of the crystal lattice symmetry by use the geometry method. Final we term the theory of rotation matrix to the computer program, and make a specific calculation. Based on the results, the method is expected to apply on the study of the symmetry group of more complex molecules.