一维三原子链的格波解及其色散关系

The lattice wave solution and its dispersion relation of one-dimensional triatomic chain

通过求解简谐近似下一维三原子链的晶格振动方程得到其格波解和色散关系,进而研究一维三原子链的晶格振动特点,讨论一维三原子链和一维单原子链色散关系的内在联系。结果表明,一维三原子链的色散关系包含一个声学支和两个光学支,频谱范围和频率禁带与原胞内三个原子的质量都有关联,在原胞内原子质量相等的情况下一维三原子链的色散关系与一维单原子链色散关系所描述的晶格振动模式相同,进而对固体物理教学中容易误解的两个相关问题进行了说明。

The lattice wave solution and the dispersion relation of the lattice vibration equation of the one-dimensional three-atom chain are obtained by solving simple harmonic approximation and then the lattice vibration characteristics of the chain are studied. The intrinsic link of the dispersion relation of the one-dimensional three-atom chain and one-dimensional single atom chain are discussed. The results show that the dispersion relation of the one-dimensional triatomic chain contains an acoustic branch and two optical branches. The spectral range and frequency band gap are related to the mass of the three atoms in the original cell. When the atomic masses of the original cell are equal, the dispersion relationship of one-dimensional triatomic chain is the same as the lattice vibration mode described by the one-dimensional single atom chain dispersion relation. Furthermore, two related problems that are easily misunderstood in solid physics teaching are explained.