无限正方晶格中线缺陷的局域化
Localisation for a line defect in an infinite square lattice
摘要：
局域缺陷模式所产生的有限线缺陷组成的几个质量，嵌入一个无限的正方形细胞晶格，分析使用的格林函数的线性叠加为一个单一的质量缺陷。本文提出并讨论了格点格林函数的几种表示形式。这个问题被减少到一个特征值系统和相应的矩阵的属性进行了详细研究，以产生关于对称和反对称模式的数量的信息。在远场的渐近展开，与长波长的均匀化。讨论了带边附近格林函数的渐近表达式。几个例子中，本征频率链接到这个系统和相应的本征模式计算各种缺陷和比较的渐近展开。无限缺陷的情况下，也被认为是一个明确的色散关系。对于线缺陷内的质量数很大的情况下，它示出的本征频率的范围可以预测使用的色散图的无限链。
英文摘要：
Localised defect modes generated by a finite line defect composed of several masses, embedded an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenisation are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.
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