Particle motion in one-dimensional crystal chains is studied with the help of the transfer matrix method. The transition from a finite to an infinite chain is analyzed. In the cases where an analytic solution is impossible, the method allows calculating the energy spectra with reasonable accuracy, based on the known cell potential. It turns out that the structure of allowed and forbidden energy bands arising in an ideal lattice contains some features that are absent in the real world. This means that the model of an ideal lattice should be extended in order to describe reality. We show that accounting for small random perturbations of periodicity may serve as such an extension. Light propagation in a layered medium (including a photonic crystal) is studied using the same method.

中文翻译：

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有限和无限一维周期链中的粒子

利用转移矩阵方法研究了一维晶体链中的粒子运动。分析了从有限链到无限链的过渡。在不可能采用解析解的情况下，该方法可以根据已知的细胞势以合理的精度计算能谱。事实证明，理想晶格中出现的允许和禁止能带的结构包含一些在现实世界中不存在的特征。这意味着应该扩展理想晶格的模型以描述现实。我们表明，考虑到周期性的小的随机扰动可以充当这种扩展。使用相同的方法研究层状介质（包括光子晶体）中的光传播。