Methodological notes

Particles in finite and infinite one-dimensional chains

 a, b
a S.L. Sobolev Institute for Mathematics, Siberian Branch of the Russian Academy of Sciences, prosp. akad. Koptyuga 4, Novosibirsk, 630090, Russian Federation
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russian Federation

Particle motion in one-dimensional crystal chain is studied with the help of the transfer matrix method. The transition from a finite to an infinite chain is analyzed. In cases where an analytical solution is impossible, the method allows the computation of energy spectra with acceptable 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. It is shown 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.

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Keywords: periodic lattice, finite lattice, transfer matrix, random perturbations, strong coupling and weak coupling approximations
PACS: 03.65.−w, 71.15.−m, 42.70.Qs (all)
DOI: 10.3367/UFNe.2019.12.038709
Citation: Ginzburg I F "Particles in finite and infinite one-dimensional chains" Phys. Usp. 63 395–406 (2020)
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Received: 23rd, April 2019, revised: 25th, October 2019, 27th, December 2019

Оригинал: Гинзбург И Ф «Частицы в конечных и бесконечных одномерных периодических цепочках» УФН 190 429–440 (2020); DOI: 10.3367/UFNr.2019.12.038709

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