Exact Normal-Mode Analysis for a Linear Lattice with “Periodic Impurities”

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The normal-mode analysis for a diatomic linear lattice of alternating point-masses of mass [formula omitted] and [formula omitted] is carried out by setting [formula omitted], where [formula omitted] is regarded as an “impurity” added to the basic mass [formula omitted]. A uniform chain containing particles of mass [formula omitted] is taken as the basic system which is modified by the addition of the impurities to alternating particles. Lagrange's equations of motion for the basic system are then modified by the inclusion of extra forces due to such impurities. A process of intermediate coordinate transformation, in which the modified equations of motion are written in terms of the normal coordinates of the basic uniform lattice rather than the original displacement coordinates, simplifies the analysis. The correctness of the analysis is verified by applying it to the systems containing several particles and comparing the results with those obtained by direct calculations. © 1969, American Association of Physics Teachers. All rights reserved.

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American Journal of Physics