On the Vibration of Disordered Linear Lattice

Abstract

The method of transfer matrix which was recently developed by Kerner to treat the problem of electronic band structure of mixed linear lattices was appliEd to the problem of vibration of atomic linear chain. It was sh.own that this metr.od aq-ords effective means of calculating the eigenfrequency-distribution of several kinds of latticEs, including disordered ones. The well-known results for regular mor.a:omic ar.d di.ato:nic lattices, and the results which were obtained by Montroll and Potts for linear la:tic treated the problem of the electronic band structure of periodic or random linear lattices containing impurity atoms by the method of transfer matrices. In this paper we show that if we treat a problem of vibration of linear atomic lattice by a similar method, we can obtain not only the same results as hitherto obtained for a regular periodic lattices or lattices containing a few impurity atoms, but also, in a comparatively simple way, a considerable amount of information as to the vibration of lattices which have more complicated structures. The problem of vibration of a lattice containing a few impurities was recently investigated by Montroll and Potts 2 >, using the method of Green functions. Though their treatment is very elegant and several important results are included therein, the calculations seem to come out considerably complicated if we attempt to apply it to lattices with more complicated structures, e.g., those containil.1g severd impurltles at arbitrarily given positions or at irregular unlmown positions. Especidly it will prove difnc:.tlt to apply this method effectively to the latter. It is true, 011 th·:: other hz:1d, that for mch random lattices there is a rigorous mathematical theory formulated by Dyson j!nd J3ellma:n 3 > 4 >, which provides us with a ~c-neral procedure of calcul:>.titl~ the disttibq.