On the Vibration of Disordered Linear Lattice

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The method of transfer matrix which was recently developed by Kerner to treat the problem of electronic band structure of mixed lincar latticcs was applied to the problem of vibration of atomic linear chain. It was shown that this method affords effective means of calculating the eigenfrequency-distribution of several kinds of lattices, including disordered ones. The well-known results for regular monatomic and diatomic lattices, and the results which were obtained by Montroll and Potts for linear lattices containing a few impurity atcms, were rederived by the matrix method in a much simpler way. The eigenvalue equations for the lattices containing several impurities at arbitrarily given pesitions were then derived. In cases where the force-constant is not altered by impurities the formulas become so simple that we can obtain the eigenfrequencies without laborious calculatiors. It is also possible to derive in such cases a simple approximate eigenvalue equation for lattices containing impurities distributed completely at random. This equation indicates that the mede of change in the distribution of eigenfrequencies is qualitatively the same as in the case of one or two impurities. Monatomic lattice with a single hole and diatcmic lattice with a single impurity were treated as simple examples. An attempt to cxtend the matrix method to the ease in which the next-nearest interaction between atoms is taken into account was also mentioned.
理論物理学刊行会の論文

On the Vibration of Disordered Linear Lattice

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