Lattice Thermal Conductivity for a One-Dimensional, Harmonic, Isotopically Disordered Crystal

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Abstract
A formally exact expression for the Kubo thermal conductivity is obtained for an infinite, one-dimensional chain of atoms which are connected by nearest-neighbor, harmonic springs of equal strength, and which are of equal mass except within a finite section of the chain which contains disordered isotopic impurities. The two infinite, isotopically pure regions are used as high- and low-temperature reservoirs which cause an energy flow through the isotopically disordered region, whose thermal conductivity is calculated. It is found that the Kubo thermal conductivity for this model is always finite. Various approximations which allow explicit evaluation of the conductivity are discussed.