The thermal conductivity of structurally disordered solids

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Abstract
A new approach to the problem of calculating the phonon component of the thermal conductivity of structurally disordered solids is presented. The basis of the method is the introduction of collective variables U(k) defined by where uα(j) denotes a Cartesian component of the displacement from equilibrium of the jth atom whose equilibrium site is at rj. For simplicity only monatomic solids are considered. The plane waves are chosen to satisfy periodic boundary conditions within a volume containing N atoms and the sum over k contains just N terms. In the case of amorphous solids the wave vectors are chosen to be within the Debye sphere. It is assumed that the waves are linearly independent. In the case of solids containing a large amount of crystalline order, for example when dislocations are present, it is not possible to give a precise definition of the upper limit in the sum over plane waves. For many practical purposes this is fairly unimportant, especially at low temperatures where only long-wavelength modes are excited. The merit of this approach is that it avoids describing a solid as a disordered lattice, which is very convenient in the case of amorphous solids and also when defects such as dislocations and interstitials are present in a lattice. The Boltzmann equation in this representation has precisely the same form as that of the standard theory of thermal conductivity. The collision terms have been calculated to first order in perturbation theory.