Expressions are obtained for the phonon density of states (DOS), lattice energy and lattice heat capacity (LHC) of an isotopically disordered anharmonic crystal. The cubic and quartic anharmonicities are taken into account besides both the force constant changes and mass difference caused by the substitutional impurities. The method of double time thermal Green’s Function (GF) is used in the development. It is shown that in the low concentration limit the LHC depends on mass and force constant changes, cubic and quartic anharmonicities and impurity-anharmonicity interactions. At low temperatures the largest contribution is found due to the defects. It is observed that the non-diagonal terms contribute significantly in the lattice energy of isotopically disordered crystal.

The theory of thermal conductivity of high temperature superconductors (HTS) based on electron and phonon line width (life times) formulation is developed with Quantum dynamical approach of Green's function. The frequency line width is observed as an extremely sensitive quantity in the transport phenomena of HTS as a collection of large number of scattering processes. The role of resonance scattering and electron-phonon interaction processes is found to be most prominent near critical temperature. The theory successfully explains the spectacular behaviour of high Tc superconductors in the vicinity of transition temperature. A successful agreement between theory and experiment has been obtained by analyzing the thermal conductivity data for the sample La1.8Sr0.2CuO4 in the temperature range 0 − 200K. The theory is equally and successfully applicable to all other high Tc superconductors

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The expression for the density of states of an isotopically disordered anharmonic crystal is investigated with the help of double-time thermodynamic Green’s functions. The cubic and quartic anharmonicities are taken into account besides the force constant changes and mass differences caused by the substitutional isotopic impurities. It is found that the density of states can be separated into defect, anharmonic and interference terms. The density of states is considerably enhanced due to the defects and impurities present in a real crystal and shows strong temperature and impurity concentration dependences which cannot be obtained by the traditional harmonic theory.

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