Theory of Lattice Specific Heat of an Isotopically Disordered Anharmonic Crystal

Abstract

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 electronic heat capacity of YBa2Cu3O7−δ superconductor

“…Where H e , H p , H e p , H A and H D represent the unperturbed electron Hamiltonian, [26][27][28] harmonic phonon Hamiltonian, 29 electron-phonon Hamiltonian, 26,30 anharmonic Hamiltonian upto quartic order 31,36 and defect Hamiltonian, [31][32][33]35 respectively and can be expressed in the form:…”

“…C(k 1 , k 2 ) and D(k 1 , k 2 ) are the parameters depending upon the changes in mass and force constant due to the introduction of impurities. [31][32][33][34][35] It is assumed that the number of impurity atoms is very small in comparison to the host atoms so that for low impurity concentration the impurity-impurity interactions would be ignored. Consider the evaluation of double time temperature dependent one electron GF:…”

Debye-Waller Factor in an Isotopically Disordered Semiconductor Crystal

“…Let us use equation of motion technique of quantum dynamics and the Dyson equation approach, Fourier transformed phonon Green function is obtained as [13,34,47,48,50]:…”

“…(10b, 13b, 13c, 15b, 15c) are evaluated by applying equation of motion technique of quantum dynamics to find the newly Fourier transformed phonon Green function G kk (ε) as [13,34,47,48,50]: (16) can be obtained again by using equation of motion technique of quantum dynamics [13,34,47,48,50] and they are substituted in Eq. (16) …”

“…This changes the frequency spectrum of host crystal. This creates gap, localized and resonance modes [31][32][33][34]. The presence of electron in crystal tries to distort the lattice causes to increase its effective mass.…”