Momentum Autocorrelation Functions and Energy Transport in Harmonic Crystals Containing Isotopic Defects

The effect of isotopic defects on the decay of the momentum autocorrelation function and on the transport of energy in a harmonic crystal is investigated. A spectral representation is obtained for the classical momentum autocorrelation function of particle $j$, ${\rho }_{\mathrm{jj}}\mathrm{}\left(t\right)\mathrm{}$. The spectral density is directly related to the normal mode frequency spectrum of the crystal. The recent investigation of ${\rho }_{\mathrm{jj}}\mathrm{}\left(t\right)\mathrm{}$ in a perfect one-dimensional crystal and a one-defect one-dimensional crystal are discussed as special cases of this general Wiener-Khinchin formula. The quantum mechanical momentum autocorrelation function of the defect particle in a one-defect crystal is treated in detail for the case in which the defect particle is very heavy. The explicit results obtained are of interest in the theory of Brownian motion. A formal relation expressing the average momentum autocorrelation function in an isotopically disordered crystal as the cosine transform of the frequency spectrum of the crystal is derived. The energy transport property is studied in terms of $〈p^2[j, t]〉$, the time-dependent ensemble average dispersion of the momentum of lattice particle $j$ when the crystal is divided initially into two regions characterized by different temperatures. A simple identity is derived, which expresses $〈p^2[j, t]〉$ in terms of the solution of a particular initial value problem of the crystal lattice equations of motion. The local temperature at lattice site $j$, which is related to the momentum dispersion by means of the definition $〈p^2[j, t]〉=M_j{k}_{B}T[j, t]$, is determined analytically in the case of the perfect and the one-defect one-dimensional crystals. The local temperature is determined numerically with the aid of an IBM-7090 computer for five isotopically disordered 100-particle one-dimensional crystals.