Theory of Lattice Thermal Conductivity of Anharmonic Crystals with Impurities. (II). Interference Effects

Abstract

The interference of the anharmonic and impurity scattering of phonons is investigated. In particular, a general formula is obtained for the resonant scattering of phonons by impurities taking into account the inelastic channel which is responsible for the temperature dependence of the total cross-section of impurity scattering. It is found that the anharmonic scattering is decreased by interference effects. The latter is connected with the finite lifetime of the phonons scattered b y impurities. The case of resonant scattering of phonons by isotopic defects is considered in detail. PaccMoTpeHa mwep(PepeHqim aHrapMoHwrecKoro

Low‐Temperature Lattice Thermal Conductivity of Nonmetallic Solids with Isotopic Impurities

“…Further it is also shown there that following Altukhov and Zavt [9], if one adds the matrix elements rather than the squares of the matrix elements, in an attempt to account for the interference term, the theoretically analyzed curves using 7-l = = v / F L + Am4 + (Bl + B,) 0 , 2 1 3 + A"m3T3I2, fall much below the experimental points near K,( T). These deviations near K,( T ) have been explained by them and a good theoretical fit was obtained by taking into account the nonlinearity in the phonon dispersion which modifies z-l as VIFL + A ( l + &4 0 4 + (Bl + B,) w2T3, where .$ is referred to as the interference factor and has been taken as a constant (.$ = 0.15) in the analysis.…”

“…Mathiessen's rule) is only approximately valid, Altukhov and Zavt [9] have studied the validity of the inverse relaxation time summation approximation but with the assumption that one should add the matrix elements of the various scattering processes rather than the squares of the respective elements. It is however seen [ll] that following Altukhov and Zavt, the analyzed thermal conductivity curves for germanium lie much below the experimental curves.…”

The Critical Phonon Scattering and Peculiarities of the Transport Phenomena in Ferroelectric Crystal

“…It is shown in [16,17] that in the one-particle approximation the thermal conductivity coefficient λ(T ) may be expressed by the dynamical Green function G ν (ω), ν = (k, s), k ≡ k is the wave vector, and s is the branch index of the phonon with the frequencies ω. The density heat current j (r, t) determined from the equation…”

“…Taking into account (1)- (4) and the fact that Im G ν (ω) has a maximum in the vicinity of ω 2 = ω 2 ν , we obtain the usual gas formula or τ -approximation (when the transport relaxation time τ T for two-phonon excitement can be approximated by one-phonon relaxation time τ ) [17] …”

“…Near the structural phase transition temperature T C the soft mode (for q ≡ k) ω q (T ) → ω 0 (T 0 ) and the spectral density ρ 2 (ω) for the critical (soft) branch of vibration 2 = ν 2 = (q, s) determined from a transport equation [8,17], can be approximated as…”