Inelastic scattering of phonons by quadrupole defects with internal degrees of freedom

We solve the quantum mechanical problem of the inelastic scattering of phonons by a quadrupole defect in a crystal lattice for the case of solid parahydrogen whose matrix contains pair complexes of H2 orthomolecules. By employing the pseudospin approximation for the operator of the energy of quadrupole-quadrupole interaction of the molecules in an orthopair we derive an effective Hamiltonian that describes the interaction of phonons with a pair quadrupole orthodefect in the lattice. We set up the scattering matrix and calculate the effective phonon relaxation time τ(ω, T) as a function of the frequency ω and the crystal temperature T. We also find that a pair quadrupole defect, which has a complicated system of levels, can be replaced by an effective two-level system with temperature-dependent parameters. The fact that a pair quadrupole orthocluster has internal degrees of freedom results in a resonant scattering peak near a certain critical temperature T0. Our estimates for H2 yield T0≃ 6–7 K. Finally, we discuss the contribution of this mechanism to the low-temperature thermal conductivity of solid hydrogen.