Mode-coupling theory for relaxation of coupled two-level systems

A Mori approach to the dynamics of coupled tunnelling units in defect crystals is presented. Transverse and longitudinal correlation functions of a two-level system are given as continued fractions; the memory kernels are evaluated in a usual decoupling approximation. Due to the random configuration of the defects on the host lattice both two-level splitting and relaxation rates show a broad distribution; the corresponding distribution function is derived for the case of a dipolar interaction of the defects. The theory covers both weak and strong coupling; the average interaction energy turns out to be the essential parameter. Dependence on frequency, temperature and concentration of the dynamical susceptibility is discussed. When passing from weak to strong coupling, the zero-temperature susceptibility shows a crossover from a constant value to a decrease with the third power of inverse defect concentration; there is quite a strong relaxational peak in the susceptibility. The theory accounts for several features observed in a recent low-frequency experiment on KCl:Li.