Abstract
The low-frequency phonon-response functions, both for transverse (phason) and longitudinal (amplitudon) fluctuations, are calculated for a displacive incommensurate system at low temperatures. The results obtained are used to calculate the spin-lattice relaxation rate governed either by transverse or longitudinal fluctuations at temperatures much lower than the Debye temperature but much higher than the energy corresponding to the Larmor frequency. It is found that direct processes determine the spin-lattice relaxation rate in both cases. The transverse spin-lattice relaxation rate is proportional to temperature $T$ and does not depend on the Larmor frequency. The longitudinal contribution is proportional to ${T}^{3}$ and does not depend on the Larmor frequency as well.
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