The second-order Raman phonon process for a multilevel spin system is shown to give a quadru-polar spin-lattice relaxation rate T x ~ varying as T 5 at very low temperatures. This relaxation rate for quadrupole spins is similar to the one discussed for a paramagnetic spin system having a multilevel ground state. The temperature dependence of Tj is discussed on the basis of some simplifying assumptions about the nature of the lattice vibrations in the Debye approximation. This type of relaxation process has been observed below 20 K in tetramethylammonium hydrogen bis-trichloroacetate for the 35 C1 Ti _1 . Below 20 K the NQR frequency in the same crystal reveals a T 4 temperature dependence due to the induced modulations of the vibrational and librational coordinates by the low-frequency acoustic phonons.