Zero-Field Nuclear Quadrupole Spin-Lattice Relaxation in the Rotating Frame

Nuclear quadrupole spin-lattice relaxation rates in different frames of reference for pure electric quadrupole coupling are evaluated in terms of spectral density functions of lattice vibrations. The rate is determined in each of three important reference frames defined with respect to (1) the axial crystalline field in the laboratory, (2) a strong rotating radiofrequency field, and (3) internal dipolar fields following adiabatic demagnetization in the rotating frame. For spin states established by pure quadrupole coupling, modification of the relaxation behavior in the different frames does not conform to the behavior obtained from spins bound in pure magnetic Zeeman states. The density-matrix perturbation formalism is applied. In order to transform the total Hamiltonian into the interaction representation of the main quadrupole interaction, a convenient representation is devised for quadratic exponential unitary operators which do not directly carry out simple rotations. The transformation permits an approach to the rigorous analysis of dynamic nuclear quadrupole resonance phenomena which usually have been qualitatively described by comparison with analogous phenomena in pure magnetic resonance. The new representation is presented explicitly for spin $I=\frac{3}{2}$. Experimental measurements of ${\mathrm{Cl}}^{35}$ pure quadrupole spin-lattice relaxation times in chlorate salts in the three reference frames are compared with the theoretically determined relaxation rates.