NMR LINE-SHAPES AND RELAXATION IN INCOMMENSURATE SYSTEMS WITH A MULTIPLE-Q MODULATION

The NMR line shape is evaluated for a superposition of one, two, and three incommensurate sinusoidal modulation waves. It is shown that the line shape does not depend on the ratio of modulation wave vectors or the angle between the modulation directions as long as the modulations are independent. For the case of two independent modulation waves of equal amplitudes, the NMR line shape has a central logarithmic singularity, where in the case of three independent modulations, singularities appear only in the derivatives of the line shapes. It is further shown that for multiple modulation waves the spin-lattice relaxation rate varies over the incommensurate line shape and that there is, in addition, a characteristic distribution of relaxation rates at each frequency shift. The possible use of these results for the discrimination between structures corresponding to the different relative phase shifts between the three coplanar modulation waves in quartzlike systems is discussed. © 1990 The American Physical Society.