The effects of thermal order-parameter fluctuations on the NMR line shape of incommensurate systems are evaluated within the mean-field Landau theory and the results are compared with the ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ and ${}_{}{}^{39}{}_{}{}^{}\mathrm{K}$ 1/2→-1/2 NMR spectra of ${\mathrm{Rb}}_2$${\mathrm{ZnCl}}_4$ and ${\mathrm{K}}_2$${\mathrm{SeO}}_4$ just below the paraelectric-incommensurate transitions. We show that thermal fluctuations do not only reduce the effective incommensurate splitting as compared to the static case but also change the shape of the spectrum. In particular they remove the δ-function-like form of the two edge singularities in analogy to the Debye-Waller factor in x-ray scattering. Two-dimensional NMR separation techniques allow for a separate observation of static and dynamic incommensurate line shapes close to the paraelectric-incommensurate transition temperature ${\mathit{T}}_{\mathit{I}}$. The static inhomogeneously broadened NMR line shape results from the static distribution of quadrupole-perturbed Zeeman frequencies, reflecting the frozen-out incommensurate modulation wave. The dynamic line shape reflects the time-dependent part of the electric quadrupole interaction resulting from phason and amplitudon thermal fluctuations of the modulation wave. Close to ${\mathit{T}}_{\mathit{I}}$ these fluctuations become so low in frequency that they influence the line shape. A precise determination of ${\mathit{T}}_{\mathit{I}}$ can be obtained from the maximum in the width of the dynamic line shape.

• Received 20 June 1994

DOI:https://doi.org/10.1103/PhysRevB.50.9729