Static quadrupolar perturbed NMR in structurally incommensurate systems. I. Electric-field-gradient tensor

Abstract

An approach for describing and characterizing quadrupolar perturbed NMR spectra in incommensurate phases in terms of structural parameters is proposed. The method is based on the superspace formalism, which is widely used in the structural analysis of phases of this type. It is shown that the modulation functions defined along the ‘‘internal’’ space, which so far have been used for describing the atomic displacements with respect to the nondistorted basic structure, can also be applied to other microscopic quantities, such as the electric-field-gradient tensor (EFG) and the NMR resonance frequencies. The use of these modulation functions is proved to be an appropriate method for describing the NMR frequency distributions in incommensurate phases. The conditions imposed on the EFG distributions and their resultant NMR spectra by the superspace symmetry of the incommensurate structure can be readily obtained. It is proved for a general case that the degeneracies and symmetries for the rotation patterns of the continuous NMR spectra in these phases correspond to those which the average structure would exhibit for its discrete lines. The relation between the modulation functions of the EFG and those of the atomic displacements is discussed in a general context, and also on the basis of Landau theory. The so-called ‘‘local’’ and ‘‘nonlocal’’ models are then compared to the present analysis. It is shown that while the first one is too restrictive, the second one implies in general a local approximation. In the subsequent work these general results are applied to the ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ NMR satellite transitions in ${\mathrm{Rb}}_2$ ${\mathrm{ZnBr}}_4$.