Analytical treatment of the variable-angle sample spinning nuclear magnetic resonance spectra in the presence of the second-order quadrupole interaction determination of the quadrupole parameters and the chemical shift tensor

Abstract

The analytical expressions for the transition probability of the central transition of the variable-angle sample spinning nuclear magnetic resonance (VAS NMR) spectrum for non-integral spins in the presence of second-order quadrupole interactions are derived on the basis of the stationary perturbation theory. The modulating parameters of the VAS NMR spectrum due to the time-dependent part of the quadrupole energy are found to be different (asymmetry parameter η > 0.1) from those obtained from the average Hamiltonian theory. Using the conditions for saddle points and local maxima, it has been observed that there is a range of values for sample spinning angle β around 32, 44 and 72 ° to obtain the single peak; the range width can indicate the value of η. Analytical expressions for the single-peak positions corresponding to β around 32, 44 and 72 ° are derived in terms of quadrupole parameters and chemical shift tensor. Further, we have described an analytical method to find the principal values of the chemical shift tensor in addition to the quadrupole parameters, using VAS NMR. As an illustration, the quadrupole parameters are computed for Na 2MoO 4 and Na 2SO 4 and compared with the values available in the literature (computed from wide-line NMR and magic-angle sample spinning (MAS) NMR data using the position of peaks and shoulders). For Na 2MoO 4 the principal values of the chemical shift tensor are found to be + 17.5, − 12 and − 6.5 ppm. The present method is accurate and easy to apply as it involves only the positions of peaks and not of shoulders. Moreover, it is suitable for all values of the chemical shift tensor.