Analytical expression for the magic-angle spinning nuclear magnetic resonance line shape in the presence of the second-order quadrupole interaction for the non-zero asymmetry parameter and large sample rotation frequency.

An analytical expression for the magic-angle spinning nuclear magnetic resonance (MAS NMR) line shape function for the non-zero asymmetry parameter and large sample rotation frequency has been derived using the time-independent part of the quadrupole transition energy based on the stationary perturbation theory. The averaging over Euler angles in the case of a powdered sample is also done analytically, which is so far implemented in the literature by a numerical simulation procedure. While the peak positions obtained from analytical and numerical averaging procedures agree well, the peak heights are significantly different, showing the need for an analytical solution for the averaging over Euler angles, as this would otherwise lead to erroneous values for the quadrupole parameters. The MAS NMR line shape function derived in this paper can be compared directly with the experimental MAS NMR spectrum for the large sample rotation frequency, and the values of the quadrupole parameters can be obtained. We have also described a method of evaluating the quadrupole parameters for the small sample rotation frequency, using the peak heights and peak separation of the central band of the experimental MAS NMR spectrum and the MAS NMR line shape function derived in the present paper. As an example, the quadrupole parameters for vitreous B2O3 are computed.