Measurement of molecular motion in solids by nuclear magnetic resonance spectroscopy of half-integer quadrupole nuclei

Abstract

An accurate and efficient formalism is presented for simulating the effects of molecular motion on satellite and central transition nuclear magnetic resonance (NMR) spectra of half-integer quadrupole nuclei. The approach is based on the principles of the density operator and the stochastic Liouville–von Neumann equation and may be applied for both rotating and nonrotating samples. The symmetry properties of nuclear spin ensembles have been used to rewrite the stochastic Liouville–von Neumann equation in the form of a linear homogeneous system of coupled first-order differential equations among the alignments and coherences. This system is highly stiff and can only be solved by methods that are sufficiently accurate and stable. The properties of Cartan–Weyl operators have been used to obtain the most efficient solution for secular interactions. The methodology has been incorporated into computer programs to simulate the effects of motion for any half-integer quadrupole nucleus. These programs include the first- and second-order quadrupole and first-order shielding interactions. The formalism has been used to calculate central transition O17 NMR spectra of representative model systems. The calculations have revealed several interesting and important properties of central transition NMR spectra that have been discussed in terms of the functional form of the line shape. The validity of the methodology has been demonstrated experimentally by simulating variable temperature central transition O17 NMR spectra of the silicate (SiO2) mineral cristobalite for both rotating and nonrotating samples. The simulations have allowed the structural and dynamical details of the α–β phase transition in cristobalite to be separated. The line shapes can only be simulated if the effects of motion are included and are consistent with a model where the oxygen atoms reorient between six different orientations. It is found that the oxygen motion is present both before and after the α–β phase transition and does not change significantly at the transition temperature. The observed changes in the quadrupole and shielding parameters are shown not to be the results of motional averaging but derive from an abrupt structural change associated with the first-order character of the α–β phase transition. The structural changes may be interpreted in terms of a model where the Si–O–Si bond angle increases slightly.