Magic Angle and Nuclear Magnetic Resonance
In nuclear magnetic resonance (NMR) spectroscopy, three prominent nuclear magnetic interactions, dipolar coupling, chemical shift anisotropy (CSA), and first-order quadrupolar coupling, depend on the orientation of the interaction tensor with the external magnetic field.
By spinning the sample around a given axis, their average angular dependence becomes:
,
where is the angle between the principal axis of the interaction and the magnetic field, is the angle of the axis of rotation relative to the magnetic field and is the (arbitrary) angle between the axis of rotation and principal axis of the interaction.
For dipolar couplings, the principal axis corresponds to the internuclear vector between the coupled spins; for the CSA, it corresponds to the direction with the largest deshielding; for the quadrupolar coupling, it corresponds to the z-axis of the electric-field gradient tensor.
The angle cannot be manipulated as it depends on the orientation of the interaction relative to the molecular frame and on the orientation of the molecule relative to the external field. The angle, however, can be decided by the experimenter. If one sets, then the average angular dependence goes to zero. Magic angle spinning is a technique in solid-state NMR spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution.
For a time-independent interaction, i.e. heteronuclear dipolar couplings, CSA and first-order quadrupolar couplings, the anisotropic component is greatly reduced and almost suppressed in the limit of fast spinning, i.e. when the spinning frequency is greater than the breadth of the interaction.
The averaging is only close to zero in a first-order perturbation theory treatment; higher order terms cause allowed frequencies at multiples of the spinning frequency to appear, creating spinning side-bands in the spectra.
Time-dependent interactions, such as homonuclear dipolar couplings, are more difficult to average to their isotropic values by magic angle spinning; a network of strongly coupled spins will make produce a mixing of spin states during the course of the sample rotation, interfering with the averaging process.
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