### The „magic angle“

Anisotropic NMR interactions follow spherical harmonics. In this regard, a special role is taken by the second Legendre Polynomial *P*_{2}(*x*)=0.5·(3*x*^{2}–1), where *x* is the Cosine of the angle with respect to the external magnetic field. This function describes the angular dependence of the most important di- and quadrupolar interactions as well as of the chemical shift anisotropy. Therefore, these interactions or anisotropies are vanishing if the relevant angle corresponds exactly to the magic angle of 54.736°. Cosequently, if, for example, the connecting vector between two coupled spins is oriented in this angle to the externam magnetic field the respective coupling has no influence on the solid-state NMR spectrum!

### Decoupling by rotation around the magic angle

Particularly in polycrystalline or amorphous samples this unique orientation ist not possible since the individual crystallites or molecules are disordered. Therefore, the sample is rotated with high frequency around an axis oriented about the magic angle. If the rotational frequency is sufficient, all interactions orthogonal to this axis are avergaed and only interactions parallel to the magic angle potentially persist. As a consequence, all couplings and interactions following the 2nd Legendre polynomial are completely removed.

An analogous picture is obtained if the magic angle is considered as the space diagonal of a cube. Rotation about this diagonal projects all three orthogonal cubic axes onto each other which causes an effecting averaging of all interactions depending linearly (i.e., in first order) on the coordinates within the unit sphere.

### Required MAS frequencies

For efficient averaging, rotational frequencies exceeding the magnitude of the anisotropic interactions are required.Therefore, efficient decoupling of the dipolar interaction between two directly bonded ^{13}C nuclei with a distance of 154 pm is achieved with a rotational frequency several-fold larger than the coupling constant of 2 kHz. ^{1}H bonded directly to ^{13}C has a coupling constant of 22 kHz; thus, heternonuclear decoupling by radio frequency pulses is used in addition to MAS. Two methylene hydrogens with a distance of 180 pm experience a similar coupling strength, however, due to the homonuclear nature of the interaction, this can only be suppressed by very fast MAS at frequencies up to the current technical limit of 110 kHz in commercial probes. For the controlled reintroduciton of the dipolar interaction (recoupling), dedicated pulse sequences may be applied. This allows for the measurement of distances and angles between nuclei.

If the MAS frequency is smaller than the magnitude of the anisotropy, its averaging is incomplete. Due to the evolution of the transverse magnetization under the modulated interaction the resulting dephasing will be periodically refocused and rotational echos will be observed after each complete MAS revolution during the free induction decay (FID). After Fourier Transform (FT), these rotational echos manifest as spinning side bands in equal distance corresponding to the MAS frequency. Since the integral area under all bands is conserved, increasing the MAS frequency does not only lead to better spectral separation of the center band (occuring at the isotropic frequency) from the side bads, but also to larger intensity due to a reduction of side bands' count and intensities.

### MAS hardware

For the implementation of MAS with frequencies up to 110 kHz modern stator-rotor systems are used. The stator is part of the NMR probe and is aligned at the magic angle inside the magnet bore. It consists of the housing, bearings, baffles and the drive unit. A radial air or nitrogen flow is injected into the space between bearing and rotor wall and holds the rotor contactless in place with minimal friction. At the same time a tangential jet drives the rotor’s turbine and spins it up to high MAS frequencies. The flow of the MAS gases is guided by baffles to the exhaust in order to prevent turbulences.

Most of the time, zirkonia (ZrO_{2}) is used as rotor material due to its hardness and ease of machining. Alternatively, a rotor made of a single crystal of synthetic sapphire (Al_{2}O_{3}) equally withstands the centrifugal forces during fast rotation. However, its brittleness makes it more prone to unplanned rapid loss of structural integrity (i.e., explosion) of the rotor. In contrast to zirconia, sapphire’s advantage lies in its transparency to electromagnetic radiation over a wide range of frequencies, making it possible to illuminate the sample inside the rotor with, for example, visible light or microwaves during MAS.

The rotor’s diameter directly limits the maximum MAS frequency. On the one hand, this belies on the dependence of the centrifugal force on the diameter; on the other hand, it has to be considered that the tangential velocity of the outside wall may approach the speed of sound of the gas medium used for bearing and drive. This limits the maximum rotational frequency particularly when sample cooling is achieved with cryogenic MAS gases. At the current time, typical rotor diameters vary between 7 mm (max. 7 kHz) and 0.7 mm (max. 110 kHz). Popular variants are 4 mm (15 kHz), 3.2 mm (25 kHz), 1.9 mm (40 kHz), and 1.3 mm (67 kHz), besides others.