Magic-angle spinning: The sample (blue) is rotating with high frequency inside the main magnetic field (B₀). The axis of rotation is tilted by the magic angle θ_m with respect to the direction of B₀.

In nuclear magnetic resonance, magic angle spinning (MAS) is a technique often used to perform experiments in solid-state NMR spectroscopy.^[1][2]

By spinning the sample (usually at a frequency of 1 to 70 kHz) at the magic angle θ_m (ca. 54.74°, where cos²θ_m=1/3) with respect to the direction of the magnetic field, the normally broad lines become narrower, increasing the resolution for better identification and analysis of the spectrum.

In any condensed phase, a nuclear spin experiences a great number of interactions. The main three interactions (dipolar, chemical shift anisotropy, quadrupolar) often lead to very broad and featureless lines. However, these three interactions in solids are time-dependent and can be averaged by MAS. The nuclear dipole-dipole interaction, between magnetic moments of nuclei averages to zero only at the magic angle, θ_m . The chemical shift anisotropy, a nuclear-electron interaction, averages to a non-zero value. The quadrupolar interaction is only partially averaged by MAS leaving a residual secondary quadrupolar interaction. In liquids, e.g. a solution of an organic compound, most of these interactions will average out because of the rapid time-averaged molecular motion that occurs. This orientation averaging in solution is mimicked by MAS of a solid. This causes the signal to become much narrower, giving rise to the isotropic value (which is of interest for structural determination of solid materials and compounds) and spinning sidebands which occur at multiples of the spinning speed and can be used to determine the chemical shift anisotropy of the nuclei.

The physical spinning of the sample is achieved via an air turbine mechanism. These turbines (or rotors) come in a variety of diameters (outside diameter), from 2.0-15.0 mm, and are usually spun on air or nitrogen gas. The rotors are made from a number of different materials such as ceramics e.g. zirconia, silicon nitride or polymers such as poly(methyl methacrylate) (PMMA), polyoxymethylene (POM). The cylindrical rotors are axially symmetric about the axis of rotation. Samples are packed into the rotors and these are then sealed with a single or double end cap. These caps are made from number of different materials e.g. Kel-F, Vespel, zirconia or boron nitride depending on the application required.

Magic-angle spinning was first described in 1958 by Edward Raymond Andrew, A. Bradbury, and R. G. Eades^[3] and independently in 1959 by I. J. Lowe.^[4] The name "magic-angle spinning" was coined in 1960 by Cornelis J. Gorter at the AMPERE congress in Pisa.^[1]

References[link]

The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, Failed to parse (Missing texvc executable; please see math/README to configure.): P_2(\cos\theta)=0 \, , and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle. This property makes the magic angle of particular importance in solid-state NMR spectroscopy.

Mathematical definition[link]

The magic angle θ_m is

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_m = \rm{arccos}\frac{1}{\sqrt{3}} = \rm{arctan}\sqrt{2} \approx 54.7^\circ

, where arccos and arctan are the inverse cosine and tangent functions respectively.

θ_m is the angle between the space diagonal of a cube and any of its three connecting edges, see image.

Magic angle and dipolar coupling[link]

In nuclear magnetic resonance (NMR) spectroscopy, the dipolar coupling D in a strong magnetic field depends on the orientation of the internuclear vector with the external magnetic field by

Failed to parse (Missing texvc executable; please see math/README to configure.): D(\theta) \propto 3\cos^2\theta - 1

Hence, two nuclei with an internuclear vector at an angle of θ_m to a strong external magnetic field, have zero dipolar coupling, D(θ_m)=0. Magic angle spinning is a technique in solid-state NMR spectroscopy which employs this principle to remove or reduce dipolar couplings, thereby increasing spectral resolution.

Application to medical imaging: The magic angle artifact[link]

Magic angle effect seen on MRI of the shoulder

The magic angle artifact refers to the increased signal on sequences with short echo time (TE) (e.g., T1 or PD Spin Echo sequences ) in MR images seen in tissues with well-ordered collagen fibers in one direction (e.g., tendon or articular hyaline cartilage).^[1] This artifact occurs when the angle such fibers make with the magnetic field is equal to Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_m .

Example: This artifact comes into play when evaluating the rotator cuff tendons of the shoulder. The magic angle effect can create the appearance of supraspinatus tendinitis.

References[link]

1. ^ Bydder M, Rahal A, Fullerton G, Bydder G (2007). "The magic angle effect: a source of artifact, determinant of image contrast, and technique for imaging". Journal of magnetic resonance imaging 25 (2): 290–300. DOI:10.1002/jmri.20850. PMID 17260400.