Magic angle

The magic angle is a root of a second-order Legendre polynomial, P2(cos θ) = 0, and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle.

In magnetic resonance imaging, structures with ordered collagen, such as tendons and ligaments, oriented at the magic angle may appear hyperintense in some sequences; this is called the magic angle artifact or effect.

OEIS: A195696 θm is the angle between the space diagonal of a cube and any of its three connecting edges, see image.

Another representation of the magic angle is half of the opening angle formed when a cube is rotated from its space diagonal axis, which may be represented as arccos −⁠1/3⁠ or 2 arctan √2 radians ≈ 109.4712°.

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.

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 width 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 produce a mixing of spin states during the course of the sample rotation, interfering with the averaging process.

The magic angle artifact refers to the increased signal observed when MRI sequences with short echo time (TE) (e.g., T1 or proton density spin-echo sequences) are used to image 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 θm.

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.