Magic angle

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This article is about the magic angle as defined in the field of nuclear magnetic resonance spectroscopy. For the magic angle as defined in the field of electron energy-loss spectroscopy see magic angle (EELS).

The magic angle is a precisely defined angle, the value of which is approximately 54.7°. The magic angle is a root of a second-order Legendre polynomial, , 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

The magic angle θ_m is

,

where arccos and arctan are the inverse cosine and tangent functions respecively.

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

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

Hence, two nuclei with a dipolar coupling 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

The magic angle artifact refers to the increased signal on sequences with short echo time (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). This artifact occurs when the angle such fibers make with the magnetic field is equal to θ_m (Bydder et al, 2007).

References

• 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 : JMRI 25 (2): 290-300. PMID 17260400.

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