Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Magic angle

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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, 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[edit]

Magic angle

The magic angle θm is

 \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.

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

Magic angle and dipolar coupling[edit]

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

D(\theta) \propto 3\cos^2\theta - 1

Hence, two nuclei with an internuclear vector at an angle of θm to a strong external magnetic field, have zero dipolar coupling, D(θm)=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[edit]

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  \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.


  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.