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

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
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
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
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). 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.