Magic angle

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, $$P_2(\cos\theta)=0 \,$$, 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 respecively.

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, in a strong magnetic field, the dipolar coupling D 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 a dipolar coupling 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 (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 $$ \theta_m$$ (Bydder et al, 2007).