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

Magic angle

The magic angle θ_m is

${\theta }_m=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\frac{1}{\sqrt{3}}\mathrm{=}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\sqrt{\mathrm{2}}\mathrm{\approx }\mathrm{54.}{\mathrm{7}}^{\mathrm{\circ }}$,

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

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, 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 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 (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.