# 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.